[unpaidmoon]

Change and mathematical scope

Negative result. No new κ. The selected conditional exponent stays 60403/20000000 (0.00302015).

On the selected k8 register file, with pinned entrances and monotone nested frames in (\mathbb{F}_2^9), no rank-at-most-8 paid profile of mass 5586 meets the u=2 nine-call moment at (a = 1/98). The ideal cell {8:698, 2:1} needs 698 rank-8 children. These entrances supply 648 monotone slots. The minimum feasible profile is {8:648, 5:32, 4:24, 1:146}. Its pooled moment, with idle {51:272, 46:1}, has certified bounds strictly above 1.

A comparison that gives each of the eight centre registers three rank-8 steps has histogram {8:664, 5:32, 4:24, 1:18}. The bare moment is below 1, and one aggregate rank-1 child pushes it above 1. That comparison is an upper model on the same eight centre indices. It is a changed path law, and the 16-child margin is absent there as well.

This is a finite certificate of that entrance model: a scoped obstruction and a failed search for a monotone producer on these pins. It is not a complete multiplication theorem. The selected exponent stays where it is.

Remaining obligations: one legal up/down/up centre trace, and an entrance count on the #4470 frames.

Task and overlap

Base commit ff3cc58e370d0f678e529246f8c73e8094e18fcf (upstream CrocSwap/integer-mult-bounds main). Dependency pins are in research/rank8-u2-entrance-obstruction/SOURCES.json: ceiling.py blob 6c0f54527d879396f689523bfde19928a9b6157c, selected word gzip blob 723decbca8995e916d6f4b783aa3e1ddd4eb47fe.

Task card complex-new-helper-inventory, priority revision 7. Scope status is refined. The card asks for a new helper inventory whose fully charged profile beats the packing ceiling. This package answers the still-open literal rank-at-most-8 producer inside the published 16-child slack and stops at the entrance obstruction.

Session r31, item 9 of the ordered list recorded in research/f2-six-difference-basis (open handoffs from the 19 October index, then open priority-table rows, then remaining catalogue cards). Item 10 of that list is #4651 and is dropped. The list is restated in the package README.

This is open question 2 of research/rank8-u2-split-boundary (#4151). Question 1 of that package, which integer splits of 896 still pass 16 extra rank-1 children, stays open. #4156 prices a plane-port screen against the same slack. #4158 records that the published selected word emits no paid child above rank 8. The monotone slot count is the remaining producer question those three leave open.

python3 scripts/find_overlap.py on 2026-10-23, terms 8:648, monotone rank-8 slot, entrance obstruction, rank-at-most-8 producer. Coverage: local notes complete at ff3cc58e370d0f678e529246f8c73e8094e18fcf (7091 head records), 307 open PRs, 4479 discussion records, both paginations complete. Closed PRs were searched through local READMEs and TASK metadata. No other note states this histogram. Same-day fork branches for the L546 six-tape bill, the R437 parent blank-return, the shared chart-1 centre visit, the SOURCE87 fibre count, and the #4091 1090-call ceiling price other rows.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B research/rank8-u2-entrance-obstruction/verify.py
python3 -B scripts/research_package.py --package research/rank8-u2-entrance-obstruction

Observed:

PASS_RANK8_U2_ENTRANCE_OBSTRUCTION slots=648 monotone=GT/0 centre3=LT/0/GT/1 ideal=LT/16/GT/17 no_new_kappa
PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
PASS_REGISTERED_RESEARCH_PACKAGE research/rank8-u2-entrance-obstruction outcome=negative-result

The checker hashes ceiling.py (86a1432870b3b21d8454106ec9162fddc0471472d9214c590a1420f2426cedfc), the word gzip (49596aa4ed4fd36cd4582fd0e8ddeb09fd80370a7d8aea1ede52a135b536d289) and the decompressed word (c2b294300760e1fd6eeab8c22c4d13e4633dca40e52cac8da0d68163a76e3ca8). It rebuilds the 704 entrance rooms, packs both histograms, and separates every merge and transfer inequality on ranks 1..8 by the pinned outward enclosures. The ideal control reproduces the published boundary: 16 added rank-1 children pass and 17 fail. The capacity census is {9:288, 8:360, 5:32, 4:24}.

Not run: Lean, Docker, the selected cold replay, and any search for a non-monotone scalar word.

Credit

The 16/17 integer boundary and the cell {8:698, 2:1} are ToikY's research/lever-integer-twins, following Nick Petalas's PR #3874. The entrance fields and the (\mathbb{F}_2^9) frame contract are the selected k8 word in research/r480-owner-optimum, as read by the frame-ceiling line (PR #1671, Gareth Davidson, using navinor's frame method). The child-cost enclosures are Douglas Colkitt's research/complex-local-word-ceiling, with Claude assistance. The coverage identity is PR #1012. The split dependence of the ideal slack is research/rank8-u2-split-boundary. The plane-port screen is research/rank8-u2-plane-port-slack.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human review and no formal-kernel acceptance are claimed.

Open next questions

1. Can one centre register legally emit three rank-8 steps, and can that word avoid every added rank-1 child? The arithmetic room of the comparison is zero: the bare moment passes and one aggregate rank-1 child fails. The first step is one explicit up, down, up frame trace and its own paid histogram. The standing monotone contract is the blocker.
2. Do the Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 productive-helper frames have at least as many monotone rank-8 slots as that invoice demands? The first step is the same entrance count on the Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 complex frames. The blocker is that the two templates must stay separate.