MANTA scientific contracts and literature mapping

This is the scientific part of the MANTA proposal. Published below identifies an algorithm described in a source; existing identifies inspected vibe-qc behavior; proposed identifies a MANTA choice or extension requiring the acceptance gates. Equations below use explicit MANTA conventions and are not transcriptions of every source’s notation or normalization.

1. What was read and what it establishes

The local reading guide supplied discovery and version information. The papers themselves, including their theory, implementation and numerical sections, supplied the algorithmic evidence. Critical extent and lattice-sum equations were also checked in rendered PDF pages. SI was inspected separately.

Source and stored version	Sections/equations used	Evidence and limit
Neese, The SHARK integral generation and digestion system, online 2022, print 2023, 10.1002/jcc.26942, journal PDF	Eqs. 11-26; Sections 3.1-3.10; Eqs. 38-50, 54	Matrix MD, folded pure transforms, segmented/general/partial-general contractions, multi-target digestion, Split-J/RI-J, radial replacement and derivatives. Its libint hybrid is evidence that one kernel family is not fastest everywhere. Its large matrices are not permission for unbounded MANTA storage.
Neese et al., The Bubblepole (BUPO) Method for Linear-Scaling Coulomb Matrix Construction with or without Density Fitting, 2025, 10.1021/acs.jpca.4c07415, journal PDF and SI ZIP	Eqs. 2-24, 29-36; Sections 2.4-2.7, Schemes 1-3, Sections 4.1-4.3, Appendix Eqs. 37-49	Molecular direct BUPO-J is outlined; the documented numerical evaluation concentrates on RI-BUPO-J. SI contains 25 XYZ geometries, not a periodic derivation or hidden exchange implementation. No periodic, exact-exchange, force or stress capability follows from this paper.
Colinet, Neese and Helmich-Paris, Improving the Efficiency of Electrostatic Embedding Using the Fast Multipole Method, 2025, 10.1002/jcc.27532, journal PDF and 18-page SI	Main Eqs. 1, 4-12, 17-21; Sections 3.2-3.7; SI Sections 1-3, 6, 8, 13	Point-charge embedding, octree M2M/M2L/L2L, shell extents, adaptive expansion order and roundoff/timing accounting. A crystal embedded in finite point charges is not a periodic Bloch J/K solver. SI and main-text adaptive-order inequality wording must be reconciled against the actual error inequality, not copied as a loop condition.
Reine, Helgaker and Lindh, Multi-electron integrals, 2012, 10.1002/wcms.78, published PDF	Gaussian products/Boys functions; Eqs. 12-21; Cartesian, Hermite, solid-harmonic and derivative sections	Reference for integral families and conventions. Its Hermite normalization is not assumed identical to SHARK or the convention below.
Asadchev and Valeev, high-l 4c integrals, 10.1021/acs.jpca.3c04574, arXiv:2307.03452v3	Eqs. 3-15 and implementation discussion	Matrix MD on accelerators and memory-aware contraction ordering. The stored text is an author manuscript. GPU versus single-CPU speedups are not MANTA expectations.
Asadchev and Valeev, memory-efficient 3c integrals, 10.1021/acs.jctc.2c00995, arXiv:2210.03192v2	Sections 2.3-2.4, multiquantal VRR/HRR and in-place storage	This is primarily an OS/HGP memory-footprint design, not the same MD-GEMM algorithm as the 4c paper. Borrow scheduling lessons; do not misattribute its recurrences.
Palethorpe and Barca, high-l GPU J engine, 10.1021/acs.jctc.5c00775, ChemRxiv v2 plus journal SI	Sections 3.2-6; Algorithms 1-4; SI kernel and J timings	MD search-tree heuristic, pruning and directional sub-batches balance registers against recomputation. This is a J engine, not evidence of exact K or periodic boundary implementation merely because a benchmark has a crystal-shaped structure. Stored main text is a preprint.
Kudin and Scuseria, periodic Gaussian DFT, 2000, 10.1103/PhysRevB.61.16440, journal PDF	Eqs. 2.2-2.10, 2.13-2.20; Section III A/C/G	Periodic FMM, density/Bloch transforms and separate atomic/cell derivative terms. Neutral electrostatics and accurate lattice sums matter even with diffuse bases. This supplies periodic precedents, not a BUPO implementation.
Kudin and Scuseria, Revisiting infinite lattice sums with the periodic fast multipole method, 2004, 10.1063/1.1771634, journal PDF	Eqs. 9-24 and 25-31; Tables I-II; Section V	Reusable lattice translation sums, renormalization recurrence, and additional stress sums. Convergence of coefficients and low-order charge/dipole treatment need explicit handling.

Additional periodic context was checked against the primary publication pages for Kudin and Scuseria 1998, periodic Gaussian FMM and arbitrary cell geometries. Kudin’s 2000 dissertation, Chapter 3, was read for its explicit discussion of neutrality, surface dipoles and the potential’s spheropole offset. It explains why matching lattice energies does not alone establish a matching potential zero. The 2004 paper also discusses the Ewald-style lattice-sum lineage, including Challacombe, White and Head-Gordon 1997; that paper was not separately available as full text in this audit and is not used here as an equation-level implementation specification.

2. Basis, integrals and density conventions

Use atomic units. Primitive Cartesian functions are unnormalized monomials

\[ g_{\mathbf a}(\alpha,\mathbf A;\mathbf r) =\prod_{i=x,y,z}(r_i-A_i)^{a_i}e^{-\alpha|\mathbf r-\mathbf A|^2}. \]

Coefficients and component factors supplied by the basis adapter carry all primitive/contracted normalization. ShellInfo coefficients must not be normalized a second time. Publish a normalization version and a component scale map; verify AO values, overlaps and pure transforms independently. General contraction is a primitive-by-channel matrix over a shared exponent set. Group equal exponent sets exactly, retaining a reversible AO index map; never merge nearly equal exponents or modify a contraction to improve batching. Mixed pure/Cartesian shells, ghosts, reordered radial channels and cross-basis overlap are part of the interface.

Pure real components are m=-l,...,+l, with the existing real-harmonic phases. Cartesian order is descending x then descending y. A pure transform acts on libint’s shell-wide axially normalized Cartesian components, which are not individually unit-normalized for l greater than one. The existing transform includes the required component factors. Multiplying them into the shell coefficients again is a wrong-answer bug. Basis functions are initially real Gaussians; complex Bloch coefficients are supported by complex contractions. GIAO, spinor and magnetic integrals need additional capabilities.

For general complex functions define

\[ (\mu\nu|\lambda\sigma)=\iint \phi_\mu^*(1)\phi_\nu(1)v(1,2) \phi_\lambda^*(2)\phi_\sigma(2)\,d1\,d2, \qquad D=CfC^\dagger. \]

Then \(\rho=\sum_{\lambda\sigma}D_{\lambda\sigma} \phi_\lambda\phi_\sigma^*\), \(J_{\mu\nu}=\sum_{\lambda\sigma}D_{\lambda\sigma} (\mu\nu|\sigma\lambda)\) and \(K_{\mu\nu}=\sum_{\lambda\sigma}D_{\lambda\sigma} (\mu\lambda|\sigma\nu)\). These reduce to the existing real-AO formulas. Real primitive integral permutation symmetry does not license eightfold equality of complex Bloch tensors: retain pair exchange and conjugation identities with their momentum labels.

For RHF, \(D=2C_oC_o^\dagger\), \(F=h+J[D]-K[D]/2\) and \(E_2=\tfrac12\operatorname{Tr}(DJ)-\tfrac14\operatorname{Tr}(DK)\). For UHF use \(J[D_\alpha+D_\beta]-K[D_s]\) in spin \(s\) and \(E_2=\tfrac12\operatorname{Tr}(D_{tot}J)- \tfrac12\sum_s\operatorname{Tr}(D_sK_s)\). Real parts remove roundoff only after checking the imaginary residual. KS exchange fractions/ranges, XC and ROHF effective-operator policy remain method-owned. Kernels never infer spin factors from the number of density panels.

3. Matrix MD and direct digestion

3.1 Equation-to-kernel mapping

For primitive pairs \(ab\) and \(cd\), set \(p=\alpha+\beta\), \(q=\gamma+\delta\), \(\mathbf P=(\alpha\mathbf A+\beta\mathbf B)/p\), \(\mathbf Q=(\gamma\mathbf C+\delta\mathbf D)/q\), \(\zeta=pq/(p+q)\) and \(T=\zeta|\mathbf P-\mathbf Q|^2\). Define derivative Hermites \(\Lambda_{\mathbf t}=\partial_{\mathbf P}^{\mathbf t} e^{-p|\mathbf r-\mathbf P|^2}\), with the Gaussian-product prefactor included once in E. The one-dimensional recurrence is

\[ E_t^{a+1,b}=\frac{E_{t-1}^{ab}}{2p}+(P-A)E_t^{ab} +(t+1)E_{t+1}^{ab},\qquad E_0^{00}=e^{-\alpha\beta(A-B)^2/p}. \]

Multiplying the x/y/z E factors includes the full Gaussian prefactor. Zero out-of-range E entries are exact zeros. Coulomb auxiliary integrals use

\[ F_n(T)=\int_0^1u^{2n}e^{-Tu^2}du,\quad R_{000}^{n}=(-2\zeta)^nF_n(T),\quad R_{\mathbf t+\mathbf e_i}^{n} =t_iR_{\mathbf t-\mathbf e_i}^{n+1}+(P_i-Q_i)R_{\mathbf t}^{n+1}. \]

Thus, with \(c=2\pi^{5/2}/(pq\sqrt{p+q})\),

\[ I_{ab,cd}=c\sum_{\mathbf t\mathbf u}E_{ab,\mathbf t} (-1)^{|\mathbf u|}R_{\mathbf t+\mathbf u}^{0}E_{cd,\mathbf u} =[E_{ab}\mathcal R_{pq}E_{cd}^{T}]_{ab,cd}. \]

This is the SHARK Eq. 21 structure and the explicit MD4 formulation of Asadchev/Valeev Eqs. 6-15, under the normalization just defined. Validate the ssss prefactor analytically before higher angular momenta. Do not copy a prefactor or a power of p from extracted PDF text without reconciling the Hermite convention. \(n_H(L)=(L+1)(L+2)(L+3)/6\) controls the Hermite panel size; compact R recurrence storage can be expanded into GEMM tiles on demand.

Published mapping	Proposed kernel/consumer
SHARK Eqs. 11-24	pair_hermite_coefficients, coulomb_auxiliary, hermite_to_pair; optionally fold pure transforms into E
SHARK Eqs. 25-28	Primitive-tiled early contraction and grouped channel GEMM; never allocate the full published giant R matrix
SHARK Section 3.4	Exact intermediate-basis transformation for partial-general contractions
SHARK Eqs. 29-37	Multi-target J/K digestion with explicit permutations, diagonal coincidences and scale factors
SHARK Eqs. 38-41	Split-J: \(d_H=E^Td\), \(v_H=\mathcal R d_H\), \(j=Ev_H\) in local pair panels
SHARK Eqs. 42-50; BUPO Eqs. 12-17	Two streamed RI passes and a metric solve, with the same pair/auxiliary transforms
SHARK Eq. 54	Range-separated Boys seed below
BUPO Appendix Eqs. 37-49	Gaussian Cartesian moments then existing-convention real solid-harmonic transform
Colinet Eqs. 17-21	Upward multipoles and adjoint downward potential propagation for an optional tree consumer
GPU MD4 and GPU J papers	Later tile/recurrence-DAG schedules, measured against identical FP64 work

2c Coulomb \((P|Q)\) and 3c \((P|\mu\nu)\) use one-function Hermite expansions on the relevant sides, with dedicated arity/prefactor tests. They are not obtained by inserting a normalized zero-exponent orbital into the user basis. S, cross-basis S, kinetic, nuclear/point-charge potential, dipole and higher multipole moments are required 1e kernels. Nuclear charge sign is applied explicitly. AO-pair and auxiliary Fourier transforms are required for periodic reciprocal partners. ECP continues through libecpint; XC grid and AO-value kernels retain their own providers. Momentum, magnetic response, relativistic corrections and Hessians are separate later capabilities.

The one-electron conventions can be checked without a four-center engine. For a primitive pair with product coefficients E, \(S_{ab}=(\pi/p)^{3/2}E_{ab,000}\). For a nucleus of charge Z at C, \(V_{ab}=-Z(2\pi/p)\sum_{\mathbf t}E_{ab,\mathbf t} R_{\mathbf t}^{0}(p,\mathbf P-\mathbf C)\), where the recurrence now uses \(\zeta=p\). Kinetic energy follows from \(\partial_x^2g_b=b_x(b_x-1)g_{b-2e_x} -2\beta(2b_x+1)g_b+4\beta^2g_{b+2e_x}\) and its y/z counterparts, with the overall \(-1/2\). Gaussian moments use polynomial insertion into overlap, and center derivatives use \(\partial_{A_x}g_a=2\alpha g_{a+e_x}-a_xg_{a-e_x}\).

Define Fourier transforms by \(\widehat f(Q)=\int e^{-iQ\cdot r}f(r)dr\). Then

\[\widehat\Lambda_{\mathbf t}(Q)=(-iQ)^{\mathbf t} (\pi/p)^{3/2}e^{-Q^2/(4p)}e^{-iQ\cdot P}.\]

The opposite FT convention requires conjugating every relevant phase, not just the final result. Test FT Hermite signs independently by differentiating the analytic s Gaussian transform. The shared periodic adapters must state which convention their raw and whitened tensor legs use.

3.2 Contractions and screening

Segmented batches group angular class, representation, kernel, derivative and similar primitive counts. Small s/p cases may use specialized loops; GEMM overhead is measured before choosing it. General contraction shares primitive work across all matching radial channels, but tiles channels/primitives to a budget. Partial-general contraction uses an intermediate basis \(\chi\) with \(\phi=\chi C\): \(D_\chi=CD_\phi C^\dagger\) and \(F_\phi=C^\dagger F_\chi C\). Transform density/target tiles locally where possible; a huge fully decontracted global density is not required. This is an exact reorganization, not an accuracy approximation.

Digestion keeps multiple right sides contiguous and fuses J and K when the same integral tile is needed. A J-only task contracts density before the second AO expansion; K’s crossed indices generally require another schedule. Permutation factors apply exactly once. Test all shell equalities separately; a generic orbit multiplier is insufficient. Non-Hermitian response targets must not be symmetrized or treated as SCF densities.

Use \(|(ab|cd)|\le Q_{ab}Q_{cd}\) with shell bounds covering every admitted component. Density-dependent screening bounds the sum of omitted target contributions using absolute density envelopes, including all requested right sides and exchange permutations. A maximum-element threshold per quartet is not a global Fock-error bound. Separate primitive prefactor screening, shell screening and final density screening in error receipts. The zero-screening path remains available. General contraction and intermediate densities need their actual transformed envelopes. Pointwise bounds on an attenuated potential alone are not a proof for signed AO products; use positive-kernel Coulomb-metric Schwarz bounds or separately derived bounds.

For \(v(r)=f/r+g\,\operatorname{erf}(\omega r)/r\) set \(\kappa=\omega^2/(\omega^2+\zeta)\) and replace

\[F_n(T)\longmapsto fF_n(T)+g\kappa^{n+1/2}F_n(\kappa T).\]

erfc corresponds algebraically to \((f,g)=(1,-1)\), with a stable numerical branch. Test zero/infinite range limits and full = erf + erfc. Physical exchange range \(\omega\) and the Ewald split parameter \(\eta\) have distinct identities. Unscreened solid-harmonic \(1/r\) translation formulas cannot be reused for erfc or erf merely by changing a label.

4. Molecular BUPO-J and RI-BUPO-J

4.1 Extents, moments and hierarchy

Published BUPO retains a coefficient-weighted contracted-pair center (Eqs. 29-30), estimates the enclosing radial extent using two inward walkers on the intercenter axis (0.01 bohr steps), sets a 2-bohr minimum, and accepts far interactions when \(R-e_A-e_B>R_{allow}\), using 1 bohr in the paper. Its K-means bubbles enclose child centers and extents, with roughly 150 shell pairs per leaf and about three children per parent in the reported settings. Diffuse objects can remain always near. These are published implementation choices, not universal accuracy certificates.

Proposed MANTA starts with that molecular algorithm as a comparison mode, then qualifies a conservative envelope for production. Signed contraction coefficients can cancel in the published center denominator. Use a documented nonnegative envelope-weighted center or a primitive-center representation when the center is ill-conditioned; this is an explicit MANTA deviation. Bounds sum absolute primitive coefficients and include angular polynomial/component factors. Locate outer envelope crossings with brackets beyond all radial maxima; an axial sample alone is not accepted as a proof of a 3D integral-tail bound. Conservative primitive-sphere enclosure is the initial fallback. Validate off-axis, nodal, signed, general and diffuse contractions.

Gaussian tails are never compactly supported. Record both a pointwise extent criterion and an integrated Coulomb-tail/penetration estimate; they have different units and meanings. A pair that cannot meet the tail budget stays near, even if its plotted density is tiny. The 2-bohr minimum is a comparison parameter, not a substitute for stable arithmetic at tight exponents.

For a real solid-harmonic basis of regular functions \(R_{lm}\) define

\[M_{lm}^{a}(C)=\int R_{lm}(r-C)\rho_a(r)dr,\quad (a|b)_{FF}\simeq M_a^T T(C_a-C_b)M_b,\quad M(C_{parent})=W(C_{parent}-C_{child})M(C_{child}).\]

Use one tested normalization for R, irregular harmonics, T and W, matching BUPO Eqs. 18-24 after conversion. Do not mix the factorial conventions in Kudin 2004 with BUPO’s scaled harmonics. M2M, M2L and L2L are distinct actions; the downward action is the adjoint of the corresponding upward moment map. Primitive pair harmonic moments at the product center terminate at \(l_a+l_b\); an atom-centered auxiliary shell terminates at its l. Contracted two-center pairs and translated distributions generally do not terminate. Finite moment termination does not eliminate Gaussian charge penetration.

Choose deterministic bubble clustering with stable IDs and recorded tie handling. Store one object exactly once at the leaf level. Parent radius is \(\max_c(|C_c-C_p|+e_c)\). Source moments accumulate contracted density blocks; retain complex moments if the supplied density/transfer requires them even though angular transformations are real. Tree depth, order caps and inability to meet a budget trigger descent/direct evaluation, never silent clipping.

For each target, an accepted parent replaces all its descendants. Otherwise descend; near leaves enter the direct MD/libint consumer. Near and far lists form a disjoint exhaustive partition of the same retained interactions. Unordered pair traversal with reciprocal accumulation, or an explicitly paired forward/adjoint schedule, is required for a self-adjoint J map.

For compact interior supports, a useful conservative kernel remainder is

\[|\delta v_L|\le\frac{\theta^{L+1}}{R-a-b},\qquad \theta=(a+b)/R<1.\]

Multiply by appropriate absolute source/target charge norms, add separately bounded Gaussian tails, and sum over accepted interactions to budget matrix or energy error. This derives from the Legendre-series remainder also used by Colinet Eq. 12; applying it to integrated signed Gaussian distributions and a hierarchy is a proposed extension, not that paper’s ready-made bound. Successive-order differences are useful estimators but not rigorous bounds. Up-translation must retain enough source moments for the requested parent order; increasing an array length cannot restore discarded child moments.

4.2 Fitting and an energy-consistent two-pass contract

Flatten real AO pairs with explicit full-pair or packed-pair weights. Let \(B_{Pa}=(P|a)\) and \(V_{PQ}=(P|Q)\) in the Coulomb metric. Exact RI-J is

\[b=Bd,\qquad Vc=b,\qquad j=B^Tc,\qquad E_J^{RI}=\tfrac12 b^TV^+b.\]

The pseudoinverse means an explicitly retained positive subspace, not unconditional inversion. The initial small-system reference uses a pivoted Cholesky or eigenprojector with documented absolute/relative cutoff. Preserve an existing source’s cutoff convention when adapting it. For large systems offer streamed V actions and preconditioned solves under a residual bound; retaining a dense \(N_{aux}^2\) metric is an admitted small-system option, not linear-scaling proof. Metric solve/conditioning error is separate from RI basis error. Do not make the metric itself BUPO-approximate in the first RI increment. A future approximate metric must remain positive on the admitted space and be used consistently in both energy and derivative.

Published RI-BUPO builds b from pair bubbles, solves V, then builds J from auxiliary bubbles. Atomic auxiliary leaves make one side’s multipoles short and terminating. The two passes have different measured sensitivities; the paper varies their orders independently. MANTA proposes a stricter paired contract: define a fixed approximate coupling \(\widetilde B\) and use

\[\widetilde b=\widetilde B d,\quad \widetilde j=\widetilde B^T V^+\widetilde b,\quad \widetilde E_J=\tfrac12\widetilde b^TV^+\widetilde b.\]

The reverse pass is the adjoint of the same approximate coupling, even if implemented using a different traversal. Different forward and reverse couplings do not generally give the derivative of this energy. Jointly choose orders/partitions or tighten a common coupling. For complex spaces use the declared adjoint and pairing, not a blind transpose. Test \(\langle x,\widetilde B y\rangle= \langle\widetilde B^\dagger x,y\rangle\) before SCF.

At fixed rank, a forward error \(\delta b\) obeys the useful estimate \(|\delta E|\le\|V^+b\|\|\delta b\|+ \tfrac12\|V^+\|\|\delta b\|^2\). An ill-conditioned fit can amplify small coupling errors. Measure pass-one and pass-two defects independently against exact B, then the coupled result; do not assume the near additivity observed in one published benchmark is universal. For exact Coulomb metric and a fixed positive retained space, RI’s variational residual interpretation applies. Arbitrary approximate B need not preserve a bound relative to exact four-center Coulomb energy, despite yielding a positive quadratic form.

A robust-fit energy \(E=(\rho|\widetilde\rho)- \tfrac12(\widetilde\rho|\widetilde\rho)\) is a possible alternative, with its own derivative and residual terms. It is not silently substituted for the stated paired-coupling functional. Charge-constrained/compensated periodic fits likewise require their own declared equations and references.

4.3 Incremental builds and error ledger

For a fixed linear J map, \(J[D_n]=J[D_0]+\sum_{i=1}^nJ[D_i-D_{i-1}]\). BUPO Eqs. 35-36 use this identity, with periodic full rebuilds (ten steps in the reported tests) and tighter restart settings. Density screening, changing orders and approximate solves introduce drift. MANTA tracks a cumulative error estimate and chain identity per J/K spin slot and whole-mesh epoch.

Rebuild on interval/budget exhaustion, changed source/tree/rank/range/symmetry, threshold tightening, restart, a large density step or drift/stagnation. Before declaring SCF convergence, perform a nonincremental build at the accepted density and final accuracy, recompute energies and commutator, and reseed both density and J/K. If that check fails, continue SCF. Full here means nonincremental at the selected approximation; a separate tight analytic/RI comparison measures BUPO error. Build-plan density adaptation is frozen during each operation and its nonlinearity is bounded/tested; it is not an exact linear-response operator unless a fixed plan is used. Energy directional- derivative and response tests use that fixed plan (or zero screening); the separate adaptive-plan test measures the error introduced by changing it.

Report individual contributions from integral roundoff, primitive/shell and density screening, Gaussian tails, multipole truncation/translation, auxiliary basis, metric rank and solve, incremental drift, periodic image/G truncation, symmetry reconstruction and SCF residual. Only proven compatible bounds may be summed as a guaranteed budget. Empirical estimators are labeled as such.

5. Proposed periodic extension

Nothing in the molecular BUPO paper supplies the following implementation. First validate MANTA kernels inside the existing same-source erfc/reciprocal and fitted builders. Then introduce the periodic bubble construction below.

5.1 Bloch and whole-mesh contract

Choose translated real AOs \(\chi_{\mu R}(r)=\chi_\mu(r-R)\) and Bloch sums with \(e^{+ik\cdot R}\). With the corresponding per-cell normalization,

\[A(k)=\sum_R e^{+ik\cdot R}A_{0R},\qquad D_{0R}=\sum_k w_k e^{-ik\cdot R}D(k),\qquad \sum_k w_k=1.\]

These signs match _bloch_sum_blocks; adapters to historical transposed pair layouts must be explicit. \(D_{0,-R}=D_{0R}^\dagger\). Do not drop imaginary parts before checking this relation. A complete uniform finite character mesh has an exact inverse transform; general quadrature weights or an IBZ list do not automatically have one. chi’s _transform_inputs already makes this distinction. Electrons satisfy \(N_e=\sum_kw_k\operatorname{Tr}[D(k)S(k)]\); weights never enter orbital normalization.

In complex chemists’ notation,

\[J_{\mu\nu}(k)=\sum_{k'\lambda\sigma}w_{k'}D_{\lambda\sigma}(k') (\mu k\,\nu k|\sigma k'\,\lambda k'),\]

\[K_{\mu\nu}(k)=\sum_{k'\lambda\sigma}w_{k'}D_{\lambda\sigma}(k') (\mu k\,\lambda k'|\sigma k'\,\nu k).\]

J uses zero momentum transfer but depends on every density block. K samples \(q=k'-k\) modulo a reciprocal vector. Store both canonical q and its integer reciprocal wrap, bra/ket k indices and exact atom-image shifts. Two q-equivalent pairs can share a metric without sharing their AO-pair phases. The existing native B convention is \(q=k_{ket}-k_{bra}\), \(K(k_{ket})=\sum_{k_{bra}P}w_{bra}B_P^\dagger D(k_{bra})B_P\); document the axis/conjugation conversion to the equations above.

Atom relabeling by an integer cell vector changes Bloch basis phases. If \(\phi'=\phi U(k)\), require \(A'=U^\dagger AU\) and \(D'=U^{-1}D U^{-\dagger}\). Nonorthogonal Cartesian symmetry requires this distinction between operator pullback and density pushforward. Preserve fractional translations through exact atom images; do not add a second global phase already present in that transport. Time reversal is an admitted antiunitary action, not an assumed property of arbitrary trial densities.

One build consumes one immutable all-k/all-spin epoch. Stream density-image and q batches, but never combine iterations from different k points. IBZ execution requires separately admitted source and state symmetry and correct star weights. Changing just one k density must affect the appropriate J/K blocks elsewhere in the mesh.

The explicit whole-mesh RI consumer stores one fixed Gamma source and one source for each ordered off-diagonal target/source pair. For target i and source j it uses q=k_j-k_i and T(k_i,k_j); its exchange contribution is w_j sum_L B_L D_spin(k_j) B_L^H. The Gamma source supplies J and diagonal exchange. The supplied pair segments have empty diagonals. This sign is opposite the direct reciprocal exchange vector interface’s target-minus-source convention. Converting that source requires negating the complete vectors, then omitting its diagonal segments, not changing q alone.

Admission checks exact reciprocal labels, inversion/reverse-transfer closure, matching retained ranks and numerical conjugacy of M, P and F F^H. It does not average matrices or silently wrap q to share a finite domain. Fixed explicit source, frame, weights and evaluated density have separate identities. H and an optional exchange scale belong to the complete Fock receipt. A method-owned singular exchange correction changes the operator source and snapshot, never the identity of an unchanged evaluated density in its frame.

The neutral all-electron RI UHF method reuses the physical one-electron and nuclear Ewald sources. Its numerical fitting split is independent of the nuclear Ewald split. The AO pair cutoffs must coincide; fitting auxiliary-image support remains separate. The existing fixed-spin solver still requires uniform full regular meshes, integer populations and virtual space in both spins. Its density closure, energy, population, gap and independent terminal Fock confirmation remain method gates. A generic Fock object’s commutator diagnostic cannot replace them. This GDF source neither changes BIPOLE’s AC/BD supports nor qualifies reduction of a fitted source from geometry alone.

The RI UHF cutoff factory generates the complete union of AO-pair image spheres, including independent unwrapped AO centers. It reuses the certified reciprocal energy-sphere enumerator and its integer-label conjugacy audit. For each off-diagonal target/source pair it negates the whole direct-exchange physical vector to obtain the fitted ket-minus-bra convention. Diagonal pair segments stay empty; Gamma owns their metric/coupling source. This conversion does not identify or share distinct finite q sources. The geometric domain identity is separate from auxiliary/fitting/projector and evaluated-density identities. Its joint plan admits generated/exported supports, every retained fit owner and the complete bounded SCF before enumeration. Auxiliary and nuclear image radii remain separate; generated spheres certify membership, not convergence with respect to their cutoffs.

Public RI-containing HF selection accepts independent exact/RI J and K choices. An auxiliary basis or an RI-J request does not implicitly replace exact K. Periodic fitting controls supply the numerical split and auxiliary image domain; the physical AO pair radius is shared with the one-electron source, and the Coulomb selector owns the metric projection controls. The job records the complete fixed fit separately from the method’s singular correction, evaluated density and canonical next orbital projector. QVF exports that evaluated spin density in the nuclear-center frame, including capped states. Public admission and serialization do not change the method’s convergence tests, infer a spatial subgroup or reinterpret BIPOLE’s finite source.

Restricted RI HF embeds the spin-summed density as two equal halves in the same immutable fitted source and solves the restricted generalized problem. Its physical Fock is H + J[D] - K[D]/2; the separately selected BvK term is -xi*S*D*S/2. Each two-electron energy, including that correction, is the weighted half trace with the evaluated total D. Restricted idempotency is D*S*D = 2*D, with the RHF leaf’s own retained-space residual and closure criteria. One restricted density/Fock is stored by the solver; admission still counts the two-spin callback and all fixed auxiliary/projector owners. Equal response channels are checked, never averaged. Restricted and singlet unrestricted methods can share a fitted source but keep distinct method identities and stopping rules. QVF stores equal spin halves of the evaluated RHF density, preserving capped states and explicit source-image transport.

Mixed exact/RI HF composes one fixed fitting target with the complementary direct reciprocal response inside the same admitted native method callback. Both see the same evaluated spin density; HCORE and the method-owned BvK correction enter once. Each component contributes its own weighted energy and retains its actual finite support. RI K uses ket-minus-bra transfers and empty diagonal pair segments owned by Gamma. Exact K uses Q=G+k_target-k_source, explicit nonempty primed diagonal segments and no implicit zero-vector insertion. Cutoff factories generate the selected convention; explicit callers must supply it. RI-J/exact-K retains only the Gamma fitting projector, whereas exact-J/RI-K retains every required transfer projector. Direct and fitted sources need not agree at finite cutoffs.

The aggregate method plan admits fitting, direct and composition owners, including repeated builds and terminal confirmation. Its fitting subplan is not the whole mixed-method memory bound. Receipts retain independent J/K source identities and optional direct-component evidence; the composed source identity does not relabel the evaluated density or AO frame. Method energy and owned Fock exports preserve this distinction. Composition grants no BIPOLE/CRYSTAL parity or symmetry admission beyond the component contracts.

Explicit fitted pure and global LDA/GGA KS now composes the same selected fixed-projector sources inside actual RKS/UKS functional callbacks. Pure KS uses only fitted J; globals independently select exact/RI J/K with at least one fit. RKS embeds total D as equal spin halves only for fitting and retains total D for unpolarized XC. UKS keeps its actual spin populations. The functional energy is nuclear plus one-electron, J, XC, method-scaled K and its separately recorded singular correction. An HF half trace of the final KS matrix is not this energy. Projectors stay fixed through confirmation; numerical fitting omega cannot stand in for a physical range kernel.

Joint admission includes retained fitting, complementary direct responses, XC panels, embedding and confirmation, without replacing the method’s solver leaf or convergence norm. The fitted child receipt is marked as a component: its embedded density and partial energy cannot be mislabeled as the complete KS state. Method density identity version 2 binds actual D, spin layout, weights and overlap frame independently of interaction source. The associated Fock snapshot additionally binds interaction, one-electron and nuclear sources. Direct KS adopts the same identity semantics without changing its numerical states or original resource counts.

Shared FockBuild exports own the evaluated D/F with a KS kind, method energy and explicit, possibly rectangular S/X frame. Shared symmetry transport checks AO-image covariance; they do not prove electronic-state, grid, fitted projector or source symmetry. Explicit fitted sources retain the full mesh and the independent BIPOLE/CRYSTAL parity boundary. Generated cutoff and joint quadrature sources now share that contract. Their geometry plan calls the actual fitted functional planner, never an HF proxy, before allocating sources or quadrature. The complete admission retains domain/grid payloads, fixed projectors, XC, complementary direct components and confirmation. Pure fitting has no K table; the selected global K controls opposite transfer orientations and Gamma versus pair-diagonal ownership. Geometry, physical source and evaluated state remain distinct. Public fitted KS jobs use these admitted native sources with resolved auxiliary/metric/fitting controls. Pure RI J retains no K; globals require explicit RI K selection. Output admission adds all Python, history, basis and serialization owners before generation. Method result types and complete energies remain KS; the partial fitted child cannot replace them. QVF records actual evaluated spin densities in the nuclear AO frame, including capped iterations and orbital-image conversion. Physical range fitting follows the same job contract with ordered identities for each resolved physical fitted/exchange source. No source reduction is implied.

5.2 Periodic Green function and disjoint partitions

For 3D cell volume \(\Omega\), choose the zero-average conducting-boundary periodic Coulomb Green function. Away from its singularities it can be written

\[G_\eta(r)=\sum_R\frac{\operatorname{erfc}(\eta|r+R|)}{|r+R|} +\frac{4\pi}{\Omega}\sum_{G\ne0} \frac{e^{-G^2/(4\eta^2)}}{G^2}e^{iG\cdot r} -\frac{\pi}{\eta^2\Omega}.\]

Here \(G\) in a sum is a reciprocal vector; \(G_\eta\) is the Green function. The constant cancels the cell-average real-space term. Its Poisson equation is \(-\nabla^2G_\eta=4\pi(\delta_{per}-1/\Omega)\). Neutrality is the default for the complete electron+nuclear system. A charged cell requires explicit uniform-background policy; otherwise refuse. Individual electron/electron and electron/nuclear terms use the same Green convention, even though each source is charged. The raw Ewald total-charge correction \(-\pi Q^2/(2\eta^2\Omega)\) is already represented by the constant above; do not add it twice. Remove nuclear point self terms consistently (\(-\eta\sum_A Z_A^2/\sqrt\pi\) in the usual split energy). Gaussian electron self interactions are not nuclear point self terms.

There are two exclusive implementations of this same limiting Green function:

1. Initial kernel adapter: direct erfc real-space shell tiles plus the matching reciprocal partner and gauge/self terms. Finite supports and cutoffs are the existing source’s. A full-Coulomb bubble far field cannot be added on top of this complete sum. Screened multipole kernels, if later used here, must be derived from screened radial derivatives.

2. Proposed periodic BUPO: select a symmetric finite near-cell set \(\mathcal N\), large enough to enclose interacting Gaussian extents, and use the exact identity

   \[G_\eta(r)=\sum_{R\in\mathcal N}|r+R|^{-1}+G_{far,\mathcal N}(r),\]
   \[G_{far,\mathcal N}=G_\eta-\sum_{R\in\mathcal N}|r+R|^{-1}.\]

   Treat interactions in \(\mathcal N\) with a molecular-style direct/bubble partition. Evaluate a lattice M2L expansion for the complementary field, obtained by differentiating the Ewald expression and subtracting those same near-cell bare interactions. Reciprocal terms thus occur once, in the tail construction. Every accepted local bubble and tail interaction has a unique source/target/image identity. Near cells and the lattice tail never overlap. Varying \(\mathcal N\) transfers terms between the two parts; it must not change the converged result.

There is an additional constraint often missed by a harmonic-only design. In a singularity-free near region \(\nabla^2G_{far,\mathcal N}=4\pi/\Omega\), so this field is not purely harmonic. Write locally

\[G_{far,\mathcal N}(r)=H_{\mathcal N}(r)+\frac{2\pi}{3\Omega}|r|^2, \qquad \nabla^2H_{\mathcal N}=0.\]

The quadratic term requires monopole, dipole and trace second moment, in addition to traceless solid harmonics. For distributions with moments \(Q_a=\int\rho_a\), \(m_a=\int r\rho_a\) and \(s_a=\int r^2\rho_a\) in a common origin, its pair contribution is

\[\frac{2\pi}{3\Omega}(s_aQ_b+Q_as_b-2m_a\cdot m_b).\]

Retain the constant potential gauge as well. Translation tests must include these polynomial terms; finite total-charge cancellation cannot justify dropping them from separately reported J and nuclear matrices. Combining electron and nuclear moments can improve conditioning, but preserve the component energy ledger. This polynomial-plus-harmonic decomposition is a proposed derivation from the declared Green function, not a BUPO paper formula.

For an alternative direct-lattice implementation, Kudin/Scuseria 2004 Eqs. 23-24 give \(S_0=L_*\), \(S_{n+1}=\mathcal U_L(S_n)\otimes M_*+L_*\), with \(\mathcal U_L(L_{lm})=3^{-(l+1)}L_{lm}\) and M2L contraction denoted by \(\otimes\). Eqs. 30-31 add cell-index-weighted stress sums. Reuse that recurrence only after converting moment conventions, excluding the identical near-cell set, and resolving low-order conditional sums/surface/gauge terms. Convergence of high-l lattice coefficients does not resolve a charged monopole divergence or a surface dipole by itself. Ewald-derived coefficients are the initial reference; direct renormalization is a later independent schedule.

In both cases, Gaussian penetration/tails, finite pair-image supports, lattice coefficient convergence, real/G Ewald truncation and multipole truncation have separate controls. Derivatives of the full Green function should be obtained before numerical truncation choices are frozen for a force comparison. Strongly skewed cells and diffuse bases can require larger near sets; runtime is not universally linear. Infinite tail coefficients may be precomputed per cell/kernel/boundary, not per density. Their workspace/order is budgeted.

The first bounded prerequisite is now multipole_lattice_tail: finite explicit Ewald domains and a contained near set, raw Cartesian M2L through total degree twelve, with the constant gauge and trace terms retained. Near images use the analytic -erf/r complement even at the origin; other images use upper incomplete gamma radial derivatives. Domain canonicalization, recurrences and result exports have count-only admission. Kernel and expansion identities are distinct. This qualifies finite coefficients, not a production periodic BUPO traversal, infinite-sum error bound, Gaussian support certificate, symmetry shortcut, complete charge ledger or nuclear self policy.

Finite-source compact Taylor remainder

lattice_tail_remainder now bounds the order-\(p\) Taylor remainder of this same finite kernel, conditional on enclosing radii \(a,b\) and absolute charge integrals \(Q_a,Q_b\) supplied by the caller. These can describe signed compact densities or finite measures. Set \(n=p+1\), \(\rho=a+b\), and, for each supplied real image, \(D=|r+R|\), \(d=\max(0,D-\rho)\), \(M=D+\rho\). Non-near images require \(d>0\). Near images have no remaining singularity. The real-image contribution is bounded by

\[B_R=\frac{2\rho^n}{\sqrt\pi} \sum_{k=0}^{\lfloor n/2\rfloor} \frac{2^{n-2k}M^{n-2k}}{k!(n-2k)!} \int_I s^{2(n-k)}e^{-d^2s^2}\,ds,\]

where \(I=[0,\eta]\) for the near \(-\operatorname{erf}/r\) complement and \(I=[\eta,\infty)\) for the other real images. This follows from the one-dimensional Taylor theorem along the actual displacement \(h\), with \(|h|\le\rho\). Differentiating \(\exp(-s^2|x+t u|^2)\), \(|u|=1\), gives terms of magnitude at most \(n!e^{-s^2d^2}(2s^2M)^{n-2k}s^{2k}/[k!(n-2k)!]\). The Taylor \(1/n!\) cancels this prefactor. Integration and the triangle inequality give the displayed bound; no harmonic-kernel assumption enters. Each supplied reciprocal term contributes at most

\[B_G=\frac{4\pi}{\Omega G^2}e^{-G^2/(4\eta^2)} \frac{(\rho|G|)^n}{n!}.\]

The constant gauge is reproduced exactly by the order-zero coefficient and has zero Taylor remainder. The pair-energy bound is \(Q_aQ_b(\sum_R B_R+\sum_G B_G)\), with separate near, non-near real and reciprocal fields. Absolute charge norms, never signed net charges, multiply the kernel bound. In particular a neutral component still has a nonzero bound.

Positive incomplete-gamma envelopes avoid subtraction and premature underflow. The lower integral uses a positive series with a geometric upper bound on its uncomputed tail, or the complete gamma as an upper bound. The upper integral starts from \(\Gamma(1/2,x)\le\min(\sqrt\pi,e^{-x}/\sqrt{x})\) and recurs upward with positive terms. A decay argument above \(10^6\) is conservatively replaced by \(10^6\). Calculations remain logarithmic through charge scaling; positive subnormal results round upward instead of becoming a false zero. Geometric and final relative guards cover routine evaluation noise, but these are not an interval arithmetic proof for the platform’s transcendental functions.

The finite-domain-compact-taylor qualification is an analytical approximation bound, separate from coefficient/moment/contraction roundoff. It does not verify the caller’s support/norm claims, bound omitted Ewald images or Gaussian tails, or certify a full Gaussian-moment substitution. Its expansion_identity matches the exact LatticeTailMatrix.source_identity; its own identity also binds both radii, both norms and the bound algorithm. No symmetry skipping, Fock producer or production periodic BUPO route is admitted by this primitive.

Finite-source Gaussian approximation ledger

lattice_gaussian_interaction pairs the identical finite coefficient matrix and compact bound with existing native PairExtent or AuxiliaryExtent envelopes. Both producers must retain full and tail absolute radial moments through the requested order. Their centers define the expansion displacement; an always_near envelope or a non-near singularity inside the combined support requires direct evaluation. The compact bound uses the full absolute masses as upper bounds on the interior masses.

Write \(Q_A,Q_B\) for full absolute masses, \(t_A,t_B\) for exterior absolute masses and \(V_A,V_B\) for global absolute Coulomb potential bounds. With \(N_{\rm out}=|\mathcal R\setminus\mathcal N|\), define

\[ B=|\mathcal N|\frac{2\eta}{\sqrt\pi} +\sum_{G\in\mathcal G}\frac{4\pi}{VG^2}e^{-G^2/(4\eta^2)} +\frac{\pi}{\eta^2V}. \]

The difference between the full and interior Gaussian interactions is bounded by

\[ E_{\rm penetration}\le B(t_AQ_B+Q_At_B) +N_{\rm out}(t_AV_B+t_BV_A). \]

This follows by writing the density-product difference as tail times full plus interior times tail. Each near \(-\operatorname{erf}(\eta r)/r\) term is globally bounded by \(2\eta/\sqrt\pi\); reciprocal cosines and the constant gauge use their absolute amplitudes. Each non-near erfc term is nonnegative and no larger than bare Coulomb, so a translated tail is bounded by its mass times the other source’s global potential bound. This derivation uses the finite complementary partition in section 5.2, including the gauge once.

If \(f_A[j],t_A[j]\) bound full and exterior radial absolute moments, replacing interior moments by full moments contributes

\[ E_{\rm moments}\le\sum_{ij}|T_{ij}| \bigl(t_A[|i|]f_B[|j|]+f_A[|i|]t_B[|j|]\bigr). \]

All Cartesian matrix entries, including traces and the constant, are used. The three fields compact_remainder, tail_penetration and full_moment_tail remain distinct and sum upward to total. A vanishing signed monopole does not replace an absolute norm by zero. Larger orders can increase a conservative bound for broad supports; a consumer must check the actual result and refine or evaluate directly when its target is not met.

Joint count-only admission precedes input reads/copies and covers both native components, reciprocal sorting, retained envelopes, the matrix and export peak. Results own their coefficient/remainder pairing. Envelope identities fingerprint the consumed numerical bounds, centers, roles and shell indices; they are not basis identities or authorization for a Fock or symmetry source. finite-domain-gaussian-approximation excludes omitted lattice/pair-image domains, floating-point coefficient/moment/contraction errors and point-nuclear self policies. Positive log-domain accumulation has conservative finite guards but is not an interval-libm certificate. Production periodic BUPO/RI-BUPO must still aggregate these bounds and bind every source before using them.

Infinite Ewald domain remainder for compact sources

lattice_domain_remainder now bounds the infinite Ewald Green function minus its explicit finite real/reciprocal domains. Both domains must be complete centered integer cubes, of radii \(N_R,N_G\) in the maximum norm; the reciprocal cube excludes zero. Canonicalization proves uniqueness, origin convention and inversion closure before maximum coordinate and count prove cube completeness. An arbitrary inversion-closed subset is refused. The identical near-cell subtraction cancels from the infinite-minus-finite difference. The result binds the existing finite kernel identity, including its near set, without changing its gauge or point-self policy.

