Classic chi RHF and MP2 convergence study

This is subsequent experimental chi development, outside the QC 0.18 and Sketch 0.6 release gates. It does not extend CCSD, molecular DLPNO or D93 qualification. An accepted finite retained Hamiltonian and a physically converged periodic calculation are different claims.

Frozen systems and gates

Use the accepted #591 H2 and LiH archives. H2 has lattice rows (8,.3,.1), (0,8.5,.2), (0,0,9) and H positions (1,1,1), (1,1,2.4). LiH has a 6-bohr cubic lattice and Li/H at (3,3,1.5) / (3,3,4.5). All coordinates are bohr. Both systems are neutral, all-electron restricted singlets, initially STO-3G with def2-universal-jkfit. No ECP is introduced.

The archived H2 controls are one-electron/pair radii 6.2/6.2 bohr, reciprocal cutoff 18 Ha, nuclear real radius 8 bohr, reciprocal radius 5 inverse bohr and Ewald alpha .6 inverse bohr. LiH differs in one-electron radius 12 bohr and reciprocal cutoff 10 Ha. Metric threshold is 1e-9. Screening remains zero; physical-pair support and the SCF Fock including Ewald exchange zero mode are unchanged. LiH active/frozen partitions zero and one are independent.

Targets are fixed before collection:

• Existing accepted-reference stationarity, metric and occupation checks stay mandatory. Request SCF energy 1e-12 Ha and gradient 1e-10; inspect the actual final-rebuild residuals, overlap spectrum, retained fit spectrum and gap.

• Full-domain/zero-PNO raw local minus same-source canonical MP2 must be at most 2e-9 Ha/cell, with residual at most 1e-11 and energy change at most 1e-13.

• An individual support plateau requires both last two successive changes below 1e-6 Ha/cell in HF and canonical MP2 at fixed other controls. This is an observed plateau, not a certified error bound or joint convergence.

• A local approximation target is 1e-5 Ha/cell raw error against the matching canonical baseline. Report PNO estimates separately; they do not satisfy the raw gate. Domain corrections are not computed.

• Independent-reference agreement at 1e-6 Ha/cell is conditional on matched and sufficiently converged approximations. Record unmatched terms instead of increasing the tolerance. Basis and mesh sensitivity alone cannot establish the complete-basis or thermodynamic limits.

Reproduction and interpretation

The payload consumes an accepted HF archive and a JSON list of named cases. Each changes mapping contains at most one control, mesh, basis or aux_basis. Empty mappings reuse the archive. Numerical outputs and new archives must be outside the checkout. Run the payload through vq with an identified Python/native runtime and its required native SHA-256. Record the complete package manifest as well as payload, archive and input hashes. A later executable does not inherit an older runtime’s scientific verdict.

One-electron AO tails, AO-pair fitting tails, fitted reciprocal resolution, nuclear real/reciprocal sums, auxiliary basis and metric threshold are separate axes. Classic fitting has no SR auxiliary-radius axis. Cell geometry is fixed in this increment; increasing character counts in one direction probes that finite mesh, not three-dimensional convergence or a molecular dilution limit. Orbital-basis changes require their own support convergence. Failed stationarity, occupation or metric checks are preserved, not repaired with clipping, extra damping or relaxed admission.

The formulation follows Nejad et al., DOI 10.1063/5.0290816, Sec. II Eqs. (1)-(8): BvK orbitals, unitary character transform, HF equations and MP2 energy per primitive cell. Sun et al., DOI 10.1063/1.4998644, Sec. II and III, explains periodic Coulomb fitting and the independent linear-dependence/reciprocal axes. The classic finite reciprocal source here is not identical to compensated GDF/MDF, and no equality is inferred from a common auxiliary-basis label. CRYSTAL23 manual pages 120, 124 and 130 separately define local MP2, SHRINK and TOLINTEG. Those thresholds do not equal chi physical radii. ORCA molecular MP2 is not a direct reference for these finite periodic cells. A matched periodic canonical implementation retaining SCF Ewald orbital energies is a more direct independent MP2 control; differences in its fitting and supports must still be quantified.

Measured bounded increment

The qualification matrix attached to #572 and #591 records all inputs, basis shells, archive and factor identities, measured conditioning, errors and refusals. Raw logs, archives and failed carrier outputs are retained outside the product checkout. There are 31 canonical cases: 29 completed and two pre-integration memory refusals. Seven of ten new prepared local solves completed; three were refused before iteration. Process exit zero is not a per-case PASS.

The canonical and new prepared-local runs used source ba3551069827ccac7afeb9b319e44e9a74aa3743, native SHA-256 d34edec5f0108f4bb8fbb15cff7f369c6d6fd1b6db3d1dc7a74a4e4ea3e641df. The classic HF/canonical/support Python code and relevant native scientific kernels are unchanged through the previously accepted de9a5d74 snapshot. This bridge does not qualify later local factor, selection or transport changes. The separate #591 note 52727 / #585 note 52751 bounded-backend PASS remains bound to de9a5d74 and native 3f52e7e9e8b0d38783cd113b7132add6dae10d2329b441ddf60b2a723c9a9011. The #580 compact source-integrity evidence is reused within its own pin. No changed scientific implementation is qualified with an older executable.

