Research: certify the cleanup minimum and handoff windows for PR #2226's 32 low lenders

[commdata2338]

Change and mathematical scope

PR2226 exhibits a successful early cleanup for 32 low lenders. This adds the fixed-alphabet necessity proof and exact timing contract needed to preserve it when compiling or retiming the word.

Every one of the 48 retained operations is necessary to restore all 32 original dirty inputs: 32 subtractions and 16 negations. For exactly k restored rows, the sharp local minimum is C(k)=k+max(0,k-16). The 96 component cases certify all 2^48 subsets. Each subtraction also requires one of 32 distinct rank-three-to-four source promotions, already included in the predecessor's bill.

Five subtractions must precede their sources' handoffs, which occur before their own lenders' handoffs: 277<-281 and 282<-285 by cut 1142; 346<-349, 357<-361 and 362<-365 by cut 1054. The original earliest schedule is valid. The latest permitted schedule has the same complete 624-row free-operator B prefix, Y+=X, all 80 handoffs, and the same frame endpoints and prefix bill. The proof covers all intermediate cut assignments, not just those two tests.

This is a finite certificate and scoped proof/audit in research/anticipatory-lender-cleanup/. It fixes the 48-operation alphabet, coefficients, original input labels, monotone frame paths, and operand withdrawal at handoff. It is not an unrestricted circuit minimum or an admission of a partial subset as a supplier. No scalar saving, new paid-profile improvement or kappa is claimed. Final inverse cleanup/private buffers, exact native phases, inverse transpose, whole-call ownership, complete supplier invoice, precision/tape and all-size/outer obligations remain unchecked.

Task and overlap

The claim is in discussion #2615. Scientific/task-selection base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (priority revision 4). Integration base: b4560d9e7216f72b73c5aa940ac099c05e33ccf9 (priority revision 5). Contribution commit: this PR's head. The final 16:40 UTC fetch found no further change after the 16:33 UTC guidance update. New AGENTS/autoresearch/catalogue/family-status guidance was read. Selected scientific inputs are unchanged; historical task provenance is intentionally retained.

The sole mathematical dependency is PR2226, exact head dacaf3d81f1936bb55b8454b1244e5d014272754, re-fetched unchanged. Necessary word/witness/block/notice bytes are retained with SHA256 and Git blob pins. The original selected word/block certificate comes from Ali Romman's complex-block-rank-optimum at 697b00e5d36297563c22aac080dd5dee398c2624.

PR2339 and this account's PR2394 concern successor T74 words and native initial compensation. PR2573 excludes additional loans in a different retired-lender class. PR2402 reports future T74 opportunities without establishing this cleanup minimum. Concurrent audits of PR2530 lender values, PR2588 frame prices, PR2575 owner maps and PR2269 private buffers have different objects and obligations.

The offline overlap helper used complete-paginated snapshots: 823 open PRs including drafts at 2026-10-15T16:37:18.169440+00:00 and 1276 Discussion records at 2026-10-15T16:27:10.714268+00:00. The changed-file snapshot at 15:59:04.742484 UTC matched only PR2226 for research/anticipatory-dirty-borrowing/*. This is metadata/path coverage, not a novelty certificate or a review of every diff.

Task card (also structured in TASK.json):

