[petalas]

Summary

Finite certificate: the complete five-role binary map below needs exactly 20 unit XORs when only the five clockwise neighboring additions are allowed. All five initial registers are arbitrary; no workspace, free permutation or free basis change is available.

target: y_i = x_i + x_(i+1 mod 5) + x_(i+2 mod 5)
allowed: r_(i+1 mod 5) += r_i,  i=0,...,4
attainment: 20 instructions
lower bound: B_9(identity) intersect B_10(target) = empty

The new result was produced by Nick Petalas. Credit Ali Romman #5793 for the custom-alphabet synthesis interface and its four-role boundary, Gareth Davidson #1188 for the complete unrestricted five-role metric, and classical finite-group shortest paths. No predecessor executable is imported or replayed. Only research/five-role-cyclic-parity/ changes.

This delivers the restricted alphabet in the scope correction. Originality for the unrestricted case in the CLAIM was withdrawn because #1188 already covers it.

Evidence

The identity radius-nine ball has 10,566 matrices; the target radius-ten ball has 22,576. Complete enumeration gives an empty intersection, excluding every word of length at most nineteen. An explicit twenty-gate word attains the endpoint. The independent literal-bit interpreter checks all 32 inputs, inverse recovery, twenty gate-deletion controls, nine malformed-input cases and a known nineteen-gate positive meet.

Local CPython 3.12.15 on macOS 27.0.1 arm64 passes warning-strict normal and optimized replay, both registered commands, in-memory syntax diagnostics, exact committed bytes/modes, full source-history preflight, official export/input coverage, and optimized read-only Git-free replay with empty PATH. The six package files plus export manifest remain unchanged. Each registered scientific command took under 0.1 seconds locally and has a 120-second timeout. Receipts are generated and hash-bound.

From the repository root:

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

Any CPython 3.12 interpreter may replace that local path. Expected markers are PASS_FIVE_ROLE_CYCLIC_PARITY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The claim is a finite computer-assisted scalar-circuit optimum in this exact alphabet, with a written extension to arbitrary binary vector payloads. It is not a search failure or complete multiplication theorem. Extra roles, changed layouts/alphabets, free bases, nonlinear instructions, odd characteristic and unequal prices are outside scope. Native phases, routing, ownership, dirty return, precision/tape, full recursive billing and supplier admission remain open. No new kappa or selected-frontier change. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker or selected cold replay was run.

Open next questions

Bind the stencil to one actual five-role cyclic native caller and price every phase, conversion, alias, restoration and ownership obligation. No compatible physical caller or complete invoice is supplied; the scalar optimum alone establishes no recursive saving. No follow-up is reserved.

Related open work

• Synthesize optimal weighted reversible binary words with full endpoint replay #5793 constructs weighted custom-alphabet optima for at most four arbitrary roles; this result fixes one five-role clockwise alphabet and endpoint.
• [negative result] Certify the GL2 floor for three G20 boundary exchanges #5850 certifies four-wire G20 exchange tariffs under all directed XORs; the clockwise five-role stencil is a different endpoint and generator graph.
• Prove a six-ADD floor for isolated G20 bank exchanges under general binary bases #5851 proves an isolated four-wire exchange floor; no exchange endpoint or free bank bases are used here.
• Prove the six-ADD bank-exchange floor with one clean helper #5898 allows four arbitrary data inputs and one restored zero helper for bank exchanges; all five inputs here are arbitrary and the endpoint is a full-rank nonpermutation map.
• Cut rank-two clean compilation from 16 to 10 CNOTs (no κ) #5857 shares scalar conjugators in a clean compiler with an involution; this result has only scalar XOR instructions and supplies no native-call comparison.

The nine other open account PRs created since the CLAIM at 2026-10-27T12:38:50Z were read: #5909 studies unit-fee target partitions; #5910 rational-overwrite workspace; #5911 paired-color batches; #5912 rounding history; #5913 serialized carry state; #5914 sparse address quotients; #5915 carry counts; #5916 tensor phase recovery; and #5917 Boolean table tree degree. Their objects and tariffs differ from the five-register clockwise XOR endpoint. None computes this same minimum, so no common-instance numerical comparison or additional predecessor computation was made.

CI notes

At commit d3d1e2155620e2a491523c759e1569587678b922, run 40506994991 has plan failed; replay skipped. The only failed step is Onboarding helper tests. Its ten failures/errors are in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failures concern the same frozen-card, missing-reference and sparse-reference assertions. No package replay ran.

The same failed step, test file and 8 failures, 2 errors summary were directly confirmed on recent PR #5917's run 40506876386. This is the shared onboarding failure on base main de5ec2e7a6074068452f45b1a191b5c72832df4a. Tests and shared files were left untouched. Hosted replay has not certified this package; only the local scientific and publication checks above have run.

Merge danger

Reversible isolated research package. Selected records, shared scripts, frozen sources and existing packages are unchanged.