All completed chi SCF cases satisfy their original admission checks. Among the initial ladders, maximum orbital residuals are 1.25e-11 (H2) and 2.10e-11 (LiH); minimum overlap eigenvalues are 0.0964 and 0.233, and minimum gaps are 0.857 and 1.442 Ha. Restored archives independently rebuild stationarity with orbital/energy tolerances 1e-9/1e-11; these are not the requested SCF stopping tolerances. Accepted reloads renew admission/process metadata and therefore state identities; retained factor identities and canonical energies agree. LiH’s smallest retained metric eigenvalue is 1.03e-9. Its fit ranks change from [67,67,68,67] to [67,68,68,68] on reciprocal refinement; that change is an explicit approximation effect, not hidden by a fixed-rank claim.

Physical-support results

All energies below are Ha per primitive cell. Each row changes only the named axis from the frozen baseline, not from the preceding row of another axis. The last-two-delta gate applies to both HF and each frozen-partition MP2 energy.

System and axis	Tested values	Largest of last two HF / MP2 changes (Ha)	Conditional verdict
H2 one-electron radius (bohr)	6.2, 10, 14, 18, 22	7.72e-8 / 4.59e-10	PASS plateau
H2 AO-pair radius (bohr)	6.2, 8, 10, 12, 14, 16	2.24e-7 / 5.48e-9	PASS plateau
H2 reciprocal cutoff (Ha)	18, 24, 30, 40, 50	1.94e-6 / 1.30e-6	FAIL plateau
H2 nuclear real radius (bohr)	8, 12, 16	4.61e-7 / 5.95e-9	PASS plateau
H2 nuclear reciprocal radius (inverse bohr)	5, 7, 9	2.63e-9 / 1.44e-12	PASS plateau
H2 metric threshold	1e-9, 1e-10, 1e-11	4.44e-16 / 1.22e-17	PASS plateau
LiH one-electron radius (bohr)	12, 16, 20; 24 refused	0.3664 / 0.004206	FAIL plateau
LiH reciprocal cutoff (Ha)	10, 20, 40	0.07258 / 0.0002231	FAIL plateau

H2 baseline HF/MP2 is -1.132616778663140 / -0.012268098263550. At one-electron radius 22 it is -1.137755593593475 / -0.012170040209412; at pair radius 16, -1.132417719904343 / -0.012280774679151; at reciprocal cutoff 50, -1.132564398130348 / -0.012288132492225. These different conditional limits must not be combined into a claimed jointly converged energy without new calculations.

LiH baseline HF is -9.224544983357706, with MP2 frozen 0/1 -0.006881336742081 / -0.005784454860660. At one-electron radius 20 these become -9.619161072997088 and -0.006075159550529 / -0.001377924462656. At reciprocal cutoff 40 they are -9.128699126182655 and -0.006985492143957 / -0.005479099158137. This is a substantial physical-support failure, despite positive overlap, a gap and stationary SCF. It is not yet a localized implementation error.

Separate approximation and reference limits

H2 character meshes (1,1,1), (2,1,1), the reused accepted (3,1,1), and (4,1,1) give HF -1.132616779, -1.125866149, -1.123515590, -1.122340390; MP2 -0.0122680983, -0.0122193617, -0.0122416009, -0.0122530190. This is UNQUALIFIED finite-size sensitivity, not a three-dimensional k limit. Changing only the auxiliary basis to cc-pVDZ-JKFIT gives H2 HF/MP2 -1.132598555239006 / -0.012277079469607. Changing only the orbital basis to 6-31G gives -1.144261966478398 / -0.016240904957639. Neither establishes auxiliary- or orbital-basis convergence. H2 cc-pVDZ and LiH radius 24 were resource refusals at the original 2/4 GiB numerical caps; their bounded replay is recorded below and does not qualify either convergence limit.

PySCF 2.6.2 periodic GDF KRHF/KMP2 uses identical cells, STO-3G, def2-universal-jkfit, charge/spin and character meshes, with exxdiv='ewald'. Its KMP2 kernel consumes the SCF orbital energies unchanged, checked explicitly; no exchange-zero-mode subtraction is used. H2 Gamma precision 1e-8, 1e-10, 1e-12 gives a stable reference (HF last changes 2.78e-11 and 4.45e-12 Ha). At 1e-12 its HF/MP2 is -1.137506045512623 / -0.012200632617494. The archived chi differences are +0.004889266849483 / -0.000067465646056. LiH (4,1,1) at reference precision 1e-10 gives HF -7.984593087472446 and MP2 frozen 0/1 -0.021004129636132 / -0.020096649324724. Chi differences are -1.239951895885260 and +0.014122792894051 / +0.014312194464064. These exceed the desired comparison target. Physical parity is UNQUALIFIED: classic finite reciprocal fitting and compensated GDF/supports are unmatched, and chi support ladders have not converged. A shared basis name is insufficient.

CRYSTAL23 public 1.0.1 was also run for the same H2 geometry, STO-3G, P1, SHRINK 1 1, TOLDEE 12. TOLINTEG ladders (8,8,8,8,16), (10,10,10,10,20), (12,12,12,12,24) produced HF -3.1711402959794, -3.4931705278661, -3.8631727592575. SCF converged, but this reference itself fails a threshold plateau and is UNQUALIFIED as a finite-BvK comparison. The drift is preserved, without attributing an unproven cause to either code. The periodic-reference conventions need reconciliation before a parity claim.

