Shared symmetry architecture and production-reduction gates

The shared group, space and diagnostic contracts are implemented, with molecular and periodic consumers. This page develops the rationale of the original 2026-09-06 design into the remaining method-integration gates. It does not enable a solver or change numerical acceptance thresholds. See the developer reference for implemented APIs, the capability status for legacy scope, and the chi symmetry reference for that consumer’s finite-source diagnostics. The October literature assessment records the new papers, inspected versions, supplements and their implementation implications.

Architecture

Use the shared symmetry module with numerical backend adapters. A unified Fock-engine refactor is not a prerequisite for these contracts. Symmetry has uses outside Fock construction: occupied/virtual spaces, density fitting, local correlation tensors, nuclear displacements and response equations. A common Fock engine is therefore neither necessary nor sufficient for this work.

The reusable object is an action of an admitted group on an explicitly named space. It is not a universal recipe for replacing every method’s computation by a scalar multiplicity. Share representation, orbit and transport machinery; require each operator and approximation to establish its own admissibility.

What can be reused now

Existing source	Reusable capability	Boundary that must remain explicit
symmetry_shared.py, cpp/src/symmetry_shared.cpp	Exact group/cocycle validation, compact block actions, orbit stabilizers, typed scatter/gather and operator/subspace audits	Shared evidence never authorizes production reduction. Native budgets are logical payload/work bounds, not fleet memory guarantees.
symmetry_core.py, symmetry_ao.py	Atom/radial matching, pure-shell Wigner rotations and compact molecular pure/Cartesian actions through l=6	Cartesian blocks share the periodic polynomial kernel and are generally not Euclidean-unitary. The legacy dense API requires pure shells beyond s.
cpp/src/periodic_ao_bloch_transport.cpp, kmesh_address primitives	Bounded native Seitz action, full atom-image shifts, Bloch phases, mesh compatibility, scalar time reversal; Cartesian and pure shells through l=6	One panel action is not an operator or state symmetry certificate. Spinor and magnetic actions require additional representations.
cpp/src/periodic_mean_field_state.cpp, cpp/src/periodic_orbital_sewing.cpp	Immutable numerical reference, metric/Roothaan audits, frozen/active/virtual mask leakage checks	Current sewing explicitly reports physical source symmetry as uncertified. A converged snapshot does not prove arbitrary-density operator equivariance.
symmetry_lattice_c.py, pair_resolved_truncation.py	Atom-pair/image action, geometric pair-distance domains, shell masks and orbit utilities	Lower-level atom matching carries fractional translations, but several callers still filter them out. Audit the actual function, not its historical symmorphic-only docstring.
bipole_symmetry_fock.py, native domain-aware J/K builders	Full masked builds, separate output/density/internal domains and representative reconstruction	Validate unreconstructed full builds first. Existing high-level mapping filters nonzero fractional translations. Changing domain membership changes the finite approximation.
periodic_k_symmetry.py	Phase-aware k-star density expansion and inverse folding; star maps restricted to an admitted operation subset, which may include glides and screws (the fractional translation travels in the atom-image shift, with no separate exp(-i k.tau) factor)	Its default star map is the zero-fractional-translation subset, for consumers that have not qualified fractional translations on their source; representative k points alone do not reduce every two-electron contraction. The multi-k GDF exchange wedge admits every operation on the SR/LR source and equal-shift operations on the classic source; other drivers’ wiring needs its own audit.
symmetry_integrals_reduced.py, symmetry_fock_reduced.py, symmetry_salc.py	Representative one-electron integrals, Gamma J/K prototype, point-group projection ingredients	Gamma, point-group and pure-shell assumptions cannot be silently generalized to multi-k chi.
symmetry_abelian.py	Complex one-dimensional irrep products, conjugates and bounded finite-ERI J/K block execution	Experimental opt-in adapter; full input tensor required, no integral skipping or production SCF routing, no antiunitary/projective character admission.
symmetry_quartets.py	Complete-source-audited molecular shell orbits, stabilizers and streamed representative-tile J/K	Experimental bounded shell schedule using full AO blocks. An initial all-ordered-tile audit is mandatory; repeated builds request representatives from the same immutable provider. No default solver or periodic routing.
symmetry_quartets_streaming.py	Pair-orbit/stabilizer enumeration without a retained quartet inventory	An opt-in schedule retaining O(order * n_shells**2) index arrays; complete source audit and arbitrary-density contractions remain quartic. Full AO transport, not irreducible-product or CC recoupling.
symmetry_quartets_native.py, cpp/src/symmetry_quartets.cpp	Native four-leg transport and arbitrary-complex-density J/K accumulation per representative tile	Provider-independent real-tile algebra. Python retains source admission, schedule and digest checks; native execution is opt-in and does not qualify a source, screen or state.
symmetry_quartets_batch.py	Independent density channels sharing each representative tile and angular transport	NumPy and native executors return per-channel J/K and residual bounds with one aggregate work receipt. Batch membership does not establish electronic-state symmetry or enable new integral sources.
periodic/chi/symmetry.py	Finite-torus actions using shared exact algebra checks, occupied-space action and native snapshot bridge	Dense diagnostic witnesses and count caps are not a scalable production representation or execution plan.
periodic/chi/pno.py	Compact translation-pair census, even-mesh stabilizers, conditional point-pair quotient	Orbit counts alone provide neither orbital phases nor local-space transport or production pair skipping.
periodic/ccm/symmetry.py	Cluster compatibility and independent small-system AO checks	Explicit supercell matrices do not scale to the target; WSSC/union-weight and chi finite-character Hamiltonians must retain separate operator identities.

Paths in this table are relative to python/vibeqc/ unless prefixed by cpp/. Reuse means tests and implementations can be consolidated incrementally; it does not mean all present contracts are interchangeable.

Why a single unconditional reduction recipe would be wrong

1. Geometry, operator and electronic state are different tests. A crystal operation can preserve the geometry while an image cutoff breaks the numerical Hamiltonian, as in the chi half-translation diagnostics. For a density-dependent Fock builder the relevant operator relation is equivariance, F[g.D] = g.F[D], with the appropriate covariant/contravariant AO transformations. Invariance of one final F[D] additionally requires an invariant density. A broken-symmetry UHF state can use a smaller subgroup than its geometry. Averaging a Fock matrix is not evidence that its producer satisfies either relation.

2. A localized orbital need not map to one other orbital. Symmetry can mix several occupied orbitals. Casassa et al. explicitly discuss the arbitrary orientations of Boys-localized MgO sp3 combinations. Their Eq. (3) allows matrix-valued actions within localized subsets. Independent pair domains and PNO thresholds need not commute with that mixing. Use a validated compatible gauge or covariant block spaces; a pair permutation is a special case, not the default theorem.

3. Different tensors have different actions. AO matrices, three-center factors, occupied-pair amplitudes and triples carry different index types, conjugations, momentum relations and stabilizers. Auxiliary-metric rank cuts, frozen-core masks, PAO/PNO/TNO projectors and near-degenerate retained spaces all need compatible transport. CCSD residual couplings and the occupied-Fock-coupled triples correction must survive reconstruction.

4. Gradients are derivatives of the same approximate energy. Force vectors and displacement representations can share symmetry machinery, but this does not supply missing Pulay, kernel, fitting, grid or correlation-response terms. A displaced geometry generally has a smaller symmetry group. An energy invariant at the high-symmetry geometry does not validate every displacement derivative, and zero forces on ideal MgO are a weak test.

Dovesi (1986), Sections 3-4, especially printed p. 1763, identifies both the reuse of symmetry relations across integral types and the loss of Fock symmetry under truncated Coulomb/exchange sums. Casassa et al. (2006), printed pp. 727-728, Eq. (3), gives the more general localized-space transformation. References: Dovesi, DOI 10.1002/qua.560290608, Casassa et al., DOI 10.1007/s00214-006-0119-z.

Shared contract and adapter responsibilities

The developer reference describes the current internal classes and functions. Their common responsibilities are:

• Group and mesh: geometry/basis identity, admitted subgroup, Seitz composition, integer atom images, reciprocal wraps, antiunitary flags and finite-torus compatibility. Preserve full integer shifts until the relevant character is evaluated. A fractional translation in a nonprimitive cell must not be discarded merely because older helpers call it nonsymmorphic.

• Space action: compact atom/shell maps and small rotation blocks provide AO actions and occupied/selected-space audits. Auxiliary, local-pair and response adapters must supply their own tensor and projector semantics. Transport blocks on demand. Avoid dense all-supercell AO matrices and explicit lists of every translated pair. Retain native allocation/work admission before materialization.

• Operator qualification: operator convention, source revision, numerical supports, screening policy and tolerances, plus state/subspace checks. Record what is established analytically and what is only numerically tested. Finite probe densities alone do not prove a universal bound. Density-dependent screening must also respect the chosen action and budget.

• Orbit execution: representatives, stabilizers, exact weights and typed scatter/gather, with consistent adjoints. Reconstruction must account for operations stabilizing a representative, not just multiply by orbit size. An admitted group and a transport map alone cannot authorize pair skipping.

• Method adapters: BIPOLE/four-center, GDF/RI, molecular SCF, chi local correlation and derivatives each declare their source and tensor contracts. Equivalent representation infrastructure must not erase different finite Hamiltonians, q=0 conventions, fitting metrics or derivative semantics.

This supports one implementation of the common mathematics with several operator adapters. It does not promise equal acceleration for all methods or systems. Identity-only execution is the correct outcome where no larger electronic/source symmetry has been qualified.

Implemented foundation

The shared native group/space interfaces and two source-distinct consumers, molecular integrals and periodic chi transport, are present. They reuse the same exact group validation and compact action contracts. Molecular Cartesian and periodic polynomial rotations also share one native kernel. The SALC consumer uses common group and conjugacy checks. Metric, selected-space, group-law and operator-leakage diagnostics build on these primitives.

The historical chi finite-source defect has a quantitative diagnosis and opt-in finite-source witnesses. The default historical source remains subject to refusal; those witnesses do not certify the full physical Hamiltonian or representative-only correlation. Similarly, a complete whole-mesh J/K producer does not by itself qualify symmetry reduction. These are distinct acceptance scopes, not reasons to reimplement the shared foundation.

Remaining production acceptance gates

1. Qualify a declared numerical source. For the selected method, identify the actual finite supports, screening, q=0 convention and density/state dependence. Validate unreconstructed J/K or fitted factors first, against an independent same-convention reference and nonsymmetric trial densities. A finite-source witness is not a default-Hamiltonian migration. Changes to support membership require separate physical/convergence acceptance. Keep scientific tolerances unchanged and retain an unreduced fallback.

2. Add and qualify method-specific tensor adapters. The bounded SR/LR GDF retained-mode diagnostic now connects actual fit batches to the shared subspace audit; it does not admit a subgroup or enable production work reduction. General GDF/RI factors, auxiliary metrics and rank cuts need index/conjugation/momentum contracts in addition to AO transport. Admit only operations preserving the chosen finite source. Reuse existing planner repairs and assess their independent acceptance before adding another implementation. Producer redesign, routing and cost decisions remain separate from the common mathematical layer.

