[krzysztofwojt]

Nick Petalas's #5114 constructs an algebra-faithful A5 action on eight binary address bits and leaves the minimum over all affine representations open. This proves eight is the minimum, including nonsplit representations and affine actions without a fixed point. It answers that PR's Open next question 2; its eight-bit upper construction retains full credit.

The new reduction covers every relevant module, rather than sampling smaller candidates. An exact quotient of F2[A5] identifies all simple modules (dimensions 1, 4, 4), with kernel powers of dimensions 35,27,17,9,0. Every module with one nontrivial composition factor is a subquotient of one of two explicit universal eight-dimensional extensions, plus fixed summands. Their actions on all 256 vectors have algebra ranks 52 and 34, with explicit nonzero annihilators. An affine action on at most seven bits homogenizes to a module of dimension at most eight with a trivial quotient, so it falls under this obstruction. The written proof includes the nonsplit, subquotient and affine-slice steps.

Only seven files in research/a5-affine-address-minimum/ are added. The complete proof is in its README; the finite quotient has a separately implemented checker. Both implementations explicitly reconcile their natural-module coordinate bases.

Validation at 6779a28938a9af1ba3d32e18ba8feb8d7cc002f8:

• CPython 3.12, standard library: registered warning-strict normal and optimized replays pass. Every group product, universal-action kernel witness and the matching regular-orbit fixture is checked exactly. There are 14 rejecting controls, plus 256 P/Q cases and 44,032 subquotient domain points per module.
• Metadata and exact source pins pass. Committed bytes and source ancestry pass for all nine package/input files.
• The official export passes its integrity check. Both direct modes pass with Git absent from PATH and all exported files read-only; all ten exported files, including the export receipt, remain byte-identical.
• Export SHA256: 824e7b6363787a4ef6bd3e92cbfda576436f850b8a0fef246931bf57f6554f4e.

python3.12 -B scripts/research_package.py --package research/a5-affine-address-minimum
python3.12 -I -W error -B research/a5-affine-address-minimum/verify.py
python3.12 -I -O -W error -B research/a5-affine-address-minimum/verify.py
python3.12 -B scripts/check_research_commit.py --package research/a5-affine-address-minimum --check-source-history

Fresh canonical base and pre-submission refresh: 20a1e31bc6009b9b3f4d084bbaadc6350e34fcba, priority revision 7. Canonical claim #5131. The pre-submission overlap refresh covers 318 open PRs, 4,290 Discussion records, 4,406 closed PRs and the complete main-tree note inventory, supplemented by targeted live PR/Discussion checks. No competing minimum claim surfaced; unpublished work and arbitrary diffs are outside that coverage.

Limitations: this is an author-checked scoped proof with exact finite ingredients, not a proof-kernel formalization or independent human acceptance. Arbitrary nonlinear address actions, optimal CNOT counts, a source-bound physical BIT caller, movement/restoration tariffs, recursive admission and a new kappa remain unclaimed. The selected frontier and frozen predecessor sources are unchanged. The remaining useful task is #5114's actual caller and complete-bill question, explicitly left unreserved.

Prepared by Krzysztof Wojt with substantial OpenAI Codex assistance, including a Luna finite-algebra implementation and proof-gap review. Credit Nick Petalas #5114; Wentao Li #4089; Cinna Davis #2246/#3914; the classical A5/SL2(F4) and module-theoretic methods; and the retained register-model lineage, including eumemic and Ali Romman. Apache-2.0.

Hosted CI is pending; the evidence above is local.