Local approximation and remaining implementation

New full-space prepared local solves agree with same-source canonical MP2 for H2 one-electron radius 18, pair radius 12, reciprocal cutoff 30 and 6-31G, and for LiH radius 20 with frozen core 1. Maximum raw error is 2.57e-15 Ha, residual 2.09e-13 and absolute energy change 2.21e-14. PNO estimates remain separate (zero for these H2 cases; 1.76e-18 Ha for LiH). H2 6-31G cutoffs TCutMKN=.01 and TCutPNO=1e-6 do not change the energy; this does not establish a useful truncation error bound.

The earlier containing-runtime LiH data remain the nontrivial truncation control. At domain .01, raw errors for frozen 0/1 are 38.844/10.548 microhartree; at PNO 1e-6 they are 46.916/11.171 microhartree, with separate estimates -39.220/-10.155 microhartree. All four raw errors FAIL the 10-microhartree target. Corrections do not turn those raw tests into PASS. Domain corrections are absent.

The new LiH radius-20 all-electron local solve refuses the older AO seed frame; replay with the delivered explicit cosine option is still needed for this new source. At reciprocal cutoff 40, both partitions refuse the retained-factor recovery audit: round-trip defect 2.53e-10 (imaginary 6.42e-11). No tolerance was changed. Replay and diagnosis on the current containing factor runtime are required before calling this a current implementation defect.

No scientific equation repair is justified by this bounded study. It identified the one-electron nuclear Fourier cache as the next workspace barrier: its 32*(oneel_cells+1)*nao**2*nG reserve refused the larger LiH support and H2 basis rungs before integration. The bounded implementation below removes that intermediate without establishing physical convergence. Next finish reciprocal and joint-support ladders, new-source local admission, matched references and multidimensional mesh/basis convergence. Production chi and D93 fitted/post-HF remain not-qualified; #572/#591 stay open and #586 stays outside this increment.

Bounded nuclear Fourier workspace

The classic chi nuclear attraction now uses the existing native weighted per-cell AO-pair Fourier contraction, with one cell and budget-derived reciprocal tiles (at most 256 vectors). It evaluates the complete finite sum Re(sum_G w(G) conj(FT_mn(G,R))) before the original real assembly, Y_lm correction, physical-pair mask and background overlap term. The native kernel contracts within shell-pair Hermite tasks. Reciprocal-tail truncation, shell screening and optional radial/power caches are disabled. No physical radius, reciprocal vector, nuclear source, Ewald convention or precision gate changes. The shared complete-cache function remains the explicit small-system oracle and the default for other consumers.

Preflight now reserves complete lattice outputs, block snapshots, Bloch matrices, support records and nuclear supports, plus the tiled phases, weights, returned contraction block and assembly scratch. Native Hermite powers, per-worker recurrences and accumulators are reserved under the existing native-workspace cap. An impossible one-vector minimum refuses before nuclear integration. The removed intermediate scaled as 32*(oneel_cells+1)*nao**2*nG; the reciprocal contraction workspace now scales with the selected tile and angular momentum, without an AO-pair Fourier cache. Complete lattice operators and reciprocal-vector/support lists remain resident. The numerical cap excludes allocator overhead, Python object metadata, external library state and downstream arrays; it is not a total process-memory guarantee.

The returned reciprocal receipt and standard output report tile dimensions, accounted peak, native reserve, removed-cache estimate and actual native call/vector counts. These execution details do not enter the immutable physical-support record. Tile accumulation changes floating-point association; complete matrix and downstream checks retain the original tolerances.

Focused source checks pass 82 tests across test_chi_fitted_controls.py, test_periodic_v_ne_ewald_ft.py and test_chi_ecp_hamiltonian.py. They compare complete matrices and complex Bloch sums on a skew cell for STO-3G, 6-31G and cc-pVDZ at tile widths 1, 7 and the planned width, with a 3e-12 absolute gate. Gamma and three-character HF/canonical MP2 agree with the full-cache oracle within the unchanged 1e-11/2e-12 energy gates. Support records are identical; guards prohibit the full-cache builder, and exact/below-minimum budgets are checked. Shared weighted-kernel and ECP assembly checks also pass.

For the archived refused inputs, the conservative preflight at the measured 18-worker setting gives the following numerical payloads (bytes):

Input	Original cap	Removed Fourier reserve	New accounted peak	Reciprocal tile
H2 cc-pVDZ	2147483648	2213971200	613361664	256
LiH one-electron radius 24	4294967296	5563014912	626432512	256

These are allocation bounds, not measured resident peaks or scientific acceptance. Each cell/tile call repeats primitive geometry and Hermite setup; no runtime speedup is claimed. Larger complete outputs, support records and later fitting/correlation arrays can still impose independent limits.

The exact formerly refused manifests now complete through vq on the verified containing snapshot, at their original 2/4 GiB numerical caps. Source/native pins are checked before and after execution. The native SHA-256 is e1a582efa866faf0f5a4fd356a0a0b5b4fa12f85bdb8b2b20ef00efa72e22c7b; the containing source is c2fa5152. Two initial checkout-bound launches refused before chemistry when an unrelated source file advanced; those operational failures are preserved separately from the successful snapshot runs. The independently owned H2 combined-support job remains untouched.

