[amaybaum]

This is correction unit S1 of the Main proof-independence work on #1029. It was authorized as a minimum-change draft with independent review (owner decisions: #1029 comments 6578246538 and 6579185876). Merge is not authorized.

What changes

1. Second-order update stated as a posit. It is added to Main's posit ledger as item (xiii):

   • the form is x(t+1) = F(x(t)) − x(t−1), the kernel's ruleDynamics form;
   • it is stated as a posit of the selected lattice model, not derived and not an axiom of OI;
   • the ledger count goes from twelve to thirteen, and the methodology range from (iv)–(xii) to (iv)–(xiii).

2. A5 (linearity) corrected at Main item (vii) and where the same claim is restated: Structure:219/:259, Substratum:100/:104/:220/:272/:274, ch02:84 and FULL:727.

   • Amplitude-scale gauge invariance yields an affine restriction: every first difference is independent of its base point. Main carries this one-line argument itself.
   • Linearity is a further, imposed restriction.
   • The claimed equivalence is withdrawn, as is the inference that A4's centre-free clause turns an affine rule into a linear one.
   • Counterexample, independently verified: x_i(t+1) = x_{i−1}(t) + x_{i+1}(t) + 1 − x_i(t−1) on a 4-site ring over Z/5. It meets A4 as stated and is affine. No translation-commuting, time-independent relabelling linearizes it, so no amplitude-scale gauge transformation does either. On a finite N-site ring with q ∤ N, no time-independent bijection of any kind linearizes it.
   • The SM lemma that classifies an already linear centre-free rule is unaffected. The refuted step at SM:240 goes to a later proof unit.

3. External results counted correctly at Main:72, :630 and :658. Besides the textbook results [15, 16], the load-bearing imports are:

   • Barandes [11, 12], for the quantum-representation layer;
   • [55], the finite prepare-and-measure obstruction;
   • canonical typicality [31], used by the §4.6 theorem.

   The posit-ledger items are declared inputs.

4. Substratum hypotheses stated in Main item (ii): a finite, independently prepared factor with product dynamics at a fixed interface; the alphabet-size hypothesis; and the O(t/τ_B) condition. The Substratum citations stay until support is secured elsewhere, which is a later unit.

Unchanged

• No Lean, LEDGER, census-registry or grade change.
• No new citation of a downstream paper (AGENTS.md, "Whole-framework scope").
• Main:274 and :590 are not edited; a later unit references posit (xiii) there.

Generated files and checks

• Rebuilt with sh ./build.sh Main Structure Substratum and sh ./build.sh --book:
  • page counts: Main 99 → 100, Structure 99, Substratum 54, FULL 560;
  • no dropped glyphs;
  • after page-number and hyphenation normalization, the PDF word changes track the source edits;
  • the ch02 and FULL edits are identical.
• Local checks: lean_manuscript_census, coverage_check, voice_check, claims_check, duplicate_check, mirror_check, citation_check, dependency_label_check --strict and staleness_check all exit 0.
• Guards and census: the R7-CTN and R7-PTR predicates are true, and proof_census --json is byte-identical to a65fe8a8.
• SC-42 integration result regenerated: one predicate is added for the changed ch02 line, giving 642 checks, verdict PASS. CI's cmp against it passes locally.

The diff is 14 files in one commit on a65fe8a8. #1036 and #1037 stay paused, and the five standing holds (#1009, #1023, #1011, #1003, #918) are unchanged.

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Generated by Claude Code

[amaybaum]

Exact-head review record: bac7999f

Verdict: READY-WITH-NITS. The review was independent. The head is bac7999f1058d66043e57d14c6e1a108386769de, a single commit on main a65fe8a8. Both were confirmed at the start and the end of the review.

Fidelity. The edits to the five .md files (Main, Structure, Substratum, ch02, FULL) are byte-identical to the independently reviewed edit set, which is Option A plus the reviewer's required changes RC1–RC7. The only other changed files are the four regenerated .tex files, the four .pdf files and the SC-42 result file.

Scientific checks.

• A5 and linearity. No sentence claims that gauge invariance plus A4 yields linearity. The finite-ring counterexample does not appear in the manuscripts.
• Counterexample, re-checked by the reviewer. It was recomputed over all 5^8 states: it is bijective, translation-commuting and centre-free, with 0 fixed points. Controls: b = 0 fixes 0, and N = 5 gives 25 fixed points.
• Posit (xiii) matches the kernel's Rule, leap and ruleDynamics exactly.
• External results. The count is right: [11, 12], [55] and [31], with [15, 16] as textbook results.
• Item (ii) matches Substratum's Theorem 24 and its corollary, with nothing strengthened.
• No downstream citations. Main gains none (AGENTS.md "Whole-framework scope").
• Mirror edits. Every mirror edit removes a sentence that asserted, or depended on, the withdrawn claim.
• No new physical premise.
• Grades unchanged. proof_census --json is byte-identical at base and head.

