DONE: least chronological batch widths and a pooling counterexample

[mihir-s-05]

DONE / 2026-10-22 UTC / priority revision 7. PR #4507, commit 98b9f9e1b96940c0c405638afa974e378436fc55, supplies a scoped batching lemma and exact finite witnesses. For #4436's fixed chronological flow and a prescribed smaller-rank library, the least replication factor is the LCM of B_e/gcd(B_e,q) over actual per-cut batches and upfront complements. At raw p7 rank 1,000, productive width drops from 17,504 to 2,524 bits, with identical normalized leaf cost. A rank-12 fixture rejects aggregate multiplier 175 because a single cut requires 700. This is author-checked conditional accounting, not native admission or a new kappa. The claim is released. The next useful task is a uniform prime-valuation bound or a source-bound application to #4483's changed-height compiler.

Result and independently inspectable evidence

• Written proof, policy and limitations.
• New implementation and exact receipt.
• All nine frozen source flow receipts; 101 literal cut fixtures; 2,304 brute-force multiplier cases; 1,074 compiled ranks and 49,474 integral batch constraints. Original final width and leaf hashes match, while new normalized costs agree exactly at every checked rank.
• CPython 3.12.0 / Windows: registered ordinary and optimized replays pass with 12 rejection controls. Source-history/committed-byte preflight and the 13-file export pass. Both exported modes run with no Git; source and receipt corruption reject, as do code mutations pooling repetitions or omitting their cost.

Reproduce from the repository root:

python -B scripts/research_package.py --package research/chronological-batch-width
python -B scripts/check_research_commit.py --package research/chronological-batch-width --check-source-history

Scope, overlap and handoff

The original claim took #4436's open question 3 at head 56003680aee2871773de408e5f01cc5dd46b144c. Canonical main was 2a13eb1827784206202db90e5dc8327cf8a250db at start and submission refresh. This explicitly changes independent-moment from invoice coverage to chronological batch divisibility and finite constructive width.

#4456 recounts the flow; #4476 checks the existing nested widths. This adds the least multiplier for the declared batching policy and smaller width witnesses. #4483 changes the stopping height and proves a different convergence gate; this package neither audits it nor claims the older exponent question remains generally open.

Final bounded overlap coverage: all tracked top-level local notes, 355 open PR bodies, 3,381 Discussion records and 3,787 closed PR bodies, with complete reported pagination at 02:39–02:41 UTC. No competing per-cut LCM construction was found in that coverage; it is not an exhaustive diff search or novelty claim.

The multiplier is minimal only for the current library and policy. The original nested compiler is valid; the counterexample concerns a tempting aggregate relaxation, not a claimed source bug. Improved asymptotic width order, global optimality, changed-height applicability and physical realization remain unproved here. The selected cold replay and native/analytic/precision/recovery obligations were not rerun. The historical orientation command stopped on a sparse-omitted selected claim file. Hosted CI is separate from local evidence.

Next questions: (1) bound prime valuations of actual per-cut moduli to decide all-rank growth; (2) bind #4483's cut inventories before applying this policy there. Stop if the schedule requires pooling across suspended parent calls without an additional proof.

Credit Jacob Alan Sussman, zxv, Cinna Davis / 1337isnot1337, Ryanson Jonathan / pinterbanget and the retained native lineage. Prepared for Mihir Srivastava / mihir-s-05 with substantial OpenAI Codex assistance; no subagents. Apache-2.0. No merge or frontier change.

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  "kind": "DONE",
  "task": "independent-moment",
  "scope": "Least replication at fixed chronological cuts, finite smaller direct-flow widths, and an aggregate-pooling counterexample",
  "base": "2a13eb1827784206202db90e5dc8327cf8a250db",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4507"
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[mihir-s-05]

DONE / 2026-10-22 08:16 UTC / priority revision 7. PR #4586, head aadb7bc59390683450b1969e4632f546929dc569, answers the all-rank upper-bound part of #4507's handoff. For every fixed rule in #4436's default-height flow, the existing chronological widths satisfy log B_N = O((1+log N)^(d+2)), where d counts primes dividing target or paid ranks. Raw p7 has d=7. This improves the inherited exp(O(N log N)) upper bound without changing the width policy, normalized leaf cost or kappa. The claim is released. Evidence is author-checked, not formal or independent human approval.

The proof, exact scope and open questions combine fractional rank descent, a prime-valuation envelope B_n | L_N^D(n), and a source-state lemma: all changes occur at products of primes in the fixed rank alphabet. New source-count primes are allowed. A concrete fixture shows why omitting paid-rank primes is wrong.

The new checker verifies 1,074 ranks, 49,474 batch constraints and 832 cutoff cases, including 642 comparisons to a strictly smaller smooth representative. Normal/optimized registered replays, ten semantic/domain controls, four external corruption/mutation controls, committed-byte checking and the final 14-file export replay pass. The exported replay has Git unavailable and socket creation disabled and preserves its bytes. The pinned #4507 generator is explicitly reused as the studied object; this is not a second independent flow implementation.

The bound is loose at finite rank and provides neither a matching lower bound nor polynomial width. #4483 changed-height applicability, complete physical/storage/precision/tape obligations and native admission remain open. The full selected replay and formal proof kernel were not run. Optional source-history preflight needs full history and was unavailable in the shallow checkout. Hosted CI stopped before package replay on four unchanged current-status/navigation tests; these reproduce locally. See the exact-head CI evidence. Local scientific passes remain separate.

Next useful work: bound actual source valuations more sharply, or pin every rank-dependent predicate of #4483 before attempting to transfer the source-state argument. No further task is claimed. Main remained f626db5c3a3d781d8f7695038776c621651f75e1; refreshed text overlap covered all 5,860 tracked local notes, 357 open PRs and 3,489 Discussion records. Credit Jacob Alan Sussman, zxv, Cinna Davis, Ryanson Jonathan and #4507's contributors. Prepared for Mihir Srivastava with substantial OpenAI Codex assistance, Apache-2.0; no subagents. No merge or selected-frontier change.

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  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "independent-moment",
  "scope": "All-rank prime-valuation and multiplicative-breakpoint bound for PR4507 default-height chronological batch widths",
  "base": "f626db5c3a3d781d8f7695038776c621651f75e1",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4586"
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Original claim, preserved as session history

CLAIM / 2026-10-22 07:50 UTC / priority revision 7. I am taking the all-rank width question left by #4507 and this thread. Canonical main is f626db5c3a3d781d8f7695038776c621651f75e1; #4507 is pinned at 98b9f9e1b96940c0c405638afa974e378436fc55.

The bounded question is whether the per-cut gcd/LCM recurrence admits a uniform width bound using strict rank contraction and the finite multiplicative rank alphabet. I will derive a prime-valuation envelope, test the source-count breakpoints against the actual default-height flow, and retain a proof or a concrete failed implication. This changes the independent-moment card to the explicit compiler-width handoff. #4519 already recounts the finite widths; repeating that census alone is not the intended contribution. #4483's changed-height policy, native admission and a new kappa remain outside this claim.

Method: a written all-rank argument plus exact integer tests on both published toys and raw p7, including a control for omitted paid-rank breakpoint primes. Stop after the stated implication or a counterexample, within this approximately one-hour session. Expected artifact: one isolated, source-pinned package/PR. Evidence is currently local-only and exploratory. Credit Jacob Alan Sussman, zxv, Cinna Davis, Ryanson Jonathan and #4507's retained contributors. Prepared for Mihir Srivastava with substantial OpenAI Codex assistance.