Replay	HF (Ha/cell)	Canonical MP2 (Ha/cell)	Orbital residual
H2 cc-pVDZ, frozen 0	-1.145851721969042	-0.024550872269520	2.94e-11
LiH radius 24, frozen 0	-9.620470608358930	-0.006072028898480	1.64e-11
LiH radius 24, frozen 1	-9.620470608358930	-0.001367774907557	1.64e-11

Both sources satisfy the original HF admission, with minimum overlap eigenvalues 0.043747/0.256404 and gaps 0.812453/1.715460 Ha (H2/LiH). The complete commands took 1.90/2.38 seconds; POSIX wait4 measured peak RSS 208.58/211.19 MiB, the largest single process, not a sum across processes. These are new-path observations, not matched speedup measurements against the formerly refused full cache. Calculation-only times were 0.22/0.72 seconds.

LiH radius 20 to 24 changes HF by -1.309535e-3 Ha and MP2 frozen 0/1 by +3.130652e-6/+1.014956e-5 Ha. All exceed the predeclared 1e-6 one-step target, so the required two-step support plateau still FAILS. H2 cc-pVDZ is one orbital-basis sensitivity point, not basis convergence. These exact manifests contain canonical calculations only; the subsequent full-space local replay is recorded below. Nonzero domain/PNO accuracy and matched independent-reference limits remain UNQUALIFIED. No correction or relaxed gate is used. The next chi-specific gap is physical support convergence and nonzero local approximation accuracy, not another V_ne full-cache workspace repair. Joint-support, mesh and reference limits above remain open; chi is experimental and D93 fitted/post-HF remains not-qualified.

Local admission on the newly admitted sources

The accepted H2/cc-pVDZ Gamma and LiH radius-24 (4,1,1) archives were reused without rerunning SCF, refitting factors or changing supports. Six full-domain/zero-PNO solves pass: bounded and prepared transport for H2 frozen 0, and for LiH frozen 0 and 1 independently. H2 uses the AO seed frame; LiH explicitly uses the existing cosine frame. Streamed factors and compact preparation are retained, with selection/factor/transport caps 64/64/256 MiB, 500 iterations maximum, energy/residual stopping tolerances 1e-13/1e-11 and damping 1. No admission threshold or canonical comparison gate changes.

Source / frozen core	Raw local MP2 (Ha/cell)	Absolute canonical error	Residual	Separate PNO estimate (Ha/cell)
H2 cc-pVDZ / 0	-0.024550872269520	3.47e-18	1.39e-17	1.04e-17
LiH radius 24 / 0	-0.006072028898473	7.55e-15	3.61e-13	2.48e-18
LiH radius 24 / 1	-0.001367774907554	2.71e-15	2.19e-13	7.05e-19

The two backends give identical raw energies, residuals, energy changes and PNO estimates, plus identical solver input snapshots, occupied selections, domains and PAO/PNO ranks. Iteration counts are 2, 13 and 11 respectively; maximum absolute energy change is 4.77e-14 Ha. H2 retains PNO rank 9; every LiH pair retains PNO rank 16. Complete factor recovery passes the original 1e-10 gate (largest defect 9.79e-12); no component or check is screened. Canonical energies reproduce the preceding accepted values within 2e-12 Ha. Corrections are reported separately and do not enter any raw gate.

The calculations use the same immutable c2fa5152 source/native snapshot identified above, with hashes checked before and after execution. Relevant chi/local/native code is unchanged through this evidence landing. The issue matrix records exact archive/input/output hashes, source identities, controls and all six results; raw output and reproducible payloads are retained. Nejad Sec. V Eqs. (42)-(43) supplies the coupled residual and per-cell energy conventions, with Riplinger and Neese Sec. II B3/C supplying PNO construction and two-sided pair transport. This is a new-source replay of the delivered method, not a new equation set or independent external-reference result.

No bounded implementation defect was demonstrated, so no method fix or native rebuild was warranted. Accepted scope is full-space local admission for these finite Hamiltonians only. LiH physical-support convergence still fails, H2 basis/mesh convergence remains unqualified, and nonzero domain/PNO errors on these sources were unmeasured at that checkpoint. The bounded follow-up below quantifies selected local truncation errors while keeping physical-support and independent-reference work separate.

LiH support extension and local truncation probes

The next bounded study starts from the accepted radius-24 LiH archive and changes only the one-electron radius. Geometry, STO-3G/def2-universal-jkfit, (4,1,1) mesh, reciprocal cutoff 10 Ha, nuclear/Ewald controls, metric cutoff, SCF Fock and independent frozen 0/1 partitions stay fixed. The declared sequence was 28, 32, 36, 40 bohr, stopping at the first two consecutive changes below 1e-6 Ha/cell for HF and both MP2 partitions, a refusal, or radius 40. Physical envelope estimates were 587/847/1189/1551 cells and numerical reserves 618/618/623/630 MB at two workers, within the original 4 GiB cap. No cap changed.

One-electron step at reciprocal 10 Ha	HF change	MP2 frozen 0 change	MP2 frozen 1 change
24 to 28	-1.348349e-4	+1.489560e-7	+1.129148e-6
28 to 32	-6.254695e-7	+1.471430e-9	+6.390402e-9
32 to 36	-8.367252e-9	+7.403091e-12	+8.872868e-11

All changes are Ha/cell. The last two steps pass the unchanged gate: a conditional one-electron plateau, not joint convergence. Radius 40 at reciprocal 10 was correctly not run after the stopping rule was satisfied. The radius-36 energies are HF -9.620606077109800 and MP2 frozen 0/1 -0.006071878463645 / -0.001366639280146 Ha/cell.

