[Lean] Prove all-length cyclic Schur elimination and indexed passes (no new kappa)

[aliromman]

This formalizes the exact Euclidean chain/closure reduction used by Rohan Guliani (@Rohansguliani)'s PR #846 and Ali Romman (@aliromman)'s PR #985. For every 0 < w ≤ n, Lean proves
the scalar system y_i = x_i + b_i y_((i+n-w) mod n) is equivalent to its
length-w closure plus reconstruction. The proof derives the endpoint
n-w+k and child stride (-n) mod w. It preserves coefficient order and
works over any semiring, without invertibility or norm assumptions.

Review files: Proof and scope · Evidence · Source pins and credit.

An explicit sequential array evaluator models the two source passes.
Lean proves its prefix and suffix invariants, that Pass A emits the
closure data and products, and that Pass C reconstructs a parent solution
from a child solution. Restriction and reconstruction preserve existence
and distinguish every parent solution. This closes an exact indexed
elimination obligation; it does not assume that obligation as a leaf
contract.

Reproduce from the repository root:

sh research/cyclic-schur-lean/verify.sh /path/to/lean-4.21.0/bin/lean

The package checks compiler commit
6741444a63eec253a7eae7a83f1beb3de015023d, uses core/Std only, and builds
in a fresh temporary directory. The final local command exited 0.
The axiom report covers all 102 theorem declarations, including generated
lemmas; all 30 named proof theorems and six reference/control theorems are
included. Only propext, Quot.sound, and Classical.choice occur.
Four mathematical controls reject wrong stride sign, wrong tail start,
an omitted coefficient, and a stale predecessor. Separate invalid-proof
probes are rejected specifically by the axiom guard.

Base: 8bddbee377e2f3ce1f1247196c4b290d55767d20. Source subjects are
PR #846 4bdb3e755ea7118170a7a34764df955880dec282 and PR #985
9df42a99dd8497b6e7d46cd2507bb89071372640; exact file pins and source
line links are included. Gareth Davidson (@bitplane)'s PR #896
4e34d42b64ff05a4f293b9748d71642e14ef273d proves a separate generic
dirty commutator and is not imported. Chafik Boukhalfa (@chafreaky)'s PR #628 and Charles Bethin (@cbethin)'s PR #764 retain their predecessor credit.

Only research/cyclic-schur-lean/ is added. The Python solvers, precision
and phase contracts, selected frontier, and upstream/ are unchanged.
Python/FIFO/tape semantics, fresh-power return times, rounding, dirty
restoration, physical costs, and the full multiplication theorem remain
separate. Global make verify, selected cold replay, and hosted CI were
not run. No new kappa, solver speedup, or new Schur algorithm is claimed.

Prepared for Ali Romman (@aliromman) with substantial OpenAI Codex
assistance. Apache-2.0. No independent human review is claimed.

Coordinator review: the full proof and package diff were read. A separate fresh build with the pinned compiler passed and reproduced AXIOMS.txt byte for byte.

The 2026-10-14 follow-up review addresses the precise theorem and source-interface request. A fresh one-worker build with the pinned compiler passed and reproduced the complete axiom report byte for byte. All four source-file digests were checked; the later PR #985 change only reduces duplicate input-validation reads. The historical pins remain intact. The review separates the proved exact array model from the still-open Python, FIFO, fresh-power, rounding, restoration and tape obligations. No proof change or new kappa is claimed.

[Mister-Meeseeks]

100-PR batch review at c4be063c0afb634db7aa02bc1876ff29640bec65.

This remains open after scoped triage. Next review step: Review the precise all-length Lean statement and its runtime/finite binding; reproduce with the pinned toolchain before formalization incorporation.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.

[aliromman]

Addressed the statement/build request in c35966d2. The follow-up review maps the exact indices, ordered products and executed array passes to the source loops, and lists the obligations outside this formalization.

A fresh one-worker build with Lean 4.21.0 at compiler commit 6741444a63eec253a7eae7a83f1beb3de015023d passed. It reproduced AXIOMS.txt byte for byte and both axiom-guard probes rejected for their intended reasons:

PASS_CYCLIC_SCHUR: 102 theorems; standard axioms only
PASS_AUDIT_GUARDS: incomplete proof and package axiom rejected

All four pinned source digests match. PR #985's later single-read validation change leaves the elimination and closure code unchanged. No source pin or theorem was changed. The result proves the exact reduction and its array model; it does not prove Python/FIFO execution, fresh-power return times, rounding, dirty restoration or tape costs. This is an author-side review with substantial OpenAI Codex assistance, not independent human review or a new kappa claim.

[Mister-Meeseeks]

Batch review at c35966d2cbf0db075918c58b0ada4ee5a6dc7dcb.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Pinned Lean 4.21.0 rebuild proves the all-length cyclic chain/closure equivalence and indexed passes over a semiring; all 102 theorem axiom reports and invalid-proof controls pass. Source-loop correspondence is reviewed, not a formal Python/tape semantics or numerical precision proof.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.