Propagation. The only unedited sites that still assert the withdrawn inference are routed to the later proof unit: SM:240, Explainer:767 and Structure:301. SM:96 (the gauge "the §4.1 linearity argument invokes") joins them. The equivalence wording, "(iv)–(xii)" and the ledger count "Twelve" have no surviving occurrences.

Generated files.

• All four .tex files are byte-identical to fresh pandoc regeneration, and their stamps match.
• Page counts: Main 99 → 100, Structure 99, Substratum 54, FULL 560. Median characters per page are essentially unchanged.
• No replacement characters appear in the extracted text, and the word diffs track the .md edits. The other differences are layout only, including a longtable header repeated at one more page break in Main.

SC-42.

• The checker reproduces the committed result byte for byte, under both -I -B and -O.
• PASS, 642/642. The one added predicate is the ch02 A5 mirror line.
• The file is not a frozen record.

Checks at the head.

• lean_manuscript_census, coverage_check, voice_check, claims_check, duplicate_check, mirror_check, citation_check, dependency_label_check --strict, staleness_check and the R7-CTN/R7-PTR guards all pass.
• git merge-tree against main is clean, and its result tree equals the head's tree.

CI at bac7999f. Run 41460684154 is green: 25 jobs passed, and 1 (A42 exclusion (${{ matrix.part }})) was skipped by its condition.

Nit N1 (commit message, not the tree). The message says the inference that A4's centre-free clause turns an affine rule into a linear one is "withdrawn". This commit edits no site that carried that inference; it remains at SM:240 and Explainer:767, which are routed to the proof unit. The PR body states the routing correctly. Whether to re-cut the commit with a corrected message is the owner's decision.

No merge is authorized. Merging needs separate exact-SHA approval from the owner.

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Generated by Claude Code

[amaybaum]

Owner decision on exact-head review N1 — choose (b), re-cut metadata only (2026-11-02)

Decision: OPTION (b). Do not merge bac7999f1058d66043e57d14c6e1a108386769de as-is.

The independent exact-head review 6579686254 found the S1 tree/science READY-WITH-NITS, with 25 CI jobs passed and one expected skip (41460684154). It correctly identifies one historical-provenance problem: the commit-message statement that the centre-free ⇒ linear inference is “withdrawn” can be read as a corpus-wide completion, yet SM:240 and Explainer:767 still carry it, with related SM:96 and Structure:301 work routed to P1.

Authorize a new minimal PR with the identical reviewed source/artifact tree and a corrected commit message, from the currently verified main, preserving the already reviewed S1 edits and unchanged scientific grades. Leave #1042 unmerged; preserve its commit, evidence, CI and review. Re-review the exact resulting SHA (including commit metadata, identical tree and base), obtain fresh green ordinary CI, and return for separate exact-SHA merge authorization; do not silently merge it.

Proposed message replacement for the inaccurate sentence:

In the passages corrected by S1, the claimed equivalence between amplitude-scale gauge invariance and linearity is withdrawn. Linearity (A5) is retained as an imposed restriction of the selected model. Remaining assertions that center freedom yields linearity are assigned to P1 and are not corrected by this commit.

All remaining commit-message claims must still accurately describe the actual new diff. No scientific redrafting, additional manuscript sites, theorem grade or hypothesis changes are approved in this re-cut. Maintain P1 ownership of SM:96, SM:240, Explainer:767 and Structure:301, and leave #1036/#1037 and all five holds unchanged.

[amaybaum]

Re-cut opened as #1043, per the owner's decision (option (b), comment 6579764705).

• New head: 290cdce3af40cd91d51ba49d5911dca4d33486b3, on main a65fe8a8.
• Tree: bec590a7…, identical to this PR's reviewed head bac7999f.
• Change: only the commit message differs. Its overstated sentence is replaced with the owner's wording.

This PR stays unmerged. Its commit, review record and CI are preserved.

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Generated by Claude Code

[amaybaum]

Administrative closure — superseded by merged #1043 (2026-11-02)

Close this unmerged S1 draft without rewriting or deleting its branch or evidence. The exact reviewed file tree at bac7999f1058d66043e57d14c6e1a108386769de was carried byte-for-byte by the metadata-corrected PR #1043, reviewed at 290cdce3af40cd91d51ba49d5911dca4d33486b3 and merged as b87733a400c563592774dad2a61425c7663b69d1. The old commit message's overbroad wording was resolved in the #1043 replacement. Closing #1042 is not a scientific cancellation, additional merge or change to S1's result. All reviews, commits and CI remain preserved.

S2 is the next bounded correction unit under MI-1. P1 still owns outstanding SM:96/240, Explainer:767 and Structure:301; held threads remain unchanged.