That exact accepted radius-36/reciprocal-10 archive became the parent for separately declared reciprocal probes 20, 40, 60 Ha. Only reciprocal support changed, with the same stopping rule and an upper bound of 60 Ha.

Reciprocal step at one-electron radius 36	HF change	MP2 frozen 0 change	MP2 frozen 1 change
10 to 20	+7.218317e-2	-8.899838e-5	+2.259265e-4
20 to 40	+2.318867e-2	-3.298356e-5	+3.631151e-5
40 to 60	+4.258098e-3	-5.879945e-6	+5.566866e-6

The reciprocal gate FAILS through 60 Ha. At the accepted reciprocal-60 parent, one-electron radii 40 and 44 were then tested independently. The HF changes are -3.229417e-12 and zero at reported precision; maximum MP2 changes are 2.55e-14 and 2.13e-16 Ha/cell. This confirms a conditional one-electron plateau at the combined point. It does not repair the failed reciprocal gate, so joint convergence is not established. The radius-44/reciprocal-60 energies are HF -9.520976139476302 and MP2 frozen 0/1 -0.006199740344550 / -0.001098834366835 Ha/cell.

All eight new HF sources and sixteen independent canonical MP2 evaluations complete without a numerical-cap refusal. Maximum HF orbital residual is 1.65e-11 and minimum overlap eigenvalue is 0.2564029. Full-space local controls pass for seven of the eight sources: 28 prepared/bounded solves satisfy the original raw/residual/energy-change gates, with maxima 7.03e-14 Ha/cell, 2.29e-12 and 9.33e-14 Ha respectively. Compatible-gauge amplitudes and residual arrays agree exactly between backends. All LiH controls use the explicit cosine frame; no frame fallback occurs.

The radius-36/reciprocal-40 source passes HF/canonical admission but both frozen partitions refuse local factor recovery in both transport backends. The retained factors violate the character-conjugate reality relation by 4.046396e-10, implying a minimum real-projection loss of 2.023198e-10, above the original 1e-10 gate. The worst retained component has magnitude 8.60e-11; it must not be screened or dropped. The dense factor oracle also refuses. An independent long-double complete Fourier calculation gives complex round-trip error 5.61e-16 but real/symmetric recovery error 2.023198e-10. Thus this is an incompatibility of the retained source with current local real-representation admission, not a tiled accumulation or transport defect. The 8 MiB and 64 MiB factor budgets reproduce the refusal. A separate 1 MiB probe refuses its 1049856-byte minimum before audit; that resource refusal is not a scientific failure. No source was projected, metric clipped or gate loosened. Canonical validity does not establish local representability.

The prior H2/cc-pVDZ and LiH radius-24 full-space controls were reused, including byte-identical amplitude/residual hashes in identical gauges. Only then were independent domain and PNO probes run on those original sources. All twelve truncated solves converge at the original residual/energy-change gates. H2 retains its complete rank-9 space at every tested cutoff, so its four raw errors of 3.47e-18 Ha/cell do not demonstrate a nontrivial truncation bound. LiH results below are microhartree/cell; the raw target is 10 microhartree.

LiH radius-24 approximation	Raw error frozen 0	Raw error frozen 1	Separate PNO estimate frozen 0 / 1
Domain .01, PNO zero	8.6674	0.12135	below 3e-12 / below 3e-12
Domain .001, PNO zero	4.8999	below 3e-9	below 3e-12 / below 3e-12
Full domain, PNO 1e-6	25.5163 (FAIL)	9.8546	-17.6928 / -7.7243
Full domain, PNO 1e-7	10.3898 (FAIL)	1.6048	-4.3909 / -1.2762

Corrections do not rescue a failed raw gate. These source-specific tests do not establish production cutoffs or transfer their errors to the new support points. No bounded method fix was demonstrated. The next implementation gap is diagnosing conjugate-character fidelity in classic retained-factor construction near small components, while preserving every retained mode and support identity. Reciprocal convergence beyond 60 Ha, repeated joint checks, matched independent references, auxiliary/basis/mesh limits and local admission of the refused source remain missing. Existing references have unmatched fitting/support limits and add no new external accuracy PASS here.

All stages ran through vq on immutable c2fa5152 with the native SHA-256 above, verified before and after execution. Exact accepted-parent archives, controls, source identities, deltas, raw refusals and compatible-gauge comparisons are in the issue qualification matrix. Primary conventions remain Nejad Sec. V Eqs. (42)-(43), Riplinger and Neese Sec. II B3/C, and Sun Sec. II-III; no scientific formulation changed. The separately owned H2 combined-support campaign was untouched. Chi remains experimental and D93 fitted/post-HF not-qualified; neither broad issue is closed.

Retained-eigenspace fidelity at reciprocal 40

The exact radius-36/reciprocal-40 archive from the preceding study was replayed on the same verified c2fa5152 Python/native runtime. Its SHA-256 is a230f28e3bde71bff007d56312be1bd905c31203a5c71470595914a4211190fc. Reconstructing the classic producer with the archived orbital basis and the actual modrho-normalized auxiliary basis reproduces every factor byte and the factor identity. A preliminary diagnostic using the unnormalized auxiliary basis did not match and was excluded. Save/reload also changes no factor byte. The original four local refusals and long-double Fourier results reproduce.

