Independent Verification and Scope Extension of the Electron-Sector Integrity Framework: Number-Sector Claims Reproduced, Spin-Sector Axis Added

Verdict (read first)

• Independently reproduced and valid: the record's core practical contribution — that unconstrained non-number-conserving VQE drifts out of the intended electron-number sector, that this is invisible to standard convergence diagnostics, and that a quadratic number penalty restores number-sector purity — is independently confirmed from the deposited statevectors using a verifier that shares no code with the engine or the record's own tools. All headline number-sector figures reproduce to the reported precision.
• Scope is incomplete (extended here, not retracted): electron-sector integrity has at least two axes — electron number and spin (2Sz). The deposited penalized cation statevector passes the record's own checklist as "SECTOR-PURE ✓" on the number axis while carrying ≈15% wrong-2Sz contamination within the target N=13 sector. A number-only penalty commutes with Sz and cannot, by construction, enforce spin-sector purity.
• Corrected statements: the "independent axis (singular)" and "general / publication-ready checklist" framing oversells a number-sector-only gate as a complete sector-integrity guarantee; and the penalized cation described as a "clean open-shell singlet" cannot be a singlet (N=13 is odd; minimum is a doublet) and is in fact a 2Sz mixture.
• This is a methods record: nothing physical is withdrawn. The practical framework survives intact and is strengthened. The correction adds the missing axis.

This is a corrected, superseding version of an earlier deposit ("Electron-Sector Integrity in Variational Quantum Eigensolvers — Computational Data and Validation Tools"). An independent audit reproduces the record's number-sector results exactly and identifies that the proposed sector-integrity checklist is incomplete on the spin axis — the same axis whose absence has separately been documented to invalidate spin-sensitive VQE results elsewhere in this project's corpus. The number-sector contribution is retained and independently verified; the conceptual framework is extended to be multi-axis.

1. What the original deposit claimed

The original record introduced a Detect→Fix→Verify framework for "electron-sector integrity" in VQE: (i) Hamming-weight P(N) analysis to detect drift out of the intended electron-number sector, (ii) a quadratic penalty H′ = H + λ(N̂ − Nₜ)² to enforce it, (iii) post-hoc number/NOON metrics to verify restoration. It demonstrated, on a Cu₅MoO₁₂ cation testbed (20 qubits, 13e/10o), that unconstrained VQE converged to a number-sector-mixed state (P(N=13)=86.69%) passing all standard convergence checks, with a 2.91 kcal/mol energy bias, fixed by λ=0.5 (P(N=13)→99.84%). It proposed a Sector Validation Checklist (P(N=target)>99%, number purity proxy <0.01, ⟨N⟩ consistency) as a general publication-quality gate, and framed electron-sector purity as a single "independent reliability axis."

2. Independent verification (what reproduces)

An independent verifier (sector_verify.py, bundled) was written that imports neither the proprietary engine nor any of the record's own tools, and re-derives every quantity directly from the deposited statevector amplitudes using the interleaved Jordan–Wigner convention specified by the record's own NOON tool (qubit 2i = orbital i α, 2i+1 = orbital i β). It was validated on a synthetic state constructed to be number-pure but spin-mixed, and correctly distinguished the two axes before being applied to the deposited data.

Number-sector claims reproduce to reported precision:

The number-sector drift phenomenon, its invisibility to standard convergence diagnostics, the Hamming-weight detector, and the number-penalty fix are independently confirmed. This part of the deposit stands without modification.

3. The scope gap (what the checklist does not see)

The same verifier additionally decomposes each statevector by spin projection 2Sz = (#α − #β) — a check the record never performs. Result:

The decisive observation: the penalized cation — which the record's own checklist classifies "SECTOR-PURE ✓ (publication-ready)" — has, within its now-pure N=13 sector, a 2Sz distribution of ≈0.845 at 2Sz=+1 and ≈0.154 at 2Sz=−1. It is an ≈85/15 mixture of the two doublet branches, not a spin-clean state. This is structural, not incidental: the penalty operator λ(N̂ − Nₜ)² commutes with Ŝ_z and therefore cannot enforce spin-sector purity regardless of λ. The number axis and the spin axis are genuinely independent — but there are two of them, and the record's checklist measures only one.

