[petalas]

Summary

Finite exact optimization certificate: the full forward carried-frame problem of #4994/#5016 has exact minimum

2201 + 412*2^p + 510*3^p + 127*4^p + 40*5^p + 8*8^p,

where p=4984891121/5000000000, so C=.0030217758. This includes all 1,339 flexible contacts and the 1,271-contact component left open by the earlier exact partial result. It answers the forward fixed-tariff part of #5091's stronger-relaxation question. No new kappa or selected-frontier change is claimed.

all legal frame assignments
  -> compatible endpoint-pair marginals
  -> 24,788 nonnegative columns / 15,524 equalities
  -> matching integral primal and exact symbolic weak dual

Evidence

• Before: Retain reproducible full-domain coupled frame screen (numerical, no new κ) #5016 supplied a numerical full-domain screen. Prove a fixed-source fractional price gap in the original full-contact relaxation #5091 showed why the original continuous relaxation could not prove the source optimal. Verify the exact small-component frame optimum and reject an all-concave extension (no new κ) #5068 already proved the restricted small-component optimum. Those are cited inputs, not replay targets.
• After: a genuine pair-consistent model has a checked integer-coefficient dual. Directed integer power bounds prove all 383 distinct nonzero reduced costs positive. Primal, dual and source objectives match coefficient by coefficient. An integral attainer changes 31 contact labels and satisfies every mapped edge.

From the repository root, using CPython 3.12:

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

All four local commands passed. The direct marker is PASS_PAIR_CONSISTENT_FRAME_OPTIMUM; both registered commands pass with warnings as errors and eight rejection controls. Each registered command takes under one second locally. Committed-byte/source-history preflight passed for all 18 package/input files. Official export integrity and eight-input coverage passed. The read-only Git-free export passed normal and optimized execution with an empty executable PATH, with unchanged bytes. The scientific verifier requires only the standard library and writes no files. Its stdout receipt is generated and binds package hashes.

Limitations

The result fixes the forward graph, scalar word, native maps, main/helper boundaries, owned-center contacts and this one historical tariff. Backward orientation and other exponents are unproved. The eight rank-eight center jobs are charged. This is a frame-bill optimum, not an arbitrary-word or complete physical-caller optimum.

The authenticated constructor and frame-contract definitions are inherited from #4971/#4994. No predecessor scientific verifier, native phase replay, selected cold replay, Lean or Docker suite ran. Physical delivery, precision, stock, compiler and all five selected framework assumptions remain separate. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Construct a separate exact certificate for the backward graph.
• Determine the exponent interval on which this dual is feasible. Verify the exact small-component frame optimum and reject an all-concave extension (no new κ) #5068 already rules out an unrestricted all-concave extension.
• Specify a changed word or boundary and its complete productive/paid invoice before further construction.

The package README and TASK.json give first steps and blockers.

Related open work

• Explore joint frame retiming after carried swaps (ongoing; no new κ) #4993 explores joint retiming with helper boundaries and owned-center contacts variable. This certificate fixes those contacts, so it does not exclude that broader search.

CI notes

At unchanged head df10a6d673cd78abdf1d001bb778c9c3a30df5e2, run 43024631324 failed the plan job in Onboarding helper tests, job 130536092134. The log reports 760 tests, exactly one failure and 24 skipped. The failing test is:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

Its AssertionError says 'make frontier7000-verify' was not found in the CI verification document. At canonical main e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json requires make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both files are inherited unchanged and are outside this PR's diff.

The same sole failing test, assertion and 760-test/24-skipped summary were verified directly in five other logs: 42905094314, 42903313790, 43023514635, 43024227699 and 43024464841.

The package replay job was skipped. CI did not execute or certify this package. The normal/optimized direct and registered scientific checks, committed-byte/source-history checks and Git-free export checks passed locally as recorded above. No fix, retrigger commit or job rerun was made; canonical documentation repair remains with the maintainers.

Credit

The new certificate and verifier were produced by Nick Petalas. Credit Cinna Davis / 1337isnot1337 #4994/#5016 for the interval model and numerical predecessor, JoyboyBrian #5091 for the stronger-relaxation question and original-model obstruction, Chris R / Clear1nsight #5068 and Krzysztof Wójt #5024 for distinct exact partial optima, Dugongue / David Ganz #4971 for signed carried swaps, Cao Yifan / intqwq for the round-three word, and the retained native/controller/compiler lineage. Source LICENSE and NOTICE files remain unchanged.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Adds one research package. No shared code, frozen predecessor, selected result or authoritative theorem is changed.