Independent audit: semantic binding of three Lean HPE Clifford pairs (no new kappa)

[commdata2338]

Change and mathematical scope

The twelve primitive circuit literals in the retained HyperplaneEnd/Exact.lean match the retained Python constructors and independently derived exact kernels on all 512 basis inputs, in both directions. The three noncold rank-seven HPE factorizations also agree exactly. This is a reproducible finite audit binding already proved identities to their source semantics.

Status: independent finite corroboration of PR #593's identities. No discrepancy was found, and no new κ or Lean theorem is claimed.

Only research/hpe-lean-semantic-binding/ is added. It holds:

• the README and derivation, the task card and the source pins;
• a bounded replay manifest, the verifier and a deterministic receipt;
• seven focused tests.

The selected construction and predecessor bytes are unchanged.

Evidence.

• The twelve source constructor outputs equal the Lean literal gate lists.
• 24 forward and inverse operators each pass all 512 columns: 12,288 columns and 6,291,456 entries, implicit zeros included. Both the literal evaluator and the retained source executor are checked against the independent kernels.
• Six composite circuits pass all columns. The three residual factorizations pass direct exact equality as well as matching fingerprints; each residual has 65,536 nonzero entries with denominator 16.
• 27 semantic controls reject, each with its exact differing coefficient retained. They cover global phase, translation, permutation direction, input character, C-sign, spectator axis, swapped wrappers, reversed order and a missing entrance inverse.

Remaining obligations.

• This is finite corroboration, not a fresh Lean build, an axiom audit, a proof of arbitrary-frame port equivalence or a full multiplication theorem. The three fixed pairs are the boundary.
• The historical 38-helper emitted words were not regenerated; their existing binding receipt is pinned as context only.
• Still inherited: scalar restoration, actual wrapper placement and tail chronology, hosting and footprints, all-size compilation, precision/tape, complete costs and the analytic transfer.
• Not claimed: a full selected supplier cold replay, human review or hosted CI success.

Task and overlap

• Task ID / scope key: formal-interface-lemma / e3ffea67887eeb4c396c, refined to HPE-three-pairs/Lean-literals-to-independent-kernels/all-512-columns. Chooser nonce 96f499b6a9e557186c7e873f3edd9551, track independent-verification, priority revision 2. The other suggestion, independent-moment, was passed over because existing endpoint engines, PR Independent verification: prove and certify the a < D/E diagnostic for the selected complex and bit ledgers (no new kappa) #1293's diagnostic and PR Independent verification: the selected frontier1094 complex rank ledger, rebuilt from its emitted program (no new kappa) #1262's complex ledger already cover its immediate scopes.

• Base and exact heads:

  • base bce54fb3f895cbe5cd4c0eb477c3afeed100190f; contribution 0d8d2c3a96346756f02563b401a042ab97198e20;
  • retained PR [Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593 fdb6dcc6d7bc29764c5b6b4d32b7fa050acf74a4 and retained PR [Lean] Hyperplane-end helpers: multi-gate cleanup tails and the cold specialisation (stacked on #593, no κ claim) #599 bc13c8a3e000442606bd2dbe8583a17649ba6c32;
  • selected PR Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 0fc329d0bec93ca7cd288eaea1295172e8789810, context only.

  There is no pending implementation dependency, and the exact consumed SHA256 values and Git blobs are in SOURCES.json.

• Question: do pair1..3, the six frame literals and the six wrapper literals faithfully port the source canonical_frame and monomial_op semantics, including phases, inverse directions and spectators?

• Exact claim audited: that the external Python semantics agree with the formal package's literal operator definitions. A theorem about the Lean definitions alone cannot establish this. The formal factorization itself stays credited to its predecessor.

• Scope status: refined. The card's theorem-to-certificate audit method is kept and bounded to these fixed instances.

• Method and domain: independently enumerate the F2^9 coordinate charts and derive Gaussian-dyadic matrix kernels. Compare them with a separate literal evaluator and with the source constructors and executor. The domain is the three fixed pairs (E of dimension 1, U of dimension 8), all 512 input/output coordinates, forward, inverse and residual products. No floating-point acceptance, and the producer's verifier is not used as the oracle.

