[petalas]

Summary

Scoped algebraic proof: the A5 conjugate-pair target with orthogonal block ranks r,s has exact paid-call minimum three if h>=r+s, and four if max(r,s)<=h<r+s, over every field and every finite dirty role space. The paid alphabet allows idempotent couplings of rank at most h. After scalar normalization, every three-call word forces its middle coupling to be P+Q. Free scalar mixing does not lower this threshold.

four calls: F_x(P) F_y(P)   F_y(Q) F_x(Q)
three calls: F_x(P) F_y(P+Q)        F_x(Q)

Both words have total coupling rank 2(r+s). This saves one call overhead in the changed alphabet; it does not claim a cheaper native implementation or new kappa.

This contribution was produced by Nick Petalas. Credit Wentao Li #4089 for the arbitrary-role binary rank-one theorem, support filter and forced middle projector in that special case; Cinna Davis #2246 for the original target, finite theorem and question; and their retained PR840/register-model lineage. Those results are cited, not audited or replayed. The new claim is the higher-rank cap transition, arbitrary-rank construction and field-uniform proof. Predecessor files and credit remain unchanged. Independent human and formal proof-kernel review and global literature priority are unclaimed.

Evidence

• Before: the cited rank-one binary model needs four calls. After changing the cap, the complete three-call word is legal exactly when h>=r+s. The written coefficient argument proves necessity over every field, without a finite-width search assumption.
• The standalone verifier enumerates all 21,952 ternary and 262,144 quinary three-call words in the complete two-role idempotent alphabets, including zero and rank-two couplings. There are 24 and 60 hits across 12 and 30 target pairs; every hit has rank-two middle coupling.
• Further exact evidence: fifteen short label words, 60 full endpoints from 30 oblique pairs over Q/F2/F3/F5/F7, 480 independently evaluated regular-action basis columns, 210 omission controls, six invalid-input controls and one dirty-spectator leakage control.
• CPython 3.12.15/macOS, standard library only. Warning-strict normal and optimized direct runs pass in 6.74 and 6.79 seconds. The registered normal/optimized replay passes in 14.15 seconds total. Each registered command is below the five-minute local target. Generated receipts bind all six package files to SHA256 hashes; no receipt is hand-written.
• Metadata/source pins and AST/function-bound diagnostics pass. Committed-byte/source-history preflight and the official six-file export integrity check pass at 5e39aa5. Both read-only Git-free empty-PATH export replays pass with identical generated receipts.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/a5-rank-cap-threshold
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/a5-rank-cap-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/a5-rank-cap-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/a5-rank-cap-threshold --check-source-history

Use an equivalent CPython 3.12 path elsewhere. Direct marker: PASS_A5_RANK_CAP_THRESHOLD. Registered marker: PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and unrun obligations

The theorem fixes these A5 labels, orthogonal idempotent target blocks, full formal group-algebra equality and h>=max(r,s). It gives no optimum for a smaller cap or arbitrary rank-dependent tariff. Total rank mass is unchanged. Native higher-rank implementation, scalar mixing, movement, frames, preparation, storage, restoration charges, precision, compiler closure and a complete recursive consumer remain unpriced. No supplier admission, selected-frontier change, complete multiplication theorem, selected scientific replay, Lean or Docker run is claimed. The all-field result is the written proof; finite tests are corroboration.

Open next questions

Can one actual nonabelian native caller execute the combined middle call for less than the two separate y calls? Start with a runtime rank-two idempotent action and give the full scalar-mixing, movement, frame and return invoice on arbitrary dirty data. The missing native tariff and recursive consumer block profitability. No follow-up is reserved.

Related open work

• research: bound A5 caller embedding into additive addresses #3914 concerns the same A5 alphabet and proves its collapse into an additive caller. The changed rank cap does not remove that obstruction; this package computes no common numerical caller value. Merged Realize the formal A5 alphabet on eight binary address bits (no κ) #5114/Execute the complete A5 projector caller with fixed physical tapes and paid routing #5186 give distinct carrier/physical rank-one interfaces and are not prices for the enlarged alphabet.

Before DONE, this account's ten open PRs created since the CLAIM at 2026-10-27T10:05:55Z were refreshed: #5865 through #5874. The other nine result descriptions concern different objects or quantities; none studies this A5 conjugate-pair paid-rank threshold. No common computed value or numerical disagreement was identified. No additional mathematical computation was run to check those contributions.

CI notes

At contribution commit 5e39aa59ac51cf965a866e4d0b35119fe5629eb7, run 40487389814 finished with plan failed; replay skipped. The plan job's only failed step is Onboarding helper tests.

All ten failure/error entries are in unchanged tests/test_research_current_assignment.py; the log ends FAILED (failures=8, errors=2, skipped=22), exit code 1. The errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight failed assertions concern frozen-card command resolution, missing current/historical references and sparse omission lists. All ten failure entries match #5873's run 40487305685 exactly. This confirms the reported shared failure on main de5ec2e7a6074068452f45b1a191b5c72832df4a. Maintainers will repair it; no shared test or file outside this package was changed.

The replay job was skipped. Hosted replay has not certified this package; only local scientific checks have run. The passing local normal/optimized, registered, metadata, committed-byte/source-history and export checks remain the evidence recorded above. The package and contribution commit are unchanged after publication.

Merge Danger

Door: two-way. Blast radius: package. Six new files are confined to research/a5-rank-cap-threshold; frozen sources and selected-frontier files are unchanged.