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Finite bit construction: one extra XOR reaches the J3 positive-envelope floor (no new kappa) - #1520
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Batch review at Merged to Reviewed scope: Complete shortest-word census gives the J3 gate/envelope Pareto frontier (7,5,5),(8,4,4), with componentwise domination for the specified replacement witnesses. All dirty columns and independent positive lifts pass; no actual hosted occurrence or full supplier guard improvement is supplied. 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. |
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CLAIM / positive-envelope follow-up to PR1520: investigate a uniform lower bound for three restored helpers adding their parity to n>=2 dirty targets, allowing arbitrary additional dirty auxiliaries and all ordered unit XORs. Under the all-ones positive lift and forward peak<=4, proposed exact cost is2n+2. This extends the finite six-register precision result through a structural argument about monotone positive rows; it does not repeat the 3x4/4x4 gate census or claim a hosted supplier. Main0fdd7d82, priority4, bit-helper-contract nonce54ac0d737844a7825d9d6cbcee723dd8; prescribed R439 star verifier passed. Scope changes from H2/K repair (already covered) to the precision-sensitive boundary. Checked local notes plus797openPRs15:52UTC/1239Discussionrecords15:22UTC; GraphQL quota prevents a newer complete Discussion snapshot, so freshness is bounded. Expected proof, attaining words and controls within1hour; currently local-only. |
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DONE: PR2623 supplies the uniform peak-four cost theorem,2n+2 for all n>=2 with any finite number of dirty auxiliaries. The proof uses monotone positive rows, not a larger census. Registered22case replay,15680dirty-input checks, fault controls, Git-free export and committed-byte checks pass; independent agent proof review clear. Credit PR1520 construction/finite frontier retained. No native hosted replacement or new supplier/kappa is claimed. #2623 |
3 participants
Change and mathematical scope
Finite bit-circuit construction and exact local cost/precision frontier; no new κ.
For the restored six-register block
(H,T) -> (H,T+J3 H), one extra XOR lowers the best seven-gate positive peak from 5 to the support floor 4, in both directions. The complete Pareto frontier(XORs,F,B)is (7,5,5), (8,4,4). All preparation/restoration is paid; there are no extra registers, frame changes, COPY/ERASE events or recursive children inside the common-zero-frame block.The census retains all 216 shortest circuits: 108 have
(F,B)=(5,5), 72 have(6,6), and 36 have(8,8). A separate algebraic support argument bounds both peaks below by four at every circuit length. The explicit eight-gate witness attains that bound. Its forward and reverse nonnegative transfer matrices are also componentwise no larger than the stated seven-gate witness, covering unequal incoming envelopes.This increases the local ADD bill by one. It does not establish a new recursive profile, full-word guard improvement, supplier admission or multiplication exponent. No selected source or
upstream/file changes.Task and overlap
ef5f39403610d231a0c27e8a6b4f08c10209bcd6.69e353410ae1e6f0555b5add9720f3e691a762b9.bit-helper-contract: Prove a semantic zero-frame D contract for a fixed R439 word #1194/Research: independently check semantic-D bridge for composed R439 word #1311 already cover the immediate H2/K substitution. This isolates precision cost at a complete local boundary.0d4c0069f1f4592833304a9ea6e61e676c566113, already proves the seven-XOR minimum for J3. The new evidence concerns every shortest word's positive lift and the exact gate/precision tradeoff; the minimum itself is corroboration.becd4a2b6fad711914fe80f4344aef0008b30a28, studies a fixed quad17 integer transfer. This package permits varying integer lifts of a different binary endpoint.OVERLAP.json; no worldwide novelty claim.The selected context remains #1094, κ = 0.001672660197076918, with #589 on #555. The complex primitive binds. Exact selected/assembly/review/source pins are retained as provenance.
Validation
Passed on CPython 3.12.2, including metadata/source pins and:
4 | <=3split; all 810,000 four-gate prefixes and 27,931 backward words retained.Expected final line:
PASS_REGISTERED_RESEARCH_PACKAGE research/bit-j3-precision-seed11 outcome=completed.The result is a finite exact certificate plus a written support/domination proof. No Lean, native multistage, full supplier, physical/common-frame, hosting, precision/tape or outer analytic replay was run. Hosted CI is separate from these local checks.
Credit
Hanson Sun (@bbMisk), with substantial OpenAI Codex assistance in research, proof, implementations, verification and writing; no independent human review. Credit @commdata2338 (#1306), @woklerx (#1194/#1311), Cinna Davis (#1341), eumemic, DreamingOfClouds, Gareth Davidson, @theon199, Sagar Thakrar, Aaron Wang, Ross Richardson and the retained supplier/review lineage. New work is Apache-2.0; predecessor sources remain unchanged.
Open next questions
The README identifies the next useful check: find one exact J3 occurrence in a pinned R439 word, propagate its actual incoming bounds through both matrices, then price the extra XOR and replay the complete guards. The blocker is that no physical hosted occurrence or full invoice has been established here.