3. Transport retained local spaces and contractions. Bind the actual occupied gauge and frozen/active spaces to PAO, auxiliary, PNO and TNO representations. Compare representative and unreduced MP2, CCSD residuals/energies, and coupled triples on small odd/even meshes. Include dense occupied mixing, fractional translations, degenerate retained subspaces, negative leakage cases and nonsymmetric trial densities.

4. Establish scalable execution. Where global factor/tensor construction remains, replace it with bounded local production data access. Validate increasing MgO meshes through vq, recording source and runtime identities, energy components, approximation settings, peak memory and actual contraction counts. Run 8x8x8 only after memory and small-mesh numerical gates pass.

5. Validate derivatives separately. Assemble chi RHF total analytic gradients using its own declared operator/support; compare with displaced-geometry total-energy finite differences and translation/rotation sum rules. Add other SCF backends and relaxed post-HF derivatives as separately validated capabilities. Symmetry transport can be shared throughout; derivative formulas remain method-owned.

Orbit-count reductions describe planner cardinalities. They do not establish CCSD(T) energy accuracy, memory feasibility or speedup. A production acceptance must include the actual data-access and contraction schedule, source/runtime identity, numerical reference and resource measurements. Keep experimental CCM/chi capabilities labelled until their method-specific gates are satisfied.

The stored molecular P0 mask now has a complete opt-in NH3/cc-pVDZ RHF acceptance calculation through vq, with identical-mask and unmasked ordered source controls, arbitrary complex non-Hermitian trial-density covariance, final evaluated-state closure and an ORCA energy comparison. The measured status records the source/runtime scope and actual work, timing and memory. This closes the bounded molecular SCF calculation gap; scalable producer certificates, method-owned dispatch and each additional source or derivative remain separate production gates.

Complex and non-Abelian tensor contracts

Keep four separate admissions: geometry/basis closure, numerical-operator equivariance, electronic-state invariance and retained-subspace covariance. For the shared coefficient action C' = U K^a C, density and operator laws are

D' = U K^a(D) U^dagger
F' = U^-dagger K^a(F) U^-1
K^a(U^dagger F_target[D'] U) = F_source[D]
U^dagger S_target U = K^a(S_source)

The last line is metric covariance, not Euclidean unitarity of a Cartesian rotation. No final-matrix symmetrization establishes the third line. Probe the raw builder on nonsymmetric densities, both spin channels and relevant complex directions. Full-source tensor selection provides stronger evidence for that fixed finite array, but still cannot authenticate the producer or its next density-dependent screening decision.

In the new finite-ERI adapter the convention is E[p,q,r,s] = (p* q | r* s). Its allowed blocks obey conj(chi_p) chi_q conj(chi_r) chi_s = 1; its J and K contractions are J[p,q] = sum E[p,q,r,s] D[s,r] and K[p,s] = sum E[p,q,r,s] D[q,r]. The common conjugate of the entire selection product is equivalent for the trivial target, but changing only one leg or using real eightfold ERI symmetry is wrong. For nontrivial operator/transition targets the variance and target character must both be declared. Scalar time reversal conjugates the tensor and its basis/labels; it is not another unitary row of the character table. Magnetic spinor corepresentations and projective little groups need separate adapters.

A general non-Abelian tensor implementation must provide all of:

1. Full irrep matrices and a common partner gauge. Express a space as direct_sum_alpha (multiplicity_alpha tensor V_alpha). Repeated copies admit dense mixing. Character traces alone do not fix the partners.

2. Clebsch-Gordan maps for each ordered, variance-aware product V_alpha^(*) tensor V_beta^(*) = sum_gamma N_ab^gamma V_gamma, with an explicit multiplicity index whenever N_ab^gamma > 1. Validate their isometry, completeness and intertwining equations against the actual representation matrices.

3. Reduced/full tensor conversions with documented normalization, partner ordering and duals. Scalar invariant operators have Schur form A_gamma tensor I_dimension_gamma in a unitary retained metric. A partner-degenerate state is a separate requirement for using this form for density-dependent Fock matrices.

4. Recoupling maps between (pq)(rs), (pr)(qs) and (ps)(qr), including conjugation, exchange signs and missing partner reconstruction. CC spin adaptation and residual terms cannot simply transpose a reduced array.

5. Resource plans: pair multiplicities are M_gamma = sum_ab m_a m_b N_ab^gamma. Reduced invariant blocks store sum_gamma M_gamma^2 scalars, versus sum_gamma dimension_gamma M_gamma^2 if all equal partner blocks are retained, before the larger unsymmetrized storage. Add CG maps, multiplicity gauges, recoupling workspace, indices and full-tensor temporaries. Charge both representations when conversion needs them simultaneously; benchmark conversion as well as contractions.

The C3v formulas in Hellmann and Gauss are a first oracle for this interface, not a substitute for it. Keep non-Abelian execution unsupported until these maps and complete small CC residual comparisons exist.

MANTA/BUPO and unified Fock interface boundary

The subsequent implemented consumer is ShellQuartetPlan, which combines spatial shell actions with the eight permutations of real molecular AO ERIs. It transports full shell blocks and therefore also admits noncommuting C3v actions, without implementing non-Abelian reduced products or CC recoupling. It enumerates every distinct ordered shell quartet once and retains its representative transporter and stabilizer. Stabilizers constrain the whole angular tensor; they are not scalar multiplicities to apply to a density.

Admission evaluates every ordered quartet of the finite provider source and compares it against the reconstructed representative, including stabilizer checks. This setup cost is explicit. Repeated builds evaluate representatives only, check their snapshots and contract every required density block. The complete-source Frobenius residual bounds each J/K reconstruction error by that residual times the density Frobenius norm, excluding arithmetic roundoff. This gate does not assume an invariant density or symmetrize a result. Source changes require new admission; incomplete or incompatible sources retain an explicit unreduced schedule.

The initial MANTA adapter consumes the P0 libint ReferenceJKBuilder source, its exact shell snapshot and a private reusable tile session. The provider owns kernels, normalization and its zero-precision screening policy. Symmetry owns the schedule, four-leg inverse-action transport and J/K digestion. Calls on one plan are serial because its workspace is private but reusable. Integral calls, requested primitive quartets, null tiles, contraction terms and transport terms have separate counts. The bounded all-shell inventory and initial complete audit are not a scalable producer-side proof or a replacement for MANTA’s production planner. The complete factory reservation precedes radial/coordinate snapshots, group construction and the retained native tile session. Stabilizers use one byte per group operation with eight permutation bits, including when many operations fix every shell. Logical reservation, retained array payload and whole-process RSS are distinct measurements.

The optional streaming schedule replaces the complete retained quartet inventory with unordered shell pairs X = {(a,b): a <= b}. For each G-orbit representative a in X, its stabilizer H_a acts on possible right pairs. Within a right-pair orbit G/H_b, these are the double cosets H_a \ G / H_b. Their cardinalities depend on H_a intersect t H_b t^-1, as in Davidson (1975), Eq. (11). They are computed from the actual permutations, never from an Abelian constant multiplicity.

Pair exchange needs another quotient. Different G pair-orbits are ordered; within one orbit, a transporter taking a to b is inverted to bring the exchanged left pair back to a, and its right partner is minimized under H_a. This is the inverse-double-coset redundancy discussed after Davidson’s Eq. (17). A separate Burnside census checks the number of quartet orbits:

N = sum_g (fix_X(g)**2 + fix_X(g*g)) / (2*order).

The exchange term uses g*g, including for non-Abelian groups; replacing it with fix_X(g) miscounts. Each yielded representative expands to at most 8*order ordered members with the full angular stabilizer. Only one such orbit is live in the iterator. Shared real pure/Cartesian actions still transport every angular leg; pair indices do not supply partner functions or non-Abelian reduced tensors.

An ordered digest of representative keys and values replaces the stored per-quartet digests. It is checked before either completed J/K target is returned; a changed representative produces no partial result. It can detect a changed generic callback only on evaluated representatives. As before, reuse requires an immutable producer; a digest is not a proof of source equivariance. Every ordered source tile and every representative stabilizer is still checked during initial admission. Neither the memory improvement nor geometry closure licenses sampled admission or screening.

Streaming regenerates the index schedule during each build, so its memory saving can cost time. The stored schedule remains the default. A future automatic policy must account for setup reuse, memory and measured execution cost without changing the admitted source or scientific tolerance.

The optional native executor compiles angular transport, permutation indexing and J/K accumulation together, with one call per representative. It consumes the existing provider tile and admitted topology; it owns no integral kernel. Inverse coefficient blocks W=U^-1 act on all four legs, then real-ERI permutation axes select the target indices. It accumulates J[p,q] += E[p,q,r,s] D[s,r] and K[p,s] += E[p,q,r,s] D[q,r] directly for arbitrary complex density matrices. Neither Hermiticity nor state invariance is assumed, and no final matrix projection repairs the result.

Native construction snapshots density and inverse blocks. Tile/member buffers are borrowed only during a serial call. The kernel validates shapes, indices, finite values and checked resource arithmetic, records executed contraction/transport terms and poisons its partial targets after a native failure. Completed vectors own their storage. Its low-level output is still only algebra: it cannot authenticate the supplied group, tensor variance, complete orbit coverage or provider. The Python adapter checks the source, representative digests and full execution census before returning either immutable J/K target. Complete admission remains on the established path.

The combined live-plan and native reservation is checked before packing angular arrays or creating the accumulator. It includes borrowed/owned density panels, targets/exports and tile scratch. MANTA’s engine inventory remains charged through the source plan. This boundary allows later MANTA tile providers to reuse the executor only after their own source admission; new fitting, range-separated or complex kernels are not implicitly enabled.

Independent densities can share this execution through jk_batch with shape (batch, n, n). Each channel retains the same contravariant density and covariant J/K conventions: D_b -> U D_b U^dagger, and U^dagger J_b[g.D_b] U = J_b[D_b], likewise for K. Channels may be complex, non-Hermitian, zero or symmetry-breaking. There is no relation assumed between channels. Each representative is evaluated, checked and rotated once; its ordered values contract into every channel before the next tile. For B densities, source calls and transport terms equal one build, while scalar products and block contractions equal B single builds.

The batch owns its density snapshots before calling the producer, returns immutable per-channel targets and computes one source reconstruction bound epsilon_source * ||D_b||_F per channel. An invalid channel, nonfinite arithmetic or late source/digest failure refuses the entire batch. Logical memory grows with B*n**2 for densities/targets, not B*n**4; all inputs, snapshots, exports and the live source plan are admitted before allocation. No channels are silently dropped or split into independent source epochs.

The method consumer owns batch order, spin normalization, exchange scaling, energy assembly and response interpretation. In an unrestricted HF example, J_total = J_alpha + J_beta and F_sigma = h + J_total - K_sigma; no extra closed-shell factor enters the batch kernel. The work receipt belongs to the complete batch, so attaching identical copies to each spin and adding them would overcount source work. Frozen Fock actions must capture the completed targets, with solver generation/spin annotations kept separate from the admitted source identity. This is a reusable linear contraction boundary, not production SCF dispatch or a nuclear-gradient implementation.

