[petalas]

Summary

A scoped algebraic construction with exact finite corroboration reduces the concrete nonunit helper handoff of #6322, Open next question 2:

- 26 ADDs; four-boundary scalar allowance 272
+ 22 ADDs; four-boundary scalar allowance 232

Two ordered, noncommuting preconditioners cancel both expensive row coefficients and one column entry. The new source/cut completions cost 62/54; both inverses are charged. The 22-ADD count matches the predecessor's separately completed boundary minimum. No weighted optimum or new kappa is claimed.

The complete new historical local word has 25,162 events, scalar allowance 217,660, 736 independent public inputs, eight owned lifetimes and peak owned storage one. It retains the rank-9 to rank-5-plus-rank-4 split and complete recursive paid/target profiles. Only research/p104-double-preconditioned-handoff/ is added.

This contribution was produced by Nick Petalas. Credit CubeJerry #6322 for the dual pairs, column-unit formula, ADD-count bound, 272-unit alternative and second arithmetic implementation; unpaidmoon #6297 for the concrete split; Ali Romman #5611/#5602 for the completion/handoff contract; and David Arias, Cinna Davis, Gareth Davidson and the retained P104/native/controller lineage. Predecessor files and notices remain unchanged. Apache-2.0.

Evidence

• Before: Cut the concrete P104 nonunit handoff from 140 to 22 gates (no new κ) #6322 reports 22 ADDs/280 units and 26 ADDs/272 units. These are cited comparison premises, not a predecessor audit. After: explicit new 22-ADD/232-unit programs satisfy both exact dual-pair equations and full inverses.
• Both retained arithmetic engines replay the new word on all 744 formal columns and 553,536 endpoint coefficients. Rational sparse rows and integer rows with binary denominators agree on the full endpoint, geometry, ownership and paid/target/scalar invoice.
• Eleven boundary omissions, two reversed-preconditioner-order controls and eight whole-word coefficient/chart/ownership/boundary faults reject. Generated stdout receipts bind source/package hashes and the candidate word.
• CPython 3.12.15, macOS arm64: warning-strict direct normal/optimized checks and registered replay pass. Each registered command was timed locally at about 1.4/1.2 seconds, below its 120-second limit. These are replay durations, not a runtime improvement claim.
• Exact committed bytes, executable modes and source ancestry pass at 9336635519f0358eb38c06a877a030268a275b14. Official export/input integrity and final Git-free, empty-PATH normal/optimized replay pass with 13 read-only files unchanged. An actual changed source byte in a separate export rejects before mathematical execution.
• Main was refreshed to 7c49fe32c55de02c2c873655c804b16e4c3fef0a; source pins resolve there. Current main/merged searches and the live open-PR corpus found no exact allowance result superseding this construction. Sparse local-note and unpublished-work coverage remain limits.

From the contribution checkout root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/p104-double-preconditioned-handoff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/p104-double-preconditioned-handoff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-double-preconditioned-handoff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-double-preconditioned-handoff --check-source-history --check-modes

Any CPython 3.12 executable may replace the absolute local interpreter path. Expected markers are PASS_P104_ORDERED_HANDOFF inserted_events=22 inserted_allowance=232, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

This is a historical scalar boundary improvement. No current-word transfer, weighted optimum, native phase/implementation, complete physical caller, stock, rounded precision/tape, compiler closure or outer admission is established. Those obligations are unrun or inherited. All five framework assumptions and the selected frontier remain unchanged. No complete multiplication theorem, independent human review or formal proof-kernel acceptance is claimed. The ordinary scalar saving does not alter the recursive power invoice.

Open next questions

• Can allowance fall below 232 under the same dual-pair contract? Start with a coefficient-sensitive lower bound including both inverse boundaries. The ADD-count bound does not decide weighted optimality.
• Does a current source retain a compatible nonunit handoff? Bind its exact prefix/suffix pair, available charts and complete native/caller bill before transferring this historical saving.

Related open work

No inspected open PR targets this exact historical nonunit pair or its coefficient-weighted allowance. Adjacent P104 supplier work has different changed boundaries:

• Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375 lifts six inputs through shared carriers and changes the recursive invoice; this package changes scalar completions at the fixed historical pair.

• Conditional κ = 0.00315087 (+3.0e-7 over #6353) #6372 contracts a joint contact block with changed native transports and productive ranks; this package retains its predecessor's paid and target profiles.

• Contract nine native helper outputs at conditional κ 0.00315057 #6353 contracts nine helper outputs; this package supplies no output contraction or comparison of its kappa.

• Compose native input lifts with the query tree at conditional κ 0.00314980 #6350 composes native input lifts with a binary query tree; this local scalar construction does not change that caller.

• Prove a paid-rank floor of 22 for three-carrier rank-two tails #6386 bounds paid frame rank for a fixed-code three-carrier merge; it computes no scalar completion allowance for these dual pairs.

• Prove exact tariffs for carriers visiting distinct entrance lines (no κ) #6388 prices a carrier's route through several entrance lines; its frame tariff differs from this coefficient-weighted scalar tariff.

• Prove exact signed idle completion for overlapping stage outputs (no κ) #6391 completes overlapping stage demands with signed idles; it changes a caller interface rather than this historical local basis.

• Prove fixed-prefix P104 tails are optimal for every concave tariff #6394 optimizes fixed-prefix P104 delivery tails under concave tariffs; its fixed-term path inventory is separate from these scalar completions.

The completion refresh inspected all 12 open account PRs created since the advance CLAIM, including this PR. None studies the same historical dual pairs or computes their scalar allowance. No shared-instance numerical disagreement was identified, and no extra computations audited another contribution.

CI notes

Both hosted checks, plan and replay (research/p104-double-preconditioned-handoff), passed at exact head 9336635519f0358eb38c06a877a030268a275b14 in run 41512429938. The completed checks were read once with gh pr checks 6392 --repo CrocSwap/integer-mult-bounds; run metadata separately confirms the matching head and successful completed conclusion. Local registered checks also passed. Hosted replay certifies execution of the registered finite checks and source bindings. The algebraic construction retains its written proof and stated limitations; independent human and formal proof-kernel review remain unclaimed.

Advance CLAIM.

Merge Danger

Door: two-way. Blast radius: isolated.

Eight new package files and opt-in replay only. Shared code, predecessor evidence and selected records are unchanged.