[petalas]

Summary

Every single four-or-more-contact A2 conjugation on the final PR5708 word, with corrections kept at their contact frames and the published route table fixed, has minimum emitted paid-query moment change exactly zero for every 0<t<1.

- PR6290 covers one to three contacts; four or more remain open.
+ All 1,503 longer-contact intervals: 1,450 missing routes, 39 strict losses, 14 neutral.

Claim type: finite certificate with a scoped algebraic sign proof. This answers the fixed-frame longer-contact part of unpaidmoon's PR6290 open question 2, at head 9d2916b07402aca2133ab315230b9c761cc9eff7. The earlier census is cited, not re-audited. No new kappa or supplier admission is claimed.

The new result and standalone certificate were produced by Nick Petalas. Credit unpaidmoon for the question and fixed-frame definition; Ali Romman for PR5708's word, row-sensitive emitter and A2 construction; Dugongue, ai-nanachi and all retained native/controller/helper/ordinary/stock sources. The elementary A2 identities and strict subadditivity are classical. No predecessor file or credit line changes.

Evidence

Before, longer-contact conjugations were an explicit uncovered domain. The new checker visits every equal-gate interval of the fixed 11,930-record scalar word without a contact cutoff and retains exactly those with at least four valid contacts. The new domain reaches 33 contacts. All 1,503 changed words preserve every one of the 685 independent scalar input columns.

Each rejection records its first missing ordered frame pair. Every emission has a reconstructed complete paid-query histogram. The seven nonzero difference shapes are strictly positive throughout 0<t<1 by positivity or strict subadditivity; the other 14 have identical histograms. The four-contact interval at zero-based scalar positions 9399 and 10815, gate ADD(560,315), attains zero.

Local CPython 3.12.15, macOS arm64, standard library only:

• Warning-strict direct normal and optimized checks pass in 5.379 s and 4.598 s. Both generated receipts are identical. Each registered command has a 120 s timeout.
• Both registered normal/optimized commands pass through the package runner, 8.931 s total, printing PASS_REGISTERED_RESEARCH_PACKAGE.
• All 519 controls pass, including 256 four-contact identities and 256 missing-correction mutants. Reversed contacts, erased operands, short intervals, a profitable signed shape and three routing cases discriminate the boundary.
• Committed-byte, source-history and executable-mode checks pass at 9064cb7abc4d8cbdee1d5d8b04c2992b97547d1e.
• Official export/input-integrity checks pass. A read-only optimized replay with an empty PATH and Git unavailable passes in 4.335 s; all ten exported files remain unchanged and its generated receipt equals both checkout receipts. A deliberately corrupted route input rejects at its SHA256 guard before the census.

The full generated candidate certificate has canonical SHA256 577323306b7b7f2547ca660e6bbef1badc953959e2f7cd20de9e1f002e254889. verify.py --output /absolute/path/outside/checkout/receipt.json regenerates it with all package and live-input hash bindings. Local receipts are evidence from these runs, not independent human or proof-kernel review.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/g20-long-conjugation-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/g20-long-conjugation-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/g20-long-conjugation-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/g20-long-conjugation-obstruction --check-source-history --check-modes

Any CPython 3.12 interpreter can replace the local absolute path. Expected direct marker: PASS_G20_LONG_CONJUGATION_OBSTRUCTION candidates=1503 missing_route=1450 emitted=53 neutral=14 strict_loss=39 min_contacts=4 max_contacts=33.

Limitations

This fixes the final PR5708 word, one conjugation, original correction frames and published routes. Missing routes are not geometric impossibility proofs. The result concerns the finite scalar endpoint and paid-query component, not the complete signed supplier invoice. Native/adjoint address action, the global caller, scalar allowance, target admission, finite compiler, precision, physical tape/storage and outer predicates were not replayed. The five framework assumptions and selected kappa 781451/250000000 remain. No uniform program, rounded multiplier or formal proof-kernel verification is supplied.

Merged PR5807 and PR5828 optimize scalar windows while preserving every query; their fixed points do not cover this changed-order conjugation class. Their changed words are outside this result. Main source pins, exact model, sign proof and bounds are in research/g20-long-conjugation-obstruction/PROOF.md, SOURCES.json and TASK.json.

Open next questions

1. Can one of the 14 new neutral rewrites expose a later paid saving? Generate one listed neutral word and enumerate its changed intervals, then price the complete emitted bill. This result permits one rewrite only; no neutral-neighbor search or caller admission is supplied.
2. Can a new native route or a moved correction frame rescue one rejected interval? Start from a recorded missing frame pair and construct its actual native action. Routes, targets, scalar fees, restoration and the complete caller must all be charged.

Related open work

• PR6290 studies the same final PR5708 word and fixed-route conjugation model at one to three contacts. This contribution answers its explicitly open four-or-more-contact case. Its existing counts and the source theorem are not rechecked as the deliverable.

• PR6306 proves a paid-moment floor for independently changed three-contact charts on a different retained-centre graph. This contribution changes scalar order on the final PR5708 G20 word. The instances and moment minima differ; no common numerical value is compared.

The completion refresh read the 15 other open account PRs created since the CLAIM, PR6298 through PR6313 excluding this PR. None studies the same final PR5708 longer-contact conjugation domain. No additional computation was run to compare another contribution's reported values.

CI notes

Both hosted checks passed at exact head 9064cb7abc4d8cbdee1d5d8b04c2992b97547d1e in run 41106842315: plan in 5m50s and replay (research/g20-long-conjugation-obstruction) in 1m52s. The completed checks were read once; separate run metadata confirms completed status, success and the matching head.

All local scientific, metadata, committed-source/history/mode and portable-export checks above also passed. Hosted replay certifies execution of the registered finite checks within their stated scope. Independent human and formal proof-kernel review remain unclaimed. No CI failure remains to report.

Merge Danger

Door: two-way. This adds one isolated research package.

Blast Radius: package. Shared code, frozen inputs and the selected frontier are unchanged.