Finite certificate: no target-feedback saving for four-helper/four-target shears (no new kappa)

[rtyu193-rgb]

Finite certificate: no target-feedback saving for four-helper/four-target shears (no new kappa)

PR1557 settled 3-by-4 and named four helpers and four targets as the next question. This package settles it: for all 65,536 binary 4-by-4 K, the fully restored dirty shear (h,y) -> (h,y+Kh) has the same minimum unit-cost single-source XOR count with and without target feedback. Same model and alphabets as the merged PR1306 package: eight arbitrary dirty registers, no extra registers, every gate charged. This is a scoped negative result, with no supplier or kappa claim.

Minimum XORs	0	1	2	3	4	5	6	7	8	9	10	11	12
K	1	16	120	560	1,820	4,416	8,452	13,040	16,002	14,372	6,329	384	24

• Maximum 12: attained exactly by the 24 matrices J − P (all ones minus a permutation matrix). Twelve direct reads are optimal there; no word of ≤ 11 gates exists in either alphabet.
• Above ten gates: 408 K need more than ten, in three orbits. The weight-11 and weight-13 orbits both cost exactly 11. The weight-13 orbit is two gates below its direct count, a saving already available without feedback.

Method (new; the coset grouping is what makes 8 registers cheap)

The shears form an elementary abelian group N. U and T_K U share a canonical key: their helper rows, plus their target rows reduced modulo the span of those helper rows. Grouping the radius-5 BFS ball by coset labels each element as T_label · anchor. A same-coset pair then gives T_{a xor b} at cost d(U)+d(V).

Only pairs whose depths differ by at most one are needed. The halves of a geodesic are themselves geodesic, so a minimal word of length L ≤ 10 always yields such a pair. That settles all K to ten gates at once:

• restricted alphabet: 3.06M ball elements;
• unrestricted alphabet: 23.1M ball elements;
• about 1 minute in total.

Beyond ten gates:

• Restricted: exact sphere 6 shows that J−I has no 11-gate word.
• Unrestricted: J−I is fixed by the diagonal S4, so an 11-gate word's first gate can be normalized to one of 6 gate-orbit representatives. No word exists.

The README states the full argument.

Evidence (research/bit-zero-frame-feedback-4x4/, 7 files)

Check	Result
verify.py --stage census	all 65,536 K, both alphabets elementwise equal; permutation invariance; attaining restricted witness for all 317 orbits replayed on all 256 dirty inputs; full certificate comparison
Positive controls	reproduces the PR1306 3×3 histogram (pinned input) and PR1557's 3×4 [1,12,66,220,495,810,1017,922,529,24] in both alphabets; normalized search finds known 11-gate words
Negative controls	radius-4 insufficiency, feedback in the restricted alphabet, truncated witness, wrong map, self-XOR, Boolean index, valid-but-longer word labeled optimal, tampered minimum
verify.py --stage hard	restricted J−I: 0 words ≤ 11, 15,852 at ≤ 12; unrestricted J−I: 0 words ≤ 11 for every first-gate orbit
independent.py (no shared code)	column-packed matrices; per-orbit direct-lookup lower bounds (no coset grouping); last-gate normalization for J−I; 65,536 witnesses transported by renaming and replayed on 256 inputs
Unregistered cross-check	an unreduced scan of all 56 first gates (about 8 min) also finds no 11-gate J−I word

Run times on the donor host: census about 2.5 min, hard about 2.5 min, independent about 1.5 min.

Exploratory (not certified here)

verify.py --shape H T shows elementwise-equal costs, all resolved within ten gates, for every other shape with ≤ 8 registers left open by PR1306/PR1557: 1×4, 1×5, 2×4, 1×6, 2×5, 1×7, 2×6 and 3×5. Transposes are equivalent by the M → M^-T duality. Together these suggest no feedback gain for any shape up to 8 registers. Certifying them, or proving a general no-feedback theorem, is the package's open next question.

Not checked / limitations

• Replay environment: run under CPython 3.14 / numpy 2.4.0 on Windows. CPython 3.12 was unavailable, so the registered 3.12 replay was not run locally; --check-only passes. The manifest pins numpy==2.4.0, which is needed because a pure-stdlib replay over 23M-element balls would not fit the time limits.
• Memory: peak memory was not measured; the largest arrays are about 63M uint64 candidates at BFS level 5.
• Out of scope: larger dimensions, other alphabets or costs, physical precision, hosting, compiler, native replay and the selected frontier.

Overlap and credit

• Overlap: a fresh find_overlap.py --refresh (394 open PRs, 350 discussion records, complete pagination) found no other 4×4 work. CLAIM posted in DONE: exact three-helper / four-target dirty shear costs #1544 before computing.
• Credit:
  • @commdata2338 (PR1306): model, substitution lemma, meet-in-the-middle setup;
  • @bbMisk (PR1557, 15e0987e): 3×4 census and this handoff;
  • @woklerx (PR1194/PR1105) and Arham Tahir (PR1311);
  • the bit-word and checker lineage listed in the README.
• No predecessor code is imported.

Substantial AI assistance: prepared with Claude Code (Claude Opus 5.5) in one donated session; no subagents or independent human review. Apache-2.0.

Feedback: Git for Windows' HTTPS helper fails with '$GIT_DIR' too big from a worktree in a deep path. Pushing from the main clone works; a note in CONTRIBUTING could save Windows contributors a detour.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at dc6a2599353b86e62b92d520df99ce452a3f8e8c.

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

Reviewed scope: Complete 65536-map, 317-orbit four-helper/four-target census and independent forward check prove equal minimum XOR counts with and without target feedback. The J-minus-permutation cases need twelve; other sizes, extra workspace and physical suppliers are outside scope.

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.