DLPNO-CCSD(T) as one engine: the correlation-source contract for molecules and crystals

Proposed 2026-10-09. This page records the design under which the molecular DLPNO-CCSD(T) engine becomes the single local coupled-cluster engine for molecules and for periodic systems, and what “production ready” means for each. It changes no number until its first increment lands, and the decisions listed at the end belong to the maintainer. Evidence for the molecular state is in the DLPNO guide and on #190 and #506.

Where the code stands

One engine, measured. python/vibeqc/dlpno/ccsd_local_solver.py and python/vibeqc/dlpno/uccsd_local_solver.py solve the closed- and open-shell DLPNO-CCSD equations in per-pair PNO spaces through the compiled kernels in cpp/src/dlpno_ccsd.cpp and cpp/src/dlpno_uccsd.cpp, followed by the triples of python/vibeqc/dlpno/triples_local.py. On 17 closed-shell molecules at the shipped defaults the engine matches ORCA 6.1.1 on CCSD and recovers more of the canonical (T). Its inputs are narrow: MO coefficients, the AO Fock matrix and the AO overlap, the reference energy and electron count, and a density-fitting object that answers mo_transform(C_left, C_right) with the three-centre factors B[P, p, q] over a real orthonormal auxiliary frame.

Historical periodic routes, none production. The toroidal MP2 and CCSD(T), CCM coupled CCSD(T), and chi MP2 and closed- and open-shell wrappers pass explicit correlation sources to their engines on a real Gamma supercell. Their finite Hamiltonians and research defaults remain distinct. The CCM union-PNO calculation stays a small-cell oracle, and the native chain is separate.

Route	What it runs	Reference and factors	Engine reuse	Label
Megacell, periodic_megacell_mp2.py	DLPNO-MP2, CCSD, CCSD(T) on an open finite supercell	molecular RHF and molecular DF	complete, through run_job	shipped, no warning, no catalog row
Toroidal, periodic_toroidal_mp2.py	the same on the Gamma BvK supercell	Gamma GDF HF and the Stage 4 compensated-cell factors	MP2 and CCSD(T) through toroidal-compcell source with explicit cyclic overlap; CCSD(T) retains pair mode	Stage 5a to 5c, no warning
CCM neutral torus, periodic/ccm/dlpno*.py	union-PNO CCSD(T), coupled CCSD(T), UMP2, UCCSD(T)	neutral GDF factors, Bloch results folded to the real supercell	coupled CCSD(T) through ccm_neutral_source and the shared engine; union-PNO and spin-orbital oracles retained	AICCM2026DevAExperimentalWarning
chi, periodic/chi/posthf.py	MP2, DLPNO-MP2, UMP2 and DLPNO-UMP2, CCSD(T), UCCSD(T) pilot	k-mesh RSGDF or GDF fit turned into an exact real torus through _BRealDFAdapter	local MP2 and CC wrappers through chi-real-torus sources with explicit cyclic overlap and compact home-auxiliary factors; UHF keeps both spin spaces; CCSD(T) retains pair mode; screens fail closed	AICCM2026DevBExperimentalWarning, library only
Native chain, cpp/src/periodic_gaussian_*, bounded_restricted_*	pair MP2, Galerkin pair CCSD, TNO coupled (T)	its own finite Gaussian reciprocal source, full Gamma-centred mesh	none; its header says it is “not the pair-specific production DLPNO algorithm”	private bindings, production_dlpno() false

The subsystems the periodic engine must respect.

• Unified Fock build. JKBuilder (cpp/include/vibeqc/jk_builder.hpp) remains a producer boundary. The shared FockOperator, OperatorFrame and FockBuild now associate fixed operators, overlap/retained spaces and evaluated method states on migrated routes. Coverage is tracked in operator consumers; this is not an all-method construction or correlation-source certification. Multi-k GDF consumers retain their actual fit caches (RSGDFFitCache, with retain_fit_cache=True) and exchange zero-mode convention. Molecular correlation can use a distinct RI auxiliary fit; that difference must stay explicit in CorrelationSource rather than being hidden by a shared Fock.

• Symmetry reduction. symmetry_reduce_k reduces the multi-k exchange cost only; the orbitals stay on the full mesh. The EWALD3D wedge drivers keep orbitals at IBZ points only, and no production helper transports them back; cpp/include/vibeqc/periodic_mean_field_state.hpp rejects IBZ-only input by contract, and periodic_correlation_admitted_reference.hpp refuses symmetry reduction until a verified unfold and orbital-sewing object exists. No correlated method consumes IBZ-only quantities.

• MANTA. manta.run_periodic_rhf and its UHF and KS siblings return per-k overlap, core Hamiltonian, Fock, coefficients and orbital energies on an explicit finite support, for full regular meshes only. Its periodic GDF tiles and retained metric projectors now exist; their correlation admission remains separate from a passed SCF calculation. The experimental molecular retained-RI bridge below consumes its actual evaluated Fock and selected factors. Production periodic correlation remains deferred.

• BIPOLE. The Ewald-split periodic SCF route in the CRYSTAL gauge, a sibling of GDF rather than a density-fitting method. No correlated code consumes BIPOLE quantities; its finite correlation source exists for tests only and reports physical_hamiltonian_certified=False. Decision D93 in docs/aiccm2026dev_b_decisions.md marks post-HF on such references as not qualified until a correlation-factor support contract exists.

The design

One engine

The molecular solvers stay the engine: PNO-space CCSD in the compiled kernels, the closed-shell (T1) and local (T0), the open-shell (T0), the options and result objects, the output and citation routes. Nothing periodic gets its own coupled-cluster loop. The CCM transcription is retired and the chi and toroidal MP2 and CC wrappers use the contract below; the native chain keeps its role as an independent oracle and as the producer of translational factors for the k-aware increment.

The contract

A CorrelationSource protocol in python/vibeqc/dlpno/source.py is the only thing the engine reads. It has three parts.

• Reference: reference kind (RHF, UHF, ROHF), per-spin MO coefficients and AO Fock matrices, AO overlap, reference energy, electron count, frozen-core count, and a real orthonormal AO frame. For a crystal this is the real BvK supercell fold of a full-mesh Bloch result, built once by the adapter.

• Factors: mo_transform(C_left, C_right) returning real B[P, p, q], the auxiliary dimension, the auxiliary-basis provenance, and an identity: the SHA-256 of everything that defines the fitted Coulomb operator (auxiliary basis, lattice options, mesh, supports, Ewald parameter, exchange zero-mode convention). Periodic adapters report the minimum-image distance function the pair screens and coupling_radius must use.

• Conventions: the exchange zero-mode convention of the reference (exxdiv, Madelung share), the Bloch normalisation, the k-mesh descriptor, and same_source, true only when reference and factors come from one Hamiltonian.

run_local_dlpno_ccsd(molecule, basis, rhf, df, options) keeps its signature as a shim that wraps its arguments into the molecular adapter, so every molecular result stays bit-identical. The .system manifest and the structured log record the source identity next to the thresholds.

Adapters:

Adapter	Reference	Factors	State
MolecularDFSource	RHFResult, UHFResult, ROHF	DensityFitting with the RI auxiliary basis	production
GammaGDFSource	run_pbc_gdf_rhf and siblings	PeriodicGammaDF from build_periodic_gamma_df with the SCF’s options	increment B
BvKSupercellSource	full-mesh run_krhf_periodic_gdf and siblings with retain_fit_cache=True, folded to the real supercell by bloch_reference.as_real_reference	the retained fit cache folded by ccm_neutral_cderi_fold or the chi home factors	increment B
MANTAReferenceSource	PeriodicMANTAResult folded to the real supercell	a GDF-type fit on the same supports until MANTA stage P3b provides factors; same_source false and refused by default	increment E
BIPOLE references	refused with the reason: no certified same-source factors		until a certified source exists
IBZ-only references	refused with the reason: no certified orbital transport		until the unfold object exists

Consistency rules

1. Same source. A production correlation energy pairs a reference with factors from the same Hamiltonian, and the output carries their identity. Mixing is a diagnostic that must be requested explicitly and is labelled as such. This is the BIPOLE and chi principle applied everywhere.

2. Full mesh at the boundary. The engine never sees an IBZ. Symmetry reduction lives in the SCF (exchange acceleration) and, later, in an unfold-and-audit step that reconstructs the full mesh with apply_periodic_ao_bloch_operation and the orbital-sewing audits before admission. Reductions inside the correlation step, such as unique pairs, are validated against the unreduced build, as the shared symmetry gates require.

3. The reference’s Fock and conventions. Denominators and the semicanonical steps use the reference’s own Fock, including its exchange zero-mode term; the factors use the bare periodic Coulomb kernel on the torus with the G=0 handling the reference used, and the manifest records both.

4. Reuse the SCF fit where it exists. Periodic adapters consume the retained fit cache rather than refitting with other parameters. The molecular adapter keeps the RI auxiliary basis, and the difference from the JK fit is documented, not hidden.

5. No new coupled-cluster loop. A new reference kind gets an adapter and an exactness ratchet (full domains equal the canonical method on the same source to within the convergence tolerance), never a transcription of the engine.

Shared-state admission and remaining integration

The remaining DLPNO Fock/block migrations belong to #580 under #472; PBC and MANTA state/source composition belongs to #577 with the existing provider owners. These are acceptance requirements for each consumer migration, not a claim that every existing adapter already carries a common build identity.

A correlation adapter must transport the actual reference Fock and its source conventions. It must preserve the Ewald exchange term, overlap metric, retained-rank projector, occupations, independent spin spaces and frozen-core partitions. Any refit or different finite kernel/support must be declared and qualified separately. same_source cannot be inferred from matching operator shapes, a shared eigensolver, or a FockBuild identity. The correlation-factor producer remains responsible for its own factor, fitting-metric and support contract.

The molecular molecular_source adapter requires an explicit successful SCF convergence status before inspecting factors. Its positional DLPNO consumers therefore refuse stopped or unqualified references at this boundary. This check does not replace source-specific residual, occupation and factor admission. For molecular RHF and independent-spin UHF, a supplied occupation vector must match the integer Aufbau partition used by the engine, including the ECP valence count. Restricted fillings are 2/0; each UHF channel uses its own 1/0 filling. Fractional, reordered, wrong-count, complex or incorrectly shaped vectors are refused before factor inspection. The absolute filling tolerance is 1e-8, as at the periodic boundary. Absent or empty vectors retain the legacy count-based contract. This check neither changes occupations nor certifies the density/Fock stationarity of a reference without such metadata. It implements the single-determinant partition assumed in Riplinger and Neese, DOI 10.1063/1.4773581, Sec. II A, Eqs. (1)-(3); it does not add a finite-temperature correlation model or a new open-shell formulation.

The occupation-admission increment at 2126c26f has bounded acceptance on its pinned containing runtime. Thirty admission checks pass, including independent RHF/UHF fillings, ECP valence counts and unchanged MP2/CCSD energies with zero or one frozen orbital. Full-space H2O/STO-3G CCSD(T), with def2-SVP-RI factors and the rotated-occupied t1 correction, agrees with the independent same-source spin-orbital transcription in both frozen partitions: maximum raw CCSD difference 8.20e-14 Ha and triples difference 1.39e-15 Ha against the original 1e-6 and 5e-8 Ha gates. Maximum final residual is 6.51e-13 and absolute energy change 8.71e-14 Ha, below the unchanged 1e-11 and 1e-13 solver thresholds. Valid explicit and implicit occupations give bit-identical molecular controls; fractional fillings are refused. This qualifies that admission change, not later whole runtimes, general open-shell accuracy, scaling or an external-reference campaign.

Molecular adapters also refuse a reference explicitly marked orbital_kind="natural", including a converged evaluated-state result with integer occupations. Density-defining natural orbitals may mix core and valence within the degenerate occupied space; their physical Rayleigh quotients are not canonical eigenvalues or an energy-ordered frozen partition. The refusal precedes factor metadata and contractions at both positional and explicit molecular source entries, and is rechecked after construction. No orbital rotation, frozen reordering or density replacement is performed. Absent or canonical metadata retains the existing legacy path and identities; this narrow refusal does not certify arbitrary unlabelled input as stationary. The explicit experimental adapter below provides a bounded canonical partition path under the shared Fock-state contract.

vibeqc.dlpno.evaluated_source.evaluated_molecular_source consumes an actual MolecularSymmetryResult and the exact process-local PreparedMolecularSymmetry handle used for its SCF. It checks geometry/basis through the existing prepared-source lease, then associates the physical FockBuild, density epoch, source, frame, energy and independent spin channels. Quartet frames name the underlying producer, whereas their qualified fixed mask has a separate source identity; both are checked. This adapter creates no new Fock or integral producer and makes no factor-certification claim.

The shared dense eigensolver enumerates the complete physical spectrum. Every original AO eigenresidual and the full metric orthogonality error must be at most 1e-11. The metric Frobenius norm of the difference between the recovered integer occupied density and the original evaluated density must be at most 1e-8. This is a full matrix check, not just an electron-count check. The occupied/virtual and, when present, frozen/active gaps must exceed 1e-8 Ha, reusing the source contract’s degeneracy tolerance. It does not arbitrarily split a degenerate boundary. The canonical coefficients, energies and occupations are immutable new snapshots; original D/F/S and evaluated energy are preserved. Frozen orbitals are the lowest canonical orbitals independently in each UHF channel. All four molecular source consumers require the count bound at construction, before factor contractions.