Target selection now accepts an immutable per-density tuple of j, k or jk. Omitting it preserves both outputs for every density. Returned J and K panels are compact, in input-channel order, with explicit j_channels and k_channels mappings. An absent target family has shape (0,n,n) and owns no matrix payload. Every input must request at least one target; no dummy channel or unrequested contraction is evaluated to complete an output shape.

For an unrestricted build, the caller can supply [D_alpha+D_beta, D_alpha, D_beta] and request ('j','k','k'). This executes three contractions per ordered integral instead of the four required by [D_alpha,D_beta] with both outputs. Integral calls, stabilizer handling and angular transport remain identical. It owns an extra total-density input, so fewer output panels do not imply a lower total memory reservation. The method still owns the addition, spin factors and physical Fock assembly.

Count-only admission charges every input density and only the requested target panels. Native construction checks a minimal one-target-per-density reservation before inspecting selectors, then checks the exact selected reservation before payload scans or snapshots. The selector itself is owned by the accumulator. Error bounds remain indexed by input channel; a compact output uses the bound at its channel mapping. Source/digest failure or a nonfinite requested contraction still refuses the whole batch. Omitting an unused target is an algebraic request, not a screening policy, electronic selection rule or strengthened source certificate.

Native selected-target execution now resolves permutation-dependent matrix strides once per member and target family, sharing matrix addresses across the requested density channels. The scalar loop visits only the selected family and preserves each output’s contribution order. It needs no permuted tile, added density snapshot or target panel. Scalar-shell tiles bypass the four-axis loops with direct matrix addresses. Both-target requests retain paired J/K accumulation. This specializes the existing contraction algebra; it does not change the admitted source, orbit schedule, error bound or cost-based dispatch policy.

This is a proposed integration contract, not another integral engine. The planned MANTA producer owns recurrence, kernel evaluation, quadrature, screening, CD/RI construction and BUPO hierarchies. Symmetry owns admitted actions, tensor variance, representative/stabilizer maps and selection metadata. Fock unification owns source selection, density/spin dispatch, energy assembly and fallback. The coordinated P0 reference provider and the MANTA engine and scientific contract are now present. They remain owned by the integral engine workstream; this adapter does not duplicate their kernels. The operator-contract and operator-consumer files remain absent here. Their workstream versions were inspected read-only for the interface names below, which remain proposed integration boundaries.

Existing/proposed boundary	Symmetry integration
Proposed KernelRequest, IntegralTileView, TensorLayout	Retain exact shell images, AO order, axes, strides and conjugation. Share small shell actions and variance-aware selection; do not attach scalar orbit weights without stabilizer resolution.
Proposed NumericalSourceContract and BuildReceipt	Bind source, support, screening, auxiliary rank and qualified symmetry-plan identities. The shared OperatorContract remains a compact declared projection, not a replacement or authentication token.
Proposed BuildRequest, DensityEpochView, prepare/execute	Qualify the complete spin/k density epoch and finite schedule. Integral tiles may expire at the next provider call; retained symmetry blocks and completed J/K panels must own their data.
Existing molecular JKBuilder seam	Future admitted providers feed direct consumers here. The complex-Abelian consumer retains dense-source allowed blocks; the quartet consumer streams tiles after a capped complete source audit. Neither is the production data model. Initial libint-backed P0 capability must not be relabeled as native MANTA kernels.
Existing HermitianOperator, FockOperator, OperatorMetadata in the Fock workstream	Method assembly creates a fixed density-epoch operator. Source receipts stay beside the method result; generation/spin/k/symmetry annotations do not establish source symmetry. Never capture a live mutable builder during a solve.
Draft FockBuild, FockChannel, SourceProvenance and OperatorFrame in the Fock workstream	A method assembles compact J/K results into physical spin/k Focks and its own complete energy before constructing an immutable evaluated state. Preserve separate source, density, energy, frame and solver-role identities. The draft is not imported or a landed dependency here; identity completeness and ownership need independent acceptance.
Existing explicit X, S, reduced_diagonal boundary	Retained-space transport must bind the actual metric frame and its column order. Supply diag(X^dagger F X) for transformed callback Davidson; do not infer it from an AO diagonal or infer partner alignment from degenerate eigenvectors.

The evaluated-state boundary must cover energy components/corrections and operator kind as well as D/F values. A source digest or build ID alone is not a symmetry certificate. A density batch constructs different operators; applying one fixed operator to multiple vectors is a different operation. Complex non-Hermitian transition densities remain valid contraction probes but cannot become physical Hermitian FockChannel states without the method’s additional requirements. Owning immutable results does not establish metric, occupied/frozen or retained-subspace covariance.

PBC integration additionally retains BIPOLE’s physical AC/BD exchange domains and wider K output support. It must not reuse the refused legacy projector. Carry actual k/quadrature weights, full AO image gauges and separate density epochs/output mappings, together with finite domains, screening and Ewald zero-mode/Madelung conventions. Validate raw unreconstructed J/K/XC before any Hermitization or symmetry reconstruction. A saved crystal Fock passing an operator/solver crossing establishes neither source equivariance nor finite-source convergence. The present molecular executor does not admit these periodic producers. Snapshot copies must be included in the producer’s memory reservation before any bounded MANTA integration.

The operator consumer inventory deliberately leaves auxiliary metrics, natural-orbital/PNO selection and non-Hermitian CC Jacobians outside Fock eigensolver policy. Preserve those distinctions. Non-Hermitian trial J/K results used here test the linear contraction; they must not be passed to a Hermitian eigensolver. Conversely, a converged projected eigensolve does not qualify the discarded space or its symmetry.

MANTA must provide a source descriptor binding kernel and conventions, geometry/radials, scalar/spin type, full finite image supports and weights, screening rules, approximation thresholds, auxiliary metric/rank or CD pivot policy, and derivative support. Batch outputs must identify their ordered legs, conjugations, auxiliary gauge and momentum labels. Return evaluated and screened shell-quartet/three-center counts, contraction counts, retained and peak workspace bytes, and timings separated by planning, evaluation, transport and contraction. Current Python ERI value counts are not native quartet-evaluation counts.

MANTA must consume candidate operations admitted on that source, compact metric-aware shell actions, representative keys with stabilizers, selection rules and typed gather/scatter destinations. Only a producer capable of honoring these requests may skip integral calls. Exposing a full array and blocking it afterward, as in the complex-Abelian increment, saves storage and contractions but no integrals. The later quartet consumer skips representative calls only after its complete finite-source audit. The same batch interface must serve fitting, SCF, correlation and response; it must not encode a symmetric SCF density as a universal premise.

For BUPO, the 2025 paper, Sections 2.4-2.6, distinguishes terminating natural-center multipoles from truncated translated expansions and builds enclosing spheres around bounded objects. A geometry symmetry does not imply that a clustering/tie-breaking algorithm produces an equivariant bubble tree. An admitted action must preserve leaf membership, expansion centers, radii/extents, child-parent relations, multipole orders, translations and each near/far decision. Rotations mix multipole components. Stabilizers act on whole multipole blocks. Unequal trees require a qualified common representation or rejection to the full source; silently rebuilding a larger symmetric domain changes the source. Density screening must obey the same orbit-consistent rule in reduced and unreduced builds (including any orbit maximum bound). A posteriori averaging cannot repair a source with unequal thresholds or bubble membership.

For multi-k batches, require Seitz composition and the lattice cocycle, unreduced integer atom-image shifts, reciprocal wraps, AO-centered Bloch phases, k-star weights and momentum conservation before any orbit execution. Preserve distinct CCM union-weight/WSSC and chi finite-character Hamiltonians, supports and weights. Auxiliary, occupied/virtual and frozen spaces, PAO/PNO/TNO projectors must intertwine the same admitted actions; dense occupied mixing can couple several pair domains. Rank cuts crossing a degenerate multiplet or noncovariant local domains must reject reduction.

The next production gates are therefore producer-side admitted integral batches and cost-based dispatch, followed by retained factor and correlation consumers. Gradient and stress receipts remain separate for every source: response, Pulay, fitting/metric, grid, cutoff/support and strain derivatives must be supplied and tested at displaced geometries or strained cells.

Fixed whole-orbit screening of the audited molecular source

QuartetScreening adds a density-independent omission policy to the stored P0 molecular quartet audit. It requires the immutable native reference producer and tile session. A custom callback’s self-declared source identity cannot justify skipping future reads. Stored plans use retained orbit norms; streaming plans can build a compact certificate during the same full audit. Primitive screening precision stays zero. A null tile contributes its observed numerical zero to the finite source audit, without becoming a certified analytic zero.

For each complete ordered shell orbit O, admission records b_O = sqrt(sum_{t in O} ||E_t||_F^2) and the corresponding reconstruction residual r_O. No dense AO four-index tensor is retained. The stored policy sorts by (b_O, representative) and omits whole orbits while their aggregate Frobenius norm stays within an explicit maximum_omitted_norm. The streaming policy visits orbits in deterministic pair-stabilizer schedule order and omits each orbit if adding its norm fits the remaining aggregate budget. It can choose a different mask and does not claim the stored policy’s smallest-first ordering. The mask is fixed before any trial density is supplied. A per-representative cutoff, average bound or state-derived permutation mask is not this policy. In particular, nonunitary Cartesian transport prevents inferring b_O from a single representative and orbit multiplicity.

Let M denote this complete-orbit mask. The screened reference is the actual ordered source M E; the executed tensor reconstructs only retained orbits. Its difference from the unscreened source has disjoint retained and omitted supports, hence the exact-arithmetic bound is

B_tensor = hypot(sqrt(sum_retained r_O^2), sqrt(sum_omitted b_O^2))
||J_exec[D] - J_E[D]||_F <= B_tensor ||D||_F
||K_exec[D] - K_E[D]||_F <= B_tensor ||D||_F

The bound against the identical screened source uses only retained residuals. Floating-point norm, transport and contraction roundoff are separate. The API does not change the source acceptance tolerance or pick an automatic screening tolerance. The method must accept the resulting density-specific matrix bounds and form its own spin-weighted Fock and energy bounds.

For coefficient action U, D' = U D U^dagger, while F' = U^-dagger F U^-1. Complete-orbit masks preserve the same tensor equivariance as the audited source within its measured residuals. Tests therefore compare U^dagger F[D'] U with F[D] on arbitrary complex, non-Hermitian trial densities. No electronic-state projection, matrix Hermitization or Euclidean-unitarity assumption enters this admission. Electronic-state and retained-subspace symmetry still require separate tests.

The certificate binds the producer, full ordered-source bytes, group and inverse shell actions, audit tolerance, actual mask and deterministic policy. Different masks identify different numerical sources; policies choosing the same mask have the same screened-source identity but different policy IDs. Result bounds follow input density order, with existing explicit compact J/K output maps. The JSON-compatible receipt excludes live source objects and densities. A Fock adapter may copy the fixed receipt into source provenance, keeping density-specific bounds beside the evaluated build. Neither record certifies an electronic state or a solver’s retained frame.

The streaming certificate retains one bit per orbit and one digest over the retained representatives. Each later build skips omitted representatives before calling the producer, reconstructs retained members, then checks that digest before exporting any target. The audit identity binds all ordered source bytes, shell actions, group multiplication, tolerance and schedule; it is independent of the requested omission budget. Equal masks within that audit keep the same source identity when the cap changes. Stored and streamed audit identities are distinct; no cross-schedule identity equivalence is claimed. An explicit screening=None still builds the full audited source.

The live schedule retains O(|G| n_shells**2) index arrays plus at most one quartet orbit. The additional mask uses ceil(N_orbits/8) bytes, so its worst-case asymptotic storage is still quartic in shell count. There are no retained per-quartet member lists, norms or hashes, and no global sort workspace. Count-only admission includes both mask construction buffers and the attached certificate even when a later build requests no screening. The complete ordered audit is still mandatory. This increment reduces retained metadata; it does not supply scalable producer-side prescreening.

MANTA’s current IntegralProvider source identity and evaluate/evaluate_batch seam does not supply a general rigorous omission certificate. Its narrower extent and BUPO ledgers cannot be relabeled as one. The eventual producer certificate must cover the actual kernel, ordered tensor legs, immutable finite domain/weights, primitive and density screening, auxiliary metric, rank/pivot policy and operation-compatible bound closure. Symmetry consumes those bounds and supplies representatives, stabilizers and mask closure; MANTA owns their computation, screening implementation and bubble hierarchy. For density-dependent screening, both reduced and unreduced builds must use the identical policy and transform its density-dependent bounds covariantly.

This implementation admits real molecular ERIs with complex densities. It does not admit complex integral sources, antiunitary operations, RI/CD factors, local projectors, periodic/BIPOLE/CCM/chi sources or nuclear derivatives. Existing Seitz/image/phase/k-weight/momentum and finite-domain requirements remain prerequisites for any periodic extension. In particular, BIPOLE physical AC/BD and wider exchange support cannot inherit a molecular mask. Reusing a certificate at a displaced geometry or on a different audit is refused. Gradients and stress need their own response, Pulay, fitting and support/mask derivative acceptance. Production routing remains disabled.

Producer bounds before integral admission

MolecularCoulombBounds binds MANTA’s existing absolute-product envelopes to an exact immutable molecular P0 source and basis snapshot. It calls the native pair-envelope producer, owns immutable Q/V arrays, and retains source and normalization identities. It implements no integral recurrence, Gaussian envelope, bubble hierarchy or alternative BUPO engine. The envelope producer owns the mathematical Q/V bounds and their numerical evaluation.

For every AO product in shell pair AB, MANTA supplies Q_AB >= integral |phi_a phi_b| and V_AB >= sup_x integral |phi_a(r) phi_b(r)| / |x-r| dr. Therefore

|(ab|cd)| <= min(Q_AB V_CD, Q_CD V_AB)
b_ABCD = sqrt(n_A n_B n_C n_D) min(Q_AB V_CD, Q_CD V_AB)
b_orbit = sqrt(sum_members b_ABCD^2)

The adapter takes outward positive products and combines every ordered member bound, including nonunitary Cartesian AO actions. Positive underflow never becomes a certified zero; overflow refuses admission. These are analytic full-Coulomb integral bounds with floating-point headroom, not interval bounds on libint arithmetic. An extent’s chosen radius is not a finite support cutoff here: only full-space mass/potential bounds are consumed. The current adapter admits only real molecular, zero-derivative, zero-primitive-screening P0.

An explicit bounds= argument on spatial streaming qualification lets the fixed schedule-order policy omit an entire orbit before reading any of its integrals when adding its bound fits the tensor omission budget. Every retained member and stabilizer is still numerically audited with the original source tolerance. Bound queries, bound construction, skipped tiles and actual integral calls are reported separately. No state density enters this decision.

A screening bound is not an equivariance certificate. The resulting plan qualifies the retained tensor, and its omission norm is an upper bound rather than a measured norm. The same Fock/energy contraction bounds apply, excluding arithmetic roundoff. An unmasked build from this partial audit is refused; callers must use the unreduced producer or perform explicit full requalification. A changed basis, source, kernel or support cannot reuse the bound certificate. Stored and completely audited streaming paths remain unchanged fallbacks.

This avoids a complete integral audit only when the envelopes can omit orbits. It still visits every ordered member’s bound, retains the compact orbit mask, and audits every retained member. Tighter kernel-specific or hierarchical bounds, producer-certified equivariance, default dispatch, periodic finite supports, fitting/rank cuts, local projectors and nuclear derivative/stress admission remain separate work. The shared FockBuild schema is unchanged; the method receipt explicitly labels retained-orbits qualification and the producer-bound error domain. A molecular certificate cannot qualify distinct CCM/chi sources.

Hierarchical producer-bound candidates

MolecularHierarchicalBounds consumes MANTA’s existing BubbleTree, native unordered BubbleTraversal and BubbleErrorModel(order=0). It implements no clustering, radial envelope, multipole kernel or BUPO digestion. Every node carries a uniform envelope over all descendant AO products. For disjoint far nodes A and B the entry bound is

min(Q_A V_B, Q_B V_A, Q_A |T_00| Q_B + epsilon_AB)

Here epsilon_AB is MANTA’s complete compact-remainder, Gaussian-penetration and full-moment-tail ledger. The interaction estimates a magnitude only; no approximate integral replaces a retained integral. Near/always-near nodes use Q/V, and a native radial overflow leaves Q/V available. Geometry alone, the compact remainder alone, or a pointwise extent threshold is insufficient. Analytic approximation bounds remain separate from arithmetic roundoff.

For a node define W_A = sum_pairs (2-delta_ab) n_a n_b. Its self block contains W_A^2 ordered AO entries and a disjoint A/B block contains 2 W_A W_B. The node’s uniform entry bound times the square root of this integer count gives a full tensor bound. The native traversal supplies a disjoint exhaustive pair partition; accepted candidate blocks cumulatively spend the fixed tensor budget. Rejected far candidates descend using MANTA’s traversal, eventually retaining a near block.

The native tree is deliberately not assumed equivariant. The symmetry plan omits an orbit only when every ordered member is covered by accepted blocks. This intersection can discard some candidates; it never completes an orbit by silently adding uncovered members. Per-member bounds retain Cartesian component sizes and the usual aggregate cap. All retained members/stabilizers still undergo numerical source qualification. A partial plan requires its exact fixed mask and preserves the unreduced/full-audit fallbacks.

The adapter stores accepted node pairs and canonical-pair ancestor chains. An orbit-local cache reuses covered bounds only for identical unordered real-ERI shell-pair pairs. It does not identify spatially related members, whose bounds can differ. The cache holds at most 128 entries, then falls back to actual coverage queries; it is discarded after each orbit. Every member still contributes its bound in the original accumulation order. Empty candidate sets need no ancestor-map lookup. Numerical candidates, whole-orbit masks, upper rounding and source identities are unchanged. Receipts distinguish native node-bound/error calls, membership lookups, integral reads and contractions. Native construction stays bounded; worst-case candidate storage remains quadratic in shell-pair count, and orbit iteration still occurs. No subquartic planning or production speedup follows from using a tree. Tighter candidates may reduce numerical work, but dense systems need measured cost-aware dispatch. The molecular full-Coulomb P0 source, shared FockBuild schema and all separate periodic/retained-space/derivative gates remain unchanged.

Molecular native-host construction increment

The next gap after the fixed-source screened SCF experiment was consumer integration. The quartet algebra already transported full shell tensors and contracted arbitrary densities, but a standalone SCF diagnostic did not make that source usable by the native method drivers or the common evaluated-state object. symmetry_molecular_hf now supplies those two explicit boundaries.

Literature-to-code capability	Existing foundation	Current consumer and remaining gap
Davidson representatives/stabilizers and Dupuis-King shell transport	P0 qualification, stored or streaming orbits, selected density channels, producer-bound omissions	Real native RHF/UHF JKBuilder consumes fixed masks; producer equivariance/support attestations and cost-aware default dispatch remain missing
Pitzer/SALC and non-Abelian partner representations	Metric-aware Cartesian/pure AO actions and full-tensor C3v reconstruction	No irreducible-product CC tensor storage, multiplicity recoupling or reduced/full CC conversion is inferred from HF admission
Complex Abelian tensor variance	Complex-density quartet and finite-ERI block contractions	Real native SCF bridge explicitly refuses complex D; complex integral and antiunitary method admissions remain separate
Cholesky/symmetry analytic gradients and periodic symmetry literature	Source, state and retained-space contracts are distinct	RI/CD ranks, local projectors, periodic finite supports, gradients and stress still need method-specific source and derivative acceptance

ScreenedMolecularJK never turns a geometry label into a numerical-source certificate. It consumes the existing exact native P0 producer through QuartetScreening, with a complete audit or producer-bound omissions and a retained-member audit. Stored and streaming schedules retain that admission. The mask is fixed over SCF iterations; method convergence cannot loosen it. Native RHF uses one fused J/K traversal. Native UHF consumes the selected (J[Dtotal], K[Da], K[Db]) batch through JKBuilder.build_jk_uhf for NOITER, ordinary iterations and the terminal rebuild. The explicit HF state adapter consumes the same selected batch. The generic joint-build default preserves the original J0/K0/K1 cache slots and call order. Stateless stability energies and Hessian response keep their separate calls. No new integral recurrence or bubble hierarchy is implemented here; producer bounds/support and their closure belong to MANTA.

The method adapter forms F = H + J[D] - K[D]/2 for RHF and Fspin = H + J[Da + Db] - K[Dspin] for UHF. Its energy uses total density for one-electron/Hartree terms and the appropriate closed-shell or spin exchange factor. Owned H/D and explicit S/X frames feed the shared FockBuild; physical and solver-modified operators retain their existing distinction. Per-density matrix/energy bounds are separate from immutable source provenance. This is an explicit molecular integration, not a migration of every native SCF result or output path to FockBuild, and not default runner routing.

A method may supply a rectangular retained frame, but doing so does not prove that the group preserves that space. No new orbital truncation or symmetry projection occurs. Multi-k Seitz/image/wrap/phase/weight and momentum contracts, CCM versus chi finite Hamiltonians, auxiliary/local projector covariance, response, Pulay, fitting and support derivatives are unchanged acceptance requirements for their respective consumers.

The joint UHF boundary returns separate physical J, K-alpha and K-beta matrices, leaving the method’s spin factors, energy, one-center corrections and shared FockBuild assembly intact. An optimized producer must admit all channel buffers and independent density-dependent bounds before work. It may reuse integrals or auxiliary factors, but must preserve the actual source and screening policy for each requested channel. MANTA’s existing implementations inherit the unchanged sequential default until they supply such an override; this consumer does not implement another integral engine. The symmetry adapter uses its immutable whole-orbit mask for every channel and accepts independent, noninvariant spin densities. Batching certifies neither electronic-state nor retained-subspace symmetry. No periodic finite source or derivative is admitted through this molecular real-AO interface.

