[petalas]

Result

Scoped written proof with an exact finite certificate. No new κ. This answers the four-contribution case of Open next question 1 in JoyboyBrian's #5490 (research/bit-three-term-twist-rigidity): can four or more off-diagonal contributions survive the dirty degree-one condition?

Take the conditions of #5490 equation (1) for one block: Σ a_j = 0 and Σ a_j·(U_j z) = 0 in Sym²(V), with V = M_m(F2) and two-sided transports U_j(z) = L_j z R_j. Let β = Σ U_j⁻¹ a_j be the role-s residue. For four contributions:

1. m = 2: β = 0 for all frames (finite certificate, exhaustive over all 46,656 normalized transport triples, all frames, not only rank-one ones).
2. m ≥ 3: β ≠ 0 only if a₁ = a₂ = a₃ = a₄ = x ≠ 0 and U₁+U₂+U₃+U₄ = 0 (written proof). In that case rank β ≤ 2·rank x.
3. Sharpness for every m ≥ 3 (written proof, checked for m = 3–6): multiplication by elements of F_(2^m) gives four distinct transports and a rank-one x with rank β = 2.

So #5490's three-term rigidity does not extend to four contributions once m ≥ 3. The escape needs equal frames mod 2 and a four-term transport relation, and the residue has rank at most 2.

The proof of part 2 uses #5490's rank-distance lemma, linear Koszul syzygies, and a classification of four-term relations Σ L_j ⊗ R_jᵀ = 0 with invertible factors.

Evidence and reproduction

Interpreter: CPython 3.12 (local run used ~/git/integer-mult-bounds/.venv/bin/python3.12). From the repository root:

python3.12 -I -W error -B research/bit-four-term-twist-escape/verify.py
python3.12 -I -O -W error -B research/bit-four-term-twist-escape/verify.py
python3.12 -B scripts/research_package.py --package research/bit-four-term-twist-escape
python3.12 -B scripts/check_research_commit.py --package research/bit-four-term-twist-escape --check-source-history

Local results at 54cb03ce2837b0c7935b8147e28507e60114990b: both verifier modes print PASS_FOUR_TERM_TWIST_ESCAPE in about 1.7 s each. The package hook prints PASS_REGISTERED_RESEARCH_PACKAGE, and the commit check prints PASS_RESEARCH_COMMIT_BYTES. The verifier checks:

• the pinned bytes of Prove three-term transport rigidity for an eight-call address-twist pattern #5490's PROOF.md;
• the m = 2 census (solution-space dimension histogram {0: 39246, 4: 7164, 8: 245, 12: 1});
• unnormalized m = 2 spot checks and the three-term m = 2 control;
• the field witnesses for m = 3, 4, 5, 6, evaluated directly as products in Sym²(V);
• 300 random m = 3 triples and a perturbed-frame negative control.

Limitations

• These are necessary degree-one conditions in one off-diagonal block. No word is synthesized, no I_m diagonal target is realized, and no other block or Exclude PR4953 seven-call address-twist pattern at second order (no new κ) #5003 second-order cross term is checked.
• Part 2 for m ≥ 3 is a written proof. Only part 1 is exhaustive, and part 3 is computer-checked for m = 3–6. No independent human or proof-kernel review.
• Lifting to odd-denominator frames and GL_m(Z_(2)) twists follows Prove three-term transport rigidity for an eight-call address-twist pattern #5490's model and is not replayed as a group-ring word.
• No native price, supplier, selected-frontier or κ claim.

Open next questions

1. Can a complete pattern use a four-contribution block's rank-two residue to reach invertible diagonal targets? First step: choose a projector pattern with one four-call block whose role-r frames agree mod 2, impose a field-family transport relation, and check every other block and the Exclude PR4953 seven-call address-twist pattern at second order (no new κ) #5003 cross terms before synthesis.
2. What is the largest residue rank for five or more contributions? First step: the exhaustive m = 2 census at k = 5 (36⁴ normalized quadruples), then a classification of five-term tensor relations.

Related open work

No open PR studies the address-twist model or #5490's residue (checked by title and body search for twist, PBQ and syzygy at submission). Rechecked after CI: the other PRs this account opened since the claim (#5976, #5977, #5979, #5980) concern power-variation roots, plane-port decoders and Pauli paid history, not this model.

CI notes

All hosted checks passed on head 54cb03ce2837b0c7935b8147e28507e60114990b: plan and replay (research/bit-four-term-twist-escape), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40565865108. The checks were read once, and the run head matches the PR head.

Credit

Produced by Nick Petalas. Builds on JoyboyBrian's #5490 (model reduction, rank-distance and three-term lemmas) and #4953 (two-sided twist model), and Krzysztof Wójt's #5003 (second-order quotient, source compatibility), with the lineage credited there.