[petalas]

Summary

Measured verifier improvement with a scoped equivalence proof and finite conformance evidence. An exact integer subset join reduces the complete positive-frame replay's median elapsed time by 71.4%, from 29.031 s to 8.290 s, in five alternating fresh-process pairs. All ten commands emit identical complete output with zero errors. No new kappa is claimed.

original: construct every rational six-label subset sum
new:      clear all label and target denominators
          join four-label and two-label integer subset sets
          retain the complete split inventory and all other checks

Only research/positive-frame-subset-join/ is added. Its adapter authenticates Josie Elliston's frozen PR #6111 checker and replaces one concrete support calculation in a fresh module. The proof covers repeated labels, signed coordinates, zero counts and different target denominators. The predecessor retains its theorem, checker, inventory and controls.

Evidence

• Five complete baseline commands take 28.521–30.837 s, median 29.031 s; five accelerated commands take 8.190–8.394 s, median 8.290 s. CPython 3.12.15, macOS arm64, 18-core shared host. Timing includes startup, source authentication, every check, output and exit. Generated BENCHMARK.json binds the code and records every return code and output hash.
• A separate cProfile run places 84.219 of 102.461 instrumented seconds in the concrete scan, dominated by Fraction operations. Instrumented times are excluded from the comparison.
• All four target decisions on 46,916 concrete splits agree with a separately written literal Fraction enumeration. The new verifier also checks 420 small rational cases and seven input/source rejections, then runs the complete accelerated source command. These are accelerator-conformance cases, not a new predecessor census.
• Direct warning-strict normal and optimized checks pass in 31.33 s and 32.96 s. Both registered commands pass; the registered invocation takes 65.21 s. Each check is below its 300-second timeout.
• Metadata, AST parsing, exact committed bytes/modes and source-history ancestry pass at 3d452329bf4e4893c0a5205eecd61c3cd6c39cb4. Official export integrity and an optimized read-only, Git-free replay with empty executable PATH pass. That replay takes 32.45 s and preserves all 16 exported files, including the manifest. Export SHA256: 2c0ffa5370ce7f829f5a0d617eb1b3418bdf20200f7cb16c55c0cbecfd10d168.

From the repository root, with the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/positive-frame-subset-join/run.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/positive-frame-subset-join/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/positive-frame-subset-join/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/positive-frame-subset-join
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/positive-frame-subset-join --check-source-history

Any CPython 3.12 installation may replace the interpreter path. Expected marker: PASS_POSITIVE_FRAME_SUBSET_JOIN_CONFORMANCE no_new_kappa. Add --baseline to run.py for original predicates. Optional timing reproduction:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/positive-frame-subset-join/benchmark.py --pairs 5 --output /tmp/positive-frame-benchmark.json

Limitations and unrun obligations

This is a single-host measurement and a scoped equivalence result for the exact pinned checker. Unknown source revisions reject. Generic inputs have six rational triples, four targets, total multiplicity at most eight and explicit bit bounds. Managed-allocation and fixed authenticated module-execution deviations are recorded in PROOF.md.

No new positive-frame theorem, native word, supplier, multiplication bound or selected-frontier change is supplied. Native/compiler/ownership/precision/analytic admission, selected scientific cold replay, Lean and Docker were outside scope and were not run. Every registered local command passed. The predecessor's scientific review status is inherited; independent human and formal proof-kernel review are unclaimed.

Open next questions

Can a separate integer formulation reduce the unchanged symbolic support-classification stage? First profile the complete accelerated command and specify that stage's exact return-value contract. No implementation, measured gain or follow-on reservation is included here.

Related open work

Completion refresh on 2026-10-29 inspected all 13 other open account PRs created since the advance CLAIM at 13:10:51 UTC. None studies this exact checker or added-positive-frame endpoint, changes its implementation, or supplies a competing result. No conflicting number was found and no extra mathematical computations were run.

Adjacent performance work has separate commands and workloads:

• Reduce complete replay planning time by 8.27% (no new κ) #6177 reduces complete replay-planning time. Its planner measurement is separate from this scientific-checker measurement; no gains are combined.
• Reduce current task-picker startup time by 4.66% (no new κ) #6179 reduces current task-picker startup time. It changes a different entry point and supplies no competing checker result.

The earlier refreshed 330-PR snapshot and immediate precommit query for #6111 also found no same-object open work. Merged #6111 is the source workload; merged #6148 and #6158 study separate mathematical extensions and are not replayed or extended here.

Base and refreshed main are 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8. Live claims, reservations, recent account work, main README terms and merged searches were checked. Remote overlap pagination was complete; sparse local-note coverage was incomplete. This is bounded coordination evidence.

Credit

This contribution was produced by Nick Petalas. Credit Josie Elliston's #6111 for the theorem, source checker, split inventory and controls; JoyboyBrian's #6104 for the two-frame question and parameterization; apoursh's #6089 for the classification and symmetries; Cinna Davis for the PR2085/2112 model; Philippe Van Velzen for the earlier floor; Nick Petalas for the determinant method; and the retained BIT/formal-model lineage. Denominator clearing and subset joining are standard techniques. The new work is their source-bound integration, equivalence argument and complete-command evidence. Predecessor notices are unchanged. Apache-2.0.

Advance CLAIM.

CI notes

Both hosted checks passed at 3d452329bf4e4893c0a5205eecd61c3cd6c39cb4 in Registered research packages run 40883034303. The plan check passed in 4m43s and replay (research/positive-frame-subset-join) passed in 5m24s. This certifies the registered normal/optimized conformance replay at the published head. All listed local checks also passed.

Hosted conformance does not certify the local performance percentage or provide independent human or formal proof-kernel acceptance. No code changed after these checks.

Merge danger

Door: two-way. Blast radius: package. One isolated opt-in package is added; shared code, frozen inputs and the selected result are unchanged.