Let \(s_R>0\) satisfy \(|An|\ge s_R|n|\), and let \(s_G=2\pi/\|A\|_F\), a lower bound for the reciprocal lattice metric. For a computed inverse \(B\), the real bound uses

\[s_R\le\frac{1-\|I-BA\|_F}{\|B\|_F},\]

and refuses a nonpositive residual bound. This follows by applying \(I=BA+(I-BA)\) to any vector; it does not assume that a computed inverse is exact. Guarded norms and outward distance rounding protect the analytical inequalities from routine evaluation noise, but are not interval arithmetic. For compact supports of radii \(a,b\), set \(B_r=|r|+a+b\), \(m=N_R+1\), and \(d=s_Rm-B_r\). Require \(d>0\) so every omitted real image is separated from the entire difference support.

The shell \(|n|_\infty=j\) contains exactly \(24j^2+2\) points. Using \(\operatorname{erfc}(x)\le e^{-x^2}/(\sqrt\pi x)\) and the decreasing shell prefactor, the omitted real kernel is bounded by

\[U_R=\frac{24m^2+2}{\sqrt\pi\eta d^2}e^{-\eta^2d^2} \left(1+\frac{1}{2\eta^2s_Rd}\right).\]

The parenthesis bounds a positive decreasing Gaussian series by its first term plus its integral, applying the same erfc inequality to that integral. For \(n=N_G+1\) and \(c=s_G/(2\eta)\), reciprocal phases have unit modulus, giving the independent bound

\[U_G=\frac{4\pi}{\Omega s_G^2}\left(24+\frac{2}{n^2}\right) e^{-c^2n^2}\left(1+\frac{1}{2c^2n}\right).\]

The pair-energy bounds are \(Q_aQ_bU_R\) and \(Q_aQ_bU_G\), using absolute charge norms even for neutral signed distributions. The constant gauge is identical in both kernels. The shell-envelope derivation is a MANTA bound on Ewald’s split, not a formula attributed to the original paper. Ewald (1921), DOI 10.1002/andp.19213690304, section IV.1, Eqs. 37-40 supplies the neutral split and regular point-self reference used by the energy tests. The separately declared zero-average component convention remains in force.

Logarithmic accumulation and conservative decay caps keep positive subnormal bounds positive; unrepresentable bounds are refused. Zero absolute charge norms give zero only after all domain and support checks. Count-only plans admit sorting, cube validation, metric work, copies and exports before input payload reads. No omitted cells are enumerated at runtime.

For translated compact sources, two corresponding domain bounds limit the finite-kernel periodicity defect by the triangle inequality through the same infinite Green function. Reciprocal terms are already periodic, so only the real bounds are needed for a pure lattice translation. Finite real cubes are not declared exactly covariant. These bounds can be added to the compact Taylor remainder, with both errors and source identities kept separate. The qualification compact-infinite-ewald-domain-remainder does not cover unbounded Gaussian tails, finite AO pair-image omission, moment/roundoff errors, automatic source support certification or a complete SCF profile. Those remain necessary before production periodic BUPO admission.

Full Gaussian infinite-domain bounds

compute_pair_gaussian_majorant and compute_auxiliary_gaussian_majorant produce immutable positive spherical Gaussian sums that dominate the absolute value of every AO component of one selected source over all space. Orbital shells support angular momentum through 6 and auxiliary shells through 7, including Cartesian and pure representations and signed contractions. Pair translation uses the same represented ket center as PeriodicPairSource. These are full-source envelopes; an extent radius is not used as compact support. Basis, selected shells, translation and actual envelope terms bind an immutable native source identity. There is no numerical envelope constructor.

For each primitive product the Gaussian product theorem and the maximum row sum of the Cartesian-to-pure transformation give

\[|p_i(x)|\le e^w\sum_n c_n r_P^n e^{-p r_P^2},\qquad c_n\ge0.\]

Choose \(\gamma=9p/16\) and \(\alpha\) just below \(p/2\). With \(M_0=1\) and \(M_n=[n/(2e(p-\gamma))]^{n/2}\), each polynomial term is bounded by \(M_n e^{-\gamma r_P^2}\). For a stored center \(C\) with guarded coordinate error \(\delta\ge|P-C|\), completing the square gives

\[e^{-\gamma|x-P|^2}\le e^{\gamma\alpha\delta^2/(\gamma-\alpha)}e^{-\alpha|x-C|^2}.\]

Thus the normalized majorant \(Q(\alpha/\pi)^{3/2}e^{-\alpha|x-C|^2}\) uses \(Q\ge e^w\sum_n c_nM_n\,e^{\gamma\alpha\delta^2/(\gamma-\alpha)} (\pi/\alpha)^{3/2}\). Auxiliary primitives need no product-center shift. Absolute primitive coefficients prevent cancellation from understating the bound. Logarithmic sums, outward charge conversion and positive underflow handling are retained. Unrepresentable decay or charge is refused. The coordinate guards are conservative floating-point headroom, not a claim of interval certification.

For two normalized majorants define \(\beta^{-1}=\alpha_a^{-1}+\alpha_b^{-1}\) and \(\zeta^{-2}=\eta^{-2}+\alpha_a^{-1}+\alpha_b^{-1}\). Their full-space convolution with the Ewald real kernel is

\[Q_aQ_b\frac{\operatorname{erf}(\sqrt\beta d)-\operatorname{erf}(\zeta d)}{d} \le Q_aQ_b\frac{\operatorname{erfc}(\zeta d)}{d}.\]

lattice_gaussian_domain_remainder applies the preceding cube-shell real bound with this effective decay and each pair of majorant centers, then adds all positive terms. Omitted images must be separated from these centers in the conservative lattice metric. This condition concerns centers, not bounded Gaussian support. Reciprocal terms use the physical \(\eta\) and the product of total majorant charges, since a Fourier transform is bounded by its absolute charge norm. The effective real decay does not change the physical kernel, its gauge, near subtraction, source identity or self convention. Canonical domain validation and sorting run once, not once per primitive pair.

This convolution and envelope derivation extends the Ewald split; it is not attributed to Ewald’s original point-charge formula. The existing Gaussian product and Ewald references apply (McMurchie-Davidson 1978; Ewald 1921, section IV.1, Eqs. 37-40). Complete smooth-charge energies include finite Gaussian self integrals, without a point-charge self subtraction.

Qualification native-gaussian-infinite-ewald-domain-remainder covers one component interaction of the full native pair/auxiliary source. It can be added to the finite Gaussian approximation ledger for matching sources and identical kernel domains. A consumer must still bind the moment source to these same native producers and account for density contraction multiplicity. It does not bound omitted AO pair images, roundoff, a whole Fock build or a selected SCF accuracy profile. Production periodic BUPO/RI-BUPO traversal, charge policy, symmetry, lower-dimensional Green functions and derivatives remain distinct gates. The shared FockBuild/JKBuilder UHF interface and BIPOLE physical AC/BD/exchange domain ownership are unchanged.

Periodic pair-density source and its adjoint

PeriodicPairSource now owns the finite orbital pair-image inventory needed by a q=0 periodic Coulomb consumer. It retains the native basis, cell, integer image set, momenta, weights, physical pair cutoff and Gaussian extent policy. The image set is unique, inversion closed and contains the origin. A physical pair cutoff is applied once to each conjugate orbit, retaining or omitting both directions together. This is independent of BIPOLE AC/BD/exchange domains and grants no alternative interpretation of those domains.

For real orbital functions, let \(p_{abR}(r)=\phi_a(r)\phi_b(r-R)\). Its partner obeys \(p_{ba,-R}(r)=p_{abR}(r+R)\). Their integrals against a periodic potential are equal. In the q=0 specialization of Sun et al. (2017), Eq. 16, DOI 10.1063/1.4998644, the corresponding density coefficients are

\[c_{abR}=\sum_k w_kD_{ab}(k)e^{-ik\cdot R},\qquad c_{ba,-R}=c_{abR}^*.\]

The second equality uses Hermiticity at each k, with no relation between different k points. Each two-member orbit therefore contributes \(2\operatorname{Re}c_{abR}\) times one representative product. A diagonal shell at the origin keeps its full AO block with real density coefficients; the imaginary antisymmetric part cancels against symmetric product integrals. No spatial subgroup or time-reversal reduction is admitted by this identity. Arbitrary nonreciprocal Fourier arguments and finite nonperiodic Green functions do not inherit the translation identity. A consuming approximation must separately establish its source, covariance and omitted-domain errors.

The source generates native shifted-shell extents without copying an image supercell basis. It streams one full Cartesian moment panel at a time about the representative pair midpoint. Full and tail absolute radial bounds are retained through the requested order, including trace moments. The forward map contracts the paired density coefficients into real leaf moments. Its charge is the signed integrated AO density of this finite pair source; it neither imposes neutrality nor adds nuclei or a background correction. Per-pair coefficient L1 norms and the resulting absolute-charge envelope remain distinct from signed monopoles.

For real local moment potentials \(L_p\), the adjoint generates the AO response \(V(k)\) with the original Bloch phases and satisfies

\[\sum_p M_p[D]^T L_p =\sum_k w_k\operatorname{Re}\operatorname{tr}[D(k)V(k)].\]

Weights enter the forward contraction, not the returned AO response. Both members of each orbit are restored in the adjoint, including their distinct AO index order. Native Hermiticity is checked without averaging matrices. Zero-weight k points retain response values. The source, evaluated density and local-potential identities are separate; an adjoint response alone is not a complete FockBuild or a method energy.

Count-only admission covers the retained source and one forward/adjoint cycle, including full moment panels, density/response copies and exports. It is not a lifetime call quota; a method must budget repeated passes and other retained objects. The existing molecular bubble tree still requires its complete triangular shell inventory and refuses these periodic images. The typed PeriodicBubbleTree now preserves the explicit entry/image mapping. The qualification finite-pair-image-moments-and-adjoint does not select periodic BUPO, exchange, fitting, SCF, nuclear self terms or an accuracy profile. Those consumers must bind this source and their own kernel ledgers.

5.3 Fitting, exchange and dimensionality

The implemented raw short-range GDF substitution retains the existing physical auxiliary sphere and AO-pair/auxiliary-segment capsule domains. MANTA replaces only contracted erfc integral evaluation. The metric phase is exp(+i q.R) and the coupling phase is exp(+i k.R-i q.T) for T(k-q,k). Shared-q batches reuse molecular tiles; source-k weights enter later method contractions, never these raw blocks. Included finite erfc zero modes are removed once by a matching jellium assembler, separately from exchange singular corrections. Optional Gaussian-envelope screening uses the existing density-independent rule, recorded in source identity; it is not a symmetry qualification. The method retains ownership of the fitted projector, charge ledger, full Fock/energy build and derivative source. A raw finite metric is not automatically a positive retained-space fit or a BIPOLE source.

The complete finite Coulomb source adds streamed auxiliary and AO-pair Fourier panels on an explicit G+q domain. It uses the same physical AO-pair cutoff as its short-range component, the existing GDF jellium kernel and no k weights. The finite erfc zero is removed exactly once for reciprocal Gamma transfers, even when the supplied primed reciprocal list omits zero. Zero in that list adds no long-range contribution. Independent bare-Coulomb reciprocal sums check converged numerical splitting. Metric and coupling identities retain both the actual SR source and the ordered reciprocal domain. These complete integral components remain unfactored; no retained subspace, response operator, exchange correction or symmetry certificate is selected by the provider.

The immutable shell-tile form is the identical finite source, with the same component identities. A separate q-channel RI consumer fixes the lower-triangle Hermitian positive-eigenvalue projector at construction, refuses significant negative eigenvalues and never refits by density. It streams B=F^H T in bounded retained-mode blocks. Gamma J uses weighted total spin density and returns an unweighted target response with energy 0.5*|rho|^2. Each K output is w_k B(k) D_spin(k) B(k)^H in the bra frame k-q, indexed by its source ket batch. It is not a complete exchange sum. Rank truncation, the exact factor and projector, raw source, weights and evaluated density have separate identities. The frame also includes the lattice and actual momenta. Metric residuals diagnose the chosen finite projected equation, not approximation accuracy. The consumer admits all repeated shell traversals jointly and retains neither complete T nor complete B.

The exact J-only baseline can contract the same raw tiles in a paired forward/adjoint schedule: b=sum_kij w_k conj(T_kPij) D_total,kij, rho=F^T b, c=conj(F) rho, and J_kij=sum_P T_kPij c_P, with E_J=0.5*|rho|^2. The two sweeps share the immutable coupling and fixed native factor, including its actual finite F^H M F; no second whitening or rank change is allowed. This factor convention requires the displayed conjugations. It implements the paired contract of Section 4 on the existing Sun periodic source, without an approximate bubble hierarchy.

Select it only for J without K when the retained-mode schedule would exceed two passes. Fixed rank/block metadata supplies this decision; density, timing profiles and geometric symmetry do not. Joint J/K and K-only keep their existing traversal. Actual coupling receipts record paired-raw, retained-mode or none; source identities remain independent of execution schedule. The original maximum-rank joint admission remains conservative. Weighted reciprocity, energy derivatives, finite-rank image transport and complete evaluated method states qualify this reordering, rather than a claim of bitwise equality or periodic RI-BUPO completion.

PeriodicGDFTiles.forward(X) and .adjoint(c) now expose that raw paired coupling as producer-owned actions. They compute sum_kij conj(T_kPij) X_kij and sum_P T_kPij c_P, respectively, under the ordinary complex Frobenius pairing. Arbitrary complex transition inputs are valid at any admitted q; the raw producer does not require Hermiticity or insert k weights, spin sums, metric solves, canonical image folding or a symmetry operation.

Each action streams the existing ordered tiles and admits one complete pass, including input/export copies and compensated output. A method must jointly admit repeated passes and its other live state. Paired RI-J supplies the weighted total spin density to the forward action and applies its original finite metric factor between actions. Source/factor/projector identities and the conservative joint RI plan retain their meanings. The new coupling receipt identifies the raw source and arbitrary input, not an SCF density.

The optional coupling_screen_error=epsilon introduces a first fixed approximate coupling using the existing ShellEnvelope, GaussianSRBound and ThreeCenterImageDomain. It is mutually exclusive with common metric/ coupling screening. Only the three-center selector receives epsilon; metric evaluation and its source identity remain unchanged. For each ordered shell triple, divide epsilon by the conservative pair/auxiliary image candidate counts. The shared selector bounds each omitted component with a positive Gaussian erfc majorant. Unit-modulus Bloch phases and the triangle inequality then give \(|\widetilde T_{kPij}-T_{kPij}|\leq\epsilon\). Tightened enumeration uses that same envelope. The finite LR sum and zero term are identical. This reuses the existing general angular/contraction bound, not an assumed Gaussian decay of a Schwarz factor. Epsilon zero retains previous identities.

With \(X_{kij}=w_k\sum_sD_{skij}\), the paired producer gives \(\|\delta b\|_2\leq e=\sqrt{m}\epsilon\|X\|_1\). Keep the exact native F, \(W=\overline F F^T\), and \(\widetilde c=W\widetilde b\). The same algebra as Section 4 applies for complex Bloch arrays:

\[ |\widetilde E-E|\leq\|\widetilde c\|_2 e+ \tfrac12\|W\|_2 e^2, \]

\[ \|\widetilde J-J\|_{\max}\leq \epsilon\|\widetilde c\|_1+ \sqrt m\,(\|\widetilde T\|_{\max}+\epsilon)\|W\|_2 e. \]

Use \(\|W\|_2\leq\|F\|_F^2\), without replacing the finite Gram matrix by the identity or re-whitening. The forward pass measures the maximum coupling magnitude while streaming existing tiles. No dense tensor or extra integral sweep is needed. Scalar bound arithmetic is outward guarded; integral, factor, phase and contraction roundoff remain separate. Zero density and zero retained rank give zero J/energy bounds. The screen is fixed before SCF, so derivative/reciprocity tests apply to the selected quadratic form. Its receipt records the unchanged finite reference coupling identity.

This first consumer admits Coulomb J only. A positive coupling-only screen refuses fitted K, including K requested with zero energy coefficient. Mixed HF composes the selected J with its existing exact exchange and correction; pure KS retains its XC path. Method receipts preserve the evaluated-density bounds without claiming density-relaxation or whole-method accuracy. The shared Fock state and output retain their original density and frame contracts. Canonical pair symmetry and operation-qualified RI block reduction remain separate. Screening supplies no new symmetry certificate.

This boundary is deliberately distinct from the existing canonical periodic BUPO tree. That tree’s finite Ewald image support is not the GDF producer’s ordered sphere/capsule support. A future approximation must qualify its partition and paired error propagation against the selected coupling before method selection; sharing a lattice or a metric is insufficient.

The optional coupling_multipole_error=epsilon now supplies an experimental ordered-image multipole approximation through that same producer. It is mutually exclusive with both screening controls. The metric, reciprocal panels and finite-zero subtraction remain exactly the selected finite source. Zero epsilon preserves earlier source identities. Positive epsilon adds the order and effective native extent controls to the coupling identity, without changing the metric identity.

For each ordered auxiliary/AO shell triple, first count its actual selected image pairs using ThreeCenterImageDomain, without storing the interaction list. Divide epsilon by this count. The second traversal uses the same sphere/capsule support. Native shifted-shell extents, absolute moments and potential bounds qualify each separated auxiliary-image/AO-pair interaction. Use the total-degree Cartesian Taylor matrix of the isolated \(\operatorname{erfc}(\eta r)/r\) kernel, with target-minus-source displacement. The shared erfc radial recurrences retain trace terms; this is not a harmonic Coulomb expansion. There is no lattice gauge or near-field subtraction in this individual image kernel.

The shared Gaussian ledger bounds the compact Taylor remainder, penetrating Gaussian tails and substitution of full moments. Accept a far leaf only when their sum fits its image allowance. An overlap, always-near extent, overflow of the bound or excessive error selects the existing direct MD integral. Both forward and adjoint reuse this density-independent decision. Auxiliary moments and shifted pair moments come from the existing native producers; no replicated basis, dense global coupling tensor or new metric is introduced.

The executed receipt sums the accepted image bounds within a shell tile and takes the maximum across tiles. Unit-modulus phases preserve this component bound. Paired RI propagates this evaluated epsilon using the equations above, the original finite factor, and the actual density. The source plan still records the requested upper limit for admission and algorithm selection. Empty far partitions therefore have zero evaluated approximation error while remaining explicitly selected multipole runs. Integral/contraction roundoff, finite domains, fitting-basis error and density relaxation remain separate.

This shell-leaf implementation is an integration baseline for hierarchical periodic RI-BUPO. It does not yet group translated products into bubbles or perform upward/downward translations. Conservative Gaussian derivative bounds can make higher orders less admissible; an order increase is not a monotonic accuracy or performance guarantee. The fixed source is compatible with shared Fock composition and full-orbit validation, but supplies no symmetry reduction authority or BIPOLE/CRYSTAL parity certificate.

These component records can feed the shared method-owned Fock build. They do not supply an exchange energy, SCF state, terminal-density criterion, BIPOLE parity claim or symmetry certificate. Whole-mesh fitted method selection, physical range-separated fits and projector symmetry/response qualification remain separate acceptance steps.

Whole-mesh RI target selection is fixed before metric construction. J-only retains Gamma and no off-diagonal transfer owners; K-only retains Gamma diagonal exchange and all selected source-to-target transfers. Absent components have no output storage or energy. The finite-source identity records J/K selection, while the evaluated spin-density and AO/Bloch frame identities remain independent. A zero coefficient is not a target selector. Each q leaf retains its conservative joint workspace admission. These components may be composed only with separately admitted sources, preserving their reciprocal orientation, finite supports and method-owned corrections. No BIPOLE support, source equality or symmetry reduction follows from sharing the completed Fock consumer.

Periodic RI uses \(V_{PQ}(q)\) and three-center tensors with the actual Bloch characters and compensated auxiliary charges of the selected producer. At \(q=0\), the chosen Green metric is semidefinite on charge modes; a fit must state how it represents/removes that mode and preserves the electron count. Use the source’s compensation or an explicit constrained solve, for example a KKT system enforcing \(Q^Tc=N_e\), only when that is the specified fit. Do not insert the constraint into an existing fit silently. RI-BUPO first accelerates q=0 J’s coupling using a paired approximation. q-dependent exchange factors remain separate. Removing an auxiliary mode by threshold can change symmetry, charge and numerical source identity.

Hartree zero average and exchange exxdiv are different operations. Full Coulomb K has an integrable \(q+G=0\) singularity whose finite-mesh treatment (such as the method’s Ewald/Madelung correction) is explicit and validated. Finite-BvK corrections can couple through overlap/density, not merely add a constant eigenvalue shift. Retain existing seam coefficients and spin factors. For erfc exchange the Fourier kernel \(4\pi[1-e^{-Q^2/(4\omega^2)}]/Q^2\) has finite limit \(\pi/\omega^2\) at \(Q=0\); it must not inherit the full-Coulomb singular correction. erf retains a singular long-range part. BUPO-J makes no exchange approximation decision and cannot make a hybrid calculation K-free.

The admitted explicit-source range-separated KS method resolves the complete functional list to \(A/r+\sum_j B_j\operatorname{erfc}(\omega_jr)/r\). For each nonzero range, the supplied nonzero Q support is extended by the finite screened zero at each diagonal k transfer. This term retains its ordinary source-k weight and has no singular correction. Only the total Coulomb coefficient \(A\) multiplies the chosen BvK probe-charge correction, \(\Delta F_{\sigma k}=-A\xi S_kD_{\sigma k}S_k/f\), where \(f=2\) for restricted total density and \(f=1\) for unrestricted spin density. Its energy is \(\tfrac12\sum_{\sigma k}w_k\operatorname{Re}\operatorname{Tr} (D_{\sigma k}\Delta F_{\sigma k})\). Strict-zero-mode omits that correction; screened-only exchange has no such choice in its actual source.

The distinction between singular Coulomb and nonsingular screened quadrature follows Sundararaman and Arias, Physical Review B 87, 165122 (2013), Section II.A. The canonical decomposition defines this particular finite source: rewriting erf as full Coulomb minus erfc retains the subtracted finite screened zero. Separately omitting the zero of both leaves, or choosing a range-dependent probe-charge prescription, would change the finite-mesh Hamiltonian. Independent contraction/SCF acceptance establishes this stated convention; it does not establish continuum accuracy or parity with a different prescription. Component receipts bind every coefficient, range, support and spin convention and keep raw K distinct from method-scaled energy.

The cutoff factory reuses the same certified nonzero G/Q spheres as HF and global-hybrid KS and records that geometric child identity. Its common Q cutoff applies to every resolved exchange range; only the method adds finite screened zeros. Canonical XC images remain separate from the physical pair translation source. Joint atomic-grid generation preserves the same evaluated Hamiltonian as explicit composition, while reserving all source owners and generation work through confirmation. Neither unchanged geometry identity nor agreement on one finite grid establishes source convergence or a symmetry certificate.

1D and 2D sources require their own Green functions, long-wavelength terms, open-direction boundary conditions and dimensional stress definitions. The 3D formula above with a long vacuum vector is a convergence experiment, not a definition of those methods. Reuse separately validated slab/wire source implementations where available; otherwise refuse until derived. There is no automatic lifting of current low-dimensional chi refusals.

6. CCM and chi adapters preserve operator identity

Store source_kind, domain selector and actual supports/weights, BvK extension, character normalization, charge ledger, fit projector, nuclear boundary and exchange seam in every build receipt. Existing routing distinguishes:

• An isolated finite cluster has free-space kernels and finite electron/nuclear sources. It may directly use molecular BUPO; no periodic completion is added.

• four-center CCM is the union-and-weight/WSSC lineage. union12, alternative weights and padded domains must remain separately named. Its known nonvariational source cannot be turned into a variational one by calling its matrix Hermitian. Preserve J/K and the historical energy expression; refuse a variational derivative claim where the source lacks that property.

• Neutral neutral-bloch and real-gamma are two representations of the same specified neutral fitted torus under their common conventions. A Fourier representation test is valid here. It does not identify that Hamiltonian with the union-weight source, chi, or infinite periodic HF.

• chi’s finite-character Hamiltonian has Wigner-Seitz tie weights forming a partition of unity over translation classes. Those are not arbitrary center-dependent multipliers on free-space ERIs. four_center, ri and rijcosx preserve their specified finite source plus their declared fit/K approximation; they are not permission to use a generic periodic surrogate.

• SECCM variants retain their parameterized two-center/self/multipole models. MANTA can only accelerate an explicitly equivalent electrostatic term. It does not replace their Hamiltonians with Gaussian ab initio integrals.

A weighted quartet functional can use BUPO only if its weights admit the required source-target factorization (or a bounded sum of such channels). Derive that factorization, including permutation/adjoint relations, or keep that part direct. A weight depending on all four centers cannot be pulled inside one density multipole by assumption. RI substitutions also need a defined same-weight metric and three-center construction; free-space fitting is not automatically a fit of the weighted Hamiltonian.

Block-circulant torus actions stream cell-pair blocks and finite characters. Small explicit supercell matrices are capped test oracles only. Source restrictions, post-HF occupied/virtual spaces and experimental labels survive provider changes. Neither an eigensolver migration nor MANTA parity closes a pre-existing physical-support gate.

7. Symmetry in generation and digestion

Use existing FiniteGroup, SpaceIdentity, BlockSpaceAction, Orbit, OperatorContract, QualificationEvidence, scatter/gather-adjoint and retained-projector audits. A MANTA plan carries actual shell-pair and quartet orbits with tensor actions on every leg. Integral permutation symmetry, spatial symmetry and time reversal are separate operations with stabilizers. Cartesian AO rotations are generally not Euclidean-unitary; use the metric contract. For localized CCM/correlation spaces allow dense mixing, not only pair permutations.

Generation computes representatives only after finite-source admission. Digestion either reconstructs bounded tiles before the unreduced consumer or gathers density panels with the adjoint action, contracts representatives and scatters target blocks. Prove the gather/scatter pairing and count stabilizers once, including coincident shells and even-mesh fixed images. A scalar orbit size cannot replace tensor rotations or antiunitary conjugation.

The method qualifies \(F[g.D]=g.F[D]\) for arbitrary trial densities with covariant/contravariant transport. Invariance of a converged Fock requires state invariance too. A density-dependent screen must be orbit-consistent; take the maximum bound over the orbit or execute it unreduced. Do not average the density or Fock to manufacture a passing residual.

Specific approximation gates are mandatory:

• Screening and finite image/G supports retain complete admitted orbits or use the subgroup preserving the actual support. Orbit-completing a cutoff changes the finite source and needs separate physical acceptance.

• Generic K-means, seeds and tie-breaking do not guarantee symmetric bubbles. Build mapped bubble families or use unreduced BUPO; retain multipole blocks with all m at each l and consistent orders over mapped nodes. Test raw covariance of the approximate source, not only the exact integral provider.

• Auxiliary rank cuts must transport the retained projector. Degenerate eigenspaces allow dense gauge rotations; splitting a degenerate multiplet by a threshold can fail admission. Do not average metric eigenvalues to repair it. Existing SR/LR GDF audits are reusable diagnostics, not production authorizations.

• Periodic truncations, atom images, q wraps and finite-character weights must satisfy exact group/cocycle laws. Subspace/operator leakage evidence remains attached when further selections are made.

Retain an identity-only/unreduced path for the same basis, finite source, fit rank, bubble approximation and numerical settings. Reduced-versus-full error is measured separately from BUPO-versus-analytic error. A qualified symmetry plan may still be slower; benchmark before selecting it automatically.

8. Derivative obligations

At a stationary variational SCF state, differentiate the selected approximate energy, including overlap Pulay terms \(-\operatorname{Tr}(W\,dS)\), core, J/K, nuclear and XC contributions. Write the locally fixed-rank RI inverse as \(G=U\operatorname{diag}(g_i)U^T\), with \(g_i=1/\lambda_i\) for retained positive metric modes and zero for discarded modes. Then

\[dE_J=c^T db+\tfrac12 b^T(dG)b,\qquad c=Gb,\quad b=\widetilde B d,\]

\[dG=U[L\circ(U^T(dV)U)]U^T.\]

Here \(L_{ij}=-1/(\lambda_i\lambda_j)\) for two retained modes, zero for two discarded modes, and \((g_i-g_j)/(\lambda_i-\lambda_j)\) between the two spaces. The retained-projector response uses the same divided difference with the retention indicator in place of \(g_i\). This is the spectral Frechet derivative (Higham, Functions of Matrices, 2008, Chapter 3). Degenerate eigenvalues within either space require no eigenvector gauge derivative. The ordinary inverse formula \(dG=-G(dV)G\), and hence \(dE_J=c^T db-\tfrac12c^T(dV)c\), holds for a full-rank inverse. For a truncated inverse it omits cross-space response unless the contraction makes those terms vanish, for example when \(b\) lies entirely in the retained space.

Use adjoints/real parts in complex form. The explicit integral/geometry derivative of b is distinguished from density response. Include orbital and auxiliary center derivatives, contraction/normalization derivatives when parameters vary, retained-projector motion and any charge constraint/compensation response. A rank crossing is a nonsmooth event requiring documented handling, not omission of \(dV\). The dense molecular RIMetricResponse primitive implements the real symmetric response and its Frobenius adjoint, with the same factorization as RIJKBuilder. It refuses cutoff distances at or below its documented floating-point spectral guard; this is a numerical admission test, not an interval error certificate.

BUPO derivatives include pair/bubble centers, multipole integrals, translations and interaction tensors. Hard screening, near/far transitions, clustering changes and order changes can introduce discontinuities. Either differentiate an explicitly smooth partition (including its weights) or qualify frozen-topology derivatives with crossing detection and convergence of the discontinuity. Freeze neither geometry-dependent centers nor basis derivatives merely because the traversal topology is frozen. Incremental state is reset for displaced geometries.

Stress additionally differentiates image vectors, reciprocal vectors/volume, Green/lattice coefficients, the polynomial/background and nuclear self terms, exchange seam, and basis/Pulay terms. State whether k points are fixed in fractional or Cartesian coordinates: under fixed fractional k, phase changes must include the compensating reciprocal transformation. Metallic smearing requires free-energy/occupation terms; q and quadrature identities must remain consistent. Low-dimensional stress cannot use an arbitrary vacuum volume as a physical normalization.

CCM derivatives also track domain weights, tied Wigner-Seitz representatives and finite-source boundaries; each source’s variational status governs what can be claimed. Post-HF forces need their own response/Lagrangian terms. Validate distorted geometries, cell strains and boundary crossings separately from energy parity. Zero forces at a symmetric equilibrium are not sufficient.

Explicit periodic atomic quadrature source

The bounded 3D atomic-grid adapter canonicalizes nuclei, reuses the shared point-centered periodic partition, and folds raw points into the canonical cell without changing weights. The resulting periodization of a finite XC image source is part of its definition, not an assertion of infinite-image accuracy. The source digest binds actual coordinates and weights along with its radial/angular/partition choices. Grid population or volume agreement cannot replace separate XC energy and spin/k potential refinement. The adapter’s modeled resource bounds include current OpenMP worker scratch and one owned export of each array. Separate KS calls must count the retained grid as external live memory. The joint grid/domain/KS plan now reserves that storage and source copies, subtracts grid work from downstream budgets, and admits all stages before point generation. It preserves the same finite Hamiltonian and does not turn SCF confirmation into quadrature accuracy qualification.

A frozen-density refinement assessment now admits all requested grid generations and XC contractions together. Basis, canonical image domain, functional, spin densities, k points, weights and AO frame stay fixed. The reference grid is an explicit input; it is evaluated first and its potential is retained while other grids are streamed. Each candidate reports signed energy and spin-population differences, and maximum-entry and Frobenius norms of every spin/k potential difference. These are source-AO matrix norms with no k weights inside them, not overlap-orthogonalized or basis-independent bounds. Only XC changes in this assessment, not the complete Fock or SCF density.

An explicitly named hybrid assessment shares this comparator while evaluating only the semilocal component. It admits the complete native LDA/GGA descriptor and records the required global exchange fraction or signed range-kernel list. At fixed density and unchanged grid-independent exchange, its difference is also the grid-induced difference of the complete physical Fock and energy. The exported reference potential itself remains partial. Exchange, singular corrections and other terms belong to the method that assembles the full build.

The fixed-state density identity excludes quadrature selection. The ordered grid/source identities and reference index identify the comparison protocol. Neither zero reference-row differences nor identical neighboring grids prove accuracy. A separately refined integration witness must qualify the chosen reference in energy and every spin/k potential. Charge agreement alone is insufficient. Finite-image and Hamiltonian errors are outside this diagnostic. It produces component evidence for a method-owned Fock build, not a second operator or solver contract. The shared grid and explicit full source are preserved; this adds neither BIPOLE/CRYSTAL parity nor symmetry admission.

Public KS snapshot and neighboring workstreams

Selected public pure, global-hybrid and range-separated KS jobs preserve the same finite joint source and method-supplied component energies. The physical Fock belongs to its evaluated density epoch; the orbital projector produced by diagonalizing that Fock is a separate state at a capped exit. Hybrid exchange energies already contain the method coefficient and restricted spin factor; the child K receipt remains unscaled. Semilocal XC and exact exchange are separate contributions, and the exchange total includes its explicitly recorded zero-mode correction. Range-separated jobs retain each signed kernel coefficient, range and raw child receipt, with separately scaled component energy and actual/requested singular conventions. A screened-only source has no active Coulomb correction. The generic Fock object’s commutator diagnostic does not replace the native method’s convergence norm, threshold or exact-confirmation gate. Source, density, Fock, quadrature and basis frame identities remain separate. A generic Hermitian KS operator may own a copy of one spin/k snapshot for diagnostics, without moving source ownership or adding copies inside the bounded native solver. A symmetry label cannot qualify the grid, image domain, electronic state or retained projector.

The shared periodic grid and midpoint nuclear kernels retain compatibility with BIPOLE’s physical definitions, but source matching and the independent CRYSTAL parity gate remain prerequisites for comparison. Selected hybrid KS uses explicit reciprocal Q spheres over the full mesh; these cannot qualify or replace BIPOLE’s independent AC/BD exchange supports. No RHF-only state contract is relabeled as KS to obtain symmetry admission. These public jobs retain the full mesh until source-specific symmetry proofs are supplied.

Physical range fitting component boundary

The dense and streamed periodic GDF interfaces accept a distinct PeriodicGDFKernelOptions physical interaction. Coulomb defaults retain their existing identities and counts. Physical erfc combines numerical erfc SR with the difference of numerical and physical erf Fourier kernels, using a stable exponential difference. Its finite Gamma coefficient is retained once. Physical erf is reciprocal-only and omits the singular zero. Receipt fields separate Gamma transfer, removed numerical zero, included physical zero and executed real-space source. RI-q admission uses Gamma transfer itself and exports the actual kernel/range; it never equates Gamma with SR subtraction.

The physical range participates in source identity. Neither interaction choice nor unused erf split controls relabel the AO frame or evaluated density. Metric projectors remain fixed and negative-eigenvalue guards are unchanged. Native ownership is still one metric or one coupling tile; conservative Coulomb admission also bounds these source variants. No complete dense generic Fock snapshot is introduced inside a bounded solve. Method-level range KS, Madelung ownership and joint solver admission are qualified separately below. Source support does not grant BIPOLE parity, response support or a symmetry certificate.

Whole-mesh physical range K now composes these fixed-q sources in PeriodicRIFockBuilder. Non-Coulomb source selection refuses a simultaneous J request: the caller must retain a separate bare-Coulomb Hartree source. Every q metric uses the same physical kernel/range, with unchanged reverse-support, metric, projector and inverse audits. The Gamma owner handles diagonal K and its zero convention; off-diagonal labels remain ket minus bra. Scalar weighted energies, evaluated-density/frame identity and owned snapshots keep the existing contract. This component supplies neither a method singular correction nor joint admission with complementary J/XC sources.

The explicit fitted range KS method now supplies that complete composition. Its functional resolves to the same canonical Coulomb-plus-erfc terms as direct range KS before fitting. Each kernel/range has its own positive-metric projector; signed coefficients scale the resulting variational responses. This defines the finite fitted functional. Fitting erf directly is not assumed equivalent to subtracting separately fitted Coulomb and erfc responses. Bare-Coulomb J remains separate, with exact/RI choices independent of K. The erfc metric source owns its finite Gamma zero. Only the resolved long-range Coulomb coefficient scales the selected Madelung energy and potential.

All retained kernel sources, metric setup, component receipts, workspaces, solver iterations and terminal confirmation are charged together. The complete energy includes H once and the actual semilocal functional energy. Partial RI Fock receipts remain labelled components. RKS embeds half its total density in each fitting spin; UKS uses the actual evaluated spin blocks. Source, AO frame and evaluated-density identities remain distinct. Exported shared FockBuild/EnergyComponents/OperatorFrame states and BlockSpaceAction transport retain the existing contracts without changing the bounded solver storage. This does not grant BIPOLE AC/BD parity or production symmetry reduction.

Generated fitted range KS preserves the same physical method. Count-only cutoff boxes and atomic quadrature plans jointly admit geometry traversal, retained grid/domain payloads, every kernel/projector and the complete solver. Geometry identity remains distinct from physical kernel and projector identity; changing functional range or metric cutoff need not change the generated support. RI K geometry has empty diagonal segments and ket-minus-bra transfers, while exact K keeps primed target-minus-source supports. Physical finite-zero and method singular-correction ownership is unchanged. Selected provenance reserves and records all actual kernel sources, grid and domain receipts. Finite-source agreement does not qualify cutoff convergence. Public fitted range-KS jobs add separate job/output admission and evaluated-state delivery. J and physical K fitting identities remain an ordered collection, including the absent J-fit case. Component evidence preserves actual physical ranges and projectors; no aggregate fit is invented. QVF retains the evaluated spin densities in the nuclear-centre gauge, including capped solves. Public selection does not expand source-specific symmetry or external parity claims.

Typed periodic bubble hierarchy and weighted adjoint

PeriodicBubbleTree accepts only a native PeriodicPairSource. Its private geometry adapter preserves the complete retained pair-image index map, including diagonal images and conjugate-orbit multiplicities. It does not relax the public molecular BubbleTree triangular-inventory requirement. Tree identity includes the pair producer’s basis, image/cutoff, lattice, momentum/weight and extent identities, plus the deterministic partition. Stable median partitioning, child-sphere enclosure, always-near propagation and disjoint unordered traversal reuse the qualified geometry algorithm. A geometric far_candidate still requires a periodic kernel error proof; separation of reference-cell centers alone is insufficient.

Let \(W_{ab}=W(C_a-C_b)\) translate full raw Cartesian moments from center \(a\) to center \(b\). First contract the entire complex Hermitian density mesh with the source’s original weights and phases. No relation between \(D(k)\) and \(D(-k)\) is imposed. Leaf and parent moments then satisfy

\[Q_n=\sum_{p\in n}W_{pn}Q_p,\qquad Q_n=\sum_{c\in\operatorname{children}(n)}W_{cn}Q_c.\]

The implementation streams one translation matrix at a time, with no global node-to-AO map or retained matrix per tree edge. For arbitrary node potentials \(V_n\), the reverse pass adds \(W_{cn}^{T}V_n\) to each child, then \(W_{pn}^{T}V_n\) to each pair, and invokes the original pair-source adjoint. It is the derivative of \(\sum_n Q_n^T V_n\) with respect to the weighted mesh inner product. Potentials supplied at multiple ancestors contribute additively; a physical interaction consumer must choose a disjoint interaction partition. Raw Cartesian trace moments remain present in both passes. These are the ordinary moment translations used by BUPO (Neese et al., 2025, DOI 10.1021/acs.jpca.4c07415), composed with the q=0 Bloch source, not a new symmetry or time-reversal reduction.

Density-dependent absolute full and tail radial moments are propagated with positive binomial bounds. For child-to-parent separation \(d\),

\[B_{n,k}\le\sum_c\sum_{j=0}^k {k\choose j}d^{k-j}B_{c,j}.\]

Here the right side defines the reported bound; at a leaf each pair bound is multiplied by the absolute component-coefficient sum from the native weighted density contraction. The same expression applies to tails because the parent ball encloses every child ball: outside the parent, each child’s tail bound applies. Absolute coefficient sums propagate without cancellation. Logarithmic positive accumulation retains subnormal bounds and refuses unrepresentable results. Floating-point headroom is not interval arithmetic.

Joint count-only admission includes the existing source reservation, tree construction/copies, one forward/reverse cycle, all node result/input and correction arrays, radial bounds, owned exports and one translation matrix. The plan’s conservative work bounds include the existing source plan; actual shell and translation counts are reported separately. No input array is read or copied until its shape matches the pre-admitted source/node dimensions. Retaining arbitrarily many independent results remains the caller’s budget.

Qualification periodic-pair-image-hierarchy-and-adjoint covers the retained finite source and its geometry/moment operations. Complete finite reciprocal Coulomb calculations validate the composition against independent ordered Gaussian Bloch sums and weighted energy derivatives. Their Fourier domain, zero mode and Taylor approximation are declared explicitly. These calculations do not qualify omitted pair images, an infinite Ewald accuracy profile, Gaussian-kernel far admission, charge/background/self selection, exchange, derivatives of moving geometry, or a production periodic SCF route. The shared FockBuild/JKBuilder and BIPOLE domain contracts are unchanged.

5.6 Direct finite Ewald response over canonical pair images

PeriodicEwaldSource retains the same immutable native PeriodicPairSource as the periodic hierarchy. It supplies the complete direct finite-domain quadratic form needed for a later kernel-admitted fallback. Its real and reciprocal integer domains are unique and inversion closed; the real domain contains zero and the reciprocal domain excludes it. An empty reciprocal domain is allowed as an explicitly incomplete finite source. Cell vectors, eta and the gauge follow Section 5.2. Shared domain validation and hashes are identical to the lattice-tail implementation. The full-kernel identity excludes a near partition, since this direct source contains none.

For retained pair-image component \(u\), let \(f_u(r)\) be its real Gaussian product and \(c_u\) its native whole-mesh coefficient,

\[\begin{split}c_{p,ij}=m_p\operatorname{Re}\sum_k w_k D_{ij}(k)e^{-ik\cdot R_p}, \qquad m_p=\begin{cases}2&\text{paired image orbit},\\1&\text{otherwise}.\end{cases}\end{split}\]

The kernel matrix is the full Gaussian integral of the finite Green function,

\[I_{uv}=\sum_{R\in\mathcal R}(f_u|\operatorname{erfc}(\eta|r-r'+R|)/|r-r'+R||f_v) +\frac{4\pi}{\Omega}\sum_{G\in\mathcal G} \frac{e^{-G^2/(4\eta^2)}}{G^2}\operatorname{Re}[F_u(G)^*F_v(G)] -\frac{\pi}{\eta^2\Omega}q_uq_v,\]

where \(F_u(G)=\int f_u(r)e^{-iG\cdot r}\,dr\) and \(q_u=F_u(0)\). The energy is \(E_J=\tfrac12 c^TIc\). Each canonical pair block is visited once. An off-diagonal block contributes its action and transpose to the two pair potentials. For a diagonal block, inversion closure makes its complete real-domain sum symmetric; applying \((I+I^T)/2\) while digesting the images implements that identity with a fixed numerical Hessian. It does not repair or discard a computed non-Hermitian Fock matrix. All off-diagonal AO components in an origin same-shell pair remain present.

Real-space terms use native contracted erfc shell quartets directly, including overlapping and identical Gaussian sources. Both shells of the source pair are translated by the same -cell @ index. No bare-minus-erf subtraction, point-charge singularity or Gaussian electronic self removal is introduced. One quartet is live at a time. Each reciprocal tile is streamed twice, first to contract \(\rho(G)=\sum_u c_u F_u(G)\), then to apply its adjoint to all pair components. No all-pair Fourier panel or global four-index array is retained. Overlap panels provide the charge and the gauge response. Real, reciprocal and gauge energies are reported separately; the constant occurs exactly once. The algorithm uses the Ewald split (Ewald 1921, IV.1, Eqs. 37-40), Gaussian MD integrals and the q=0 Bloch contraction of Sun et al. (2017), Eqs. 16-17, with the declared component convention.

The original phase map scatters pair potentials to complex \(J(k)\). Its weighted derivative is

\[dE_J=\sum_k w_k\operatorname{Re}\operatorname{Tr}[dD(k)J(k)],\qquad E_J=\tfrac12\sum_k w_k\operatorname{Re}\operatorname{Tr}[D(k)J(k)].\]

Neither positivity nor a relation between \(D(k)\) and \(D(-k)\) is assumed. An unrestricted consumer supplies total density \(D_\alpha+D_\beta\) for J; exchange remains separate. Source and density identities match the original pair map; the new source also binds the finite kernel. Moment order, extent radii or reciprocal batching do not change these exact Gaussian integrals. They are not multipole approximations.

