[mihir-s-05]

PR #4507 constructs smaller chronological batch widths but leaves their all-rank growth bounded only by the inherited exp(O(N log N)) estimate. This package proves, for every fixed rule in #4436's default-height flow,
log B_N = O((1+log N)^(d+2)),
where d counts distinct primes dividing target or paid ranks. For raw p7, d=7, giving B_N = exp(O((1+log N)^9)). This answers the upper-bound part of the #4503 handoff. No new κ.

The proof combines strict fractional rank descent, the exact prime-valuation recurrence for B_e/gcd(B_e,q), and a source-state lemma: the default-height flow changes only at products of primes from the fixed rank alphabet. In particular, B_n | L_N^D(n), where L_N is the LCM of source totals through rank N. This is a new bound on #4507's existing width sequence, not a new batching policy or a finite-rank improvement. The bound is loose: at p7 rank 1,000 its binary exponent is 303,840, while the existing actual width has 2,524 bits.

Only research/chronological-width-growth/ is added. Start with its proof and scope and exact receipt. The new valuation/breakpoint checker uses an authenticated copy of #4507's flow/width generator as the object being studied; it does not claim a second independent implementation of that generator.

Validation on CPython 3.12.0 / Windows, standard library only:

• 1,074 compiled ranks and 49,474 actual batch constraints: exact valuation equality, fractional rank descent, source-LCM envelope, and all pinned final width hashes.
• 832 cutoff cases, including 642 nontrivial comparisons with a smaller smooth representative; full flow equality also checked across each floor-cutoff interval.
• Ten rejection controls, including a paid-prime omission counterexample, new width primes outside the rank alphabet, and failure when the depth factor is dropped.
• Registered ordinary and optimized replays pass. Four external source/receipt/code corruption controls reject.
• Final-commit byte preflight and 14-file export integrity pass. The exported verifier passes with -I -O -B, Git unavailable, socket creation disabled, and unchanged payloads.

python -B scripts/research_package.py --package research/chronological-width-growth

Limitations: the universal conclusion comes from the written proof, not extrapolation of the finite checks. Constants depend on the fixed rule. No matching lower bound, polynomial width, changed-height #4483 application, generation-time bound, physical/native admission, or new normalized leaf cost is proved. The optional source-history preflight requires full history and was unavailable in the shallow checkout; all pins are at the canonical-main base. The orientation-only baseline stopped on a sparse-omitted selected claim file. No selected cold replay, formal proof kernel, independent human review, or hosted-CI pass is claimed.

Started from canonical main f626db5c3a3d781d8f7695038776c621651f75e1, priority revision 7, unchanged on the final fetch. Refreshed overlap covered all 5,860 tracked local note paths, 357 open PRs and 3,489 Discussion records with reported complete pagination at 08:07–08:08 UTC on October 22. #4519 already recounts finite widths; #4476/#4521 cover different old-width/expanded-count obligations. This explicitly changes the independent-moment card to the live compiler-width handoff.

Credit Jacob Alan Sussman, zxv, Cinna Davis, Ryanson Jonathan, Mihir Srivastava's #4507 and the retained native/outer lineage. Prepared for Mihir Srivastava / mihir-s-05 with substantial OpenAI Codex assistance, without subagents or independently launched reviewers. Apache-2.0. Selected frontier unchanged.

Hosted CI update: the planning job failed on four unchanged current-status/navigation tests before package selection; the package replay was skipped. The same four failures reproduce locally. See the exact-head CI evidence. Local scientific passes are separate.

[mihir-s-05]

Hosted status for exact head aadb7bc59390683450b1969e4632f546929dc569: workflow 39746391044 failed in the Onboarding helper tests planning step, before package selection; research replay was skipped.

The four failures are all in the unchanged tests/test_research_current_status.py: current CI navigation still names the historical replay, the decimal assertion expects different Markdown bolding, the history banner is absent in the tested prefix, and an old heading contains “selected.” These reproduce with the narrow local test module; the tests, affected documentation and selected record are byte-identical to canonical main f626db5c3a3d781d8f7695038776c621651f75e1. No shared-code/navigation fix is bundled into this scoped mathematical contribution.

The registered normal/optimized package checks and the official committed-export replay passed locally as documented above. This is not a hosted scientific replay pass. The optional full-history preflight and the sparse-limited orientation command remain separately disclosed.