This initial path admits real, full-rank, all-electron RHF/UHF with nonempty active and virtual spaces in each channel. Complex or rank-deficient frames, empty spin spaces and degenerate boundaries remain unsupported. Rank loss needs explicit compatibility with downstream PAO retained-space thresholds. The adapter is dense, reports canonical retained bytes, and advertises no hard memory or scaling bound. It retains the prepared handle for source revalidation and shares the original evaluated build. Separate correlation DF remains same_source=False; immutable MANTA factor/support/rank transport is still separate work. The source kind is molecular-evaluated-df, with experimental status and build/frame/density/source/partition diagnostics in provenance["evaluated_reference"]. There is no automatic run_job dispatch or change to the ordinary molecular identity contract or periodic adapters.

Bounded containing-runtime acceptance covers C1 ordered H2O RHF and OH UHF in STO-3G with def2-SVP RI factors, independent frozen=0/1 and full-space MP2/UMP2/CCSD/UCCSD. Eight evaluated-state calculations agree with independent generalized-eigenvector controls to 1.12e-16 Ha. Maximum raw canonical error is 4.49e-13 Ha, correlation residual 2.06e-12 and absolute energy change 9.38e-14 Ha, below the unchanged raw and 1e-11/1e-13 solver gates. PNO/other corrections and triples are recorded separately. All 157 focused checks and eight exact legacy controls pass. Original reference and factor arrays are unchanged. The new canonical snapshots retain 504/768 bytes for these RHF/UHF fixtures; measured adapter Python allocation peaks are below 24 KiB. These small fixtures are not a native memory, scaling, general open-shell or external-reference qualification. Evidence binds the changed consumer to accepted immutable producer/native operands; later producer implementations and factor bridges need their own containing acceptance.

Dense molecular, Gamma GDF and folded BvK CorrelationSource objects bind a construction-time fingerprint of reference Fock, density, coefficients, spectrum and occupations, energy/spin metadata, explicit overlap, conventions and declared provenance. The four Python MP2/UMP2/CCSD/UCCSD entry points reject changed states or lost convergence before factor work, including directly constructed sources. Rebuild the source after an intentional reference change. Fingerprints use bounded array chunks and retain no reference-array copies; factor tensors are not read. Provider object replacement is rejected, but internal factor mutation and concurrent mutation during a solve are not certified. Gamma GDF and BvK adapters require explicit successful input SCF convergence; missing status is refused. The retained fit, Ewald exchange term and frozen partitions are unchanged. Compact cyclic sources also bind sealed overlap and Fock identities and immutable occupied-projector audit records at construction. Translation selection and admission recheck this binding before using the state, without densifying cyclic operators or reading factors. Existing metric, projector, frozen-partition and covariance checks still apply. Other periodic sources and full stationarity/occupation qualification remain separate.

The molecular adapter also advertises its numerical reference in provenance["reference_state"], with schema molecular-reference-v1. Its streamed digest includes array names, shapes, dtypes and exact values for restricted and independent spin arrays; exact energy, ECP core count and filling metadata are included separately. Equal contents and dtypes yield equal identities across buffer strides. Different reference states no longer share an identity merely because geometry, basis and rounded energy match. This changes newly constructed molecular identities without rewriting old artifacts or changing the shared contract version or periodic identities. It reuses the private mutation fingerprint and adds one streamed reference pass at construction. Factor contents, stationarity and evaluated-HF producer attestation remain separate requirements; same_source=False is unchanged. Directly constructed legacy sources still receive their existing mutation guard, but do not gain this adapter-specific numerical provenance implicitly.

Bounded water/OH STO-3G controls with the retained def2-SVP RI factors and independent zero/one frozen-core partitions preserve MP2/UMP2 and CCSD/UCCSD scientific results exactly against the previous identity adapter, including triples and iteration traces. The maximum raw same-factor canonical error is 4.48e-13 Ha; maximum final residual and energy change are 2.06e-12 and 9.38e-14 Ha under the unchanged 1e-11 and 1e-13 gates. PNO and other corrections are excluded from the canonical comparisons. The 118 bounded checks also preserve dense/compact mutation guards and occupation refusals. Complete source construction peaks at about 131 KiB of traced temporary allocation for a 16 MiB strided complex reference. The added streamed pass has a measured cost; this is no speedup, external-reference campaign or whole-method production qualification.

At a stopped SCF state, density-defining orbitals and physical Rayleigh quotients need not diagonalize the physical Fock. A consumer requiring a stationary HF reference must refuse that state or explicitly implement its own supported nonstationary formulation; it must not silently replace the reference density or spectrum. Semicanonicalization preserves the declared occupied, virtual and frozen partitions. Validate generalized residuals and metric orthogonality in the retained space, occupied projectors and degenerate subspaces, rather than requiring identical eigenvector columns.

The molecular UCCSD rotated-occupied t1 correction now solves each active occupied spin block through the shared dense Fock boundary. The named orthonormal coordinates include the admitted source identity, frozen count and spin; the solver checks the actual block residual and orthogonality. Alpha and beta rotations stay separate even for degenerate eigenvalues, and empty spin blocks are skipped. Each solve snapshot is released before the next spin, without retaining a warm subspace or entering the native residual loop. The compiled-PNO UCCSD preparation path also sends each nonempty retained pair-PNO Fock block through that boundary, adding the active occupied pair indices to the source/frozen/spin coordinate label. It releases the result before the next spin or pair. Pair-density diagonalization, occupation cutoffs and the largest-occupation fallback are unchanged; they determine the retained space before its Fock solve. When singles PNOs are enabled on the compiled route, their retained Fock blocks also use the shared boundary with source/frozen/occupied/spin labels. Density selection, its singleton fallback and optional PAO projection are unchanged. The default full singles space needs no additional solve. Each singles result is released before the next occupied orbital. The density helper retains its old boundary for callers without a named Fock frame. Both compiled and union routes use the cached spin-PAO builder, which names its retained Fock frame by source, frozen count, atom domain and spin. Its PAO overlap eigensolve, strict rank cutoff, projection tails and empty-space policy remain local; only the already orthonormal Fock block enters the shared solve. The result is released before returning, while the existing cache keeps the lifted spin-virtual PAO coefficients. Union pairs and singles now reuse these coefficients for each atom-domain and spin instead of rebuilding the same PAOs. Combined-spin column ordering and empty-domain refusals are unchanged. Retained union pair-PNO and enabled singles-PNO Fock rotations also use the shared boundary. Their named frames include the source, frozen partition and pair or occupied index. Pair coordinates retain the combined alpha/beta space; singles coordinates are combined without PAOs and use the occupied spin with PAOs. Density selection, degenerate spin mixing and the singleton fallback are unchanged. The named density helper copies only the retained rotation to a NumPy-owned buffer to preserve lifting roundoff, then releases the solve result before the multiplication. This also applies to its compiled-singles caller. Default full union singles now use that same checked helper once per calculation, with a source/frozen/combined-spin frame. Identity-density ordering and the full virtual rank are preserved. Occupied orbitals share only the resulting coefficients and energies; each singles amplitude remains independent. The rotated-occupied t1 compatibility path also uses the shared solve for each retained TNO Fock block, naming the source, frozen count, rotated occupied triple and combined alpha/beta coordinates. Triple density selection, its singleton fallback and the existing screening refusal are unchanged; each snapshot is released before the retained-space lift. The Python local-TNO branch now checks its retained Fock block too, covering union T0 and both iterative T1 residual routes. It names the local occupied triple and triples mode while preserving the pair-union SVD rank, density selection and screened triple list. Each snapshot is released before the retained and precursor lifts. Native TNO kernels, other PAO callers and remaining C2 construction/state migrations retain their existing roles. This is the current independent-UHF compatibility correction, not a change to the single-set ROHF/QRO formulation in Saitow et al., DOI 10.1063/1.4981521, Sec. II A.

Restricted DLPNO-MP2 pair PAOs and the occupied-domain dipole-screening prepass also use the named retained Fock solve. Their frames include the source, frozen count, atom domain and pair or occupied index. Full occupied projection, including the frozen columns, stays with the PAO producer; overlap-rank selection, screening estimates and PNO thresholds are unchanged. Only the retained orthonormal Fock block enters the checked solve, and its snapshot is released before returning the PAO coefficients.

Restricted CCSD pair-PAO cache fills now use this boundary for both the semicanonical and LMP2 PNO guesses. Their frame names the source, frozen count, atom-domain cache key and restricted spin. The existing cache shares that frame between pairs and between guess stages without retaining solve snapshots. Occupied projection, overlap rank, screening and PNO construction are unchanged. Extended residual-domain PAOs also use a named checked Fock solve, with the extended atom-domain key distinct from the pair-domain key. The existing domain cache retains only coefficients, transformed factors and Fock blocks, sharing them across occupied coupling groups without retaining solve snapshots. Atom unions, occupied coupling sets and residual contractions are unchanged; the full-virtual route still uses its identity frame without an extended PAO solve.

Restricted CCSD retained pair-PNO Fock rotations also use the checked boundary, distinguishing intermediate LMP2 and final PNO frames by source, frozen partition, pair, guess and stage. Pair-density selection and its largest-occupation fallback remain outside the solve. A writable copy of the small rotation preserves the original lift; the solve snapshot is released before that lift. The optional discarded-PNO complement now uses the same boundary with a distinct discarded-space frame. Its selection, denominators and one-shot tail residual are unchanged. Empty complements perform no solve; final kept and discarded PNOs span the selected parent space. With an LMP2 guess that parent is the retained intermediate PNO space, so the diagnostic does not restore the earlier guess truncation. The tail estimate is reported separately and never added to the production energy.

Reuse source-matched canonical/full-domain and accepted molecular/PBC controls for each changed path. A matching external reference is useful only with the same Hamiltonian, fit and zero-mode conventions. These obligations do not turn shared operator code into periodic correlation accuracy evidence. Native production periodic correlation remains outside the bounded #557 release.

What production means

A route is production when all of the following hold, and the result object, the .out block and the docs say so with numbers.

1. An exactness ratchet against the canonical method on the same source.

2. A matched accuracy measurement at the shipped defaults against canonical references and against ORCA where ORCA can run the same inputs.

3. Measured cost against molecule or cell size, stated as measured; no scaling claim before the measurement exists.

4. Same-source identity in every artifact; refusals that name their reason.

5. Documentation, changelog fragment and citation routes for every published algorithm the route executes.

Molecular closed shell meets 1, 2, 4 and 5; 3 is the open C20/C40/C60 ladder on #190. The 17-molecule TightPNO table is now complete: CCSD / CCSD(T) mean absolute errors are 0.140 / 0.194 kcal/mol against canonical DF-CCSD(T), including C10H22 at +0.595 / +0.831 kcal/mol. Matched-build triples timings and n-butane threshold probes are in the guide; they do not establish CCSD size scaling or change any default. The (T1) screening fix and its acceptance are tracked separately in #506, now closed.

Molecular open shell has its full-domain exactness tests, source artifacts and citation routes, but its broader accuracy gate is open: HCO/cc-pVDZ returned non-finite local (T0) corrections at NormalPNO and TightPNO after CCSD convergence in the 2026-10-10 campaign. The native kernel now checks the triple-space SVD for finiteness, orthonormality and its retained range, and retries a failed decomposition with Jacobi SVD; the solver rejects any remaining non-finite correction with an error. The closed-shell local (T0) kernel reproduced the same basis-dependent failure on an algebraic pair-space control and shares the checked union builder and non-finite-result policy. All nine radicals converge with finite energies at both presets in the rebuilt-core check; canonical C6/C8 radical references are still missing. The broader radical accuracy table and cost gate remain open. The periodic GDF route has the H2 exactness evidence described under B1 and B2 and source identity in the library and runner artifacts. Its broader canonical comparisons, matched accuracy on crystals and cost against cell size remain open; it is not a production route.

Aligned tracker work map

The development tracker umbrella is #508, DLPNO molecular and periodic production readiness: shared engine and aligned work breakdown. Its claimable issues cover the remaining implementation and acceptance work; the landed increments below remain the technical history. Read current issue notes and claim a bounded scope before editing shared interfaces. Existing owners, jobs and accepted evidence remain authoritative.