Qualification finite-canonical-pair-ewald-j is deliberately finite. Canonical folding is exact against a periodic potential, while a finite real Ewald sum has a covariance defect. This response is the exact derivative of the declared folded finite quadratic form, not proof that an arbitrary short real domain equals an ordered infinite Bloch calculation. Domain enlargement and separate native Gaussian bounds remain necessary. Omitted AO pair images, full-cell charge/background selection, whole-method accuracy, nuclear energy, exchange and production SCF selection are not certified here. No interval roundoff bound is claimed. Kernel-admitted bubble traversal and aggregation of these error sources remain required before production periodic BUPO.

Count-only admission precedes domain copies and all density payload reads. It reserves the retained pair source, input/output/correction arrays, domain sorting, shell snapshots, two local coefficient/potential panels, one quartet or Fourier tile and independent exports. The existing full-basis Fourier plan conservatively bounds each streamed shell panel; its work reservation can exceed executed work. It creates no corresponding global panel cache. All five resource caps are enforced and actual shell/primitive calls are reported separately. Shared FockBuild/UHF, physical BIPOLE AC/BD and exchange domains, and source-qualified symmetry dispatch remain unchanged.

5.7 Bound-admitted finite periodic BUPO response

PeriodicBupoSource retains a native PeriodicBubbleTree and PeriodicEwaldSource with exactly matching pair-source identities. The partition is fixed at construction by geometry, the finite kernel, moment order and an absolute integral tolerance. It never depends on the current density. This composes the BUPO separation of Neese et al. (2025, DOI 10.1021/acs.jpca.4c07415) with the Ewald convention of Section 5.6 and the q=0 Bloch coefficients of Sun et al. (2017), Eqs. 16-17.

A geometric far candidate is tested against every retained real image. The support spheres must be separated at all singular images, with a roundoff guard on translated-center separation. A near set contains the origin, is unique and inversion closed, and is a subset of the direct real domain. For candidate nodes A and B, the approximate matrix is

\[T_{AB}=\sum_{R\in\mathcal N}T^{1/r}_{AB}(R) +T^{G_{\mathcal R,\mathcal G}-\sum_{R\in\mathcal N}1/r}_{AB}.\]

The native Gaussian interaction ledger supplies compact Taylor, penetration and full-moment tail bounds for each term. Their positive sum must be no larger than integral_tolerance; otherwise the traversal refines, eventually falling back to exact finite Gaussian Ewald integrals. A nonrepresentable far field also refines. Every unordered canonical pair interaction occurs once in either an admitted far block or a direct block.

Node error envelopes are density-independent maxima over descendant AO components, translated with the binomial theorem. They differ from the density-weighted bounds returned by tree.contract. If their interaction bound is \(\delta_{AB}\), it bounds the error of each descendant component integral uniformly. With \(L_A=\sum_{u\in A}|c_u|\) and the same definition for B, disjoint off-diagonal far blocks give

\[|\Delta E_J|\le\sum_{(A,B)\in\mathcal F}\delta_{AB}L_A L_B,\qquad |\Delta v_u|\le\sum_{(A,B)\in\mathcal F,\ u\in A}\delta_{AB}L_B +\sum_{(A,B)\in\mathcal F,\ u\in B}\delta_{AB}L_A.\]

Ancestor errors add down to the leaves. Absolute Bloch phases are one, so scattering these component errors supplies a uniform-over-k AO matrix j_error_bound. Both canonical conjugate arms contribute, including the two arms of a nonzero-image diagonal AO component. Positive logarithmic accumulation keeps underflowed nonzero bounds positive. The bounds exclude floating-point roundoff and do not constitute interval arithmetic.

The forward pass aggregates full raw Cartesian Gaussian moments. For each far block, the local potentials receive \(T_{AB}Q_B\) and \(T_{AB}^TQ_A\) and the energy receives \(Q_A^TT_{AB}Q_B\). The transposed hierarchy and original pair adjoint recover J. No trace projection is valid for the nonharmonic complementary Ewald kernel. Direct blocks reuse the same erfc, reciprocal and gauge primitives as Section 5.6. Near plus tail contains each reciprocal and gauge contribution exactly once. With no accepted far blocks, the source calls the existing complete direct builder.

The density-independent partition therefore defines one fixed symmetric finite quadratic form. Its J is linear in D and satisfies the weighted first derivative and half-trace identities of Section 5.6. Recomputed partition identities must match construction. The receipt separates direct and far energies, reports independently evaluated trace energy, and binds the pair source, tree, finite kernel, partition, tolerance and density. For a mixed partition, the nested direct receipt has a distinct subset source identity and qualification finite-ewald-direct-partition-subset. Its real/reciprocal/gauge energies and trace concern only direct blocks; its charge is that of the complete retained source.

Count-only joint admission includes retained producers, Gaussian envelopes, worst-case direct blocks, two traversal/admission passes, forward and reverse moments, corrections and owned exports. One M2L matrix and local shell or Fourier panels are live; there is no per-edge matrix cache or global four-index tensor. Conservative work counts are reservations, not FLOPs or measured RSS. Many retained result copies remain the caller’s responsibility.

Qualification finite-periodic-bupo-j compares the approximation to exactly the finite canonical kernel supplied by the matching direct source. integral_tolerance is a component-integral threshold, not an SCF or total-energy accuracy profile. The receipt’s density-dependent aggregate bounds must be assessed separately. Infinite Ewald/domain and omitted pair-image errors, canonical finite-domain covariance, full-cell charge and nuclear policy, RI fitting and production selection remain separate work. Total alpha plus beta density supplies UHF J; exchange, shared FockBuild, physical BIPOLE domains and qualified symmetry dispatch are unchanged.

5.8 Retained-source infinite-Ewald error aggregation

PeriodicBupoOptions.bound_infinite_domain=True adds the full Gaussian domain bounds of Section 5.2 to the same finite response from Section 5.7. It does not change the partition, moments, J or energy. The default remains the finite-source mode, which also admits non-cubic finite domains.

The native consumer constructs a full Gaussian majorant from each retained pair’s original immutable basis, selected shells and represented translation. The caller cannot inject a numerical envelope. Canonical domain validation, complete-cube checks and conservative lattice metrics are prepared once. Every pair of majorant centers must be separated from the omitted real images, even if the current density is zero. Construction refuses a domain that cannot establish this condition. Majorants require linear storage in the retained primitive-pair inventory; component interaction bounds stream without a quadratic cache.

Let \(\epsilon^R_{pq}\) and \(\epsilon^G_{pq}\) be the uniform full-Gaussian real and reciprocal domain bounds for every AO component of retained pair entries p and q. They refer to the physical Ewald eta and the exact same finite real/reciprocal domains as the direct source. Near subtraction and the constant gauge cancel in the infinite-minus-finite difference. With \(L_p=\sum_{u\in p}|c_u|\), the consumer reports

\[B_E^X=\tfrac12\sum_p \epsilon^X_{pp}L_p^2 +\sum_{p<q}\epsilon^X_{pq}L_pL_q,\qquad X\in\{R,G\},\]

\[B_{v,p}=\sum_q(\epsilon^R_{pq}+\epsilon^G_{pq})L_q.\]

These bounds use full signed component coefficient norms, including original mesh weights and canonical multiplicities. The diagonal energy factor is one half; the diagonal potential factor is one. Each component of pair p receives \(B_{v,p}\), then the absolute original Bloch scatter adds both conjugate arms to a uniform-over-k AO matrix. A nonzero-image diagonal AO component contributes twice. Positive logarithmic sums/products preserve nonzero underflowed bounds. Separate normalized coefficient scales avoid premature underflow in the density norm.

The result’s domain_j_error_bound and nested receipt.domain are present only when the option is enabled. The nested receipt reports real/reciprocal energy bounds, their sum, actual pair/majorant counts, minimum effective decay and omitted-center distance, and source/majorant/domain/kernel/density identities. Its source identity excludes the unrelated near partition. finite_energy_error_bound and finite_maximum_j_error_bound retain the Section 5.7 approximation bounds. Overall energy_error_bound and j_error_bound add finite BUPO and infinite-domain errors outward. Density changes scale these bounds without changing the computed Hessian.

Joint count-only admission adds majorant production, construction and build domain passes, coefficient norms, retained envelopes and owned AO bound arrays. It occurs before reading near payloads or copying density arrays. Domain geometry is not rebuilt for every primitive or pair interaction. The conservative work model includes worst-case component evaluations; it is not measured work or a production scaling claim.

Qualification periodic-bupo-j-with-retained-source-ewald-bound compares the finite approximation with the infinite zero-average conducting Coulomb kernel for the same retained canonical Gaussian source. Canonical folding is exact in that periodic limit, so the domain bound also covers its finite kernel covariance defect. This does not establish that the finite pair-image inventory equals an untruncated AO density. Pair-image omission, roundoff, full electronic+nuclear charge/background/self policy, RI fitting, automatic accuracy-profile selection and geometry derivatives remain separate work. The derivation reuses the full Gaussian convolution bound and Ewald1921 IV.1 Eqs.37-40, not a compact-support assumption or point-charge self term. Shared FockBuild/UHF, physical BIPOLE domains and qualified symmetry are unchanged.

5.9 Full AO pair-image omission bounds

PeriodicBupoOptions.bound_pair_images=True additionally bounds all omitted AO pair translations for the infinite zero-average conducting Coulomb kernel. It requires the retained-source infinite-domain bound of Section 5.8. The finite calculated values and density-independent partition are unchanged. This is a native source-bound envelope, not a user-supplied error estimate.

For one primitive of angular degree l and exponent a, absorb its Cartesian polynomial into half its Gaussian using

\[r^l e^{-a r^2}\le H_l(a)e^{-a r^2/2},\qquad H_0(a)=1,\quad H_l(a)=\left(\frac{l}{ae}\right)^{l/2}\ (l>0).\]

The maximum absolute row sum of the Cartesian-to-shell transformation bounds every pure or Cartesian component. For primitive exponents a,b, absolute contraction coefficients and these row norms give a positive amplitude M. Their product is bounded by a spherical Gaussian of exponent \(\alpha=(a+b)/2\), product center P and absolute integral

\[q(n)=M H_{l_a}(a)H_{l_b}(b) (\pi/\alpha)^{3/2}e^{-\mu|A-B-Ln|^2},\qquad \mu=\frac{ab}{2(a+b)}.\]

For a plain s-s pair the polynomial is constant, so no absorption is necessary. Let A be the largest primitive exponent with nonzero coefficient in the basis. The original half-Gaussian model already has maximum exponent A, from that primitive paired with itself. Choose \(t=\min(1,A/(a+b))\), which lies in \([1/2,1]\). For a constant polynomial, reducing both primitive exponents by t gives a pointwise majorant with no extra prefactor. Its product mass has \(\alpha=t(a+b)\), \(\mu=tab/(a+b)\) and amplitude \(M(\pi/\alpha)^{3/2}\). This uses the exact product exponents when \(a+b\le A\) and otherwise the tightest common scaling that stays within the original kernel ceiling. This keeps the common kernel bound unchanged while reducing or preserving every positive retained and omitted mass, so image J/energy enclosures cannot worsen, including mixed angular bases. The maximum-exponent scan fits within the existing primitive-work reservation. Contraction coefficients still enter through absolute primitive products, including signed contractions. Angular pairs keep the half-Gaussian envelope above. Gradient and Fourier-contour producers retain their separate half-Gaussian formulas; the derivative of an s Gaussian has a nonconstant polynomial.

All primitive contributions add positively. No signed cancellation, extent radius or asymptotic compact-support assumption enters this envelope. The weaker separation decay is intentional: it removes image-dependent angular polynomials while still giving a convergent all-image sum.

Let N be the largest absolute integer coordinate in the original admitted image inventory, before applying the physical pair cutoff. Complete the integer cube of radius N, recognize retained entries through the original pair involution and sum every missing entry inside it explicitly. This handles sparse input images, interior holes and cutoff exclusions. Every image outside the cube is missing. Retaining the original image inventory lets a larger declared cube tighten the omitted tail even when its extra pairs fail the physical cutoff. This changes the error enclosure and admitted work without changing the finite Hamiltonian. Never subtract one unrelated upper bound from another to estimate the missing source.

For the admitted lower real metric s, put \(a_0=\mu s^2\), \(\tau=|A-B|/s\) and \(u=N+1-\tau\). Each shifted one-dimensional Gaussian sum is bounded by \(\Theta=2+\sqrt{\pi/a_0}\), from the two nearest samples and the integral of the remaining monotone tails. Thus the omitted 3D sum is at most \(\Theta^3\). If \(u>0\), the union of three omitted coordinate slabs gives the additional bound

\[6\Theta^2 e^{-a_0u^2}\left(1+\frac{1}{2a_0u}\right).\]

Use the smaller of these bounds. The second follows by bounding each half-line by its first term plus its integral and applying the erfc inequality from Section 5.2. It tends to zero with N. Guarded lower distances, positive logarithmic sums and conservative decay caps preserve nonzero underflowed bounds; they do not provide interval roundoff certification.

The result is an ordered AO component mass bound \(T^r_{ab}\) for the retained inventory and \(T^m_{ab}\) for its entire missing complement. For off-diagonal shells the reverse image inventory gives the transposed matrix. For a same-shell nonzero image both images enter, including twice on a diagonal AO component. The native matrices are independent of the density and k.

To propagate these masses to the infinite Coulomb action, let \(\alpha_{\max}\) bound all envelope exponents. Convolution of two normalized Gaussians has exponent \(\beta\le\alpha_{\max}/2\). Its periodization is uniformly at most \((2\sqrt{\beta/\pi}+1/s)^3\), using the shifted Gaussian sum above. Since \(\int_{\mathbb R^3}\operatorname{erfc}(\eta r)/r\,dr =\pi/\eta^2\), a uniform bound on the real Ewald interaction is

\[C_R=\frac{\pi}{\eta^2} \left(2\sqrt{\frac{\alpha_{\max}}{2\pi}}+\frac1s\right)^3.\]

Use \(V_0\le s^3\le\Omega\) as a conservative volume lower bound and the admitted reciprocal metric g. With \(t=2\sqrt\pi\eta/g\), positive reciprocal coefficients and unit phase modulus give

\[C_G=\frac{4\pi}{V_0g^2}(3t+3t^2+t^3),\qquad C_0=\frac{\pi}{\eta^2 V_0},\qquad C=C_R+C_G+C_0.\]

Here \(3t+3t^2+t^3\) bounds the nonzero integer Gaussian sum: each centered one-dimensional sum is at most \(1+t\), and \(|n|^2\ge1\) bounds the reciprocal denominator. \(C_0\) bounds the constant zero-average gauge. This proves \(|B(f,h)|\le C Q_fQ_h\) for any signed functions dominated by these Gaussian sums, independently of their centers. This derivation extends the Ewald1921 IV.1 Eqs.37-40 split; the bound is not attributed to the point-charge paper.

Define \(W_{ab}=\sum_k w_k\max(|D_{ab}(k)|,|D_{ba}(k)|)\). The maximum equals either magnitude for Hermitian D and also covers the small accepted Hermiticity residual. Set \(Q_r=\sum_{ab}W_{ab}T^r_{ab}\) and \(Q_m=\sum_{ab}W_{ab}T^m_{ab}\). Then

\[B_E^{\rm image}=C(Q_rQ_m+\tfrac12 Q_m^2),\]

\[B_{J,ab}^{\rm image} =C\left[Q_mT^r_{ab}+(Q_r+Q_m)T^m_{ab}\right].\]

The second expression includes missing target pairs as well as missing density. Bounding only the retained target response would be insufficient. Absolute Bloch phases are one, so the J bound is uniform over the original mesh, including zero-weight points. These expressions have the expected quadratic energy and linear J scaling without changing the fixed Hessian.

receipt.pair_images binds the native pair/kernel/density identities and reports the completion cube, actual counts, lattice metrics, three kernel bounds, density mass bounds and resulting errors. Owned result arrays expose retained_pair_mass_bound, omitted_pair_mass_bound and pair_image_j_error_bound. They are None when disabled. Overall bounds add finite BUPO, retained-source Ewald-domain and pair-image errors outward; the nested domain receipt continues to describe only its own contribution. Joint admission includes completion enumeration, membership lookup, primitive envelopes, contraction and owned arrays before near payloads or density copies. No dense supercell tensor or infinite image enumeration is allocated.

Qualification periodic-bupo-j-with-full-pair-image-ewald-bound compares with the full AO pair-image electronic Coulomb source under the declared zero-average kernel. Small-domain bounds can be very loose. Compare both total J and energy bounds with the desired target and enlarge domains or refuse that target; this option does not choose a production accuracy profile. This electronic-only receipt excludes roundoff, full electronic+nuclear charge/background/self policy, exchange, RI fitting, automatic SCF selection and geometry derivatives. Section 5.10 supplies the matched charge composition. The shared Fock/UHF, BIPOLE physical domains and symmetry interfaces are unchanged.

5.10 Matched electronic and point-nuclear electrostatics

PeriodicElectrostaticSource composes a native PeriodicBupoSource with finite point positions and signed charges. Both electronic options bound_infinite_domain and bound_pair_images are required. The constructor uses that source’s basis, canonical AO images, original k mesh and weights, physical eta, real and reciprocal cubes, and zero-average conducting kernel. The existing pair-midpoint nuclear cutoff driver has a different finite support and cannot supply these components. Native McMurchie-Davidson potential and Fourier kernels are reused without changing either interface.

Let the electronic density be positive electron number density, \(Q_e\) its retained charge, \(Q_n=\sum_A Z_A\), \(Q=Q_n-Q_e\), and \(c=-\pi/(\eta^2\Omega)\). The energy and its weighted density derivative are

\[E=\tfrac12 B(\rho_e,\rho_e)-B(\rho_e,\rho_n)+E_{nn}, \qquad \delta E=\sum_k w_k\operatorname{Tr}[\delta D(k)(J(k)+V_n(k))].\]

\(E_{nn}\) includes the ordered real point sum with \((A,A,0)\) omitted, \(\frac12\sum_{G\ne0}\lambda_G|\rho_n(G)|^2\), point self removal \(-\eta\sum_A Z_A^2/\sqrt\pi\), and \(cQ_n^2/2\). Gaussian electronic self interaction is finite and receives no point-self subtraction. The three gauge terms are \(cQ_e^2/2\), \(-cQ_eQ_n\), and \(cQ_n^2/2\): their sum is \(cQ^2/2\). A separate background energy must not be added again. This is the conducting three-dimensional Ewald split in Ewald (1921), IV.1 Eqs. 37-40, DOI 10.1002/andp.19213690304. There is no surface dipole term or lower-dimensional boundary claim.

The default neutral policy admits a density only when \(|Q|+Q_m\le\epsilon_Q\), where \(Q_m\) is the full omitted-image charge bound from Section 5.9. Retained \(\operatorname{Tr}(DS)=Q_n\) alone is insufficient. The alternative uniform-background policy explicitly records compensating retained charge \(-Q\) and the same uncertainty. Both policies use the same zero-average kernel; the choice changes admission and identity. The overlap and point potential are immutable owned exports. Charge admission precedes the electronic build. Charge tolerance is positive and finite. It is a physical admission tolerance, not a floating-point interval guarantee.

For point pairs use the domain proof in Section 5.8 with effective real exponent eta; for a Gaussian majorant of exponent alpha against a point use \(\eta_{\rm eff}=(\eta^{-2}+\alpha^{-1})^{-1/2}\). The Gaussian/Gaussian convolution’s second variance is absent. The omitted real domain must be separated from each retained pair of centers; unsupported domains refuse independently of the density. Reciprocal bounds use absolute point charge and the same Gaussian lattice shell bound. Gauge and point-self terms cancel between finite and infinite domains. Thus

\[B_{nn}=\tfrac12\sum_{AB}|Z_A Z_B|\,\delta_{AB},\quad \delta_p=\sum_A|Z_A|\,\delta_{pA},\quad B_{en}^{\rm domain}=\sum_p L_p\delta_p.\]

\(L_p\) is the absolute canonical density coefficient sum. Scattering \(\delta_p\) through both canonical arms gives the uniform AO potential bound. For missing electronic images use the Section 5.9 mass envelopes and replace its Gaussian/Gaussian real kernel constant with

\[C_{R,\rm point}=\frac{\pi}{\eta^2} \left(2\sqrt{\alpha_{\max}/\pi}+\frac1s\right)^3, \qquad C_{\rm point}=C_{R,\rm point}+C_G+C_0.\]

This is the periodized normalized Gaussian supremum times the integral of erfc(eta*r)/r; a point contributes no convolution variance. With \(Z_{\rm abs}=\sum_A|Z_A|\), the missing-source energy and missing-target potential errors are

\[B_{en}^{\rm image}=C_{\rm point}Z_{\rm abs}Q_m,\qquad B_{V,ab}^{\rm image}=C_{\rm point}Z_{\rm abs}T^m_{ab}.\]

These bounds are derived here; they are not attributed to the point-charge paper. Positive logarithmic sums preserve nonzero underflow. The final energy bound adds the electronic, point-domain and both attraction errors; the final matrix bound adds electronic and nuclear potential errors. No image cancellation, k/-k relation or neutrality is used to shrink an error.

plan_periodic_electrostatic admits combined source storage, shell calls, primitive products, scalar and kernel work before point payload copies. Count-based limits include exported arrays, density copies, integral/Fourier scratch, point-pair sums and envelope work. Identical or unresolved periodic nonzero point charges refuse. Zero charges are ignored in interactions. Representatives are not silently wrapped; enlarge the finite real cube when center separation prevents its error proof. Signed or fractional charges are allowed; positive atomic numbers represent physical nuclei.

Qualification full-pair-image-electronic-point-electrostatics refers to this total electrostatic component. values is J+V, excluding kinetic, exchange and XC terms, and is not a complete HF/KS Fock matrix. Roundoff, production SCF accuracy-profile selection, RI-BUPO, gradients and stress remain outside this increment. The shared Fock/UHF, BIPOLE physical-domain and qualified symmetry contracts stay unchanged; exporting this component does not establish a new symmetry certificate.

5.11 Bounded periodic HF with matched BUPO electrostatics

run_periodic_bupo_hf composes the immutable source of Sections 5.7-5.10 with native Bloch kinetic integrals and exact finite reciprocal exchange. Its scope is neutral all-electron 3D RHF or collinear UHF, with integer spin occupations on the original uniform full mesh. Both spins use the same frozen source, physical image inventory, pair cutoff, Ewald domains and partition. No density-dependent repartition or symmetry reduction occurs during SCF. The physical system must match the source lattice and shell centers modulo that lattice. The driver creates point nuclei from its atomic numbers and uses the source’s matched attraction and nuclear energy, including self and gauge terms. It does not import the older midpoint-sphere nuclear support.

Put \(H=T+V_n\) and \(D_t=D_\alpha+D_\beta\). The spin Fock and total energy are

\[F_\sigma=H+J[D_t]-K[D_\sigma]+\Delta F_\sigma,\]

\[E=E_{nn}+\sum_k w_k\operatorname{Tr}(D_tH) +E_J[D_t]-\tfrac12\sum_{\sigma k}w_k \operatorname{Tr}(D_\sigma K[D_\sigma])+E_{\rm corr}.\]

RHF uses one total density, a factor one half in front of its exchange matrix, and exchange energy \(-\frac14\sum_k w_k\operatorname{Tr}(DK[D])\). These are the existing HF spin conventions, with the fixed periodic BUPO quadratic functional replacing only J. The Gaussian Bloch exchange panels follow Sun et al. (2017), Sec. II, DOI 10.1063/1.4998644, with an explicitly supplied finite reciprocal domain. The periodic BUPO functional is the project derivation above; Neese et al. (2025), Eqs. 29-36, do not supply periodic HF, exchange or an SCF convergence proof.

For BvK-ewald, the original full mesh defines the Born-von Karman cell. The existing full-Coulomb correction is \(\Delta F_\sigma=-\xi S D_\sigma S\), with the RHF factor one half, and \(E_{\rm corr}=\frac12\sum_{\sigma k}w_k \operatorname{Tr}(D_\sigma\Delta F_\sigma)\). See Sundararaman and Arias (2013), Sec. II.A, 10.1103/PhysRevB.87.165122. There is no extra short-range erfc zero-mode subtraction in this full reciprocal exchange path. strict-zero-mode applies no such correction. Neither choice certifies convergence of the finite exchange domain.

Every evaluated density must satisfy neutral charge admission and positive finite electrostatic energy and maximum matrix error targets, both default \(10^{-8}\). The driver fails immediately when a target is exceeded. It records the last and maximum observed bounds, so a final acceptable density cannot conceal an earlier failed target. The source is never silently expanded, repartitioned or assigned a looser target. Qualification electrostatic-and-one-electron-images-at-evaluated-density;finite-exchange also includes the S/T image bounds and admission in Section 5.12. Exchange, basis, state-relaxation and roundoff errors remain excluded. This is an explicit bounded SCF consumer, not an automatic whole-HF accuracy profile.

The existing bounded solvers retain the common overlap projector, fixed occupations, coupled spin epoch and last evaluated D/F/E state. Capped runs export that evaluated density, not the next density reconstructed from the returned eigenvectors. Numerical convergence additionally requires a full native Fock and energy rebuild at the same final density. Confirmation is not a second SCF iteration. Scalar history excludes matrix snapshots.

plan_periodic_bupo_hf combines source, kinetic, exchange, Madelung, solver, scalar history and owned-export storage and work under all five resource caps before exchange payload copies. Integral counters report actual native integral calls; the larger planned limits also cover envelope and solver work. Source and density identities, complete energy components, nested charge/error and partition receipts accompany the full Fock export. Spin-major original-mesh output maps into the shared FockBuild contract. Its unreduced frame carries no symmetry certificate and changes no BIPOLE physical domain. Automatic public job/profile selection, q-aware RI-BUPO, external SCF parity, gradients and stress remain separate requirements.

5.12 Full AO-image overlap and kinetic bounds

bound_periodic_one_electron_images consumes the immutable original PeriodicPairSource. It bounds every omitted translation in overlap and kinetic matrices, including sparse interior holes and physical-cutoff exclusions. It uses no Coulomb kernel, eta, auxiliary fit, density or Bloch phase cancellation. The complete integer cube and exterior Gaussian sum are the native construction in Section 5.9, with the same checked lower lattice metric. This is a new bound derived here from the Gaussian and kinetic integral conventions, not an Ewald or BUPO formula.

For a Cartesian monomial of total degree l, Section 5.9 supplies \(|p_l(r)|e^{-a r^2}\le H_l(a)e^{-a r^2/2}\). Its gradient obeys

\[|\nabla[p_l(r)e^{-a r^2}]| \le [l r^{l-1}+2a r^{l+1}]e^{-a r^2} \le G_l(a)e^{-a r^2/2},\]

\[G_l(a)=lH_{l-1}(a)+2aH_{l+1}(a),\]

with the first term omitted for l=0. The Cartesian polynomial gradient norm is at most \(l r^{l-1}\) because the sum of squared component powers is at most \(l^2\). Absolute contraction coefficients and maximum representation row sums then bound each pure or Cartesian component and its gradient.

For one primitive pair, let C include these coefficient/representation factors, \(p=(a+b)/2\), \(\mu=ab/[2(a+b)]\) and \(d_n=|A-B-Ln|\). Bounds on the overlap and kinetic contribution of translation n are respectively

\[q_S(n)=C H_l(a)H_m(b)(\pi/p)^{3/2}e^{-\mu d_n^2},\]

\[q_T(n)=\tfrac12 C G_l(a)G_m(b)(\pi/p)^{3/2}e^{-\mu d_n^2}.\]

The second uses the exact localized-Gaussian identity \(T_{ab}=\frac12\int\nabla\phi_a\cdot\nabla\phi_b\), obtained by integration by parts with vanishing Gaussian surface terms. Both use the same decay and exterior lattice sum. This is an integral-error enclosure, not a new nuclear-gradient or stress implementation.

Summing only the disjoint missing images gives nonnegative AO matrices \(B^S,B^T\) with \(|\Delta S_{ij}(k)|\le B^S_{ij}\) and \(|\Delta T_{ij}(k)|\le B^T_{ij}\) uniformly over all k. The original pair involution restores both ordered arms, including nonzero diagonal images. No relation between states at k and minus k is assumed. Logarithmic positive sums retain a positive subnormal enclosure instead of rounding a nonzero bound to zero. Roundoff is still excluded from the scientific bound.

For the total evaluated spin density and original nonnegative weights, put \(W_{ij}=\sum_k w_k\max(|D_{ij}(k)|,|D_{ji}(k)|)\). Then

\[|\Delta E_T[D]|\le\sum_{ij}W_{ij}B^T_{ij}.\]

Periodic BUPO HF computes these immutable envelopes once after joint admission. It checks maximum overlap and kinetic matrix errors before SCF (default targets 1e-10 and 1e-8), and the positive density-weighted kinetic energy error at every evaluated state (default 1e-8 hartree). Targets must be positive and finite. The last and maximum observed kinetic energy bounds accompany the existing electrostatic ledger and confirmation. A target failure refuses; it never changes the fixed source or projector.

plan_periodic_one_electron_images admits the retained pair source, primitive gradient envelopes, both error matrices and owned exports under all five caps. Counts include finite completion enumeration and omitted primitive evaluation. The enclosing HF plan additionally budgets this setup and its retained matrices. Receipts bind the pair source, cube and metric, report maxima and counts, and declare roundoff_included=False.

These are bounds at fixed evaluated D, not a guarantee about the relaxed SCF state, overlap-rank stability or basis completeness. Exchange, including its overlap-dependent BvK correction, remains a separately finite source unless the explicit certification in Section 5.13 is requested. No whole-HF accuracy profile, public source selection or symmetry certificate follows from the new ledger. Existing Fock, BIPOLE and symmetry contracts are unchanged; q-aware RI-BUPO and the rest of the capability matrix remain.

Complete RI methods: canonical-density image admission

Periodic RI HF and KS can request three positive image targets in PeriodicOneElectronAccuracyOptions: maximum overlap entry, maximum kinetic entry and kinetic energy in hartree. All zero disables this optional ledger. The same shared image producer constructs B^S and B^T from the actual finite Bloch translations and physical shell-pair cutoff. Integer lattice labels, Cartesian reconstruction and both ordered cutoff arms are checked. A retained block absent from the physical Bloch source is a refusal, never a hidden change of domain. Integral/phase roundoff is excluded.

An explicit completion_cube_radius chooses the enclosure evaluation cube. It need not contain every retained image: the exterior positive Gaussian sum then includes some retained terms, which overbounds the missing sum. Increasing this radius does not change S, T, the fitting domain or the Hamiltonian. The legacy source-only overload still uses its original inventory radius.

The bounded solver’s shared canonical-overlap helper supplies the uniform entry envelope H[k,i,j] and total L described in the automatic RI density sections below. Both apply to every audited occupied subspace, including restricted-total density and an empty spin channel. Thus, before SCF,

max(B^S) <= overlap_error
max(B^T) <= kinetic_error
min(sum_kij H[k,i,j] B^T[i,j], L max(B^T)) <= kinetic_energy_error

The last inequality holds for the same finite canonical-overlap density family, with fixed spin occupations and original uniform full-mesh weights. It does not compare densities or overlap ranks obtained from different physical domains. The existing log-positive trace producer prevents a nonzero subnormal enclosure from becoming zero. Automatic RI coupling selection and image admission reuse one canonical density analysis; an explicit RI density override keeps its own provenance unchanged.

Count-only plans admit pair-source construction, Gaussian bounds, the canonical analysis, both error matrices, owned exports and output records before copying inputs. Source, envelope, targets and result have separate identities. Complete HF, pure KS, global hybrid and physical range compositions retain their Fock source identities and energies when this ledger is enabled; exact J with RI K is included. Non-RI public jobs refuse this option. Nuclear, two-electron, metric/solve, arithmetic and relaxed-state errors remain outside this certificate. No symmetry reduction or whole-method accuracy claim follows.

5.13 Full reciprocal and AO-image exchange bounds

PeriodicExchangeBoundSource certifies the full-Coulomb exchange convention of Section 5.3 at fixed original k points, weights and evaluated spin density. It consumes the immutable pair source, the actual ordered Q vectors and offsets, and an explicit reciprocal completion-cube radius. It does not apply the q=0 pair involution to an exchange panel. The involution is used only to recover the absolute retained and omitted ordered-image amplitudes. The Gaussian Fourier and Bloch conventions are those of Sun et al. (2017), Section II.A, Eqs. (3), (10) and (16), DOI 10.1063/1.4998644. The error majorants below are a project derivation, not a published MDF approximation.

For a primitive pair of degrees l,m, write P=a+b. Shifting the entire Gaussian Fourier contour by \(-iQ/(2P)\) gives the exact Gaussian factor \(\exp[-Q^2/(4P)]\) and polynomial coordinates shifted by the same amount. The absolute Cartesian polynomial is bounded by \((|r-A|+|Q|/(2P))^l(|r-B|+|Q|/(2P))^m\). Expand the binomials, absorb spatial powers in half Gaussians with Section 5.9’s H functions, and use

\[\sup_{q\ge0}q^d e^{-q^2/(8P)}=(4Pd/e)^{d/2},\]

where the d=0 factor is one. Each primitive image then obeys

\[|F_{ab,n}(Q)|\le A_{ab,n}\exp[-Q^2/(8P)],\]

\[A_{ab,n}=C(2\pi/P)^{3/2}e^{-ab|A-B-Ln|^2/(2P)} \sum_{u=0}^l\sum_{v=0}^m {l\choose u}{m\choose v} H_{l-u}(a)H_{m-v}(b) \frac{(4P(u+v)/e)^{(u+v)/2}}{(2P)^{u+v}}.\]

C includes absolute normalized contraction coefficients and representation row sums. Summing the disjoint retained and missing images with Section 5.9’s checked metric gives AO matrices R and M. A common positive \(\alpha\le1/(8P)\) for every primitive yields \(|B_{ret}(Q,k)|\le R e^{-\alpha Q^2}\) and \(|B_\infty-B_{ret}|\le M e^{-\alpha Q^2}\), entrywise and uniformly in k. For an entirely s-type source, no Fourier polynomial is absorbed and the stronger common \(\alpha\le1/(4P)\) applies.

For each target/source transfer \(\delta=k_t-k_s\), let

\[c_{ts}=\sum_{Q\in G_{ret}+\delta}\frac{4\pi}{\Omega Q^2} e^{-2\alpha Q^2},\qquad h_{ts}=\sum_{Q\notin G_{ret}+\delta}\frac{4\pi}{\Omega Q^2} e^{-2\alpha Q^2}.\]

Both exclude Q=0 exactly, matching full Coulomb exchange. Native upper bounds enumerate the declared completion cube, including sparse holes. The exterior uses the shifted Gaussian slab bound and \(|Q|\ge\sigma_G(R_G+1)-|\delta|>0\). An insufficient separation refuses. The reciprocal metric takes the stronger of the checked inverse-residual bound and a guarded Gershgorin lower bound on the reciprocal Gram matrix; the latter needs no approximate eigenvector or lattice-axis assumption. The checked real metric cubed bounds volume from below. Supplied vectors must lie on their transfer lattice, inside the declared cube, be unique, nonzero and closed under transfer reversal. No support is silently added.

With \(A=R+M\) and nonnegative source weights, the two disjoint K errors are

\[B^{image}_t=\sum_s w_s c_{ts} [R^T|D_s|M+M^T|D_s|R+M^T|D_s|M],\]

\[B^{reciprocal}_t=\sum_s w_s h_{ts} A^T|D_s|A.\]

Their sum bounds the full-image, full-reciprocal K error for arbitrary Hermitian D, including signed densities. The exchange energy error in each spin channel is bounded by half the weighted absolute density contraction with this matrix. Positive logarithmic matrix products retain nonzero subnormal bounds; zero density produces exactly zero. No four-index tensor or all-q AO panel is retained. Joint plans admit constructor image/cube work, source storage, density copies, positive cubic contractions and owned exports under all five resource caps before numerical payload reads.

Canonical occupied-space exchange enclosure

bound_occupied_subspaces(S, n_alpha, n_beta, scf) reuses the bounded solver’s canonical overlap and occupied-vector Gram audits. It requires the source’s original weights to equal \(1/N_k\). Its returned density envelope has the stronger spin-separated property

\[H_{kij}\ge \frac{1}{N_k}\sum_\sigma |D_{\sigma,kij}|, \qquad L_k\ge\sum_{\sigma,ij}|D_{\sigma,kij}|.\]

Indeed each spin contributes its own Gram bound times the product of the two canonical row norms. Summing those positive bounds before applying the triangle inequality encloses the sum of absolute spin entries, including an empty spin channel. It is not merely a bound on a possibly cancelling spin sum. The retained canonical space, occupations and algebra audits stay fixed.

Let \(P_k=N_k H_k\), rounded outward, with each transpose pair replaced by its positive maximum. The existing exchange enclosure is a positive linear map \(\mathcal B\). Therefore \(\mathcal B(P)\) bounds the omitted-image plus omitted reciprocal error in every spin K throughout this density family. It also bounds the error for the total density. With \(B_k=\mathcal B(P)_k\),

\[|\Delta E_x|\le \frac12\sum_{kij}H_{kji}B_{kij},\qquad |\Delta E_x|\le\frac{1}{2N_k}\sum_k L_k\max_{ij}B_{kij}.\]

The implementation intersects the outward positive contractions. Its entry contraction uses the slightly larger symmetrized P, preserving the inequality. The energy is the sum of unscaled spin exchange energies. Restricted total D uses one half of the corresponding total-density K response; the returned per-spin enclosure still bounds that response. A hybrid coefficient requires its absolute method-owned scale. These are fixed-density-family component errors, not changes between relaxed states or different retained spaces.

bound_density_envelope accepts an existing native canonical envelope to avoid a second overlap analysis in future method consumers. Its qualification is conditional on that declared envelope; it does not prove that the supplied overlap equals the physical pair source’s overlap. It checks dimensions, rank, occupations and positive bounds, and accepts no user array in place of the typed envelope. Source and envelope identities are distinct. A certificate identity binds both and the resulting matrix/energy limits. No evaluated SCF density or Fock snapshot is manufactured from the positive envelope.

Count-only plans cover analysis, envelope copies, logarithmic contractions, source storage and owned exports before overlap values are copied. The already-built source has no additional shell or primitive integral calls. The two energy limits and the image/reciprocal matrices remain inspectable. The existing evaluated-density path retains its numerical results and identity. The standalone API does not select physical supports, impose complete RI method accuracy, or include the BvK correction or Madelung numerical error, or certify erfc exchange, fitting/metric error, roundoff, state relaxation or symmetry reduction. Those contributions remain separate required work.

Complete RI exchange accuracy admission

RI HF and RI full-Coulomb global hybrids can now impose separate positive matrix and energy targets through PeriodicExchangeAccuracyOptions. The selected K must be the independent exact reciprocal producer. Fitted K, pure KS and physical range kernels explicitly refuse this option; their missing error terms cannot be certified by a Coulomb component alone.

The method reuses its single canonical density analysis for automatic RI coupling, optional S/T image accuracy and exchange accuracy. It reconstructs the exact finite Cartesian AO-image source with the same shared integer-label, roundtrip and both-arm cutoff audit used by S/T admission. Reconstruction must not retain a block absent from the actual Bloch producer. The original ordered exchange Q vectors, transfer offsets and uniform full mesh enter the shared exchange-bound source unchanged. An explicit AO completion cube must contain every retained pair image; omitted inventory labels may lie outside. This keeps the retained Fourier amplitude enclosure valid. The standalone producer’s omitted optional radius retains its previous source and values.

Write \(B^K\) for the uniform positive exchange bound and \(P=N_kH\) for the outward density envelope. For fixed finite overlap S, image bound \(B^S\), and the method’s fixed BvK coefficient \(\xi\), define

\[B^C=|\xi|\,[|S|P B^S+B^S P|S|+B^S P B^S].\]

The admitted method matrix bound is \(B=s(B^K+B^C)\), where \(s=|a_x|/2\) for restricted total density and \(s=|a_x|\) for separate spin densities; HF uses \(a_x=1\). Omitted-zero-mode calculations have \(B^C=0\). The energy bound intersects the positive entry contraction \(\frac12\sum_{kij}H_{kji}B_{kij}\) with \(\sum_k L_k\max_{ij}B_{kij}/(2N_k)\). Scaling and positive products are performed in logarithmic arithmetic so a nonzero subnormal coefficient is not silently dropped. This compares the same finite-canonical-space density family under complete AO/reciprocal K and, when selected, complete overlap in the fixed BvK correction. It does not change the Madelung convention.

Both targets must pass before the RI builder and SCF iteration are created. Failure refuses; it neither enlarges an in-flight source nor changes the Fock/energy equations. The receipt keeps method scale, fixed coefficient, uncorrected exchange and overlap components, combined limits and distinct source/envelope/overlap/certificate identities. Automatic RI density provenance remains separate from an explicitly supplied RI density limit. Count-only plans cover reconstruction, source construction, envelope reuse, positive correction work, retained results and Python output before input copies. Child budgets reserve the already admitted outer live state once.

Madelung numerical error, finite fitting/metric error, other Hamiltonian components, arithmetic, mesh convergence, overlap-space changes and state relaxation remain outside this ledger. Physical source selection and whole-method automatic RI-BUPO accuracy are still incomplete. Unified Fock states, BIPOLE/CRYSTAL conventions and source-qualified symmetry permissions are unchanged by enabling this optional certificate.

Explicit periodic BUPO HF certification uses bound_exchange=True and exchange_bound_cube_radius. It scales K bounds by one half for restricted total density and by one for each unrestricted spin. For the declared fixed Madelung coefficient, the additional overlap-dependent correction is bounded entrywise by

\[s|\xi|[|S||D|B^S+B^S|D||S|+B^S|D|B^S].\]

This follows by expanding \((S+\Delta S)D(S+\Delta S)-SDS\). The coefficient and original BvK cell remain method-owned as in Section 5.11. The sum of uncorrected and correction errors must pass the positive finite matrix and energy targets at every evaluated density, default 1e-8 in atomic units. The last and maximum observed bounds and native source/density identities are recorded. Failure refuses before exchange evaluation and does not enlarge the physical domains, change the Hamiltonian, or alter the SCF state.

bound_exchange=False preserves the explicitly finite exchange route and its prior qualification. With certification enabled the qualification is electrostatic-one-electron-and-exchange-at-evaluated-density;fixed-mesh-and-madelung. This still excludes Brillouin-zone quadrature error, Madelung evaluation error, basis incompleteness, relaxed-state and overlap-rank stability, and roundoff. It is not an automatic whole-method accuracy profile or a symmetry certificate. Range-separated exchange, q-aware RI-BUPO, public job selection and the remaining capability matrix require their own method qualification.

5.14 BUPO HF reciprocal cutoff selection

The exchange-domain factory consumes the immutable electronic BUPO source from Sections 5.7-5.11 and the full exchange certificate from Section 5.13. It selects physical vectors by the shared certified reciprocal-energy predicate, for every centered target-minus-source mesh fraction. Gamma excludes the integer zero vector. Non-Gamma channels retain the existing predicate’s small numerical boundary padding; a new approximate norm test is not used. This is the same Gaussian Bloch exchange formulation of Sun et al. (2017), Section II, DOI 10.1063/1.4998644, with project-owned finite support and error accounting. It is not a new published periodic BUPO approximation.

For each transfer, certified integer box edges bound all candidate labels. Exact integer wraps map these edges into the original raw-transfer label frame: q_raw = q_centered + wrap, n_raw = n_centered - wrap. The mapped edges’ maximum absolute component defines the exchange certificate’s completion cube. Counting accepted vectors never shrinks it. The certificate therefore accounts for both unselected vectors inside the cube and the exterior tail. Centered opposite fractions can differ from exact negatives at a half-grid boundary; their integer partner map includes the corresponding reciprocal shift. In the raw frame it is n -> -n. Equal accepted counts and the shared opposite-source predicate audit establish closure before any exchange calculation. Empty spheres can be represented, but their missing-exchange bound must still pass the unchanged HF targets before evaluation.

Count-only planning reserves all transfer plans, counting, conjugacy audits, replay, domain hashing, native and owned export storage, and complete HF work. All five caps precede vector enumeration. Source-specific electronic, AO-image, tree, near/far and nuclear domains remain unchanged. The factory requires exchange certification and derives only the completion radius; it does not alter density-dependent tolerances or adapt the Hamiltonian. Result identities bind the original source, lattice, mesh, cutoff, radius, offsets and executed vectors. Numerical HF state remains the same as an explicit-vector run over those arrays, including nonconverged states and fixed-Madelung energy corrections.

