[unpaidmoon]

Change and mathematical scope

Scoped proof. No new κ. The selected conditional exponent stays 781451/250000000.

The selected ordinary formula reads one field, ordinary_BIT.after_cover. The thinned controller writes it once, from C1=0.00345 (69/20000). The one-weight delayed-cover proof keeps before > after, with before-cover C0=3459/1000000. The formula is strictly increasing, and

ordinary(C0) = 8613294599/2500000000000

is strictly below the recorded PR5679 component

C = 3507410738409/1000000000000000.

Every rational cover strictly below C0 therefore misses C. The unique cover that hits C, 3507256898809/996000000000000, lies strictly above C0, so the positive-gap comparison does not apply to it.

The second source assignment authenticated here is the retained delayed-cover bridge: after = Q(11, 4000), stored as C1_after_cover. That key is not after_cover. The selected formula at 11/4000 is 6847884599/2500000000000, which also misses C. Emitting that weight beside 69/20000 would drop the one-weight sentence.

Evidence class: scoped proof under the single-writer, single-C1, and positive-gap hypotheses. The rational ray below C0 is infinite and is not enumerated. This does not change the selected frontier.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840 on canonical CrocSwap/integer-mult-bounds main. Priority revision 8. Catalogue card supplier-boundary-audit. Scope status: refined.

The card asks for one resolved gap on a latest supplier candidate. Merged #6957 censused the single emitted after-cover. This package takes only open question 1 of that note. The caller question from the same note is merged #6958 and is not retaken. The PR5708 preimage remains with research/g20-component-stock-gate.

Overlap on 2026-11-10, with 339 open upstream PR bodies and 7,007 discussion records: the preimage appears in discussion #1166 comments 19492491 and 19507126, and in no open upstream PR body. The phrase "second after-cover" appears only in the #6957 open question.

Package: research/g20-second-after-cover-obstruction.

Validation

Python 3.12, standard library only.

python3 -B research/g20-second-after-cover-obstruction/verify.py
python3 -O -B research/g20-second-after-cover-obstruction/verify.py
python3 -B scripts/research_package.py --package research/g20-second-after-cover-obstruction

Observed locally: PASS_G20_SECOND_AFTER_COVER_OBSTRUCTION and PASS_REGISTERED_RESEARCH_PACKAGE with outcome negative-result. The optimized command matches. Checks raise SystemExit.

Hashes and Git blobs are checked before any Fraction evaluation. Controls reject an unbound call, a second after_cover key, a bridge key renamed to after_cover, a second C1 sentence, and a second controller writer.

Not run: the controller replay, the bridge script, the outer ledger, either component caller, native maps, an odd-prime search, the odd-prime avoidance list, Lean, Docker, and make frontier6030-verify. The comparison 2**26 > 256/(C0-C1) is evaluated only at the kept gap 9/1000000. At the preimage the denominator is negative.

Credit

The component is Ali Romman's PR #5679. The selected pricing and claim are DreamingOfClouds' PR #6030. The one-weight contract is Cinna Davis's research/query-thinned-controllers. The bridge weight is the retained source in research/kappa-rebuild/bit. The single-emission census is merged PR #6957.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. The odd-prime avoidance of fixed denominators and rank witnesses was not executed. It is not the blocker: every cover strictly below C0 already misses C. First step, for a later single cover that restores a positive gap and is checked only for the fallback surcharge: name one odd prime above 2^26*N^5 and the denominator list it avoids. Blocked by the size of the compiled N.
2. If before > after and 256/(C0-C1) are dropped, the unique preimage makes this formula equal C. That drops the one-weight proof. First step: write the replacement for those two sentences, then read whether bit_saving is still admitted. Equality with C would still leave bit_saving = 17/5000. This package does not retake the caller census of Record that the caller outside the selected pricing has no bit_saving input #6958.

Workflow feedback (optional)

The open question's first step, naming the second assignment before arithmetic, matched the bridge weight already pinned as a controller input. A link to the feedback discussion is enough: #1200