The dominant discrepancy is localized to the double-precision retained eigenspaces at conjugate quarter characters. This is a forward-accuracy limitation near the retained/discarded spectral boundary, not evidence of a wrong primitive integral or an arbitrary eigenvector phase. With q = k_ket - k_bra, the actual finite-source convention is

\[M_{PQ}(q)=\sum_{G+q\ne0} w_{G+q}\widehat\chi_P(G+q)^* \widehat\chi_Q(G+q),\qquad T_{P,\mu\nu}(q;k_j)=\sum_{G+q\ne0}w_{G+q} \widehat\chi_P(G+q)^*\widehat{\mu\nu}_{k_j}(G+q),\]

\[W(q)=U_{\rm ret}\lambda_{\rm ret}^{-1/2}U_{\rm ret}^{\dagger}, \qquad L(q;k_j)=W(q)T(q;k_j).\]

Real primitive functions, inversion-matched reciprocal support and conjugate Bloch phases require M(-q)=M(q)*, T(-q;-k_j)=T(q;k_j)* and hence L(-q;-k_j)=L(q;k_j)* in these original auxiliary coordinates. Reciprocal relabeling also applies at self-inverse characters. Sun et al., DOI 10.1063/1.4998644, Sec. II Eqs. (13), (16)-(18) and the spectral truncation discussion after Eq. (21), supply the Fourier and fitting conventions; Nejad et al., DOI 10.1063/5.0290816, Sec. II Eqs. (2)-(3), supplies the unitary finite-character transform. This classic finite sum does not adopt Sun’s compensated GDF/MDF source.

For the failing conjugate pair, measured maximum absolute discrepancies are:

Boundary	Conjugation discrepancy
Matched reciprocal vectors	0
Auxiliary Fourier values	3.14e-16
Native AO-pair Fourier values	4.44e-16
Complete metric	8.88e-16
Complete three-center tensors	1.79e-15
Retained metric projector	9.17e-8
Canonical inverse square root	2.86e-3
Retained factors	4.046396e-10
Complete within-transfer Coulomb blocks	1.40e-12

Both partners retain 68 of 69 modes. The smallest retained eigenvalue is 1.0349134e-9 and the largest discarded value is 9.8547819e-10, on opposite sides of the unchanged 1e-9 threshold; their gap is 4.94e-11. The retained condition number is 8.58e9. No rank-count mismatch occurs. Substituting only the whitening difference reproduces 4.0463962e-10 in the factors; substituting only the three-center difference contributes at most 5.33e-13. Gamma and the self-inverse half character have factor defects 1.33e-14 and 6.75e-12.

Diagonalizing the same stored binary metrics at 45 decimal digits, followed by extended-precision matrix assembly, reduces the conjugate projector and factor discrepancies to 2.91e-13 and 1.76e-12 without changing ranks. Extended precision in the final multiplication alone changes factors by at most 2.78e-13. Extended-precision reciprocal accumulation and alternative LAPACK drivers were separate sensitivity probes, with unchanged ranks; they were not substituted into an accepted source. The primary localization is therefore the retained eigenspace calculation, rather than final multiplication, serialization, reciprocal ordering or local tiling. This does not establish a general error bound for every cutoff, basis or implementation of eigh.

Factor equality is not an operator test. Complete cross-pair Coulomb blocks within each transfer were compared, including every auxiliary component and all AO pairs. An independent spectral contraction agrees with the original canonical-gauge blocks within 1.05e-12. The 45-digit diagnostic factors change original entries by up to 3.19e-10 and complete blocks by up to 1.87e-12. Thus they are not identical archived data, and an exactly source-preserving auxiliary rotation has not been established. No representation transformation or producer repair is installed. In particular, small operator discrepancies do not authorize bypassing the original factor-entry gate.

Independent complete AO four-index contractions reproduce the archived J/K matrices within 3.11e-15 and canonical MP2 within 1.05e-17 Ha/cell across the radius-36 reciprocal-20/40/60 sources and both frozen partitions. The 40-Ha HF energy remains -9.525234237860450, with canonical MP2 frozen 0/1 -0.006193860399434 / -0.001104401232773 Ha/cell. Both providers still admit the neighboring 20/60 archives for frozen 0/1 and refuse the 40 archive. These are admission replays, not repeated convergence ladders or new external accuracy claims. Earlier prepared/bounded local energies and residuals on the accepted neighbors are reused within their original pin. The refused source has no local energy, residual or energy-change PASS.

The read-only fidelity payload reproduces archive admission and complete conjugate-block measurements. factor_conjugation_measurements distinguishes fixed-gauge entry fidelity from unitary-invariant Coulomb fidelity without projecting or screening. Array controls cover auxiliary rotations, degeneracies, self-inverse characters, reordered columns, small components, cross-pair blocks and rotation across a narrow retained/discarded gap. Existing factor-provider threshold, null-component, ordering and deliberately inconsistent-source checks retain the unchanged gate. The issue matrix preserves the producer trace, higher-precision payloads, intermediate arrays, runtime pins and all diagnostic failures separately from accepted measurements.

The remaining implementation requirement is a precision-controlled retained spectral projector and inverse-square-root construction, with complete operator and rank-boundary checks. Any changed producer must generate a new source identity and its own consistent HF state before prepared/bounded local MP2 qualification against independent same-source canonical values. Neither changing the LAPACK driver to pass this case nor averaging partners is justified as a general repair by these measurements. The exact archived source remains refused. Complete supports, retained modes, the original 4 GiB cap, Ewald Fock convention and frozen 0/1 semantics are preserved. Reciprocal convergence beyond 60, joint convergence and matched external accuracy remain separate obligations. Chi stays experimental and D93 fitted/post-HF stays not-qualified.