The complete RHF/UHF calculation still reports evaluated-density integral bounds rather than basis, mesh, relaxed-state or roundoff accuracy. This factory is a prerequisite for public BUPO source selection; it does not complete automatic AO/Ewald selection, whole-method accuracy profiles, q-aware RI-BUPO or operation-qualified symmetry admission.

5.15 Complete native BUPO HF source preparation

prepare_periodic_bupo_hf constructs the pair-image source, bubble tree, Ewald source, fixed near/far partition and certified exchange spheres from the physical basis, neutral all-electron 3D system and full regular mesh. The source policy explicitly supplies integer cube radii, pair cutoff, Gaussian extents, moment order, hierarchy controls, Ewald eta and partition targets. It composes the existing producers in Sections 5.7-5.14 without changing their algorithms, domains or error definitions. Each nuclear center must still match its original basis center modulo a lattice image.

Preparation executes scientific work. A later stage’s plan depends on earlier immutable source contents, so this API is not a count-only dry run. It first admits generated labels, momenta, physical-input copies and metadata. It then plans and admits each source stage before constructing it. All five cumulative limits retain the earlier work charges. The live memory reservation sums the existing stage peak models, including their shared parent and workspace reservations. This deliberately overcounts storage; it is a conservative ownership model, not measured allocation or RSS. The final exchange-domain plan includes generation and complete bounded HF under the remaining work caps and the accumulated live reservation. Failure at any stage releases the partially constructed source.

The prepared handle owns physical-input and policy copies, source objects and exchange support. It contains no density, Fock matrix, orbitals or SCF history. Every run starts a new electronic state over those fixed inputs. One active run lease per handle prevents simultaneous workspace ownership from exceeding its admission; separate handles need separate budgets. Each result reports both source preparation and run costs, conservatively charging preparation even when an existing handle is run again.

The receipt identifies the original pair, tree, Ewald and BUPO sources and the copied source policy. Its physical identity also binds the nuclear input, exchange support, spin convention and zero-mode correction. Solver iterations and accuracy targets do not become physical-source identity. The returned evaluated density, Fock and energy retain the existing shared Fock consumer contract, including capped states. The prepared source grants no operation, support-equivariance or roundoff attestation for symmetry. It does not alter BIPOLE domains or establish CRYSTAL parity.

Both source bounds and full exchange bounds are mandatory. Every evaluated density must pass the existing electrostatic, S/T and exchange targets. This establishes explicit source construction and a selected HF API; ordinary job dispatch is described in Section 5.16. Automatic whole-method profiles, q-aware RI-BUPO, mesh and relaxed-state accuracy remain separate capabilities.

5.16 Public BUPO HF job dispatch

The public job consumer uses the prepared source of Section 5.15 without changing any numerical kernel, integral target, fixed domain or evaluated state. Typed PeriodicBupoIntegralOptions declares matched integer-cube Ewald domains separately from radial integral options. The existing CoulombBuildOptions.bupo supplies moment, extent and tree controls. The route admits only neutral all-electron 3D RHF/UHF, exact reciprocal K and full regular meshes. Both source certificates and exchange bounds remain mandatory. Geometric symmetry and producer qualification remain distinct.

Before source construction, the runner reserves output/history storage, policy/provenance copies, primitive shell serialization and their modeled work. A count-only pair plan supplies the primitive census. Complete source preparation then runs once under the remaining budget and keeps its live source reservation through SCF and output. The provenance distinguishes the conservative preparation admission from actual-support result modeling; complete-job scalar/kernel totals include the output work reservation. The native memory model already includes caller/output live reservations.

Output discovery does not execute scientific preparation. Its estimate is the declared complete-job memory cap, explicitly an upper reservation for admitted execution, not a source-content prediction or feasibility verdict. All scientific stage checks still run during execution. This preserves cheap dry-run artifact discovery without silently doing integral work.

Restricted results remove the native singleton spin axis at the job export boundary. Spin-resolved UHF, empty beta, nonconverged density/Fock/energy and QVF AO-gauge conversion retain the existing physical state contract. The job does not reconstruct the evaluated density from terminal orbitals. No post-HF, gradient, symmetry, RI-BUPO or whole-method accuracy capability is granted by this dispatch increment.

Ordered GDF auxiliary-image hierarchy

The optional coupling_multipole_hierarchy refines the ordered finite GDF coupling approximation. For each ordered pair image, the existing capsule selector supplies the complete auxiliary image inventory. The native bubble producer groups that inventory using its deterministic median tree, with leaf capacity one, branching three and maximum depth 32. No canonical periodic AO-pair source replaces this support, and no second selector is used.

For auxiliary shell component P, node v carries the complex raw Cartesian moments M_v(P) = sum_T exp(-i q.T) W(c_T-c_v) M_P. The shared translation producer evaluates W; child moments translate upward and are reused at their parent. The accepted contribution is M_v T_erfc(c_v-c_pair) M_pair^T, followed by the unchanged exp(i k.R) pair phase. Both paired actions consume these same density-independent tiles. The finite metric, retained factor, reciprocal panels and zero correction are unchanged.

The shared bubble envelope bounds each descendant distribution separately. Its translated full and tail moments are maxima, never phase-cancelled sums. An accepted N-image node therefore contributes at most N times the shared erfc Gaussian bound. The allowance remains the requested shell-tile epsilon divided by the total selected image count. Overlap, always-near flags, unrepresentable interaction bounds and excessive error force descent; terminal refusals use direct MD. The enclosing sphere and the translated tail envelopes are essential: applying an isolated-shell bound at a node center is invalid.

This uses the upward translation and auxiliary hierarchy of Neese et al., DOI 10.1021/acs.jpca.4c07415, section 2.3 equations 23-24 and section 3.2, with raw Cartesian total-degree erfc expansions instead of harmonic Coulomb moments. Order stays fixed during translation; the explicit node remainder controls the resulting error. The paper’s automatic order increments and density-compressed two-tree implementation are not yet implemented here.

The uncached route retains one pair image’s selected tree, radial envelopes and complex moment panels at a time. Admission includes tree construction, at most 2N-1 error queries, upward translations, moment contractions and simultaneous workspace. There is no full auxiliary-by-pair tensor or rectangular image interaction list. Bounded partition reuse and compressed hierarchical actions are described below; the streaming route remains the numerical reference. Counters distinguish represented images, accepted blocks, nodes and translations. This source does not certify symmetry reduction, BIPOLE or CRYSTAL parity. Existing approximation exclusions, including RI exchange and neutral correlated-fit exports, remain in force.

Producer-owned periodic RI Gaussian preparation

PeriodicGDFTiles owns an optional bounded preparation cache. Auxiliary entries are keyed by source shell; pair entries are keyed by the ordered bra and ket shells and the pair-image ordinal from the shared unscreened GDF selector. A cached pair verifies its shifted ket center on reuse. The coupling-only multipole mode forbids integral screening, so every auxiliary shell observes the same ordered geometric pair inventory. The source owns copied bases, cell and controls for its entire lifetime; no global basis or image registry participates.

The same native extent and moment functions produce each entry. A mutex serializes extent/support-count insertion and an entry-local call_once publishes each moment panel. Published numerical arrays are immutable. Moments remain lazy, preserving always-near/direct behavior and avoiding unneeded moment evaluations. A failed initialization is not published. Separate producer instances have independent caches; access order, density and caller mutations do not alter their numerical source or partition.

One integer per shell triple retains the exact selected-image count used in the existing epsilon allocation. It stores no image interaction decisions. The actual capsule selector still visits every integral/image needed by a tile. There is no full T/B, replicated basis, or auxiliary-by-pair image interaction list. Node charges and fields remain call-local. A separately bounded cache can retain trees and certified decisions as described below.

Let I bound pair-image candidates, S be the orbital shell count, A the auxiliary shell count, n and m the AO/auxiliary dimensions and H the number of Cartesian moments. Admission reserves ISS pair slots, A auxiliary slots, ASS support-count scalars and H*(Inn+m) real moment elements, including map/entry ownership and extent vectors. This capacity is live in every parent RI/Fock plan even before materialization. Cold numerical work remains reserved on all calls, together with bounded lookup/publication work; a warm cache never permits bypassing admission before input inspection.

coupling_multipole_reuse changes execution/storage policy only, so it does not change the coupling source identity. Cold, warm, streaming and dense paths must agree with the same numerical partition. The finite metric and paired contraction remain unchanged. Method receipts distinguish newly built moments/envelopes and reused data. These counters do not turn approximation bounds into arithmetic-error or symmetry certificates. The published algorithms and their limits remain those of the shared extent, raw moment, BUPO translation and erfc producers cited above.

Compressed periodic RI far actions

For one accepted auxiliary node and ordered pair image R, write its coupling as exp(i k.R) A I P^T, where P is the real native AO-pair moment panel, I is the same real erfc translation matrix and A is the complex auxiliary moment panel produced by phase-weighted upward translations. The shared finite selector and error queries choose the same density-independent antichain as the materialized tile route.

The raw forward action forms z = P^T sum_k exp(-i k.R) X_k. Each accepted node receives field I z. Transposed native translations propagate this field downward, and each auxiliary image contributes exp(+i q.T) M_aux f. The raw adjoint starts from M_aux^T c, translates these charges upward with exp(-i q.T), accumulates I^T z_node in pair moment space, then projects once with P and distributes exp(+i k.R) over the output mesh. Ordinary transpose and complex conjugation have distinct roles; the two maps remain adjoints under the complex Frobenius product for arbitrary complex inputs.

This is the input-moment/output-field organization of Neese et al., DOI 10.1021/acs.jpca.4c07415, equations 29-36, with MANTA’s already qualified ordered periodic support, nonharmonic erfc kernel and fixed shared partition. The existing molecular method’s canonical pair weights do not transfer to this ordered Bloch source. The finite metric and paired RI energy derivative are unchanged; no approximate metric, weight folding or new Fock owner appears.

The far map is not reconstructed for its magnitude diagnostic. At each auxiliary leaf let b be the componentwise maximum absolute row of its moment panel. Upward sum |W| b_child bounds every row of A without phase-cancellation credit. Likewise p bounds each row of P. Then b^T |I| p bounds every entry of that far block. Sum these positive bounds over accepted blocks/images and add the measured maximum of the near/reciprocal/zero residual tile. Guarded positive accumulation supplies the existing maximum_coupling upper bound used in fixed-density RI J error propagation. The approximation ledger is unchanged; arithmetic and phase roundoff remain explicitly excluded.

Per-action charges, downward fields, compensation and magnitude envelopes have admitted storage and work. Structural activity follows accepted nodes, including for zero density; it does not change the partition. Immutable source preparation can be shared across concurrent calls, but compressed inputs and fields are call-local. No source retains a density or metric solution. Dense tile exports continue through the full materialized producer path.

This compresses the actions of the selected auxiliary hierarchy, but retains one tree per shell triple and pair image. Global AO-pair bubble aggregation, tighter automatic accuracy, source-qualified symmetry and the rest of P3b remain unfinished. Bounded cross-pass partition reuse is described below. Existing J-only exchange, correlated-fit export and physical-kernel restrictions stay in force.

Source-owned bounded periodic RI partitions

Partition reuse is an execution policy over the same qualified numerical source. A cache key comprises auxiliary, ordered bra and ket shell indices, and the pair-image ordinal supplied by the exact unscreened GDF selector. The source owns copied geometry and immutable approximation settings. Reuse verifies the selected-image count and shifted ket center; no canonical Ewald inventory or symmetry label substitutes for the ordered support.

Each immutable entry contains the shared AuxiliaryImageBubbles tree and an ordered antichain of certified far-node matrices/bounds or direct leaves. Construction uses the unchanged BubbleErrorModel, strict separation test, native erfc interaction and per-selected-image tolerance. Replay preserves traversal and compensated contraction order, represented-image factors and the evaluated error sum. Gaussian moments remain independently prepared; input charges, fields and magnitude envelopes remain private to each action. No density, metric solution, full T/B or rectangular image tensor is cached.

A source-local mutex serializes lookup, bounded FIFO eviction and complete entry publication. Failure never publishes a partial entry. Oversized entries stream without retention. Before construction, a conservative bound covers the shared tree and at most 2N-1 decision records with H-by-H Cartesian interaction matrices. Actual retained accounting includes tree storage, decision-vector capacity, matrix elements and per-entry/map bookkeeping. All arithmetic for storage and work admission uses checked counts.

The source plan reserves the configured capacity plus 8 KiB bookkeeping, and a separate full-capacity workspace for an in-flight or evicted borrowed entry. Existing producer workspace, cold certification and translation costs remain admitted, with additional lookup/copy work. Parent RI/Fock plans compose these bounds before numerical input inspection. Cache history never relaxes admission, changes the mathematical source identity or authorizes symmetry.

This is bounded reuse of the published BUPO translation/certification path already mapped above, not a new periodic approximation or a claim that the molecular paper specifies a FIFO cache. Global AO-pair aggregation, automatic accuracy and the remaining capability matrix still require implementation.

Call-wide periodic RI charges and fields

The optional global-action policy factors repeated moment contractions out of the per-pair auxiliary-tree traversal. For ordered pair image r, form z_r = P_r^T sum_k exp(-i k.R_r) X_k once. If f_a,r,T is the auxiliary leaf field obtained from the unchanged accepted interactions and downward translations, accumulate h_a = sum_r,T exp(+i q.T) f_a,r,T with compensation. The forward far contribution is y_a = M_a h_a, projecting each auxiliary shell only once after all pair images. Near, reciprocal and finite-zero terms remain in their original residual tile and are added separately.

The adjoint starts from one M_a^T c_a per auxiliary shell. The existing phase-weighted upward translations and real erfc interactions contribute fields to each ordered pair r. Sum those fields across every auxiliary shell before projecting with P_r and distributing exp(+i k.R_r). These operations are the conjugate transpose of the forward map. Canonical molecular pair weights, mesh weights and metric solves do not enter this raw coupling stage.

This uses the same input-moment/output-field equations of Neese et al., DOI 10.1021/acs.jpca.4c07415, equations 29-36, with the qualified periodic phases and finite masks described above. The pair key is the ordered bra/ket shell and exact producer image ordinal; the auxiliary key is its AO offset. Entries borrow immutable source-owned moments and verify their owners. All charges, fields, compensation and phases are action-local; no density or metric solution becomes part of source preparation or partition retention.

With I candidate pair images, S orbital shells, A auxiliary shells, H moments and K mesh points, additional admitted storage is 8192 + I*S*S*(2048 + 96*H + 64*K) + A*(1024 + 64*H) bytes. Checked counts also reserve map work for A*I*S*S + A visits. Existing cold producer, compressed-action and partition-cache allowances remain in force; saved contractions do not discount admission. This is an upper bound, not measured residency, and is composed into parent RI/Fock plans before data inspection. Exact, leaf, unprepared, materialized and reciprocal-only routes do not activate it.

The execution policy changes snapshot identity and summation order, not the mathematical coupling identity, Gaussian error ledger or fixed metric. It provides no roundoff, symmetry or external-program parity certificate. A true global pair-bubble traversal remains required: pair-dependent auxiliary capsule masks cannot be replaced by the canonical Ewald pair inventory.

Producer-qualified global pair geometry

The shared GDF pair tree preserves the original ordered sphere selector and the exact pair-dependent auxiliary capsules. For bra shell s, choose an integer frame f_s from its fractional position (roundoff-close integers are stabilized before taking the floor). Translate both endpoints and the Gaussian extent by -L f_s. This changes their representation only. For auxiliary shell a with frame f_a and canonical image t, its original image for that pair is t_original = t + f_s - f_a.

The raw coupling phase factors exactly as exp(i k.R - i q.L f_s) * exp(-i q.L(t-f_a)). Their product is the original exp(i k.R - i q.L t_original). The first factor belongs to the pair input/output stage and the second to the auxiliary stage; the nonzero-q bra-frame term must not be dropped. This source mapping permits global spatial aggregation without substituting a different pair inventory. Neither a reduction certificate nor any new Fock normalization follows.

For a node, enclose all first endpoints around A and all corresponding second endpoints around B with a common displacement bound delta. Every descendant segment is within Hausdorff distance delta of segment AB. Therefore the distance d from an auxiliary point to AB gives uniform capsule bounds d-delta and d+delta. A node is wholly inside if the upper bound is strictly below the cutoff, and wholly outside if the lower bound is strictly above it. Roundoff guards turn uncertain cases into refinement. Single-pair leaves evaluate the original shared double-precision predicate in the original source frame. These masks are exact discrete support decisions, separate from Gaussian extent radii and the erfc approximation ledger.

Every selected auxiliary image lies within (pair cutoff + auxiliary cutoff) of at least one canonical bra center. The existing conservative integer-box enumerator supplies a superset; integer labels are deduplicated and exact masks remove empty images. This preserves skew-cell and image-frame support without retaining a rectangular membership tensor. The union capacity is the orbital shell count times the expanded image-box bound. Admission includes its ordered set, result/export copies, per-query traversal and the source-owned preparation. Tree/endpoint construction is bounded by the shared depth and candidate pair inventory. All resource caps apply before preparation is materialized.

The shared native bubble builder, source-owned Gaussian extents and published BUPO translations remain those mapped above. This increment supplies qualified global pair geometry and masks. The consumer below retains its source identity and separately certifies interactions over wholly included nodes; geometry does not supply an approximation or accuracy certificate.

Global ordered pair-tree RI-BUPO coupling

coupling_multipole_pair_tree defines a new fixed approximate short-range coupling. It reuses the above source frames/masks, native bubble error model, Gaussian panels and Neese equations 29-36. Each complete auxiliary-image union is traversed against the global ordered pair tree. An accepted block must be wholly inside its finite capsules and separately pass the native erfc Gaussian interaction bound. Otherwise it refines; an unaccepted leaf uses its original source-frame integral.

For an accepted node b, forward charges are the sum of translated pair moments times the pair-phase-folded input. The interaction maps that charge to the auxiliary field with the conjugate auxiliary phase. The adjoint uses the same fixed interaction matrix, unconjugated auxiliary phase, transposed translations and unconjugated pair phases. Both share the same antichain; all input, charges and fields are call-private. Dense exports expand the same interactions. No density-dependent accept/reject decision enters this map.

Let N be the largest actual count of admitted (R,T) interactions over source shell triples. A block is admitted only if its uniform per-descendant error is at most epsilon/N, with a conservative roundoff guard on that allocation. The evaluated coupling bound sums accepted errors down the shared pair tree, then sums original R images only within each orbital shell pair. If E[a,b] is that positive sum for auxiliary shell a and orbital shell pair b, the uniform bound is max(E). The earlier N times largest interaction bound is retained as an independent ceiling. Refining masks ensures every original interaction appears once. Exact near, reciprocal and finite-zero terms have no Gaussian approximation charge. Their arithmetic error remains separate.

For a forward input X, let x[b] be its L1 norm over the momenta and AO components of shell pair b. Each output in auxiliary shell a has error at most e[a] = sum_b E[a,b] x[b]. The maximum of e bounds the action component error; sqrt(sum_a size(a) e[a]^2) bounds its Euclidean error. The adjoint instead accumulates e[b] = sum_a E[a,b] ||c[a]||_1, with Euclidean bound sqrt(sum_b nk size(b) e[b]^2). No cancellation or density-dependent partition selection is used. Both action bounds are also capped by their independently valid uniform counterparts. The RI energy/J ledger uses the forward Euclidean bound with the same fixed retained metric. At Gamma, average the two raw-pass bounds by the triangle inequality, including the transposed input/output.

The coupling magnitude envelope follows the same block reduction: positive absolute moment/interaction envelopes propagate to leaves, and only images of the same shell pair are summed. Exact near magnitudes join that block; unrelated auxiliary outputs or shell pairs contribute through a maximum. The reciprocal/zero residual retains its separately evaluated maximum. The new ledgers change neither coupling values, partitions nor source identities. One auxiliary shell is processed at a time using admitted per-node and per-shell-pair scratch; no auxiliary-by-pair ledger is materialized. Positive additions round outwards; the scalar guards are not an integral-roundoff proof.

At reciprocal Gamma, qualify the literal finite support under (mu,nu,R,T) -> (nu,mu,-R,T-R) before completing its conjugate orbit. For short-range coupling S, define S_H[k,P,i,j] = (S_raw[k,P,i,j] + conj(S_raw[k,P,j,i]))/2. Its forward action is half the raw action on X plus the conjugate raw action on conj(transpose(X)); its adjoint is half the raw action on c plus the conjugate transpose of the raw action on conj(c). Thus both maps represent the same linear coupling for arbitrary complex inputs. Since the exact finite short-range source has this symmetry, convex averaging preserves the same uniform approximation bound. The unchanged reciprocal/zero terms are added afterwards and retain their own domain requirements. This does not repair an unqualified exact source or grant a reduction certificate.

Each ordered pair must have its reverse integer image, and every included auxiliary image must map to an included counterpart. Floating-point cutoff ties that fail this literal test are refused. A transfer merely within the older Gamma classification tolerance must also be a reciprocal representative within roundoff. The coupling identity records this completion rule; the exact reference coupling, metric, retained projector and J-only qualification retain their existing owners. Fixed-rank variational checks and final Fock Hermiticity gates remain in force.

Admission includes global geometry, the error model, local moment state, Gamma orbit lookup and two sequential raw passes, a bounded shared partition cache and a complete uncached antichain workspace. Entries borrow the common tree. Source, dense export, action and RI parent limits apply before copies and preparation, without discounting work for expected cache hits. No full T/B or rectangular pair-image membership array is retained. Uniform envelopes and positive magnitude sums remain deliberately conservative; useful-accuracy profiles and broader production crossover are still open.

Isolated erfc derivative and interior-moment bounds

The shared erfc_gaussian_interaction now tightens two independent steps in the isolated erfc Taylor ledger. The finite Ewald-complement bound retains its existing formula. Both reuse the native Taylor matrix, Gaussian extents and positive upper-incomplete-gamma recurrence. This is a MANTA derivation for the nonharmonic erfc kernel; the molecular harmonic remainder is not substituted.

For a directional derivative of order \(n=p+1\), let \(r=n-2k\), \(m=n-k\), \(d=D-a-b>0\) and \(s\ge\omega\). Instead of bounding the polynomial by \((D+a+b)^r\) while bounding the Gaussian at \(d\), keep their common radial argument \(z\):

\[\begin{split} \sup_{z\ge d} z^r e^{-s^2z^2} =\begin{cases} (r/(2e))^{r/2}s^{-r},&s<\sqrt{r/2}/d,\\ d^r e^{-s^2d^2},&s\ge\sqrt{r/2}/d. \end{cases} \end{split}\]

For \(r=0\) only the second branch applies. Multiply by \(s^{2m}\) and integrate over \([\omega,\infty)\). The first branch is a positive integral of \(s^n\) on a finite interval; the second is \(\Gamma(m+1/2,d^2\max(\omega,\sqrt{r/2}/d)^2)/(2d^{n+1})\). The native upper-gamma recurrence bounds that tail. Taking the smaller of this envelope and the earlier separate polynomial/decay envelope is valid term by term. The compact remainder retains the factors \(2^{r+1}(a+b)^nQ_AQ_B/(\sqrt\pi k!r!)\) from the directional Taylor theorem. Logarithms and expm1 avoid premature overflow and cancellation at the branch boundary. Existing geometric guards and positive-subnormal handling apply.

The Gaussian ledger can additionally replace the worst-case radius/mass factor using existing full absolute radial moments through order \(p\). For either source, define

\[ U_A(j)=\min_{0\le k\le\min(j,p)} a^{j-k}f_A[k],\qquad \mu_n=\sum_{j=0}^{n}\binom nj U_A(j)U_B(n-j). \]

On the interior sphere, \(|x|^j\le a^{j-k}|x|^k\), so \(U_A(j)\) bounds its absolute radial moment. The triangle and binomial inequalities then bound the integrated \(|x-y|^n\) by \(\mu_n\). Thus the Gaussian compact remainder is the conditional radius/mass remainder times \(\min(1,\mu_n/((a+b)^nQ_AQ_B))\), rounded upwards. No order-\(p+1\) native moment is required, even at \(p=12\). Signed contractions use absolute producer envelopes, and the full Gaussian tail-penetration and full-moment-substitution terms remain unchanged. Gaussian tails never become compact support.

LatticeGaussianError.compact_moment_scale exposes this factor (one for the unchanged lattice-complement ledger). Its compact_remainder is the tightened Gaussian compact contribution; LatticeGaussianInteraction.compact.total retains the separate conditional radius/mass bound. The reported total sums the tightened compact term and both unchanged Gaussian-tail terms. These are approximation bounds, not interval certificates for arithmetic roundoff.

Both additional radial evaluation and moment-cap work are admitted before payload evaluation. The isolated compact and Gaussian error identities use version two. All periodic GDF multipole consumers share this producer and bind ordered-erfc-multipoles.v2;joint-radial-and-interior-moment-bounds in their approximate coupling identity, because tighter acceptance can change the fixed partition. Exact metric/reference identities, finite image support, Bloch conventions and Gamma completion remain unchanged. Dense exports, forward and adjoint actions still use the same source-qualified partition. No density-dependent selection or new Fock/symmetry permission is introduced.

Lossless global partition storage

The global pair-tree cache keeps the exact existing Taylor interaction, error and node decision. With total order p and Hermite row degree d, only (p-d+1)(p-d+2)(p-d+3)/6 columns can be nonzero. Packing retains every allowed binary64 coefficient verbatim and checks that omitted positions are zero. There are binomial(p+6,6) stored values rather than binomial(p+3,3)^2; at order six this is 924 rather than 7056. No radial coefficient is recomputed and no sparsity threshold is used. A single zero-filled dense buffer restores the original matrix before the same forward, adjoint, magnitude-bound and export contractions. Allowed signed zeros are retained.

The shared source owns the fixed tree and bounded partition cache. Entries retain only their accepted antichain, not a full-tree-sized vector reservation. The global action’s existing complete-entry workspace permits constructing an entry before cache eviction. Actual allocated coefficient/vector capacities determine the retained charge. An entry above the cache cap is returned to the current action without publication or duplicate construction; entries that fit evict only enough FIFO storage for their actual charge. Immutable borrowed entries and concurrent calls retain the existing separate workspace contract. The legacy auxiliary-image route keeps its prior reserve-before-build policy.

Packing, expansion and validation work is explicitly admitted. The conservative dense-antichain memory bound remains, covering compact construction and the temporary metadata move; the existing single-interaction workspace covers expansion. No expected cache hit reduces admission. Cache history/capacity does not enter numerical source identity. This storage change grants no new method, approximation, BIPOLE/CRYSTAL parity or symmetry-reduction permission.

Componentwise fixed-metric periodic RI error transport

The shared PeriodicGDFTiles.forward and adjoint results retain approximation_bounds in the same shape/order as their numerical output. Each entry bounds that output’s omission relative to the producer’s fixed finite reference. Exact sources return zero entries. Approximate sources also return reference_magnitude_bounds[P], an upper bound on the largest absolute exact-reference coupling for auxiliary component P over all momenta and AO pairs. Exact sources leave this second diagnostic empty. Both Python exports are owned read-only copies; they neither change source identity nor retain an auxiliary-by-pair tensor.

For the global pair tree, auxiliary-shell a and orbital-shell-pair b already have omission E[a,b] and approximate-magnitude M[a,b] ledgers. Forward entries retain sum_b E[a,b] ||X[b]||_1; adjoint entries retain sum_a E[a,b] ||c[a]||_1. The short-range exact-reference column envelope is max_b (M[a,b]+E[a,b]), shared by components of auxiliary shell a. At Gamma, average the component envelopes along the same two raw passes and output transpose as the numerical action. Average the auxiliary column maxima as well; transposing the AO pairs preserves that maximum. Exact reciprocal/zero tiles add their absolute component magnitudes to the short-range envelope. Other routes use their existing uniform omission and direct/factored far magnitude bounds. Independently valid scalar ceilings still cap the entries.

Let b~ be the approximate forward result, F the actual retained metric factor, rho~=F^T b~, c~=conj(F) rho~, and e the componentwise forward omission vector. Positive triangle propagation gives

u_L = sum_P |F_PL| e_P
g_P = sum_L |F_PL| u_L
|delta rho_L| <= u_L,   |delta c_P| <= g_P
|delta E| <= min(sum_P |c~_P| e_P, sum_L |rho~_L| u_L)
             + 0.5 sum_L u_L^2
|delta J|_max <= adjoint_error(c~) + sum_P C_P g_P

Here C is the producer’s exact-reference column envelope. These inequalities follow directly from the paired quadratic RI expression described above and the triangle inequality. They do not assume exact eigenvector orthogonality, replace F with an ideal inverse, change rank/projector, refit a metric or use cancellation. The previous scalar Euclidean/Frobenius bounds remain separate valid bounds; the reported energy/J bounds take the smaller result. Positive long-double accumulation and outward binary64 guards protect these envelopes, not integral arithmetic. Rank zero and zero input retain exact zero bounds.

The action plan adds conservative storage for component/column vectors, Gamma mates, residual workspace and binding exports. The RI parent jointly admits the two retained action results and factor-propagation vector. Scalar/kernel plans include positive propagation work; cold and warm admission is identical. No extra source sweep, dense coupling, density-selected partition or Fock path is introduced. The finite source, paired maps, shared Fock, exact K/XC and source-qualified symmetry remain unchanged. These bounds concern fixed-density coupling error only: basis/domain, arithmetic, metric solve and SCF relaxation errors remain separate. Automatic whole-method accuracy is still unfinished.

Fixed-envelope automatic periodic RI coupling accuracy

PeriodicRIAccuracyOptions selects a fixed Gamma Coulomb pair-tree source before SCF. Both maximum-entry J and Coulomb-energy targets are positive, with an explicit density envelope L. The envelope is sum_k w_k sum_ij |sum_spin D_spin,k,ij| <= L, in the admitted AO basis; it is not the electron count in a nonorthogonal basis. Each evaluated density, including the initial and final confirmation densities, must pass this check. No partition or tolerance changes during SCF.

For producer-owned exact-reference auxiliary column magnitudes C, evaluated uniform coupling omission epsilon and the actual retained factor F, define

r_Lmode = sum_P |F_P,Lmode| (C_P + epsilon) L
u_Lmode = sum_P |F_P,Lmode| epsilon L
c_P = sum_Lmode |F_P,Lmode| r_Lmode
g_P = sum_Lmode |F_P,Lmode| u_Lmode
J_bound = epsilon sum_P c_P + sum_P C_P g_P
energy_bound = 0.5 L J_bound

The first term bounds the approximate-adjoint omission at the approximate coefficient; the second bounds exact-adjoint transport of the coefficient error. The paired Hermitian Gamma quadratic expression gives the energy inequality. These triangle inequalities extend the paired RI contractions of Neese et al., DOI 10.1021/acs.jpca.4c07415, equations 29-36; the envelope selection policy is an implementation choice. No ideal inverse, cancellation or additional orthogonality assumption is used. Positive accumulation and outward guards protect the envelope, not the underlying integral arithmetic.

A zero-density action obtains source-owned magnitudes and the evaluated coupling omission without constructing a dense coupling tensor. If either target fails, the shared producer creates a tighter immutable source from the same basis, cell, domains and physical kernel. Metric/reference identities must match, and the original metric factor remains unchanged. With r = min(J_target/J_bound, E_target/E_bound), a failed trial proposes epsilon_next = epsilon_requested * min(0.5, 2*r). This implementation heuristic probes above the linear target estimate because the discrete partition can have an evaluated omission far below its requested tolerance. It reduces the request by at least half, but assumes neither linear scaling nor monotone certificates. Only the evaluated bounds accept a trial.

Underflow clamps the proposal to the smallest positive double. If no smaller positive request exists, selection reports unmet targets instead of passing a zero tolerance to the shared producer. Zero finite-coupling omission does not certify integral arithmetic. One through 32 attempts are admitted in advance, including worst-case work and simultaneous old/new source storage. Exhaustion returns an error. The heuristic need not find the loosest valid partition or the fastest implementation.

The certificate records targets, density envelope, selected tolerance, both bounds, attempts, setup work, source/reference and certificate identities. Per-density component and scalar bounds remain available when tighter. Automatic accuracy admits J-only fitting with independent K/XC. It gives no new BIPOLE boundary convention, CRYSTAL parity or symmetry permission. Only coupling error relative to the same finite metric/source is covered. Basis/domain error, metric eigensolver error, arithmetic and density relaxation remain separate; automatic whole-method accuracy is still unfinished.

Canonical-overlap density envelope for automatic RI selection

Complete bounded HF/KS methods may request automatic coupling selection with zero density_l1_limit. Raw RI builders still require a positive value. The method derives L from the same immutable overlap, numerical projection, rank cutoff and eigensolver as its bounded SCF leaf. The shared numerical helper is independent of MANTA physical sources and adds no symmetry claim.

Let X be the actual retained canonical overlap transform at one k point. For a spin channel with m occupied columns, the solver constructs D = X U_occ U_occ^H X^H. Its reduced eigenvector audit tests every entry of U^H U against identity with absolute tolerance a and relative tolerance r. For r below one, Gershgorin and the diagonal sum give

g_m = (1 + m a)/(1-r),  d = (1+a)/(1-r)
||U_occ||_2^2 <= g_m,   ||U_occ||_F^2 <= m d
A = (sum_i ||X_i,:||_2)^2
Q = ||X||_F^2,   H = min(Q, ||X||_1 ||X||_infinity)
||D||_entrywise-1 <= min(g_m A, n min(g_m Q, m d H))

The first branch uses row Cauchy; the second uses positivity, ||D||_entrywise-1 <= n Tr(D), and the Frobenius/spectral product bounds. A zero occupied channel contributes zero. Sum alpha and beta bounds, then average over the uniform full k mesh. Restricted density uses the same two equal occupied channels. These inequalities are an implementation derivation, not a new fitted Hamiltonian. Positive accumulation/outward conversion protects the envelope; the integral and solver arithmetic retain their own scope.

The helper audits overlap projection, negative eigenvalues, a common retained rank and occupied-plus-virtual capacity using the existing solver routines. Its count-only plan admits overlap traversal, one eigenanalysis per k point, norm accumulation and owned per-k exports before reading values. Parent HF/KS plans include the added setup work and storage. The returned bound is a fixed source-selection input, independent of HCORE, later Fock matrices and density history. Per-density enforcement remains mandatory, so an exceeded bound is a failure, never a request to change partitions mid-SCF. Explicit positive limits and all raw-builder, exchange, BIPOLE and symmetry guards remain intact.

Componentwise density envelopes for automatic RI coupling selection

The shared canonical-overlap helper also returns a positive entry envelope H[k,i,j] = (sum_spin g_m) ||X[k,i,:]||_2 ||X[k,j,:]||_2 / nk, with zero contribution from an empty spin channel. Cauchy and the same audited occupied Gram bound imply |w[k] sum_spin D[k,i,j]| <= H[k,i,j] for the uniform full mesh. Keep the independently derived total L1 bound L as an additional constraint. Neither bound selects a density, Hamiltonian or symmetry action.

An explicit positive L may also carry density_entry_limits, a finite nonnegative row-major vector over the entire weighted spin-total AO-pair input. The raw builder owns a snapshot and checks both constraints at every call. A complete method with L=0 derives both bounds; an explicit entry override with L=0 is rejected. The count-only method plan copies no entry vector and admits its eventual ownership and up to two source probes per selection attempt before physical setup. Native receipts and managed output identify the additional constraint by a digest; the numerical helper exports an owned read-only (nk,n,n) array for reproduction and explicit controls.

The existing coupling producer now returns reference_action_bounds for approximate sources. In forward order it bounds |T_exact|^T |input|; in adjoint order it bounds |T_exact| |input|. These are positive triangle actions, valid for every complex input dominated entrywise by the supplied magnitudes. Shared pair-tree shell-block ledgers include near magnitudes, far moment envelopes and omission bounds. Gamma conjugate-orbit averaging transports these envelopes with the same input/output permutations as the paired map. Exact reciprocal/zero residual magnitudes are added separately. Other coupling routes retain their own uniform omission/far envelopes. Only one auxiliary shell’s block ledger is live; no auxiliary-by-AO-pair tensor is retained. Storage, scalar work and exported arrays are admitted.

For the fixed H, let the shared forward probe provide per-auxiliary error e, exact-reference absolute action B, and exact column maxima C. Intersect the bounds with e <= epsilon L and B <= C L. For the actual retained metric factor F, define positive vectors

R = |F|^T (B+e), U = |F|^T e
c = |F| R,       g = |F| U
J_bound = min(epsilon sum(c), adjoint_error(c)) + C.g
E_bound = min(L J_bound/2, R.U + U.U/2)

The additional adjoint probe supplies the first term through the same partition and producer error ledger. It uses the fixed coefficient envelope c, never a current SCF coefficient. The proof is the paired-map identity delta J = delta T c_tilde + T_exact delta c and the quadratic RI energy identity. Every trial is accepted only by evaluated bounds. Retained source, metric, factor and reference identities still obey the original selector contract. Explicit L without entry limits keeps the earlier scalar selector.

This refines finite-source coupling allocation only. Overlap truncation, basis/domain, metric solve, arithmetic and density relaxation remain separate. Independent exchange/XC, unified Fock, BIPOLE/CRYSTAL conventions and qualified symmetry retain their existing owners and admission rules. Useful whole-method accuracy and production performance still require their full acceptance.

5.17 Nuclear bounds for the physical midpoint-sphere source

The existing RI nuclear producer uses a sphere about each ordered AO-pair midpoint, not the common real cube of Section 5.10. The new standalone bound_periodic_nuclear certifies this exact finite convention against the infinite 3D zero-average Ewald potential. It consumes the original Cartesian AO and G supports, k order, nuclear positions, alpha and pair/nuclear cutoffs. Its physical-source identity is generated by the existing nuclear producer’s shared identity routine. No integral, Fock state or symmetry permission changes.

The kernel and phase conventions follow Ewald (1921), IV.1, Eqs.37-40, DOI 10.1002/andp.19213690304, and Sun et al. (2017), Sec.II.A, Eq.16, DOI 10.1063/1.4998644. The finite-support enclosure below is a derived extension; neither paper claims this particular bound. The shared native Gaussian majorants absorb contraction signs and Cartesian/spherical components before the nuclear estimate.

For a normalized envelope Gaussian of exponent a and charge Q, set \(\eta_*=(\alpha^{-2}+a^{-1})^{-1/2}\). Its real interaction with a unit point at distance d is bounded by

\[Q\min\left[2\sqrt{a/\pi}, \frac{\exp(-\eta_*^2d^2)}{\sqrt\pi\,\eta_*d^2}\right].\]

The first expression applies at d=0. The second follows from the exact Gaussian erfc convolution \([\operatorname{erf}(\sqrt a d)-\operatorname{erf}(\eta_*d)]/d\) and the Gaussian erfc tail inequality. For two points use the second expression with \(\eta_*=\alpha\); the same-point self term is excluded, and an unresolved distinct point collision refuses qualification.

For each ordered AO-pair arm and nucleus, recenter an integer completion cube about the physical midpoint. Obtain retained image labels from the shared PairNuclearImageSelector, including its boundary tolerance. Sum the above bound only over cube holes. Add the shared logarithmic Gaussian-point exterior-shell bound outside the cube. Rounded shifted centers carry an outward positional uncertainty. The exterior may include retained physical images; this positive overcount is safe. An exterior that cannot be separated from the majorant centers refuses. Completing the opposite arm independently avoids assuming that a rounded physical sphere authorizes a new symmetry relation.

The reciprocal omission is bounded by

\[C_G=\sum_{G\notin\mathcal G,\,G\ne0} \frac{4\pi}{\Omega|G|^2}\exp[-|G|^2/(4\alpha^2)].\]

The shared reciprocal Gaussian producer evaluates missing labels inside an independent completion cube and bounds its exterior. Physical supports may be noncubic, have interior holes, or extend beyond that cube. Their labels must be unique, inversion closed and agree with the Cartesian vectors to roundoff. A retained pair’s reciprocal bound is \(Q\sum_A|Z_A|C_G\). Summed ordered shell maxima give real and reciprocal AO matrices uniform in k.

The omitted-AO-image contribution is \(B^{\rm image}_{ij}=C_{\rm point}\sum_A|Z_A|M_{ij}\), reusing the existing Gaussian image mass M and the periodized Gaussian-point kernel bound from Section 5.10. That shared kernel includes real, reciprocal and gauge terms. The retained source’s constant \(+\pi\sum_AZ_A S/(\alpha^2\Omega)\) has no additional nuclear-domain error; its missing AO images are already in M. The total matrix bound is the positive sum of all three components.

For nuclear repulsion, the same selector, recentered holes and point tail apply to each ordered nuclear pair with weight \(|Z_A Z_B|/2\). The reciprocal error is bounded by \((\sum_A|Z_A|)^2C_G/2\). Self removal and the uniform-background constant remain exact shared terms, with no second gauge contribution. All positive sums and products use outward logarithmic arithmetic and retain a positive binary64 subnormal floor.

Count-only admission covers pair-source reconstruction, Gaussian majorants, selector labels, image and reciprocal completions, both ordered arms, nuclear pairs and owned matrix exports under all five resource caps. Immutable receipts distinguish physical source, pair source, image envelope and certificate identities. Options are returned by value; arrays are owned and read-only on export.

Complete RI RHF/UHF and pure/global/range-separated KS can impose separate positive matrix, attraction-energy and nuclear-repulsion targets before SCF. PeriodicNuclearAccuracyOptions inherits the proof cubes and leaves physical supports fixed. The method reuses its single native canonical overlap-space analysis for all enabled RI, S/T, exchange and nuclear certificates. Let \(H_{k,ij}\) bound the weighted spin-summed density entries in that space, let \(L_1\) be its weighted entrywise norm bound, and let \(B_{ij}\) be the nuclear matrix enclosure. The one-body contraction has no half factor:

\[ |\Delta E_{\rm en}| \le \min\left( \sum_{kij} H_{k,ji} B_{ij},\ L_1\max_{ij}B_{ij}\right). \]

The shared positive trace producer evaluates both bounds; their minimum and the outward sum with the nuclear-repulsion bound are recorded separately. The envelope is not an evaluated SCF density. All three targets must pass before the Fock provider starts. Native count-only admission includes the envelope, contraction scratch, nested producer, retained receipts and owned exports under all five resource caps before array copies. The certificate’s physical nuclear identity must equal the method’s nuclear receipt. This adapter also supports exact-J/RI-K because nuclear bounds do not depend on the exchange kernel. Its scalar provenance retains targets and exclusions.

The optional selector below can enlarge physical supports. Whole-method error remains open. Floating integral arithmetic, basis and k convergence, finite fitting/metric errors, overlap-rank stability and state relaxation remain excluded. Unified Fock, BIPOLE/CRYSTAL conventions and source-qualified symmetry continue to use their existing physical producers.

5.18 Automatic physical support selection before RI SCF

select_periodic_supports is a bounded source selector over the existing physical AO-pair sphere, nuclear midpoint sphere and reciprocal kinetic cutoff. PeriodicSupportRequest carries S/T and nuclear component targets, with optional full-Coulomb exchange targets and the method’s exchange scale. It is independent of an HF or KS solver. The public RI job adapter supplies the correct restricted/unrestricted and hybrid coefficient. Range-separated and fitted exchange retain their existing exchange-certificate refusals.

Every trial uses the shared cutoff-domain generator, one canonical overlap density envelope and the shared S/T, nuclear and optional exchange bound producers. A trial may return unqualified component values; ordinary complete method admission still throws on a failed target. Failed S/T grows the pair radius. The largest normalized nuclear matrix/energy component selects pair, real or reciprocal growth, while nuclear repulsion compares its real and reciprocal parts. Exchange compares its image/correction and reciprocal components. All selected radii grow by growth_factor; reciprocal kinetic cutoff grows by its square. This deterministic policy does not promise a minimal or optimal support. Attempt, physical support, completion-cube and resource limits are explicit; exhaustion cannot start SCF.

Proof cubes are enlarged to enclose conservative physical support boxes and never shrink the requested proof radii. Physical cutoffs change the operator; proof cubes only change its enclosure. The selected fitting pair cutoff follows the physical AO-pair cutoff. Auxiliary cutoff, fitting omega, metric projector, coupling-accuracy policy, Ewald alpha, BvK convention and XC grid remain their independent controls. Existing finite-source conjugacy and metric failures propagate; the selector never projects away such a failure or silently adjusts the auxiliary source.