Work	Tracker issue	Dependency boundary
Reference admission and periodic parity contracts	#572	Consume the common PBC accuracy matrix #573 under #593; add correlation-specific source and Fock admission
Shared Fock-state and remaining pair eigensolves	#580	DLPNO C2 consumers of #472; reuse #577 state/symmetry evidence, preserve libint and bounded native paths
MANTA molecular and periodic correlation sources	#581	Consume producer-owned tiles, retained metric and evaluated Fock; reuse #576, coordinate with #519
Localization, domains and pair/PNO accuracy	#582	Same admitted finite Hamiltonian; independently measure each approximation
Repeated inverse recovery in the factor audit	#583	Continue D17 with complete audits and bounded payload
Reference phases and greedy occupied preparation	#584	Continue D15/D16 without changing seeds, global residuals or frozen projectors
Retained transport links and projected pair workspace	#585	Preserve stable cross-pair contractions and the compiled solver
Translation-native shared CCSD residuals	#586	Consume admitted MP2/source/transport contracts; compare every term with the unreduced engine
Translation-native named perturbative triples	#587	Scoped CCSD acceptance, explicit triples definition and independent nonzero controls
Independent-spin compact periodic sources and solvers	#588	UMP2 first; CC/triples consume #586/#587; no implicit ROHF/QRO conversion
Symmetry unfolding and pair reduction	#589	Shared source, density and retained-space qualifications, then correlation-specific audits
Molecular open-shell accuracy and triples	#590	Reuse closed-shell acceptance and #190 cost work; distinguish UHF from ROHF/QRO references
Complete route accuracy and cost qualification	#591	Accept delivered rows now; consume #573/#574/#576 evidence and add correlation-specific checks
Retained periodic source-factor storage	#592	Shared producer tile/export contract; distinct from consumer audit workspace

Molecular C20/C40/C60 scaling remains #190, and reproducible QVF local triples options remain #558. Canonical chi reference streams remain #492, #518, #521 and #523. Closed scalar-ECP acceptance in #530 is reused only for its qualified scope. Toroidal exchange/seam acceptance remains #566/#568; the BvK-Ewald exchange decision has already been implemented. The amplitude equations retain the SCF Fock and its zero mode, as confirmed by the maintainer’s 2026-10-23 decision in #508. Comparisons and artifacts must identify that convention; no stripped-Fock option is implied.

PBC umbrella #593 owns the ground-state CRYSTAL23 accuracy and full-SCF speed goal. DLPNO consumes its matched cell, basis/ECP, spin, k-point, support and gauge records rather than creating another SCF reference catalogue. CRYSTAL23 mean-field agreement does not establish correlated energy accuracy: use a method-capable independent reference or justified same-Hamiltonian limits, with the comparison boundary stated. MANTA component bounds likewise do not establish whole-method or post-HF qualification. Integral provider, Coulomb algorithm, exchange approximation and method remain separate choices.

The native BvK and periodic coupled-cluster milestones retain production DLPNO work beyond the bounded QC 0.18 PBC/MANTA release gate #557. These issues do not expand that gate or remove experimental warnings. A child dependency names a required interface or acceptance result for its scoped stage, not a wait for every umbrella to close. Source delivery, affected method validation, production qualification and release status stay separate; the umbrella closes only on the measured production criteria above, not on issue counts or source landing alone.

Increments

Status 2026-10-11: A, B1, the B2 fold, the B2b front door and C are on main; the periodic routes carry their exactness evidence and are not yet production routes (see the gates above).

• A. Contract and molecular adapter. Landed: vibeqc.dlpno.source, the molecular adapter and the shim, the source keyword on both engines, the source identity in the manifest and the structured log, and tests that every molecular DLPNO result is bit-identical before and after. No periodic behaviour changed.

• B1. Gamma-point GDF route. Landed: vibeqc.periodic_dlpno.run_periodic_dlpno_ccsd and gamma_gdf_source. Two facts shaped it: the k-mesh GDF driver retains its fit at Gamma only with fit_source="classic" (the default source’s Gamma fast path keeps nothing), and the Gamma eigenvectors come back complex with arbitrary phases, so the adapter realifies them block by block and re-diagonalises the real Fock inside each block. Evidence: the Coulomb energy rebuilt from the factors equals the SCF’s, and full domains reproduce canonical CCSD on the same source. Still open for B1 to count as production: the matched accuracy table and the measured cost.

• B2. BvK-supercell route. Landed: bvk_supercell_source and run_periodic_dlpno_ccsd(kmesh=...). The SCF runs on the unreduced Gamma-centred mesh in cyclic order with its classic fit retained; the per-k overlap, Fock and every retained (k_i, k_j) factor are inverse-transformed to the real torus with the finite-character kernels the chi line owns (Theorem 1 of periodic/ccm/bloch_reference.py), the real supercell Fock is re-diagonalised in the canonical orthogonal frame and its spectrum checked against the union of the per-k spectra, and the engine solves one real supercell problem with the minimum-image distance of the supercell. Evidence: on the H2 chain in STO-3G the folded (1,1,2) and (1,1,3) meshes reproduce the explicit Gamma supercell to 2e-13 Ha in the reference energy and 1e-15 Ha in the correlation energy, with triples, and full domains equal the canonical CCSD on the folded source. Follow-up for the maintainer: the fold kernels are private helpers of periodic/chi/posthf.py; promoting them to a shared periodic module is a small move that touches a file the chi sessions edit every few days, so it was left for an explicit decision.

• B2b. The run_periodic_job front door. Landed: run_periodic_job(method="dlpno-ccsd(t)") for 3D GDF references at Gamma and on full meshes, through the two periodic adapters, with the minimum-image distance in the pair screens and the shipped thresholds rather than the pre-#140/#448 pins the toroidal and chi routes carry. The reference runs on the k-mesh GDF engine with fit_source="classic" and its fit retained; the result lands on the .out, the .system manifest, the structured log, the citations (route keys periodic-gdf-dlpno-ccsd and periodic-gdf-dlpno-ccsd(t) plus the runtime-gated triples kernel) and the QVF. Refused before any SCF work, with the reason: BIPOLE and symmetry-reduced references (decision 2), functionals, dim < 3, smearing, gradients and Hessians. Open shells use increment C below. The provenance helpers both runners share live in vibeqc.dlpno.provenance. For a universal JK fitting basis shared by both codes, pass --aux def2-universal-jkfit to the parity script. Its legacy spelling --aux def2-universal-jk resolves to that same catalog name for every Vibe-QC call, including the explicit supercell check, and for PySCF. Per-zeta def2-{svp,tzvp,tzvpp,qzvp,qzvpp}-jk selections retain their Vibe-QC names and use the corresponding PySCF -jkfit aliases. These per-zeta records match the auxiliary blocks those PySCF aliases load. The script resolves the periodic driver’s default and checks PySCF can load every selected auxiliary element before running Vibe-QC. Missing reference blocks fail explicitly; the script never substitutes a default. Evidence measured on 2026-10-10 with examples/regression/parity_periodic_dlpno_vs_pyscf.py (H2 molecular crystal, STO-3G, same def2-SVP-JK auxiliary): E_HF and the full-domain CCSD(T) correlation energy per cell equal PySCF KRHF/GDF and KRCCSD(T) to 0.00 µHa on (1,1,1), (1,1,2) and (2,2,2) when PySCF keeps the Ewald exchange term in the CC Fock (keep_exxdiv=True), and in def2-SVP the (T) correction per cell on (1,1,2) agrees to 0.001 µHa; the fold equals the explicit Gamma supercell to 1e-13 Ha; the shipped NormalPNO thresholds sit 23 µHa (1,1,2) and 4 µHa (2,2,2) per cell above full domains. Still open: the exactness ratchet against the canonical supercell CCSD(T) of the chi and CCM lines, the n-alkane chain as the 1-D polymer limit of the molecular table, accuracy at the shipped thresholds on real crystals, and cost against cell size.

  Toroidal and chi migration: toroidal DLPNO-MP2 and CCSD(T), chi DLPNO-MP2, UMP2/DLPNO-UMP2 and the unrestricted CC pilot pass the cyclic overlap through CorrelationSource. The closed-shell CCSD(T) migration preceded the MP2 and unrestricted migration. No wrapper replaces the global molecular overlap function during the solve. The MP2, UMP2 and unrestricted pilot engines keep their molecular positional calls and also accept a source keyword. Their results and wrapper results name the source and its identity. The identity hashes the actual Fock, overlap, orbitals and stored factors, with the lattice, mesh and exchange convention. Chi retains its compressed home-auxiliary factors; the identity does not require translated AO copies. Validated classic fit reuse reports same_source=True; a separate chi correlation refit or the toroidal compensated-cell fit reports false.

  The unrestricted source preserves separate alpha and beta coefficients, Fock matrices, orbital energies and electron counts on the common cyclic AO metric. It does not use the restricted real-torus source constructor. Frozen-core spaces, the total-to-per-cell normalisation, exchange zero-mode convention and admission guards are unchanged. Wrappers copy caller options before applying route controls and engines preserve the supplied reference, including when a calculation raises.

  This migration preserves the historical thresholds and pair-mode residual. Under PNO truncation that residual projects coupled amplitudes into the target pair before contraction, unlike the source-space contractions of Riplinger and Neese 2013, Sec. II C, Eqs. (26)-(30). Their Eq. (30) also makes the metric dependence of cross-pair PNO overlaps explicit: each contraction must use the metric of its own reference. Production settings use the shared pno residual; correcting a historical residual is a separate scientific change. The finite BvK representation agrees with the construction discussed by Ye and Berkelbach 2024, Sec. 2, Eq. (1); it does not implement their fragment LNO approximation or the unique home-cell pairs of Nejad et al. 2025, Sec. II, Eqs. (11)-(16), and Sec. IV C. Translational reduction remains increment D. Removing global metric replacements checks calculation isolation, not periodic production accuracy. Bounded tests of the finite Hamiltonian do not establish broader crystal accuracy or cost scaling.

  CCM migration (decision 1): ccm_dlpno_ccsd_coupled now delegates pair construction, iteration and triples to the shared engine through periodic.ccm.dlpno_source.ccm_neutral_source. The adapter retains the cyclic AO overlap and Fock, uses the supercell minimum-image distance, and localises only active occupied orbitals. Its identity hashes the actual factors, Fock and overlap. same_source is true only for retained factors or an identical explicit array; a legacy refit or unverified explicit replacement reports false. The experimental label remains.

  The API keeps its Mulliken domains, legacy PNO density, unscreened pair list, uncorrected PNO tail and local (T0) default. The residual now contracts terms in the source amplitude’s PNO space, following Riplinger and Neese 2013, Sec. II C, Eqs. (26)-(30), instead of first projecting every amplitude into the target pair. This fixes lost pair-interaction contributions under truncation; it intentionally changes truncated energies. triples_mode="dense" uses the shared exact oracle, which diagonalises the active occupied and virtual Fock blocks even for a pre-localised source. The union-PNO driver remains a separate reference.

  Regression evidence: on the four-cell H2 chain in STO-3G, with tcut_pno=1e-4, full atom domains, Pipek-Mezey occupieds and full occupied coupling, the retired loop’s cyclic CCSD correlation energy differs from the full-virtual residual by 42.34 µHa. The shared PNO engine agrees within 1 nHa at the same retained pair spaces. The tests also cover a frozen occupied orbital, canonical CCSD(T) at full domains, and a folded (1,1,2) Bloch reference. These check the adapter and residual, not production accuracy on real crystals.

• C. Open shell on crystals. Implemented through the same Gamma and BvK adapters for run_kuhf_periodic_gdf and run_local_dlpno_uccsd. Both spin Fock matrices and occupied spaces are retained independently; overlap and the retained fit are shared. On an N-cell mesh the multiplicity is 1 + N * (multiplicity_cell - 1), and frozen orbitals scale by N. Integer occupations, a global occupied-virtual gap in each spin, a frozen-active gap when freezing core orbitals, full orbital rank, and both folded spectra are checked before the engine runs. The library and run_periodic_job select UHF for open shells; artifacts and citation routes record the reference and spin counts. The open-shell accuracy set remains the molecular control; this wiring does not qualify the periodic production gate.

  Focused checks in tests/test_periodic_dlpno_gamma.py use H3 doublets in STO-3G at Gamma and on (1,1,2): full-domain UCCSD and (T) agree with the dense spin-orbital reference on the same factors within 1e-8 Ha, both with zero frozen orbitals and one frozen orbital per cell. The folded result also matches an explicit Gamma supercell within 1e-6 Ha in HF and CCSD correlation and 1e-7 Ha in (T). The singlet UHF limit, empty beta occupied space, independent spin projectors, occupation refusals and front-door artifacts are covered. These are adapter correctness checks, not a real-crystal accuracy set.

  Literature comparison: Saitow et al. 2017 (doi:10.1063/1.4981521) use shared ROHF/QRO spatial orbitals and NEV-PNOs (Secs. II A and II B). Our engine retains independent UHF alpha/beta spaces and two-sided pair densities. Ye and Berkelbach 2024 (doi:10.1021/acs.jctc.4c00936) use the same finite BvK equivalence but a spin-restricted fragment LNO approximation. Nejad et al. 2025 (doi:10.1063/5.0290816) exploit translationally equivalent local orbitals and unique home-cell pairs; that reduction belongs to D. C changes the reference adapter and dispatch, not the molecular amplitude equations or PNO ansatz.

• D. k-aware local CCSD(T). Unique home-cell pairs and translational factors L(q, P) from the native chain’s producers feed the compiled PNO-space kernel without a supercell expansion. This is the roadmap’s native BvK milestone; the production integration needs B’s evidence. Compact chi auxiliary storage alone does not supply the pair reduction.

