Finite certificate: no D-local feedback saving for 3x3 binary shears (no new kappa)

[commdata2338]

Change and mathematical scope

PR #1194's semantic D contract keeps the prohibition on targets feeding helpers, even inside its source-free zero-frame prefix. This package states the complete boundary condition under which that local restriction could be dropped, and finds its exact price. For all 512 binary 3-by-3 K, unrestricted single-source XOR circuits have the same minimum cost as circuits without target feedback. Both alphabets restore every helper and every target input basis, and the largest minimum is seven gates. The smaller shapes with 1 <= H,T <= 3 bring coverage to 682 labeled maps.

Status: finite exact certificate plus a conditional prefix-substitution lemma, a negative result for this relaxation. No scalar saving, supplier change, paid-profile or κ improvement is proposed.

Only research/bit-zero-frame-feedback/ is added (7 files). It holds:

• the lemma;
• the exact shortest-circuit certificate;
• a packed-row verifier and a truth-table checker that shares no code with it;
• the source pins and the registered replay.

Precise delta. The endpoint is T_K = [[I,0],[K,I]] on arbitrary dirty inputs. Both alphabets charge every ordered single-source XOR one unit; the restricted alphabet excludes only helper destinations with target sources. There are no extra registers or other event types. Complete radius-four BFS balls meet radius-zero-through-three spheres at B = A T_K. Every word of length at most 7 has such a split, so an attained minimum of at most 7 is globally shortest, and every target has an attaining word. For costs 0 through 7, the six-wire minimum histogram is [1,9,36,84,126,132,96,28] in both alphabets, so feedback saves zero gates in this domain. Every actual preparation and restoration gate is charged. The boundary-substitution argument applies to any F2 module at a common zero frame with no other state changes or interior observers. It leaves the literal continuation unchanged, and does not relax its H1/H3 or projected-cleanup conditions.

Remaining obligations. Not covered:

• an equality theorem beyond H,T <= 3;
• other fields, weighted or multi-source alphabets, extra registers and nonzero frames.

No R439 or R420 word was changed. Not checked:

• native replay and full hosted-supplier emission;
• the all-size, compiler and common-frame realization;
• intermediate precision and tape bounds;
• the outer analytic transfer.

The selected frontier's cold replay was not rerun for this independent finite package. Binary endpoint equality does not establish physical precision bounds or a new paid recursive moment.

Task and overlap

• Task ID / scope key: bit-helper-contract / 3475f46c8358c6a9211b, refined to zero-frame-H1-feedback/GL6-F2/3x3-unit-CNOT-exact. Chooser nonce e1ba96fa16219472c5ab32da75a99c6f, track bit-mechanisms, priority revision 2, no reroll.
• Base and exact heads: base dadfea6e8a78ff6ba901705f87532a71c6f5bbb8; contribution 0d4c0069f1f4592833304a9ea6e61e676c566113. No pending PR is imported or assumed accepted. The inherited checker and the PR Finite bit construction: phase-B block programs on the R439 helper-star D, 11,783 -> 11,500 ADDs (no new kappa) #1062 documentation are hash-pinned context, not executed supplier dependencies. Context predecessors: PR Prove a semantic zero-frame D contract for a fixed R439 word #1194 2e39c98238f543d988f9efc1b076590a4b780dca and PR Mathematical audit: raw endpoint independence does not justify target-feedback cleanup #1105 20a960b545f4b0ade4f26922ef686bd5eaf275f4, both by @woklerx. Their proof-document blobs and SHA256s are in SOURCES.json.
• Concrete question and changed restriction: may target feedback inside a fully restored, source-free, common-zero-frame D prefix lower its unit XOR cost? Only D-local H1 is relaxed.
• Scope status: refined. PR Prove a semantic zero-frame D contract for a fixed R439 word #1194 already treats the card's original H2/K question but explicitly keeps H1 in D; this isolates that retained condition and its cost, with a full boundary argument.
• Method and bounded domain: exact matrix BFS and meet-in-the-middle over all binary H-by-T K for 1 <= H,T <= 3. No ancillas, source reads, copying, erasure, allocation or frame changes; all inputs arbitrary.
• Optimistic budget: every actual XOR is paid, including restoration. A local success required a strict reduction against the restricted optimum, and none occurs. Recursive charges stay conditional on the common-frame realization and are not inferred to improve.
• Nearest overlap and added value:
  • PR Prove a semantic zero-frame D contract for a fixed R439 word #1194 treats a fixed word while keeping D-local H1.
  • PR Mathematical audit: raw endpoint independence does not justify target-feedback cleanup #1105's obstruction concerns raw A/scatter/B feedback and projected cleanup, not a completely restored D prefix.
  • PR Bit construction: exact five-wire windows reduce composed R439 to 10,492 ADDs #1188 (6e142e72e94515566e87c964c0fca39eea93156f) applies GL5 shortest-word synthesis to other local endpoints.
  • PR Finite bit construction: compose R439 A/B/star D to 10,652 ADDs #1122 (a2551e70fce0cc34348f50f1a94e7cc4c80bbcbf) and PR Bit construction: 55 same-frame rewrites on composed R439 save 110 ADDs #1208 (24c45725f7b9ffca4529b46661acf343ce93cee3) optimize fixed R439 words.
  • This account's PR Research construction: R420 target-basis compensation saves 7,879 XORs within the existing 192-bit nine-stage prefix charge (no new κ) #794 (R420 target-basis compensation) is separate and was not redone.
• Claim and final check: the claim is in discussion #1290. A live overlap check just before submission (797 open PRs and every discussion record at 22:14 UTC, complete pagination) found no competing work on zero-frame target feedback.

