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Scoped negative result: common rounding of Gaussian helper gauges - #1490
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Add a scoped scalar-cost theorem, exact one-helper census, literal-circuit cross-checks and source-bound negative controls. Prepared with substantial OpenAI Codex assistance; no new kappa or supplier admission.
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Batch review at Merged to Reviewed scope: Common rounding of all static Gaussian-unit helper gauges is proved for the stated scalar lowering and boundary class, with full dirty-column and independent exact census controls. It preserves scalar-cost dominance, not arbitrary precision admissibility or every possible Gaussian lowering. 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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Change and mathematical scope
Extending static helper rescalings from real powers of two to all units of
Z[1/2,i]does not improve the unconstrained scalar-cost optimum under the specified shared-product ADD lowering and separate two-real-register boundaries. The new common-rounding proof givesC_G >= (C_0 + C_1)/2 + 4 N_odd, including shared ADD edges between gauged helpers. It does not assert preservation of an additional precision or physical-admission constraint under rounding.The exact census covers all 237 cold/FULL, unwrapped, non-center, no-COPY/clear helpers of the selected PR1094 k8 word, both orientations, every integer exponent and all four unit phases. The best odd Gaussian factors save 58 scalar primitives per orientation; the known real-dyadic factors from @woklerx's #1161 save 332. This is a scoped negative result with attained finite odd-class witnesses, not a new real-dyadic record or kappa claim.
Task and overlap
Only
research/k8-gaussian-unit-rounding-seed24/is added. Base:410026dc5513b87db92ae98571d1d5984c69a983; OS-random assignment seed 24, complex supplier. Exact source/blob pins, the theorem and exclusions, all minima, primitive-stream hashes, controls and open next questions are included. The selected record andupstream/are unchanged.Scope status: changed relative to the
complex-new-helper-inventorycatalogue card: this follows the user's permitted complex-supplier scalar-circuit scope. It closes one coefficient-class extension while leaving the paid inventory and central moment bottleneck unchanged. The task card records all three candidates and the scope change. The new Gaussian-unit/common-rounding argument differs from PR1161's real one-helper search and PR1214's real pair search.Overlap was checked against a complete 823-open-PR snapshot, a fresh complete 173-discussion snapshot, and the newest 50 updated open PRs covering changes since the initial snapshot. Two full-list refreshes returned HTTP 500; the bounded fresh update check found no matching external scope. Exact timing and coverage limits are in TASK.json. Upstream main advanced to
ef5f39403610d231a0c27e8a6b4f08c10209bcd6; all declared live inputs, selected record and replay runner remain byte-identical to the base.Validation
The registered replay passed locally on Python 3.12.13 in 64.5 seconds. It checks all 712 complex input columns for baseline and both winners, 464 six-column dirty primitive fixtures, 66 boundary pairs, 16 rejection controls, and a separately implemented literal-lowerer grid. The independent grid covers 13,684 integer evaluations and 1,115,016 incident terms. Optimized read-only relocated runs pass; source/certificate/census corruptions reject; equivalent gzip/JSON encodings are accepted. Python 3.14.6 optimized replay, Ruff, whitespace and selected-record integrity checks also pass. Complete positive envelopes remain 52/68 bits; the 192-bit charge and paid profile stay unchanged. These are local results; hosted CI is separate.
Scope excludes temporal gauges, a different Gaussian ADD lowering, cross-macro product reuse, shared boundaries across helpers, and fusion with original SCALEs. Native/global admission, a new assembly and the inherited all-size/compiler, precision/tape, recovery and analytic obligations remain unverified by this package. The written scalar theorem is not a Lean multiplication theorem. No arbitrary joint real-dyadic optimum is claimed.
Credit
Credit Sagar Thakrar's PR696 lowering; @woklerx's #1161 (
d210969d3158a1e6c505e41f67dd84e555c88564); Hanson Sun's #1214 (26162c56ec9c50a80e49949f36d4dd9a137a76a6); and icekylinx, Jeremy Kothe, Rohan Arun, Douglas Colkitt and the retained supplier lineage. No pending predecessor implementation is imported. Hanson Sun (@bbMisk), with substantial OpenAI Codex assistance in the mathematics, implementation, verification and writing; separate implementations by the same AI-assisted author, not independent human review. Apache-2.0.Open next questions
The package questions identify two ways to change the theorem's premises: reuse products across two ADD macros, or share a four-real-register boundary across two helpers. Each needs an exact dirty-state word and complete frame, restoration and precision charges; neither is claimed to work.
Sprint claim: #1163 (comment)