Cost routing within one qualified molecular source

The next production increment adds an ordered fallback for the exact fixed mask and an opt-in measured route choice. It does not add a new symmetry identity. Davidson stabilizers and Dupuis-King full-shell reconstruction already supply the reduced candidate. MolSym’s RHF supplement Table S1 motivates measuring execution: fewer representatives need not mean less time. The ordered candidate evaluates every retained ordered tile through the same MANTA P0 session and contracts the same selected density channels. It skips omitted tiles and applies no angular reconstruction or density projection.

Producer qualification remains deliberately narrow: exact immutable native ReferenceJKBuilder construction fixes real, molecular, full-Coulomb, derivative-zero, primitive-precision-zero support and shell normalization. The adapter checks each tile’s provider, kernel, range parameter and identity. The qualified basis and complete or retained-only numerical audit bind the mask. A copied source string, geometry group or caller declaration cannot qualify another producer. The new qualification record describes this existing evidence; it is not a producer-issued algebraic equivariance seal and does not eliminate the retained quartet audit.

For each requested channel signature, calibration contracts fixed arbitrary nonsymmetric trial densities through both routes, alternating their order. The discrepancy must satisfy the retained reconstruction residual times the density norm plus the unchanged source tolerance. The omission allowance cannot excuse cross-route disagreement because both paths omit the same entries. Calibration is separate from physical build counts and records all integral, contraction, transport, logical memory and timing costs. At most five native HF signatures and five paired probes per signature are retained. Unmeasured signatures, no measured advantage and insufficient expected reuse choose ordered execution. Native reconstruction wins only if the fastest ordered sample minus the slowest reduced sample, times expected remaining builds, exceeds the entire measured calibration time. This conservative heuristic is not a future timing guarantee. Already-paid qualification cost is reported separately; pre-qualification dispatch remains missing.

Execution decisions live in MolecularJKReceipt, outside the fixed screening receipt and FockBuild source. Numerical contraction order can change FockBuild bytes and hence its identity; source/policy identity and physical agreement, not forced equality of state hashes, certify compatibility. The user still selects the source, omission cap, frame and scientific tolerances. No public method default, complex-source SCF, RI/CD, BUPO, periodic, CCM/chi, retained projector, CC residual or derivative admission follows from this routing. MANTA must eventually supply versioned immutable kernel/domain/screening and auxiliary-rank descriptors plus operation-specific support/equivariance attestations. Its bounds and bubble hierarchy remain producer-owned. Periodic admission must retain complete Seitz/image/wrap/phase/weight/momentum data and the distinct finite Hamiltonians before any cost comparison is meaningful.

Source-owned molecular method entry and output state

The explicit public HF entry closes the gap between a callable JKBuilder and a usable method result. It constructs and qualifies the actual P0 source, checks basis centers against the molecule, validates overlap/Hcore covariance, and invokes the existing native RHF/UHF loop. Exact method-local J/K snapshots now supply the returned evaluated density in the shared FockBuild when their source, channel order and density bytes match. A miss uses a terminal cold build. No solver rewrite, state symmetrization, different finite support or implicit engine substitution is involved. Capture work and any fallback terminal contraction are reported separately. Separate native RHF J and K calls are joined only on the same source and exact density bytes; UHF retains its complete spin-channel order.

At an iteration cap the native host’s next-step orbitals can describe a different density from the one whose energy was evaluated. The wrapper owns physical D/F/energy and diagonalizes X.T S D S X only to obtain an exact output representation with occupations. It preserves fractional densities; diag(C.T F C) supplies honestly labeled Rayleigh energies. Reconstruction and metric checks establish output consistency, not state stationarity, canonical eigenpairs or symmetry of a truncated/degenerate retained space. The actual retained-frame bytes enter the existing frame identity, while source, density and build identities remain distinct.

Fock unification supplies the existing run_fields/QVF seam and method-declared orbital labels. MANTA supplies the exact source, envelopes and hierarchy; this entry adds no producer kernels or algebraic attestations. PBC methods, CCM/chi Hamiltonians, fitted/complex sources, local/auxiliary projector transport, correlation and analytic derivatives remain outside this admission. Zero-copy native evaluated-state capture and pre-qualification cost selection remain production work. Method-local exact J/K reuse already avoids the additional terminal build in the admitted cases, while retaining an explicit cold fallback and independent FockBuild ownership.

Explicit prepared sources now amortize the existing qualification over independent molecular states. The preparation owns the native source and fixed mask, immutable one-electron matrices and exact geometry/basis signature; it contains no electronic or retained-space state. Each method rebuilds its own retained frame and creates fresh physical counters. Only five bounded scalar route profiles may be reused. Preparation, newly measured routes and physical execution have separate costs, avoiding fictitious savings from omitted setup accounting. A process-local lease protects the shared native workspace; changed nuclear/radial content refuses before integral work. Reusable evidence is not an algebraic equivariance attestation for an unseen source. That producer contract, and cost admission before a fresh audit, remain necessary for scalable default routing.

The standard molecular job now accepts that explicit source request and retains physical FockBuild provenance in its normal outputs. An executable QVF carries operations, error policy, resource bounds and cost policy, never an audit seal or process-local prepared handle. It reconstructs and qualifies its own producer before execution. This makes the existing measured route usable without moving MANTA source authority into the job parser. Unsupported method compositions and unqualified legacy retries refuse.

Output consumers must distinguish density-defining natural orbitals from canonical Fock eigenvectors. Occupation order does not define a spectral gap: Rayleigh quotient tables are labeled, canonical frontier diagnostics are suppressed, and symbolic frontier cube requests refuse. This is an output representation correction, not a new electronic-state symmetry assumption. QVF coordinate unit round trips can change floating-point bytes: a fresh source identity is expected, and validation must use the reloaded numerical source. The MANTA producer contract and all periodic domain/Seitz/momentum admissions remain independent of this molecular interface increment.

The cost-routing boundary now distinguishes admission of a numerical source for ordered execution from admission of angular reconstruction. A fixed mask may be constructed without integral reads from an exact immutable molecular P0 producer and native full-space omission bounds. It authorizes evaluation of every retained ordered tile only. The existing retained-member/stabilizer audit remains required for representative reconstruction. Geometry, an asymmetric bubble hierarchy or a finite-domain certificate alone supplies neither that audit nor producer-issued equivariance authority.

This distinction enables an explicit ordered route and an early auto-routing cost proof for short expected use: when the number of expected calls cannot repay even the ordered halves of paired calibration, neither calibration nor the angular audit is performed. The mask, scalar tolerance and state remain unchanged. Source identity binds producer plus actual mask independently of audit evidence and cost policy; the evaluated Fock object preserves that separation. Longer-use auto routing still measures both qualified executors. Production default routing still needs producer operation/support/roundoff attestation for cheap reduced admission. MANTA owns that producer contract; this consumer adds no kernels, extents or bubble implementation.

The molecular fitted-source increment now consumes MANTA’s exact retained metric factor and one owned three-center snapshot. Complete complex-Abelian AO and auxiliary spaces are adapted without changing the producer rank cut. Selection acts on conjugated AO and auxiliary characters, with full auxiliary partner mixing and inverse AO variance retained. Actual allowed three-index blocks drive arbitrary-density J/K; an unreduced contraction of the identical selected factors is available. Full-source selection/reconstruction error is measured before reduced work is permitted. The shared native RHF/UHF bridge and FockBuild consume this fixed source without asserting electronic-state invariance. Setup still evaluates all ordered fitting tiles.

MANTA remains responsible for factor export, symmetry-compatible screening, operation/support and roundoff authority, and bubble bounds. A future retained factor export can replace the snapshot traversal only when it identifies the same finite source and metric policy. The consumer must not infer such permission from the geometry or a periodic domain certificate. No changes are made to Seitz/image/wrap phases, k-star weights, momentum constraints, CCM/chi supports or Hamiltonians. RI derivative acceptance still requires metric/fitting/Pulay/response terms, moving supports and independent displaced geometry checks; this density-contraction increment supplies none of them.

For ill-conditioned but retained native RI factors, ideal metric orthonormality must not substitute for the finite producer source. The auxiliary audit now preserves G = F^dagger M F, checks transported G and coefficient containment with a QR coordinate solve, and records the defect from identity separately. A unitary polar correction of the assembled irrep rotation preserves F F^dagger; it does not whiten F, recut M or repair the electronic density. Partner relations, leakage, isometry and final tensor error still pass the original thresholds. This removes false rejections of large JK-fitting bases without weakening source qualification.

Prepared RI factor routing now uses the same measured payback rule as the quartet consumer. The choice is between dense and blocked contractions of identical selected factors. Five bounded per-channel scalar profiles may be reused across fresh method states; neither probe timing nor executor choice enters source or evaluated-state identity. Fixed nonsymmetric paired probes must agree before selection, short-use requests can prove probe repayment impossible and skip all probe work, and unknown signatures retain dense execution. Preparation remains separate from physical contraction costs. This is explicit developer-level routing, not default method/job dispatch or permission to omit source qualification. MANTA producer work and periodic finite-domain admission are unchanged.

RI source admission at the molecular method boundary

The next literature-to-code gap after measured RI routing is integration, not new tensor mathematics. Melega Eqs. (3)-(5) and Kitsaras/Stopkowicz preprint Eqs. (16)-(17) already motivate the implemented finite factor identity and conjugated Abelian products. Their existing source and version limitations remain as documented above. Davidson/Dupuis-King quartet execution continues as a separate qualified source; neither the gradient paper nor geometric characters grant an RI gradient, CC or periodic acceptance claim.

The coherent consumer increment is explicit RHF/UHF and executable job/QVF admission through the existing prepared source and FockBuild. A new immutable RI request specifies auxiliary basis, complete complex characters and native rank controls. Actual one-electron tensors must pass covariant invariance; source preparation and repeated SCF costs are accounted separately. No solver, integral kernel, metric factorization or state projection is duplicated.

MANTA continues to own finite integral generation, metric rank and factor construction. The remaining producer capability is a qualified immutable factor export, with identity, finite support, rank/cutoff policy and screening closure evidence, so the consumer can avoid an independent complete coupling snapshot. BUPO hierarchy and omission bounds must bind that same source; an AO geometry action or cost profile cannot substitute for this evidence. The consumer must still test arbitrary trial densities and compare actual J/K, not only a converged symmetric state. Native bounded block execution and automatic workload estimates remain separate performance increments.

Fock unification consumes the existing evaluated result, with a distinct RI source identity and fitting-specific setup receipt. RI and quartet bridges now share method-local exact-input J/K capture. Its key is the qualified selected-source identity, channel order and exact density bytes; neither spatial equivalence nor converged-state symmetry permits a match. Capture copies and comparisons are reported separately from contractions. Periodic Parity retains its own finite-domain, Seitz/Bloch, momentum and k-star qualification: this molecular entry grants no multi-k, CCM or chi admission. Likewise non-Abelian partner/multiplicity resolution, local/frozen retained spaces, response, fitting/Pulay/support derivatives and stress still require their separate contracts and numerical evidence.