Consequently:

• The framing "electron-sector purity is an independent reliability axis" (singular) and the checklist "for publication-quality VQE" overstate a number-sector-only gate as a complete sector-integrity guarantee. A state can pass every line of the proposed checklist and still be spin-sector contaminated at the ~15% level.
• The README's description of the penalized cation as a "clean open-shell singlet" is incorrect on two counts: N=13 is odd, so a singlet is impossible (minimum multiplicity is a doublet); and the deposited statevector is not a spin-pure doublet but a 2Sz mixture.
• The neutral basin B state is drifted on both axes jointly (N=14, 2Sz=−2; P of the intended N=13 ∧ 2Sz=0 block = 0.000000), confirming the engine's drift is joint number+spin, not number-only — consistent with the record's "sector-drifted" labeling of that case but showing the drift is broader than the number-only framing implies.

4. The extension (the genuine new contribution)

This correction does not weaken the record; it completes it. Electron-sector integrity is multi-axis: enforcing and verifying the electron-number sector (the record's contribution, independently confirmed) is necessary but not sufficient. Spin-sector purity (2Sz, and ideally the full Ŝ² multiplicity) is a second axis that requires its own enforcement (e.g. an Ŝ_z and/or Ŝ² penalty, or a spin-adapted ansatz) and its own checklist line. A complete sector-integrity gate must report, at minimum, P(N=target), the joint P(N=target ∧ 2Sz=target), and a spin-purity proxy within the target N — not P(N) alone.

This is the same lesson encountered independently elsewhere in this project's corpus, where enforcing particle number without spin allowed spin-sensitive VQE results to converge to spin-artifacts while passing number-based checks. Documenting it here, with the deposited penalized cation as a concrete worked example that passes the record's own gate while spin-contaminated, makes the framework genuinely general rather than asserted-general.

5. Limitations and honest scope

• The spin-axis result is established at the 2Sz level (proven, decisive: the +1/−1 split within N=13 is large and unambiguous). A full Ŝ² spin-multiplicity analysis (a state can be 2Sz-pure yet not an Ŝ² eigenstate) is the natural deeper check and is not performed here; it is flagged as future work. The core finding — number-pure, 2Sz-mixed, checklist blind to it — does not depend on the Ŝ² analysis and stands on the joint (N, 2Sz) decomposition alone.
• This correction does not assert that the number penalty worsens spin purity. The raw within-N=13 spin-purity proxies (0.112 unconstrained → 0.261 penalized) are conditioned on different populations and are not directly comparable. The proven, calibrated statement is narrower and sufficient: the number penalty does not, and mathematically cannot, enforce 2Sz purity, and the penalized state remains 2Sz-mixed.
• Downstream implication (stated as caveat, not finding): any deposit that cites this record's checklist as its sector-integrity guarantee inherits this scope gap on the spin axis. Records making spin-sensitive claims should not treat number-sector validation as sufficient.

6. Files in this deposit

The independent verification re-derives all results directly from the carried-over statevectors; the deposit is self-contained and re-runnable (python sector_verify.py --dir <files>). The proprietary VQE engine is not included (consistent with the rest of this project's corrected deposits).

7. Citation and supersession

This version supersedes and corrects the prior deposit. The number-sector framework, statevectors, penalty mechanism, and detector are retained and independently verified — cite these as before. The conceptual claim should be cited in its corrected form: electron-sector integrity is multi-axis; the original checklist validates the electron-number axis only and is necessary but not sufficient; spin-sector (2Sz) purity is a second axis requiring separate enforcement and verification, demonstrated here on the deposit's own penalized cation statevector.