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Research: sharp clean selector encoding width and volume screen - #1924
Conversation
Exact single-host clean-subspace width and phase specialization of credited spectral methods. No new kappa or frontier change. Prepared with substantial OpenAI Codex assistance.
The minimum is for C_d with unchanged external spectator factors; it does not minimize active recursive rank under arbitrary spectator reassignment. Keep the exact theorem, and state explicit changed-contract examples.
Pascal averaging resolves all positive widths: optional phase always needs one extra axis, and fixed-phase exceptions are exactly the two small-width residue classes. Corroborate the written proof with independent exact count oracles.
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Batch review at Merged to Reviewed scope: All-size eigenspace/Pascal proof gives sharp literal-host width n+s+1 or n+s+2, including phase classification.4096 width cases,12 full rational intertwiners and five relocation/corruption tests pass. Clean complement is required; inherited full-volume upper-bound pricing loses by more than2x and is not a universal runtime lower bound. 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. |
2 participants
Change and mathematical scope
For q=2^s same-direction C_n lanes, characterize exactly when one i^u C_d child admits an injective clean invariant encoding. The sharp least literal width has a closed formula. Put N=n+s. With any fourth-root global phase, it is always N+1. With the exact phase it is N+1 for every n>=3; at n=1 exactly selector residues s mod8 in {0,6,7} pass; at n=2 exactly {0,1,5,6,7} pass. The remaining exact-phase cases need N+2. A uniform Pascal-averaging/character proof resolves all positive widths, separately from the finite checks. Sixteen width-one lanes need d=7 with the exact phase, or d=6 with an allowed -1 phase. An explicit dyadic Walsh encoding and left inverse preserve every arbitrary complex payload with a zero complement. Dirty complement generally rotates and is not restored.
The minimum width and the following budget screen are for the literal C_d host, with no independently variable spectator factor. Common spectator fibers extend only as E tensor I with the same factor on both sides. Arbitrary spectator reassignment is outside the theorem: PROOF.md shows an unchanged-volume width-one example and a separate twofold-volume width-two example. This does not minimize a supplier’s physical recursive rank.
Within that contract, the inherited full-volume child-budget ratio is 2^(d-n-s)(d/n)^sigma > 2 for 0<sigma<1, even with every encoder/decoder/phase/preparation expense waived. This compares available upper-bound expressions, not optimal runtime. Scoped algebraic specialization and negative budget result; no physical supplier, 5% construction-gate success or new kappa. Only research/clean-selector-embedding/ is added; selected frontier unchanged.
Task and overlap
Base 0b1cccf; OS nonce 9a396c07a6d107acdb9a76567c1010d0; revision 3 complex-rank-shape-decoder / scope_key 02b31ebd160542c86299. Scope changed after one pre-implementation overlap pivot: Arham Tahir’s PR1920, head 057b214, covers full-space reciprocal encoding and explicitly excludes this clean injective premise. Claim/handoff.
PR665, head d0e4f1e, already gives the general eigenspace-capacity/Walsh method for a different two-host split. The delta is the sharp single-host repeated-lane minimum width and phase dependence. PR679, head dce137e, and Ross Richardson’s 8a3900c audit cover monomial routing/volume and motivate the changed premise. Exact source pins and credited scopes are in SOURCES/PREDECESSORS/TASK.
Entry baseline, assigned ceiling control and lever table pass. Final refresh at 13:54–13:55 UTC covered 573 open PR bodies including drafts, 873 local notes and directly read relevant Discussions. Numbered pages are non-atomic; no complete Discussion census or novelty certificate.
Validation
Registered CPython 3.12.14 replay passes, including 131 independent binomial/character counts, 4096 width cases, 12 complete Gaussian-dyadic fixtures (9440 intertwiner entries and 3360 Gram entries), seven semantic/domain controls, and five actual subprocess export/corruption/output tests. Normal and optimized no-Git exports pass; source/numeric/boolean-alias receipt corruptions reject; occupied output remains unchanged. 18 onboarding tests, exact source/Git pins and staged diff check pass. Separate bounded Codex proof/code reviews and a code-disjoint 576-case recurrence check agree.
PROOF.md is written all-size mathematics, distinct from these finite checks. Full selected cold replay, native/physical/hosting/production precision admission, all-size tape compiler, Lean, hosted-CI success and independent human review are not claimed.
Credit
Douglas R. Colkitt/PR665 for spectral capacity and Walsh encoding; Greer Guthrie/PR679 for kernel, dirty padding and volume; Arham Tahir/PR1920 and PR1605; Ross Richardson’s contract audit; Jose Manuel(jmbarrancoml) and publishing account Cinna Davis(1337isnot1337)/PR1779; Josie Elliston/PR1817; icekylinx and the Rohan Arun/Jeremy Kothe/Hao Yan retained selected lineage. Prepared by ToikY with substantial OpenAI Codex assistance and bounded proof/review agents; Apache-2.0. No priority claim for the general spectral method or full-space trace obstruction.
Open next questions
README/TASK give exact continuations: prove one selected caller’s clean invariant entry state and a complete encoder/decoder bill, price dirty-complement repair, or name a different recursion/adapter contract that legally discounts the full-array budget. No selected occurrence or such implementation is supplied here.