[amaybaum]

Source-correction unit SC-32, from gate #946. The owner classed the SM Theorem 3 finding as critical.

Route decision pending (owner). #965 and #966 cover the same theorem. An independent review found:

• this branch's route is correct and matches what SM's Theorem 3 is about, namely the update rule's center coupling C;
• WIP: SM Theorem 3 restricted converse — proof, exact controls, and scope repair #965's first-order on-site lemma is correct, but it is a different theorem: it is the first step of Proposition 3b here, and it is credited in the record;
• WIP: guarded seven-file source-repair draft for SM Theorem 3 #966's proposed SM text changes what the theorem is about, leaving SM §4.11 (the Higgs argument) without support and making Theorem 13(ii)'s citation of Theorem 3 circular.

#966's Lean docstring fix (OI_Structural_Core.lean:413) and its structural_chain_probe.py C3 docstring and print fix are correct. They are not applied here, pending the route choice.

The flaw

SM §4.3 (main :300) displays "Center independence … is equivalent to exact chiral symmetry". Both lines of the proof prove one direction. The Remark's own rule, C = 2(1−d) at α = 1, is chiral but center-dependent, so the displayed converse is false.

The correction

• Lemma 3a (SM:L-3a, new). K(C,α) = B² − μ with μ = C − 2(1 − dα), on even periodic extents (≥ 4 for the coefficient identification).
• Theorem 3 (SM:T-3). At α = 1/d, a center-free rule is the square of the massless staggered operator and has exact chiral symmetry. A mass m ≠ 0 gives C = ±m², with a₀ = 1.
• Proposition 3b (SM:P-3b, new). The converse within Lemma 3a's family at fixed α = 1/d, for λ ≠ 0, characteristic 0 and even extents ≥ 4.
• Remark.
  • Off α = 1/d the two conditions separate, in both directions. Within the family, the center-free rule at α = 1 is massive: m² = 2(d−1) in the form (B−m)(B+m).
  • The family hypothesis is load-bearing. F = K T₀⁻¹ Π₋ + T₀ Π₊ is local, anticommutes with ε and squares to K for every (C, α).
• §A.34. No equivalence is displayed, and each direction is stated separately.

Downstream. Every dependent (SM :236, :296, :543, :563, :595) uses the proved direction on the α = 1/d branch. The book and Explainer sites carry α = 1/d through SC-17 (#963 at e316cf92).

Coverage. SM:T-3 goes from K2 to K1, because Lean certifies only the unit-coefficient factorization and the anticommutator identity. This downgrade needs the owner's acceptance. SM:L-3a and SM:P-3b are added at K1.

Record. verification/audits/manuscript/intro-corpus-audit-20261026/tracks/SC32-sm-theorem3/, containing the README and theorem3_exact.py. The script runs exact rational checks for d = 1, 2, 3 at extents 4 and 6, with countercontrols and the G6 family control. Its output is byte-identical under -O.

Verification

• Independent review: ACCEPT WITH CHANGES at 97d5b37c; REVISE (one word) at 35720a21; ACCEPT at 107d5d5e. The reviewer re-verified Lemma 3a and the F construction with their own exact code.
• Local checks: the release gate passes except lean-axioms, which runs in CI. coverage_check passes (130 statements), as does edge_rigidity_probe. The added-lines and secrets scans are clean.
• Build: SM is rebuilt at 148 pages.

Merge order. SM :296 sits one blank line below SC-28's SM :294 edit (#962), so the two must be checked together when both land.

Draft: merge only at an exact SHA the owner approves, by normal merge commit.

Related: #946, #965, #966, #963, #962.

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[amaybaum]

Closed unmerged.

• The defect (SM Theorem 3's false converse) is fixed on main by the owner-chosen Route A, Workstream D: consolidated minimal source corrections #1019 D-3 9a6a2f18: first-order onsite equivalence with a kernel witness per direction, ChiralCenterRecovery.lean, LEDGER SM:T-3 K3.
• This PR's α = 1/d Lemma 3a / Theorem 3 / Proposition 3b (K1/K1/P) would weaken that K3 statement and is not adopted.
• Its mathematics remains open research at Research backlog: advance downgraded and conditional scientific claims from corpus audit #970 orphan 25 (re-cut needed), which cites this branch at 6d32ce51.
• theorem3_exact.py was replayed on 2026-11-03 and is byte-identical.

Branch corpus/sc32-sm-theorem3 is kept.

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