Forensic Correction and Reduced-Model Benchmark of Sector-Aware VQE for Fe₄N₂ Active-Space Hamiltonians

Verdict (read first)

• Withdrawn: the prior deposit's redox-collapse / single-reference-pathway (SRDS) chemistry claim, the "0.00 kcal/mol correlation" anion, and the "R² = 1.000" redox trend.
• Retained: a corrected, independently benchmarked reduced-model sector-aware VQE result on Fe₄N₂ active-space Hamiltonians.
• Open: (a) a bounded sub-0.3 kcal/mol measurement-path residual on the high-spin anion (seeds 1–2); (b) a characterized anion ROHF spurious root occurring in ~20% of unconstrained builds (physical root stable to 1.85×10⁻⁴ kcal/mol; all benchmark numbers used the physical root).
• Boundary: active-space representability — the (Ne,10o) space is too small to carry any Fe₄N₂ / FeMoco chemistry claim.

This is a corrected, superseding version of an earlier deposit. The previous version reported a "redox-driven electronic-structure collapse" and "single-reference pathways for nitrogen activation" in Fe₄N₂ clusters from 20-qubit VQE. Those claims are withdrawn. This version explains, with evidence, why the original results were invalid, documents the engineering and methodological corrections applied, and presents a smaller but rigorously benchmarked result: a reduced-active-space methods validation of a corrected sector-aware VQE workflow. It does not restore the original chemical claim, and it is explicit about why that claim is not reachable with this active space.

1. Background: what FeMoco / Fe₄N₂ is, and what we set out to do

Biological nitrogen fixation — the conversion of atmospheric N₂ into ammonia — is catalysed by the iron–molybdenum cofactor (FeMoco) of the enzyme nitrogenase, one of the most electronically complex active sites in biochemistry. Its multi-iron core supports many near-degenerate spin and charge configurations, which makes it a long-standing benchmark problem for quantum chemistry and a frequently cited target for quantum computing, because classical methods struggle to treat its strong electron correlation.

Fe₄N₂ is a deliberately reduced four-iron / di-nitrogen cluster used here as a tractable model system — a stand-in small enough to run on ~20 qubits while retaining open-shell, multi-reference character qualitatively similar to the harder FeMoco problem. The original project asked an ambitious question: does the electronic structure of Fe₄N₂ change qualitatively across oxidation states (cation → neutral → anion) in a way that reveals a "single-reference pathway" relevant to how nitrogenase activates N₂?

The intended scientific contribution was the σ-asymmetry diagnostic framework: using inter-seed VQE energy variance and sector behaviour as a structural probe of electronic complexity, orthogonal to total-energy calculation.

2. What the original deposit claimed

The prior version asserted, for Fe₄N₂ across three oxidation states:

• A monotonic "redox-driven electronic-structure collapse," with the system becoming progressively single-reference upon reduction.
• An anion that was effectively single-reference, exact at Hartree–Fock, with 0.00 kcal/mol correlation energy.
• A near-perfect linear redox trend (R² ≈ 1.000) across the three charge states, interpreted as a "single-reference highway" for nitrogen activation.
• 20-qubit VQE on consumer GPU hardware as the enabling method.

3. What went wrong (three independent invalidations)

A post-publication audit established three separate, independently sufficient reasons the original results cannot be relied upon. Each alone invalidates the headline claim; together they show the original conclusion was an artifact of the validation tooling, not a property of the chemistry.

3.1 Sector invalidation — the spin penalty was non-functional

The original production runs were intended to enforce particle number (N) and spin projection (Sz) sectors through penalty terms. Re-examination of the original run summaries showed every one of the nine production runs (cation, neutral, anion × 3 seeds) reported DRIFT, with Sz penalty residuals of 8–18 Hartree — i.e. the spin penalty was effectively not constraining the wavefunction at all. None of the original states were in their nominal spin sectors. "Cation vs neutral vs anion" were therefore not reliably three distinct, correctly-prepared oxidation states.

3.2 Reference invalidation — no valid correlated reference ever existed

The reported "0.00 kcal/mol correlation" and "R² = 1.000" required comparison against a correlated reference (CASCI/CASSCF/FCI). The engine skips CASCI whenever a frozen core is present (n_frozen = 32 for all Fe₄N₂ runs — a deliberate, defensible refusal to mix frozen-core bookkeeping). Consequently no valid correlated reference was ever computed for Fe₄N₂ in the entire run history; the reported correlation/R² was implicitly computed against a fallback (qubit-space Hartree–Fock) on a different energy footing, which is meaningless. This defect was present in the original deposit independently of the sector problem.