• Nearest work:

  • PR Research audit: verify all selected cold-wrapper amplitudes (no new kappa) #702 (27db45ac6befa99d6c06be920a93845023e81937) audits fourteen historical rank-four cold wrappers, and PR Independent verification: quadratic-phase certificate for a selected cold wrapper #712 (4773b879a1087540f5919c726ec8c670dadfa0c5) gives a quadratic certificate for one of those.
  • PR Research audit: exact phase criterion for selected PR1094 residual charts #1255 (add1e2673ef000c097ec0fd70058e644af048d6b) checks physical chart phases.

  This audit covers the three noncold rank-seven Lean pairs and their semantic port. The retained recipe-binding checker compares records with the pair literals; it is credited, not replaced.

• Claim and final check: the claim is in discussion #1394. A live overlap check just before submission (817 open PRs and 141 discussion records at 00:28 UTC, complete pagination) found no competing work.

Validation

From the repository root on Python 3.12:

python3 -I -B research/hpe-lean-semantic-binding/verify.py
python3 -B scripts/research_package.py --package research/hpe-lean-semantic-binding --check-only
python3 -B scripts/research_package.py --package research/hpe-lean-semantic-binding
make selected-record-check

The registered command runs the replay and the complete focused test file in about a minute. The seven tests cover:

• the C-square and inverse identities, and the 256-column input-character sign change;
• rejection of malformed literals, sources and receipts, with JSON type and duplicate-key checks;
• overwrite refusal and a read-only isolated optimized relocation.

Normal replay is read-only. The registered replay and the metadata/pin check were run again on the final commit before submission, and both passed.

PASS_HPE_LEAN_SEMANTIC_BINDING pairs=3 operators=24 operator_columns=12288 factorizations=3 controls=27 no_new_lean_build no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/hpe-lean-semantic-binding outcome=completed

Credit

• Jeremy Kothe/jeremykothe2 (PR [Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) #593), for the exact Clifford formalization.
• DreamingOfClouds (PR [Lean] Hyperplane-end helpers: multi-gate cleanup tails and the cold specialisation (stacked on #593, no κ claim) #599 and Lemmas A/G), for the wrapper recipes.
• icekylinx, for the retained Clifford engine.
• Chafik Boukhalfa/chafreaky, Zhihao Chen/jacklightChen and the retained realization contributors.
• The nearby independent audits by BigCactusLabs (Research audit: verify all selected cold-wrapper amplitudes (no new kappa) #702), Joshua Snider/jsnider3 (Independent verification: quadratic-phase certificate for a selected cold wrapper #712) and Gareth Davidson/bitplane (Research audit: exact phase criterion for selected PR1094 residual charts #1255) keep their credit.

New work is Apache-2.0; original notices are unchanged.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the research, the derivation, the implementation, 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:

• Can the source-to-formal gate semantics be proved for a shared serialized circuit grammar? That would shrink the manual semantic-translation trust boundary. First step: a small formal interpreter for the five gate tags, plus an explicit correspondence theorem for the permutation and C gates. Blocker: a formally specified Python execution semantics or a separately verified shared compiler, not more finite examples.
• Can one current emitted helper be bound end-to-end to this finite audit? That would join the operator check to an actual invocation. First step: take a pinned emitted word containing one of the three certified pairs, trace both wrapper records and their actual event positions, and verify the unit row and the remaining label and hosting hypotheses. Blocker: a raw pair match alone does not discharge those obligations.

[Mister-Meeseeks]

100-PR batch review at 0d8d2c3a96346756f02563b401a042ab97198e20.

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

Reviewed scope: Independent exact Gaussian-dyadic semantic binding for three fixed Lean HPE pairs, both directions and every 512-dimensional basis column. Literal/Python/kernel comparisons and 27 mutations pass; no new Lean kernel build or all-helper/caller admission is inferred.

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.