Producer factor ownership before reduced generation

The molecular RI producer now exports the actual retained fitted coefficients with a native identity and executed-work receipt. This resolves the ownership gap between MANTA’s exact metric factor and the symmetry consumer’s independent raw coupling snapshot, and removes that raw array from default preparation. It does not resolve the remaining algebraic/roundoff certificate for symmetry representatives or screened factor generation. The existing ordered source has no declared finite-precision pair-exchange policy, so the exporter keeps its complete ordered coupling pass. Treating its geometry as permission to mirror missing values would silently change the numerical source.

The consumer can compare the native export with the explicit ordered-tile fallback and independent integrals without changing cutoffs or final source error limits. Both selected and dense execution consume one identified owned factor payload. MANTA owns integral evaluation, rank selection and export; the shared symmetry layer owns spatial/retained-space qualification and complex block adaptation; FockBuild owns the evaluated method state. Periodic Parity continues to require its full image/domain/momentum contracts separately.

Native execution of qualified RI blocks

The next execution increment compiles the selected factor blocks into an immutable native layout. It adds no source authority: the existing factory still checks the producer export, retained auxiliary Gram and rank, complex irrep products and aggregate factor-selection error before compilation. The selected numerical source identity excludes this execution choice.

The blocked route contracts one auxiliary panel at a time, without a dense rank-by-AO-by-AO reconstruction or exchange intermediate. It evaluates J = sum_A B_A tr(B_A^dagger D) and K = sum_A B_A D B_A^dagger for arbitrary complex nonsymmetric D. Complete factor blocks may be rectangular and irrep groups may be empty. AO density/operator transport remains metric-aware and outside the contraction kernel. No conjugation of D, state projection or antiunitary operation is introduced.

The native snapshot costs one additional selected-factor copy. Count-only resource admission precedes descriptor/payload reads, copying and contraction. Actual complex products and auxiliary-panel contractions are reported; the existing cost router compares complete AO calls against the same-factor dense fallback, including transport and native dispatch. The former NumPy blocked path remains an explicit low-level reference. MANTA still owns factor generation, bounds, screening and bubble hierarchies. This change does not remove any source integral calls or qualify periodic/CC/derivative consumers.

Route calibration follows method demand

Automatic route selection must not spend work on channels the actual solver never requests. The method now obtains a profile at the first physical request for each signature; the qualified source and physical density remain unchanged. The probes are still nonsymmetric and use the identical numerical source, so an invariant state cannot substitute for operator qualification. Only successful scalar profiles may be shared by prepared sources.

Native NOITER supplies an exact one-evaluation workload and therefore skips new probes. This local cost bound must not overwrite a profile or policy for a future iterative calculation on the same prepared source. An iterative SCF cap is not a convergence forecast or a general bound on response and line-search calls. Such estimates and producer operation/screening closure remain separate production requirements. FockBuild ownership, MANTA integral and hierarchy ownership, and periodic finite-domain admission are unchanged.

MANTA’s fixed-metric periodic coupling-only screening now supplies a selected coupling identity, an unscreened finite reference identity and fixed-density J/energy approximation bounds. Its paired forward/adjoint maps use the same ordered image partition; the metric, rank and finite factor remain fixed. The current consumer admits Coulomb J and refuses fitted K under a positive coupling-only screen. See the producer contract. This is useful producer evidence, but it is not an operation certificate. Symmetry reuse must additionally bind each admitted operation to the actual ordered support and selection policy, auxiliary and AO actions, reciprocal and atom-image transport, and an operation/pair error budget. Integral, factor, phase and contraction roundoff remain outside the current screen bound. Representative generation therefore still requires separate closure and arithmetic evidence; neither the paired map nor its scalar error bound allows the consumer to omit an orbit audit. The producer owns that evidence and hierarchy implementation; symmetry consumes it through the shared source contract without reimplementing screening.

Complete matrix irreps and reduced invariant maps

The experimental symmetry_products layer now resolves ordinary unitary representations using complete matrix projectors. It requires a complete inequivalent set of matrix irreps in the admitted group’s exact operation order. It checks group multiplication, unitarity, character orthogonality, sum d_gamma**2 = |G|, integer product multiplicities and conjugate closure. Nonzero lattice cocycles and antiunitary operations refuse. Complex variance conjugates the relevant matrix representation; no realification substitutes for a complex irrep.

Columns have order (irrep, multiplicity, partner). For each irrep, P00 selects multiplicity seeds and Pi0 generates partners. Full intertwining and orthogonality checks validate the resulting basis. Its identity includes the chosen multiplicity gauge. Two bases with matching characters therefore cannot silently exchange reduced arrays.

An invariant map between two resolved spaces has the form T = U_left direct_sum(A_gamma tensor I_dgamma) U_right^-1. The actual finite inverse is retained; numerical orthogonality does not license replacing it by the adjoint or inserting an ideal Gram matrix. The factory reads the full supplied tensor, retains a specified partner of each multiplicity block, reconstructs every partner and checks the complete Frobenius residual at the unchanged explicit tolerance. It does not average or Hermitize the source. The residual bounds an arbitrary-input application by epsilon * ||x||_F in these orthonormal coordinates; arithmetic roundoff remains separate. AO users must carry their metric-aware density/operator transformations and corresponding norm factors themselves.

Composition executes only A_gamma B_gamma in multiplicity spaces, without an extra irrep-dimension factor or a full intermediate. Its inner gauge must match exactly; an explicit recoupling map carries different gauges and leg permutations. Full conversion restores all partners before a four-index re-sort, as required by Hellmann-Gauss’s missing-element discussion. This bounded reference path retains dense basis matrices and uses dense audits; their storage and transformation costs cannot be called free. Full CC spin adaptation/residuals, local/frozen projectors, periodic domains, gradients and stress are not supplied by this algebra.

The QC0.18 candidate contains this experimental module but has no method caller of it. Existing quartet/RI routing, Fock construction and MANTA/PBC sources do not depend on its product resolution. Completing non-Abelian infrastructure is therefore not a release prerequisite. After release, first qualify an actual producer through the shared source contract, including operation closure, screening and arithmetic bounds; then optimize a consumer that stays in reduced coordinates. Compare complete elapsed work, including source qualification, basis transport and reconstruction, before changing dispatch. Dense audits remain required until that producer evidence exists.

Native MD finite-source admission

This producer increment reuses the existing Dupuis-King shell transport, stabilizers, whole-orbit masks and native block contractions with MANTA’s immutable DirectJKBuilder. It adds a numerical source, not another group algorithm or integral engine. Exact native type, full-Coulomb kernel, zero range parameter, complete shell snapshot and per-tile precision/source receipts are checked. Every ordered quartet and stabilizer is audited before representatives may reconstruct J/K. The resulting mask and provenance are specific to this native MD arithmetic; libint qualification cannot stand in for it.

The existing molecular HF bridge consumes that certificate without projecting trial densities or selecting an electronic irrep. Its shared Fock result keeps the MD source, physical density, orbital metric and retained frame distinct. The ordered fallback uses the same native producer and mask. Existing cost routing must measure that producer and channel signature, including transport and full output construction; an orbit count is not a routing decision. This adapter does not yet use the coupled-coordinate product algebra above.

This is explicit experimental molecular admission, not general producer certification or a default backend change. MANTA still owns kernel evaluation, bubble hierarchies, arithmetic evidence and operation-compatible screening bounds. Native MD cannot inherit the libint-only pre-audit omission adapters. Operation-bound support and arithmetic certificates remain necessary to avoid the full audit. Periodic, fitting, range-kernel, derivative and antiunitary sources require their own admission. The frozen QC0.18 routes remain independent.

Cost of streamed quartet metadata

A representative schedule must avoid repeated metadata work as well as integral and contraction work. The measured NH3/cc-pVDZ native-MD path spends most of its profiled execution time expanding shell images in Python. The shared schedule therefore separates representative enumeration from expansion, applies an already qualified fixed mask before expansion and creates only unique image descriptors. Stabilizer masks, member order, source digests and full numerical audits remain unchanged. No quartet-sized cache or new MANTA kernel is introduced. Compare complete setup and SCF cost on identical sources; a faster scheduler alone does not establish a production crossover against a stored-tensor or direct native method.

Native streamed execution now expands one selected orbit inside the existing quartet accumulator. It receives only the admitted shell action on the four representative indices, preserves identity-first and first-occurrence member order, checks the orbit-stabilizer census, and reuses the existing metric-aware AO transport and selected-channel contractions. The full Python numerical source audit still precedes this path. Compiled index algebra cannot certify an integral source or replace producer support, screening and arithmetic evidence. An older runtime retains the checked Python image path.

One orbit’s temporary map, duplicate set and member rows fit the existing metadata reservation; no quartet inventory or evaluated-tile cache is added. Matched complete-method timings must include qualification and reconstructed outputs. This optimization changes neither Fock/state identities nor the MANTA producer/session and periodic source contracts. The experimental product algebra and broader producer certification remain separate from release scope.

The native source bridge consumes MANTA’s existing IntegralProvider and Workspace interfaces for exact immutable libint/MD molecular full-Coulomb sources. It owns a separate workspace and consumes each borrowed tile before reuse. It retains the canonical representative digest, complete call/value census and source identity checks before exporting targets. The complete Python qualification audit still supplies the symmetry evidence; neither native ownership nor this execution bridge supplies a producer certificate. There is no new recurrence, hierarchy or integral cache in symmetry.

The extra workspace and its conservative work reservation are optional. Admission reserves screening, target and method/capture buffers first; if the remaining budget cannot hold the native source bridge, the existing checked Python reader remains valid with the same native contraction. Source and physical-state identities do not include this resource decision. Compare complete elapsed costs and resource use before changing public routing.

Producer evidence for finite molecular support

The next producer-certification component is MolecularSourceSupport, issued by MANTA from the immutable native libint or MD provider. This is a source admission increment, with no new integral mathematics. Davidson representatives and Dupuis-King angular reconstruction still consume the shared group and metric-aware shell actions. The missing permission is arithmetic equivariance of the actual finite source; shell support closure alone cannot supply it.

The certificate binds the provider source identity, ordered operations and shell destinations, declared geometry tolerance and native constructor policy: real molecular four-center full Coulomb, derivative zero, no images, primitive precision zero, axial shell normalization and no density-dependent screening. It checks each shell destination bijectively, requires exact exponents, contracted coefficients and angular/pure-shell conventions, and measures the transformed-center residual. Finite support means the finite shell inventory, not a spatial cutoff on Gaussian functions. Nonzero center residuals remain subject to the complete numerical operator audit.