Count-only admission precedes domain arrays. The five joint resource caps include every discarded trial, the selected trial and retained scalar history. Initial count-only preparation is conservatively charged by its whole scalar/kernel trial plan. Trial counters are admitted upper bounds, not measured execution counts. The public job deducts these charges before admitting the final method, and keeps the history live beside its output. Dry runs reserve the declared complete-job memory cap and defer numerical selection. No domain or operator changes during SCF.

The final complete method recomputes its normal component certificates; overlap and nuclear source identities and S/T, nuclear and exchange certificate identities must equal the selected trial. This guarded adapter currently repeats final source preparation. It does not yet transfer a prepared-source cache from the selector. Scalar history, component values, growth decisions, targets, limits and identities are exported in provenance. Unified Fock consumes the same final source, and no new symmetry permission or BIPOLE/CRYSTAL convention follows from successful selection.

Qualification covers only requested component targets in the final finite canonical density family. It excludes finite RI fitting/metric error, integral arithmetic, canonical-rank stability, state relaxation, basis/k convergence and XC quadrature. Existing automatic RI coupling selection is separate and still executes at the final physical source. A complete automatic RI-BUPO total-error profile and the remaining capability matrix remain unfinished.

5.19 Nested periodic RI metric-factor error

PeriodicRIAccuracyOptions.metric_reference_cutoff optionally extends the Gamma J-only pair-tree certificate to a finer absolute positive-mode cutoff. The reference cutoff must be finite, nonnegative and no greater than the resolved selected cutoff. None preserves the existing coupling-only contract, numerical operator and certificate. Both cutoffs act on the same lower-triangle canonical metric and the same stored eigensystem. Negative and zero modes remain excluded; this never replaces a failed metric audit.

Sun et al., DOI 10.1063/1.4998644, Section II.B, Eqs. 20-23 motivates the positive-mode fitting space. The paired RI-J map follows Neese, DOI 10.1021/acs.jpca.4c07415, Section 2.1, Eqs. 9-11. We reuse the existing finite GDF source and its metric; we do not introduce Sun’s mixed-basis transformation or Neese’s molecular boundary condition. The following inequalities are derived here for the stored factors and the already qualified periodic coupling envelopes.

Let F contain the selected columns and H contain the additional positive columns admitted by the reference cutoff. The reference factor is their concatenation, so the paired inverse action differs by conj(H) H^T exactly as an algebraic factor identity. It does not assume numerically exact orthogonality or F^H M F = I. The existing shared probe supplies C, an exact-reference coupling column maximum, and R, a bound on the forward action for every admitted weighted spin-total density X. Use R=C L or the tighter componentwise intersection with the positive source action.

z = |H|^T R
metric_J_bound = C^T |H| z
metric_E_bound = min(z^T z / 2, L metric_J_bound / 2)
combined_J_bound = coupling_J_bound + metric_J_bound
combined_E_bound = coupling_E_bound + metric_E_bound

The nested exact-reference fitted energy increases by ||H^T T_exact^H X||^2/2, regardless of density definiteness. The selected approximate coupling additionally contributes the existing two-sided coupling error. Triangle addition yields the combined bounds; the same fixed density family is enforced at every build. Positive accumulation and outward guards enclose the scalar bound arithmetic, including subnormals. An empty removed space or zero-entry density family has exactly zero metric contribution. No evaluated density is used to select or change a metric.

Target admission now uses the combined bounds when the reference is set. The coupling selector may refine its existing multipole partition within the admitted attempt limit, but the selected metric rank remains explicit. It refuses unmet targets before SCF. The receipt keeps both components, selected/reference ranks, absolute reference cutoff, discarded-factor and reference identities. The raw builder exports an owned discarded-factor copy for independent inspection. The optional storage and scalar work are charged before allocation; no extra coupling traversal or full T/B tensor is introduced. Parent RI HF/KS plans inherit these reservations.

approximation_j_bound and approximation_energy_bound on an evaluated Fock receipt remain coupling-only. Combined family bounds are the accuracy receipt’s j_bound and energy_bound; output explicitly identifies the additional metric-truncation component and its reference. Physical sources, unified Fock ownership, BIPOLE/CRYSTAL conventions and symmetry permissions are unchanged. This is not the error against an unfitted Coulomb operator. Finite fitting-basis and image/reciprocal-domain errors, metric eigensolver error, floating integral/contraction arithmetic, rank stability and state relaxation remain excluded. The full automatic RI-BUPO goal remains open.

5.20 Full-Coulomb fitting enclosure from shared response maps

PeriodicRIFittingBoundSource compares a fixed PeriodicRIQBuilder with an existing PeriodicBupoSource that supplies both infinite-Ewald-domain and omitted-AO-image bounds. Both must have identical orbital basis, cell and literal momentum frame; RI must be literal Gamma Coulomb. The reference owns immutable weights. This compares with the full zero-average Coulomb J on that orbital basis and mesh. It does not alter either physical source, metric factor, Fock owner, BIPOLE/CRYSTAL convention or symmetry permission.

Sun et al., 10.1063/1.4998644, II.B, Eqs. 20-23 and Neese, 10.1021/acs.jpca.4c07415, Section 2.1, Eqs. 9-11 define the fitted source interpretation used above. The new comparison below is a derived linear-map enclosure. In particular, independently truncated periodic metric and coupling domains need not form an exact Coulomb Hilbert projection. No variational residual, positive Schur complement or monotonic fitting-energy assumption is used.

For weighted spin-total X(k)=w(k) sum_spin D(k), probe a real Hermitian basis Eii, (Eij+Eji)/2, and i(Eij-Eji)/2 at every nonzero-weight k. Call the existing RI and BUPO builders on scaled probes and divide out the scale. For each output entry let a and b bound the magnitude of the finite RI/BUPO difference and the BUPO remainder respectively. Diagonal inputs use these directly; off-diagonal inputs use hypot(a_real,a_imag) and the corresponding hypot(b_real,b_imag). For Xij=u+iv, Cauchy bounds each paired contribution by 2 |Xij| hypot(a_real,a_imag), and similarly for b. Thus each row is bounded by its maximum coefficient times the fixed L1 limit, or by its positive contraction with the optional fixed entry limits. Keep the tighter bound separately for the two terms, then add them.

This spans arbitrary complex Hermitian spin-total densities, including indefinite inputs and broken time reversal. Zero-weight input rows and zero-entry subspaces are skipped; all output momenta remain included. The energy bound is the smaller of half the L1 limit times the maximum J bound and half the fixed entry-envelope contraction with the per-entry J bounds. This bounds the Coulomb energy at the same density, not the energy difference between separately converged states.

The certificate records the selected RI, full reference, pair source and family identities, finite discrepancy, reference remainder and combined bounds. Its build validates fixed weights and density limits and delegates the unchanged J build to the retained RI owner. Optional J/energy targets refuse the certificate before it can be consumed. Probe scales also respect any existing RI accuracy family; unrepresentable scales refuse explicitly.

Count-only admission covers at most n_k n_AO^2 complete probe pairs, both retained producers, scratch, owned exports and one consumer build. The explicit probe cap and all five resource caps apply before allocation. There is no stored AO-four-index response, but the work is expensive and is not a linear-scaling fitting algorithm. This is a bounded qualification path; automatic auxiliary/domain selection and complete-job attachment remain to be implemented. It encloses the chosen stored RI operator’s fitting, rank, coupling and finite-domain discrepancy together. Do not add those component bounds again. Floating integral/contraction arithmetic, rank stability and density relaxation are excluded. No total SCF energy accuracy or completion of the full MANTA matrix follows.

5.21 Shared full-Coulomb source preparation

PreparedPeriodicBupoSource separates electronic reference construction from the existing PreparedPeriodicBupoHF method consumer. Both use one sequence of the original pair, bubble tree, Ewald and BUPO producers. No integral formula, extent or domain estimate, multipole partition or Coulomb convention changes. The source factory owns no nuclear, exchange, occupation or SCF policy. Charge/spin restrictions remain method-owned.

The generator and each producer are admitted cumulatively before that stage allocates. The source plan preserves the existing conservative sum of stage peaks. Its remaining-budget copy includes all retained source reservations and subtracts preparation work under all five resource caps. RI certification separately admits its probe and consumer cost; passing that remaining budget deliberately overcounts some shared parent peaks rather than omitting them. The generated uniform weights and literal native Cartesian mesh preserve the source identity used by complete jobs.

The returned immutable source retains its original producers, including both full Ewald-domain and omitted AO-image bounds. Consumers retain the same source directly, without a second reference implementation. Existing HF preparation and final numerical/source identities are unchanged; only the generic preparation wrapper has its own identity. This is staged construction, not a numerical-free dry run or a new accuracy proof. Automatic complete RI admission, useful source selection and the full remaining method/error matrix stay required.

5.22 Complete RI full-Coulomb fitting admission

Optional PeriodicRIFittingAccuracyOptions supplies one immutable prepared BUPO reference and fixed J/energy targets. The RHF/UHF and pure/global/range KS planners jointly reserve reference construction, count-only maximum Hermitian probes, canonical density analysis and every later family check. The certificate count planner depends only on admitted RI/reference plans; it does not run probes or assert target success during dry-run planning.

After overlap construction, the method derives the same canonical fixed occupation envelope already used by physical component accuracy. Its weighted entry bounds and L1 limit are passed to the Section 5.20 proof. The existing PeriodicRIFockBuilder shares its actual Gamma owner with that certificate; no nominally equal metric, coupling or momentum frame is reconstructed. Positive finite targets must pass before the solver is entered. Every later build validates the fixed family, then uses the unchanged numerical RI J/K contraction. K, XC, nuclear and symmetry ownership and BIPOLE/CRYSTAL conventions are unaffected.

The reference preparation and probe plan are separate admitted-work records; evaluated physical Fock counters keep their original scope. Parent reservations include the full certificate/reference storage even when the solver does not own those objects. Cumulative preparation work and all five caps are enforced before certificate construction. Existing coupling/metric certificates retain their own family provenance; the new full-Coulomb proof is not added to them as another independent error term.

This reuses the Sections 5.20-5.21 producers and proof without extending their arithmetic, rank-stability or relaxation scope. It is automatic pre-SCF admission for explicit reference domains, not useful automatic auxiliary/reference selection or a total SCF-energy accuracy guarantee.

5.23 Bounded full-Coulomb metric selection

An optional metric search retains the Section 5.22 canonical density family, auxiliary basis, fitting domains, coupling producer and prepared BUPO reference. Start with the requested RI absolute/relative cutoff. When the combined full-Coulomb J or same-density energy target fails, reduce the resolved cutoff by a specified factor in (0,1), clamped at a nonnegative absolute floor. Subsequent candidates use this absolute cutoff and zero relative cutoff. At most 31 refinements follow the initial candidate. The floor and attempt count can refuse a request; neither constitutes an accuracy certificate. Negative-mode validation is unchanged.

Each candidate constructs a positive metric factor through the original RI producer, sharing the same coupling source. It is compared against the same full-Coulomb reference using all Hermitian probes. The first candidate meeting both targets is retained directly as the actual Gamma Fock owner; its certificate and source identities describe that owner. Rejected certificates cannot build a density. No SCF iteration or relaxed state is used to choose the metric. Source identities are refreshed after a factor change, and no new symmetry or Fock composition authority is introduced.

This uses the existing Sun2017 Section II.B fitting representation and Section 5.20 triangle enclosure. It does not assume that error decreases monotonically as rank increases, or that independently truncated fitting and reference sources form an exact variational projection. Every trial must pass the complete combined bound. All allowed probes, factorization work and rejected trials are admitted in advance under the five resource caps. Replacement permits two concurrent RI owners; trial receipts and serialized copies have explicit reservations. Plans remain conservative when the search stops early. The fixed-cutoff default is unchanged.

The initial search supports fitted Coulomb with independent exchange and without a separate coupling/metric component-accuracy policy. Those combinations refuse during count-only planning rather than silently changing a second contract. The combined certificate already includes coupling error. The existing fixed-cutoff policies remain available. Auxiliary basis and domain selection, automatic reference preparation, arithmetic and rank stability, relaxation and the full method matrix remain required. This selection does not imply total SCF-energy accuracy.

5.24 Automatic shared BUPO reference selection

An optional PeriodicRIFittingReferenceOptions replaces the explicit prepared reference in Section 5.22. It constructs the original Section 5.21 electronic source inside the actual Gamma RI owner, using the method’s orbital basis, cell, native full mesh and canonical density family. An unset initial pair cutoff inherits the selected coupling source cutoff. No integral formula, Fock composition, BIPOLE/CRYSTAL convention or symmetry authority changes.

Every candidate is probed against the same selected RI map using Section 5.20. A complete J/energy pass accepts immediately. Otherwise, reference selection continues until its J remainder is at most reference_fraction of the J target and its Section 5.28 entrywise energy enclosure is at most the same fraction of the energy target. This conservative trace bound only guides selection; it is not another additive certificate. Optional Section 5.23 metric selection then uses that immutable reference. The final combined J and same-density energy bounds must both pass before SCF. No monotonic improvement or variational fitting identity is assumed.

After a rejected reference, pair, real and reciprocal cube radii increase by one up to maximum_cube_radius; the pair cutoff increases by growth_factor (with a next-cutoff minimum of one bohr), capped at maximum_pair_cutoff_bohr. The multipole tolerance decreases by ten. The policy permits at most 31 refinements. Failure at a resource, attempt or domain limit refuses the calculation; a limit does not assert accuracy.

The eventual tree depends on numerically admitted pair-source values. The count-only complete plan therefore reserves a declared search budget, not an asserted prediction of the eventual reference cost. Its memory and four work caps are included in the parent plan. Each shared preparation stage, probe and metric replacement is subsequently admitted against the remaining reservation. Rejected work is never refunded. Rejected numerical sources are released before preparing the next reference; receipts remain. The reservation includes retained RI/reference owners, bounded receipt and family storage, and both RI owners during refactorization. Output reports declared caps and cumulative admitted costs, rather than presenting the caps as measured work. Dry runs construct no reference and claim no target success. The physical Fock counters retain their original scope.

The Sun2017 Section II.B representation, Neese2025 Section 2.1 motivation and original periodic Ewald/image proofs remain those of Sections 5.20-5.23. Automatic reference selection also supports fixed-cutoff fitted K and the separate component certificate; metric refinement keeps its existing J-only restrictions. Auxiliary and physical fitting-domain selection, arithmetic/eigensolver/rank and relaxation ledgers, and all other unqualified MANTA capabilities remain required. This is a fixed-density Coulomb bound.

5.25 Bounded physical fitting-domain selection

PeriodicRIFittingDomainOptions adds a bounded search over the actual Gamma GDF coupling and metric source. The orbital and auxiliary bases, cell, AO frame, mesh, weights, physical Coulomb kernel, split parameter and coupling approximation controls stay fixed. Complete RHF/UHF and pure, global-hybrid and range-separated KS admit this policy for fitted J with independent exchange. Fitted K and a separate component accuracy policy refuse domain selection, because replacing their shared producer would change another contract. The existing input pair-cutoff agreement is preserved; only the certified J owner receives the selected supports. Nuclear, S/T, exchange, XC, unified Fock composition and symmetry authority do not change.

The shared PeriodicGDFTiles.with_domains factory preserves every original numerical reciprocal vector in its original order and appends missing nonzero vectors from a complete integer cube in lexicographic order. It requires a reversal-closed Gamma source and nested pair/auxiliary image radii. Its count plan admits the union upper bound, label/vector generation, producer snapshots and work before allocating the enlarged vectors. The original GDF metric and coupling implementations evaluate every candidate; there is no second integral formula or imported metric. Basis-space fitting error remains part of the existing Section 5.20 enclosure.

The default policy permits three refinements, multiplying the pair and auxiliary image radii by 1.5 with a next-radius minimum of one bohr and caps of 32 bohr. The reciprocal cube increases by one, capped at eight. Optional initial radii must contain the original source; the initial cube defaults to zero. These are bounded search heuristics, not a published error estimator or a monotonic-convergence assumption. The Sun2017 Section II.B representation and Neese2025 Section 2.1 factorization motivation remain those cited in Sections 5.20-5.24. A valid initial finite metric is required; negative-mode refusal is never replaced by damping or a repaired metric.

The reference is prepared/selected once by Section 5.24 or supplied explicitly. Every physical domain begins with the originally requested metric controls and may run Section 5.23 metric refinement. Both combined full-Coulomb bounds must pass on the same canonical density family. The first passing numerical owner and certificate are retained directly for SCF. All failed trials remain in provenance. An attempt, physical-domain, metric or resource limit refuses before SCF. Section 5.27 adds optional selection among auxiliary basis candidates.

Complete count-only plans reserve the largest permitted fitting source for all solver/Fock calls, before values or density analysis. Each actual candidate is checked against that capacity. The search separately reserves a declared memory/four-work cap. With automatic reference selection its effective caps are the componentwise minimum of the two requested caps, with the larger backend margin. Preparation, generation, metric factors, probes, rejected work, bounded history and simultaneous original/current/ replacement owners are charged cumulatively. No work is refunded. The explicit reference’s preparation is charged once. Search caps are upper reservations, not measured operation counts or timing predictions.

A zero-refinement policy with the selected radii/cube and metric reproduces the accepted source without changing complementary physical inputs. Output separates requested supports, selected supports, maximum SCF capacity, actual selected-source plan and all trial source/certificate identities. The certificate still excludes contraction arithmetic, eigensolver/rank stability and density relaxation. It is not a whole-method energy bound.

5.26 Shared primitive-product auxiliary candidates

vibeqc.auxiliary_generation provides a shared, bounded basis metadata producer. plan_product_auxiliary_basis counts resources and prepare_product_auxiliary_basis returns an owned basis, plan and receipt. The construction follows the same-center primitive-product exponent sums and even-tempered motivation in Stoychev, Auer and Neese (2017), DOI 10.1021/acs.jctc.6b01041, Section 2, Equations 5-7. The implementation is independently written. It does not implement the paper’s effective-exponent, valence/ECP or angular-pruning heuristics and must not be identified as ORCA AutoAux.

For each atom and product angular degree L = l1 + l2, all same-atom orbital shell pairs contribute the minimum and maximum primitive exponent sums. Coefficients do not prune this envelope. The geometric grid starts at the minimum and extends through the first beta power covering the maximum; rounding at the endpoint may require one further interval. Beta defaults to two. Level zero keeps the coarse endpoints; levels one through eight successively insert dyadic midpoints in log-exponent space. Every coarser exponent remains bitwise present. Generated shells are Cartesian: their homogeneous polynomial space contains products of either pure or Cartesian orbital angular functions. This angular statement does not assert exact radial fitting of every off-grid exponent or cross-center product.

An optional seed retains every shell, coefficient and order as an exact prefix. Only exact same-center, same-degree, single-primitive seed matches with nonzero coefficient suppress generated duplicates. Pure s/p shells span the corresponding Cartesian space; pure d and higher shells do not replace Cartesian trace components. Generated primitives use the existing libint normalization convention. Unrepresentable normalization, nonfinite ranges, indistinguishable exponents, mismatched centers or an angular cap that would omit a product channel refuse. No metric repair is introduced.

Planning admits shell/function counts, modeled peak bytes and work before candidate construction. Defaults are 4096 shells, 32768 functions, 256 MiB and one billion modeled work units. Counts conservatively include possible seed duplicates, input snapshots, output constructors, normalization, identity/duplicate scans and receipt exports. The orbital angular limit is 16 and generated limit is configurable up to 32, default eight. These are metadata capabilities; each integral consumer retains its own limits. The receipt records input/output content identities, exponent spans, controls, counts and an explicit candidate-only qualification. Source matching ignores display names and retains the numerical basis frame.

Selected periodic RI jobs accept the prepared object directly as CoulombBuildOptions.auxiliary_basis. They verify its orbital identity, reserve its complete peak/work in addition to existing job costs, pass its owned BasisSet to the existing GDF producer, and route generation provenance/citations through standard output and QVF. The plain .basis remains available to other basis consumers without this job-level receipt. No integral formula, metric authority, unified Fock composition, BIPOLE/CRYSTAL convention or symmetry permission changes.

Generation alone has no fitting or energy accuracy certificate. Optional Sections 5.20-5.25 certification still evaluates the actual numerical source on its fixed density family, including finite metric and fitting effects. A larger auxiliary basis need not improve that certificate monotonically. Section 5.27 adds bounded selection among these basis candidates. Arithmetic, eigensolver/rank, relaxation and whole-method ledgers remain unfinished.

5.27 Automatic auxiliary selection under the fitting certificate

PeriodicRIFittingAccuracyOptions.auxiliary_selection enables a bounded search over the original seed, then the shared Section 5.26 candidates in increasing refinement order. PeriodicRIFittingAuxiliaryOptions captures the initial generator controls, generation limits, maximum refinements and search budget. The default is three generated levels starting at zero; the largest requested level must not exceed eight. This order is a search heuristic, not a monotonic error or least-cost assertion.

The shared ProductAuxiliarySequence plans all candidates before building any basis metadata and owns the resulting immutable candidates. Its plan reserves all retained candidates, construction scratch and temporary atom metadata together, with cumulative generation work. Complete periodic jobs prepare it automatically. Direct native complete-method callers first use prepare_periodic_ri_auxiliary_selection; count plans require a captured sequence matching the numerical orbital and seed identities. Preparation cannot certify an integral error or guarantee that later source admission will pass. Metadata construction still requires the generator’s matching atom-centered input frame; this does not broaden AO gauge support.

The shared GDF plan_auxiliary_domains and with_auxiliary_domains methods retain the original reciprocal vector order and union convention while changing the admitted auxiliary basis. Existing integral formulas and finite metric positivity/rank authority are reused. A separate replacement guard checks the generated candidate and unchanged orbital identity; the existing physical-domain replacement guard still forbids an auxiliary identity change. No metric repair or alternate Fock composition is added.

One supplied or automatically selected full-Coulomb reference and one canonical density family apply to every basis trial. Each basis starts at the original requested physical fitting domains and metric controls. Optional domain expansion and metric refinement run inside that basis trial. Without domain selection, fitting supports remain fixed. The first actual source passing both J and same-density Coulomb energy targets is retained unchanged for SCF with its certificate. Exhausted candidates, metric/domain limits or resource caps refuse before SCF.

Planning takes componentwise maximum source costs over every candidate and the allowed domains before integral values or density analysis. Shared reference, domain and auxiliary searches use the componentwise minimum of their requested resource caps and the largest backend margin. Generation, source construction, factors, probes, rejected work and bounded history remain charged. Concurrent original, replacement and reference owners are reserved; no rejected work is refunded. A dry run is resource admission, not evidence that an accuracy target passes.

Provenance separates requested seed and selected auxiliary counts, maximum SCF capacity, selected-source plan, candidate/generation/source identities, all domain/metric trials, and requested/effective/charged search budgets. The selected generator receipt and citation use standard output and QVF. A fixed selected basis, reference, domains and metric reproduces the accepted owner. The policy requires fitted J, independent exchange and no separate component accuracy policy. Unified Fock, BIPOLE/CRYSTAL source conventions and existing symmetry permissions retain their authority.

This extends the finite fitting/metric certificate, not its mathematical scope. Arithmetic, eigensolver and rank stability, density relaxation, other energy components and the remaining capability matrix are still outside this acceptance. Conservative fixture targets are not recommended production tolerances or a whole-method energy guarantee.

5.28 Separate fitting and reference energy enclosures

The fixed-family certificate also reports finite_difference_energy_bound and reference_remainder_energy_bound. Let a_x and b_x be the existing entrywise bounds for the finite RI-minus-BUPO map and the BUPO-to-full-Coulomb remainder, respectively. They already include the same Hermitian probes, weighted density entry caps and L1 cap used for the combined certificate. For either vector v, the same-density trace inequality gives

\[ B_E(v)=\min\left(\tfrac12 L_1\max_x v_x, \tfrac12\sum_x\min(H_x,H_{x^T})v_x\right). \]

Only positive-weight k entries enter the sum. Without entry caps, use the first expression alone. The shared outward positive arithmetic preserves zero and guards compound sums. The two new receipts report B_E(a) and B_E(b). The combined energy certificate still uses B_E(a+b). Taking a minimum between two upper estimates does not make this formula linear or imply an ordering relative to B_E(a)+B_E(b). The component bounds are separate enclosures, not an additive measured-error decomposition. Integral values, density families and final target checks remain unchanged. Count admission includes the extra contraction work; no response tensor or additional source owner is retained.

Automatic reference readiness now compares B_E(b) with the requested reference fraction of the energy target, alongside the existing reference J criterion. This replaces the former looser scalar expression L1*max(b)/2 used for Section 5.24 readiness. A reference can therefore be retained when its energy envelope is adequate even if that scalar estimate is too large. Passing this stage only permits the fitting/metric/domain search to continue. The actual selected source must still pass both combined full-Coulomb targets before SCF; reference fractions do not weaken or replace those targets.

The finite/reference energy bounds appear with the final certificate and trial provenance. The underlying full-Coulomb fitting proof, shared producer identities, unified Fock, BIPOLE/CRYSTAL convention and symmetry permissions are unchanged. These bounds still exclude arithmetic, rank stability and density relaxation, and do not certify the full SCF energy.

5.29 Bounded shared reference actions

PeriodicEwaldSource and PeriodicBupoSource expose plan_batch(count,  budget) and build_batch(densities, budget) for one to 32 independent complex Hermitian densities with shape (batch, momenta, AO, AO). Admission precedes input copies and numerical work. Memory covers simultaneously live inputs, responses and exports, coefficient panels, compensated sums, reciprocal panels, bounds and receipts. Work retains conservative cold scalar reservations plus batch setup; rejected search work is never refunded.

The Ewald producer streams each real tile, Fourier panel and overlap panel once across a batch. Each density retains the scalar reduction order, charged electronic gauge and source identity. Direct BUPO partitions also share each Gaussian majorant-pair evaluation, then apply the same positive bound arithmetic and pair-image envelopes independently. No four-index, all-probe response tensor or quadratic pair-bound cache is retained. Approximate BUPO partitions explicitly use the scalar fallback for each density; this increment does not accelerate their multipole actions.

Batch receipts count actual source passes, kernel evaluations and domain majorant evaluations. Per-density receipts retain logical scalar counts for comparison; summing those counts would double-count shared evaluation work. Physical values, energies, density identities and bound components have the same meaning as scalar results. Returned arrays own their copies.

Fitting options use probe_batch_size=8, admitted between one and 32. One selects the unchanged scalar reference path. The fitting consumer holds only a bounded batch and at most one pending real-probe envelope when a batch boundary splits a real/imaginary pair. The original hypot, positive sum and certificate identity order are preserved, including partial batches, zero entry caps and zero k weights. Final and trial provenance reports reference_batches, reference_source_passes, reference_shell_calls and reference_majorant_pairs.

This reorganizes the existing Ewald1921 and Sun2017 producer evaluations; it changes no integral formula, fitting proof, error target, source family, unified Fock composition, BIPOLE/CRYSTAL convention or symmetry permission. The stored-integral and fixed-density scope of Sections 5.20 through 5.28 remains in force, including the arithmetic and relaxation exclusions.

5.30 Reference contributions and directed refinement

The existing shared BUPO Gaussian majorant producer already separates real and reciprocal lattice remainders. Its scalar and batch consumers now retain separate positive AO response envelopes, alongside the finite multipole envelope before adding domain and pair-image omissions. The original total response reduction remains unchanged. Extra live arrays, exported copies, positive contractions and receipt history are included in count admission.

The fitting consumer applies the same normalized Hermitian probes, complex hypotenuse bound, weighted entry caps and L1 cap to each reference component. It reports maximum J and Section 5.28 trace energy enclosures for finite approximation, retained-pair real-domain omission, retained-pair reciprocal omission and omitted pair images. No ordering between the sum of separately minimized bounds and the combined bound is assumed. Components are distinct from the finite RI-minus-reference discrepancy. For a fixed source, the original combined certificate, digest and final target checks are unchanged.

PeriodicRIFittingReferenceOptions.selective_refinement defaults to true. After a failed reference-readiness check, consider only components with a control that can still advance. Select the largest of each component’s J or energy bound divided by its target and the reference fraction. Ties use approximation, real domain, reciprocal domain, then pair images. Advance only that control: reduce tolerance by ten, grow the corresponding lattice cube by one, or grow pair cube and cutoff within their caps. Record the action on the next trial. The default allows twelve component steps; the explicit refinement cap remains authoritative. False preserves the former joint schedule.

This is a deterministic bounded greedy policy, not an optimality or convergence theorem. Saturation or exhausted trials still refuse before SCF. The existing combined reference readiness and full fitting target tests remain authoritative; a component estimate alone never accepts a source. Rejected work is charged and the actual first passing RI owner is retained.

The derivation reuses the Ewald1921 and Sun2017 producers and Section 5.20 fixed-family proof; no new integral formula or variational fit assumption is introduced. Unified Fock, BIPOLE/CRYSTAL conventions and symmetry permissions are unchanged. Stored-integral, fixed-density scope and the arithmetic, rank and relaxation exclusions remain in force.

5.31 Stored metric spectrum and retained rank

Each periodic RI owner audits its existing Hermitian eigensystem before constructing the same retained factor. The shared certify_hermitian_spectrum leaf accepts a finite exactly Hermitian complex128 matrix A, ascending real values L and a square approximate eigenvector matrix Q. Count-only admission precedes array copies and numerical inspection. Cubic interval work and linear norm scratch are included in the periodic source plan; no additional quadratic retained object or coupling tensor is introduced.

Outward binary64 operations enclose R=AQ-QL and G=Q*Q-I. Entry magnitudes use |real|+|imag|; the maximum row/column sum bounds each spectral norm. Write r >= ||R||2 and e >= ||G||2. When e<1, sigma_min(Q) >= sqrt(1-e) >= 1-e. The all-column theorem of Rump and Lange2023, Sections 3 and 4, Eq. (3.2), Eq. (4.2) and Theorem 4.1, therefore pairs every exact eigenvalue with an approximation within delta=r/(1-e). Every bound operation rounds outward. This conservative implementation does not use the paper’s cluster-specific algorithms or assume that a union of individual residual intervals covers the whole spectrum. See DOI 10.1016/j.cam.2023.115332.

The exact adaptive cutoff max(absolute, relative*max(0,lambda_max)) has its own interval. Every retained eigenvalue interval must lie strictly above that interval, and every discarded interval strictly below it. If these separations have positive minimum m, Weyl perturbation theory gives the strict radius m/(1+relative): every additional Hermitian matrix perturbation with spectral norm smaller than this radius preserves the selected rank, including recomputation of the relative cutoff. Empty retained/discarded spaces and repeated eigenvalues within a space are allowed. A cutoff crossing or unproved full rank of Q reports an unresolved certificate, never a repair.

The API returns spectrum_certified, rank_certified, rank, residual and orthogonality bounds, eigenvalue error, cutoff interval and rank radius. When a qualification flag is false its downstream bounds are unavailable; zero is a sentinel, not a zero-error claim. Unsupported arithmetic modes or unrepresentable interval operations refuse. The implementation requires round-to-nearest IEEE binary64 with gradual underflow and forbids fast math. It neither changes global rounding mode nor relies on BLAS for the proof.

The scope is the exact stored canonical Hermitian metric. It does not bound errors in integral construction, retained eigenvectors/factors, inverse or Fock arithmetic, overlap rank, density relaxation or total SCF energy. The existing full-Coulomb fixed-density bounds retain their arithmetic exclusions. The radius is diagnostic in this increment, not a new automatic rejection policy. Source/factor identities, unified Fock composition, BIPOLE/CRYSTAL conventions and symmetry authority are unchanged. The real metric-response heuristic guard remains separate and is not promoted to a verified bound.

5.32 Rank-aware fitting admission

PeriodicRIFittingAccuracyOptions.metric_rank_perturbation optionally requests a finite nonnegative spectral-norm perturbation margin for the actual selected Gamma Coulomb metric. PeriodicRIFittingBoundOptions exposes the same contract for raw fixed-family consumers. None retains the earlier acceptance policy; zero requires a resolved rank even though no positive margin is requested. The shared Section 5.31 certificate must report rank_certified and its strict rank_stability_radius must exceed the requested margin. Equality refuses. A perturbation whose norm is at most the requested margin then preserves the selected rank. The value is a caller-declared margin, not an estimate of the unknown error in Gaussian integrals or the metric itself.

Fixed sources refuse before SCF when the condition fails. Bounded metric, physical-domain and auxiliary searches use it together with the unchanged full-Coulomb J/energy targets, retaining the first actual owner that passes all conditions. The existing refinement order, cutoff floor, domain/basis caps and cumulative budgets remain authoritative. Neither rank nor radius is assumed monotone. Exhausted searches refuse; no eigenvalue shift, absolute value repair, alternate factor or post-SCF retry is introduced.

Reference readiness remains a J/energy reference-quality decision. An unqualified provisional rank cannot force the reference producer to refine its lattice domains. Final admission still checks the selected Gamma owner through the rank condition. Each reference, metric, domain and auxiliary trial, and the final fitting receipt, exports the requested margin, actual spectrum and qualification. The full-Coulomb certificate digest continues to identify its original map/family/targets; the rank contract is separately reported against the same ri_source_identity. An unenforced rejected raw trial cannot be consumed later. Plans include scalar checks, receipt copies and all existing reserved failed trials before numerical work.

This reuses the Rump/Lange2023 and Weyl proof in Section 5.31 and the shared Sun2017/Neese2025 fitting/reference producers. It grants no additional bound on eigenvectors, factor/inverse or Fock arithmetic, no integral-error estimate, no overlap-rank guarantee and no relaxed total-energy certificate. It applies to Gamma fitted J, not every exchange-transfer metric. Unified Fock, BIPOLE/CRYSTAL conventions, physical source identity and symmetry permissions are unchanged; roundoff_included in the J/energy certificate remains false.

5.33 Stored factor inverse error

certify_hermitian_inverse(A, values, vectors, factor, options, budget) bounds ||F F* - C||2, where F is the actual stored factor and C is the positive retained inverse of the exact stored canonical Hermitian metric A. It reuses Section 5.31 once, requires certified adaptive-cutoff rank, and does not compute another eigensystem. Complete periodic RI owners export the result as metric_info.inverse_certificate; final method provenance carries it alongside the actual source and factor identities for each metric owner.

Write the sorted approximate eigenvalues as mu, Q for the approximate vectors, D for the diagonal retained inverse of mu, R = A Q - Q diag(mu), and e for the verified orthogonality error. Let alpha be the lower enclosure of the smallest retained exact eigenvalue, beta the smallest retained mu, dA the upper enclosure of the largest discarded exact eigenvalue, and dB the largest discarded mu. For a nonempty retained space, the certificate requires alpha > 0 and, when a discarded space exists, alpha-dB > 0 and beta-dA > 0. All arithmetic in these bounds expands outwards.

In an exact unitary eigenbasis V of A, put W = V* Q. Entrywise, [V*(C Q-Q D)]ij = (f(lambda_i)-f(mu_j)) Wij and [V*R]ij = (lambda_i-mu_j) Wij, where f is the retained inverse on each certified spectrum. The finite divided difference is bounded by L = max(1/(alpha beta), 1/(alpha (alpha-dB)), 1/(beta (beta-dA))). The mixed terms are omitted for a full retained space; a repeated eigenvalue inside a space uses the same inverse derivative or zero. Thus ||C Q-Q D||2 <= L ||R||F, without dropping higher-order terms. This is a finite intertwining identity, not a first-order Frechet error estimate.

Using C-Q D Q* = C(I-Q Q*) + (C Q-Q D)Q*, square Q gives ||I-Q Q*||2 = ||I-Q*Q||2 <= e, ||Q||2 <= 1+e, and ||C||2 <= 1/alpha. The reported eigensystem contribution is therefore e/alpha + (1+e) L ceil(sqrt(n)) residual_bound. The integer square-root ceiling conservatively converts the spectral residual bound to Frobenius norm without relying on a rounded square root. No eigenvector orientation or within-space eigenvalue separation is assumed.

An independent interval contraction encloses every entry of F F* - Q D Q* using the actual stored F, Q and mu. The maximum absolute row/column sum bounds its spectral norm, including factor normalization rounding. Adding this factor_product_bound to the residual and orthogonality terms gives inverse_error_bound. A certified empty retained space has an empty factor and exactly zero inverse/error. An unresolved rank gives qualified=false; its downstream zero fields are unavailable, not zero-error claims. Unrepresentable interval arithmetic refuses.

The shared plan reserves public copies before payload access and adds a second bounded cubic contraction with linear norm scratch. Periodic owners reuse their existing eigensystem and factor storage; they do not retain a new dense inverse. No physical factor, projector, Fock source, symmetry permission or BIPOLE/CRYSTAL convention changes. This certificate is a numerical diagnostic, not a new selection target.

The construction combines the residual enclosures of Rump/Lange2023, Sections 3 and 4, with spectral divided differences described by Higham2008, Chapter 3 and the author’s Frechet-derivative exposition. The finite identity and conservative norm conversion above are the implementation’s explicit extension; no first-order formula is used as a finite bound. The new citation feature is manta_hermitian_inverse.

This measures an absolute inverse-operator error, which can be conservative for nearly dependent auxiliary functions. It does not bound errors in the stored integral metric, coupling integrals, Fock contractions, overlap rank, relaxed energy, XC, basis or k sampling. The existing full-Coulomb fixed-density certificate still reports roundoff_included=false. A small inverse bound is not by itself a certified SCF energy tolerance.

5.34 Coupling-projected metric action bounds

The optional metric_action_j_error and metric_action_energy_error controls in PeriodicRIFittingAccuracyOptions and PeriodicRIFittingBoundOptions require a fixed-family numerical metric contribution below both targets. Both zero disables the new policy; otherwise both must be positive and finite. The bound compares the exact product of the actual stored RI factor with the exact retained inverse of the stored metric, applied through the same stored explicit coupling tiles. It is separate from the full-Coulomb fitting difference, which already compares the actual factor’s map to its reference. These bounds are not added to that difference or counted twice.

For a coupling column t, define b = F* t, x = F b, e = t-A x and G = F* A F-I. Let C be the exact retained inverse, P its exact retained projector, and Pd = I-P. The finite identity for two columns p and q is

tp* (F F*-C) tq = bp* G bq + xp* Pd tq + tp* Pd xq - xp* A Pd xq - ep* C eq.

It follows by substituting e and using C A = A C = P. No first-order remainder is omitted. For full rank the three discarded-space terms vanish. The action audit reuses the Section 5.33 certificate and its lower retained eigenvalue alpha. It encloses A F and every entry of G using the shared binary64 interval arithmetic. If gj bounds the absolute row sum of G, then diag(g) +/- G is positive semidefinite by diagonal dominance. Consequently |bp* G bq| <= sp sq, where sp^2 = sum_j gj |bjp|^2.

The exact b and e are enclosed per coupling column. Complex componentwise absolute sums bound Euclidean norms; outward sums and products propagate them. Square-root bounds use exact powers of two from the binary exponent, not an assumed libm square-root error. For partial rank, Section 5.33 gives ||Pd F|| <= sqrt(inverse_error_bound), since Pd C Pd = 0. Thus an upper bound lp on ||Pd xp|| is obtained from this factor bound and the projected column norm. A stored-metric row norm bounds ||A||. The scalar pair bound is

sp sq + ||ep|| ||eq||/alpha + lp ||tq|| + ||tp|| lq + ||A|| lp lq.

An independent fallback is inverse_error_bound ||tp|| ||tq||. The discarded space estimate can be conservative; the implementation never assumes a rounded projector is exact. A certified empty retained space has exactly zero metric action. An unresolved spectrum/rank remains unqualified.

Each positive outer-product term is contracted against the fixed weighted spin-total density family. For any scalar column envelope v, both L max(v) and sum(hq vq) bound its contraction when L and component bounds h are available. The implementation also uses both constraints jointly. For every nonnegative breakpoint lambda, the elementary dual inequality

sum(vq dq) <= lambda L + sum(hq max(vq-lambda,0))

holds whenever 0 <= dq <= hq and sum(dq) <= L. Sorting v in descending order and accumulating capacities selects a breakpoint near the primal allocation optimum. That selection need not be exact: the inequality holds for every selected lambda. The dual sum, positive differences and products are enclosed outward with the same binary64 arithmetic as the metric audit. The result is the minimum of this enclosure and the two preceding valid bounds. No rounded unused capacity is treated as an exact remainder. Zero component limits, ties, nonbinding total limits and near-equal capacities remain valid. Heap sorting and linear scratch are included in public and aggregate admission plans; sorting work scales as N log(N).

Each output keeps the smaller of the projected and fallback bounds. The energy bound applies the same joint support function to the resulting J entry envelopes and then multiplies by one half. This is a half-trace inequality, not the exact maximum of the quadratic energy error. This remains valid for complex indefinite Hermitian densities, zero component limits and zero-weight k points. There is no positivity or variational-fit assumption on independently truncated Coulomb integrals.

HermitianInverseAction provides reusable admitted column/family APIs. Columns are tied to their immutable owner; mixing owners refuses. Public array copies and all cubic setup, quadratic column work and family reductions are admitted before payload access. Periodic fitting reads one AO shell-pair panel across auxiliary shells via the existing PeriodicGDFTiles.evaluate API. It retains only scalar column envelopes and final entry bounds, never a complete auxiliary/AO-pair tensor. The actual factor and eigensystem remain owned by the existing RI producer; no alternate Fock factor is substituted.

The projected policy supports explicit tile actions, including coupling screening and ordered multipole tiles. Section 5.35 extends metric admission to ordered compressed/global moment representations using norm bounds. Pair-tree actions retain the guard; their existing approximation controls remain available. Actual contraction arithmetic remains excluded in all cases.

Fixed sources reject unmet targets before SCF. Existing bounded metric, domain and auxiliary searches require these targets along with the original full-Coulomb and optional rank targets, and retain the actual first passing owner. Reference readiness remains a separate full-Coulomb reference-quality decision. No monotonicity of the metric action error is assumed. Exhaustion refuses, and a rejected raw trial cannot later be consumed. Final and trial metric_action provenance reports targets, qualification, bounds, column and producer work counts beside the same physical RI source identity.

Rump2010 Section 10.8, Eq. (10.52) supplies the inverse-norm/residual principle; Rump/Lange2023 supplies the verified retained spectrum. The quadratic and discarded-projector identities above are this implementation’s explicit finite extension, not a copied Cholesky verification algorithm. The original Sun2017/Neese2025 coupling and periodic conventions are unchanged. Citations are emitted under manta_hermitian_inverse_action when this policy is used.

The bounds cover metric eigensolver and factor-construction errors relative to exact stored metric/coupling entries. They do not include integral construction, actual coupling/Fock contraction roundoff, overlap rank, relaxed SCF energy, XC, basis or k sampling. The separate full-Coulomb certificate still reports roundoff_included=false. Unified Fock, BIPOLE/CRYSTAL partitions and symmetry authority are preserved.

5.35 Compressed coupling norms for metric action

For ordered compressed auxiliary hierarchies, define T as the exact real arithmetic graph of the actual stored residual tile entries, auxiliary and pair moment panels, translation matrices, interaction matrices and complex phases. This is a precise comparison object; a separately materialized, rounded dense tensor need not equal T entrywise. Global-panel reuse evaluates the same graph algebraically and shares this bound. The global pair-tree extension below covers its distinct gathering and Gamma orbit averaging.

The producer propagates positive component envelopes on its existing upward auxiliary traversal. Leaf panels use their maximum absolute entry per moment component and the componentwise absolute sum of each actual stored phase. Each parent applies u_parent += abs(W) u_child. For each selected far block, the positive contraction sum_ij u_i abs(I_ij) p_j, multiplied by the maximum stored pair-phase component sum, bounds every auxiliary-row/AO-pair entry. Here p is the maximum absolute pair-panel entry per component. Adding all accepted far blocks and the absolute real/imaginary components of each stored residual tile gives an entry bound. Summing these over auxiliary rows gives c_q >= ||T_q||_2 via the L1 norm. All envelope sums and products use the shared outward binary64 primitives and their arithmetic guard. No computed phase is assumed to have exact unit modulus; no cancellation is credited.

Let eta be the verified Section 5.33 bound on ||F F*-C||_2, where C is the exact retained inverse of the stored metric. Then

|T_p* (F F*-C) T_q| <= eta c_p c_q.

Using the joint density support H from Section 5.34 gives J_p <= eta c_p H(c) and E <= H(J)/2, with outward arithmetic throughout. An unresolved inverse certificate remains unqualified. A certified empty retained inverse and factor give exact zero. Supplied norm envelopes must be finite and nonnegative. These statements require no density positivity or variational-fit assumption.