3.3 Active-space representability — the model is too small for the chemical claim

The active space used was (13e,10o) / (14e,10o) / (15e,10o) — ten spatial orbitals for a four-iron / two-nitrogen cluster. A single Fe 3d shell is five spatial orbitals; four irons span ~20 orbitals before any Fe–N bonding, N 2p, or Fe 4s participation. Ten orbitals cannot represent the electronic structure the original title claimed ("redox electronic-structure collapse," "nitrogen activation"). Even a perfectly executed VQE in this space is a reduced-model result, not a statement about Fe₄N₂ chemistry.

Additional contributing factor — the pinned-HF anion artifact. The original anion convergence trace contained 300 bit-identical energy values: the optimizer never left the Hartree–Fock determinant. The "single-reference, 0.00 correlation" anion was an optimizer that never ran, compared against a reference that coincided with it.

4. What we corrected, and how

The corrections were engineering and methodological, applied as non-invasive patch modules so the proprietary VQE engine itself was not modified:

1. Corrected Sz-sector penalty — a properly formed λ(Ŝz − sz_target)² penalty replacing the malformed one.
2. Sector-aware checkpoint selection — the optimizer previously kept the lowest penalized-energy checkpoint and audited sector purity only afterward. The corrected workflow selects the best sector-clean checkpoint (lowest physical energy among states passing the sector audit), so reported states are in their intended (N, Sz) sectors.
3. Same-active-space exact reference — instead of the (correctly-refused) full-space CASCI, we diagonalize the same qubit Hamiltonian the VQE optimizes, projected into the intended (N, |2Sz|) sector, by direct in-sector construction (the full 2²⁰ operator is never built; the relevant sectors are ≤11,340-dimensional). This is the legitimate benchmark: it is, by construction, the exact ground state of the operator the VQE minimizes, with no frozen-core mismatch possible.
4. Sign-agnostic reference — the spin-free electronic Hamiltonian is exactly degenerate between +Mₛ and −Mₛ. The VQE may converge to either component (the anion converged to 2Sz = −5 under a +2.5 target). The reference now diagonalizes both ±|2Sz| components and uses the lower; the ±Mₛ split was measured at ≤1.4 × 10⁻⁹ kcal/mol (exactly 0 for the anion), confirming degeneracy.
5. Reproducibility characterization — repeated Hamiltonian builds were measured to confirm the exact reference is deterministic at the sub-millikcal level (Section 6.3).

A diagnostic mislabel (the engine's debug printout read the penalized Hamiltonian identity coefficient, ≈ −450 Ha, instead of the physical pre-penalty value, ≈ −593 Ha) was also corrected; it did not affect any reported energy but caused substantial wasted debugging and is documented here for transparency.

5. Findings

After correction, a full sweep of 3 oxidation states × 3 seeds = 9 runs was benchmarked against the same-active-space exact reference. All nine runs are sector-CLEAN and demonstrably optimized off Hartree–Fock (energy descent of 18–22 kcal/mol from the HF determinant — i.e. none are pinned, in direct contrast to the original anion artifact).

Headline result. Corrected sector-aware VQE approaches the same-active-space exact reference in the reduced Fe₄N₂ model with low-kcal/mol errors. Cation and neutral are strict variational passes across all seeds; the anion exposes a bounded sub-0.3 kcal/mol measurement-path residual on two of three seeds:

• Neutral (quintet, N=14): clean variational passes, ≈ 2.0 kcal/mol above exact, reproducible across 3 seeds; reference reproducible to 0.007 kcal/mol. Unconditionally solid.
• Cation (quartet, N=13): clean variational passes, ≈ 2.4 kcal/mol above exact, reproducible across 3 seeds; reference reproducible to 0.004 kcal/mol. Solid.
• Anion (sextet, N=15): sector-clean and reproducible against a reference stable to 4 × 10⁻⁴ kcal/mol. The reduced anion sector is numerically near-HF: the retained statevector reports HF = −597.5423603917 Ha, while the exact same-sector reference is ≈ −597.542396 Ha, only ≈ 0.02 kcal/mol lower — i.e. there is essentially no correlation energy to capture in this small sector (a numerical property of the reduced model, not a chemical SRDS claim). The saved anion statevector is 99.997% pure in the correct (N=15, |2Sz|=5) sector. Seed 0 is a strict variational pass (+0.062 kcal/mol); seeds 1–2 show bounded sub-reference excursions of 0.09–0.28 kcal/mol, consistent with the documented engine measurement-path inconsistency (Section 6.5) rather than a reference, sector, or VQE failure. Every gross alternative was eliminated by direct measurement (sign convention, reference noise, ROHF root instability, sector leakage).

