[petalas]

Summary

Add an opt-in exact backend for the selected P104 scalar transpose checkpoint. Claim type: verifier-engineering improvement with a scoped correctness proof and finite exact corroboration. No new κ or selected-frontier change.

bound every dyadic numerator row
choose an injective integer radix
replay whole packed rows and compare every endpoint coefficient

On the frozen PR5599 word, five alternating fresh-process pairs reduce the median complete checkpoint command from 0.5282 to 0.1770 seconds, a 2.98× speedup. The checkpoint function itself falls from 0.4840 to 0.1301 seconds. This is one endpoint checkpoint, not a whole-supplier replay speedup. All existing source files and official reproduction commands remain unchanged.

The bounded dyadic trace planner, new backend, independent behavior checks and measurements were produced by Nick Petalas. Credit Ali Romman #5599/#5544/#5545, Pierre-Loic Doulcet #5565, David Arias, Cinna Davis, Gareth Davidson and the retained P104/native/controller/compiler lineage. Classical Kronecker substitution is not new mathematics. Sagar Thakrar's #1781 already applies packed integer matrices and signed-radix injectivity to Clifford recipes, following the retained Lean comparator. This contributes the new arbitrary-dyadic scalar-trace planner/application, including COPY and zero-row operations; it does not claim that predecessor method.

Advance CLAIM. Base canonical main 19fdb086ea96af9a9f87936a2b888e10af4ed810, priorities revision 8, unchanged at submission refresh. The autonomous-run playbook stopped at the first verified improvement.

Evidence

• Before: the dense checkpoint repeatedly scans individual column coefficients, including 5,492,230 parity-generator calls in a separate profile. After: a proved l1 bound makes one exact integer per row an injective representation. No probabilistic fingerprint or cached PASS result is used.
• Independent chronological Fraction matrices pass 4,841 words and all 22,804 literal coefficients. All 4,841 altered endpoints reject; fourteen further controls cover domains, growth/memory bounds, an unsafe-radix collision and an altered complete-word endpoint.
• The retained workload has 25,475 events, 756 columns and 571,536 endpoint coefficients. The planned radix is 111 bits. The new endpoint checkpoint returns the unchanged original summary.
• Local CPython 3.12.15, macOS arm64: warning-strict normal and optimized direct checks pass in 0.93 and 0.96 seconds; the complete registered invocation passes in 2.00 seconds. Each registered command is below its 120-second limit. Ruff E/F/I/B/UP/SIM/RUF, formatting, syntax/function bounds and staged whitespace pass.
• PASS_RESEARCH_COMMIT_BYTES validates all eleven package/input files and source ancestry at 2380159a2dc8c37ed51a4ef7c77ec0a212b7fc7d. Official export and declared-input integrity pass. Normal/optimized Git-free replay with empty PATH and read-only files agrees exactly and leaves all twelve exported files unchanged. Two disposable mutations reject changed input bytes and a changed recorded benchmark result.
• BENCHMARK.json is generated by the committed harness and bound to input/code hashes. All ten fresh processes completed the same checkpoint with zero errors. Dense complete-command range is 0.5242–0.5317 seconds; packed range is 0.1657–0.1779. Median process peak RSS rises from 41,222,144 to 46,465,024 bytes. Other work was running on the 18-logical-CPU host; samples alternate sequentially and none is discarded. Profiles identify row arithmetic as the remaining packed cost, with syntax/Fraction conversion and bound planning also payable.

From the repository root, use the exact interpreter path below or another CPython 3.12 installation:

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

The last command is an optional new five-pair timing experiment. Registered checks certify exact behavior and the existing generated receipt's bindings/consistency; they do not repeat timing. Sparse checkouts need this package and its two declared inputs under research/p104-helper-subspace-splits.

Limitations and unrun obligations

The backend checks the complete scalar transpose projection. Frames, erasure legality, ownership, native phases, physical callers, pricing and compiler conditions remain the surrounding verifier's responsibility. No new audit or admission of PR5599 is claimed. The new backend remains opt-in and is not installed into a complete supplier replay.

Explicit limits are 1,024 roles, 60,000 scalar events, 4,096 bits of planned growth and 128,000,000 packed coefficient bits. Python allocation is documented as a bounded managed-runtime deviation. Linux timing, the full selected cold replay, rounded/native execution, stock, tape, analytic recovery, Lean and Docker are unrun. No whole-multiplier theorem, physical runtime improvement or improved κ follows. Independent human and formal proof-kernel review are unclaimed. Every registered check ran locally; none is deferred solely to CI.

Open next questions

Install the backend at one complete successor replay's literal transpose call site while retaining all surrounding controls, then compare the complete scientific output. The missing integration and whole-replay measurement are explicit; no additional tuning or construction search is included here.