PeriodicGDFTiles.coupling_norm_bounds owns the traversal and fixed source partition. HermitianInverseAction.norm_family owns inverse-error propagation and the shared density support; periodic fitting only composes these APIs. The audit runs a zero forward action with additional scalar envelopes and reuses magnitude panels. It leaves normal forward/adjoint actions, source identity, actual RI factors, unified Fock and symmetry authority unchanged. Count plans reserve the extra traversal, exported column norms and outward work. Aggregate fitting admission conservatively retains explicit-panel reservations as well. No dense coupling tensor is materialized by the audit.

Neese2025 Eqs. (29)-(36) supplies the compressed gather/translate/scatter structure; Sun2017 Eqs. (16)-(17) supplies the periodic convention. The positive DAG enclosure and norm-only metric propagation above are this implementation’s explicit extension, using the verified inverse result rather than asserting a new integral algorithm. Integral construction, compressed/Fock contraction roundoff and relaxed SCF energy are excluded. This is not a complete arithmetic ledger or a claim of useful accuracy for arbitrary production systems.

Global pair-tree stored graph

For each leaf, the maximum absolute pair-panel coefficient per moment is multiplied by the maximum real/imaginary component sum of its stored Bloch phases. At a parent, the componentwise maximum of abs(W) u_child bounds any one descendant column. This is a maximum across different leaves, not their charge sum. Each accepted auxiliary interaction contributes abs(phase_aux) sum_ij a_i abs(I_ij) u_j uniformly to its represented leaves, where a is the componentwise maximum of the stored auxiliary panel.

These positive interaction envelopes propagate down the existing tree and are summed over original images of each AO shell pair. Exact near blocks contribute their maximum stored absolute entry times both stored phase bounds. Multiplication by the auxiliary shell size and summation over shells then bound each column’s auxiliary L1 norm. All positive sums and products use the same outward binary64 primitives as the ordered audit. The source partition, packed total-degree interactions and selected finite masks stay unchanged; the audit reuses the existing magnitude workspace.

At Gamma, the producer’s conjugate-completed coupling is T(k,mu,nu) = (T0(k,mu,nu) + conj(T0(k,nu,mu)))/2. Apply that same convex average to the two column bounds with the AO indices transposed on the mate. Add the norm of the stored reciprocal/zero residual after this averaging. The extra column and mate buffers fit the existing norm-audit reservation; the count plan charges outward traversal work before execution. No auxiliary-by-AO-pair tensor is allocated in production.

The resulting column norms enter the unchanged inverse-error and joint density-family support formulas above. This admits pair-tree metric-action targets without certifying errors in integral construction, actual gathering, translation, scattering or Fock arithmetic. Source symmetry admission and evaluated Fock/solver policy remain separate authorities.

5.36 Executed paired metric arithmetic

The shared certify_paired_metric_roundoff leaf accepts the actual stored factor F, forward-action output x, computed projection r, reconstruction c and Coulomb energy e. Its reference is exact arithmetic on these stored F and x: r0 = transpose(F) x, c0 = conjugate(F) r0, e0 = sum_j (Re(r0_j)^2 + Im(r0_j)^2)/2. This is the paired RI convention of Section 5.34, not an adjoint substituted for the transpose.

Outward complex interval products and sums enclose r0. Those intervals then propagate through the stored conjugated factor to c0 and through the squared norm to e0. Subtracting the actual returned r, c and e gives per-component and maximum absolute error bounds. Complex errors use the sum of absolute real/imaginary component errors. The replay does not assume a BLAS reduction order, ignore cancellation or assume every product is normal. It uses the existing round-to-nearest/gradual-underflow guard and fails closed on unrepresentable enclosures. An empty factor and exact zero outputs have exact zero arithmetic error.

plan_paired_metric_roundoff admits work and storage before copying Python payloads. Native callers borrow F; the periodic q/Fock plans conservatively reserve the maximum-rank audit alongside their existing workspaces. The audit does not recompute or substitute a numerical factor, change a density or alter physical source identities. Complete paired RI builds requesting metric-action certification report metric_roundoff_evaluated and separate metric_projection_roundoff_bound, metric_coefficient_roundoff_bound and metric_energy_roundoff_bound values. Raw q builds may request the same audit with audit_metric_arithmetic=True. Retained-mode and fused J/K paths report unevaluated; the request does not force a different contraction route.

These are actual-input, post-build arithmetic bounds. They are not the pre-SCF fixed-family inverse error and are not folded into its target or the full-Coulomb discrepancy twice. The coefficient error alone is not a J error. The propagation below applies when an admitted fitting owner retains the required source norm. Forward/adjoint coupling arithmetic, spin/k-weighted density construction, final J/K/Fock assembly, integral construction and relaxed SCF energy remain excluded. Consequently roundoff_included=false remains correct for the full-Coulomb receipt, and the wider executed-arithmetic work package remains open.

Metric arithmetic contribution through the stored adjoint

The fitting owner retains metric_action_adjoint_norm_bound, the maximum of the outward auxiliary L1 column norms already computed during metric-action admission. Explicit sources reuse their streamed column certificates; ordered, global-panel and pair-tree sources reuse the producer’s stored-graph norm audit of Section 5.35. No new integral sweep, coupling tensor or numerical factor is needed. The plan charges the scalar scan and outward propagation.

Let epsilon_c bound the maximum component of the executed coefficient error, and let N bound every stored coupling column’s auxiliary L1 norm. The exact stored adjoint obeys max_p |sum_a T_ap delta_c_a| <= N epsilon_c. An outward multiplication reports metric_j_roundoff_bound and sets metric_j_roundoff_evaluated only for an audited paired metric result belonging to this same admitted source. Complex component L1 bounds dominate the complex modulus and remain valid under cancellation. Zero coefficient error yields exact zero contribution. The norm is immutable across builds and physical source/snapshot identities remain unchanged.

This is the metric substep’s contribution through the exact adjoint, not a bound on rounding in the executed coupling adjoint. Raw q builds without a fitting certificate, retained-mode and fused J/K paths leave propagation unevaluated. It remains a separate post-build diagnostic; inverse-action pre-SCF targets, full-Coulomb roundoff_included=false, forward-input errors, weighted density and final Fock arithmetic are unchanged. This advances #624 without closing its remaining executed-coupling and assembly scope.

5.37 Executed periodic coupling arithmetic

The producer’s optional audit_roundoff=True action replays outward complex intervals alongside the existing forward or adjoint schedule. Its reference is exact arithmetic on the declared stored representation of Section 5.35: explicit residual tiles, stored moment panels, translations and interactions, near blocks and actual stored phases. Kernel, moment, phase and residual-tile construction errors remain separate. This is not exact arithmetic on ideal integrals or on a rounded exported compressed tensor.

For ordered/global panels, the replay folds phases and input into pair charges, propagates auxiliary hierarchy charges or fields, then performs the existing panel projections. Global actions retain interval charges/fields beside their source-owned panel references. Pair-tree actions replay upward and downward translations on the same nodes and accepted antichain, including near blocks. Gamma completion applies the same conjugation, AO transpose and convex average to the interval outputs. The residual tile contraction is then included in the same output interval. No new partition, density-dependent selection or dense auxiliary-by-AO-pair tensor is introduced.

All products and sums use the existing binary64 outward interval primitives. The final interval is subtracted from the actual returned numerical output, so the bound includes its compensation/reduction order without assuming a normal-only relative-error model or assigning an exact unit modulus to a stored phase. Per-output roundoff_bounds and maximum roundoff_bound use complex component L1 error; roundoff_evaluated distinguishes evaluated zero from an absent audit. Arithmetic mode guards, nonfinite checks and enclosure overflow fail closed. Exact zero input has zero output error.

plan_coupling_roundoff preflights all five caps before input copies. It reserves interval vectors and temporary products beside existing action/tree workspaces, Gamma mates and binding exports; its work allowance covers the outward replay. Numerical values, source/snapshot identity and integral passes remain unchanged. Complete paired RI builds requesting metric arithmetic also audit their actual forward and adjoint inputs, reporting coupling_forward_roundoff_bound, coupling_adjoint_roundoff_bound and coupling_roundoff_evaluated through the shared q/Fock receipts and provenance. Retained-mode and fused J/K routes do not claim this paired audit.

The two action bounds are separate actual-input contributions. Section 5.38 composes forward error through the fixed metric and adjoint; the adjoint bound alone is not their sum. Spin/k-weighted input construction, integral/source construction, final J/K/Fock assembly and relaxed SCF energy remain outside this ledger. Pre-SCF inverse targets and full-Coulomb roundoff_included=false remain unchanged. This advances #624 without closing its composition and assembly requirements.

5.38 Paired periodic J and energy arithmetic composition

For the actual stored spin-summed, k-weighted input d, let the exact stored coupling graph define x0 = conjugate(T) d. The paired reference is rho0 = transpose(F) x0, c0 = conjugate(F) rho0, J0 = transpose(T) c0, and E0 = norm(rho0)^2 / 2. F is the existing stored retained factor. This extends the finite paired convention of Sun2017 Eqs. 16-17 and Neese2025 Eqs. 29-36 with the outward arithmetic of Sections 5.36-5.37; it does not replace the inverse-approximation reference of Section 5.34.

The executed forward action supplies componentwise complex L1 bounds eta on x minus x0. certify_paired_metric_action accepts these nonnegative finite bounds alongside the actual x, projected vector, coefficient vector and energy. It encloses each exact input in a complex rectangle of radius eta about the stored x, then replays both fixed-factor products and the squared norm with outward intervals. Subtracting the actual numerical witnesses yields combined forward-plus-metric projection, coefficient and energy bounds. It includes all propagated input and metric arithmetic, including higher-order terms. It does not add the metric-only diagnostic a second time. Zero input radii reuse the original point replay; zero rank consumes no uncertain input and has exactly zero error for zero numerical outputs. Shape, finite, arithmetic mode, resource and overflow guards fail closed.

Let epsilon_c be the maximum combined coefficient bound, N the same-source auxiliary L1 column norm admitted in Section 5.36, and epsilon_adj the executed adjoint error at the actual computed coefficients from Section 5.37. Then max_p abs(J_p - J0_p) <= N epsilon_c + epsilon_adj. Both final operations round outwards. The energy bound is the combined metric energy bound because the numerical energy is computed before the adjoint. The same argument uses complex component L1 norms, which dominate complex modulus.

Q receipts report forward_metric_roundoff_evaluated, forward_metric_coefficient_roundoff_bound and forward_metric_energy_roundoff_bound. The fitting owner combines these with its existing source-qualified norm and actual adjoint bound, setting paired_roundoff_evaluated, paired_j_roundoff_bound and paired_energy_roundoff_bound. Shared Fock receipts and standard provenance carry all fields. Raw q calls without a fitting norm expose only the combined metric diagnostic. Retained-mode and fused J/K paths remain unevaluated.

plan_paired_metric_action charges interval input storage and widened products in addition to metric replay; the q plan includes both old and combined audits and their live outputs. No new source sweep or dense coupling tensor is needed. The existing numerical schedule, source, metric rank, physical evaluated state, Fock/eigensolver and symmetry admission stay unchanged.

The comparison begins at the actual stored weighted input and stored graph. Spin summation and k weighting, integral/phase/moment/source construction, final J/K/Fock assembly and relaxed-SCF error remain separate obligations under #623-#627. Existing pre-SCF inverse-action targets still apply to their original contribution. Full-Coulomb roundoff_included=false remains mandatory. These post-build paired bounds do not by themselves establish automatic whole-method accuracy or close #624.

5.39 Stored-density spin and k weighting

The paired arithmetic comparison can now begin at the supplied stored spin densities D_s and k weights w, before spin summation and multiplication. For each AO-pair entry p at k, the reference is u0_p = w_k sum_s D_s,p, evaluated in exact arithmetic on those stored operands. An outward complex interval sums the original spin entries and multiplies by the stored weight. Subtracting the actual numerical weighted input u gives a per-entry complex L1 error eta. The numerical summation and multiplication order are unchanged. weighted_input_roundoff_evaluated and weighted_input_roundoff_bound report this input step and its maximum error.

The shared producer forward/adjoint APIs accept optional input_errors, a contiguous float64 vector of nonnegative finite complex L1 radii. Supplying it requires audit_roundoff=True and matching flattened input length. The existing interval graph starts from rectangles about the actual input, with radius eta in each real/imaginary component. Zero radii preserve the point replay. The Gamma mate receives the conjugate/permuted input intervals, including the AO transpose for a forward density, before its normal action. All explicit residual, ordered/global-panel, pair-tree near and hierarchical paths use these same intervals. Numerical values and source/input/snapshot identities stay unchanged; input_error_included identifies the extended comparison scope on the action receipt. The caller supplies the input-error hypothesis; the producer verifies shape, arithmetic mode, finite values, resources and enclosure overflow, not an external error model’s derivation.

The returned forward enclosure covers both input uncertainty and executed coupling arithmetic. Its rectangle contains the original stored input too, so it remains a valid, possibly wider, actual-input rounding bound. The paired metric audit of Section 5.38 consumes these per-RHS enclosures once. The fitting owner then combines its coefficient enclosure with the existing same-source norm and actual-input adjoint error. The reported paired J and energy bounds therefore enclose the exact stored paired graph at u0, without adding the input contribution or metric arithmetic twice. coupling_forward_input_error_included and paired_input_roundoff_included mark the extended scope on q/Fock receipts and standard provenance. They are false on unevaluated routes. Raw q calls without the fitting norm still cannot claim the composed J bound, though their forward/metric audit includes input weighting. Retained-mode and fused J/K execution are not replaced to obtain an audit.

Resource plans charge input rectangles, Gamma copies, binding error arrays and the spin/weight interval work. No new source sweep, partition or dense coupling tensor is added. This uses the paired convention and outward arithmetic references in Sections 5.36-5.38. Source/kernel/support, retained factor, Fock/eigensolver and symmetry authority remain unchanged.

These bounds begin at stored D_s and w; they do not certify the orbitals or operations that produced D_s, or uncertainty in the physical inputs. Integral/source construction, final J/K/Fock accumulation, other energy terms and relaxed-SCF error remain excluded. Full-Coulomb roundoff_included=false and existing pre-SCF inverse-action targets remain unchanged. #624 now requires final assembly and its remaining non-paired/K routes; #623-#627 and the full capability matrix retain their other obligations.

5.40 Evaluated physical Fock and energy assembly

For an admitted paired arithmetic path, the native HF/KS method adapter now retains a shared interval audit of its stored physical components. With stored method density D_s, k weights w, one-electron H, Coulomb J, semilocal Vxc, exchange components K_t with signed method coefficients a_t, and actual stored zero-mode correction C, its exact reference is F_s = H + J + Vxc_s - sum_t a_t K_ts + C_s. The energy reference is Tr_w(D H) + EJ + EXC + Enuc

• (1/2) sum_t a_t Tr_w(D K_t) + (1/2) Tr_w(D C). Here EJ, EXC and Enuc are the method’s actual stored component energies, and Tr_w is the real spin-summed, k-weighted matrix trace. RKS direct exchange uses a_t equal to half the physical exchange fraction; fitted K built from equal spin halves uses the full fraction. Range components retain their signed coefficients. HF uses the corresponding stored K and unit coefficient. The audit consumes the method’s actual component arrays, with no new source calculation, partition, eigensolve or numerical contraction in the physical path.

The existing outward binary64 interval operations enclose these expressions at exact stored operands, including density weighting and trace arithmetic. This reuses the verified-computation basis in Sections 5.36-5.39 and Rump (2010), DOI 10.1017/S096249291000005X. The audit then compares that enclosure with the actual public evaluated Fock and method energy. Thus the reported complex entrywise L1 maximum covers both method assembly and the numerical solver’s callback reconstruction and Hermitian projection. Capped states use the last evaluated state; converged states retain the existing confirmation. No additional confirmation call is introduced. The shared Fock/eigensolver policy, physical/source identities and source-specific symmetry admission remain unchanged.

assembly_roundoff is a method receipt, separate from the partial RI fitted_fock. Its evaluated, fock_bound and energy_bound describe only this stored-component comparison. paired_coulomb_included marks composition with the accepted paired J/energy bounds of Section 5.39: combined_fock_bound = outward(fock_bound + paired_j_roundoff_bound), combined_energy_bound = outward(energy_bound + paired_energy_roundoff_bound). No forward, metric, weighting or adjoint term is added a second time. The paired graph begins at the stored source spin-density embedding, whereas the assembly traces use the public method density. Errors in generating those densities, including restricted-to-source embedding, remain outside the paired source certificate. The combined comparison holds the stored H, K, Vxc, EXC, Enuc and correction values fixed; it replaces only the stored Coulomb response/energy by their exact stored paired graph values.

Persistent interval F and weighted-density arrays and all audit work are reserved in complete HF/KS driver plans before SCF. The raw diagnostic plan_periodic_assembly_roundoff(n_basis, momenta, spins, budget) admits the single-exchange comparison. certify_periodic_assembly_roundoff accepts complex128 H/J[k,n,n], D/K/Vxc/C/F[spin,k,n,n], float64 weights[k], a signed exchange scale, stored EJ/EXC/Enuc and the actual total energy. It admits counts before copying arrays or inspecting payload values, requires finite operands, and refuses arithmetic overflow or unsupported rounding modes. This diagnostic has no independently supplied paired-source certificate and therefore returns assembly-only bounds. The actual method may add multiple signed exchange components through the same helper.

The audit is active only when the paired Coulomb arithmetic is qualified; other routes explicitly leave evaluated=false. Integral/phase/moment/source construction, the actual K/XC/correction construction, density generation, rank/relaxation errors and non-paired execution remain separate obligations. Pre-SCF inverse-action targets and full-Coulomb roundoff_included=false are unchanged. #623-#627 and the full capability matrix remain unfinished.

5.41 Executed direct periodic exchange contraction

The existing finite reciprocal exchange producer optionally audits its actual stored Bloch Fourier panels B, input density channels D and real reciprocal factors f. For each state s and target k its exact stored-input reference is K0_s,k = sum_(source,Q) f_(source,Q) B_Q^dagger D_s,source B_Q. The factor is the actual stored coefficient used by the numerical producer, including its source k weight, Fourier kernel and cell-volume division. Its construction is held fixed. The interval audit encloses D B and then B^dagger (D B), scales by that stored factor and accumulates over the existing support. It compares the resulting intervals with the actual compensated output K. The numerical Eigen products, panel batching, compensation and source schedule are unchanged. This uses the same Sun2017 Fourier convention and shared Rump2010 outward arithmetic as the earlier paired and assembly audits.

audit_roundoff=True on plan_periodic_exchange or compute_periodic_exchange admits the interval response and scratch storage, per-entry error output and cubic audit work before payload copies or values. The plan marks roundoff_requested. The result’s roundoff_bounds array has shape [density,k,AO,AO] and encloses each entry’s complex L1 difference from K0. The receipt reports roundoff_evaluated, its maximum roundoff_bound, and roundoff_scope=stored-bloch-panels-and-reciprocal-factors. Unaudited calls return no entry bounds and keep the scope flag false. There is no additional Fourier evaluation or four-index AO tensor. Empty or zero-factor support and zero densities retain exact zero bounds. Bare Coulomb, erf and erfc use their original reciprocal and zero-mode conventions.

The producer also encloses each reported channel energy against E0_s = -(1/2) sum_k w_k Re Tr(D_s,k K0_s,k), using exact stored target weights and densities. energy_roundoff_bounds includes both K-contraction and final weighted energy-trace arithmetic. It is a complete producer energy diagnostic, not an extra term to add to the method assembly bound of Section 5.40.

Complete RI HF/KS methods requesting the existing metric-action accuracy policy admit this audit for their complementary direct exchange components. When the paired Coulomb path qualifies, the existing assembly helper consumes those K-entry bounds and the actual signed exchange coefficients a_t. It reports direct_exchange_roundoff_included, direct_exchange_fock_bound and direct_exchange_energy_bound. For each Fock entry the added radius is sum_t |a_t| eta_t,entry. Its energy contribution is (1/2) sum_(t,s,k,i,j) |a_t| |w_k D_s,k,j,i|_1 eta_t,s,k,i,j, with outward bounds on each term. The maximum entry radius is added once to the combined Fock bound, and the energy contribution once to the combined energy bound, after the accepted paired Coulomb and assembly terms. The producer energy-trace bound is not added again because the assembly audit already compares the reported method energy with exact traces at stored K. Restricted/global/range scaling stays method-owned. Pure KS adds no direct-K term, and unaudited or fitted K is not silently relabelled as direct exchange.

This extends the combined comparison to exact direct-K contraction at stored B and f. Fourier integrals, Bloch phase and reciprocal-factor construction, density generation and restricted source embedding, correction and XC source construction, fitted/retained/fused K arithmetic, rank and relaxed-SCF errors remain separate obligations. Full-Coulomb roundoff_included=false and pre-SCF inverse-action targets are unchanged. No Fock object, solver policy, BIPOLE/CRYSTAL source convention or symmetry permission changes. #624 and full MANTA #617 remain open.

The raw certify_periodic_assembly_roundoff diagnostic also accepts optional exchange_errors, a finite nonnegative float64 [spin,k,n,n] array of complex L1 radii about its stored K. The caller owns the validity of that hypothesis. It uses the same propagation and records the direct-exchange scope flag; zero radii recover the assembly-only numerical bounds. Admission precedes payload inspection. Exact rational perturbed-K references qualify signed coefficients, spin channels and the weighted trace composition independently.

5.42 Stored-input exchange zero-mode correction arithmetic

The existing BvK exchange correction is C0 = -a m S D S, with exact interpretations of the stored overlap S, source spin density D, Madelung coefficient m and method exchange scale a. The shared outward arithmetic from Section 5.40 encloses both matrix products and the product of the two stored scalar coefficients, then compares each entry with the actual compensated correction. It does not replace S with its adjoint or change the physical correction order. Restricted HF audits its actual half-density source channel and propagates the radius using the public total density; KS retains its method-owned restricted or independent-spin exchange scale.

The method receipt and standard provenance expose exchange_correction_roundoff_included, exchange_correction_fock_bound and exchange_correction_energy_bound. Given per-entry complex L1 radii eta, the Fock contribution is max eta, and the energy contribution is the outward sum (1/2) sum_(s,k,i,j) |w_k D_s,k,j,i|_1 eta_s,k,i,j. These contributions are added once to the combined assembly, paired Coulomb and direct-K bounds. The assembly-only reference still holds its stored C fixed and already includes the correction trace arithmetic; the construction radius must not be added twice. Strict-zero-mode and pure KS have no correction contribution. Converged and capped evaluated states follow the same existing state and terminal-confirmation contracts.

The raw plan_periodic_correction_roundoff and certify_periodic_correction_roundoff APIs expose this same comparison for S[k,n,n], D[spin,k,n,n] and actual C[spin,k,n,n], all complex128. The diagnostic returns float64 radii of D’s shape and accepts one or two spin channels. Memory and cubic arithmetic caps are checked before copies or payload inspection; parent HF/KS plans charge audit scratch and every callback’s work. The optional correction_errors argument of certify_periodic_assembly_roundoff consumes finite nonnegative radii with that shape; the caller owns their validity. Zero radii preserve the previous assembly-only numerical bounds. No additional physical-source sweep or four-index tensor is introduced.

This extends the reference only to exact arithmetic on stored S, D, m and a. Overlap-integral and Madelung-source construction, restricted density embedding, fitted/retained/fused RI arithmetic, XC, rank and relaxed-SCF errors remain outside it. Full-Coulomb roundoff_included=false, source identities, pre-SCF inverse targets, shared Fock/eigensolver, BIPOLE/CRYSTAL and symmetry policies are unchanged. Exact rational independent four-index expansions and complete-method controls qualify the new contribution. The Rump2010 reference and shared arithmetic assumptions from Section 5.40 apply. #624 and #617 remain open.

5.43 Retained-mode periodic RI arithmetic

The optional audit_metric_arithmetic build request also audits the existing retained-mode route. At each streamed mode block, the exact stored-input reference is B = F^dagger T, with the actual retained factor F and the exact values of the tiles T returned in that sweep. Shared outward binary64 arithmetic encloses the factor contraction without another source sweep or a full auxiliary-by-AO-pair tensor. The physical compensation, block size, route selection and source identities are unchanged.

For J, rho_L = sum_(s,k,i,j) w_k conj(B_L,k,i,j) D_s,k,i,j, J_k,i,j = sum_L B_L,k,i,j rho_L, and E_J = (1/2) sum_L |rho_L|^2. The interval comparison includes spin addition, stored k weights and all retained-mode products/reductions, including energy accumulation. For K, K_s,k = w_k sum_L B_L,k D_s,k B_L,k^dagger, with its existing target/ket q-channel convention. The audit encloses both matrix products, the weight and the reduction across mode blocks. The Rump2010 arithmetic assumptions of Section 5.40 and the Sun2017 source conventions of the producer apply.

PeriodicRIQResult.J_roundoff_bounds and K_roundoff_bounds contain complex L1 entry radii with the respective J/K shapes when audited; unaudited arrays have empty leading dimensions. The receipt separately reports retained_j_roundoff_evaluated, retained_k_roundoff_evaluated, their maximum retained_j_roundoff_bound and retained_k_roundoff_bound, and the absolute retained_energy_roundoff_bound for E_J. Zero-rank or zero-density results retain exact zero bounds. The existing count-only plan reserves interval mode blocks, response arrays, spin/weight inputs, scratch and cubic contraction work at maximum rank before construction. The audit allocates no unbounded coupling owner and executes no additional integral evaluation.

Complete RI HF/KS with the existing metric-action request propagate the retained J and E_J radii through the shared evaluated assembly ledger. retained_coulomb_included distinguishes this contribution from paired_coulomb_included; the two routes cannot be counted simultaneously. The retained contribution is included once in combined Fock and energy bounds, followed by any admitted direct-K and BvK-correction arithmetic. The physical J/K build, native eigensolver and terminal/capped state contracts are preserved.

Raw fused K radii refer to one q channel. Section 5.44 composes these radii through the complete fitted-K cross-q accumulation and evaluated method assembly; the present section defines only the individual channel reference. Tile/integral construction, factor/eigensystem and rank perturbations, restricted density embedding, XC and relaxed-SCF errors remain excluded. No metric-only or paired bound is added again to the retained reference. Full-Coulomb roundoff_included=false, #624 and the full #617 matrix remain open, with source construction and whole-method accuracy separate.

5.44 Whole-mesh fitted exchange arithmetic

The existing ordered-q RI Fock builder optionally encloses accumulation of its stored Gamma and transfer-channel K values. For each target spin, k point and AO entry, let c_q be the computed channel value and e_q its retained complex L1 radius from Section 5.43. Outward interval addition encloses sum_q c_q. Comparing the actual compensated sum with that enclosure gives an accumulation radius e_sum. The resulting whole-mesh radius is the upward sum e_sum + sum_q e_q. Each channel error enters once; no auxiliary tensor, extra integral sweep or replacement physical contraction is introduced. The Rump2010 arithmetic assumptions of Section 5.40 apply.

PeriodicRIFockBuilder.build(audit_arithmetic=True) returns these radii in K_roundoff_bounds[spin,k,i,j]. The receipt records fitted_exchange_roundoff_evaluated and the maximum entry radius fitted_exchange_roundoff_bound. An unaudited result has an empty leading dimension. The K radius does not depend on the requested exchange scale. Zero rank or zero density retains zero error. The existing count-only Fock plan reserves the interval accumulator, channel and result radii and all ordered-q summation work before constructing sources. Physical source, projector, frame and density identities are unchanged.

Complete RI HF and global/range KS reuse the existing metric-action request and evaluated assembly audit. A fitted exchange component with signed coefficient a contributes |a| e_K to the Fock radius and (1/2) sum_(s,k,i,j) |w_k D_s,k,j,i| |a| e_K,s,k,i,j to the energy radius. Outward arithmetic also encloses coefficient multiplication, density pairing and final assembly. Range components retain their separate physical kernels, projectors and coefficients, including negative coefficients. The method owns the BvK correction and its separate Section 5.42 contribution.

assembly_roundoff.fitted_exchange_roundoff_included, fitted_exchange_fock_bound and fitted_exchange_energy_bound distinguish this contribution from direct-exchange errors. Combined bounds include the selected Coulomb route, direct and fitted exchange contributions, and the correction once each. The diagnostic raw assembly function uses fitted_exchange=True to classify supplied exchange_errors accordingly. The raw Fock-builder K certificate alone does not certify its energy or Fock assembly; that further claim requires the method assembly receipt.

The reference holds the input density, weights, coupling tiles and retained factors fixed. Integral/source construction, eigensystem and rank changes, restricted density embedding, XC construction and relaxed-SCF errors remain excluded. Full-Coulomb roundoff_included remains false. This does not close #624, #626 or full #617, change shared Fock/eigensolver or source-qualified symmetry policy, or establish BIPOLE/CRYSTAL source equivalence. The known reference-budget refusals in #643 remain separate.

5.45 Full-Coulomb fitted exchange on a fixed density family

certify_periodic_ri_exchange_fitting consumes an existing ordered-q RI Fock owner and the existing periodic pair source, finite reciprocal exchange producer and full-image/full-reciprocal exchange enclosure. It compares their linear K actions; it does not assume a positive or variational fit residual between independently truncated metric and coupling integrals. Sun2017, Section II.B, defines the periodic fitting convention. The existing exchange source and tail contracts retain their physical meaning and zero-mode policy. Rump2010 outward arithmetic encloses probe subtraction and propagation.

The family is explicitly unweighted: sum_(s,k,i,j) |D_s,k,i,j| <= L, with optional symmetric nonnegative B_k,i,j bounding |D_alpha,k,i,j| + |D_beta,k,i,j|. D must be exactly Hermitian. This is not a bound on the potentially cancelling signed spin total. Canonical solver bounds may be converted from their weighted convention with outward products; taking the maximum of two transposed entry bounds supplies a symmetric upper envelope without projecting a physical density or granting symmetry reduction. The source basis, literal cell, k frame and weights must agree exactly.

At each source k, the probe basis is E_ii, P_R = E_ij + E_ji and P_I = i(E_ij - E_ji), for i < j. There are n_k n_AO^2 probes, independent of spin count. The existing build_exchange component action reuses the actual Gamma and transfer owners and their retained factors. It omits J and its full-Coulomb fitting-family check; existing provider-specific admission still applies. Its source identity distinguishes the component action from a full Fock build. No second metric/source owner or Coulomb algorithm is introduced.

For each output entry, the finite difference coefficient includes the outward difference of the computed RI and finite-reference probe responses plus both accepted contraction-arithmetic radii. The reference coefficient is supplied separately by the existing full-image/full-reciprocal bound on that same probe. For either coefficient family, an off-diagonal pair with real/imaginary bounds a_R and a_I has coefficient c = sqrt(a_R^2 + a_I^2), computed outward. Its component-family contribution is B_ij c; its coefficient per unit matrix L1 norm is c/2. Diagonal probes contribute B_ii a and coefficient a. The minimum of the component sum and L times the largest normalized coefficient bounds each spin response. The finite and reference bounds are added once. No AO^4 response tensor is stored.

The unscaled spin exchange-energy error is bounded by (1/2) sum_(k,i,j) w_k B_k,i,j e_k,i,j, or by (1/2) L max_(k,i,j) w_k e_k,i,j when smaller. Finite-map discrepancy and reference remainder retain separate matrix and energy contributions. The result records the RI, pair, finite reference and tail-reference identities, probe counts, declared family and certificate identity. Strict paired matrix and energy targets refuse before returning an unqualified result.

Count-only preflight reserves both producers, a previously attached J certificate’s retained memory, all probe evaluations, tail preparation, interval propagation and bounded response scratch before reading vector payloads. It conservatively uses the total reference-vector count as a per-channel ceiling. Source tables and work are not hidden behind a dense fallback. The result owns its bound arrays; validate_density checks the family before a consumer uses the certificate.

This component bounds the exact stored-input RI map against the full-Coulomb exchange reference at fixed density and mesh, with q=0 omitted. Probe arithmetic is included; integral/Fourier/source construction, factor and rank perturbations, restricted density embedding and SCF relaxation are not. An executed K response adds its own Section 5.44 arithmetic radius. Method coefficients, BvK corrections and energy/Fock assembly retain their separate owners. Physical range kernels and XC remain unsupported by this certificate. Automatic complete-method attachment and useful selection remain #626/#627; this does not close #623-#627 or the full #617 matrix, change source-qualified symmetry, or establish BIPOLE/CRYSTAL parity. Existing #643 failures remain.

5.46 Preconstruction exchange-certificate admission

plan_periodic_ri_exchange_fitting_capacity consumes the shared RI capacity plan, basis and reference image/vector counts before either reference-pair construction or numerical vector payload access. The original pair-source planner supplies maximum entries and retained bytes; the shared exchange-tail planner uses those capacities. The certificate aggregation is the same code used by the realized-owner planner. Capacity admission additionally includes reference-pair preparation work and peak storage, and explicitly reserves any retained J-fitting certificate. The enclosing method still charges RI setup. All five resource caps and checked integer products apply before construction.

A capacity plan is not a scientific source identity or execution permission. The actual certificate still verifies orbital basis, cell, momenta, weights, full-Coulomb K ownership, reference completion and the Hermitian density family. A capacity can exceed the realized plan because screening removes pair entries; no capacity receipt certifies a physical matrix or relaxed energy. This seam is required for automatic complete-method admission; its existence alone does not attach the Section 5.45 certificate to SCF or supply useful accuracy targets.

5.47 Pre-SCF fitted exchange attachment

Periodic RI RHF/UHF accept optional PeriodicRIExchangeAccuracyOptions through exchange_fitting_accuracy, or public PeriodicFittingOptions.full_coulomb_exchange_accuracy. Both positive targets refer to the raw full-Coulomb K matrix and its spin-summed exchange energy, with the reciprocal zero mode omitted. They do not include BvK/source corrections, other Fock components or nonlinear SCF relaxation.

The complete method admits the Section 5.46 capacity, including the fixed reference image and reciprocal cubes, before allocating sources. Shared cube enumeration retains the existing BUPO ordering. The reference uses the original pair, finite reciprocal K and full-image/full-domain tail producers; it is not a second Fock implementation. Reference support remains fixed during SCF.

The existing canonical overlap preparation supplies weighted entry bounds on (|D_alpha| + |D_beta|) / nk. Multiplying these and the weighted L1 bound by nk with outward rounding gives the component’s unweighted family. Taking the maximum of transposed entry limits preserves exact Hermitian symmetry without narrowing the family. RHF embeds each spin as D/2 before family validation; UHF retains independent spins. Every evaluated and terminal-confirmation density must remain within the admitted family. No observed SCF density chooses the certificate envelope or fitting projector.

The certificate is built against the actual method-owned RI source after any admitted J fitting setup and before the first SCF callback. Unmet targets refuse before a result is produced. The same certificate and source identity are retained in output. Dry-run admits counts without claiming numerical acceptance. Every executed K response requests its separate arithmetic radius. That radius and evaluated assembly arithmetic are reported separately from the raw component targets; they must be composed before claiming a computed Fock-error tolerance. Input/factor construction, restricted embedding, rank perturbation and relaxation retain their previous exclusions. Source preparation counters conservatively charge the admitted certificate work, including reference preparation.

Joint J/K policies retain separate projectors, density conventions, certificates and reservations. The enclosing method admits simultaneous lifetime costs once; child source plans remain local so certificate composition does not count the parent’s entire allocation as another child owner. Assembly and BvK arithmetic workspace/work are charged when K accuracy activates their audits. There is no change to the shared orbital eigensolver, Fock state, symmetry permission or physical exchange convention. Section 5.48 extends this attachment to global hybrids; physical range and XC accuracy remain separate work.

5.48 Global-hybrid fitted exchange attachment

Periodic RI global-hybrid RKS/UKS consume the same Section 5.47 helper and actual method-owned PeriodicRIFockBuilder. The existing canonical overlap family is converted by the shared helper; restricted KS embeds the total density as two D/2 spin blocks and unrestricted KS preserves both spin blocks. Every evaluated and terminal-confirmation density is validated against that fixed family before requesting the existing fitted K response and its separate arithmetic audit.

Targets and qualification retain the raw full-Coulomb K convention, with q=0 omitted. For physical exchange coefficient a, output additionally records outward bounds |a| times the raw matrix and spin-summed exchange-energy bounds. This scaling applies to signed coefficients admitted by the existing functional API. It does not include coefficient multiplication or assembly arithmetic, BvK/source corrections, source/factor construction, restricted embedding, rank perturbation, XC, other Fock components or SCF relaxation. Those retain their separate ledgers and exclusions. A global-hybrid component bound is not a total Fock certificate.

The enclosing KS plan reserves certificate construction, simultaneous J evidence, canonical family, response/assembly/correction audits and repeated density checks before constructing owners. Child plans consume local reservations after global admission; they do not recursively add the entire enclosing method allocation. Exact-J/fitted-K and fitted-J/fitted-K use existing source admission. Pure KS, exact K and physical range-separated KS refuse this optional certificate. A count-only dry run still supplies no numerical acceptance.

No new Fock, eigensolver, XC, fitting or symmetry implementation is introduced. Source identities, retained projectors, physical coefficients, source corrections and terminal evaluated states stay with their shared owners. Useful automatic bounds, physical range and XC remain requirements under #626/#627; this optional attachment does not enlarge the qualified 0.18 controls.

5.49 Physical range exchange component enclosures

The shared PeriodicExchangeBoundSource accepts kernel="erf" or "erfc" and a finite nonnegative omega, in addition to its unchanged Coulomb default. This extends the Section 5.13 absolute image/reciprocal enclosure, not the physical method or fitting projector. For nonzero Q,

\[0\le v_{\mathrm{erf},\omega}(Q),v_{\mathrm{erfc},\omega}(Q) \le 4\pi/Q^2.\]

The existing Coulomb Gaussian majorant therefore bounds every retained nonzero-Q image error and every omitted nonzero-Q contribution for either physical range kernel. It deliberately gives up their extra range decay. At positive range, erfc has the finite zero coefficient \(\pi/\omega^2\); include it in the retained coefficient sum when that Q is supplied and in the omitted coefficient sum otherwise. The same conservative cell-volume bound applies. This makes a missing screened zero explicit in the error ledger instead of silently identifying it with the singular Coulomb zero. Coulomb, erf and zero-range erfc still refuse a supplied Q=0. At zero range, erfc uses the Coulomb nonzero enclosure, while erf vanishes identically.

The Fourier convention is Sun et al. (2017), Section II.A; singular versus screened zero-mode treatment follows Sundararaman and Arias (2013), Section II.A and the physical range boundary above. The new inequality and placement of the finite zero are enclosure constructions, not claims from those papers that an arbitrary truncated RI residual is positive. No range-dependent Madelung prescription is introduced. Every certificate identifies its actual physical kernel, range, supports, source and density convention. Default Coulomb source/certificate identities and arithmetic remain unchanged.

certify_periodic_ri_exchange_fitting now consumes a same-frame physical Coulomb, erf or erfc K owner and invokes that owner’s existing exchange action and the matching shared direct reciprocal response for each Hermitian probe. Both responses retain their separate probe arithmetic enclosures. The shared range tail enclosure supplies the remainder to the full physical interaction on the fixed mesh. The fixed Hermitian spin-family aggregation is unchanged; no Coulomb projector or variational residual is substituted for a range fit. The existing fitted-source guard still requires strictly positive physical range; zero-range limits above apply to the tail source, not to a newly admitted RI fit.

Count-only admission remains a conservative common capacity, including all explicit reference vectors and finite zeros, retained J evidence where present, probe work and output storage. Actual construction still validates kernel, range, frame, reversal and support. This raw component API does not attach range bounds to a complete KS method or combine signed functional terms. Automatic method attachment and useful bounds remain required under #626/#627. Source construction, factor/rank perturbation, XC, method correction/assembly and relaxed-SCF errors retain their exclusions. Shared Fock/eigensolver, PBC/BIPOLE/CRYSTAL and symmetry contracts remain authoritative.

5.50 Pre-SCF signed range exchange admission

Periodic fitted range-separated KS accepts the explicit optional range_exchange_accuracy policy. Its PeriodicRIExchangeAccuracyOptions controls reference support and two positive targets on the sum of physical exchange components. The existing full_coulomb_exchange_accuracy public control retains its raw Coulomb meaning; the policies are mutually exclusive. Pure/global KS, HF and exact range K refuse the new range policy.

The method resolves its original signed Coulomb-plus-erfc components before admission. Every component keeps its actual RI owner, physical range, projector, source identity and coefficient. A single shared canonical overlap preparation supplies the spin-density family before the first SCF call. Each component uses the same Section 5.47 family conversion and Section 5.49 physical certificate. Restricted KS embeds total D as two D/2 blocks; UKS retains independent spins. Every evaluated and terminal density must remain in every admitted family.

For component coefficient \(a_c\), entry bound \(B_{c,ij}\) and spin-summed exchange-energy bound \(E_c\), compose with outward arithmetic:

\[B_{ij}=\sum_c |a_c|B_{c,ij},\qquad E=\sum_c |a_c|E_c.\]

Admit only when the maximum entry of B and E meet the requested targets. Raw component certificates are built without individual target enforcement; the combined targets are enforced before SCF. This avoids arbitrarily dividing a method budget among physical components. Signed coefficients remain unchanged in the computed functional. Cancellation does not reduce the certified bound. The reference includes the finite screened zero for erfc and omits the singular Coulomb zero. The count-only helper takes an explicit physical kernel/range so its generated-reference capacity includes the appropriate number of vectors.

The enclosing method reserves all physical sources, component certificates, reference preparation, retained J evidence, canonical family, composition and serialized exports, per-density checks and applicable response/correction/ assembly audit work. After that admission, child construction uses local caps; it does not recursively count the whole parent allocation. Exact-J/fitted-K, joint J/K policies and multiple distinct ranges retain their existing sources. An impossible target refuses before an output state; dry-run remains count-only.

The aggregate receipt binds ordered component certificate identities and signed coefficients, exports their separate raw evidence and reports the outward sum. Each actual fitted response requests its separate arithmetic enclosure. Target bounds exclude response/assembly arithmetic, fixed-Madelung/source corrections, source/factor construction, restricted embedding, rank perturbation, XC and other Fock components, and nonlinear SCF relaxation. The resolved long-range coefficient alone continues to scale the original singular correction.

This is automatic admission of the physical exchange-component sum, not useful whole-method accuracy or a new Fock/eigensolver. The existing method, PBC/BIPOLE/ CRYSTAL and source-qualified symmetry owners remain authoritative. Useful bounds, XC and the remaining error ledgers/matrix requirements stay open under #626/#627 and their existing dependencies. The bounded #557 release controls do not change.

5.51 Positive Fourier-decay groups for exchange references

The optional fourier_decay_bins control partitions the existing positive primitive Fourier envelopes of Section 5.13. It does not partition or replace the physical integral, fitted metric, Bloch source or evaluated Fock. One bin is the original algorithm and preserves its values and identities. The allowed range is one through eight; unsupported counts refuse before reference payloads. Both raw exchange-bound sources and RI exchange accuracy policies reuse this native producer, including signed range-separated method admission.

Let M be the largest orbital primitive exponent, including any zero-coefficient primitive, and let P=a+b for a primitive pair. Assign the term to the largest j below the requested bin count for which P <= 2M/2^j. The first bin includes all pairs that cannot move to a later bin. Empty bins contribute zero. The original absolute coefficients, angular polynomial envelope, normalization, image membership and image-tail proof remain unchanged. For bin j choose a guarded lower decay alpha_j <= 2^j/(16M), strengthened to 2^j/(8M) when the entire source is s-type. These are the same contour-shift inequalities from Sun2017 II.A and Section 5.13, with a tighter maximum P in each positive sum. This grouping and its cross-term bound are a project derivation.

Let R_j and M_j bound retained and omitted image amplitudes in bin j, and A_j=R_j+M_j. The retained and omitted reciprocal sums c_jl and h_jl use decay alpha_j+alpha_l, with the same physical kernel, transfer, weights and supports as before. The image error is bounded by the sum over j,l of

\[c_{jl}[R_j^T|D|M_l+M_j^T|D|R_l+M_j^T|D|M_l],\]

and the reciprocal error by the sum of

\[h_{jl}A_j^T|D|A_l.\]

Each term is nonnegative; signed primitive contractions are already enclosed by their absolute coefficients. No cancellation or positivity of a truncated RI residual is assumed. Finite erfc zero is assigned by the original support; its value is independent of the decay. The native implementation retains the original bound and takes the smaller valid bound separately for image and reciprocal errors. Thus optional grouping cannot enlarge reported bounds. The legacy diagnostic amplitude/reciprocal arrays continue to describe the valid original global envelope. The plan records the bin count; the bound source identity additionally binds every group decay, amplitude and reciprocal coefficient. Physical pair, RI and finite-reference identities are unchanged.

