[petalas]

Summary

Scoped proof with a constructive opt-in implementation: share the encoded radix traversal in fresh_rows.quotient, retaining every division by the original small constant q. This answers Open next question 2 of Brian's #4794.

original: m^2 traversals, each containing w-1 small-q divisions
candidate: one traversal containing m^2 divisions at each of w-1 yields

The Count division sequence, returned tape, divisibility domain and full Count record invoice agree. Successful residue savings are exactly (m^2-1) rho(q,w,L), where rho is one complete validated radix traversal with cleanup. Divisor widths and peak simultaneous residue inventory are unchanged. ToikY #4025 supplies loop interchange at the directory port; the new step is the actual fresh-row quotient port and its complete local invoice.

Only research/r437-shared-radix-quotient/ is added. Produced by Nick Petalas. Credit David Arias #2753, ToikY #4025, ywz65 #2590/#2566, Chris R #4062, Tekalign #4077, Brian #4794 and the retained caller/stock/construction lineage in the README.

Evidence

• Before, q3/m2/w2 residue reads/writes/moves are (7256,1376,7456); after, (1814,344,1864). Both Count invoices are (495,111,456), including output disposal, and both residue peaks are nine with an unrelated dirty word present.
• Eight exact accepted binary-tape cases include q2/3/5, dimensions 1/2/3/64, exponents 1/2/3/4 and the real GL64 low factor at w1. Eleven rejected inputs preserve the first failed predicate and division trace. Four controls detect omitted/extra division, skipped validation and an undercharged retained traversal.
• Warning-strict normal and optimized local checks pass with identical generated source-bound receipts. Each registered command takes about 0.5 seconds locally and has a 60-second limit. Package metadata/replay passes; all 17 committed package/input files and their canonical source ancestry pass preflight. Official export integrity and normal/optimized read-only Git-free replay pass, with all 18 export files unchanged.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/r437-shared-radix-quotient
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/r437-shared-radix-quotient/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/r437-shared-radix-quotient/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/r437-shared-radix-quotient --check-source-history

Expected verifier marker: PASS_R437_SHARED_RADIX_QUOTIENT cases=8 rejections=11 controls=4.

Limitations

The harness supplies the consumed context attributes with actual Ring and Count implementations and exact GL cardinalities. Full Context construction, Machine.node, the downstream 9475 guard, Program/payload, recursion and admitted-root composition remain unrun. The interval includes quotient execution and returned-Count disposal; context/input construction and borrowed-state disposal lie outside it. Malformed modulus contexts are excluded, preserving the predecessor's reported cleanup limitation. No exception-time saving formula is claimed.

Exact source operations do not establish a physical-machine tariff, full supplier, new kappa or multiplication theorem. No predecessor package verifier, Lean build, Docker run or historical suite is run. Independent human and formal proof-kernel review are unclaimed. The written proof covers valid fixed contexts; finite checks corroborate that proof.

Related open work

• Certify batched R437 Count quotient and explicit setup tradeoff #4794 studies this same fresh-row quotient using the enlarged constant q^(m^2). This PR implements its explicit small-q shared-traversal alternative. No full paid crossover comparison is claimed; its reported numbers are cited rather than recomputed.
• Execute R437 nonzero-high leaf payload on two complete fibers #4793 executes the R437 nonzero-high leaf payload and consumer. It concerns the downstream payload obligation; this PR stops at the quotient port and computes no corresponding payload count.
• Prove native-history action coupling and reduce the QQ16 comparison #4172 studies native-history allocator/codec coupling and the QQ16 comparison. It is a separate caller-cost boundary and supplies no common quotient invoice compared here.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T21:49:35Z were refreshed. The nine other submissions, #4844, #4845, #4846, #4847, #4849, #4850, #4851, #4852 and #4855, concern center-output compression, a CRT batch bill, UT3 tariffs/couplings, fixed-factor workspace, weighted reversals, S3 rigidity, elimination downsets and block-four leakage. None studies this fresh-row quotient invoice or supplies a competing value for these fixtures. No additional numerical comparison was run.

Open next questions

1. Compose this adapter with an actual admitted caller, preserving the stock guard, complete payload and cleanup.
2. Price the small-q and large-divisor alternatives under one physical control, amortization and storage contract.

CI notes

Both hosted checks passed on exact head 2877e4920338d9a2e56aa960bdb5663f8f96b2d8 in run 39861270310: plan in 3m9s and registered package replay in 4m28s. The replay executes both warning-strict normal and optimized commands. Run metadata confirms that head and a successful conclusion. Local registered replay, committed-byte/source-history preflight and read-only Git-free export replay also passed. Hosted execution is bounded automated verification; independent human and formal proof-kernel review remain unclaimed.

Advance claim. Canonical source baseline is 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7, refreshed unchanged before publication. Sparse-note and unpublished-work coverage is incomplete.

Merge Danger

Door: two-way. Blast Radius: isolated research package. The adapter is opt-in; shared sources and the selected frontier are unchanged.