  D1. Finite-torus pair transport and MP2 contractions. The bounded experimental modules vibeqc.dlpno.translation and vibeqc.dlpno.translation_mp2 establish the real, restricted algebra for supplied translation-covariant occupied and pair spaces. They do not change a solver default or add a CC amplitude equation. One AO virtual basis is retained per row of the compact translation-pair plan; ordered placed pairs resolve to that row, a common cell translation and an endpoint-reversal flag. The input occupied coefficients must already be in the declared cell-major gauge. A support permutation, orbital-centre label, or invariant occupied density alone does not establish that gauge. The boundary checks the actual occupied translates, the covariance of the AO overlap and Fock, and the orthonormality and occupied-space exclusion of each pair basis. Optional frozen occupied coefficients carry complete bands in the same gauge; pair bases exclude the full active and frozen occupied union.

  With P_L shifting AO coefficients by the finite cell displacement L, the overlap between representative pair p translated by A and pair q translated by B is

  \[O_{pA,qB}=V_p^T S P_{B-A}V_q.\]

  Cross-pair Fock blocks replace S by F, and occupied couplings are \(f_{iA,jB}=C_{i0}^T F P_{B-A}C_{j0}\). All cell arithmetic is modulo the mesh. A source amplitude contracts as \(O T_q O^T\); endpoint reversal transposes \(T_q\), not the overlap. These are the finite-torus forms of the metric-weighted cross-pair overlaps in Riplinger and Neese 2013, Sec. II C, Eq. (30), and the occupied-coupling terms in Nejad et al. 2025, Sec. V, Eqs. (42)-(44). Nejad’s Sec. IV C, Eqs. (37)-(41), supplies the PNO construction, a separate layer from these contractions.

  For an unweighted ordered spatial-pair energy \(q_p=\sum_{ab}K^p_{ab}(2T^p_{ab}-T^p_{ba})\), a representative contributes \(m_p(2-\delta_{i0,jL})q_p/N\) per primitive cell, where \(m_p\) counts its unordered placed-pair orbit and N is the number of cells. Most off-diagonal representatives have weight 2. A same-band pair at a nonzero self-inverse displacement (\(2L=0\)) has \(m_p=N/2\) and weight 1, as does a home diagonal. The same-band stabilizer must preserve the pair span, and the supplied tensors must respect endpoint reversal in that span. Treating every non-home pair as weight 2 would double count the antipodal pair on an even mesh. Energy orbit weights never multiply the residual coupling terms: each occupied band and cell enters those sums once.

  translation_mp2_residual evaluates one residual per representative, including both occupied diagonal terms. translation_pair_energy_per_cell returns the already normalized primitive-cell energy. Both take the pair exchange blocks and amplitudes in the plan’s representative order; pair virtual Fock blocks need not be diagonal. The tensor consistency checks do not authenticate their Coulomb-factor provenance or establish same_source. The API does not build an SCF reference, select PAO/PNO domains, screen pairs, add discarded-space corrections or iterate to a correlated solution. It requires real restricted references and an explicitly validated translation gauge; it does not identify arbitrary localized orbitals with Wannier translates, repair phaseful or reordered inputs, or constrain independent UHF spaces to a shared restricted frame. D2 below connects these spaces to an admitted source and a representative PAO/PNO builder. Neither this algebra nor its finite-Hamiltonian comparisons establishes periodic production accuracy or cost scaling.

  D2. Source-bound representative PAO/PNO construction. vibeqc.dlpno.translation_builder.build_translation_pair_pnos consumes a real restricted CorrelationSource, the declared mesh, active occupied coefficients and optional frozen occupied coefficients. The active and frozen projectors must each match their source subspace, independently; equality of their union cannot excuse mixing core and active orbitals. Both spaces need complete bands in D1’s explicit translation gauge. Source mesh, finite-torus normalization, convergence, electron count including ECP metadata, and AO dimensions are checked. The builder uses the source’s actual overlap, Fock and factor provider.

  pair_ao_domains supplies one explicit AO-index set per representative; the default uses every AO. The builder projects out the complete active-plus-frozen occupied union and retains the full AO tails of each selected PAO column. It reuses the molecular Gram orthogonalization and Fock semicanonicalization, constructs \(K_{ab}=(ia|jb)\) with the source’s mo_transform, and forms the semicanonical MP2 seed. This follows Riplinger and Neese 2013, Secs. II B1 and II B3, Eqs. (23)-(25), and Nejad et al. 2025, Sec. IV C, Eqs. (37)-(41). It does not yet implement Nejad’s OSV-selected pair domains or periodic local density fitting.

  For generic factor providers, the untruncated PAO exchange block is recomputed at every translated origin and compared with its representative before PNO truncation. This audits the contracted Coulomb operator rather than demanding an identical auxiliary gauge, without expanding compact chi home-auxiliary storage into translated AO copies. D4 below adds a constructive covariance path for a specific immutable factor representation. Both paths preserve the source’s same_source value; covariance does not establish that correlation factors are the SCF fit.

  The density convention is explicit: pno_norm="mp2" selects the shared Riplinger normalization, and the default tcut_pno=0 retains the full admitted PAO space. Positive cutoffs use the existing strict occupation comparison and largest-PNO fallback. The retained PNOs are recanonicalized, and their K blocks and semicanonical seeds are rebuilt. Empty PAO spaces or a retained space that breaks the self-inverse pair’s translation stabilizer are refused, without silently adding PNOs.

  The distinction between published densities matters. With \(u=2T-T^T\) and \(\delta=\delta_{i0,jL}\), Riplinger Eq. (23) uses \(\widetilde T=2u/(1+\delta)\) and \(D=\widetilde T T^T+\widetilde T^T T\). Pinski et al. 2015 (doi:10.1063/1.4926879), the paragraph after Eq. (18), also uses the contravariant-amplitude density; it does not justify the historical bare-amplitude legacy density \((TT^T+T^TT)/(1+\delta)\). Nejad Eqs. (38)-(39) write \(D=2(Tu^T+T^Tu)\) without that diagonal divisor. For the same T this equals the shared mp2 density off diagonal and is twice it on a true home diagonal. A same-band nonzero displacement is not a diagonal pair. The builder keeps the named shared convention and records it rather than silently importing a different occupation scale or threshold.

  TranslationPNOBuild carries the pair spaces, K blocks, semicanonical amplitudes, PAO/PNO ranks and occupations, and a consumed-input identity alongside the original source identity. pno_correction_per_cell is the orbit-weighted difference between semicanonical PAO and retained-PNO energies. It estimates PNO truncation within the selected PAO domains; it does not recover omitted AO domains or solve occupied-Fock coupling. These are prepared inputs and a correction, not a converged MP2 result. D3 below connects these prepared inputs to automatic frame/domain selection and a compiled iterative MP2 solver. Public wrapper defaults, unrestricted equations and historical numerical conventions are unchanged.

  D3. Automatic frames, population domains and iterative MP2. vibeqc.dlpno.translation_driver.run_translation_dlpno_mp2 is an explicit experimental entry point for an admitted real restricted finite-torus source. It connects select_translation_occupied, select_translation_pair_domains, D2’s builder and solve_translation_mp2. It does not change any molecular or periodic job-runner dispatch. The default tcut_mkn=0 and tcut_pno=0 retain full AO domains and the admitted virtual space; positive thresholds request separate domain and PNO approximations.

  The occupied selector constructs the active and frozen frames independently. For either source subspace C, its metric projector is \(Q=CC^T S\). It projects unit-S-norm AO seeds, selecting home-cell seed families together with all their translates. A deterministic greedy rank/conditioning test chooses these families. If their columns form B, symmetric Löwdin orthogonalization gives

  \[W=B(B^T S B)^{-1/2}.\]

  The seed frame intertwines AO and column translations; its Gram matrix and principal inverse square root commute with the column translations. W therefore supplies the required phase-free cell-major gauge without changing the source projector. Actual orthonormality, translation and separate active/frozen projector agreement are audited. n_frozen counts spatial orbitals over the whole torus, must contain complete bands and selects the source’s first occupied columns. The bounded selector refuses a projector for which its greedy common-AO seed search cannot find a sufficiently conditioned frame. This does not establish that no other Wannier gauge exists, nor does the construction optimize spatial localization.

  Domain selection evaluates home-orbital gross Mulliken populations on all atoms of the finite torus, \(q_{iA}=\sum_{\mu\in A}W_{\mu i}(SW)_{\mu i}\), and retains an atom when \(|q_{iA}|>T_{\rm CutMKN}\). Zero is an explicit full-domain setting. Only home-orbital masks are thresholded; other masks are their exact cell translates, and representative pair domains are the endpoint union. Empty orbital domains are refused. Atom AO groups are read from repeated cell-major source shell metadata, or supplied explicitly via home_atom_aos for model sources. The builder retains all AO tails of the selected projected columns. It does not add a domain-omission correction or silently grow a domain.

  This gauge and domain policy have a deliberately limited literature relationship. Riplinger and Neese 2013, Sec. II B1, motivates the initial Mulliken atom criterion and orbital-domain union, but its final PAO-tail domain enlargement and extended CC pair domains are additional steps. Nejad et al. 2025, Sec. III A, Eqs. (17)-(21), uses a Gamma density-Cholesky and per-k polar diabatic starting gauge, then optimizes the fourth-power Bloch-IAO Pipek-Mezey metric in Sec. III B, Eqs. (22)-(23). Its Sec. IV uses integral-screened PAO domains and OSVs. The present projector/Löwdin frame and Mulliken masks do not reproduce those localization or screening algorithms. Ye and Berkelbach 2024, Sec. 2, Eqs. (3)-(5), likewise requires translation-related orbitals spanning the occupied space, but uses PM orbitals or IAOs and a fragment LNO approximation. Its IAOs also span part of the virtual space. D3 remains restricted; it does not implement Saitow’s shared ROHF/QRO and NEV-PNO construction or an independent-spin UHF quotient solver.

  The compiled solver iterates only representative amplitudes with a damped Jacobi update. It evaluates D1’s full raw residual, including every placed occupied coupling and both occupied diagonal terms, and tests the energy and residual on the same evaluated amplitude state. Orbit weights enter the primitive-cell energy only. There is no Fock cutoff, denominator floor or DIIS step. This is the real finite-torus MP2 equation of Nejad Sec. V, Eqs. (42)-(44), with explicit cross-pair overlap transport; it is not a new CC residual or a transfer of the historical CC residual convention. Nonconvergence remains visible in the result. The semicanonical pno_correction_per_cell is reported separately from the iterated energy and does not enter its residual.

  AutomaticTranslationMP2Result retains occupied_selection, domain_selection, pair_build and solution, with consumed-input provenance. Source identity and same_source keep their original meaning; deterministic selection and factor-covariance audits do not authenticate a correlation refit as retained SCF factors. D4 below removes selected dense preparations and repeated transport work. The route remains bounded to small experiments and establishes no production threshold accuracy, optimized locality or measured scaling.

  D4. Compact factor and prepared transport paths. vibeqc.dlpno.translation_factors.TranslationHomeFactors snapshots real home-auxiliary factors in an immutable representation. The full logical factor is defined, rather than sampled, by

  \[B_{tP,\mu\nu}=B_{0P,\mu-t,\nu-t}.\]

  Translating both orbital inputs permutes the complete auxiliary-cell sum, so the contracted exchange block is translation covariant by construction. pair_exchange streams that sum one auxiliary cell at a time. It neither constructs translated AO-factor copies nor retains the full logical auxiliary MO transform. The compatibility mo_transform still returns that complete transform for existing consumers.

  The builder’s factor_audit="auto" uses this construction only for the exact immutable implementation on a matching mesh and AO dimension. A generic provider, including one declaring itself periodic, keeps the all-origin audit. factor_audit="all-origins" forces the reference path for either kind. Provenance distinguishes the paths and records the immutable factor identity. max_translation_error=None means no numerical origin audit was run; it is not a measured zero error. The factor construction proves covariance, not retained-SCF provenance or a physical Coulomb boundary condition, and never promotes same_source.

  PAO preparation now uses a selected-domain, low-rank form of the same projection. With C the complete active-plus-frozen occupied union, \(H_D=C^T S E_D\) and \(A_D=E_D-C H_D\), the domain Gram matrix is \(G_D=S_{DD}-H_D^T H_D\). Its Fock matrix is evaluated as

  \[F_D=F_{DD}-(FC)_D H_D-H_D^T(FC)_D^T +H_D^T(C^TFC)H_D.\]

  Subtraction in \(G_D\) can lose relative precision for nearly dependent PAOs. After the existing rank cutoff selects X, a retained-space Cholesky refinement uses \(V_0=E_D X-C H_D X\) and the metric action \(S V_0=S_{:,D}X-(SC)H_D X\). This uses selected full AO overlap columns and shared SC, without a dense S-times-pair-basis product. It refines the retained metric without a second rank cutoff or threshold change.

  In PAO basis construction, shared occupied contractions replace a full virtual projector and repeated full-AO metric/Fock contractions for every domain. Both Fock cross terms remain; no diagonal occupied-Fock approximation is used. ProjectedPairBasis retains the local and occupied-tail coefficients of \(V_p=E_{D_p}A_p-C B_p\) after PNO rotation. Full AO tails and the existing linear-dependence and PNO thresholds are unchanged. This is the PAO definition in Riplinger and Neese 2013, Sec. II B1, and Nejad et al. 2025, Sec. IV A, Eqs. (26)-(28), evaluated in a different order.

