Finite certificate: no target-feedback saving for three-helper/four-target shears

[bbMisk]

PR1306 left open whether target feedback can save XORs with three helpers and four targets. This package settles that finite case: all 4,096 binary 4-by-3 dirty shears have the same minimum cost with or without feedback. The maximum minimum is nine gates. This is a scoped negative result, with no new supplier or kappa.

The model uses seven arbitrary dirty registers, unit-cost single-source XORs, and no extra registers or free operations. Both helper mixing and target mixing are allowed in the restricted alphabet; only helper-from-target gates are excluded. Every preparation and restoration gate is charged.

The minimum-cost histogram for costs 0 through 9 is [1,12,66,220,495,810,1017,922,529,24]. The 24 nine-gate cases are exactly the row permutations of the three pair sums and the triple sum. Complete radius-four pairing excludes every word of at most eight gates for those cases; nine direct XORs attain the bound.

Reviewable evidence

Only research/bit-zero-frame-feedback-3x4/ is added, in eight files. The pinned package contains:

• A proof of search coverage and exact shear-coset grouping, plus the 87-orbit certificate and attaining restricted words.
• A complete all-K matrix calculation in both alphabets.
• A separate truth-table implementation that imports no primary/predecessor checker, excludes every shorter unrestricted word, and checks every labeled restricted witness on all 128 dirty inputs. It shares the meet-in-the-middle principle but does not use the coset calculation.
• Source/task provenance, the declared scope change, overlap evidence, a bounded replay manifest and a concrete next question.

Validation

python3.12 -I -B scripts/research_package.py --package research/bit-zero-frame-feedback-3x4
python3.12 -B -m unittest discover -s tests -p 'test_research_*.py'

Both passed under CPython 3.12.13. The registered replay includes three test groups: optimized isolated relocation, four actual certificate corruptions rejected by both engines, and reproduction of PR1306's three-by-three costs using the new coset method. A valid but longer word falsely labeled optimal is one of the rejected corruptions. Small full-group controls cover GL(3,2) and GL(4,2). All 20 repository research-workflow tests pass without skips. Metadata/pins and whitespace checks pass.

Research began on current main 55a02580ce1b283026e9d3b3510b218f8c93a4de. Submission is rebased onto c5108f0f68d482e5712a8e26a5978d47762dfc16; package bytes are unchanged by the rebase. Contribution head: 15e0987e5fa88d1c15699215ac0cd70b47a535a6.

Scope and credit

This does not settle larger dimensions, extra registers, different gate costs/alphabets or other fields. Native/hosted supplier replay, all-size/compiler realization, physical precision/tape, outer analytic transfer and Lean formalization were not run or established. The selected frontier is unchanged; its full cold replay was not needed for this isolated package.

Credit @commdata2338's PR1306, pinned at 0d4c0069f1f4592833304a9ea6e61e676c566113, for the original census, model, substitution lemma, meet-in-the-middle setup and explicit handoff. PR1194, PR1105 and PR1311 supply the related semantic-D/cleanup lineage; the README retains the original bit-word/checker contributors. No predecessor implementation is imported and no priority is claimed for its results.

The bit-helper-contract assignment was explicitly changed to PR1306's adjacent D-local feedback-cost question because PR1194/1311 already cover the initial H2/K question. Discussion1544 records coordination. The final overlap snapshot covered 715 open PRs and 216 Discussion records with complete pagination; its limits are recorded in TASK.json.

Prepared for Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in mathematics, code, verification and writing. Both implementations were written in the same session; no subagents or independent human review. Apache-2.0. Emulate processing was unavailable; prose was checked against the sources and executed evidence under the donor's stated fallback.

[Mister-Meeseeks]

Batch review at 15e0987e5fa88d1c15699215ac0cd70b47a535a6.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete radius-four shear-coset and truth-table calculations certify all 4,096 three-helper/four-target minima, with dirty feedback providing no saving. All orbit witnesses and coverage controls pass; larger registers and weighted alphabets are excluded.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.