[petalas]

Summary

Scoped mathematical proof with a finite exact certificate: the frozen joint BIT/complex role ledger from Matarener's #5298 needs exactly 406 disjoint cross-side paid-job merges to pass the complete signed moment at C=7/2000. This is a new accounting threshold, with no physical supplier or new kappa claimed. This contribution is produced by Nick Petalas.

sort each paid list in descending order
select the largest k ranks on each side
pair corresponding ranks and replace each pair by its sum
retain every productive target and every unmerged paid job

Monotonicity and a supermodular rearrangement proof make this pairing optimal for every 0<alpha<1. At alpha=1993/2000, every matching with at most 405 pairs fails. A 406-pair witness passes: 222 pairs (72,63), two (72,56) and 182 (64,56).

Evidence

• Before: Joint bit+complex supplier cannot beat the average of the two p16 invoices under the whole-job premise (negative result) #5298 leaves cross-side merging outside its whole-job premise and gives example merge models. Those source results are cited, not audited or replayed.
• After: exact rational bounds give optimal margins [-0.197285824316,-0.197285824315] at 405 pairs and [0.088428560458,0.088428560459] at 406. Every target and unmerged paid rank stays charged.
• Both registered commands and direct warning-strict normal/optimized runs pass on CPython 3.12.15/macOS arm64. The commands took 2.45 and 2.63 seconds locally. The replay enumerates 13,200 partial matchings over 1,200 small cases and checks eight controls.
• Independent integer-power comparisons certify all 145 needed rank bounds and separately certify both moment signs. The verifier prints a generated receipt bound to package hashes.
• Metadata/source pins, exact committed bytes, source-history availability, official export integrity, and both read-only Git-free empty-PATH exported replays pass. AST parsing, function lengths and git diff --check pass. No separate linter was available on PATH.

From the repository root, use the recorded interpreter or an available CPython 3.12 path:

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

Expected marker: PASS_JOINT_CROSS_MERGE_THRESHOLD with minimum_merges: 406.

Limitations

The result is sharp only for the stated optimistic disjoint-pair model and exact historical ledger. All roles are eligible, each original job appears in at most one pair, and merged jobs cannot merge again. Restrictions on legal pairings or additional fees can consume the positive margin. Changed targets, splitting, repeated merges and changed invoices require a different calculation.

No joint native word, physical merge, dirty-state restoration, compiler/controller, tape bill, all-size theorem or supplier admission is supplied or checked. The benchmark 7/2000 is not a current admission threshold. The selected frontier stays unchanged. Independent human and formal proof-kernel review are unclaimed.

Sources and credit

Matarener's #5298 at e63b0a32ed2c021a8f4344e5aeb7e9123ff2eaef supplies the full role ledger, whole-job boundary and illustrative merge models. Its RESULT.json is retained verbatim and authenticated by SHA256, so no pending Git object or network fetch is needed for replay. James Reed/jwr404 supplied the scope correction; zxv's #4470 and ai-nanachi's #4814/#4760 lineages supplied the side invoices. Ali Romman's #5341 crossing results are separate cited context. The retained Zhihao Chen, RaD/hipotures, icekylinx and eumemic controller and moment work keeps its credit. The classical rearrangement method itself is not claimed as new.

Related open work

• Joint bit+complex supplier cannot beat the average of the two p16 invoices under the whole-job premise (negative result) #5298 studies the same frozen joint invoice under the whole-job premise and supplies nonoptimal illustrative merge variants. This contribution changes that premise to disjoint cross-side pairs and proves the exact minimum at a fixed benchmark; it does not compare or re-check the predecessor's root values.

• Prove the sharp query-price threshold for the #5554 path reduction (no new κ) #5936 changes the query tariff in Research: the productive-helper path (#4440 lineage) can't pass κ ≈ .00304 without cheaper helpers #5554's separate productive-helper path invoice. This package fixes Joint bit+complex supplier cannot beat the average of the two p16 invoices under the whole-job premise (negative result) #5298's joint BIT/complex ledger and changes only disjoint cross-side merging; no computed quantity or numerical comparison is shared.

The post-CI scan of this account's open PRs created since the advance CLAIM found no additional PR studying this frozen joint invoice.

Open next questions

• Can one native child realize a witness (72,63) pair with both arbitrary payloads, required returned states and every fee? Begin with one complete producer/consumer contract. That physical contract is missing.
• Can an eligibility graph derived from an actual joint word still support 406 effective merges? Derive its legal edges and fees first, then solve the finite weighted matching problem. No physical edge relation is supplied here.

CI notes

Local checks and hosted CI pass at ea2153947df50738577d511c29fff016bd8f1366. Both plan and replay (research/joint-cross-merge-threshold) passed in run 40523722699. Hosted CI confirms the registered package checks; it does not supply independent scientific or proof-kernel acceptance.

Merge danger

Door: two-way. This adds one isolated research package and leaves shared implementation and the selected record unchanged.

Blast radius: package. The finite accounting threshold does not admit the predecessor or a joint native supplier.