Count-only admission charges original plus cross-bin work, with at most 1+B^2 work units per original envelope unit. It reserves additional group matrices, reciprocal tables, log-domain intermediates and result copies. Existing owners and backend margin are retained once, not multiplied by the bin count. Actual budget limits remain authoritative. Every RI certificate continues to enclose the same canonical Hermitian family and report finite-fit and reciprocal/image remainder contributions separately. Method targets, response arithmetic, correction, source/rank/XC and relaxation exclusions keep their existing meaning. This sharper reference bound does not establish useful automatic whole-method accuracy or complete the remaining capability matrix.

5.52 Fitted-exchange contribution to the fixed-density HF ledger

The optional RHF/UHF component ledger consumes the existing Section 5.47 fitted-exchange energy enclosure at unit HF exchange coefficient. For the same canonical density family, kinetic, nuclear, Coulomb and exchange energy discrepancies are disjoint terms; the triangle inequality permits an outward sum of their stored nonnegative bounds. The original spin-summed exchange bound already includes its one-half trace convention. No extra restricted factor or reweighting is introduced. Simultaneous direct and fitted exchange certificates claim the same component and refuse composition.

Automatic attachment exports canonical_density_envelope_identity for the original weighted entry/L1 envelope, using the existing kinetic/nuclear/J byte convention before the outward conversion to unweighted spin limits. The raw probe API leaves this identity empty. The ledger requires the actual whole-mesh RI source, the canonical-family identity shared by other requested canonical components, and the literal zero-mode-omitted Coulomb qualification. Missing or incompatible identities, invalid bounds, unqualified receipts and unsupported method scaling refuse the combined ledger. The existing numerical certificate digest is unchanged. Identity construction is charged before execution: 512 additional modeled bytes, 4096*(nknAOnAO+8) scalar terms and eight times that kernel allowance.

This composes the previously accepted Sun2017/Neese2025 response-map and spin-family contracts of Sections 5.45-5.47; it does not extend their proof. The reference is full Coulomb K with its reciprocal zero mode omitted. Source/BvK corrections, executed response/assembly arithmetic, restricted embedding arithmetic, source construction, factor/rank and overlap perturbations, basis/BZ errors, XC and nonlinear relaxation remain excluded. The closed-shell relaxation delimiter remains an uncertified scale estimate. Section 5.53 extends aggregation to KS; useful automatic whole-method accuracy remains subsequent work; no new Fock, eigensolver, physical source or symmetry owner is created.

5.53 KS fixed-density exchange composition

The existing Sections 5.48 and 5.50 certificates may enter the same fixed-density component ledger as Section 5.52. For global-hybrid coefficient a and raw spin-summed exchange energy enclosure b, |a|b encloses the method component error. Binary64 multiplication is rounded outward, with exact zero preserved; an unrepresentable finite result refuses qualification. The raw certificate already contains its trace/spin convention, so no further restricted factor is applied. Its signed coefficient must match the actual method receipt and its original canonical family must match the other canonical components.

For physical range components, Section 5.50 already encloses the signed sum by summing |a_i|b_i outward. Composition validates the ordered complete physical kernel/range/coefficient list against the functional requirements and each actual RI owner. Every child must identify the original canonical family, be qualified for the stated physical reference and have no standalone target claim. The native aggregate must enclose the outward energy sum; composition then adds that aggregate once, never its children. A competing direct/global exchange claim, a missing physical component or incompatible source/family refuses the ledger. An erf/erfc child must retain a strictly positive physical range; the screened zero convention stays with its existing producer.

This is triangle-inequality composition of the previously accepted Sun2017/Neese2025 and Sundararaman/Arias2013 physical-response contracts, not an extension of their component proofs. Scaling and scalar summation enclose stored bound arithmetic only. Executed coefficient multiplication, response, assembly and restricted embedding arithmetic, source construction, BvK/source corrections, rank/overlap perturbations, XC/grid, basis/BZ and relaxation retain their exclusions. KS receives no relaxation delimiter estimate. The existing shared Fock, eigensolver, source and symmetry APIs remain authoritative; no new provider or quadrature is created.

5.54 Conditional overlap rank under omitted images

PeriodicOneElectronAccuracyOptions.overlap_image_rank is an optional pre-SCF guard for periodic RI HF and KS. It consumes the existing full AO image-overlap entry bounds, not a user-supplied perturbation estimate. The fixed-source method refuses an unresolved guard before SCF. Physical-support selection marks that trial unqualified and grows the AO pair support within its existing limits. Defaults, physical kernels, fitting owners, symmetry permissions and the shared Fock/eigensolver conventions are unchanged.

Let S be the stored finite overlap, C its actual shared canonical Hermitian projection, and B the producer’s nonnegative omitted-image entry bounds. Outward interval subtraction encloses each entry of C-S, including imaginary diagonals. The maximum row or column sum of B+|C-S| gives t >= ||Delta S||2 for the declared image perturbation relative to C. The implementation reuses the bounded solver’s exact projection, Jacobi, negativity and residual checks. Section 5.31 then certifies its complete eigensystem and the strict radius r, including movement of an adaptive relative cutoff. Every k point must have t<r and the same retained rank as the already prepared canonical density family. Equality, an unresolved spectrum or a rank mismatch refuses. This is the existing Rump/Lange2023 and Weyl argument; it introduces no alternative orbital solver or tolerance repair.

The raw bound_periodic_overlap_image_rank producer accepts stored overlaps, entry bounds, the expected rank and explicit shared SCF controls. Its plan_periodic_overlap_image_rank admits copies, Jacobi and interval work, per-k certificates and output storage before inspecting payloads. Complete methods reserve that work in their one-electron plans; Python separately reserves per-k dictionaries and serialization. Rejected support trials retain their charges. Typed receipts record per-k spectra, perturbation and projection bounds, the minimum radius and an identity covering the stored inputs and rank policy. The combined image certificate includes that identity; selected and final sources must still match.

Qualification is conditional on the image-bound model. Gaussian integral, phase and source-assembly arithmetic remain excluded. This does not prove stability against all overlap errors, preserve occupied subspaces or certify the density family after changing the overlap. It does not bound overlap contributions to an energy, orbital response, density relaxation or total relaxed energy. Global ledger exclusions therefore remain in place. Source perturbations of the auxiliary metric and rigorous construction arithmetic remain under #623/#625. MANTA remains experimental.

5.55 Conditional fitting metric rank under assembly roundoff

PeriodicFittingOptions.metric_assembly_rank=True enables an optional pre-SCF guard in the existing periodic RI transfer owners. The GDF tile producer retains the same short-range integrals, phases, Fourier panels, weights, compensated accumulation and physical source identity. Alongside that route, shared outward complex intervals enclose the exact sum of stored short-range phase/integral products and stored weighted Fourier products, including finite-zero subtraction. The interval sum spans the short-range to long-range compensation reset. Complex L1 distances from the final stored entries give nonnegative per-entry error radii. This uses the Rump2010 verified-arithmetic principles already used by the shared spectral producers; it does not treat evaluated transcendental or Gaussian operands as exact physical integrals.

Both matrices in the perturbation argument use the established RI lower triangle and real diagonal. The upper triangle is its conjugate, rather than an average of independently assembled entries. Symmetrizing the lower-triangle radii and summing outward gives t >= ||Delta M||2. The existing spectral certificate from Sections 5.31-5.32 must resolve the same retained rank as the actual factor, with t strictly below its rank-stability radius. This includes movement of the adaptive relative cutoff under the existing Rump/Lange2023 and Weyl argument. Equality or an unresolved certificate refuses before SCF; no cutoff adjustment, alternate factor or fallback repairs the request.

The tile receipt exports entry radii, their maximum and an identity covering the physical source, stored matrix and radii. Every RI metric receipt exports the requested/qualified flags, perturbation bound and certificate identity. Source and factor identities remain unchanged. Optional source planning charges interval arrays, output copies, lower-triangle norm workspace and conservative audit work; Python reserves additional per-channel provenance storage. Dense raw GDF APIs refuse this option because they do not implement its audit. The tile producer alone reports assembly radii; rank qualification belongs to its RI consumer. Default behavior remains unchanged.

The proof is conditional on stored primitive integrals, Fourier transforms, phases and kernel weights. Their construction, contractions and pure transforms, phase/volume/weight evaluation, source truncation and screening remain excluded. Rank stability does not certify retained-space rotation, inverse response, Fock/energy assembly, density relaxation or total relaxed energy. This increment adds no general bulk CRYSTAL parity, speedup or production-scale qualification. Global ledger exclusions remain; #623/#625 and the broader #617 programme stay open. MANTA remains experimental.

5.56 Finite inverse response to conditional metric assembly error

The opaque shared HermitianInverseCertificate now provides bound_perturbation(t, budget). Given its actual stored Hermitian matrix A, it bounds the difference between the exact retained spectral functions of A and B=A+E for any Hermitian E with ||E||2 <= t. Both spectra use the same absolute/relative cutoff policy. The existing spectrum proof must establish t strictly below its rank-stability radius; an unresolved or crossed cutoff returns an unqualified result. Nonfinite/negative inputs and unrepresentable arithmetic refuse. No eigensolve, factor change or dense inverse is added.

Let alpha be the verified lower retained eigenvalue of A, g its verified retained/discarded gap, and beta=alpha-t. Weyl gives a lower retained value beta for B and a mixed-space gap at least h=g-t. All bounds expand outward. For nonempty rank require beta>0; partial rank additionally requires h>0. The inverse divided difference is bounded by L=max(1/(alpha beta),1/(beta h)), omitting the second term at full rank. A power-of-two ceiling for sqrt(n) gives T >= ||E||F from t. The finite intertwining identity in exact unitary eigenbases U of B and V of A is [U*(f(B)-f(A))V]ij = f[b_i,a_j] [U*E V]ij. Consequently ||f(B)-f(A)||2 <= L T. The inverse function is 1/x on each retained spectrum and zero on its discarded spectrum. Repeated eigenvalues within either space do not require an eigenvector orientation or internal gap. This is the same Higham2008 Chapter 3 divided-difference argument as Section 5.33, extended to two exact spectra; no first-order remainder is discarded.

For the exact retained projectors, the mixed divided difference is bounded by 1/h, giving ||P(B)-P(A)||2 <= min(1,T/h). Empty and full retained spaces have zero projector response. Empty rank also has zero inverse response. This projector bound concerns exact spectral projectors, not the rounding of the exported numerical projector. Adding the existing Section 5.33 stored-factor inverse error outward gives a bound on ||F F*-f(B)||2 for the actual stored F. It is a bound on its exact matrix product, not an executed Fock, J, K or energy contraction.

The scalar plan reserves 4096 bytes and conservative 512 scalar/4096 kernel operations before inspecting t; periodic RI plans include that workspace and work. Python separately reserves the per-channel result dictionaries and serialized copies. There is no dimension-sized new allocation or second spectrum audit. Public certificates remain immutable and carry the producer’s dimension, rank, retained minimum and gap metadata.

With metric_assembly_rank=True, each actual RI transfer owner feeds its Section 5.55 assembly norm bound into this shared operation. The resulting assembly_response receipt records qualification, t, T, beta, h, L and projector, inverse and stored-factor-plus-source bounds. Its audit identity is included in the assembly receipt; physical source, factor and evaluated state identities remain unchanged. Rank admission keeps its original rule. A response can remain unqualified even when rank is resolved, and its zero fields then mean unavailable rather than zero error. The response is a conditional diagnostic, not a new automatic acceptance target.

The input t still excludes primitive integral, contraction/pure-transform, Fourier, phase and kernel-weight evaluation, source domains and screening. The new inverse response does not cover coupling perturbations, propagation to J/K/Fock/energy, overlap or occupied-space changes, density relaxation or total relaxed energy. Global ledger exclusions stay in place. No general parity or speedup follows. MANTA remains experimental and #623/#625/#617 retain their remaining obligations.

5.57 Conditional metric-source contribution to fixed-density J and energy

PeriodicRIFittingAccuracyOptions.metric_action_include_assembly is an explicit opt-in for the existing metric-action J/energy targets. It requires positive metric-action targets and PeriodicFittingOptions.metric_assembly_rank. The default remains false. The raw fitting-bound options expose the same choice. A missing source proof or an unqualified finite inverse response cannot satisfy the action target. Existing selected-source and search owners retain their physical sources, rank rules, density families and budget limits. An unresolved assembly rank still refuses at its existing source guard; this option does not add a recovery path across a rank boundary.

Let C(A) and C(B) be the exact truncated inverses of the stored canonical metric and its conditional exact stored-operand assembly reference. Section 5.56 gives eta >= ||C(B)-C(A)||2. For the declared stored coupling columns t_p, let n_p >= ||t_p||2. Then the fixed coupling response obeys |Delta K_pq| <= eta n_p n_q. For the existing weighted spin-total density family |d_p|<=h_p, sum |d_p|<=L, use the shared positive support function s(v)=sup sum v_p |d_p|. The source contribution is bounded entrywise by J_p <= eta n_p s(n) and in energy by E <= 0.5 s(J). All products, sums and support evaluations round outward. This reuses the Section 5.34 norm-family producer, including its bounded dual support solve; it introduces no independent density weighting or Coulomb convention.

HermitianInversePerturbationCertificate.norm_family exposes this source-only contribution. It uses inverse_error_bound, not the already combined factor_inverse_error_bound. The periodic consumer adds its entries and energy outward to the existing stored-factor action certificate, which may use a sharper residual proof. Thus the stored-factor error is counted once. The combined action target compares the actual stored-factor map with the conditional assembled-metric inverse, at fixed coupling and density.

Explicit sources reuse the admitted column audit. Supported compressed and pair-tree sources reuse their existing metric-column norm producer, without materializing a full auxiliary/AO-pair tensor or changing the graph. Planning adds one norm-family contraction, source/combined entry arrays, identity work and output storage. Python includes the added trial metadata in its export reservations. No additional primitive source traversal is needed solely to propagate the metric perturbation.

Receipts distinguish stored_j_bound, stored_energy_bound, assembly_j_bound, assembly_energy_bound, requested/qualified flags and an assembly-action identity. That identity binds the actual RI source, assembly response, coupling norms and fixed density limits. Existing combined j_bound/energy_bound drive admission in every existing fitting selection loop. The raw API exports both combined and source-only per-entry bounds. The numerical J, Fock, factor and physical source remain unchanged.

This is an additional metric-action target, not a replacement for or a sum with the separate full-Coulomb fitting certificate. The global component ledger is unchanged because those reference objects need separate accounting. The source contribution excludes primitive/Fourier/phase/weight construction, metric-domain and screening errors, coupling construction/perturbations, executed J/Fock/energy arithmetic, K response, overlap/occupied-space changes and nonlinear relaxation. No total relaxed-energy, bulk parity or speedup claim follows. MANTA remains experimental; #623/#625/#627/#617 stay open.

5.58 Conditional metric-source response of ordered exchange channels

The raw PeriodicRIExchangeFittingOptions and complete-method PeriodicRIExchangeAccuracyOptions expose paired optional metric_assembly_matrix_error / metric_assembly_energy_error targets. Zero disables this separate component; positive targets require the existing metric_assembly_rank proof on every actual K channel. Invalid, missing or unqualified proofs refuse. These are raw physical-component targets, including when used by a signed range method: each child must satisfy them separately. They are not targets on the signed aggregate or on relaxed energy.

For target k and source l, retain the actual ordered q owner and its exact truncated inverse response eta_kl from Section 5.56. Gamma supplies diagonal channels, with the corresponding k-dependent coupling columns; off-diagonal channels retain their individual source, metric and range kernel. The existing coupling norm producer supplies n_kl(mu,a) >= ||T_kl(:,mu,a)||2. It may use a conservative complex/auxiliary L1 envelope. For each spin, the fixed-source map is the existing B D B* contraction (Sun2017, q-dependent fitting; Section 5.44). The operator-norm inequality gives

c_kl(mu,nu,a,b) = w_l eta_kl n_kl(mu,a) n_kl(nu,b)
|Delta K_sigma(k,mu,nu)| <= sum_l,a,b c_kl(mu,nu,a,b) |D_sigma(l,a,b)|.

With unweighted spin-absolute density caps h and total L, each output bound is the minimum of L max_l,a,b c and sum_l,a,b h(l,a,b) c when entry caps are present; otherwise use the first expression. The spin-summed energy bound is the minimum of 0.5 L max_k,mu,nu(w_k bound_k,mu,nu) and 0.5 sum_k,mu,nu h(k,mu,nu) w_k bound_k,mu,nu. Symmetric entry caps justify the trace transpose. All elementary products and sums round outward; zero source weights contribute zero. There is no extra restricted-spin factor. This encloses all Hermitian densities in the declared family, including indefinite ones, with positive, negative or zero perturbation responses.

The producer streams one norm pass per actual metric owner. Gamma norms are retained once; pair norms and coefficient loops are streamed without an auxiliary-by-pair or four-index coefficient tensor. Plans charge all additional source passes, norms, contraction/hash work and receipt/export storage before reading payloads. The separate identity binds actual RI and assembly identities, every norm, weights, family limits and returned bounds.

Receipts export metric_assembly_requested, metric_assembly_qualified, metric_assembly_matrix_bound, metric_assembly_energy_bound, per-entry bounds, channel count and identity. Complete-method provenance records them under the raw exchange component’s metric_assembly object. Existing fitting certificates, signed aggregate, physical factors, source/state identities and global ledger remain unchanged. Do not add this response to those unlike reference objects without a separate composition proof.

The response compares exact C(A+E) and C(A) at fixed stored T, so stored-factor error is explicitly excluded rather than counted again. Primitive/Boys, contraction/pure transforms, Fourier/phase/weight evaluation, coupling-source construction, metric screening/domain perturbations, executed K/Fock/energy arithmetic, BvK correction, XC/grid, overlap/occupied-space response, basis/BZ and relaxed density remain outside this component. Physical erf/erfc owners keep their existing finite/singular zero conventions. This grants no new symmetry reduction, total-energy accuracy, bulk parity or speedup. MANTA remains experimental and the broader #617 scope stays open.

5.59 Exact stored-vector Bloch phase in metric construction

The existing metric_assembly_rank audit now includes evaluation of the short-range metric Bloch phase exp(i q.R), including the dot product, for exact stored binary64 vectors q and R. This replaces the stored-phase premise of Section 5.55 for that term only. It preserves the actual numerical phase, stored metric, factor, physical support and evaluated SCF state. Fourier-panel phases, coupling phases, integral evaluation and construction of q/R remain outside this increment.

The shared certify_bloch_phase(momentum, translation, evaluated_phase, budget) returns an immutable rectangular enclosure of the exact phase and an absolute complex-L1 error bound for any finite supplied witness. It makes no correctness assumption about the witness’s math library. First enclose all three products and their sum using the existing outward binary64 interval operations. Reject nonfinite inputs or unrepresentable intermediate endpoints. Scale this dot interval by an exact power of two, x = angle / 2^s, chosen to put its magnitude at about 1/16 or below. Zero is kept exact. There are at most 1028 squarings for a finite dot interval.

Compute degree-24 cosine and degree-25 sine Taylor polynomials with interval recurrences and outward division by exact positive integer denominators. Taylor’s theorem applied at degrees 25/26 bounds the omitted terms by |x|^26/26! and |x|^27/27! respectively, because every real sine/cosine derivative has magnitude at most one. Bound those remainders from the next term’s interval and enlarge both components. This is a finite Taylor/Lagrange proof, not an empirical error estimate or an assumed libm ULP bound. No pi reduction or transcendental function is used by the proof. Square the complex enclosure s times, intersecting each result with [-1,1] + i[-1,1], which contains every exact unit-modulus phase. For very large or uncertain angles this can become a broad enclosure; it never silently becomes a precision claim. Compare the actual stored phase with that rectangle to bound its error. Exact point/witness equality has zero error, including Gamma.

This reuses the verified arithmetic conventions of Rump2010 and the Bloch phase convention of Sun2017 Eq.16; the Taylor remainder derivation above states the additional assumptions explicitly. The shared diagnostic is count-admitted with 4096 bytes plus external/backend storage, 262144 scalar and 2097152 kernel operations. The periodic metric planner charges that worst-case work for every possible short-range shell-image phase and adds workspace before evaluating any payload. The proof uses round-to-nearest binary64 and gradual underflow, checked by the existing arithmetic guard.

The actual short-range metric assembly multiplies its stored integral by the exact-phase enclosure. It then follows the unchanged interval sum and long-range/finite-zero boundary. The tile receipt records metric_phase_included, maximum_metric_phase_error, and a versioned assembly certificate identity. The RI metric receipt carries assembly_phase_included, exported as assembly_rank.phase_included. The existing canonical lower-triangle perturbation norm feeds strict rank admission and finite inverse/projector response; the optional J and raw K component targets automatically consume this expanded proof. No new reference object, eigensolver, rank repair or signed/global ledger sum is introduced. The overall source-construction flag remains false because other construction terms are still missing.

Independent validation uses 800-digit AGM pi with argument reduction and scalar sine/cosine series, rather than binary phase squaring, and reconstructs signed contracted short-range metrics from stored integral tiles with those phases. This proves the new phase contribution, not the integrals themselves. Primitive/Boys/radial/contraction/pure-transform evaluation, Fourier and coupling phases, kernel weights, q/R generation, screening and omitted domains, XC/grid, overlap/occupied-space/density response and total relaxed energy remain separate. MANTA stays experimental; #623/#625/#617 remain open.

5.60 Reciprocal metric weight construction from stored prefactor

The optional metric_assembly_rank proof includes reciprocal metric weight construction, in addition to Section 5.59’s short-range phase. Its reference uses exact stored momentum components, numerical range eta, physical range omega and reciprocal prefactor C. C is the actual stored 4*pi/volume value; evaluating pi, the cell determinant and C remains excluded. Stored auxiliary Fourier panels remain conditional operands. No numerical weight, physical support, source identity, factor or SCF operation is replaced.

For squared momentum t and the existing nonsingular branch, the reference weights are C/t * exp(-t/(4*eta^2)) for Coulomb, C/t * exp(-t/(4*omega^2)) for erf, and their difference for erfc. The finite-zero correction is -C/(4*eta^2) for Coulomb and C/4 * (1/omega^2 - 1/eta^2) for erfc. Erf has no finite-zero correction. The physical erfc finite zero mode and numerical SR subtraction are thus still included once. Equal stored ranges give an exact zero erfc reference, even when the evaluated finite-zero expression has a rounding residual. These are the existing Sun2017 range decomposition and zero conventions.

The shared certify_periodic_gdf_weight diagnostic returns an immutable interval and a bound for any finite supplied evaluated weight. It encloses the sum of three stored-vector squares, range squares, divisions and final products/subtractions using the existing outward binary64 arithmetic. The existing singular threshold is the stored binary64 value 1e-24. An interval straddling it refuses; a wholly singular interval has zero weight. Selection of a transfer’s Gamma/finite-zero branch remains with the source. The finite-zero diagnostic requires a zero vector. Nonpositive denominator intervals, nonfinite inputs and unrepresentable intermediate endpoints refuse.

For exp(-x), x is a nonnegative interval. Scale by a power of two to at most about 1/16, evaluate a degree-24 Taylor polynomial with outward integer division, and enlarge by the order-25 Lagrange remainder. Every derivative of exp(-x) on the nonnegative axis has magnitude at most one, so the next term’s absolute interval bounds that remainder. Square at most 1028 times, intersecting with [0,1] after every step. There is no correctly rounded exp/expm1 premise and no transcendental call in the proof. Exact zero arguments remain exact. Underflow is enclosed with gradual-underflow arithmetic; a conservative enclosure may cause a downstream target refusal. This follows the shared Rump2010 verified-arithmetic convention, not an empirical ULP allowance. Subtracting the two decay intervals covers either sign of the erfc difference, including nearly equal ranges.

The weight plan reserves 4096 bytes plus external/backend storage and 262144 scalar / 2097152 kernel operations for both possible exponentials. The metric planner charges each reciprocal vector and the possible zero term, plus interval panel storage, before payload evaluation. The unchanged numerical weights multiply the unchanged Fourier panels; separately, the ideal assembly uses the enclosed weights. Version-3 assembly identities, metric_weight_included, maximum_metric_weight_error and assembly_rank.weight_included identify this extension. The canonical lower-triangle rank/inverse producers and optional J/raw-K component targets consume it without adding a new eigensolver, Fock path or global ledger term.

Independent Decimal exponentials and exact stored-scalar rational arithmetic check all physical kernels, both zero conventions, equal/near-equal ranges, underflow, large momenta and arbitrary returned witnesses. Stored-panel Fourier Gram reconstructions and finite-zero contraction checks exercise the actual metric producer. These certify the stated weight contribution, not Fourier amplitudes or the primitive integral reference. Prefactor and vector generation, primitive/Boys/radial/contraction/pure transforms, Fourier and coupling construction, domains/screening, BvK corrections, XC/grid, overlap/occupied-space/density relaxation and total relaxed-energy accuracy remain separate. MANTA stays experimental and #623/#625/#617 remain open.

5.61 Reciprocal prefactor from the stored lattice

The optional metric assembly proof now removes Section 5.60’s stored reciprocal-prefactor premise. Its reference is 4*pi/abs(det A) for exact stored binary64 lattice entries A and mathematical pi. The numerical lattice determinant, pi value, weights, source, factor and SCF state remain unchanged. Construction of the lattice entries and reciprocal momentum vectors remains outside the reference. This grants no new symmetry or coordinate equivalence.

The shared interval implementation encloses the three cyclic cofactor terms of the determinant. A strictly negative determinant interval is reflected; a strictly positive one is retained. An interval containing zero refuses, including when a floating determinant happens to be nonzero. Overflow in intermediate products also refuses. The proof does not repair the cell or substitute a determinant algorithm into the numerical path.

Pi is enclosed using the identity pi = 16*atan(1/5) - 4*atan(1/239). For each exact rational argument x, enclose its first 40 alternating odd powers sum((-1)^j*x^(2j+1)/(2j+1), j=0..39). The alternating-series remainder has magnitude at most x^81/81; enlarge the interval by that quantity. The rational input divisions, powers, sums and final linear combination all use shared outward arithmetic. The tangent addition identity and the first-quadrant value establish the Machin identity. No libm pi, arctangent or correctly rounded constant is assumed. This follows the Rump2010 verified-arithmetic conventions already used by the weight proof. Outward division of four times this pi interval by the positive volume interval encloses the reciprocal prefactor.

certify_periodic_gdf_lattice_weight(lattice, momentum, omega, physical_kernel, zero, evaluated_weight, budget) uses this enclosure in the existing weight producer. The earlier certify_periodic_gdf_weight remains a diagnostic explicitly conditional on its supplied stored prefactor. The actual metric path constructs its prefactor enclosure once and shares it across all reciprocal and finite-zero terms. The common weight plan reserves 8192 bytes plus external/backend storage, 278528 scalar and 2228224 kernel operations, including both possible exponentials and prefactor proof. The metric planner conservatively charges that work per possible weight, plus the enlarged workspace, before payload evaluation.

Version-4 assembly identities, metric_prefactor_included and exported assembly_rank.prefactor_included distinguish the expanded reference. The same lower-triangle rank, finite inverse/projector and optional J/raw-K bounds consume it. Original physical source and fitting/signed/global reference ledgers remain unchanged; the complete-source flag remains false. The weight proof still uses the owner’s actual Gamma/finite-zero convention and strictly refuses unresolved momentum pole classification.

Independent AGM pi and Decimal determinants validate all three physical kernels, finite-zero terms, skew cells, orientation reversal, cancellation and lattice scales 1e-100 through 1e100. Contracted stored-Fourier Gram and finite-zero assembly witnesses now use the lattice-derived prefactor. They do not establish primitive or Fourier accuracy. Lattice/vector construction, primitive/Boys/radial/contraction/pure transforms, Fourier and coupling construction, source domains/screening, BvK corrections, XC/grid, basis/BZ, overlap/occupied-space/density relaxation and total relaxed-energy accuracy remain separate. MANTA and the unfinished #617 programme remain experimental; shared Fock/PBC/BIPOLE/symmetry ownership is unchanged.

5.62 Auxiliary Fourier construction enclosure

The optional metric audit now encloses the full auxiliary Fourier evaluation for exact stored Gaussian exponents, centers, normalized contraction coefficients and Cartesian-to-pure representation coefficients. This replaces the stored auxiliary-panel premise in Sections 5.55-5.61. It does not cover generation of those basis/representation data, AO-pair Fourier construction, short-range Coulomb integrals or reciprocal-vector generation. Supported angular order and normalization are unchanged: auxiliary Cartesian or pure shells through l=7. No new physical provider or symmetry permission follows.

The shared certify_auxiliary_fourier_panel(auxiliary, wave_vectors, budget) returns the unchanged evaluated Fourier result, a rectangular enclosure and an absolute complex-L1 error bound for each element. Its plan admits basis, vector, shell, primitive, scalar, kernel and storage costs; the binding admits counts before copying vector payload. The same internal shell enclosure is consumed by the actual metric producer, with no second numerical integral route or replacement of the evaluated panels.

For a primitive of exponent a at center A, use the exact Fourier seed (pi/a)^(3/2) * exp(-q.q/(4a)) * exp(-i q.A). The dimensionless Cartesian moments obey h_0=1, h_1=-i q/(2a) and h_n=(-i q/(2a))*h_(n-1)+(n-1)/(2a)*h_(n-2) independently on each axis. Integration by parts against the decaying Gaussian proves this recurrence; its surface term vanishes. The product of three moments multiplies the seed and stored contraction coefficient. This is the same exact Gaussian polynomial integral as the existing MD/Hermite Fourier formulation (Neese2023 and Sun2017), not an assumption about the rounded MD E matrix.

Shared interval routines enclose the squared norm, divisions, pi and exp. The existing phase proof uses exact stored centers, thus also covering the numerical producer’s rounded product-center/phase arithmetic in this single-center case. Square-root endpoints are bounded by two finite binary searches on a power-of-two-scaled mantissa: one certifies an upper squared endpoint below the argument, the other a lower squared endpoint above it. Each search takes at most 64 steps, preserves its initial bracket and returns a proven endpoint. It assumes no libm square-root accuracy.

Outward complex products and sums enclose the primitive moment recurrence, signed contraction and application of the shared stored representation matrix. The Cartesian identity transform does not change the reference. Unrepresentable intermediate enclosures refuse; there is no clamping of physical integrals or weakening of a target. The reusable pi/exp/division proofs moved into a shared internal helper without changing their arithmetic. Only the certificate computes these enclosures; evaluated numerical values continue through the existing Fourier producer.

The construction plan conservatively charges per primitive/vector moment, transcendental and contraction work, per shell/vector phase and transform work, the existing numerical producer, exported enclosures and workspace. The metric planner includes every possible reciprocal/finite-zero panel, full auxiliary interval storage and the proof workspace. Version-5 metric assembly identity, metric_fourier_included and assembly_rank.fourier_included identify the expanded reference. The Gram assembly uses these enclosures on both auxiliary factors with its existing weight enclosure. Shared rank/inverse/J/raw-K consumers propagate the result; physical source/factors/states and fitting/signed/global ledgers stay intact.

Independent 250-digit finite Gaussian-moment sums use factorial coefficients rather than the native recurrence, AGM pi, Decimal sqrt/exp and the earlier independent phase oracle. They cover l=0..7, Cartesian/pure shells, signed coefficient scales, diffuse/tight primitives and deep cancellation. The representation data are deliberately shared conditional operands; this does not validate their construction. Existing quadrature and independent native Fourier references remain separate checks of the physical convention. Basis/representation and lattice/vector data generation, short-range integral and coupling construction, screening/domains, stored-factor/executed Fock/energy arithmetic, BvK, XC/grid, basis/BZ and overlap/occupied/density relaxation remain outside a total relaxed-energy claim. MANTA remains experimental; #623/#625/#617 and the broader capability matrix stay open.

5.63 Two-center metric integral construction enclosure

The optional periodic metric audit now encloses short-range two-center Coulomb integral construction, in addition to Section 5.62 auxiliary Fourier construction and the earlier phases and weights. Exact stored exponents, normalized contraction coefficients, centers, representation coefficients, image translations and reciprocal vectors define the reference. The audit includes rounding of center-plus-image addition. Generation of those input data, three-center coupling and AO-pair Fourier construction, omitted source domains/screening and relaxation remain separate.

plan_metric_construction and certify_metric_construction expose the same proof for full, erf and erfc two-center tiles through auxiliary l=7. The certificate contains the existing FittingTiles numerical result, lower and upper element enclosures and absolute error bounds. The numerical MD producer is unchanged. The internal shell enclosure also feeds the actual periodic GDF metric producer; it is not an alternative numerical integral route.

For stored primitive exponents p and q, outward arithmetic encloses rho=pq/(p+q), center displacement, T=rho|A-B|^2 and the prefactor 2pi^(5/2)/(pq*sqrt(p+q)). Pi and square root reuse Section 5.62 proofs. The Boys integral F_n(T)=integral_0^1 u^(2n)exp(-Tu^2) du uses the positive series exp(-T)/(2n+1) times the sum over k of (2T)^k/product_(j=1..k)(2n+2j+1). The first omitted term divided by one minus the next ratio bounds the positive tail once that ratio is below one; all subsequent ratios decrease. At most 2048 terms are admitted for T upper endpoint at most 512. A relative-tail comparison only chooses the stopping index: the explicit outward remainder, not that comparison, certifies error. Positive downward recurrence supplies lower orders.

For larger arguments the integral over [0,infinity) bounds F_n from above by Gamma(n+1/2)/(2*T^(n+1/2)), also capped by 1/(2n+1); the lower endpoint is zero. If the T interval includes zero, the latter elementary upper bound alone applies. This branch is deliberately conservative and may prevent a requested rank/action target from qualifying. It is not an asymptotic equality or a silent accuracy fallback. No arbitrary large-T series truncation is used.

With kappa=omega^2/(omega^2+rho), erf seeds are kappa^(n+1/2)F_n(kappaT). Erfc seeds are the difference from F_n(T), intersected with nonnegative values. This follows the same physical integral as Neese2023 Eq. (54); the evaluated producer’s positive-interval quadrature is retained. The proof can widen in deep cancellation, which remains visible to downstream admission. Zero and infinite range limits are explicit.

Single-center Cartesian Hermite coefficients have the finite exact form E_(l,t)=l!/[((l-t)/2)!t!2^la^((l+t)/2)] for matching parity and t<=l, and are zero otherwise. Products over axes provide the sparse expansion. The usual McMurchie-Davidson Coulomb derivative recurrence consumes interval seeds (-2rho)^n F_n(T), with the ket derivative parity sign. Outward sums and products enclose primitive contraction, signed stored coefficients and application of the shared stored Cartesian-to-pure matrices. This follows McMurchieDavidson1978 and Neese2023, using Rump2010 verified arithmetic. The proof does not certify generation of basis or pure-matrix coefficients.

Plans account for finite series, derivative cache, sparse Hermite products, primitive/shell work, transformation and diagnostic output storage before evaluation. Periodic planning charges every candidate metric image. Invalid or unrepresentable intermediates and budget exhaustion refuse; no numerical integral, cell or density is repaired. The complete-source flag stays false.

Version-6 metric assembly identity and assembly_rank.integrals_included identify the enlarged construction reference. Existing shared rank/inverse and optional J/raw-K consumers propagate the perturbation without changing physical source/factor/state identities or original fitting/signed/global ledgers. Independent high-precision alternating radial integrals and finite closed Cartesian derivative sums test the proof against a different scalar and derivative evaluation, with stored representation data conditional. Existing libint comparisons remain independent physical-integral checks. No total relaxed-energy bound, automatic whole-method accuracy, bulk CRYSTAL parity, speedup, new symmetry permission or production-ready claim follows.

5.64 Positive short-range radial enclosure

The Section 5.63 short-range audit now encloses the positive integral integral_theta^1 u^(2n)exp(-Tu^2) du directly. It no longer subtracts two Boys enclosures for finite positive erfc range. The evaluated MD quadrature is unchanged, and the ordinary Boys enclosure remains unchanged for full/ long-range integrals and other producer certificates. Coverage, exact stored operand assumptions and the remaining exclusions of Section 5.63 stay intact.

Let kappa=omega^2/(omega^2+rho), theta=sqrt(kappa) and c=rho/(omega^2+rho)=1-kappa. Compute c directly rather than subtracting rounded kappa from one. Choose a stored fraction f in (0,1] of this squared interval and partition it into at most nine panels with squared width c*f/N. A scalar choice approximately caps the total exponential gain at 64. Its rounded accuracy is not an assumption: explicit outward checks admit each panel gain at most eight, otherwise evaluation refuses. Small outward guards on the count avoid unstable decisions at exact panel boundaries.

For a panel [a,b], its width h is evaluated as (b^2-a^2)/(a+b); this retains very narrow intervals even when a and b round alike. The last endpoint of a complete interval is exactly one. With u=a+hy and y in [0,1], the integral is hexp(-Tb^2) times integral_0^1 (a+hy)^(2n)exp(Q(y)) dy, where Q(y)=Th*(1-y)(a+b+hy) is nonnegative and at most the admitted panel gain. Expand exp(Q) in positive powers. Products of the linear factors use nonnegative Bernstein coefficients: multiplying a degree-d polynomial by a linear factor with endpoints L0,L1 gives coefficient i equal to ((d+1-i)/(d+1))L0c_i + (i/(d+1))L1c_(i-1). The integral of a Bernstein polynomial is the mean of its coefficients. Every operation and coefficient is enclosed outward.

After term k-1, the omitted positive tail is bounded by the integral of term k divided by 1-g/(k+1), when g/(k+1)<1, where g bounds Q. All subsequent term ratios are smaller. At most 64 terms per order and panel are admitted. The relative-tail check only chooses a stopping index; the explicit geometric remainder always supplies the certificate, including at the iteration cap. There is no assumed quadrature or libm accuracy and no cancellation between full and long-range Coulomb terms.

If f<1, the remaining squared interval is c*(1-f), and the exact remaining width is c*(1-f)/(1+b). Its integral is bounded by that width times exp(-Tb^2), using u^(2n)<=1. When T and b are positive, an independent upper bound exp(-Tb^2)/(2Tb) follows by extending the Gaussian integral to infinity and using u/b>=1. The smaller outward upper bound is added as a nonnegative tail interval for every order. Truncation never silently discards the tail. Thus large T and narrow erfc intervals need neither the earlier full-Boys subtraction nor an unqualified asymptotic equality.

The proof is a verified evaluation of Neese2023 Eq. (54), with Rump2010 outward arithmetic; the polynomial identities above give its derivation. It adds no new physical integral source. The construction plan charges nine panels, 64 terms per order, Bernstein products and integral/remainder work before evaluation. Overflow, unresolved panel bounds and resource limits refuse. Version-7 metric assembly identity identifies the sharper algorithm; rank/inverse/J/raw-K still consume the same construction reference and bounds. Neither the original fitting ledger nor physical source/factor/state identity changes. Independent high-precision alternating radial integrals, half-line Gaussian limits and finite derivatives test the proof, including severe cancellation, large T, sub-ulp widths and panel boundaries. Coupling and basis/vector-data construction, domains/screening and relaxed response remain outside a total-energy or automatic whole-method accuracy claim.

5.65 Cartesian-to-pure coefficient construction

The shared auxiliary Fourier and two-center metric construction proofs now include generation of the Cartesian-to-pure representation coefficients. They no longer treat the rounded representation matrix as exact. Exact stored normalized basis coefficients, exponents, centers and vector data remain the reference inputs; their generation and all coupling/domain/relaxation exclusions remain separate. The numerical MD transforms are unchanged.

plan_representation(l, pure, budget) admits fixed storage and work before certify_representation returns the existing evaluated matrix, lower/upper coefficient enclosures and absolute errors. Both construction proofs consume the same internal enclosure. Supported l=0..7, Cartesian order, real row order m=-l..l and common axial Gaussian normalization are unchanged. Cartesian identity coefficients are exact. This is coefficient construction evidence, not authority to omit finite operator audits or assume angular equivariance.

The proof uses the same mathematical solid harmonics as the existing producer: DLMF 14.30.1 and 14.30.2, expressed through derivatives of the Legendre polynomial. The complex phase convention is converted to real sine/cosine components, with the Condon-Shortley sign removed and Racah rather than unit-sphere normalization. The coefficient scale is sqrt((2 if m>0 else 1)(l-m)!/(l+m)!). The finite expansion of (x+iy)^m times the differentiated polynomial, homogenized by powers of x^2+y^2+z^2, supplies the remaining rational coefficients. This matches the existing convention; it does not redefine the basis.

All factorials through 14! are exactly representable integers below 2^53. Outward multiplication, division and signed summation enclose the finite polynomial coefficients; the verified square-root routine bounds the scale. No libm square-root accuracy assumption enters. Products with these intervals now replace products with point-valued stored transform entries in both the auxiliary Fourier and two-center metric proofs. Count-only plans charge the coefficient proof per shell/panel or metric shell pair and include storage for the coefficient intervals. Existing work/storage limits refuse before evaluation. No numerical source or requested accuracy target is modified.

The metric version-8 assembly reference and assembly_rank.representation_included identify the expanded reference. Existing rank/inverse/J/raw-K consumers propagate it through the shared metric bound; original fitting/signed/global ledgers and physical state identities remain intact. The complete-source flag stays false.

Independent witnesses generate Legendre coefficients by an exact rational three-term recurrence, differentiate and homogenize them, then use high-precision square roots. They avoid the producer’s factorial Legendre coefficient formula. The integral and Fourier witnesses now use these mathematical coefficients, not the rounded native matrix. Existing SciPy spherical-harmonic comparisons check real convention and normalization independently. None of these checks certifies numerical angular transport, omitted operator members or geometry mismatch in the separate symmetry machinery. Basis/vector data generation, AO-pair Fourier/three-center coupling construction, domains/screening and relaxed response remain outside automatic total-energy accuracy claims.

5.66 Orbital-pair Fourier construction component

plan_pair_fourier_construction and certify_pair_fourier_panel extend the shared Fourier proof to products of two stored orbital Gaussians, each with l=0..6, in pure or Cartesian form. The numerical compute_fourier_panel is unchanged. The certificate returns that panel and entrywise real/imaginary lower/upper bounds and a conservative absolute error, in (q,bra,ket) order. The internal shell producer returns the same enclosures for future coupling consumers. This is a separately admitted component diagnostic, not a complete three-center construction certificate or a new SCF accuracy admission.

For positive exponents a,b and exact stored centers A,B, define p=a+b, P=A+(b/p)(B-A), and rho=a*b/p. The Gaussian product theorem gives a common seed (pi/p)^(3/2) exp(-rho*|A-B|^2-q.q/(4*p)) exp(-i*q.P). Product-center, squared-distance, exponent and prefactor arithmetic are all enclosed. The existing phase proof now accepts an interval angle internally; no rounded product center is treated as exact. The public point-vector phase contract and its arithmetic are unchanged. Pi, exponential and square-root proofs reuse Sections 5.59-5.62 without a libm accuracy assumption.

Expand each Cartesian pair about P using finite binomial products of P-A and P-B. For each axis, normalized Fourier moments obey h0=1, h1=-iq/(2p) and hn=(-iq/(2p))h(n-1)+(n-1)/(2p)*h(n-2). Outward complex arithmetic encloses all moments through order 12, polynomial products, signed primitive contractions and both exact mathematical pure transforms from Section 5.65. Binomial coefficients through order six are small exactly representable integers. Intersections with known positivity and unit-circle bounds retain the exact reference. Overflow or unresolved positive denominators refuse; underflow is retained by outward endpoints.

This is the Gaussian-product and Fourier convention of Neese2023 and Sun2017 (DOI 10.1002/jcc.26942 and 10.1063/1.4998644), with Rump2010 verified arithmetic. The explicit derivation above bounds the same finite Gaussian integral rather than introducing a different physical source. Independent witnesses shift a Gaussian contour and sum its finite even central moments in high precision; they do not use the native h recurrence. Exact rational Legendre transforms, separate high-precision phase/elementary evaluation and existing real-space Gauss-Hermite integration check convention and values.

Count-only planning precedes vector copying and evaluation. It includes the existing numerical panel, all certificate arrays, shell metadata, contraction and transform workspace, at most 13 moments per axis, at most 49 binomial products per Cartesian axis pair, and shared phase/elementary/representation proof work. Storage, scalar, kernel, shell and primitive caps remain strict; conservative reservations are not measured allocation or speedup claims. Empty vector panels retain their declared shape without primitive evaluation.