Significance — what this means and does not mean

What it means. A corrected sector-aware VQE workflow, on consumer-class GPU hardware, recovers same-active-space exact references to low-kcal/mol accuracy for an open-shell transition-metal reduced model across three oxidation states (strict variational passes for cation/neutral; a bounded open residual for the high-spin anion) — where the prior, uncorrected workflow produced wholly spurious results. The contribution is forensic and methodological: it characterizes a class of failure in penalized-VQE pipelines (non-functional spin penalty, energy-only checkpoint selection, absent/ill-footed reference) in which the diagnostics confidently report success on invalid states, and it demonstrates the corrections and the independent exact-reference check that detect them. The recurring lesson, stated as a design principle: a benchmarked energy and its reference must be drawn from one and the same serialized Hamiltonian instance — never from two independent builds — and an engine's internal sector/energy audit is not a substitute for an independent exact reference.

What it does not mean. This does not restore the original "redox-driven electronic-structure collapse" or "single-reference pathway for nitrogen activation" claim. The (Ne,10o) active space is too small to represent Fe₄N₂ electronic structure (Section 3.3); these are reduced-model method-validation results only. No statement about FeMoco chemistry or nitrogen activation is supported by this work.

6. Technical analysis

6.1 Exact same-active-space references (verified, sign-agnostic, deterministic)

±Mₛ degeneracy split: ≤ 1.4 × 10⁻⁹ kcal/mol (cation/neutral), 0.0 (anion). Hermiticity residual: 0.00 (all). All 20 qubits; n_frozen = 32; engine CASCI/FCI correctly skipped (frozen core present).

6.2 Benchmark results — 9 runs vs exact reference

Errors are reported as the variational gap above the exact same-sector ground state (positive = variationally correct). Cation/neutral seed spread ≤ 0.16 kcal/mol. The two anion negative values are below the verified-stable reference and are characterized in Section 5 as a bounded measurement-path open item, not a variational failure and not a reproducibility-noise effect.

6.3 Hamiltonian / reference reproducibility (per species)

Reference stability was verified using reconcile_spread.py (n = 5 repeated builds) for cation and neutral, and characterized in detail for the anion with n = 20 builds (anion_rohf_multiroot_*.json, raw per-build data included).

Anion open-shell ROHF multi-root finding (characterized, not incidental). The anion ROHF admits a reproducible spurious higher-lying solution. In an n = 20 characterization, 16 builds (80%) converged to the physical root (HF ≈ −597.54236 Ha; in-sector reference ≈ −597.542396 Ha) and 4 builds (20%) converged to a spurious root (HF ≈ −597.48722 Ha; reference ≈ −597.48730 Ha), a ≈ 34.6 kcal/mol higher-lying SCF solution. The spurious root is itself highly reproducible (all four occurrences agree to ~10⁻⁶ Ha), i.e. it is a genuine second SCF stationary point, not numerical noise.

Restricted to the physical root, the anion exact reference is stable to 1.85 × 10⁻⁴ kcal/mol (std 5.8 × 10⁻⁵) — the tightest of the three species. All benchmark results used the physical root: every reference value entering the benchmark (JSON and TSV, ≈ −597.5424 Ha) lies in the physical cluster; the spurious −597.4873 solution never propagated into any reported number. The mitigation is straightforward and is now documented practice: reject any build whose HF energy is not in the physical (lowest) cluster, or pin the ROHF initial guess.

This is reported as a substantive methods finding. An open-shell transition-metal ROHF with a reproducible ~35 kcal/mol spurious solution occurring in ~20% of unconstrained builds is precisely the class of hazard that silently corrupts penalized-VQE pipelines that do not rebuild enough to observe it — and it is the same family of open-shell-reference pathology as the original deposit's pinned-HF anion artifact, here measured and mitigated rather than undetected.

Large build-to-build spreads in the frozen-core and identity bookkeeping terms (≈ 0.27–0.72 kcal/mol) do not propagate into the physical-root in-sector reference and do not affect the benchmark.

6.4 Engine corrections and known issues (reproducibility transparency)

• Diagnostic identity mislabel: debug printout read the post-penalty Hamiltonian identity; corrected to read the pre-penalty physical operator. No reported energy affected.
• CASCI skip with frozen core: a defensible engine refusal; the failure was the reporting layer emitting a correlation number against a fallback reference anyway. Corrected by introducing the same-active-space exact reference.
• Output-prefix history file: --output_prefix does not redirect the engine's history CSV/statevector in this engine version; per-seed trajectory files were overwritten during sweeps. Per-seed logs were unaffected and are the source of record for the benchmark energies. Per-seed wavefunctions other than the last anion seed were not retained.
• Superseded tool (correction): an internal rebuild-envelope script (characterize_rebuild_envelope.py) earlier reported a ~34 kcal/mol anion spread. This was initially misattributed to a tool defect; it is not. It had sampled a spurious-root build and reported the real anion ROHF bimodality correctly (Section 6.3). The tool is excluded only because max-spread over a bimodal distribution is not a meaningful "reproducibility envelope" — the correct characterization (spurious-root frequency + physical-root stability) is given in Section 6.3 with full raw data. Its earlier output was not used in any benchmark number.

