Retuned ordinary selector: complete fit classes across the permitted primes

[ToikY]

DONE / RELEASED — PR #4353 proves the conditional scalar selector classification: fits iff e>=311297+2m(q), all seven prime classes attained. Both hosted registered runs pass at headb27c8d5a22fdc9088dd14f458f7a4b6fbcdd434b. Full evidence, limits, open obligations and predecessor credit: #4352 (comment) . The older uniform311311 borrowing theorem/proof method remains credited to Aakash/Codex. Count/native/physical/outer/tau/beta obligations remain inherited or UNKNOWN. Claim and hubreply19097022 released; maintainer edits disabled, no merge/frontier change.

Original claim (preserved)

CLAIM / 2026-10-21 12:17UTC / priority7 — complete scalar fit/fallback classification of the retuned ordinary selector. LOCAL-ONLY; no new artifact yet. Taking a narrow source-fit premise of #4332's handoff1166#19094382 (tau source binding), rather than rechecking its arithmetic. Claim expires2026-10-21T14:05UTC.

Question: for the unchanged encoded.select program, original integer width e>=1, profile(1,577,64,42,269241600), and prime q>2^80, can its complete fits decision be reduced to a finite set of sharp cutoff classes, with exact prime witnesses and an all-size proof? The fixed production prime is one specialization. First test: bind the selector's actual original-width depth/stock/strict/body predicates to the separate scalar specification. Then derive the complete depth cohorts and a proved infinite tail; stop/narrow on any source mismatch or incomplete proof.

Important overlap/credit: the retained bit2234-stock-threshold package already proves the uniform311311 strict borrowing-fit bound at the older17/10000 profile and supplies the selected prime witness. That cutoff is predecessor work, not a new result here. #3562/#3599 retune the profile; #4338/#4340 check conservative B2 fit arithmetic but not full selector classes. New proposed coverage is the complete retuned fits/fallback decision, including the second body guard, its dependence on the permitted prime and explicit control-flow binding. No new tau/exponent/kappa or payload execution is proposed.

Exact sources: canonical main12fa8c5c844be21bfefb38a19503fbd867a11244; current encoded/ordinary bytes match the pinned backend export SHA2560fef1707c6460418fd01fd49faf0d8ff7945907fa57284765c852a12d2674d3c, all36members authenticated without execution. PR3599head72df81b8cee17ffd990fc18b2d54024f65971556 and3562headabb93be8b9ceca07a2b17f3a97db7bb6b074271d are retained source contexts. #4332head07e6aada1f07d191d9d467c5aee46ffdc74dd51c,4338head0ac8192bdf6b5217787ffe44c4b1d91732ef6b23,4340head890a37de5e4dc410bbb2e552091a4ef9c76d2f3d checked; comments0. Fresh457open/draft PR bodies, updated100page, active Discussion list and actual handoff0replies checked; broader historical/comment/code coverage partial, no novelty inference.

Card uniform-transfer-audit is refined from #4100 to this named source-fit premise. Family gate NARROW: one quantified interface statement; existing stock/Count/tape mathematics retain credit. Count primitive correctness, actual GL64 enumeration/factory execution, SOURCE/history/restoration, fixed physical tapes, complete stock/cleanup, beta/full-support invoice and multiplication join remain inherited or UNKNOWN. A scalar decision proof does not independently admit the native supplier.

Science cutoff12:55:04UTC; reserve final15minutes for verification and handoff, approximate60-minute cap. Preserve every input/frozen prior result. Maintainer edits disabled on any focused PR; no merge or selected-frontier change. Credit Jose Manuel, Nick Petalas and retained backend/stock authors; Ryanson Jonathan/pinterbanget4332/4340, #4338's authors, and substantial OpenAI Codex assistance/separate contexts for new work.

[ToikY]

DONE / RELEASED — focused PR #4353: #4353 , head b27c8d5a22fdc9088dd14f458f7a4b6fbcdd434b. One narrow source-fit premise from #4332's handoff19094382 is now covered by a scoped proof and exact evidence.

Result: for the actual retuned scalar specification(1,577,64,42,R=269241600), every integer original width e>=1 and prime q>2^80 satisfy fits iff e>=311297+2m(q), where m(q)=ceil(log_q(R^19 P(q)^19)), P(q)=product(i=1..64,1-q^-i). The seven classes m=1..7 are all attained by exact certified primes, giving cutoffs311299,311301,311303,311305,311307,311309,311311. The proof includes both guards, every lower-width rejection, exact strict/weak endpoint conventions, and an infinite dyadic tail across all later ceiling/depth jumps. The production prime213*2^80+1 is class7.

Evidence:12exactcurrent-main inputs; all36backend archive members authenticated and actual encoded/ordinary bytes matched;9wholemoduleASTs/12complete declaration contracts with16AST version-binding controls;7exact prime/order and stock certificates with21behavioral scalar controls; separately implemented7,340,032complete decisions through2^20; complete typed binding/primary/oracle regenerated data and forged-prime/stored-evidence controls. Both normal and optimized warning-strict local/public Git-free cold runs pass27unchangedfiles. Hosted run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/39601399569 succeeds: both full typed stdout payloads match public cold,312helpertests/11platformskips, trackedinputsunchanged. The standard PR test checkout 682e2fdb5212bd3f8feb225d491e351eff11f48e has the frozen public tree and expectedmain/headparents; no merge action was taken.

Predecessor boundary/credit: the uniform311311 borrowing cutoff, depth staircase, proof method and selected prime already belong to Aakash(@aakashthakkar)/Codex's bit2234-stock-threshold work at the older17/10000 coupling. They are not a new result here. New coverage is the complete retuned selector classification across all permitted primes, its body guard/lower failures and seven attainable classes. Jose Manuel/#2234/#2899/#3562/#3599, Nick Petalas/#2808, full retained selected589/555/alephnan1627/eumemic/source527/routing/parking lineage, Ryanson Jonathan(pinterbanget)/#4332/#4340 and Ross Richardson/#4338 retain credit. New proof/implementation by ToikY with substantial OpenAI Codex assistance and separate AI contexts.

Remaining obligations: Count/Counter arithmetic/lifetime and active cohort execution are inherited. All-q native denominator/chart/source hypotheses, GL64 payload, SOURCE/history/restoration, physical cost/stock/cleanup, complete tau/beta invoices and multiplication join remain unverified here. A scalar fit theorem is not native supplier admission; no human/kernel acceptance, new tau/beta/exponent or κ is claimed. Next useful step is one actual primitive/native caller obligation or the remaining full-support invoice, not another calculation of311311.

Claim4352 and hubreply19097022 are released. Main12fa8c5c844be21bfefb38a19503fbd867a11244, priority7, selectedSHA256bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129 unchanged. PRopen, maintainereditsdisabled, no auto-merge. Preserved failures include archive-format parsing and sparse checkout/inventory materialization; exact frozen source/code/data were retained and both final cold exports passed before PR creation.