The exact stored normalized basis data and wave vectors define this component reference. Their generation, lattice/image translation, periodic Bloch sums, three-center short-range integrals, reciprocal coupling products and sums, zero-mode terms, screening/domain truncation and coupling action propagation remain separate. No metric, RI-BUPO, Fock, symmetry or automatic-selection certificate silently acquires this coverage. Existing component budgets and complete-source flags are unchanged. This diagnostic establishes neither total relaxed-energy accuracy, bulk CRYSTAL parity, numerical equivariance nor production readiness for the full MANTA programme.

5.67 Streamed three-center integral construction

ThreeCenterConstructionTiles(auxiliary, bra, ket, request, budget) returns separately owned construction certificates for individual ordered (P,mu,nu) shell tiles. plan_three_center_construction counts the complete requested shell sweep and its maximum live tile storage before copying basis data. Auxiliary l=0..7 and each orbital l=0..6, mixed pure/Cartesian shells and Coulomb, erf and erfc kernels retain the existing FittingTiles conventions. Range parameters must be finite and nonnegative; full Coulomb requires zero. No global three-index tensor is allocated. Numerical tiles and their physical source identity are unchanged; lower/upper/error arrays have the same shape.

The reference is the exact integral of the stored normalized Gaussians. For orbital exponents b,c at B,C, p=b+c, P=B+(c/p)(C-B), and the product decay is exp(-bc/p|B-C|^2). All product-center and decay arithmetic is enclosed. For auxiliary exponent a, rho=ap/(a+p), displacement D=A-P and T=rhoD.D. The primitive prefactor is 2*pi^(5/2)/(a*p*sqrt(a+p)) times the product decay and stored signed contraction coefficients. Shared verified elementary and representation producers cover pi, exponentials, roots and pure transforms.

The one-dimensional Hermite raising identity is E(i+1,j,t)=E(i,j,t-1)/(2*p)+(P-B)*E(i,j,t)+(t+1)*E(i,j,t+1); raising the other orbital uses P-C. The seed E(0,0,0)=1 leaves the common decay in the prefactor. Auxiliary coefficients use the same identity with zero center shift and a unit partner. Outward arithmetic encloses every coefficient; zero terms are omitted only when both interval endpoints are exactly zero.

The existing metric radial producer supplies Boys, erf and positive erfc seed enclosures through order 19 via a thin shared kernel wrapper. Its implementation and ordinary Boys interface are unchanged. Seeds multiplied by (-2*rho)^n enter the standard Cartesian Coulomb derivative recurrence; ket derivatives carry (-1)^(tx+ty+tz). Successive interval contractions of the Hermite kernel with orbital and auxiliary coefficients avoid a four-axis Cartesian/Hermite loop. Three successive mathematical representation transforms and signed primitive sums produce the final tile enclosure. The error compares the actual numerical tile to that enclosure.

This is the McMurchieDavidson1978 / Neese2023 Gaussian-product formulation (DOI 10.1002/jcc.26942), with the radial conventions of Sections 5.63-5.64 and Rump2010 outward verification. Independent witnesses expand the Cartesian product polynomial about P, use closed factorial Hermite coefficients and finite Cartesian derivative sums with high-precision alternating radial integrals. They avoid both native raising recurrences. Existing libint three-center comparisons check the numerical source separately. Selected maximum-order Cartesian witnesses supplement all-angular-axis, mixed, contracted, signed, underflow and range-limit cases.

Admission charges basis copies, numerical and proof tiles, three transform stages, sparse Hermite tables, derivative/kernel scratch, all primitive products, radial work and output copies. Shell, primitive, scalar, kernel and memory caps refuse before evaluating a tile; invalid arity, indices, angular roles, derivatives or nonfinite input refuse. Numerical and proof scratch scale with a shell tile, not the number of global tensor entries. Conservative reservations are not measured allocation or performance claims.

This streamed component supplies a prerequisite for complete periodic coupling construction. It does not yet compose with pair/auxiliary Fourier, reciprocal weights, Bloch/image sums or zero-mode terms into a coupling action bound. Exact stored centers do not include errors in constructing translated image centers. Normalized basis/vector generation, periodic domains/screening, BvK, XC/grid, basis/BZ, overlap/occupied/density relaxation and total relaxed energy remain separate. Existing metric/Fock/SCF budgets and complete-source flags are unchanged. No symmetry reduction, bulk CRYSTAL parity, general speedup or automatic whole-method accuracy permission follows.

The shared scalar fitting planner now explicitly rejects derivative requests. Its capability query describes several producers and previously admitted a first-derivative request that the scalar tile evaluator did not execute. DerivativeTiles remains the supported first-derivative producer. Metric and three-center construction diagnostics inherit this corrected refusal; ordinary integrals and their scientific results are unchanged.

5.68 Finite periodic coupling construction and paired actions

PeriodicGDFTiles.plan_coupling_construction(budget) admits the complete finite construction proof and one unchanged numerical coupling action. certify_coupling_tile(p, mu, nu, budget) returns one owned numerical tile and flattened real/imaginary lower/upper/error arrays in its native order. certify_forward(input, budget) and certify_adjoint(input, budget) return the unchanged numerical action plus the corresponding output enclosures. Inputs are exact stored complex128 values; C/F contiguous layouts agree. All five work/storage caps are checked before binding input copies.

The reference is the unscreened literal finite integer image set selected by the shared sphere/capsule producer for the supplied cutoffs. Selection uses its existing rounded geometry and finite-zero convention. For each selected integer triple n, the proof encloses the exact product L*n and the addition of each stored center. It then encloses the Gaussian integral on those exact centers. This distinguishes an exact finite-set integral from an assertion that the selected set is a mathematically complete geometric domain.

For a tile, the reference is the sum of the real-space erfc term sum_R exp(i*k.R) sum_T exp(-i*q.T) (P_T|mu nu_R)_erfc, the reciprocal term sum_G w(G+q) conj(auxFT(P,G+q)) sum_R exp(i*k.R) pairFT(mu,nu_R,G+q), and the existing finite-zero correction when applicable. Shared three-center, orbital-pair/auxiliary Fourier, Bloch-phase, radial, representation and lattice weight producers enclose all terms. Coulomb, physical erf and physical erfc keep their existing decompositions and zero-mode conventions. A reciprocal pole classification whose interval crosses the threshold refuses. No new radial algorithm, image selector or symmetry reduction is introduced.

Exact finite-source forward and adjoint contractions are respectively conj(T):X and T*c, including arbitrary complex, non-Hermitian inputs. Per-output error is the outward complex L1 distance from the actual numerical value to its reference rectangle. Therefore this single certificate includes construction, omitted/screened/multipole terms and executed contraction error for the admitted numerical route. It must not be added to overlapping old approximation or roundoff budgets. Screened, explicit, compressed, global and pair-tree actions are compared directly with the same reference, rather than assuming their real-arithmetic maps or Gamma averaging are identical.

source_identity identifies the actual numerical coupling; reference_identity identifies the physical finite reference and selected integer supports; certificate_identity additionally binds actual values, input snapshot and bounds. reference_terms counts selected SR auxiliary images and LR pair images per reciprocal panel, not scalar operations or unique images. The numerical receipt retains its original plan and counters; the separate construction plan reserves the additional proof work. Only shell tiles, Fourier panels and action vectors are retained, never the global coupling tensor. Conservative component-plan multiplicities and explicit folding/ geometry/contraction charges bound work and live scratch. Those reservations are not measured allocations, costs or speedup evidence.

The formulation reuses Sun et al. 2017 (DOI 10.1063/1.4998644), Ye and Berkelbach 2021 range-separated Gaussian fitting, McMurchieDavidson1978 / Neese2023 Gaussian products and Rump2010 verification from preceding sections. Independent witnesses use a brute-force finite image census, exact Decimal lattice products, closed polynomial Gaussian moments/derivatives and high-precision scalar phase/weight evaluation. They do not consume the native interval endpoints. Separate numerical-source and paired-action regressions preserve the shared Fock/PBC/symmetry contracts.

This remains an experimental finite-source diagnostic. Existing SCF/Fock admissions and metric budgets do not automatically consume it. Stored normalized basis, lattice and reciprocal-vector generation, selection/domain completeness and infinite tails, fitting-basis incompleteness, BvK, XC/grid, basis/BZ, overlap/occupied/density relaxation and total relaxed energy remain separate. No bulk CRYSTAL parity, general speedup or useful automatic whole-method accuracy follows. Subsequent composition must account for coupling and metric response without counting the same error twice.

5.69 Shared screened primitive source

MANTA’s dense and streamed real-space three-center paths now use the same PrimitiveTripleScreen as libint values and gradients. Previously they shared shell/image decisions but retained different primitive sets within contracted shells. This caused the retained-source comparison in #687 to fail at 5.65e-11 for a requested omission budget of 0.01; it was not a new coupling-certificate error or an unbudgeted primitive-epsilon screen.

The existing Gaussian/angular envelope of Ye and Berkelbach 2021 SI S4 assigns each image term a fraction of the shell budget, then divides that fraction by its primitive-triple count. Each omitted primitive obeys the same absolute envelope; all survivors retain their original signed normalized coefficients. No fragment is renormalized. If every primitive survives, the original contracted evaluation is retained; if none survives, no numerical integral call is made. The selector, bounds and traversal are shared without changing the libint implementation. Zero screening and multipole routes keep their previous numerical paths.

Screened source identities now include shared-budgeted-primitive-triples.v1; unscreened source identities and the exact finite reference from Section 5.68 are unchanged. Common and coupling-only screening both use the shared primitive selector; the latter leaves the metric unchanged. Split survivors can increase numerical shell invocations while reducing evaluated primitive products, so neither fewer shell calls nor general speedup is promised.

Count admission reserves one call/transform per possible primitive triple, selector envelopes/survivors and copied primitive-shell storage, plus phase accumulation work. This applies before dense or streamed numerical allocation. Actual counters report executed numerical calls and primitive products. Existing five resource caps, source ownership, Fock/PBC/symmetry interfaces and omission targets remain intact. Independent libint comparisons retain their original tolerance; complete calculations exercise contracted screened sources. These are finite-source checks, not total relaxed-energy or bulk CRYSTAL parity claims. MANTA and the broader accuracy programme remain experimental and unfinished.

5.70 Joint finite-source Coulomb response

PeriodicRIQBuilder.plan_finite_coulomb and certify_finite_coulomb combine Sections 5.56, 5.68 and 5.69 on the same immutable Gamma owner. This is an explicit producer certificate for a fixed weighted density family, before integration into automatic complete-job admission. It does not modify the selected factor, coupling, Fock builder, SCF, symmetry state or global ledger.

Let A contain the columns of the declared stored coupling graph, B the exact unscreened finite coupling reference of Section 5.68, C the exact spectral inverse of the stored lower-triangle metric and C0 the spectral inverse of its qualified finite assembly reference. F is the actual stored factor. The certificate bounds the difference between A* F F* A d and B* C0 B d. The periodic implementation’s conjugation convention gives the conjugate metric form with the same norms. This is not an assertion that either finite fitted map equals the infinite full-Coulomb operator.

For n_j >= ||A_j||_2, e_j >= ||B_j-A_j||_2, define b_j = n_j + e_j and s(v) = sup sum_j v_j |d_j| over the declared weighted spin-total family sum |d_j| <= L, optionally |d_j| <= h_j. The existing outward dual support calculation in manta_ri_inverse_action.cpp is reused. With eta_F >= ||FF* - C||_2, eta_M >= ||C0-C||_2, and rho >= ||C||_2 + eta_M >= ||C0||_2, the separate entry bounds are:

• stored factor: eta_F n_j s(n);

• metric source: eta_M n_j s(n);

• coupling source: rho (e_j s(b) + n_j s(e)).

The last inequality follows from B* C0 B - A* C0 A = (B-A)* C0 B + A* C0 (B-A). The shared bound_hermitian_coupling_family implements it with outward arithmetic. The total J radius is the outward sum of the three terms. Each energy term is bounded by s(J_term)/2, and their outward sum bounds the energy discrepancy. Arbitrary complex family members are allowed; no positivity, variational fit or cancellation premise is used. Finite metric response retains the accepted spectral/rank assumptions and primary references of Section 5.56, without a second eigensolver or inverse proof.

Explicit sources stream certified shell tiles and accumulate auxiliary L1 column norms and construction-error radii. Compressed, global and pair-tree sources reuse the existing graph-norm producer. Each unit-column forward construction certificate is joined to the same executed action’s arithmetic replay: |B-A| <= |B-executed| + |executed-A|. The outputs must match exactly. Only column envelopes, action vectors and shell tiles are retained; no global auxiliary-by-AO-pair tensor is stored. Shell/primitive counters report the numerical producer calls; reference_terms separately counts construction terms, including repeated column proofs. This general compressed route repeats the construction proof per column and is not a speedup claim.

Count-only admission reserves every repeated pass, replay, norm/family proof, source retention and owned export before numerical allocation. All five resource caps apply. PeriodicRIFiniteCoulombOptions declares L, optional entry caps and either both positive J/energy targets or neither. Impossible targets, invalid families, nonliteral Gamma/non-Coulomb sources, missing or unqualified metric assembly response and common integral screening refuse. Coupling-only screening is supported. Common screening is excluded because the metric assembly proof does not enclose omitted metric terms.

The metric reference retains its existing exact stored translation-vector contract, including enclosed center addition; the coupling reference uses exact lattice-times-integer translations. Generating the metric translation vectors remains outside that metric proof. Thus the joint reference records both finite conventions; it does not silently claim a common exact-lattice metric/coupling source. The metric lower triangle and adaptive rank policy are preserved. Source, reference, family and certificate identities bind the actual owner and bounds. Exact stored normalized basis and vector inputs remain assumptions.

The source-only inverse response is used once. Its combined factor_inverse_error_bound must not be added again. Coupling radii already include screening/multipole discrepancy and construction error; older approximation budgets must not be added. Executed input/contraction/Fock and energy arithmetic, domain completeness/infinite tails, full-Coulomb fitting error, K, BvK, XC/grid, basis/BZ, overlap/occupied/density relaxation and total relaxed energy remain separate. This certificate is not automatically added to the existing full-Coulomb BUPO ledger, whose reference differs. MANTA and #617 remain experimental; useful whole-method automatic accuracy, bulk CRYSTAL parity and general speedup are not established here.

The integration also repairs the inactive pair-tree flag on internal residual panels after shared primitive-selector validation was introduced. Real-space pair-tree contributions are supplied separately; clearing that residual flag restores the original numerical action without changing public source options.

5.71 Complete-method finite-source Coulomb admission

PeriodicRIFittingAccuracyOptions.finite_coulomb_j_error and finite_coulomb_energy_error are optional paired positive targets for the Section 5.70 reference. They are independent constraints inside the existing pre-SCF fitting selection, not additive corrections to its full-Coulomb BUPO reference or its stored-coupling metric-action reference. Both zero preserves the previous policy, work plans and provenance. Nonfinite, negative or one-sided targets refuse. The selected Gamma source must fit Coulomb J, have an unscreened metric and request its assembly proof.

The actual immutable Gamma owner produces the joint certificate on the canonical overlap-derived weighted density family. The native receipt retains that audit certificate plus the independently requested targets. A numerical target failure can be retained as a rejected search trial; it cannot be used for SCF. Existing metric, physical-domain and auxiliary selection loops require both the old full-Coulomb targets and the new finite-source targets to pass. No monotonicity of either error with cutoff, domain or basis is assumed. Failure at the allowed attempt/domain/cutoff limit refuses before SCF.

Count admission reuses the same coupling-construction and joint-response planners as the actual producer. Only requested finite-source policies carry this extra count metadata through complete-method capacity planning. Source construction, repeated compressed column proofs and every discarded trial are charged. Certificate vectors retained in metric/domain/auxiliary/reference history have a separate bounded allowance. Searches retain the existing cumulative work ledger; they do not refund rejected trials. The accepted physical source, canonical family and Fock state remain immutable during SCF.

The selected result and every evaluated trial export the finite reference, source, family, metric and certificate identities, separate factor/metric/ coupling/total bounds and proof work reservations. The old full-Coulomb certificate identity and bounds do not absorb these terms. The final qualification means all requested constraints passed, not that their sum is a total relaxed-energy guarantee.

All Section 5.70 exclusions remain: metric translation generation, stored input and basis generation, infinite domains, full-Coulomb fitting, K, executed Fock/energy arithmetic, basis/BZ/XC/BvK and occupied/overlap/density relaxation. The metric and coupling translation reference conventions remain distinct. This admission does not make the conservative bounds useful automatically, establish bulk parity or imply a speedup. MANTA remains experimental.

5.72 Residual finite metric-source action

For explicit coupling panels, the Section 5.70 joint producer now reuses the shared HermitianInverseAction column witnesses and Section 5.34 quadratic stored-factor bound. The shared perturbation_family method tightens the source-only metric response when the certified retained rank is the full auxiliary dimension. It retains the spectral norm-family bound for truncated metrics and whenever the resolvent margin cannot be certified positive. Compressed coupling graphs retain their existing norm-family proof.

Write M for the exact stored Hermitian metric, C for its spectral inverse, F for its stored factor, and a_j for a stored coupling column. Interval products enclose y_j = FF* a_j and r_j = a_j - M y_j. With P the exact retained projector, C a_j = P y_j + C r_j. The new column field inverse_action_norm therefore bounds ||C a_j||_2 by the smaller of ||C||_2 ||a_j||_2 and ||y_j||_2 + ||(I-P)y_j||_2 + ||C||_2 ||r_j||_2. Complex L1 envelopes upper-bound the Euclidean norms. The existing discarded leakage certificate supplies the middle term, which is zero at full rank. No new eigensolver, eigenvector alignment or auxiliary truncation is used.

For full rank and a Hermitian source perturbation E with ||E||_2 <= t, the finite resolvent identity gives

(M+E)^-1 - C = -C E (I+C E)^-1 C.

If 1-t||C||_2 > 0, then with z_j >= ||C a_j||_2,

|a_i* ((M+E)^-1-C) a_j| <= t z_i z_j / (1-t||C||_2).

The implementation uses outward arithmetic for the margin and coefficient, then the existing density-family support function s to form J_i <= t z_i s(z)/(1-t||C||_2) and E <= s(J)/2. Each J entry is the minimum of this bound and the existing spectral source-response bound. The adaptive rank-stability proof remains mandatory, including relative-cutoff movement. A failed rank proof remains unqualified; zero-rank and partial-rank cases retain their original response semantics. This is a finite bilinear use of the inverse-norm/residual principle in Rump (2010), Eq. (10.52), DOI 10.1017/S096249291000005X, with the resolvent remainder retained exactly. It is not a derivative approximation or a total relaxed-energy estimate.

The periodic producer streams one auxiliary-by-orbital-shell-pair panel. It obtains every entry and construction radius from the existing coupling tile certificate, without constructing a full auxiliary/AO-pair tensor. Column witnesses, panel scratch and original-order reference hashes are reserved by the shared finite-source planner before allocation. Original reference hash order, source identities, rank, physical supports, canonical density family, Fock ownership and symmetry permissions remain unchanged. The bound and its certificate identity may change. The separate stored- coupling metric-action consumer retains its existing assembly norm bound; this tightening is consumed by the joint finite-source certificate.

All Section 5.70 exclusions remain, including coupling-source conservatism, metric translation generation, basis/input generation, infinite domains, full-Coulomb fitting, K, executed Fock/energy arithmetic, basis/BZ/XC/BvK and occupied/overlap/density relaxation. MANTA remains experimental.

5.73 Finite coupling response from inverse-action envelopes

HermitianInverseAction.coupling_response bounds the coupling-only difference B* C0 B - A* C0 A, with the exact finite reference metric inverse C0 from Section 5.70. It consumes columns belonging to the same immutable action, a finite metric perturbation radius t and Euclidean coupling-error envelopes e_j >= ||B_j-A_j||_2. Rank stability remains mandatory. This method does not include the separate stored-factor or metric-source response a second time.

Let C be the exact stored metric inverse and eta_M >= ||C0-C||_2 its qualified finite source response. Section 5.72 provides z_j >= ||C A_j||_2; define rho = ||C||_2 + eta_M. At any stable retained rank, including discarded-projector rotation,

||C0 A_j||_2 <= min(rho n_j, z_j + eta_M n_j),

where n_j >= ||A_j||_2. At full rank with positive certified margin m = 1-t||C||_2, the exact resolvent also gives ||C0 A_j||_2 <= z_j/m. Let w_j be the smallest available outward bound. No full-rank identity is applied across a truncated projector. Zero rank remains zero only when the same adaptive classification is proved stable.

Writing D = B-A, the exact finite expansion is

B* C0 B - A* C0 A = D* C0 A + A* C0 D + D* C0 D.

Hermiticity of C0 gives the adjoint action without assuming positive D or positive density. With the existing declared-family support function s,

J_i <= e_i s(w) + w_i s(e) + rho e_i s(e).

The quadratic term is retained for all inputs, including A = 0. Each entry is the minimum of this bound and the original Section 5.70 norm triangle; the family energy bound is s(J)/2. The finite spectral bound and resolvent retain metric/coupling cross terms without a derivative approximation. This combines the Section 5.72 residual principle of Rump (2010), Eq. (10.52), DOI 10.1017/S096249291000005X, with the finite Gram expansion already used in Section 5.70. Published inverse-norm principles are reused; the explicit finite coupling-family composition is the implementation described here.

plan_hermitian_coupling_response admits both coexisting family passes, including the perturbation certificate and support reductions, before payload conversion or allocation. The joint periodic planner uses this same plan. Explicit sources reuse their existing certified shell-pair columns and construction radii, without another integral pass. Compressed sources retain the original declared-graph norm certificate. Stored factor, metric, source, family, rank and physical reference identities remain unchanged; the coupling bound and joint certificate identity may improve.

This is an independent constraint in finite-source admission. It is not added to the full-Coulomb BUPO fitting ledger, whose reference differs. The exact stored metric-translation and exact lattice/integer coupling- translation conventions remain distinct. Input/basis and metric translation generation, infinite domains, full-Coulomb fitting, K, executed Fock/energy arithmetic, basis/BZ/XC/BvK and occupied/overlap/density relaxation remain excluded. It supplies no new screening or symmetry permission, total relaxed-energy guarantee, bulk parity or speedup. MANTA remains experimental.

5.74 Exact finite metric lattice images

The optional metric assembly proof now encloses exact lattice-times-integer translations for every selected metric image, matching the coupling reference convention of Section 5.68. MetricImageDomain.each_index shares the original selector, ordering and rounded membership decisions with its numerical and derivative consumers. Its existing each adapter preserves those consumers. The numerical shifted shells and phases remain unchanged.

The shared enclosed_image and enclosed_shift compose exact stored lattice entries, integer labels and stored centers. The existing metric radial and representation proof accepts these interval centers; bloch_phase_enclosure uses the same enclosed translation. Thus both the integral displacement and phase include lattice multiplication and center-addition roundoff. No new radial, Fourier, Fock, eigensolver or symmetry implementation is introduced. The interval shell overload preserves the original molecular exact-center API. This reuses the Sun2017 finite-source, McMurchieDavidson1978 Gaussian and Rump2010 verification assumptions already documented in Sections 5.63 and 5.68.

The metric receipt exports metric_lattice_images_included; the RI owner exports assembly_lattice_images_included and requires it for assembly rank qualification. The assembly certificate namespace advances to v9 and the joint finite reference namespace to v2, explicitly binding exact lattice images. Raw source, metric, factor, density and Fock identities are unaffected; assembly response and dependent finite-source certificate identities change with the reference. Do not compare the old rounded-image certificate as if it had the same exact reference. The existing lower-triangle reconstruction and adaptive-rank stability conditions remain mandatory.

Count admission adds 128 scalar terms and eight times that kernel-work allowance per candidate metric shell image for the interval lattice product. Existing metric proof scratch includes the two three-coordinate interval stack records. All five caps still refuse before numerical construction. The shared phase and Gaussian proof costs remain charged by their existing planners.

This closes only metric image-vector generation for the selected finite set. The set is still chosen by rounded geometric filters; completeness of a mathematical cutoff domain is not proved. Reciprocal-vector and stored basis/ input generation, source screening, infinite tails, full-Coulomb fitting, K, executed Fock/energy arithmetic, basis/BZ/XC/BvK and occupied/overlap/density relaxation retain their exclusions. The metric and coupling now use the same lattice-product convention, but their physical support and fitted reference still do not establish a total relaxed-energy bound or bulk CRYSTAL parity. No performance claim follows. MANTA and the full #617 programme remain experimental and unfinished.

5.75 Evaluated finite Coulomb response

Finite-source admission now also requests the existing stored-map arithmetic audit at every evaluated density. The native fitting owner composes one arithmetic proof with its immutable finite-source certificate. For numerical J_hat and E_hat, exact stored-map J_A and E_A, and the exact finite fitted reference J_B and E_B of Sections 5.70-5.74, the triangle inequalities are

|J_hat-J_B| <= |J_hat-J_A| + |J_A-J_B|, |E_hat-E_B| <= |E_hat-E_A| + |E_A-E_B|.

The second terms are the existing fixed-family finite-source totals. The first terms come from either retained-mode Section 5.43 or paired-action Sections 5.38-5.39, including exact stored spin addition and k weights. The two arithmetic routes are exclusive. For paired actions without a separate metric-action request, existing finite column norms provide an adjoint L1 norm via ||A_j||_1 <= sqrt(n_aux) ||A_j||_2; the existing forward/ metric coefficient radius and executed adjoint radius are then propagated once. No source sweep, inverse, physical source or symmetry rule is added.

The fitting owner verifies source identity and outward membership of the exact weighted spin density in the declared finite family. Its interval spin sum, weight product, Euclidean radius and L1 sum do not assume the rounded input vector represents those exact values. Unresolved or exceeded family limits refuse. This is a conservative sufficient admission, with no tolerance repair. Shared per-build count admission reserves the additional validation/composition work (4096 scalar units per entry plus 4096 for composition) and 4096 bytes of receipt/scratch storage alongside existing arithmetic buffers. Shared Fock receipt storage retains its existing per-channel margin.

finite_coulomb_response binds source, finite reference, source certificate, actual stored density and weights. It exports disjoint source/arithmetic J and Coulomb-energy bounds, outward sums, arithmetic route and a content identity. The native Fock owner preserves the same receipt; RHF/UHF and RI RKS/UKS provenance expose it separately from their existing component ledger. Repeated substep metric, approximation, full-Coulomb or assembly bounds are not added. Rump2010 outward verification and the previously qualified Sun2017 RI maps supply the existing mathematical assumptions.

The original pre-SCF finite targets remain source-only budgets. An evaluated response bound is not automatically a new target guarantee or a relaxation certificate. It covers the raw computed J and Coulomb energy for the stored spin inputs. Restricted-density embedding, final Fock/total-energy assembly, reciprocal and basis/input generation, finite-set completeness, infinite domains, full-Coulomb fitting, K, basis/BZ/XC/BvK and occupied/overlap/density relaxation remain separate. The full-Coulomb ledger is not summed with this different reference. MANTA remains experimental; no total relaxed-energy, bulk CRYSTAL parity or general speedup claim follows.

5.76 Enclosed compressed columns for finite metric response

Let A be the exact stored compressed coupling graph and a_hat_j its executed unit-action column. The existing producer replay supplies per-entry complex modulus radii r_pj containing A_pj-a_hat_pj. The shared HermitianInverseAction.column_enclosure encloses each input by the rectangle [a_hat.real-r,a_hat.real+r] x [a_hat.imag-r,a_hat.imag+r] with outward endpoints. Empty or zero radii use the original exact-point graph. Invalid, nonfinite or negative radii refuse before certification.

Interval propagation of F* A_j, A_j-M F F* A_j and F F* A_j yields the same factor-score, residual, discarded-space and inverse-action scalar witnesses used by Sections 5.34, 5.72 and 5.73. Each witness retains its action-owner seal. The finite identities apply to every column in the input rectangle; interval dependency loss widens the bounds but cannot omit a source term. The Rump2010 residual/inverse principles and finite quadratic terms are unchanged. Full-rank metric response can use the resolvent; stable truncated and empty ranks retain their spectral treatment. No numerical inverse or new eigensolver is introduced.

The compressed finite-source certificate passes existing unit-action centers and replay radii to this shared API. Its construction discrepancy still adds that replay radius once to connect the exact stored A to the exact finite coupling reference B. Using the radius to enclose A in a coefficient witness is not a second additive source-error ledger. Factor, metric and coupling terms keep the same telescoping references as Section 5.70. The reported column norm can take the minimum of the graph-norm and enclosed-column proofs. The evaluated response of Section 5.75 remains source plus one executed-action arithmetic bound.

Preconstruction admission now reserves a column proof per existing streamed unit action and the shared two-pass coupling response. Column planning adds 128 scalar units and 256 bytes per auxiliary entry for the input rectangles, including public binding copies; its existing quadratic allowance covers interval products. No additional integral sweep or full auxiliary-by-AO-pair tensor is stored. Source, rank, Fock, PBC and symmetry semantics are unchanged. The exact finite reference identity is unchanged; certificate identities bind the new bounds. Target guards and all physical, input, assembly and relaxation exclusions remain. This does not establish total relaxed energy, general performance or bulk CRYSTAL parity; MANTA remains experimental.

5.77 Factored paired Coulomb arithmetic response

For the actual stored coupling T and retained factor F, use the periodic convention A=conj(T), P=F^T A, b=A d, rho=F^T b and c=conj(F) rho. The ideal stored-map response is J=A* c=P* rho. Let rho_hat and c_hat be the actual executed intermediates, and J_hat the executed adjoint action. The following exact finite identity separates the two metric stages:

J_hat-J = (J_hat-A* c_hat) + A* (c_hat-conj(F) rho_hat)            + P* (rho_hat-rho).

The first term is the existing executed adjoint replay error. The shared paired metric replay now also returns a Euclidean projection error radius and a coefficient reconstruction infinity-norm radius. The projection error includes existing forward-action and exact stored spin/weight input errors; reconstruction compares c_hat with exact conj(F) rho_hat using the actual stored rho_hat, independently of that input error. No projected factor is assumed orthogonal or numerically exact.

For nu >= max_j ||A_j||_1, pi >= max_j ||P_j||_2, eps_r >= ||rho_hat-rho||_2, eps_c >= ||c_hat-conj(F) rho_hat||_infinity and the existing adjoint arithmetic radius eps_a, an outward bound is

||J_hat-J||_max <= pi eps_r + nu eps_c + eps_a.

The immutable finite-source owner supplies pi during its already admitted column sweep. Shared column witnesses expose both ||F* t||_2 and ||F^T t||_2 envelopes, preserving complex transpose/adjoint distinctions. Explicit tile columns T use the former; compressed forward columns conj(T) use the latter, including their existing replay radii. Only their scalar maximum is retained; no complete projected coupling map or new integral pass is stored.

The fitting owner takes the minimum of this bound and its previous coefficient norm propagation. Both proofs enclose the same exact stored map. They are not added. Existing source, metric-only, approximation and full-Coulomb ledgers remain separate. Coulomb-energy arithmetic is unchanged. Empty rank and zero input retain zero bounds; nonfinite inputs, unsupported arithmetic modes, source mismatches and resource overages retain their strict refusals. This reuses Rump2010 outward verification and the paired RI conventions of Sun2017 and Neese2025 Eqs.29-36, with the displayed finite identity specifying the extension.

Column plans add 512 scalar units per squared auxiliary dimension and 128 per auxiliary entry for transpose projection and norms. Paired metric plans add 256 scalar units per factor entry, 128 scalar units and 128 bytes per auxiliary or retained-mode entry for reconstruction and norm/receipt storage; existing parent admission consumes those shared plans. The original scalar receipt margin covers composition fields. Finite certificate namespace v2 binds pi; the mathematical reference and numerical source identities remain unchanged. Shared Fock receipts and HF/KS provenance carry the projected audit flag and its bound. The Section 5.75 evaluated response consumes the tighter arithmetic radius once. Source-only targets retain their original meaning.

These are raw J/Coulomb-energy bounds at exact stored inputs. Restricted-density embedding, final Fock/total-energy assembly, input/basis/reciprocal generation, image-set completeness, infinite domains, full-Coulomb fitting, K, basis/BZ/XC/ BvK and occupied/overlap/density relaxation remain excluded. MANTA remains experimental; no total relaxed-energy, bulk CRYSTAL parity or speedup claim.

5.78 Scaled Euclidean metric-column envelopes

The residual/resolvent identities of Sections 5.34 and 5.72 require vector 2-norm envelopes for the projected column, residual and inverse action. Earlier proofs used absolute component sums, which are valid but can lose a factor proportional to the square root of the auxiliary dimension. The shared column owner now also evaluates scaled sums of squares of the real and imaginary rectangle endpoints. For a rectangle z, each component is the maximum absolute value of its two endpoints, so every enclosed complex vector is covered, including uncertain compressed columns.

Maintain a scale s and an upper bound u on the squared norm divided by s^2. For a nonnegative component x greater than s, update u to 1 + u (s/x)^2 and s to x; otherwise add (x/s)^2 to u. Every division, product and addition rounds outward. Zero components require no division. This avoids squaring the unscaled physical component. Subnormal ratios are still expanded outward under the existing gradual-underflow requirement. The L1 sum remains an independent fallback.

The square root of u retains the existing power-of-two upper bound. A libm square root, moved two representable values upward, may tighten it only if an outward product verifies that the candidate’s squared lower endpoint is at least u. No accuracy assumption about libm authorizes that candidate. The resulting outward product s times the accepted root bounds the Euclidean norm. The owner takes the minimum with its previous envelope. This is a verified norm evaluation under the Rump2010 arithmetic assumptions already used by the finite residual proof, not a new numerical RI algorithm.

The residual and solution use these Euclidean envelopes. Discarded-factor leakage uses the projected Euclidean norm; both complex transpose and adjoint projection norms are enclosed separately. The public coupling_norm retains its L1 interpretation, including its existing componentwise adjoint uses. Factor-score, spectral rank guards, finite resolvent denominator, quadratic coupling terms, source/reference identities and density families are unchanged. The certificate content binds the resulting tighter bounds. Empty rank keeps its previous spectral fallback. No physical integral or SCF operation changes.

The shared column plan adds 512 scalar units per auxiliary entry for the four scaled norm accumulations and verified roots. It needs only constant scalar scratch within the existing allowance; parent plans consume the shared count. No integral sweep or auxiliary-by-AO-pair tensor is added. Fock, PBC and symmetry ownership and all input, domain, fitting, exchange, assembly and relaxation exclusions remain. These finite-source bounds do not establish total relaxed-energy accuracy, bulk CRYSTAL parity or a general speedup.

5.79 Finite Coulomb response through evaluated assembly

Sections 5.70 and 5.75 define an exact finite fitted Coulomb reference and bound its J and Coulomb energy at the stored embedded spin density and weights. The existing assembly audit compares the evaluated Fock and energy with exact arithmetic on the supplied component matrices and scalars, including the actual reconstruction and Hermitian projection. Their common intermediate is the supplied executed J and Coulomb energy. Rump2010 outward triangle composition therefore gives

finite_assembly_F <= assembly_F + finite_response_J

finite_assembly_E <= assembly_E + finite_response_E.

The new reference replaces only the supplied Coulomb matrix and energy by that exact finite reference. H, XC matrix and energy, signed exchange matrices and coefficients, exchange correction matrices, nuclear energy, physical assembly density and weights remain the exact stored inputs. Thus the energy expression is the exact stored one-electron trace plus exact finite Coulomb energy, supplied XC and nuclear energies, and exact weighted exchange and correction traces. This is an explicitly mixed reference, not a bound against the complete physical Hamiltonian or a relaxed SCF solution.

Use the assembly-only radii, not combined arithmetic radii. The latter already include Coulomb arithmetic and may replace other stored components by their own exact numerical graphs. Adding them here would double-count Coulomb and silently change the reference for exchange or corrections. The optional new receipt remains separate; original assembly and combined arithmetic fields retain their meaning. The unlike full-Coulomb fitting ledger is not added.

The shared audit binds the immutable finite response to its source, zero transfer frame, actual spin density and weights. For restricted methods it reproduces the actual binary64 half-density embedding and records that input identity. The embedding’s discrepancy from ideal halving remains excluded. The public raw assembly API accepts a native RI Q result, checks its J and Coulomb energy against the supplied components, and refuses missing or mismatched source, frame, shape, density or weights. It cannot attach an unrelated bound using a caller-supplied scalar radius.

A reference identity hashes the held inputs, operation sequence, exact finite source reference and embedded-density/weight identities. A separate certificate identity also binds the source-response certificate, actual Fock/energy witnesses and all radii. Changing a witness changes the certificate; changing a held component changes the reference. Supplying additional independent exchange/correction arithmetic errors does not change this held-component reference or its certificate.

The shared assembly plan adds 4096 bytes for the optional receipt and digest state, plus 4096 constant and 512 per-entry scalar units for identity and composition work. Parent HF/KS plans consume this allowance before payload copies or chemistry. The adapter passes the existing Gamma result to the same audit; no extra integral evaluation, source, Fock builder, density epoch, eigensolver or symmetry authority is introduced.

This closes evaluated assembly for the specified finite-Coulomb reference. It does not change pre-SCF source-only targets. Non-Coulomb construction and response, ideal restricted embedding, input/basis/reciprocal generation, image completeness, infinite-domain/full-Coulomb fitting, basis/BZ/XC/BvK and occupied/overlap/density relaxation remain separate. MANTA stays experimental; no total relaxed-energy accuracy, bulk CRYSTAL parity or speedup follows.

5.80 Coulomb density embedding at the physical stored input

Restricted adapters supply two binary64 half-density spin blocks to the RI producer. Normal halving is exact, but halving a subnormal or a value near the normal boundary can lose a bit. Section 5.79’s original reference used that stored embedding. The new assembly reference instead uses the exact total of the physical stored spin density, while the source response still refers to its identified embedded input. Non-Coulomb components remain held fixed.

For each restricted entry D, reproduce h=fl(D/2). The sum h+h is exactly representable for every finite binary64 D under the existing gradual-underflow and round-to-nearest requirements. If h+h equals D, the difference is exactly zero, including signed zero. Otherwise the shared outward interval operations enclose D-(h+h), its multiplication by the stored k weight, and the sum of complex magnitudes. This gives delta >= ||d-d_tilde||_1 for the exact weighted physical and embedded totals. Independent spin blocks and a direct one-spin source need no embedding and have delta zero.

The finite source owner already bounds each stored coupling column and its distance to the exact finite reference column. Their outward sum gives a reference envelope beta >= max_j ||B_j||_2. Its existing reference inverse bound gives rho >= ||C0||_2, where C0 is the selected exact finite metric spectral inverse. The finite certificate now retains beta; its evaluated response exports beta, rho and the admitted embedded-density L1 limit L. All are source-owned and identity-bound, not caller-supplied error estimates.

For the Hermitian finite-reference Gram map K=B* C0 B, every entry has magnitude at most g=rho beta^2. The exact finite identities then imply

||J(d)-J(d_tilde)||_max <= g delta

|E(d)-E(d_tilde)| <= g delta (L + delta/2).

The quadratic term is retained. This is an input perturbation of the same finite reference, using Rump2010 outward arithmetic and the previously stated RI Gram convention; it is not a new source approximation. No assumption about variational cancellation or density signs is needed. L bounds the embedded input already validated by the finite source owner; the physical input need not be separately admitted to that family for this finite difference bound.

Compute the products only for nonzero delta and nonzero reference envelopes, so exact embedding and a zero Gram map add exactly zero without demanding an unnecessary large Gram-bound product. Unsupported arithmetic and overflowing nonzero bounds still refuse. Empty retained rank has zero reference inverse and hence zero embedding response. The assembly receipt adds this response once to its existing Coulomb-plus-assembly bound. It reports the weighted embedding radius, exactness flag, separate J/energy terms and inclusion of the physical stored Coulomb density.

The finite certificate namespace advances to v3 and the raw evaluated response to v2 to bind the reference envelopes. The exact finite source reference and source-only target meanings are unchanged. Assembly reference/certificate v2 uses the held physical density and exact finite reference; its reference no longer depends on the embedded-density identity, which remains recorded in the source response and certificate chain. Other held components, final witnesses and weights remain bound as before.

Finite-source preparation adds 128 scalar units per density entry to its shared bookkeeping plan. Assembly adds 16 bytes per k point for weights, 512 constant and 512 per-entry scalar units for the embedding proof. Existing parents use these plans before payload copies. No numerical source, integral sweep, Fock/eigensolver or symmetry permission changes. The assembled reference now includes Coulomb embedding; non-Coulomb embedding/construction/response, input and physical-domain completion, basis/BZ/XC/BvK and occupied/overlap/ density relaxation remain excluded. This is not total relaxed-energy accuracy.

5.81 Euclidean finite coupling-construction error envelopes

The shared coupling response in Sections 5.73 and 5.76 accepts a Euclidean bound for each column perturbation B_j-A_j. Previously its periodic producer summed the supplied component moduli into an L1 envelope. That is safe but loses auxiliary-dimension information. For component radii e_p >= 0,

||B_j-A_j||_2 <= sqrt(sum_p e_p^2) <= sum_p e_p.

The producer now uses the same scaled outward sum-of-squares helper as Section 5.78, retaining the original L1 result as an independent fallback. The helper verifies its root candidate by an outward squared lower endpoint; no correctly rounded libm or normal-only product assumption is introduced. This reuses Rump2010 outward arithmetic and the existing finite Gram response.

For explicit coupling, e_p is the existing tile construction modulus. For compressed moment graphs, it is the outward sum of the construction radius relative to the executed forward action and its existing replay radius. The latter addition is performed per component before the Euclidean norm, so the exact stored graph remains the intermediate reference. No compressed construction or replay error is omitted or counted twice.

The refined column error enters the existing coupling-response certificate, its finite quadratic term, total source J/energy bounds and reference-column envelope. The stored-factor and metric-only families, coupling L1 norm fields, projected norms, numerical maps, retained rank and exact finite reference identities keep their meanings. Finite certificate v4 identifies the new error aggregation; source-only targets and the separate full-Coulomb ledger remain unchanged. This is a tighter component enclosure, not a relaxation bound.

Preparation reserves 32 additional bytes per density entry for streamed norm state and 256 scalar units per auxiliary/density-entry pair, with the existing kernel multiplier. No extra integral traversal or full auxiliary-by-density error tensor is retained. Shared HF/KS parents consume this plan before payload allocation; exact/one-short caps retain strict refusal semantics. The source, Fock/eigensolver, PBC/BIPOLE and symmetry owners are unchanged. Physical source completion, non-Coulomb errors and nonlinear relaxation remain excluded, as in Sections 5.79 and 5.80. MANTA remains experimental.

5.82 Terminal finite Coulomb assembly targets

Complete RI RHF/UHF and KS fitting policies optionally specify paired finite_coulomb_assembly_fock_error and finite_coulomb_assembly_energy_error. Both zero disables the check; otherwise both must be positive finite values. An existing paired finite-source target is required, so pre-SCF preparation must supply the qualified source response. Malformed policies refuse during public or native planning before chemistry.

These targets use exactly the physical-density, other-components-held reference of Section 5.80. Assembly provenance sets source_targets_only false when this terminal target is requested; raw source provenance remains source-only. After the shared solver returns its actual terminal state and the existing confirmation/assembly audit finishes, the method compares its maximum-entry Fock and energy bounds with the requested targets. Equality passes; either strict excess refuses. Missing or unqualified finite physical-density evidence refuses. A stopped unconverged state is checked too; a successful target check neither asserts convergence nor replaces the normal convergence/confirmation indicators. Failure does not restart SCF or silently refine an already selected source.

The shared check records targets_requested, targets_met, fock_target and energy_target in the finite assembly receipt. A target certificate digest binds the existing evaluated certificate and both requested values. Reference, numerical source, physical density and actual witnesses do not change. Disabled targets preserve the previous certificate identity. The raw certify_periodic_assembly_roundoff API exposes the same check through finite_coulomb_fock_target and finite_coulomb_energy_target after constructing its receipt from the immutable evaluated Coulomb result and supplied witnesses.

The shared assembly plan charges 256 extra scalar/kernel units for validation, comparison and target digest; the existing receipt allowance covers its fixed fields. There is no new integral sweep, solver, Fock builder, density family or symmetry permission. Existing pre-SCF J/energy targets remain source-only and continue to guide only their existing bounded preparation searches. The new terminal policy is a separate non-relaxed admission against a delimited finite reference, not a bound for non-Coulomb construction, infinite-domain physics, exchange/XC source error, basis/BZ/BvK or occupied/overlap/density relaxation. No total relaxed-energy, bulk CRYSTAL parity or speedup claim follows.