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[Lean] Hyperplane-ended helpers: Lemma A with an arbitrary terminal frame (no κ claim) - #593
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…ved in Lean 4 Generalizes Lemma G's extended labels so an entering helper may end at any frame U (wrappers keyed by (E, U)), derives Lemma A's (F) from the exact factorization T_U T_E^-1 = L (C (x) id) R, and applies Lemma A unchanged (hpe_block/family/streams/word). factor_iff_pre formalizes review finding F1. code/check_hpe_pairs.py checks the instance hypotheses on the ck/atm3/jp8 HPE words (38 helpers, 3 (E,U) pairs each). No sorry, no new axioms. Prepared with Claude Opus 5.5 (Anthropic) in omp; submitted by jeremykothe2.
Proves T_U T_E^-1 = L (C_7 (x) id) R exactly over C, global phase included, for the three (E, U) pairs used by the ck/atm3/jp8 HPE words (the nine word/pair instances), and states hpe_word_exact: hpe_word for the tools' operators (canonical_frame, the "E:U" recipe monomials, fourier(7)) with no (N) hypothesis. Clifford.lean gives clifford_exact.py's five gates exact C-linear semantics with inverses, and proves cL_eq_of_check: a packed Z[i] evaluation of both circuits on u(x) = B^x (all columns at once, base-B digits made unique by an entry bound) implies equality of the operators. Exact.lean ports canonical_frame and monomial_op, transports to C^128 (x) C^4, and discharges the instances with decide +kernel (no native_decide; check1-3 depend only on propext). check_hpe_pairs.py now also requires each word's pair to be one of the Lean-proved pairs. Prepared with Claude Opus 5.5 (Anthropic) in omp; submitted by jeremykothe2.
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The twelve frame and wrapper literals in HyperplaneEnd/Exact.lean, and their inverses, match the retained Python constructors and independently derived exact Gaussian-dyadic kernels on all 512 basis inputs (12,288 columns), and the three rank-seven residual factorizations hold exactly, including global phase and spectator coordinates. 27 semantic controls reject. Independent finite corroboration of PR #593's identities; no new Lean theorem or kappa. Adds research/hpe-lean-semantic-binding/ only. Submitted by @commdata2338 with substantial AI assistance (OpenAI Codex research and implementation; packaged with Claude Code).
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- The producer's lower_child check (signed Z4 tableau plus column 0) is complete but not independent; K2 is a control on this package's own comparator, not a gap in the producer's check. Lemma G's I-tr and check_adapters.py ran on four other words. - Scope is record level for CLIFFORD-RECIPES.json; identity adapters are 866 per orientation (352 entry + 514 exit). - Gate-order equivalence holds on all 224 adapters (counted in RESULT). - B3 accepts only the selected record or checkpoints/frontier1094.json; B8 uses a path pin in the source manifest. - Tests (26): mutated child/adapter records rejected by evaluate(), receipts cover each pair once, measured digit bound on the 16-gate identities and a beta = 2K collision, I5/I6/I7 and fixture-pin negatives, calibration reproduced. - Credits: HyperplaneEnd Lean (jeremykothe2, CrocSwap#593), Lemma G (DreamingOfClouds), CrocSwap#433 H6 identities, PR425 (jacklightChen), CrocSwap#1029/CrocSwap#662 engines.
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Summary
No κ claim. This is a Lean 4 proof package for the hyperplane-end step used by PR591, PR592 and PR596.
research/hyperplane-end-lean/proves that Lemma A's realization still holds when a dirty entering helper ends at an arbitrary terminal frame U, not only FULL. With the extended labelsT_(prew,j) = R_{E,U} T_EandT_(done,j) = L_{E,U}⁻¹ T_U, and wrappers keyed by the pair (E, U), the exact factorization (N)T_U T_E⁻¹ = L(C⊗id)Rimplies Lemma A's hypothesis (F). Lemma A then gives the helper's streamχ⁻¹(C⊗id)χof itself for any U (hpe_streams,hpe_word). The FULL-ended helpers of #484 are the instance U = ⊤.factor_iff_preformalizes a review finding: (F) holds iff the pre wrapper is R_{E,U}. So a wrapper chosen by entrance alone, as in the original code, is wrong for U-ended helpers.Update (commit fdb6dcc): (N) is now proved in Lean
HyperplaneEnd/Clifford.leanandHyperplaneEnd/Exact.leanprove the factorization (N)T_U T_E⁻¹ = L(C⊗id)Rexactly for every (E, U) pair the hyperplane-ended words use. The 9 word/pair instances (ck, atm3, jp8 × 3) come down to 3 distinct identities. They are checked bydecide +kernelon a packed ℤ[i] circuit evaluation (check1–check3, axioms[propext]); a soundness lemma (cL_eq_of_check) lifts each check to an equality of complex operators.hpe_word_exactishpe_wordwith no (N) hypothesis: it requires only(ent h, term h) ∈ pairs.Checks (fresh clone of
fdb6dcc, Windows, Lean v4.21.0)sorry,admit,axiom,native_decide,unsafe,opaqueorexternin 17 files.lake exe cache get+lake build: Build completed successfully.#print axioms: all 28 results are withinpropext,Classical.choiceandQuot.sound(check1–check3:[propext]only).code/check_hpe_pairs.py --mutantson the ck, atm3 and jp8 hyperplane-ended local runs: PASS, every mutant rejected.CI=truein the environment, building Mathlib from source (--no-cache) fails in Mathlib's ownCauSeq/Basic.lean:440sayscheck at this pin. Use the Mathlib cache (the default), or unsetCI.Scope
The climb recipes, the unit row and (O′)/(W) remain hypotheses here, checked exactly by the PR591/PR592/PR596 verifiers. This package doesn't prove a multiplier or κ theorem. PR #599 (DreamingOfClouds) extends it to multi-gate tails and cold helpers.
AI disclosure and authorship: Claude Opus 5.5 (Anthropic), running in the omp coding harness, wrote this package. Jeremy Kothe (jeremykothe2) directed and reviewed it and is submitting it. It builds on the Lemma A and Lemma G Lean packages and their authors, credited in NOTICE.md.