Related open work

• Add all-width native certificate for the selected #5599 word (native_all_columns; no new kappa) #5974 checks the selected PR5599 native maps at all widths. This accelerates its scalar transpose projection and makes no native-map or all-width claim.

• Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994 searches helper-split capacity on PR5907. This changes no helper inventory and computes no split capacity.

• Compress the ordinary events of the PR5907 P104 word: 25,998 to 25,861, same invoice #5975 compresses the PR5907 scalar word. This preserves the frozen PR5599 word and changes the checkpoint implementation.

• Save twenty P104 scalar units and two events through dirty-output sharing (no new κ) #5999 changes a PR5907 scalar window. This emits no changed word and computes no scalar allowance saving.

• Save six more scalar units after P104 sign normalization (no new κ) #5969 removes a scalar ADD on PR5599/Simplify the accepted P104 word by 974 scalar primitives per call #5750. This checks the unmodified PR5599 endpoint and supplies no cumulative circuit saving.

• Bound rounded error on the two #5833 P104 words: 20-bit guard, envelope below 97B (no new κ) #5964 bounds rounded errors on PR5833 words. This concerns exact dyadic endpoint equality and supplies no rounded-error guard.

• Classify P104 lattices invariant under local Clifford gates (no κ) #5971 classifies invariant P104 lattices. This changes no grid or scalar alphabet and claims no lattice classification.

• Independent replay, invoice, native and funding audit of #5827 and #5907 (no new κ) #5960 audits the PR5827/PR5907 chain. This supplies a new backend, without auditing that chain or transferring its conclusions.

• Conditional κ 0.00309871 from a prepared P104 split helper #5606 changes a P104 helper's storage and invoice. This preserves every helper and paid event.

• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 changes helper lifetimes on an older P104 word. This changes no lifetime or boundary.

• Prove one local rank-5 deletion per FULL lifts the selected complex invoice to the BIT saving (no new κ) #6010 prices hypothetical changes to the selected invoice. This changes no recursive invoice and computes no threshold.

• Certify six-bit stochastic precision for the selected P104 prefix (no κ) #6026 studies stochastic precision on the same selected P104 word's first scalar prefix. This backend checks its exact complete transpose endpoint and computes no stochastic error or precision-bit certificate.

• Prove exact chronological helper spill and repair rank (no new κ) #6019 determines chronological helper-replacement spill and repair rank. This preserves the source word and makes no replacement or repair-rank claim.

• Prove sharp two-role dyadic helper-basis conditioning (no new κ) #6016 optimizes a two-role dyadic helper basis. This fixes the complete scalar word and computes no condition-number optimum.

• Exclude finite-menu adaptivity for exact helper endpoints (no new κ) #6015 excludes selection among finite menus of linear helper endpoints. This evaluates one fixed endpoint and proves no menu lower bound.

• Prove zero-preserving polynomial helpers cannot reduce linear call minima (no new κ) #6017 extracts linear call profiles from polynomial helper words. This changes the endpoint representation used by a verifier, not the word or its paid calls.

• Classify fixed nonlinear role recodings under exact ADD compatibility (no new κ) #6025 classifies nonlinear role recodings compatible with exact ADDs. This uses the existing linear coordinates and claims no nonlinear recoding result.

• Give exact fixed-weight packed-bit repair counts (no new κ) #6013 counts fixed-weight repairs in a packed-bit gadget. Its digit-gadget probability differs from this exact packed matrix representation; no shared repair count or timing is computed.

The completion refresh listed all fifteen other open PRs from this account created since the advance CLAIM at 2026-10-27T21:57:29Z. The adjacent scopes are distinguished above. None reports a competing timing for the pinned transpose checkpoint, and no additional computations were run to compare another contribution's numbers.

No additional computations were run to compare those results. Merged #5757's native quadratic-state acceleration retains the old transpose checkpoint and is complementary; its complete-replay speedup was not rerun. Main README, merged-PR, live-claim and local reservation searches found no competing packed dyadic transpose backend within their stated coverage. Those searches do not cover unpublished work or prove historical novelty.

CI notes

Both hosted checks passed at exact head 2380159a2dc8c37ed51a4ef7c77ec0a212b7fc7d in run 40592283445: plan in 4m36s and replay (research/packed-dyadic-transpose) in 2m15s. The completed PR check list was read once. Run metadata confirms completed success and the same head as the PR.

Hosted replay exercises both registered normal and optimized correctness commands, including source and generated-benchmark bindings. It does not repeat the timing experiment or establish a Linux speedup. All recorded local checks, committed-byte/source-history checks, export integrity and read-only Git-free checks also passed. The scoped proof, local performance claim and unrun physical/full-supplier obligations retain their stated limits; independent human and formal proof-kernel acceptance remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Nine new files in one opt-in research package; shared code, frozen sources and selected-frontier records are unchanged.