Precision-controlled retained construction

The follow-up classic producer implements the bounded arithmetic repair identified above. metric_precision_digits=45 computes the retained projector and canonical inverse square root at 45 and 65 decimal digits from the same binary metric, with independent residual, orthogonality, threshold-distance and complete matrix-function checks. It preserves the original double-precision retained mask or refuses. The default remains unchanged. This is a new numerical factor source, not a unitary relabeling of the archived factors. The required conjugation relation and the Sun and Nejad equations in the preceding diagnosis remain applicable; no conjugate averaging or modified fitting equation is used.

The radius-36 reciprocal-40 case and neighboring 20/60 controls were run through the queue with the original 4 GiB resource cap, geometry, orbital and auxiliary bases, mesh, supports, 1e-9 metric cutoff and SCF Ewald-Fock convention. Each case generated a new consistent HF reference before correlation. Executed support records and retained masks match their respective originals exactly; the four ranks remain 67/68/68/68. All 69 auxiliary coordinates, including small components, remain present. Serialization reloads every factor exactly.

At reciprocal 40 the new HF energy is -9.525234237860147 Ha/cell. The maximum factor-conjugation defect is 1.592e-12, complete Coulomb-block conjugation defect 2.088e-13 and measured real round-trip loss 7.960e-13. The unchanged 1e-10 gate admits the new source. The old archive is still refused; it has not been overwritten or assigned the new identity.

Frozen core	Same-source canonical MP2	Prepared and bounded raw MP2	Absolute difference
0	-0.006193860399437228	-0.006193860399437972	7.45e-16
1	-0.001104401232774522	-0.001104401232764818	9.71e-15

Energies are Ha/cell, with corrections kept separate. The largest residual is 7.45e-13 and energy change 9.28e-14, satisfying 1e-11 and 1e-13 respectively. Prepared and bounded amplitude and residual arrays agree exactly. All twelve solves across reciprocal 20/40/60, both frozen partitions and both backends pass the raw 2e-9 energy criterion. These neighboring controls test the changed producer; they do not extend the reciprocal convergence ladder.

Independent complete AO four-index contractions agree with canonical MP2 within 4.34e-18 Ha/cell across all six cases, and with complete J/K matrices within 4.45e-16/3.56e-15. At reciprocal 40, the new operator changes J/K on the new density by at most 6.76e-14/8.00e-13 relative to the archived operator. Those changes are recorded rather than treated as an identical source. The issue qualification matrix records the new archive and factor identities, the pinned native kernel and containing Python producer hashes, precision receipts, full operator controls and queue jobs. The optional numerical dependency was mpmath 1.4.1 for these measurements.

Focused controls exercise conjugate and self-inverse characters, reordered meshes and auxiliary coordinates, degeneracies, narrow retained boundaries, failed eigensystems, inadequate budgets, small components and serialization. The option does not promise that every metric will become admissible: uncertain rank or arithmetic convergence still refuses, and downstream gates still apply. Reciprocal convergence beyond 60, joint convergence and matched external accuracy remain separate obligations. Chi remains experimental; D93 fitted and post-HF remains not-qualified.

Precision-controlled LiH reciprocal extension and joint check

The qualified radius-36/reciprocal-60 precision-controlled archive was reused as the exact parent. The declared reciprocal sequence was 80, 100, 120, 160, 200 and 240 Ha, stopping at two consecutive absolute changes no larger than 1e-6 Ha/cell for HF and both canonical frozen-core partitions, a source or resource refusal, or 240 Ha. Every point generated a fresh HF state with metric_precision_digits=45 and its own retained source. Geometry, STO-3G/def2-universal-jkfit, (4,1,1) mesh, pair support, metric threshold, other supports and SCF Ewald-Fock convention stayed fixed. The original 4 GiB cap and all local admission and iteration gates were retained.

Reciprocal step at one-electron radius 36	HF change	MP2 frozen 0 change	MP2 frozen 1 change
60 to 80	+1.014371e-3	-1.479003e-6	+1.185867e-6
80 to 100	+3.972848e-4	-5.585755e-7	+4.766575e-7
100 to 120	+1.071222e-4	-1.609408e-7	+1.110835e-7
120 to 160	+4.625848e-5	-6.901095e-8	+4.449639e-8
160 to 200	-5.752892e-6	+8.166219e-9	-9.474176e-9
200 to 240	-9.829820e-6	+1.423563e-8	-1.234219e-8

Changes are Ha/cell. The reciprocal gate fails through 240 Ha because HF does not meet the criterion. The HF changes reverse sign, so these results do not justify monotone extrapolation. MP2 convergence alone does not qualify the combined HF/correlation result. At radius 36 and reciprocal 240, HF is -9.519426685692014 and canonical MP2 frozen 0/1 is -0.006201985473081949 / -0.001097038079060192 Ha/cell.

