[amaybaum]

MI-1 unit H2 (#1029). Implementation authorized in #1029 comment 6585855086. This is a draft; merging needs separate owner approval of an exact SHA.

Base: main at 7b389414a71e08486738ff7069930c9e09acff98. Head: 5008d916d2028b67c3fe96fdc964f3783dab294f.

Finding

SM §4.1 (SM:240) and the Explainer (Explainer:767) state that the center-free branch reduces the affine rule from the amplitude-scale-gauge lemma to a linear one. That step is refuted under its own hypotheses. On a ring of four sites over $\mathbb{Z}/5\mathbb{Z}$, the rule $x_i(t+1)=x_{i-1}(t)+x_{i+1}(t)+1-x_i(t-1)$ is reversible, commutes with translations, and is center-free and affine. It has no fixed point. A linear map fixes the zero state, and a relabelling carries fixed points to fixed points, so no time-independent relabelling makes this rule linear. Structure:301 and SM:96 cite the step by name, so they get wording changes only.

Edits (four source lines)

Site	Change
SM:240 (§4.1)	The lemma yields an affine rule. Linearity (A5) is a further restriction on the selected lattice model, not a consequence of center independence; the witness and its fixed-point argument appear inline (R1).
SM:96 (§2.7)	"the §4.1 linearity argument" → "the §4.1 argument for an affine rule"
Explainer:767	Same correction, with the "independent of base point" wording (R2) and an existence statement only
Structure:301	"necessity argument for linearity" → "argument for an affine rule"

Preserved: the affine lemma, A5 as an explicit posit, SM:224/228, Main:726(vii), and every result conditional on linearity. Nothing is claimed for infinite lattices, site-dependent shifts, q∣N cases or time-dependent relabellings. SM, Explainer and Structure .tex/.pdf are regenerated with unchanged page counts (150, 71, 99). There is no LEDGER, census-registry, grade or sc42 change. The book needs no mirror edit: its only corresponding passage (ch02:84 = FULL:727) already states the affine form.

Not in this PR (routed per owner decision)

• The SM:96 / Structure:301 / Substratum:274 status inconsistency and related terminology go to a separate corpus-consistency unit.
• Revision-history wording near SM:240 goes to a later editorial unit.
• The SC17 audit README goes with SC-17 (Explainer): Bell at Main §3.3's scope, the dynamics' form fixed up to one coefficient, chirality without the memory claim #963.
• This PR does not resolve the existing merge conflicts in SC-17 (Explainer): Bell at Main §3.3's scope, the dynamics' form fixed up to one coefficient, chirality without the memory claim #963, SC-32: SM Theorem 3 as the proved direction at α = 1/d, with the restricted converse separate #969 and SC-34: cycle ergodicity on energy shells — map named at every site; false for the exact linear rule, unsupported for the rounded rule #971. SC-17 (Explainer): Bell at Main §3.3's scope, the dynamics' form fixed up to one coefficient, chirality without the memory claim #963 carries a compatibility note.

Verification at head 5008d916

• Local checks. lean-manuscript, coverage, voice, claims, duplicate, mirror, citation, dependency-label --strict, staleness and R7-CTN/R7-PTR all pass. SC-42 --verbose output is byte-identical to the checked-in result. proof_census --json is identical to base. The build reported no dropped glyphs.
• Independent exact-head review: ACCEPT, no blocking findings.
  • The diff equals the reviewed edit set exactly.
  • The rebuilt .tex files are byte-identical to the committed ones.
  • The witness was checked over all 5^8 two-slice states. It is bijective, translation-commuting, center-free and affine, and has 0 fixed points. Positive controls (q∣N) find fixed points.
  • No surviving center-free ⇒ linear statement was found in the papers, the book chapters or FULL.
  • Register is clean.
• CI. Pull-request run 41507105419: success, 24 jobs passed and 1 skipped, the matrix placeholder. This is a pull-request run, not an exact-head attestation.

Related: #1049 (linear vs affine research question; planning only).

🤖 Generated with Claude Code

https://claude.ai/code/session_013T6pfKuv5mi3MXtKjH2tZb

[amaybaum]

Owner exact-head decision and verified normal merge — H2 #1050 (2026-11-02)

APPROVED AND MERGED at exact head 5008d916d2028b67c3fe96fdc964f3783dab294f, from prior main 7b389414a71e08486738ff7069930c9e09acff98.

Evidence and gates

• Independent exact-head verdict ACCEPT, no blockers, recorded in the PR description at this head; the reviewed four-line edit set with R1 fixed-point contradiction and R2 base-point-independent wording matches the PR diff.
• PR CI 41507105419: success, 24 jobs passed and one expected skip. Workflow head_sha verified as 5008d916d2028b67c3fe96fdc964f3783dab294f.
• Reverified open draft PR, unchanged head, clean mergeability and main at 7b389414a71e08486738ff7069930c9e09acff98. Marked ready, then performed normal merge with exact expected-head SHA protection. GitHub reported merged=true.
• Verified GitHub merge commit 981a54606f189a160385d7be4af41dcb22de73b4 as current main, with parents 7b389414a71e08486738ff7069930c9e09acff98 and 5008d916d2028b67c3fe96fdc964f3783dab294f; PR MI-1 H2: center independence gives an affine rule, not a linear one #1050 is closed/merged. Exactly nine changed files: SM, Explainer, Structure each with .md, regenerated .tex and .pdf.
• Post-merge main CI 41510606950 was queued when checked; this record does not claim it passed.

Scientific disposition: The refuted microscopic center-free affine ⇒ linear inference is withdrawn in its verified finite-ring scope; the valid affine result, A5 as explicit selected-model posit, and the already-linear conditional results remain intact. No assertion is made for infinite lattices, arbitrary q or N, or physically admissible site-/time-dependent transformations. SM:96 and Structure:301 reference wording changed without treating the entire cited discussion as refuted. No Main, book, FULL, Lean, grade, census or ledger edits.

Next governance: H2 is complete. Start P1 only under a distinct bounded approval; the O3/O4 book unit remains queued after P1 under the separate full-book audit/proposal authorization. #1049 remains research-planning only, #1036/#1037 and five holds unchanged.

[amaybaum]

Post-merge record. Merged by the owner as 981a54606f189a160385d7be4af41dcb22de73b4.

• Merge commit: parents are 7b389414a71e08486738ff7069930c9e09acff98 (main) and 5008d916d2028b67c3fe96fdc964f3783dab294f (the reviewed head). Its tree, 6282b817, is identical to the reviewed head's tree.
• Main push CI: run 41510606950 succeeded, with 25 jobs passed and 1 expected skip.
• Before merge: the independent exact-head review at 5008d916 was ACCEPT with no blocking findings, and pull-request CI run 41507105419 was green.

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