[petalas]

Summary

A scoped conditional all-rank proof answers codyw912's #4638 Open next question 1. Let d count primes in the target/paid rank alphabet and let r be the rational rank of its prime-valuation vectors. Independent prime coordinates give an injective projection of the multiplicative rank semigroup, sharpening the plateau count and the inherited width bound:

#4638: log B_N = O((log N)^(d+1))
Here:  log B_N = O((log N)^(r+1)), r <= d

For targets {6:1,36:1} and paid ranks {1:1,6:1}, the proved upper-bound degree decreases from three to two. This changes no compiler sequence and claims no matching lower bound. Only research/semigroup-rank-width/ is added. No new kappa or selected-frontier change.

This contribution was produced by Nick Petalas. Credit codyw912 #4638 for the theorem and explicit exponent-lattice question, Mihir Srivastava #4507 for chronological least batching, zxv #4436/#4483 and Jacob Alan Sussman for the compiler, Ryanson Jonathan and Cinna Davis for the retained flow/native lineage, and #4586/#4588 for earlier growth/checking context. Elementary linear algebra and unique prime factorization supply the projection argument. Predecessor results are cited, not audited or replayed as the deliverable.

Evidence

• Before: the published exponent box counts all d prime coordinates. After: the written proof uses r independent coordinates while retaining the entire all-prime source-count divisor.
• CPython 3.12.15 normal and optimized warning-strict checks pass 1196 fixtures, 7641 enumerated member occurrences, 186 fixtures with r<d, thirteen rejecting controls and two semantic comparisons. Semigroup multiplication agrees with an independent integer-membership recurrence; exact rational ranks agree with independent nonzero-minor enumeration. The {6,36} example has exactly seven semigroup elements through cap 46656.
• Each registered command took about 0.13 seconds locally, below its 120-second limit. Three disposable optimized-code mutations reject: insufficient coordinates, omitted products and a wrong width charge. In-memory static compilation passes.
• Registered metadata/replay and committed-byte/source-history checks pass for all six files at 6f87acd840f31a092238ecf7e5c455289430878a. The verifier generates a receipt binding its computed results and all six file hashes; no receipt is hand-written.

Exact reproduction from the repository root:

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

Substitute another CPython 3.12 path on another machine. The standard library is sufficient. A direct run can create a fresh hash-bound receipt with --record /absolute/scratch/path/receipt.json.

Limitations and unrun obligations

The all-rank conclusion rests on the written proof and #4638's published default-height source, contraction and divisor premises. Finite tests corroborate the new counting lemma. No predecessor scientific replay, selected supplier replay, proof-assistant build or independent human review was run. No actual selected inventory is claimed to have r<d. The abstract degree-reduction example is not a physical native supplier. Native semantics, allocation, dirty restoration, precision, stock, fixed-tape movement, uniform physical compilation and outer transfer remain inherited obligations. The selected five framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Related open work

No other open PR studying this default-height multiplicative-rank width refinement was found in the refreshed open list and targeted searches. The completion refresh read all seven other open account PRs created since the advance CLAIM: #5961 studies adaptive minor probabilities; #5963 cyclic register tariffs; #5964 rounded P104 error guards; #5965 an Eisenstein stage-count tail; #5966 theta-mask encoded-width construction; #5967 Clifford tensor thresholds; and #5968 four-call observation adapters. None computes this chronological width bound or a conflicting shared value. No extra comparison computations were run.

Adjacent #4192 studies sequential tape scans for a Beneš permutation compiler, rather than this chronological width recurrence; no shared numerical comparison is claimed. Merged #4638 is the direct predecessor. This account's merged #4892 proves fixed-library divisibility minimality and supplies no growth-degree bound.

Open next questions

1. Extend the counting refinement to the fixed adaptive-height theorem after adjoining M, preserving its decrement and nonemptiness hypotheses.
2. For a named default-height rule, retain source valuations and batch gcd cancellation to test whether the remaining logarithmic factor can be reduced.

The package records concrete first steps and blockers. Work stopped at the first verified degree refinement.

CI notes

At commit 6f87acd840f31a092238ecf7e5c455289430878a, both hosted checks passed in run 40538966780: plan in 6m35s and replay (research/semigroup-rank-width) in 1m23s. The completed check list was read once and the run head matches the PR head. The replay executes both registered normal and optimized commands. CI certifies the bounded automated checks; independent human and formal proof-kernel acceptance of the written all-rank proof remain unclaimed.

Merge Danger

Door: two-way.

Blast radius: isolated. Six new research-package files; no shared implementation or selected-frontier changes.