[amaybaum]

Scope

Active proof/correctness work following the SM Theorem 3 finding in #963 / #946. Head: d2e01a3a2670dab1a0f613d2dcfb7193e2608e1a.

Only the audit directory changes. No manuscript, Lean module, coverage classification, or workflow is edited. This does not authorize a merge or change #918's gates.

Mathematical result

For a fixed chiral operator D and involution epsilon, over the real or complex numbers, a commuting on-site perturbation C preserves {D+C,epsilon}=0 iff C=0: the anticommutator is 2 C epsilon.

Separately, for scalar C=m epsilon+p I, both nonzero parity eigenspaces imply m=p=0. Both parity sectors are necessary for coefficient identification, not for the operator-level converse. Operator-valued coefficients can cancel; the generic Lean identity does not itself rule that out.

The report contains a complete author-derived proof and proposed restricted theorem wording. This preserves the valid restricted equivalence without identifying it with a general microscopic-to-observer theorem.

Exact checks performed

• 83 scalar and 243 commuting-block rational cases.
• Four explicit staggered lattices: extent 4, dimensions 1–4. The independently constructed square has diagonal -2d while the first-order operator remains chiral.
• Ten hypothesis-removal/boundary countercontrols.
• Normal and optimized outputs byte-identical; Python compilation succeeds.
• Published Git blob hashes match the local report, executed script, and result bytes.

Files: REPORT.md, restricted_converse_probe.py, RESULTS.json under verification/audits/manuscript/sm-theorem3-direction-20261027/.

Evidence level / outstanding work

Author mathematical proof plus bounded exact computation. No new Lean compilation or axiom result, no independent review, no full repository-gate result, and no full propagation audit is claimed. Existing Chiral.center_anticommutator was inspected as source, not rerun.

Independent review must check the operator/parameter distinction, lattice assumptions, scope of the candidate manuscript statement, and corpus dependents before any manuscript correction or formal-status upgrade. Exact-SHA merge approval remains with the owner.

[amaybaum]

Exact-head independent-review handoff

Object: d2e01a3a2670dab1a0f613d2dcfb7193e2608e1a, specifically REPORT.md, restricted_converse_probe.py and RESULTS.json under verification/audits/manuscript/sm-theorem3-direction-20261027/.

This is a writer-prepared handoff, not an independent review, an acceptance, or a reviewer assignment. No independent review was present in the PR discussion when checked. The author proof and proposed manuscript replacement stay pending review.

Questions the independent reviewer should resolve

1. Operator converse. Verify both directions of {D+C,epsilon}=0 iff C=0 under epsilon^2=I, {D,epsilon}=0, [C,epsilon]=0 and additive two-torsion-freeness (or a real/complex algebra). Check that commutation is not inferred merely from an arbitrary lattice update being called center-free. On-site blocks commute with site parity, but zero diagonal alone does not establish chirality.
2. Coefficient identification. Separately verify scalar C=m epsilon+p I implies m=p=0 when both parity eigenspaces are nonzero. Confirm the one-sector and operator-valued-coefficient cancellation examples. Do not require two parity sectors for the weaker conclusion C=0.
3. First-order versus second-order scope. Verify the constructed B=2D, its anticommutation, and the independently constructed second-difference square on the stated even periodic lattices. Confirm that the nonzero diagonal of B² is not being identified with the particular time-update coefficient in SM's remark, a Dirac mass, or a general projection theorem.
4. Reproduction and controls. Run the exact pinned probe normally and with python -O; compare both outputs byte for byte with retained RESULTS.json. Inspect whether each countercontrol actually violates the intended hypothesis and witnesses the claimed failure. Expected retained hashes: script 0ddc0d9dd905d9aa59f8e5dabc5e837b6c04497325279bdef60a9bb275481d60, result e9031340988e11ed0db9d1433fcaaae4f4b1e122a90edbe0ec52c386baec77c7. Finite tests support the construction; the universal argument must stand on its proof.
5. Formal-evidence boundary. Compare the precise statements of Chiral.center_anticommutator and Chiral.mass_square with the new claims. Source inspection or a green Python result is not a new Lean certificate. Any later formalization requires its own actual kernel and axiom evidence.
6. Manuscript and propagation. Check the proposed restricted theorem against SM §4.3, its remark and actual branch definitions. Identify concrete dependent sites across the core papers, book/FULL, Explainer, roadmap, probe descriptions and coverage ledger; do not treat this report as a completed propagation audit. The valid restricted equivalence must be preserved without promoting it to an arbitrary microscopic-to-observer equivalence.

Please report separately: mathematical verdict, computational reproduction, manuscript-scope verdict, required fixes, optional refinements and dependency/propagation findings. Each verdict must name this exact head. This PR has made no manuscript correction and does not close the underlying source finding by itself.

The standing independent-review and exact-SHA merge gates remain unchanged.

[amaybaum]

Closing without merge: the owner chose SM Theorem 3 Route A (Workstream D audit on #946; owner decision of 2026-10-31).

• The kernel now proves this PR's first-order lemma in a stronger form. main's OIBridge/ChiralCenterRecovery.lean proves it over any ring without additive 2-torsion. Both directions are kernel witnesses: scalar_center_coefficients_zero_of_chiral and scalar_center_chiral_of_coefficients_zero, with onsite_chiral_iff for site-dependent coefficients.
• Route A states exactly that result. Draft Workstream D: consolidated minimal source corrections #1019, commit 9a6a2f18, phrases SM Theorem 3 as "vanishing of the onsite terms ($m=p_0=0$) is equivalent to exact chiral symmetry". It also restores the separate-bridge qualification for the second-order center-independence reading.
• No manuscript or verification reference from this PR is still needed. It changes no manuscript files.

The branch and history are kept as research provenance.

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