Capability	Current evidence	Still required
Native molecular source and shell support	Producer-issued immutable certificate; exact radial matching and explicit center tolerance	General source/bubble/support policies and their operation compatibility
Finite numerical operator equivariance	Complete ordered-tile and stabilizer audit, including nonsymmetric trial-density acceptance	Producer arithmetic enclosure before any audit can be waived
Fixed screening	Existing whole-orbit mask and qualified omission bounds	Producer-certified adaptive screening and bubble closure
State and retained spaces	Existing independent Fock, spin and retained-projector admissions	Method-specific periodic, correlation and derivative acceptance

The factory consumes this evidence before quartet evaluation and retains it with the source/session. It does not replace the group law, AO metric action, finite-Gram inverse or source reconstruction audit. The certificate is scoped to the live immutable producer and cannot be attached to another provider. Its identity is qualification evidence, not a new physical Hamiltonian or Fock-state identity. Older runtimes retain the complete numerical audit.

MANTA owns the certificate and numerical source implementation; symmetry owns operation composition and angular transport. Fock unification consumes the existing qualified source/state interface. PBC Parity still requires actual Seitz/image/wrap/phase, k weights, momentum conservation and support evidence; no molecular certificate admits a periodic, RI, BUPO, CCM or chi source. Antiunitary actions, gradients and stress remain separate. This component is experimental, introduces no default route and is not a release prerequisite.

Native consumption of certified shell maps

MolecularSourceSupport retains the exact shell-destination inventory that it qualified. The shared stored and streamed plan constructors compare their maps with this inventory before source evaluation. Reusing a source with a different operation inventory requires fresh qualification, or an explicit source without that certificate followed by the complete numerical audit. The certificate does not acquire authority over angular matrices or group composition: those checks remain with the shared representation and source audit. Its evidence identity is unchanged by retaining the already hashed map.

The native source accumulator can bind this certificate once, before reading any representative. Binding checks the live provider, operation/shell counts, complete destination map, identity map and execution mode against the existing accumulator. Execution then receives representative keys only and obtains their four mapped shell indices directly from the immutable native inventory. It uses the same orbit expansion, transport and contraction code in the same order. Explicit maps cannot override a bound certificate; a failed binding or key poisons partial targets. The native bridge owns the certificate for its lifetime.

This removes per-quartet Python map construction and copying. It does not change which integrals or contractions execute, grant arithmetic equivariance, remove any source audit, or project a trial density. Certificate storage and inspection remain explicit costs. Older runtimes, sources without certificates and the existing resource-limited checked-reader path retain their fallbacks. MANTA retains producer ownership; Fock state/source identities and all periodic support, Seitz and phase obligations remain separate.

Dropping an ordered-tile audit still needs a bound for the numerical source and the actual finite transport. A sufficient per-image triangle bound has four separate contributions: the member’s producer arithmetic error, the representative’s arithmetic error multiplied by the transport norm, finite angular-transform/application error, and physical geometry mismatch. The last term matters even when the measured center residual passes the geometry gate. Exact radial identity and precision zero do not bound Boys-function evaluation, contraction, underflow or pure-shell conversion error. A future producer certificate must supply those terms for its declared kernel and support, while the shared layer supplies metric-aware transport and finite-inverse factors. The resulting tensor residual must bound arbitrary-density J/K and fit the unchanged source tolerance before any member read can be omitted. Its setup, storage and contraction cost must be measured against the full-audit route.

Four-center construction evidence from the producer

The next arithmetic component is MolecularQuartetConstruction, an explicit experimental diagnostic for one contracted s-shell quartet of an immutable native molecular Coulomb provider. It reuses the MANTA metric producer’s verified Boys reference and elementary enclosures. McMurchie and Davidson (1978), DOI 10.1016/0021-9991(78)90092-X, give the Gaussian product/Hermite foundation; Rump (2010), DOI 10.1017/S096249291000005X, gives the verified arithmetic framework. This is a derived construction-error bound, not a new integral algorithm or an inferred accuracy property of either library.

For exact stored exponents and centers, let p=a+b, q=c+d, P=(aA+bB)/p, Q=(cC+dD)/q and rho=pq/(p+q). Each primitive reference is

\[ (ab|cd)=\frac{2\pi^{5/2}}{pq\sqrt{p+q}} \exp[-ab|A-B|^2/p-cd|C-D|^2/q]F_0(\rho|P-Q|^2). \]

Outward binary64 operations enclose product centers, squared distances, decays, prefactors, the shared radial reference, signed stored contraction coefficients and accumulation. Gradual underflow and round-to-nearest are required. The shared Boys proof uses a positive series with an explicit tail; its large-argument gamma-integral fallback can be wide. It does not assume libm accuracy. If the reference lies in [L,U] and the existing producer returns x, the outward maximum of |x-L| and |x-U| bounds that actual finite tile’s construction error, even if x lies outside [L,U]. A null tile is x=0 and does not erase the positive reference bound. Numerical evaluation remains on the existing IntegralProvider path.

This reference treats the stored normalized basis data as exact. Their construction is excluded. Pure and Cartesian s functions coincide; no higher angular conversion is admitted. Other angular quartets, range-separated or complex kernels, image shifts and derivatives refuse this diagnostic. Count admission includes proof primitives, bounded scalar/kernel work, one actual native shell call, its additional workspace and retained evidence.

The certificate retains its exact live provider and quartet key. It cannot be constructed or edited in Python and cannot validate another provider, even with identical content. This is arithmetic evidence for one evaluated tile, not group, state, subspace, screening or equivariance authority. It deliberately leaves arithmetic_equivariance_certified false. Fock unification can keep its existing source/frame/state identities; periodic Seitz, image, wrap, phase, weight and momentum contracts are unchanged. No QC0.18 route depends on this new diagnostic.

The remaining audit-reduction work is substantive: bound angular shells and finite action/inverse application, physical geometry mismatch and complete operation-compatible source/screening coverage, then compose arbitrary-density J/K residuals within the unchanged tolerance. A quartet diagnostic that reads the actual tile cannot eliminate that read. Its proof cost must be included before a future hierarchical producer certificate or reduced pipeline can claim a useful saving. Full ordered and stabilizer audits remain mandatory.

Angular four-center construction reference

The experimental MolecularAngularConstruction extends the explicit tile construction diagnostic to contracted molecular Coulomb shells through l=6, subject to the selected producer’s own angular capability. A reference backend compiled with a lower limit must reject that basis before certification. The scalar s-shell API remains separate and compatible. This does not yet provide source-wide equivariance permission or remove any ordered/stabilizer source audit.

Both Gaussian pairs use the same enclosed Hermite raising coefficients as MANTA’s three-center construction. The kernel uses the shared radial reference and finite Coulomb derivative recurrence. The ket Hermite degree supplies its parity sign; two contractions precede signed primitive accumulation. Four successive tensor-axis transformations apply shared enclosed Racah coefficients in native last-index-fast order. Stored normalized exponents, coefficients and centers define the reference; their generation remains outside the certificate. The Gaussian product and Hermite foundation remains McMurchie-Davidson (1978), DOI 10.1016/0021-9991(78)90092-X, and the rectangular contractions follow Neese (2023), DOI 10.1002/jcc.26942. These outward reference constructions are derived bounds, not claimed accuracy properties of the numerical library.

For every returned element x, the certificate encloses its exact reference in [L,U] and reports the outward maximum of |x-L| and |x-U|. It compares the actual immutable native provider tile, including a null tile interpreted as numerical zero. The result retains its provider, source identity, ordered quartet key and shape. It cannot qualify another provider or another numerical source.

Admission counts all primitive products and reserves the derivative cache, Hermite expansions, kernel, intermediates, transforms, returned arrays and an additional native workspace before evaluation. Work limits remain conservative caps, not measured operation savings. The diagnostic necessarily reads the actual tile; uniform producer bounds without those reads remain separate work. Complex integral kernels, range separation, images and derivatives remain excluded. Existing complex-density J/K, finite inverse, Fock/state and periodic source contracts remain unchanged. This experimental API is not default routing or a release prerequisite; numerical acceptance must be recorded in the status document before it is described as validated.

Experimental exact scalar sign source

An opt-in producer route targets the existing single-channel molecular MANTA MD source with scalar s shells and exact diagonal coordinate signs. This is a finite-program equivariance proof, distinct from the construction diagnostics above: it does not claim an integral accuracy bound or assume a libm ULP bound. Geometry detection alone cannot enable it. Acceptance requires exact radial arrays and primitive order, bijective shell maps, zero translations, componentwise exact mapped centers and current round-to-nearest binary64 with gradual underflow. A zero geometry norm is insufficient for exact admission.

In the executed McMurchie-Davidson pair construction, each Gaussian product center changes by its coordinate sign. Fixed-order squared distances and scalar Gaussian seed exponents remain the same. For l=0, no directional Hermite recurrence executes. Consequently the Boys and elementary functions receive identical arguments, and the scalar E-R-E products, signed primitive weights, compensated summation order and s-shell representation transforms are identical. Numeric zero signs need not be identical. Every spatial class must still supply an evaluated finite value; the proof does not establish source finiteness by looking only at its metadata.

The existing spatial schedule also includes eightfold real-ERI index permutations. Their finite arithmetic is not certified by this proof. The consumer therefore reads one actual tile for each distinct spatial class inside every combined orbit, retaining the original permutation residual accumulation. Only spatially related duplicate reads are removed. Scalar AO inverse actions must be exactly 1 in each mapped shell block; ordinary group laws and metric compatibility remain separate admissions. All-s AO permutations are a special case, not justification for treating general Cartesian or pure actions as Euclidean unitary.

Representative digests remain checked on execution. A typed immutable native certificate binds the live provider and operation maps; execution revalidates its floating-point environment before a compiled source bridge can bypass Python reads. Fixed-mask screening uses the same member order, scalar values and norm accumulation as the complete audit. The first route excludes producer omission bounds, so no spatial class is declared finite without an actual evaluation. Group enumeration and residual bookkeeping remain real costs even when native tile calls are avoided. The small spatial-class cache and its lookup work are explicitly reserved.

General rotations, coordinate permutations, nonzero translations, angular shells, channel regrouping, libint sources, complex kernels, antiunitary operations, periodic images and derivatives remain outside this certificate. Unsupported sources retain the audited route. No electronic-state symmetry is assumed; arbitrary nonsymmetric complex densities and broken spatial states still exercise the same J/K map. Covariance uses the AO coefficient and dual transformations, with permutations only in this restricted domain. Gradients, stress, fitting and retained-space transport retain their own acceptance requirements. The bounded acceptance and measured costs are recorded in the status document. Producer maintenance must preserve this finite operation graph or invalidate the capability. In particular, geometry-dependent primitive reordering or a new channel schedule needs a new proof and regression evidence.

The public prepared molecular source carries producer selection separately from qualification and execution selection. Native MD can use a complete numerical audit or its narrowly admitted scalar sign certificate; neither choice imports reference-producer omission bounds. Public RHF/UHF receives the existing shared J/K source object. Qualification evidence distinguishes an exact spatial proof plus permutation audit from a complete numerical audit. Periodic, response and post-SCF consumers retain their own admission gates.