The predeclared joint check used that accepted 240-Ha source as parent for one-electron radii 40 and 44. HF changes are -3.2241e-12 and -3.5528e-15; maximum MP2 changes are 2.511e-14 and 1.501e-16 Ha/cell. These two steps pass, so the conditional one-electron plateau is established and larger radii were not run. The last reciprocal increments were then repeated at radius 44:

Reciprocal step at one-electron radius 44	HF change	MP2 frozen 0 change	MP2 frozen 1 change
160 to 200	-5.752892e-6	+8.166219e-9	-9.474176e-9
200 to 240	-9.829820e-6	+1.423563e-8	-1.234219e-8

The enlarged one-electron support does not cure the reciprocal drift. Joint convergence is not established. At the combined radius-44/240-Ha point, HF is -9.519426685695242 and canonical MP2 frozen 0/1 is -0.006201985473074758 / -0.001097038079035077 Ha/cell. This joint test concerns the one-electron and reciprocal axes only; pair support, nuclear/Ewald, orbital/auxiliary bases and mesh have not thereby been converged.

All ten fresh HF sources and forty full-space local solves pass their own admission and iteration gates. Both frozen partitions use both prepared and bounded transport, with maximum raw local/canonical difference 9.254e-15 Ha/cell, residual 7.116e-13 and energy change 9.525e-14. Amplitude and residual arrays agree exactly between transports in the same gauge. PNO estimates remain separate from raw energies; no correction rescues a gate failure.

Independent complete four-index AO/MO contractions reproduce all twenty canonical energies within 1.388e-17 Ha/cell and complete J/K matrices within 1.333e-15/5.552e-15. Serialization preserves all factors exactly. The four retained ranks remain 67/68/68/68, with all 69 auxiliary coordinates present. Precision receipts preserve the original mask for each newly assembled metric. Executed records confirm that one-electron changes do not alter fitted pair or reciprocal supports. Each new archive has its own source identity and consistent HF state. The qualified containing runtime and original native kernel were checked before and after the queue job; unrelated main changes are outside this numerical acceptance.

The issue matrix retains exact parent/archive hashes, controls, source and native identities, complete support records, arithmetic receipts and all results. No resource cap, admission tolerance, fitting threshold or producer implementation changed in this study. The H2 campaign was untouched. The next LiH requirement is resolving reciprocal HF convergence beyond this nonmonotone 240-Ha bound and repeating the joint check before claiming matched external accuracy. No extrapolated limit or production default is inferred. Chi remains experimental and D93 fitted/post-HF remains not-qualified.

Beyond 240 Ha: numerical plateau and resource qualification failure

The exact accepted radius-44/240-Ha precision-controlled archive was reused for the declared reciprocal sequence 320, 400, 480, 640, 800, 960, 1280 Ha. The same two-consecutive-step 1e-6 Ha/cell gate applies to HF and both canonical MP2 partitions. Geometry, bases, mesh, other supports, retained thresholds, precision policy and SCF Ewald-Fock convention stayed fixed.

Reciprocal step at one-electron radius 44	HF change	MP2 frozen 0 change	MP2 frozen 1 change
240 to 320	-1.146238e-5	+1.695629e-8	-1.333799e-8
320 to 400	-4.965842e-6	+7.306999e-9	-5.716070e-9
400 to 480	-2.097826e-6	+3.113096e-9	-2.365239e-9
480 to 640	-1.446701e-6	+2.154430e-9	-1.617441e-9
640 to 800	-4.168738e-7	+6.190080e-10	-4.677559e-10
800 to 960	-1.423500e-7	+2.117693e-10	-1.592016e-10

The last two steps meet the numerical criterion, so 1280 Ha was not run. At reciprocal 960, enlarging the one-electron radius from 44 to 48 to 52 changes none of the three energies at reported precision. Repeating reciprocal 640/800/960 at radius 52 gives the same last two increments. Thus these measurements show a numerical plateau on both axes around radius 52/960 Ha. The combined energies are HF -9.519447217669379 and canonical MP2 frozen 0/1 -0.006201955111482168 / -0.001097061742733958 Ha/cell.

However, qualification under the original 4 GiB resource cap fails. The first multi-case collector was interrupted after four completed points to avoid retaining all prior support records in memory. The resumed collector stored those records separately but its measured largest-process peak was 6559.55 MiB. It completed numerically, not within the requested resource cap. An isolated-process replay then reproduced the 320- and 400-Ha source identities and energies. The measured peaks were 3290.06 and 4367.19 MiB respectively; the harness stopped at 400 rather than accepting the over-cap result. Isolation alone therefore does not resolve the excess. The exact allocation responsible has not been localized to producer, checkpoint or diagnostics. A requested queue allocation and numerical-payload reserve are not measured process-RSS guarantees. No larger cap was substituted.

The diagnostic run contains ten fresh HF states and forty prepared/bounded local solves, with raw canonical differences at most 9.222e-15 Ha/cell, residuals 7.086e-13 and energy changes 9.522e-14. Complete independent four-index contractions agree with canonical within 8.674e-18 Ha/cell and J/K within 1.777e-15/5.330e-15. Factors reload exactly; ranks remain 67/68/68/68 with all 69 auxiliary coordinates. Transport amplitudes and residuals agree exactly and corrections remain separate. These numerical checks do not override the resource failure or qualify the combined point for production. The original containing runtime and producer hashes were verified, including after the interrupted collector.

The next bounded requirement is to localize and reduce peak memory in a single fresh-source qualification, preserving every support record and retained component, then replay the final reciprocal and joint checks within 4 GiB. No producer, gate, support, fit or rank was changed here. H2 was untouched. Matched external accuracy and other support/basis/mesh convergence remain separate obligations. Chi stays experimental; D93 fitted/post-HF stays not-qualified.