Research: determinant-balanced cyclic leaves with exact dirty restoration (no new kappa)

[VictorLiwentao]

Change and mathematical scope

PR628's additive cyclic-inverse leaf requires dirty commutators to form products. This adds an explicit in-place leaf for a real rational cyclic recurrence with position-dependent coefficients of absolute value at most 1/2. One borrowed scalar balances the closure determinant; four rounded shears make the pivot pair exactly reversible on the integer grid. An ordered factor product then uses two evaluations and two exact inverses, restoring one arbitrary dirty array, the dirty scalar and the source exactly.

Status: research construction / recursive interface; no new kappa. The proof gives the all-length identity, exact-coefficient error bound, precision/storage accounting and complete work recurrence. Exact coefficient entries can grow with cycle length. Complex roots, truncated preparation, tape layout, common-label compilation and a supplier invoice remain outside this contract.

The seed-55, 55-factor / length-5 fixture uses 220 stabilized factor executions and 2,660 coordinate updates, with one dirty array plus one dirty scalar. The pinned original dispatcher uses 3,232 additive leaf calls and six arrays. The leaf contracts and prices differ; this is not a runtime comparison.

Only research/balanced-cyclic-lift/ is added. Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Consumed PR628 head: 0768f97d06c623b8211e651543c66135796b62bf. Original source/proof/notices are retained unchanged with Git and SHA-256 pins. The selected PR626/PR543 frontier and upstream/ are unchanged.

OS-random seed 55 assigns recursive interfaces. No pinned coordination issue exists; the narrow claim is local. The final refresh covered 621 open PRs through #802. PR714/730/744/758/784/788 retain different echo-tree or clean-storage contracts; PR764 studies a different scan word, and PR796 audits nonarchimedean root valuations. Their context and exact inspected heads are credited, without importing their code. The immediate pre-creation check also read Rafael Guevara's #803 degree-controlled inverse bound; it is a separate analytic result and is not consumed.

Credit Chafik Boukhalfa / chafreaky and PR628's retained predecessor notices. Prepared for Wentao Li (VictorLiwentao, @Is__Victor) with substantial OpenAI Codex assistance, under Apache-2.0. No literature-priority or independent human-review claim.

Validation

python3 -I -O -B research/balanced-cyclic-lift/verify.py

Expected: PASS_BALANCED_CYCLIC_LIFT, with 1,560 factors, 7,420 full columns, 4,680 rounded factor cases, 315 products and 25 controls.

• Separate dense rational Gauss-Jordan references; exhaustive small factor coverage in both orientations; noncommuting signed products through 55 factors; 220-bit dirt; exact restoration and negative-call cancellation.
• Mathematical/admission controls include every single-shear omission in a nonzero fixture, factor order, unpaired-pivot collision, missing restoration and aliases. The actual pinned predecessor dispatcher is counted through 128 factors with a leaf stub; its numerical kernel suite is not replayed.
• Ordinary and isolated optimized replays pass on macOS arm64 / Python 3.13.2; the isolated optimized replay also passes on Python 3.9.6. An offline copy in a path with spaces passes and rejects source drift, certificate integer-to-Boolean corruption and an instruction-sign mutation.
• Selected-record integrity and staged whitespace checks pass.
• One root make verify on an untouched exported base passes seven historical groups, then stops at gen4-bank-verify with missing SymPy (exit 2). No full-suite, supplier cold-replay, Linux/remote CI or proof-assistant pass is claimed.

Finite replay supports the written scoped proof; it does not formally verify the multiplication theorem.

[Mister-Meeseeks]

100-PR batch review at ef34bd5b32fcf2d5640dae90b5b8f28488738d5a.

This remains open after scoped triage. Next review step: New reversible cyclic leaf and error/composition contract require proof review.

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.