[petalas]

Summary

Prove exact polynomial productive widths for one homogeneous chronological compiler. Claim type: scoped written mathematical proof with exact finite corroboration. No new kappa or selected-frontier change.

W rank-m productive targets; P rank-one paid positions
one complement batch; separate rank-one exits
B_n = lcm(S_n, B_(n-s_n))
default height: B_n = W^floor(log_m n)
logarithmic adaptive height: B_n <= n^(log_m W + g log_M P)

The new step retains actual gcd cancellation, solving the existing width recurrence rather than tightening a generic count of rank products. It includes empty-root forests. At fixed height two with m=2,W=2,P=3, the rank-10 chain has source totals 8 and 18 and exact width 72; replacing LCM by maximum fails. The default formula is sharp on ranks m^h.

This contribution was produced by Nick Petalas. It takes codyw912 / PR4638, Open next question 2 of research/adaptive-chronological-batching. Credit Mihir Srivastava / PR4507 for chronological batching, zxv / PR4436 and PR4483 for the forest, Jacob Alan Sussman for integer compilation, and Cinna Davis and Ryanson Jonathan for their retained native/flow lineage. PR5962 and JoyboyBrian's PR6086 supply different semigroup/shell-count improvements; their results are cited, not audited. LCM arithmetic and positional notation are classical; no literature-wide priority is claimed.

Advance CLAIM. Scientific base and refreshed canonical main are 7c49fe3, priority revision 8, selected PR6030. Only the six-file research/homogeneous-gcd-width package is added.

Evidence

• Before, the source leaves actual gcd savings open and supplies general quasipolynomial/subexponential envelopes. After, PROOF.md derives the complete specialized source/cut list, exact chain-LCM solution, default closed form and polynomial adaptive divisor. This does not replace the generic mixed-rank theorem.
• A new literal ordered-word/paid-path enumerator agrees on 2,052 forests, including 1,422 empty-root cases and 139,536 enumerated leaf cuts. All exact rank invoices agree. No predecessor scientific verifier is imported or run.
• A literal-batch compiler checks 27 default width tables. The recurrence and chain expression agree at 121,440 coordinates across 480 default, fixed and adaptive tables. Twelve domain rejections, seven semantic control groups, four capacity controls and three disposable optimized-code mutations pass.
• CPython 3.12.15 on macOS arm64, standard library only. Warning-strict normal and optimized commands pass in 1.99s and 1.97s locally. The complete registered replay passes in 4.27s; each registered command has a 120-second limit. These are verification-duration observations.
• Committed-byte, executable-mode and ancestral-source-history checks pass at 9df1128. Official committed export integrity and declared-input coverage pass. Normal and optimized standalone replay also pass in a read-only Git-free export with empty PATH, identical generated receipts and all seven exported files unchanged. Receipts bind the six package files; no receipt is hand-written.

From the worktree root, using the exact local interpreter path or another CPython 3.12 installation:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/homogeneous-gcd-width/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/homogeneous-gcd-width/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/homogeneous-gcd-width
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/homogeneous-gcd-width --check-source-history --check-modes

Expected markers are PASS_HOMOGENEOUS_GCD_WIDTH, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The standalone checker writes no files.

Limitations and unrun obligations

The all-rank statement rests on the written proof and cited conditional chronological/native contracts. Finite arithmetic corroborates its stated domains. The symbolic inventory has one productive rank and only rank-one paid positions; the selected mixed-rank supplier is outside this theorem. The adaptive polynomial exponent is an upper bound, not an optimality claim. The default formula is exact only for this batching policy.

No native realization of this inventory, physical caller, finite generation-time bound, practical cutoff, storage extent, precision, stock or outer admission is provided. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. The selected cold replay, native suites, Lean and Docker were not run. All registered package checks ran locally.

Open next questions

• Bind one literal physical native word to this inventory and price its complete caller. A useful realized inventory is missing; polynomial width alone does not improve the selected exponent.
• Find a valuation invariant retaining gcd savings for one actual mixed-rank profile. The one-complement/rank-one-exit reduction fails there, so this proof supplies no general mixed-rank polynomial claim.

Related open work

No open PR computing this homogeneous chronological width sequence was identified in the refreshed live searches, main/merged searches and reservations. Cached overlap reported complete pagination for 555 open PRs and 6,599 discussion records. Sparse local notes, arbitrary historical diffs and unpublished work remain coverage limits.

• PR6384 studies source-capacity gates for a capped-flow compiler. This result solves a width-divisibility recurrence; it computes no competing capacity ratio or compiler saving.
• PR6390 sharpens higher-moment integer split charges. This result retains the chronological batching policy and computes no split-charge optimum.

The completion refresh inspected the 15 other open contributions created since the advance CLAIM, PR6605 through PR6611 and PR6613 through PR6620. None studies this homogeneous chronological width sequence or computes a conflicting value for the rank-10 example. No extra computations rechecked another contribution.

CI notes

Both hosted checks, plan and replay (research/homogeneous-gcd-width), passed at exact head 9df11284bcc652a272014f8d19ee7e0bc642ed6d in run 41634549541: plan in 4m30s and replay in 1m32s. The completed checks were read once. Run metadata confirms success and the matching head.

Hosted replay certifies execution of the registered finite checks and metadata checks, not independent mathematical or proof-kernel acceptance of the written theorem. All local registered, publication and export checks above passed. The completion refresh found no competing calculation of this width sequence; the limitations above remain.

Merge Danger

Door: two-way. Blast radius: isolated.

Six new package files. Shared implementations, frozen predecessor evidence and selected records are unchanged.