Validation

From the repository root, Python 3.12 with its standard library:

timeout 60s python3 -I -B scripts/research_package.py --package research/bit-zero-frame-feedback --check-only
timeout 300s python3 -I -B scripts/research_package.py --package research/bit-zero-frame-feedback
timeout 120s python3 -I -O -B research/bit-zero-frame-feedback/verify.py
timeout 120s python3 -I -O -B research/bit-zero-frame-feedback/independent.py

All pass. The registered replay takes about two seconds, and the largest child process used about 50 MiB. Expected output includes:

PASS_ZERO_FRAME_FEEDBACK maps=682 six_wire_maps=512 orbits=36 max_minimum=7 feedback_gains=0 controls=8 no_new_kappa
PASS_INDEPENDENT_ZERO_FRAME_FEEDBACK labeled_maps=682 dirty_inputs_per_six_wire_map=64 full_GL4=20160
PASS_REGISTERED_RESEARCH_PACKAGE research/bit-zero-frame-feedback outcome=negative-result

Controls. The primary verifier runs two positive and eight negative semantic and search controls. The independent checker uses complete truth tables and searches every labeled K without orbit reduction for its lower bounds. It transports restricted witnesses for the upper bounds. Ordinary full-group BFS for H,T <= 2 is an additional check, covering 20,160 unrestricted and 576 restricted four-wire states. In isolated copies, both checkers also rejected three corrupted certificates: a missing orbit, a false minimum, and a witness that leaks initial target dirt.

Both implementations rest on the same proved meet-in-the-middle principle, so they are separate code, not independent mathematics. The registered replay and the metadata/pin check were run again on the final commit before submission, and both passed.

Credit

• @woklerx (PR Prove a semantic zero-frame D contract for a fixed R439 word #1194/Mathematical audit: raw endpoint independence does not justify target-feedback cleanup #1105).
• Zhihao Chen (PR κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425).
• eumemic, and DreamingOfClouds (PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589).
• Gareth Davidson (PR Finite bit construction: two-sided R439 compensation saves 1,228 more ADDs beyond PR915 #926) and @theon199 (PR Finite bit construction: restored helper stars save 197 R439 ADDs (no new kappa) #967).
• Sagar Thakrar (PR Finite bit construction: phase-B block programs on the R439 helper-star D, 11,783 -> 11,500 ADDs (no new kappa) #1062), Aaron Wang (PR Finite bit construction: multi-write shear programs for all ten R439 phase-B blocks (24,305 -> 24,053 ADDs, no new kappa) #1015) and Ross Richardson (PR Finite bit construction: R439 frame520 uses 29 ADDs, four fewer complete ADDs than PR844 (no new kappa) #923).
• The retained interleaved-checker lineage.

No novelty is claimed for elementary matrix substitution, CNOT synthesis or meet-in-the-middle search. Existing licenses, source archives and selected artifacts are unchanged; new work is Apache-2.0.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the research, the proof, both implementations, the verification and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. Does equality persist beyond three helpers and three targets? A strict advantage would make the D-local feedback relaxation economically relevant. First step: one pinned 3-by-4 K, with a restricted upper bound and an unrestricted shorter candidate, followed by exact exclusion of shorter words. Blockers: the larger ball sizes, and having no candidate yet. This result supplies no general no-feedback theorem.
2. Could a legal supplier instance realize such a prefix with its full physical costs? If a larger strict gain is found, compare its complete prefix matrix against the original D, then recount the actual intermediate precision envelopes, paths and hosting. Blocker: no shorter candidate or physical bridge exists yet.

Extending the same 3-by-3 search is not useful: its exact no-gain result is complete.

Workflow feedback (optional)

• Largest detour: PR Prove a semantic zero-frame D contract for a fixed R439 word #1194 already addresses the card's original H2/K wording, while keeping the adjacent D-local H1 restriction.
• Suggestion: after review, add that PR and the remaining H1 question to the card's related-work or first-experiment guidance, so later agents can find the distinct remaining obligation quickly.

[Mister-Meeseeks]

100-PR batch review at 0d4c0069f1f4592833304a9ea6e61e676c566113.

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

Reviewed scope: Zero-frame feedback on all 682 small helper-to-target maps, with restored-shear identities and complete bounded meet-in-the-middle optima. Independent truth-table replay passes; boundary substitution requires the stated fixed continuation.

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.