  The default native solver path prepares its fixed cross-pair overlaps and coupling routes before iteration. Given \(P_L C=C R_L\), the compact pair form yields the exact overlap

  \[O_{pq}(L)=A_p^T S_{D_p,D_q+L}A_q-B_p^T R_L B_q.\]

  This selected-block expression is the normal preparation path. A conservative estimate of roundoff and the measured occupied Gram, projection and gauge defects detects targets whose links risk cancellation. Their links instead use \(M_p=A_p^T S_{D_p,:}-B_p^T(SC)^T\) and contract M with translated source coefficient cell blocks. This uses selected full AO overlap columns and shared SC; it avoids a full dense S-times-pair-basis product and a copied translated coefficient array. It changes no pair space, rank cutoff or admission tolerance. The additional AO-row contraction occurs only during preparation. TranslationTransport.stable_links counts these links and is recorded as stable_selected_column_links in transport and solver provenance.

  Both evaluations represent \(V_p^T S P_L V_q\). The prepared solve retains pair-space links and fixed pair/Fock data, so no AO translation or AO metric contraction is performed inside a Jacobi iteration. This follows the metric overlap of Riplinger Sec. II C, Eq. (30); that section also describes storing amplitude-independent pair overlaps before iteration. It does not adopt the paper’s pair screening or its CC amplitude equations. Nejad Sec. V, Eqs. (42)-(44), remains the residual equation, including every placed occupied coupling and the existing energy weights.

  vibeqc.dlpno.translation_transport.prepare_translation_transport returns a reusable TranslationTransport, matched to its consumed spaces by an input identity. The solver accepts transport=... or prepares one with transport_backend="prepared", the default. transport_backend="ao" retains the independent AO reference path. Builds lacking projected-basis metadata contract AO pair coefficients directly; their prepared iterations still operate only in pair spaces. The default max_transport_bytes=256*1024**2 covers retained native numerical/route storage and a conservative preparation workspace, not total process memory. Caller arrays, binding copies, allocator overhead and later amplitude/residual workspaces are excluded. Exceeding the budget refuses preparation without an automatic backend change. Link/coupling counts and retained, workspace and required bytes are recorded in transport provenance.

  This factor path is not Nejad’s periodic local density fitting in Sec. VI, Eqs. (45)-(57), which uses local auxiliary domains, Coulomb metrics and charge projection. Its Sec. VI C also distinguishes translation symmetry of corrected integrals from individual origin-dependent lattice and surface terms. D4 consumes already formed finite-torus factors and preserves their source and zero-mode conventions. Ye and Berkelbach’s finite BvK equivalence and translation-related occupied spaces remain relevant, but no fragment LNO approximation or new localization algorithm is introduced here.

  These are specific storage and reuse changes, not a linear-scaling implementation. For M total AOs, N cells and home auxiliary rank a, the retained home factors still require \(aM^2\) numbers, versus \(NaM^2\) for explicit auxiliary-cell copies. D5 below adds an explicit cyclic-block representation for source overlap/Fock and streams occupied-projector checks. Compatibility pair AO coefficients remain dense. The streamed pair contraction still visits all N auxiliary cells. Prepared links require \(\sum_{\rm links}r_p r_q\) numbers and retain unscreened occupied couplings; full domains may retain large pair ranks.

  D5. Cyclic-block metric and Fock preparation. vibeqc.dlpno.translation_operators.CyclicAOOperator(home_row_blocks, mesh) supplies an immutable real operator through all of its home-row displacement blocks, shaped (N, b, b) for N cells and b AOs per cell. Its convention is

  \[A_{R,T}=A_{0,T-R}=B_{T-R},\qquad (AX)_R=\sum_\Delta B_\Delta X_{R+\Delta},\]

  with displacement arithmetic modulo the declared mesh. All N blocks are retained. The representation does not introduce a distance cutoff, an open-boundary matrix, a cell-normalization factor or a change of AO ordering. Block actions, transposed actions and selected columns provide the metric and Fock operations without constructing an \((Nb)\times(Nb)\) AO matrix.

  Symmetry requires \(B_{-\Delta}=B_\Delta^T\), including symmetric blocks at self-inverse displacements. For Fourier coefficients proportional to \(e^{ikR}\), the reciprocal symbol is \(\widehat A(k)=\sum_\Delta e^{ik\Delta}B_\Delta\). The overlap is positive definite exactly when every symbol on the full finite mesh is positive definite. Admission checks those symbols; it does not infer positivity from the home block, clip eigenvalues or shift the metric. This follows the unitary finite Fourier transform and block-diagonal Bloch overlap/Fock in Nejad et al. 2025, Sec. II, Eqs. (2)-(6). Its Fock transformation in Eqs. (15)-(16) likewise preserves the reference operator. Our cyclic Gamma-centred mesh conventions and treatment of even-mesh self-inverse cells remain unchanged; the paper’s odd-grid examples are not a new admission rule.

  The source remains real and restricted, and must explicitly supply both source.overlap and reference.fock as matching block operators. Dense sources remain supported. There is no automatic conversion or migration of periodic adapters, and no implicit array expansion. The legacy CorrelationSource.overlap_matrix() and transport_backend="ao" refuse block sources; the prepared transport path consumes their blocks directly. The caller remains responsible for supplying the actual finite-source operators. Structural covariance and positivity do not establish a different Fock as source-equivalent or authenticate retained SCF factors. Consumed-input identities record the operator representation, while source identity, same_source, electron counts, frozen-core partition and exchange-zero-mode conventions keep their existing meanings.

  Operator actions now serve source admission, automatic occupied-frame selection, population domains, PAO construction, pair-space validation and native transport preparation. Active and frozen projector comparisons and hashes stream AO rows instead of retaining full projector matrices. Their existing maximum-entry comparison is preserved. Occupied-frame selection forms candidate seed Gram matrices as needed rather than a full AO-seed Gram matrix. Selected columns supply the D4 retained-space metric refinement and stable transport path; these preserve full occupied-projection tails and the existing PAO/PNO ranks and tolerances. The metric-weighted pair transport remains Riplinger and Neese 2013, Sec. II C, Eq. (30), and the MP2 residual remains Nejad Sec. V, Eqs. (42)-(44). Ye and Berkelbach 2024, Sec. 2, Eq. (1), supplies the same finite unit-cell/supercell equivalence, not a new approximation to the interaction or a fragment LNO step.

  Each metric/Fock operator stores \(Nb^2\) numbers rather than \(N^2b^2\). This removes the dense AO overlap/Fock dependency from the explicit block-source preparation path; it does not make the whole calculation sparse. Source and selected occupied coefficients, pair AO coefficients, occupied Fock, pair Gram matrices and home-auxiliary factors still use dense arrays. Full AO domains can still create projected matrices of dimension Nb, filled by row blocks without a second source AO matrix, and all-displacement block actions do not screen work. Existing dense adapters still allocate their original matrices. D6 below adds an explicit chi adapter that supplies these operators directly from its periodic reference. D5 adds no production accuracy, timing or scaling claim.

  D6. Direct chi Bloch-to-block source preparation. vibeqc.periodic.chi.translation.prepare_chi_translation_source runs a fresh restricted chi RI-HF reference with the accepted-reference SCF controls and prepares a CorrelationSource for the experimental translation driver. The source carries cyclic overlap and Fock operators and compact home-auxiliary factors. It does not construct dense finite-torus AO overlap or Fock matrices. This is a separate explicit entry point with unchanged periodic-wrapper and job-runner defaults. D7 below adds optional ChiAcceptedReference reuse.

  With the Bloch convention in D5, its home-row transform is

  \[B_\Delta=\frac{1}{N}\sum_k e^{-ik\cdot R_\Delta}A(k), \qquad A_{R,T}=B_{T-R},\qquad A\in\{S,F\}.\]

  The inverse sign and single factor of \(1/N\) follow from Nejad et al. 2025, Sec. II, Eqs. (2)-(6), with BvK orbital coefficients in Eqs. (9)-(10). The forward symbol in D5 recovers the original Bloch operator. Admission requires the full Gamma-centred cyclic mesh and uniform weights, compatible time-reversal partners, real home-row blocks within tolerance, and positive-definite overlap symbols. Per-k orbital spaces must be full rank and satisfy the supplied generalized Fock eigenproblem. The adapter refuses unsupported matrices or fillings. The existing global overlap-rank tolerance and transpose average after admission are preserved, with no new metric cutoff, phase averaging or Fock shift.

  Real canonical finite-torus orbitals are formed from time-reversal partner eigenspaces with block operator actions. Arbitrary orbital phases and rotations within degenerate eigenspaces do not require coefficient-by-coefficient conjugate matching. The preserved occupied and virtual spaces have a global gap; a nonempty frozen/active split requires its own global gap and complete bands. n_frozen is the total number of frozen spatial orbitals on the torus, and the caller passes that same count to run_translation_dlpno_mp2. Reference energy and electron counts remain extensive, while the MP2 driver reports energy per primitive cell with its existing normalization. The supplied SCF Fock retains the existing BvK exchange-zero-mode term.

  fit_source="classic" reuses the canonical factors only after the existing chi admission validates the executed source, factor digest, auxiliary frame, mesh and numerical support; this is the same_source=True case. fit_source="auto" retains the existing separate SCF and classic correlation fits and records same_source=False. Equal requested cutoffs or constructive factor covariance do not promote a refit to retained-source identity. The home-auxiliary transformation preserves the supplied finite fit and does not introduce Nejad’s local auxiliary domains, chargeless projection or surface correction from Sec. VI, Eqs. (45)-(65).

  Nejad’s Sec. III A, Eqs. (17)-(21), motivates treating the Bloch gauge explicitly, but this realification is not its diabatic Wannier or optimized IBO procedure. Ye and Berkelbach 2024, Sec. 2, Eq. (1), and Sec. 2.1, Eqs. (3)-(5), supply the finite-representation equivalence and occupied-span requirement, not their fragment LNO approximation. Nejad’s odd Monkhorst-Pack grids do not replace our full cyclic even/odd mesh contract. This adapter remains restricted, real and three-dimensional; it does not widen UHF, ECP or explicit fitted-support admission.

  Canonical orbital coefficients remain dense, as do the downstream occupied and pair coefficients, projected domain matrices and compact home factors described above. Removing dense source S/F folds is a representation change, not a new scientific approximation or a measured cost claim. D7 below adds generic GDF and accepted-reference block adapters; smaller remaining orbital/factor payloads and qualified localization and screening remain implementation gaps. Periodic DLPNO production accuracy is not established by this increment.

  D7. Attested generic GDF inputs and reusable chi block references. vibeqc.dlpno.translation_gdf.gdf_translation_source(system, basis, result, mesh=..., aux_basis_name=None, n_frozen=0) prepares a block source from an existing direct multi-k GDF RHF result. It does not run SCF or build a replacement fit. Admission requires the executed classic source descriptor and its retained canonical AO-pair factor mapping, with matching numerical digest, orbital/auxiliary bases, mesh and finite support. A legacy cache object alone is not evidence that the factors are those used in SCF. The initial route refuses eigenmode auxiliary frames rather than inferring an unrecorded auxiliary rotation. same_source=True follows only after this attestation.

  The adapter preserves the executed physical_pairs or historical_index_ball support policy. It admits BvK-ewald exchange zero mode and keeps the source Fock and one-electron support. It does not silently replace a historical finite fit with the physical-pair fit, strip the exchange shift, refit the interaction or change normalization. Cartesian-only k-point data are supplemented with derived fractional coordinates on a detached input view and checked against the requested complete cyclic mesh. The caller’s result stays unchanged. D6’s full-mesh, realification, global-rank and band-gap guards apply, and n_frozen counts spatial orbitals over the entire torus. UHF, ECP, COSX, SR and explicit fitted-support sources, explicit reciprocal tails and symmetry-unfolded references remain outside this generic adapter.

  prepare_chi_reference(..., matrix_representation="cyclic-blocks", n_frozen=...) builds a process-local accepted handle with the direct block S/F transform and compact home factors. The existing default is matrix_representation="dense"; legacy consumers continue to require dense handles. prepare_chi_translation_source(..., reference=handle) validates the original geometry/basis, controls, process/runtime, state, factor and metadata seals. Fractional and Cartesian k points, numerical tensors and their representation are included in consumed state validation. A block handle also binds the exact total frozen count and reuses all transforms. A dense handle is migrated from its sealed Bloch state directly to block S/F, with a fresh audit of the requested frozen partition; its existing compact home factors are reused. This migration does not undo dense storage already retained by the original handle.

  Returned sources have detached orbital/energy arrays and provenance; only immutable block operators and compact factors are shared. Reusing an auto handle reuses its already separate correlation fit and remains same_source=False. Its correlation_fit_status="reused" describes an execution stage, not equality to the SCF fit. Accepted reference provenance records the seal, consumed state and real-state identities, original matrix representation, and whether the Bloch transform was reused or built from the sealed Bloch state. Reuse is process-local, with no disk cache or relaxed source admission.

