[petalas]

Summary

Scoped written proof with exact finite corroboration: five rank-at-most-one address-twist contributions have residue rank at most two for every m >= 4, and five nonzero frames attain two at every such dimension. This answers the m >= 4 rank-one part of Open next question 2 in the older #5978. No new kappa is claimed.

five nonzero frames with zero sum
  span 4 -> all transports coincide -> residue zero
  span 3 -> a cancelling triple and pair -> residue zero
  span 2 -> multiplicities 3,1,1 -> credited four-term rank-two bound

The model is V=M_m(F2), U_j(z)=L_j z R_j with invertible L_j,R_j, sum a_j=0, and sum a_j*(U_j z)=0 in Sym^2(V) for every z, with polynomial squares retained. The residue is beta=sum U_j^-1(a_j). Zero frames reduce directly to the cited four-term theorem. Attainment splits one frame of its rank-two witness into two nonzero rank-one frames with the same transport.

This contribution was produced by Nick Petalas. JoyboyBrian #5490 supplies the dirty degree-one conditions and transport rank-distance lemma. Nick Petalas #5978 supplies the four-term theorem, field witness and five-term question. Krzysztof Wójt #5003 supplies the inherited second-order quotient and source compatibility; the earlier Cinna Davis, Ali Romman, eumemic and huxint lineage remains credited. Those results are cited premises, not replay deliverables. Only research/bit-five-term-residue-bound/ is added.

Evidence

Before, #5978 leaves five or more contributions open. After, the written case analysis proves the sharp rank-two bound for five rank-one contributions at m >= 4.

Local CPython 3.12.15, standard library only, at e7b75e61e129c5142840e7f7a072f185d26337ef:

• Enumerate all zero-sum multisets of five nonzero vectors spanning F2^d, d=1,2,3,4. Counts are 0,3,28,168. Check the types and every pointwise kernel-basis syzygy for all 199 multisets in ambient dimension six. The written proof supplies the universal quantifiers.
• Check new five-frame rank-two witnesses at m=4,...,8 on every input basis matrix, all frame ranks and inverse transports. Omitted-term, changed-frame and changed-transport controls reject; the square-retention control passes.
• Warning-strict normal and optimized direct checks pass, each timed locally at 0.05 seconds. The generated hash-bound receipts agree. Both registered commands and metadata/pins pass.
• Committed bytes/source ancestry, eight-file export integrity and normal/optimized read-only Git-free export replay pass. Exported proof corruption and receipt-overwrite controls reject. Syntax and function-length checks pass. The worktree is clean.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/bit-five-term-residue-bound/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/bit-five-term-residue-bound/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/bit-five-term-residue-bound
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bit-five-term-residue-bound --check-source-history

Expected markers: PASS_FIVE_TERM_RESIDUE_BOUND, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. On another machine use its CPython 3.12 executable. Default verification writes nothing; optional --receipt requires a fresh external path.

Advance CLAIM. Canonical main refreshed before publication remains e54523517d0f03e58352e647c403a5af3e525a57; selected #7000 and affected guidance/source files are unchanged. The card has changed scope from a native alphabet search to this local obstruction. Overlap checks covered current main, open and merged PRs, live claims and the older question's recent use. This is provisional overlap evidence, not a novelty guarantee.

Limitations

The result concerns one block's necessary equations. It does not exclude complete words, other blocks or direct diagonal contributions. Five-term m=2 and m=3 cases, arbitrary frame ranks, constant-input second-order cross terms, native twist costs, physical realization, compiler/precision/tape obligations, a recursive invoice and the complete multiplication theorem remain outside scope. No selected-frontier change, predecessor scientific replay, Lean, Docker, independent human or formal proof-kernel review is claimed.

Open next questions

• Can five rank-one frames have residue rank three at m=3? Start with a rank-one five-circuit in a four-dimensional frame span; rank distance no longer identifies transports in that branch.
• What is the first contribution count permitting full-rank residue for one named m>=4? Begin with six-frame multiplicities, then impose every remaining block, cross term and physical charge before word synthesis.

Related open work

No open PR on the two-sided address-twist degree-one residue was found in the publication search. #5490 and #5978 are merged cited predecessors. Concurrent polynomial two-gap #7191 studies a different paired-star insertion contract and supplies no value for this residue.

This account's open PRs created since the CLAIM at 2026-11-11T09:58:44Z were rechecked before DONE. Apart from this PR, the submissions concern Gaussian reader coefficients, paired-star polynomial insertions, sparse rounded-inverse lattices, three-shear COPY floors, cyclotomic precision, coupled Jordan-block optima, compact switches and query-cache stock. None computes this address-twist residue quantity; no additional scientific computations or numerical comparisons were run.

CI notes

CI did not certify this package. At head e7b75e61e129c5142840e7f7a072f185d26337ef, run 43000937833, plan job 130456595356 failed in Onboarding helper tests. Its log reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

The AssertionError says that 'make frontier7000-verify' was not found in the CI verification document. At canonical main e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names that command, while docs/ci-verification.md still describes selected #6030 and make frontier6030-verify. Both files are unchanged by this PR; the complete diff adds only this package. Package replay was skipped after plan failed.

The failed-step logs were also read for run 42905094314, run 42903313790, run 43000211746, run 43000676071 and run 43000790178. Each reports the same sole failing test and missing-command AssertionError, with 760 tests and 24 skipped. This confirms the shared repository-level failure; no job was rerun and no fix or retrigger commit was pushed. Local package evidence above remains the available verification.

Merge danger

Door: two-way. Blast radius: one isolated research package and its opt-in replay. No shared implementation or selected record changes.