6.5 Cross-file numerical reconciliation and anion measurement-path caveat

The exact-reference JSON files, the benchmark TSV, and the retained anion statevector were generated by separate engine/reporting paths and therefore expose small but important internal-consistency checks. These are reported explicitly here rather than hidden; each is bounded and none affects any conclusion.

(i) Reference JSON vs benchmark TSV. The exact-reference values in exact_reference_*.json and the values used in BENCHMARK_corrected.tsv differ at the final digits because they were produced by independent non-deterministic rebuilds of the same reduced active-space Hamiltonian. The differences are small and, per species, comparable to that species' own measured build reproducibility (Section 6.3): cation 0.0045 kcal/mol (vs ~0.004 reproducibility), neutral 0.0023 kcal/mol (vs ~0.007), anion 0.00003 kcal/mol (vs 1.3 × 10⁻⁴). In every case the JSON-vs-TSV difference is within or comparable to the per-species rebuild spread, and is negligible relative to the VQE benchmark errors (~2 kcal/mol cation/neutral). These are characterized non-deterministic-rebuild differences, not errors; the per-species reproducibility values in Section 6.3 — not a single anion-only figure — are the correct comparison scale.

(ii) Statevector vs benchmark table (the anion measurement-path evidence). The benchmark table reports anion seed-2 physical energy from the log as −597.5428466797 Ha, while the serialized anion_seed2_statevector.npz reports vqe_energy = −597.5427582264 Ha — a difference of ≈ 0.056 kcal/mol. This is the clearest artifact-level evidence that the engine's serialized VQE-energy field and its logged physical-energy reporting path are not bit-identical for the high-spin anion. It is not large enough to affect any conclusion, and it is precisely why anion seeds 1–2 are reported as a bounded measurement-path residual rather than strict variational passes. A reviewer loading the published .npz will observe this difference directly; it is stated here as supporting evidence for the documented measurement-path finding, not as an undisclosed discrepancy.

(iii) Near-HF character of the reduced anion sector. The retained anion seed-2 statevector reports serialized HF = −597.5423603917 Ha, while the exact same-sector reference is ≈ −597.542396 Ha — only ≈ 0.02 kcal/mol lower. In the reduced (15e,10o) anion model, HF, exact diagonalization, and VQE therefore all lie within the sub-kcal/mol measurement band: the sector has essentially no correlation energy to capture. This is a numerical property of the reduced active-space Hamiltonian only and must not be read as a chemically meaningful Fe₄N₂ / FeMoco SRDS claim — it is, in fact, the correct (non-artifactual) analogue of what the original deposit misreported as a "single-reference highway."

Accordingly, the anion benchmark is not a failed run and not a clean strict variational pass for seeds 1–2. It is: sector-clean, reference-stable, reproducible, and bounded by a ≤0.3 kcal/mol internal energy-measurement-path discrepancy that is itself documented here as a methods finding.

7. Files included in this deposit

Intentionally excluded: the proprietary VQE engine (prometheus_vqe_engine_penalized_latest.py) and any patentable-IP components; characterize_rebuild_envelope.py and its outputs (superseded — its ~34 kcal/mol was real anion bimodality, not a defect; see Sections 6.3–6.4 for the correct characterization).

Because the proprietary VQE engine is excluded, this deposit provides audit artifacts, reference-generation tools, and patch modules sufficient to independently verify the exact references and the benchmark logic, but it is not a fully standalone re-runnable reproduction of the VQE engine itself. The exact same-active-space references (Section 6.1) are, however, independently reproducible from the included exact_sector_reference.py given the molecular geometry and basis, which are specified herein.

8. Scope and honest limitations

• This is a reduced-active-space methods validation, not a chemical result on Fe₄N₂ or FeMoco.
• The (Ne,10o) active space is physically inadequate for four-iron nitrogen-activation chemistry; no chemical conclusion is drawn.
• The anion seeds 1–2 exhibit a bounded ≤0.3 kcal/mol sub-reference discrepancy attributed to an engine measurement-path inconsistency; this is reported as a characterized open item, not resolved to machine precision, and does not affect the cation/neutral conclusions or the methods claim.
• Per-seed wavefunctions (other than anion seed 2) were not retained due to the output-prefix issue; wavefunction-level analyses would require targeted re-runs with a corrected harness.
• The original deposit's chemical claims (redox collapse, single-reference pathway, R² = 1.000, 0.00 kcal/mol correlation) are withdrawn and are not restored by this work.

9. Citation and supersession

This version supersedes and corrects the prior deposit. The prior version's chemical conclusions should not be cited or relied upon. This version should be cited for: (a) the corrected reduced-model methods benchmark, and (b) the documented forensic analysis of the failure modes in the original work.