  The scientific distinction is between changing a tensor representation and changing its finite operator. Nejad et al. 2025, Sec. II, Eqs. (2)-(10) and (15)-(16), and Ye and Berkelbach 2024, Sec. 2, Eq. (1), justify the unitary finite Bloch/BvK transform. They do not authenticate a cached fit or make a refit source-equivalent. Riplinger and Neese 2013, Sec. II C, Eq. (30), likewise requires the metric of the actual pair spaces for cross-pair transport; stored transport is reusable only for its admitted inputs. D7 changes source preparation and reuse, not that pair algebra, PNO thresholds or MP2 residual conventions.

  A bounded generic-GDF comparison uses all-electron RHF H2/STO-3G, a 1.4-bohr H-H bond centred in a cubic 6-bohr cell, the (2, 1, 1) cyclic mesh, physical_pairs classic factors, BvK Ewald exchange, no frozen core and full AO/PNO spaces. The focused regression compares three contractions of the same executed finite Hamiltonian:

  MP2 representation	Correlation energy (Eh/cell)
  Block source and representative translation solver	-0.00970871869654531
  Dense finite-torus canonical contraction	-0.009708718696547205
  Independent character-space contraction	-0.009708718696547193

  The largest energy difference is \(1.90\times10^{-15}\) Eh/cell and the maximum raw iterative residual is \(3.32\times10^{-14}\). Rebuilding SCF Coulomb energy from the retained factors gives zero recorded discrepancy for this case. This is an internal finite-Hamiltonian comparison; external reference accuracy, mesh/basis convergence and broad periodic qualification remain open.

  A separate accepted-reference chi check at the same geometry, basis and mesh retains chi’s own finite support. It gives -0.00970871869562221 Eh/cell through the block translation solver and -0.009708718695624109 Eh/cell through its dense finite reference, a difference of \(1.90\times10^{-15}\) Eh/cell with maximum raw residual \(3.32\times10^{-14}\). This checks reuse within the chi source; the generic and chi comparisons do not establish equivalence between their distinct fitted operators.

  Real folded canonical coefficients remain dense. D9 below streams their complex preparation from Bloch data. Compact home factors, compatibility pair AO coefficients and unscreened pair links remain substantial payloads. Reducing these, qualifying localization/screening and extending separately attested source coverage are the next implementation gaps. The new adapters establish neither production accuracy nor cost scaling.

  D8. Retained classic chi block adapter. D6-D7 cover conventional GDF and ChiAcceptedReference blocks. The subsequent vibeqc.periodic.chi.local_mp2.build_chi_translation_source consumes an admitted classic physical-pair RI RHF ChiHFReference. Direct Fourier home-row overlap/Fock blocks preserve the actual SCF Hamiltonian and its BvK exchange zero mode. No dense supercell AO overlap/Fock is built. Active and frozen occupied subspaces are separately realified and their original complex metric projectors audited in streamed rows. The native compact home-auxiliary transform must recover every complex input factor block on a forward transform. No refit, SCF replay, rank change or historical torus source is used.

  run_chi_local_mp2 connects this source to D3-D5 with independent domain and PNO cutoffs, iteration controls and explicit experimental output. Raw energy and the semicanonical discarded-PNO estimate are separate; no domain correction or pair screening is added. Scalar ECP metadata is preserved without claiming #530 acceptance. Unsupported spin, SR, reduced/fractional/overlap-truncated and non-real spaces refuse. Dense orbital, factor, pair and transport payloads remain; the exposed budget covers prepared native transport only. The chi local-MP2 guide specifies the math, memory scope, source checks and reproducible vq cases. Scientific acceptance and D93 qualification remain separate from delivery.

  D9. Streamed canonical orbital realification. The D6-D7 adapters now form each complete degenerate eigenspace directly from the primitive Bloch coefficients. For an orbital labelled by k and band p, its finite-torus AO column is

  \[C_{R\mu,kp}=N^{-1/2}e^{ik\cdot R}C_{\mu p}(k).\]

  This is Nejad et al. 2025, Sec. II, Eqs. (3) and (9), evaluated only for the columns currently needed. The adapter sorts orbital indices and energies, without constructing the full complex folded coefficient array or its globally reordered copy. Phase panels are formed only for the current eigenspace. The dense and streamed entry points use the same real-span rank test, metric orthogonalization, Fock rediagonalization and eigenvalue tolerance. A complete degenerate space is never split to reduce memory. Frozen/active and occupied/virtual gap guards, complete-band counts and arbitrary complex gauges are unchanged. Final orthogonality and stationarity checks visit every orbital in b-column panels, including overlaps between different eigenspaces, with the same maximum-entry residual definitions.

  For M = Nb total AOs and largest degenerate space d, the constructed complex coefficient block has Md entries instead of M squared. Phase panels have Nd entries. Final validation uses Mb-entry panels instead of another full S-times-C array and square Gram/residual workspaces. These are array dimensions, not process memory or wall-time bounds; eigensolvers and block actions have additional workspace. Broad degeneracy can still make d large. Provenance records max_folded_orbital_columns and orbital_validation_columns.

  The default source snapshots and accepted-reference seals retain the real M-by-M canonical array. Selection reads its active and frozen occupied columns; PAO preparation projects their complete union; factor contractions and transport then consume occupied and pair AO coefficients. The prepared native solver receives pair-space data. D10 below replaces that retained array with optional compact occupied Bloch projectors across admission, selection and seals. This increment changes no PAO domain, PNO density, factor fit, transport overlap, residual, energy normalization or same_source claim.

  Riplinger and Neese 2013, Sec. II C, Eqs. (26)-(30), still requires the actual pair metrics and mixed-pair contractions downstream. Ye and Berkelbach 2024, Sec. 2, supplies the finite BvK equivalence, not a reason to introduce fragment LNO truncation here. Saitow et al. 2017, Sec. II, assumes one ROHF/QRO orbital set for its NEV-PNO construction; it does not justify replacing independent UHF spaces. Molecular and unrestricted adapters retain their existing calls and orbital spaces. This storage change does not establish production accuracy or scaling.

  D10. Compact occupied projectors across source consumers. compact_occupied=True on chi translation preparation, generic GDF adaptation or chi block-reference preparation retains independent frozen and active home-row metric projectors instead of the real M-by-M canonical array. For each occupied partition X,

  \[Q_X(k)=C_X(k)[C_X(k)^\dagger S(k)C_X(k)]^{-1}C_X(k)^\dagger S(k).\]

  The Gram inverse removes normalization roundoff within each partition; it cannot rotate frozen orbitals into the active space. The inverse finite Fourier transform yields Q[R,T]=Q[0,T-R]. These metric projectors are generally nonsymmetric and are not transpose averaged. This is the occupied projection of Riplinger and Neese 2013, Sec. II B1, in the Bloch convention of Nejad et al. 2025, Sec. II, Eqs. (3) and (9). It does not implement their optimized Wannier gauge.

  Preparation still performs the complete D9 real canonical audit, including virtual orbitals and all existing gap, rank, orthogonality, stationarity and reality guards. Both compact projectors are compared against that audited canonical split in streamed AO row blocks. The maximum entrywise projection error, occupied orthogonality defect and imaginary residual are retained and must also meet the consumer’s tolerance. Compact data bind the mesh, electron count, frozen count and exact overlap/Fock identities. The compact reference exposes no mo_coeffs fallback. D11 below removes the temporary canonical array used by the initial D10 implementation.

  Selection applies each projector to the same normalized AO seed families. The seed Gram is the corresponding submatrix of \(Q_X^T S Q_X\), computed from cyclic symbols without assuming exact numerical idempotence. Greedy scores, tie order, linear dependence threshold and symmetric Lowdin construction remain unchanged. The resulting real frame passes the existing metric, projector, partition and translation-gauge checks. Cyclic covariance is constructive, so compact projectors require no all-origin covariance scan. Provenance and accepted-reference seals hash their home blocks and audit metadata; they do not expand AO projectors or restore canonical coefficients. Reuse shares only immutable projectors and operators while detaching mutable reference state, energies and provenance. Block handles require the same explicit occupied representation and frozen partition.

  Retained projector storage is two N-by-b-by-b real arrays rather than an M-by-M real canonical array; the no-frozen case currently retains a zero frozen array. Full one-dimensional energies remain available. This is a retained-storage reduction, not a peak preparation-memory bound: selected occupied frames, PAO/pair coefficients, home factors and solver payloads remain. D11 addresses the temporary canonical audit. D9’s literature distinctions, experimental scope, independent-spin boundaries and unchanged downstream metrics/residuals still apply. Existing defaults remain unchanged.

  D11. Replayed complete-spectrum admission. Compact preparation now uses a replayable eigenspace realifier shared with the default dense adapter. It stores sorted energies, indices and eigenspace boundaries, and produces one complete real block at a time. Its real-span Gram rank test, metric orthogonalization, projected Fock diagonalization and eigenvalue tolerance remain unchanged. All occupied and virtual blocks are visited. Degeneracy and the global frozen/active and occupied/virtual gap rules are unchanged.

  The audit replays blocks to evaluate every entry of \(C^T S C-I\), including entries between distinct eigenspaces, and every AO entry of \(FC-SC\epsilon\). It retains the original maximum-entry residuals and 1e-8 admission threshold. The occupied-only Gram defect is recorded separately for the subsequent consumer tolerance. Neither per-k checks nor a unitary-invariant norm replaces these global checks.

  Frozen and active projector comparisons accumulate \(\sum_X C_X[\mathrm{rows}]C_X^T S\) one AO row tile at a time, where X runs over the complete eigenspaces in that partition. Every AO row is compared with its compact projector at the original maximum-entry tolerance. Agreement of the occupied union, or of the home row alone, cannot certify a frozen split. This retains D10’s numerical comparison; it adds no all-origin covariance scan or localization approximation.

  The assembled M-by-M canonical array and the full occupied audit frame are absent. Workspace contains a fixed number of M-by-d eigenspace arrays, real-span Gram/eigensolver work of order d squared, and b-by-M row tiles, where d is the largest complete degeneracy and M = Nb. Broad degeneracy can still require quadratic workspace. Replay trades work for storage: B spectral blocks require B squared block reconstructions for the global audit, plus N times the occupied block count for the projector comparison. D11 originally repeated both eigendecompositions on each reconstruction; D12 below caches bounded reconstruction recipes. orbital_audit="replayed-complete-eigenspaces" and orbital_audit_block_evaluations expose every reconstruction; the existing column-width diagnostics remain available. No time or total-memory scaling claim follows from eliminating the assembled array.

  D10’s cited Bloch/projector formulas and scientific scope still apply. Bounded regressions compare the replayed audit with the assembled reference, including complex phases, large degenerate blocks, virtual residuals, cross-eigenspace overlaps and frozen-space mixing. Default dense, molecular and independent-spin consumers keep their assembled outputs. Downstream occupied and pair storage remains a separate gap.

  D12. Bounded eigenspace reconstruction recipes. Compact preparation now retains small transforms for repeated visits to the same complete eigenspace. For a d-column complex block, let R contain its real and imaginary columns. The recipe consists of the retained Gram eigenvectors V, square roots w of their eigenvalues, projected Fock rotation U and resulting energies. Each visit reconstructs \(C=(RV/w)U\) in the original multiplication and division order. No AO coefficient block is cached, combined into a full canonical array or retained on the accepted reference.

  The four owned, immutable recipe arrays require \(8(3d^2+2d)\) payload bytes. Their total payload is bounded by one primitive Bloch coefficient tensor, \(16Nb^2\) bytes for N cells and b primitive AOs. Admission is first fit without eviction; a complete recipe that exceeds the remaining budget is recomputed on every visit. The budget covers retained numeric payload, not Python metadata or transient eigensolver workspace. Large degeneracies are never split or silently admitted to exceed this budget.

  Recipes belong only to one preparation call over its private Bloch coefficient snapshot and immutable cyclic operators. They are cached only after the unchanged real-span rank and Fock eigenvalue checks pass. Replays still reconstruct and check finite input columns, and every global overlap, AO stationarity and separate frozen/active projector comparison from D11 still runs. Returned energies are detached from the recipe. Default dense, molecular and independent-spin consumers do not enable this cache. The Bloch-transform orbital_audit_recipe_cache record reports the policy, byte budget, stored payload, entry count, hits and fresh realifications; the existing block-evaluation count still includes both fresh and cached reconstructions.

  This reschedules the same D9/D10 literature-based transformations; it introduces no new scientific approximation or orbital gauge. Bounded checks compare against the assembled reference, forbid dense fallback, count fresh decompositions, exercise insufficient cache budgets and preserve failure and caller-state guards. Global block reconstruction and contraction work, uncached wide eigenspaces, and downstream dense occupied/pair storage remain concrete optimization gaps. D14-D15 below address retained pair storage and compact chi selected-frame preparation. D13. Retained-Bloch Coulomb contractions for chi local MP2. The D8 adapter now selects TranslationBlochFactors by default. It borrows the sealed executed canonical AO factors, including null modes, metric ranks, physical supports and source identity. The shared pair builder calls its bounded pair_exchange; PAO/PNO construction and the iterative solver are unchanged. Explicit factor_provider="dense" retains the former home-auxiliary oracle, with no automatic fallback.