Task ID / proposed scope key: complex-new-helper-inventory / f3b0895a4f9f1cc62e75; PR2226/32-low-lenders/48-tail-subsets/monotone-frame-and-withdrawal-cuts.
Base commit; exact predecessor and dependency heads: original main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; final main b4560d9e7216f72b73c5aa940ac099c05e33ccf9; PR2226 dacaf3d81f1936bb55b8454b1244e5d014272754.
Concrete object and question: all retained-order subsets and full-cleanup retimings of the fixed 48 early-tail gates restoring 32 low lenders.
Scope status: changed from new-interface synthesis to an independent necessary-boundary audit; the predecessor checked one successful schedule, not the all-subset minimum and source-withdrawal intervals.
Existing restriction being changed (or exact claim audited): replace one positive witness by a complete fixed-alphabet classification and chronological safety contract. No changed coefficients, frames, lenders or compensation are allowed.
Method; explicit domain: exact rational rows, free noncommuting loan operators, finite GF(2)^9 frame sets; 704 original/624 remaining inputs; 96 local cases covering 2^48 subsets; 1846 B events and 80 loans.
Nearest overlaps and added value: PR2226 original construction; PR2339/2394 native initial prefix; PR2573 different retired-lender exhaustion; PR2402 later T74 leads. New coverage is cleanup necessity and retiming preservation at the pinned original source.
Declared milestone: audit.
Optimistic full cost budget; required observable improvement: no new supplier budget claimed. Required output is the exact fixed-alphabet minimum, 32 inherited H1 promotions, and safe timing intervals. The prefix bill is explicitly not the complete supplier invoice.
Unresolved consumer/restoration costs; expansion decision: full native phases/final cleanup, ownership, precision/tape and all-size costs are unknown here. Conclude the audit; no dependent expansion follows without a changed premise and complete boundary/cost allowance.
Success evidence; falsification and time cap: typed source-bound receipt, independently formulated row arithmetic, whole-prefix equality and actual chronological mutation rejection; bounded 60-minute session, single CPU, standard library.
Predecessor authors; artifact/command: Cinna Davis, Ali Romman, Thomas DiFiore and retained PR1042/1094 lineage; python3 -I -B scripts/research_package.py --package research/anticipatory-lender-cleanup.

Chooser provenance: nonce a6f46cd88e711b738859e407ddbec9cd, complex-mechanisms, second suggestion; no reroll. The current family-expansion gate is recorded in TASK.json. This local closure does not justify another implementation campaign with unknown consumer costs.

Validation

python3 -I -B scripts/research_package.py --package research/anticipatory-lender-cleanup --check-only
python3 -I -B scripts/research_package.py --package research/anticipatory-lender-cleanup
python3 -I -O -B research/anticipatory-lender-cleanup/verify.py

CPython 3.12.3/Linux, standard library, about 2 seconds for direct replay. The metadata/pin check and registered optimized replay pass after rebasing. RESULT.json records all 64 independently bound lender/source rows, the complete subset distribution, all timing intervals, and the two complete prefix comparisons.

136 semantic rejection cases cover omitted/reordered cleanup, early frame visits, source withdrawal, equal-rank wrong axes, wrong original-input labels, and missing/wrong-sign compensation. 34 positive schedule/basis cases pass. A separate Git-free read-only relocated package passes both normal and optimized Python without changing any file. Source-byte corruption, proof-relevant boolean-to-integer receipt mutation, duplicate JSON keys and overwriting an existing recorded result reject. The format-patch applies to an isolated index at final main and reconstructs the committed tree exactly.

The catalogue's ceiling positive control and selected-baseline binding check passed during orientation. No predecessor full replay, selected supplier cold replay, native phase/dual execution, final cleanup, hosted CI or independent human review is claimed. Successful finite checks are not a formal verification of the full multiplication theorem.

Credit

Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.

Credit Cinna Davis (@1337isnot1337)/PR2226 for anticipatory borrowing, its 48-gate cleanup, original successful schedule and logical construction; Ali Romman for the fixed-block extraction and certificate; Thomas DiFiore for block discovery; Jeremy Kothe, Rohan Arun, Hao Yan, Chafik Boukhalfa, icekylinx, Jonathan Capalbo and the retained PR1042/1094 provider/frame/controller lineage. Original notices and source bytes are preserved. The new result is the subset/retiming certificate, not the existing gate count or borrowing mechanism. New proof and code are Apache-2.0.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the derivation, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

The README and TASK.json identify two bounded next steps: change one cleanup source/frame/reference premise and fully price the replacement, or source-match a native prefix and verify both orientations plus final cleanup. This account's PR2394 concerns different successor words and cannot be silently substituted. Global ownership and a favorable completed bill remain blockers. This is a completed audit, not a decision to expand that construction family.

Workflow feedback (optional)

The upstream refresh caught priority 5 during the session; historical scientific pins remained stable while the new audit/expansion distinction was recorded. Future loan witnesses could expose cleanup-source withdrawal alongside the lender deadline, making the five earlier-read constraints directly available to schedulers.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG