[petalas]

Summary

For q regular p-cycle stages over every field, an arbitrary invertible noncentral boundary J needs exactly q floor((p-1)/2) ordinary rank to reverse the stage, even with any finite identity-stage dirty workspace and unrestricted mixing with it.

rank(J-I) <= r
    minimum complete reversal-defect rank
        = max(0, q*(p-gcd(p,2))-2r)

Partial permutation reflections attain every budget. Different input and output boundaries have sharp total rank floor q*(p-gcd(p,2)). Claim type: scoped proof with finite exact corroboration. No new kappa or selected-frontier change is claimed.

This answers one specified boundary question from Ali Romman's #6168 noncommuting-map direction. It changes the constant-in-stage premise to a regular cyclic coordinate representation. It does not recheck or depend on the predecessor query-tariff theorem. Adam Gall's #1941 already gives concrete cyclic-reflection permutations and swap counts for even cycles; the new step is optimality over all invertible boundaries, the residual-rank tradeoff and arbitrary-field workspace stability. Standard rank inequalities and permutation facts retain their classical status.

Only research/cyclic-reversal-rank-tradeoff/ is added. New proof and checker are produced by Nick Petalas. Credit Ali Romman, Adam Gall, Cinna Davis and the retained native/query/caller lineage. No predecessor code is copied or imported. Base: canonical main a5bff3bf133e482541c09eba4e773884bdb2b1b7, priority revision 8. Advance claim.

Evidence

Before: #6168 explicitly leaves noncommuting native maps outside its cyclic tariff. After: this particular regular-cycle boundary has an exact all-field rank allowance, with explicit attaining maps and a complete workspace contract.

At head eb26f36eebcd06c18974ea8e14b08e0a53d71524, CPython 3.12.15, macOS 27.0.1 arm64, standard library only:

• 5,544 partial-reflection constructions; 1,560 independent dense finite-field cases over F2,F3,F5,F7,F11; 5,912 two-sided permutation boundaries pass.
• Complete small binary censuses enumerate 132,096 matrices, including 40,656 invertible boundaries. They include an arbitrary dirty-workspace case and characteristic dividing the period. Actual minimum defect ranks agree with the theorem.
• Controls cover a wrong low-rank frame, a reversing boundary mixing dirty workspace, a nonfaithful stage and 256 dense/bit-matrix comparisons.
• Direct normal and optimized warning-strict checks pass in 0.49 s and 0.50 s locally, with identical generated source-hash receipts. These are check durations, not a performance claim. Both registered commands pass under their 120 s bounds.
• Committed bytes, executable modes and source-history ancestry pass for all six package files. Official package export and declared-input integrity pass. Optimized replay in a read-only Git-free export with an empty executable PATH produces the identical receipt; all seven exported files retain their bytes.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/cyclic-reversal-rank-tradeoff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/cyclic-reversal-rank-tradeoff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cyclic-reversal-rank-tradeoff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cyclic-reversal-rank-tradeoff --check-source-history --check-modes

Expected markers: PASS_CYCLIC_REVERSAL_RANK_TRADEOFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Universal coverage follows from PROOF.md, not finite extrapolation. The binding-only selected baseline passed; no selected scientific cold replay was run.

Limitations and unrun obligations

The charge is ordinary rank(J-I), and the defect is measured on the complete matrix, including all workspace rows and columns. The theorem assumes the stated direct sum of regular cyclic permutations and identity workspace. A merely nonfaithful order-dividing-p action is outside the formula. No promised input subspace or norm-error guarantee is asserted.

This rank is not identified with Clifford symplectic rank, recursive child rank or execution cost. Native realization and phases, productive helper targets, concave recursive prices, complete caller, compiler, precision, tape, storage and movement remain open. Predecessor scientific suites, native/outer replays, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed. All five selected framework assumptions remain.

Open next questions

• Bind one literal native noncentral map to this regular-cycle action, including phases and spectators, and derive its actual paid profile. A source-specific realization is still missing.
• Price a complete caller using a noncentral boundary, including both boundaries, restoration and all productive target changes. The rank floor supplies no passing signed invoice or physical word.

Related open work

• Price the p19 coded-reverse chain: conditional κ = 0.0033999964 #5440 studies the p19 coded-reverse P104 caller and its conditional invoice. This package instead proves ordinary boundary ranks for a regular cyclic coordinate stage and computes no common kappa or paid profile.
• Certify conditional kappa 0.0033999964 via the p19 word without the coded overlay (complex 0.007578535, BIT 0.0034, zero field bill) #5270 studies the uncoded p19 controller and physical binding. This package supplies no attachment or numerical comparison to that controller.
• Prove odd cyclotomic Clifford rank cap, sharp at degree four (no κ) #6192 studies whole-bank odd-cyclotomic Clifford responses and their native/symplectic rank caps. This package studies rank(J-I) for reversing a regular cyclic coordinate stage; no common numerical result is computed or compared.

The fresh live open-PR and claim checks found no other PR studying this regular-cycle boundary-rank quantity. Main README/formula searches and merged PR searches included general cycle counts and workspace forms. The overlap snapshot covered 310 open PRs and 5,898 Discussion records; sparse local-note coverage was 85 of 8,623 records. These checks are bounded overlap evidence, not a novelty certificate.

CI notes

Both hosted checks passed at exact head eb26f36eebcd06c18974ea8e14b08e0a53d71524 in run 40911293566: plan in 4m38s and replay (research/cyclic-reversal-rank-tradeoff) in 1m28s. The completed check list was read once and both entries report pass.

Hosted replay certifies the registered normal and optimized package checks at this head. The listed local mathematical, publication, source-history and export checks also pass. Independent human and formal proof-kernel review remain unclaimed.

The completion refresh read this account's open PRs created since the advance CLAIM at 2026-10-29T16:13:06Z. The adjacent #6192 is listed above; no other submission computes this regular-cycle reversal optimum. No shared numerical disagreement was found, and no extra computations were run to check another contribution.

Merge Danger

Door: two-way.

Blast Radius: scoped.

Six new files in one self-contained research package. No shared implementation, selected record, upstream archive or predecessor evidence changes.