  With the finite unitary convention of Nejad et al. 2025, Sec. II, Eqs. (2)-(10), the occupied/virtual contraction for momentum transfer q is \(X_q=N^{-1}\sum_{j-i=q}c_i^{\dagger}L_{ij}c_j\). The auxiliary-cell transform is \(B_t=\sum_q e^{iqt}X_q\). For real pair coefficients, character orthogonality yields \((ia|jb)=N\sum_{qP}X_{q,Pia}X^*_{q,Pjb}\). The shared energy normalization remains separate. The chi guide derives the phases, conjugations and factors of N, and specifies the complete source-recovery and covariance audits. Every AO/auxiliary entry is checked at the existing 1e-10 tolerance in primitive-AO-pair/auxiliary tiles; no full folded tensor enters preparation, provenance, validation or the pair contractions.

  A separate max_factor_bytes control covers serial provider numerical payload, including temporary transforms, dtype conversions, returned buffers and a bounded phase cache. The implementation does not call a native factor binding. The cap excludes the retained source, reference admission, caller inputs, previously returned buffers and downstream work; it is not a total-memory bound. D17 below removes the initial quadratic minimum audit workspace. Bounded measurements record both reduced allocation and the slower complete Python audit.

  Riplinger and Neese 2013, Sec. II B3, Eqs. (23)-(25), and II C, Eq. (30), and Nejad Sec. V, Eqs. (42)-(44), still define the shared PNO and residual conventions. This increment adds no local fit, integral screening, optimized Wannier procedure or domain correction. D9-D12’s orbital audits and reconstruction recipes are not duplicated. Scalar ECP metadata and separate frozen projectors retain D8’s admission scope. Source tests compare both providers with the independent same-source canonical limit and compare raw energies, residuals and separate PNO estimates. Scientific vq acceptance remains separately recorded; chi is experimental and D93 fitted/post-HF is not-qualified. D14 addresses retained selected occupied and pair-AO storage; the audit’s minimum character tile is addressed by D17 below.

  D14. Shared selected columns and recipe-only pair AO storage. Chi local MP2 defaults to compact_storage=True; the generic shared driver/builder opt in explicitly. The phase-free gauge gives \(C_{R\mu,Ti}=C_{(R-T)\mu,0i}\) with finite-torus modulo arithmetic. TranslationOccupiedColumns retains immutable active and frozen home columns separately. Every reconstructed column is compared with the original selection at the existing maximum-entry tolerance. Complete source-projector admission then consumes that reconstruction, including the frozen partition; covariance alone does not establish same-source identity. Selection, source inputs and pair spaces share those buffers.

  The existing ProjectedPairBasis supplies \(V_p=E_{D_p}A_p-C B_p\). ProjectedPairCoefficients reconstructs individual PAO/PNO bases by summing the translated occupied home families, retaining all tails. Every PAO and rotated PNO is compared entrywise with the dense algebra. Full orthonormality, occupied exclusion, self-inverse stabilizer and factor covariance checks keep their original tolerances. No all-pair AO list is retained or passed to pybind. The native prepared transport accepts empty AO coefficients only alongside complete validated recipes. Its existing stable-link evaluation reconstructs one source pair and preflights an additional largest AO rectangle within its transport cap. Complete recipe/source hashes seal reuse without AO reconstruction.

  Nejad Sec. II after Eq. (13) states the translated-column identity; Sec. IV A, Eqs. (26)-(28), gives BvK PAOs. Riplinger and Neese Sec. II B1 supplies occupied projection and II C Eq. (30) supplies cross-pair overlap. This storage change reuses those equations and D4’s stable evaluation; it adds no localization optimization, cutoff, refit or new solver. Dense coefficient storage remains an explicit independent oracle. Source checks cover complex Bloch phases, nonorthogonal bases, odd/even meshes, frozen partitions, ill-conditioned domains, budget boundaries, raw iteration/residual agreement and the independent canonical limit. Bounded measurements report retained bytes and timing without claiming linear scaling or total-memory bounds. Full selector Gram matrices, transient full selected frames, SC/FC, projected domain matrices and the prepared link graph remain separate gaps, with the selected-frame preparation addressed by D15 below. Scientific vq acceptance remains pending and separately recorded; chi stays experimental and D93 fitted/post-HF stays not-qualified.

  D15. Cyclic occupied selection and streamed preparation. Compact chi preparation decomposes the translated normalized-AO seed Gram into character blocks. Its complete spectrum is the union of those blocks; greedy scores, global relative conditioning, candidate/tie order and the symmetric Lowdin gauge are unchanged. Home columns use the inverse transform of \(Q(k)E_JD^{-1}G(k)^{-1/2}\) with normalization \(1/N\). Complete degeneracies remain in the matrix function; frozen and active partitions are independent. The chi guide derives the spectrum, normalization, inverse root and real reconstruction, including self-inverse characters. The dense selector remains an explicit small-system oracle.

  The chi adapter evaluates its original real-span Gram spectrum with conjugate-character blocks, avoiding the full folded occupied span. It compares all entries of the normalized span projector with the original complex source projector. Canonical source metric/stationarity admission and the other adapters’ D9-D12 replay remain unchanged. Selection, domain preparation, PAO construction and source provenance consume streamed columns or home buffers. All global selected Gram, separate source-projector, frozen-cross and generator checks remain exhaustive at their original tolerances. Full selected C, SC and FC are absent, including the native transport boundary. Stable transport reconstructs one occupied column and one source pair at a time and charges them to its existing cap.

  max_selection_bytes preflights controlled numerical array payload for the compact occupied-source/selector phase, with explicit charges for eigensolver inputs/outputs, transform copies, home snapshots, reconstruction and audit buffers. Native library internal scratch, caller arrays, canonical admission and downstream pair work are excluded; this is not a process-memory cap. Infeasible budgets refuse with no dense fallback. The bounded ladder reports retained and traced preparation bytes and time. Occupied Fock, projected domain matrices, individual pair bases, amplitudes and retained transport links remain dense; all-entry replay can cost more time as the mesh grows.

  Nejad Sec. II Eqs. (2)-(10) supplies the unitary convention and translation identity; Riplinger and Neese Sec. II B1 supplies the PAO projection. The derivation reschedules those operations without optimizing a new Wannier gauge, screening tails or changing the PNO/residual conventions. Source checks retain independent same-source canonical MP2, raw residuals and separately reported corrections. Scientific acceptance remains pending; chi is experimental and D93 fitted/post-HF is not-qualified. The next bounded gap was exhaustive occupied-audit replay cost, addressed by D16; D17 removes the minimum quadratic factor-audit tile, while the retained transport graph remains separate work.

  D16. Complete occupied audits through verified cyclic blocks. Immutable selected home columns and cyclic overlap now give the complete Gram symbols \(c_X(k)^\dagger S(k)c_Y(k)\) and separate projector symbols \(c_X(k)c_X(k)^\dagger S(k)\). Every displacement block is inverse transformed before evaluating the original maximum real-entry norm, with the identity subtracted only from the zero-displacement same-partition Gram. The active and frozen projector targets remain independently source-bound. Admitted overlap asymmetry is preserved; no metric adjoint or idempotence shortcut changes a contraction.

  The chi source realifier checks its actual complete character table and ordering. Its real conjugate-pair Gram uses separate real/imaginary entry maxima, not the complex symbol magnitude. Both original and real-operator metric actions, complete self-inverse blocks and all complex source-projector displacement entries remain checked. Translation is the verified permutation identity over these complete blocks. Unverified tables, dense coefficients and custom representations retain streamed replay. Near a floating-point admission boundary, the original arithmetic determines acceptance at the unchanged tolerance. The manual derives both original representations and their norms.

  A single preparation context reuses one overlap and independent active/frozen symbol tensors plus at most sixteen scalar measurements. Exact source, mesh, operator, partition and column-content identities bind reuse. The selection ledger charges all intermediates, phase snapshots, cache payload and outputs before allocation. The cache expires on return or exception; no full selected C/SC/FC, translated seed Gram or dense AO projector returns. The native transport boundary retains its independent streamed checks. Provenance reports measured defects, formulations, displacement work, cache limits/hits and threshold-boundary replay. Equivalent checks are not described as skipped or as new scientific acceptance.

  The bounded benchmark separates source realification, selected-source admission, selection and complete occupied preparation. It records time and storage regressions as well as improvements, with operation guards instead of timing assertions. The independent canonical MP2 comparison, raw residuals, domains, PNO spaces and separately reported corrections remain in the focused tests. Nejad Sec. II Eqs. (3), (9), (10) and the translation identity after (13), and Riplinger/Neese Sec. II B1, supply the same equations as D15. No optimized Wannier gauge, screening or new scientific scope is introduced. Chi remains experimental and D93 fitted/post-HF remains not-qualified; scientific vq acceptance remains separate. Repeated greedy candidate-Gram construction is the next occupied-preparation cost; one-pair reconstruction and the retained transport graph remain. D17 addresses the factor audit workspace.

  D17. Character/displacement-tiled retained-factor audit. The streamed factor provider now reconstructs the complete retained Fourier audit using bounded phase rectangles and linear d*N*p slabs. For each target character tile I it recomputes every inverse displacement block B, checks its imaginary entries and separately transformed reversed AO block, then forward transforms the same real transpose average. Every I,J,P,mu,nu entry is compared with the original immutable source, including canonical null modes. Every character/origin/shift phase identity is still measured. No sampling, occupied projection, factor compression or energy proxy enters admission; the 1e-10 tolerances and real-representation scope are unchanged.

  Nejad Sec. II Eqs. (2)-(10) gives the Fourier convention. The manual derives the two inverse sums and forward recovery. Pair Parseval and complex MO contractions use bounded phase blocks too. A one-entry cache is bound to the sealed source, mesh and character ordering; it cannot retain a complete table even under a large budget. Dense factors remain an explicitly selected small-system oracle. The source’s own quadratic retained-factor storage and the earlier reference/occupied phase table are outside this provider workspace and remain separate limitations.

  max_factor_bytes preflights phases, cache replacement/snapshot copies, inverse/forward partial sums, conversions, outputs and iterator allowance. The minimum audit reservation is the iterator allowance plus 320*N bytes. Contractions additionally reserve their unavoidable outputs; an insufficient cap refuses without a dense fallback. Output records tile dimensions, work and cache counters, measured defects and accounted peak. Recomputing inverse blocks for each target tile trades runtime for the smaller workspace. Measurements report this tradeoff, with no linear-scaling or total-memory claim.

  Complete-table comparisons include late-tile and null-component defects, threshold cases, reordered characters, odd/even multidimensional meshes, complex contractions, nonorthogonal references and frozen partitions. The shared PNO/residual conventions of Riplinger/Neese II B3 and II C remain unchanged; raw energies, residuals, separate PNO estimates and the independent full-space same-source canonical limit remain focused checks. Scientific vq acceptance is separate; chi remains experimental and D93 fitted/post-HF remains not-qualified. The next factor gap is repeated inverse recovery work, addressed in part by D18 below. D19 removes the complete reference phase table and reuses greedy occupied-Gram contractions; the retained transport graph remains separate work.

  D18. Bounded reuse inside the complete factor audit (#583). After the D17 tile plan, spare workspace and cache capacity can retain the first audited displacement slab for one AO pair and auxiliary tile. The immutable snapshot contains the original real transpose average after measuring its imaginary and symmetry defects. Subsequent target character tiles reuse it in the same recovery sum order; every retained source entry, null component and phase-action identity is still compared. Other slabs retain D17 recomputation, and minimum budgets keep that exact path.

  The additional reservation is 16*N*d*p, covering snapshot construction and concurrent lifetime. Only one slab exists, d remains capped at eight and below N for non-Gamma meshes, and the slab is released before the next AO/auxiliary tile or return. Source, mesh and character ordering bind reuse. No complete character, home-factor or quadratic audit array is restored. Cache work counters and separate accounted/traced measurements accompany the unchanged complete-table, threshold, frozen-partition and residual regressions. The benchmark records the limited reduction; repeated work outside the first slab remains. The Fourier formulation and cited Nejad equations are unchanged. This scheduling improvement does not qualify the experimental chi method or expand the PBC/MANTA release gate.

  D19. Bounded reference phases and greedy primitive-Gram reuse (#584). Compact chi reference/occupied preparation generates bounded phase blocks, including forward recovery, real-span audits and threshold replay. It keeps complete reduction axes and no phase cache. Immutable source, mesh and character-order bindings replace the mandatory complete phase table. Independent active/frozen selectors reuse the principal submatrices of D^-1 Q(k).H S(k) Q(k) D^-1, preserving every conditioning eigenvalue, candidate order and tie envelope. Boundary decisions replay the original contraction; winning scores and final symmetric Lowdin frames use it too. The partition-local primitive Gram is released on success or failure. The selection ledger preflights its payload and bounded phases, including live construction/copies and outputs. The manual gives the derivation and excluded library scratch/source/downstream storage. No full selected frames, translated-seed Gram or AO projector is restored. Generic inputs retain exhaustive checks. Scientific acceptance remains separately pending; chi remains experimental and D93 fitted/post-HF not-qualified. This source increment does not expand #557’s bounded 0.18 production release gate.