Exact diagonal angular actions

For an exactly diagonal spatial operation with entries (sx, sy, sz) in {+1,-1}, every monomial of a real solid harmonic has the same coordinate parity. In the existing m=-l,...,l real ordering, cosine rows (m>=0) carry sx^m sz^(l-m) and sine rows (m<0) carry sx^(abs(m)-1) sy sz^(l-abs(m)). The Wigner adapter evaluates these signs exactly instead of passing this discrete subgroup through Euler angles. Only exact input matrices take this branch; nearby rotations are not snapped. This preserves Cartesian/pure metric conventions and introduces neither a complex-conjugation operation nor antiunitary time reversal.

Exact representation matrices are a prerequisite for an angular producer proof, not that proof itself. The MD scalar sign certificate still refuses angular shells. Hermite/auxiliary recurrences, packed matrix products, signed contractions and pure-shell conversion must all be qualified before angular source reads can be omitted. Geometry, finite operator, electronic state and retained-subspace admissions remain distinct, including for periodic users of the Wigner adapter. No Seitz/image/Bloch/source certification is inferred.

Angular finite-program certification boundary

Component parity propagates through the explicit MD Axis and Auxiliary recurrences and through a fixed sequence of multiply/add operations. However, the production build can delegate E-R-E and pure-shell matrix products to a vendor BLAS. A mathematical matrix-product identity and passing samples do not by themselves certify that delegated finite program. An angular source certificate must identify and qualify its contraction schedule, including backend/thread dispatch, before claiming an exact zero equivariance error. The existing scalar certificate uses identical scalar inputs under spatial signs; it does not infer sign covariance of an angular BLAS contraction.

Two producer-owned options remain: qualify the actual fixed contraction schedule, or introduce an explicitly distinct finite source that canonicalizes axis signs before evaluating each ordered quartet. The latter requires exact sign selection from ordered center coordinates, identical canonical inputs, producer-derived component parities, and explicit structural zeros for odd components when every quartet center on an axis is exactly zero. It must bind its own source identity and preserve the original producer as a fallback. The explicit canonical-sign source below realizes this separate option. It is checked against independent integrals and complete method references, with reduced/unreduced comparisons using that identical new source. It is not post-SCF matrix symmetrization, a geometry certificate or permission to skip ERI permutation audits. MANTA owns these integral/contraction choices; shared symmetry owns group and action binding, source admission, orbit planning and consumers.

Complete ordered execution has a separate admission path from symmetry reduction. When omission is exactly disabled, the immutable native producer owns every value that execution evaluates; no source-equivariance claim is needed to reconstruct a value because none is reconstructed. The shared schedule still validates geometry, radial support and AO group relations, but its complete-source receipt has equivariance_qualified=false. A zero omission statement is exact for this all-kept inventory and does not assert integral accuracy or zero arithmetic error. Reduced executors cannot consume this receipt. This provides a source-preserving fallback while producer arithmetic certification and compatible screening continue independently.

The complete ordered molecular fallback may execute its entire immutable producer inventory inside the shared native source bridge. Admission requires an unused identity-only accumulator and exact full ordered census; it conveys no group equivariance or omission authority. The MANTA provider owns integral evaluation and workspace requirements, while the shared accumulator owns J/K contractions. A second workspace is included in cost admission, with the checked reader retained when unavailable or over budget. Fock consumers keep the existing ordered result and qualification interfaces. This molecular execution option does not extend PBC, fitting, response or derivative source certification. See the measured native complete-source comparison in symmetry status.

Distinct canonical-sign angular source

CanonicalSignJKBuilder defines an explicit finite molecular Coulomb source. For each ordered quartet and coordinate axis, the sign of the first nonzero center coordinate selects a common reflection of all four centers. Exact zero coordinates become positive zero. All radial channels, primitive order and shell ordering remain unchanged. For any admitted diagonal sign operation, the resulting canonical kernel inputs are identical. The original MD kernel, including its representation transform, evaluates those inputs once per call. The producer then restores Cartesian monomial or real-harmonic component signs. If every center has zero coordinate on an axis, odd total component parity is a structural zero. Without this stabilizer rule, a small computed residual would violate the exact sign action.

This constructs a new source and does not prove covariance of the original vendor BLAS execution. Its identity includes the original basis/source and canonicalization convention. IEEE round-to-nearest and gradual underflow are required at construction, execution and certificate validation. The existing producer remains the default and explicit fallback. There is no coordinate permutation, translation, ERI permutation folding, global integral cache or density projection. Basis copies, parity workspace, source-support inspection and component transport remain counted costs.

The producer certificate supplies its exact component signs as well as shell maps. The shared layer checks every AO action against those signs before replacing spatial duplicate reads. It transports cached spatial-class tiles by diagonal signs, audits every distinct index-permutation class and checks stabilizers including permutations. The complete ordered route evaluates the same new source without reconstruction. Thus a reduced/full comparison does not silently compare different finite Hamiltonians.

This is an experimental molecular Abelian subgroup capability, not general non-Abelian source certification. C3v rotations, complex Abelian actions, antiunitary operations, periodic Seitz/Bloch domains, RI/BUPO, CCM/chi, correlation and gradients/stress retain their separate requirements. It adds no release prerequisite. MANTA owns the canonical source and parity convention; shared symmetry owns group/AO binding and audit scope; Fock consumers retain the existing qualified source/state interfaces.

Exact zero screening of the canonical source

The first omission proof for the distinct canonical source is deliberately separate from analytic Q/V envelopes. A quartet whose four centers are exactly at the canonical origin and whose total shell angular degree is odd has odd component parity on at least one of the three fixed axes. Every component is therefore a structural zero. Pure harmonics have the same total degree parity as their Cartesian monomials. This proves a whole tile zero, not merely a representative norm or a small sampled value. A nonzero coordinate, including a subnormal coordinate, does not satisfy the predicate.

Canonical source v2 implements this rule before calling the MD kernel. That ordering is essential: a proof used to omit a call cannot silently bypass an overflow or exception in an earlier kernel evaluation. The versioned source identity records this change of finite program. The original producer remains unchanged. The native predicate admits only this immutable source, validates quartet indices and arithmetic environment, and evaluates no integrals. The native ordered builder separately reports requested and actually evaluated primitive quartets, including zero evaluations for a proven zero tile.

MolecularExactZeroBounds binds the producer rule to its source identity. The shared screening adapter requires zero omission budget and checks every ordered member before dropping an entire orbit. Nonzero or unproved tiles retain their normal audit. The spatial-sign certificate can be combined with this exact-zero proof, while ERI permutation and stabilizer checks still apply to retained tiles. Receipt fields identify exact omitted zeros and explicitly state that no nonzero values are bounded. This does not qualify the analytic Q/V envelopes as bounds on angular floating-point source errors.

No density projection, threshold relaxation, derivative scope, periodic admission or change to Fock source/state interfaces follows. General finite angular envelopes and hierarchical screening remain future producer work.

Exact-zero bound accounting distinguishes total constructor admission from additional adapter workspace. The source plan already reserves its immutable producer. Shared qualification may charge only additional_bytes when the bound and source retain the identical producer object. Equal source hashes on separate builders do not prove shared storage and retain both reservations. The constructor still admits the full producer plus adapter, and the native execution workspace remains separately charged. This accounting correction confers no new source certificate or screening authority.

The canonical producer v3 extends whole-tile zero selection to exactly coincident centers at any position. For a common center C, inversion of both electron coordinates about C multiplies the four Gaussian factors by (-1)^(l_a+l_b+l_c+l_d), while the Coulomb kernel and integration measure remain invariant. Every Cartesian or pure component therefore vanishes for odd total angular degree. The producer returns that zero before evaluating the kernel, so the proof applies to its explicitly versioned finite program. This does not assert that translating a retained floating-point tile leaves its values unchanged. The source identity includes the new zero rule.

Center equality uses direct coordinate comparisons, never a norm or geometry tolerance. Distinct centers separated by a subnormal-scale displacement remain unproved. Existing whole-orbit closure checks and all retained-value audits continue to apply. Spatial-sign source certificates still require their own zero-translation admission; a common-center parity rule is not a certificate for general affine operations, periodic sources or derivatives.

After native sign-source and exact component-action admission, qualification uses diagonal sign multiplication for covariant tensor transport. Identity legs do no arithmetic. Every stabilizer and permutation comparison remains; this is not an exemption from source auditing. General Cartesian and other uncertified actions continue through the metric-aware dense transport. Executed transport calls, scalar products and stabilizer checks are recorded separately from integral requests. Conservative admission budgets are retained.

Canonical-sign producers also participate in the existing molecular quartet construction diagnostics. The reference remains the exact stored normalized basis, before any finite canonicalization or structural-zero rule. Comparing an evaluated canonical tile with that interval measures its construction error, including changes made by those explicit source rules. Evidence is bound to the actual immutable producer object and identity, with existing angular, primitive, memory and work admissions intact.

These diagnostics evaluate the requested tile. They cannot justify omitting that evaluation, certify all source tiles, or prove operator equivariance. Nonzero pre-evaluation screening still requires a producer-owned enclosure of the evaluated finite tile, including arithmetic and representation errors and exceptional behavior. Exact-integral Q/V envelopes or a construction error measured after evaluation do not supply that missing contract.

Proposed finite producer for signs and ERI permutations

The remaining index-permutation audit cannot be waived for the current canonical-sign producer: its finite kernel need not commute with index permutations. A proposed distinct producer would instead select one canonical input under the direct product of eight real ERI permutations and eight Cartesian-axis sign operations. Existing producers and defaults stay intact. This proposal does not apply the eight real permutations to complex AO ERIs.

Canonical ordering must use immutable shell contents, not basis indices: angular representation, ordered exponents and coefficients, and transformed coordinates with signed zero normalized. Content ordering ensures that a spatial shell permutation does not change the selected kernel input. All minimizing images must be retained when descriptors tie. Selecting just the first minimizing descriptor is insufficient for repeated-shell components.

For each requested component tuple, map it through every minimizing image, select the smallest canonical flattened component index, and recover its phase from the exact Cartesian/pure component parities. If minimizers at that index imply both signs, return exact zero. Otherwise return the selected finite kernel entry with the unique recovered sign. This defines the new finite source; it does not assert permutation symmetry of the raw kernel. Under an admitted operation, the minimizing-image set is relabeled, preserving the selected component index and changing only the required phase. Conflicting phases remain conflicting. This is the proposed finite-program proof of the combined action, including stabilizers.

An independent synthetic-kernel prototype passed 512 combined-action checks on Cartesian/pure p-only, mixed s/p/d and repeated d/p quartets. The deliberately non-equivariant kernel also made descriptor-only canonicalization fail. This is algebra evidence, not integral accuracy, source certification or a timing benchmark. No native producer or audit waiver implements this proposal yet.

Native implementation must admit the 64 descriptor candidates, bounded list of minimizing images, component selection and live kernel/output storage before allocation. It must report actual kernel evaluations and canonicalization work, then validate same-source reduced/ordered J/K and complete SCF against independent integrals and matched references. Construction enclosures can quantify deviations from the exact stored-basis reference. A producer-issued combined-action certificate may replace permutation reads only after those checks; support, arithmetic environment, source identity and state admissions remain separate. This does not supply nonzero magnitude bounds, general rotations, antiunitary symmetry, periodic phases or derivative certificates.