[petalas]

Summary

Construct an explicit eight-bit binary affine carrier for the formal A5 alphabet, with an injective extension to F2[A5]. This is a finite exact construction with a scoped algebraic proof, produced by Nick Petalas. No new kappa is claimed.

one natural F4^2 copy: 16 addresses, faithful group action, nonzero formal kernel
2 natural copies:     256 addresses, regular orbit of size 60, faithful F2[A5] action

The generators corresponding to x=(12)(34) and y=(135) have literal address CNOT words of lengths 4 and 12. Complete in-place data permutation programs use 360 and 510 bit XORs per 256-entry bit array, preserving arbitrary dirty data and using no extra data bit. The construction answers the address-carrier part of Wentao Li's #4089 handoff and Cinna Davis's #3914 nonabelian escape route. Their endpoint theorem and caller obstruction are cited premises; predecessor searches and verifiers were not replayed.

Scientific base: 81e9bd6. Contribution head: d165f73. The package retains the exact original card, scope refinement, source pins, generated hash-bound result and overlap coverage.

Evidence

• Before: the natural four-bit action kills the nonzero formal norm element on all 16 addresses.
• After: the identity matrix has 60 distinct orbit images, giving 60 independent coefficient columns and injectivity of the whole formal group algebra.
• Exact local replay checks the 60-element isomorphism and all 3,600 multiplication pairs, both full orbit inventories, every address-circuit input, every symbolic dirty-data input of both swap programs, and eleven semantic controls. Eight controls omit individual address CNOTs. Corrupted generated receipts reject under normal and optimized execution.
• CPython 3.12.15/macOS, standard library only. Warning-strict normal and optimized registered checks pass, each timed at about 0.02 seconds locally. Metadata/source bindings, committed-byte/source-history preflight, official export integrity and both read-only Git-free export replays pass. Selected metadata orientation also passes; no selected scientific replay was run.

From the repository root:

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

Direct success marker: PASS_A5_EIGHT_BIT_CARRIER. Registered success marker: PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations and unrun obligations

Two natural copies are necessary and sufficient only within the specified replication family. There is no minimum over all affine representations or optimal CNOT claim. Address-coordinate CNOT counts and data-array XOR counts are separate tariffs. Fixed table generation/storage/reads, actual tape movement, arbitrary scalar role mixing and rank-one-projector adaptation, source/target frames, ownership, routing, precision, compiler closure and recursive/outer admission are not supplied or priced as free. No native supplier, full multiplication theorem, selected-frontier change, human acceptance or formal proof-kernel verification follows.

Credit Wentao Li #4089, Cinna Davis #2246/#3914, eumemic, Ali Romman and the retained register-model lineage. The classical A5=SL2(F4) isomorphism, regular-orbit criterion, finite-field arithmetic and XOR-swap construction retain their mathematical credit. Independent human and formal proof-kernel review are unclaimed.

Related open work

• research: bound A5 caller embedding into additive addresses #3914 proves collapse into an additive address caller; this construction instead gives a concrete nonabelian binary affine carrier. It does not claim to repair or admit that original caller.

Open next questions

• Bind one actual BIT caller and rank-one-projector call to this carrier, including all scalar role changes, address movement, frames and restoration. A source-bound native tariff is missing.
• Decide whether another algebra-faithful binary affine A5 representation uses fewer than eight address bits. Begin with six- and seven-bit module candidates and their full group-algebra kernels; no such classification is supplied.

CI notes

CI pending at d165f73. Local registered normal/optimized checks and publication/export checks pass. No hosted CI result is claimed yet.

Merge Danger

Door: two-way. Blast radius: package. Seven new files are confined to research/a5-eight-bit-carrier; shared code, frozen sources and selected-frontier files are unchanged.