  D20. Raw reversed-AO audit reuse beyond the first home slab (#583). After D18’s unchanged tile plan and optional averaged first-R slab, another separately budgeted immutable slab can retain raw complex B_nu,mu[S,R] for the first S tile and all R. Later displacement and target tiles reuse these values for the original symmetry comparison and real transpose average. The raw cache is never averaged or realified. Full imaginary, symmetry, retained-source recovery and actual phase-covariance checks remain exhaustive, including null components and threshold cases.

  Each optional slab reserves 16*N*d*p extra bytes; both and the largest phase snapshot must fit the cache ceiling. Their lifetimes end at the AO/auxiliary tile boundary. No quadratic audit array or full canonical array returns, no tile shrinks to fit reuse, and tighter budgets preserve D18 or D17 recomputation. Complete-table and frozen-partition regressions continue to check the unchanged Fourier, PNO and global residual equations. Separate work counters and measurements quantify the remaining repeated inverses. This scheduling increment uses the same Nejad and Riplinger/Neese formulation cited above; it does not qualify chi, D93 fitted/post-HF, or expand the PBC/MANTA release gate.

• E. MANTA. #581 consumes the existing producer-owned periodic GDF tiles, retained metric projectors and fitted J/K capabilities where their correlation export is qualified. A Fock action alone is not a certified correlation-factor source. Preserve source/kernel/support/rank identity; a separate fit remains explicitly different-source. #592 tracks bounded retained-factor access, while #576 owns complete-method MANTA controls.

  The explicit experimental molecular bridge is vibeqc.dlpno.manta_source.manta_molecular_source. It admits the actual evaluated RHF/UHF result and prepared molecular RI handle through the shared canonical-partition gates above. Its factors are the immutable AbelianMolecularRI blocks that built the physical Fock. A raw producer export precedes symmetry selection and is not substituted for those selected factors. The bridge binds the underlying producer/coupling, metric-factor identity and rank, selected source/support, normalization, evaluated build/density/frame and independent frozen partitions. It uses same_source=True only for that exact retained source. Separate correlation fits still use the different-source adapter; no new fitting option is added.

  SymmetryRIBlocks::transform performs C_left.adjoint() * B_A * C_right on each selected block, accumulating in the original auxiliary order. The factorization is the retained version of Melega et al., DOI 10.1021/acs.jpca.5c08579, Eqs. (3)-(5); this operation is a basis change of those factors, not a new metric factorization. It uses one block panel and the requested MO output, with no dense AO-factor reconstruction or missing-block completion. Native plans admit byte and work extents before input scans/output allocation, preserve complex conjugation, and report executed scalar products and block contractions. The bridge adds source, coefficient transport and real-output-copy costs to its per-call logical admission. The original source preparation retains its own dense metric/factor costs; this is not a whole-method or RSS bound.

  The real correlation engines require exactly zero imaginary components in the retained factor gauge and coordinate transforms. There is no tolerance-based realification. Generic complex native transport does not establish complex DLPNO support. Producer export, finite full molecular support and unchanged metric rank are checked; periodic, mixed, unqualified or different prepared sources are refused. The selected finite tensor’s omission, reconstruction and auxiliary-gauge bounds remain in provenance, with arithmetic roundoff explicitly excluded by the producer receipt. Preparation timing and caller resource ceilings do not define Hamiltonian identity. This bridge has no automatic run_job route and does not change QC0.18/#557 or periodic deferral.

  Bounded containing acceptance covers C2v H2O RHF and OH UHF, STO-3G with def2-SVP RI, independent frozen=0/1 and full-space MP2/UMP2/CCSD/UCCSD. All 168 checks pass. Eight new molecular calculations have maximum raw canonical error 4.48e-13 Ha, residual 2.08e-12 and absolute energy change 9.54e-14 Ha. Corrections and triples are separate. Independent materialized selected-factor consumers agree within 1.60e-16 Ha; same-density J/K controls agree within 1.07e-14, including independent libint couplings with the unchanged native metric factor. Retained ranks are 76/62, with 1143/740 selected factor elements versus 3724/2232 full AO elements. The generic evaluated and legacy controls also pass on the rebuilt native runtime. Reduced-rank, complex transport/refusal, ownership and pre-work budget checks pass without changing a numerical gate. This is bounded molecular bridge acceptance, not nonminimal-basis convergence, general open-shell accuracy, scaling or production periodic qualification.

  A subsequent nonminimal row qualifies C2v water/cc-pVDZ (24 AOs), with cc-pVDZ-JKFIT and the original retained rank 116. Geometry in angstrom: O (0,0,0.1173), H (0,+/-0.7572,-0.4692); charge zero, singlet, no ECP, Boys occupieds, independent frozen=0/1. Six focused checks pass, including both nonminimal frozen MP2 partitions and the existing minimal RHF/UHF ownership controls. Four full-space MP2/CCSD calculations have maximum raw canonical error 7.10e-14 Ha, residual 3.96e-12, and energy change 9.17e-14 Ha. The existing raw gates remain 1e-6 Ha for MP2 and 1e-8 Ha for CCSD, with residual 1e-11 and energy change 1e-13 Ha. Independent canonical amplitudes and the full spatial triples sum agree with local (T1) within 1.45e-14 Ha, below the existing 1e-9 Ha triples gate. Nonzero canonical triples are -0.00306137378016 and -0.00303911481001 Ha for frozen=0/1. No local correction enters a raw gate.

  Independently materialized selected-factor consumers agree within 1.20e-15 Ha. Same-density J/K comparisons, including independent libint couplings with the unchanged native metric factor, agree within 2.36e-14. The selected source stores 18,438 factor elements versus 66,816 in a full AO array; its tensor bound is 3.19e-12, excluding arithmetic roundoff. Source, evaluated reference, selected factors and both frozen partitions remain unchanged throughout the controls.

  The public NormalPNO and TightPNO threshold selectors, with the unchanged semicanonical PNO guess and (T1) defaults, give the following positive errors against this same finite-source canonical reference, in kcal/mol. These are separate approximation measurements, not raw exactness results.

  Frozen orbitals	Threshold selector	CCSD error	CCSD(T) error
  0	NormalPNO	0.25656	0.27468
  0	TightPNO	0.23859	0.25458
  1	NormalPNO	0.01531	0.01750
  1	TightPNO	0.00210	0.00207

  Threshold names follow Liakos et al., DOI 10.1021/ct501129s, Table 1; they do not select the paper’s separate full-LMP2 option in Sec. 2.2. PNO corrections, screened-pair contributions and triples are recorded separately. Subsequent fixed-source cross controls show that both selectors already use all atoms and rank-19 PAO domains on this case. Full domains reproduce the default energies bit for bit, while full PNOs recover the accepted full-space endpoints. Thus the domain contribution and its mixed term with PNO truncation are zero on these operands. Keeping all diagonal PNOs reduces the all-electron CCSD(T) errors only to 0.27364/0.25356 kcal/mol; it leaves the frozen-core results unchanged. The domain/PNO separation record gives the twelve controls, orbital and pair dimensions, exact pins and limits. This does not justify changing the diagonal cutoff or generalizing from a single molecule; off-diagonal pair-density/coupling qualification remains open. Archived ORCA 6.1.1 canonical CCSD(T), at this geometry/basis and frozen=1, gives -76.241041178819 Ha. Full-space MANTA RI CCSD(T) differs by -51.42 microhartree. This is an external accuracy observation, not an exactness PASS: ORCA uses direct integrals and a 1e-5 residual threshold, while this route uses the stated retained fit and stricter solver gates.

  On one Apple M5 Max thread, profiled source preparation/SCF take 0.074/0.096 s. Full-space CCSD(T) correlation takes 0.859/0.384 s for frozen=0/1; Normal/Tight take 0.575/0.646 s and 0.343/0.353 s respectively. Localization, factor transforms, domains, pair setup, amplitudes and triples are measured separately in the acceptance record. The entire multi-control job peaks at 646.14 MiB RSS, including imports and reference controls; this is not per-method memory or a hard bound. These tiny timings establish neither speedup nor scaling. Exact runtime/source pins and the preserved reporting-only failed triples carrier are recorded on #581/#591. This single basis row does not establish basis convergence, general open-shell accuracy, complete production qualification or a new release gate. Further periodic work remains deferred.

• F. Symmetry. The unfold-and-audit admission step, then unique-pair reduction inside D, each validated against the unreduced build.

  D20. Bounded native numerical transport links (#585). Explicit transport_backend="bounded" reuses compact home columns and complete projected pair recipes in the existing compiled Jacobi solver. Native admission still checks every link, discarding each numerical block immediately. Immutable owned metric/recipe snapshots reconstruct links during iteration; a solve-local single-entry cache releases on return or failure. Prepared links remain the default oracle. Routing, both occupied coupling legs, reversal and half-cell stabilizers are unchanged.

  The stable selected-column contraction of D4 remains exact in its original evaluation order, including admitted asymmetric metrics and all projection tails. This is a storage rescheduling of Riplinger and Neese II C Eq. (30) and Nejad V Eqs. (42)-(44), not new screening or a changed residual. The bounded cap includes snapshots, conversions, route metadata, replacement, reconstruction and link contraction workspace, with Python preparation charged separately inside the same cap. Source and downstream arrays, library scratch and interpreter overhead are outside its stated scope. Numerical links no longer persist, but metadata, occupied Fock, projected domain matrices and amplitudes remain separate storage gaps. Recomputing stable links is an explicit runtime tradeoff. A subsequent bounded increment caches one stable target metric across a target row. The cached expression is exactly admission’s \(A_p^T E_{D_p}^T S-(SCB_p)^T\), without reassociating asymmetric-metric or occupied-tail operations. Source reconstruction and cell-block accumulation remain unchanged. It releases the old target before replacement and reserves one maximum target rectangle plus its descriptor in workspace_bytes, including coexistence with nonstable link work. Native source ownership and solve-local lifetime prevent cross-source reuse; detached link() access remains uncached. Construction/hit/eviction counters and target_metric_cache_bytes expose the recomputation/storage tradeoff. The prepared oracle and all native admission checks remain unchanged. See the manual and benchmark. Scientific acceptance remains separate from source verification; chi is experimental and D93 fitted/post-HF is not-qualified. #557 is unchanged.

  D21. Shared native translation CCSD residual and energy (#586). An internal restricted evaluator now supplies translated operands to the molecular compiled PNO residual contractions. Complete representative singles/doubles residuals and raw energy are evaluated at arbitrary amplitudes; there is no second CC equation set and no molecular-supercell execution in this path. Native transport owns the complete pair inventory, stable overlap reconstruction and energy weights; occupied residual sums remain unweighted. Complete strong-pair inventories, compact recipes and sealed real factors are the initial scope. Independent frozen partitions, nonlinear singles terms, exchange/reversal and self-inverse sewing remain intact. MP2 discarded-PNO corrections are separately named, never included in the CC residual. CC overlap scratch and operand reconstruction have their own reported numerical reservation; they do not inherit MP2’s memory claim. Iteration/DIIS, screened weak-pair treatment and production dispatch remain future increments. See the contract and derivation. Scientific acceptance remains separate; D93 and #557 qualifications are unchanged.

Decisions

Taken by the maintainer on 2026-10-10 (#508), following the recommendations of this page.

Confirmed by the maintainer on 2026-10-23 (#508): periodic amplitude equations keep the SCF’s Fock with the Ewald exchange zero-mode term, so a finite-mesh correlation energy is the one PySCF computes with keep_exxdiv=True; PySCF’s default strips the term, which raises every occupied orbital energy by the Madelung constant inside the amplitude equations and lowers the correlation energy per cell by 4.6 to 8.8 mHa on the H2 crystal (STO-3G and def2-SVP). The two conventions share the thermodynamic limit and differ in how fast a finite mesh reaches it. Keep the SCF Fock (one Hamiltonian for reference and correlation), with no stripped-Fock option, and report the convention on every artifact, as the route does now.

1. The CCM transcription ccm_dlpno_ccsd_coupled is retired in favour of the engine; the union-PNO driver stays as the small-cell oracle.

2. BIPOLE and IBZ-only references are refused for correlation by default, with the reason in the error and in the receipt, not run with a warning.

3. Increment B’s production front door is run_periodic_job with method="dlpno-ccsd(t)" on the GDF route. The AICCM experimental warnings stay on the AICCM and chi routes only.

4. The molecular adapter accepts the SCF’s JK fit cache as an opt-in factor source for a same-source molecular run; the RI auxiliary basis remains the default and the measured convention.

Sources

Riplinger and Neese 2013; Riplinger, Sandhoefer, Hansen and Neese 2013; Riplinger, Pinski, Becker, Valeev and Neese 2016; Saitow, Becker, Riplinger, Valeev and Neese 2017; Guo et al. 2018 and 2020; Sun, Berkelbach and Chan 2017 for the periodic fitting conventions; Nejad, Zhu, Tew et al. 2025 for the native BvK lineage. All are in the citation database and fire through the engine’s routes.