[JoyboyBrian]

Result

Changed hypothesis for bit-new-register-model: free per-register address twists B -> P B Q (P, Q in GL_m(Z_(2))) between paid calls, together with free binary mixing and any number of restored dirty roles. The three-register bound (#468) and all its successors explicitly exclude these. Twisted XORs put operators in matrices over the skew group ring F2[M_m(Q)] ⋊ G, a nonabelian operation family on an abelian address group.

• Theorem A (written proof, PROOF.md §3). Suppose every frame has odd denominators and every twist reduces mod 2 to a one-sided map (X -> P̄X for all twists, or X -> XQ̄ for all twists). Then any exact word needs Σ_s rank_F2(C_s mod 2) paid calls. For three targets invertible mod 2 that is nine, and the direct word attains it. One-sided twists therefore do not escape the obstruction. The proof extends Prove the odd-denominator mW call floor with arbitrary auxiliaries #2110's square-zero map: twists are equivariant for it, and one-sidedness keeps each call rank one in a suitable flattening.
• Proposition B (finite exact). For two-sided twists, take σ(B) = S B S^T, a simultaneous swap of axes 0 and 1. An explicit seven-call word for diag([I],[I],[I]) agrees with the target in the full square-zero image M_3(Λ ⋊ G) in all 5040 call orders. Exact skew-ring products show that none is a real identity. Hence no square-zero argument (Theorem A, Prove the odd-denominator mW call floor with arbitrary auxiliaries #2110) can prove nine once two-sided twists are free. Whether they actually beat nine is open. No seven-call word and no sub-nine bound are claimed.

This is a scoped proof plus a finite relaxation witness, not a complete theorem for two-sided twists. There is no supplier, no native price for a relabelling, no κ change and no selected-frontier change.

Task and overlap

• Card bit-new-register-model (track bit-mechanisms), priority revision 7, base cef4b52f2a59009865bf8eabde1be3f159063703. The scope is refined to the twist model. The reason is recorded in TASK.json.
• CLAIM: Handoffs and requests for independent checks #1166 (comment). The board was rechecked before submission and has no crossing claim.
• Distinct from Prove six-call two-SWAP cost with arbitrary address spectators #4939 and Address spectators never help coordinate swaps: t swaps cost 3t for every n (no new κ) #4950 (mihir-s-05, jwr404). Those compute coordinate-SWAP costs in M_R(F2[GL_n(F2)]), where the address group itself is nonabelian. Here the address group stays abelian and the nonabelian part is the twist action.
• Distinct from petalas's Prove exact unequal-width query capacity on bipartite masks (no new κ) #4909–Classify shared native maps for three subspaces (no new κ) #4919 and from the existing dirty/clean-workspace packages, which keep "no free address maps". Q8, D16, dihedral, S3, A5 and SL2(F3) packages change the address group instead.

Test plan

• python3.12 -B research/bit-address-twist-register-floor/verify.py: PASS in about 6 s. Checks the direct nine-call word with 0/1 dirty helper; rejects a deleted call and a mutated frame; bounds the flattening on 240 seeded random one-sided words, with an opposite-flattening control exceeding on 108; runs all 5040 witness orders (relaxation match, 0 exact); rejects altered-target, untwisted and left-only witness controls.
• The same verifier under python3.12 -O -B and python3.12 -I -W error -B: PASS.
• python3 scripts/research_package.py --package research/bit-address-twist-register-floor --check-only: PASS_PACKAGE_METADATA_AND_PINS.
• python3.12 -B scripts/research_package.py --package research/bit-address-twist-register-floor: PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=partial.
• python3.12 -B scripts/check_research_commit.py --package research/bit-address-twist-register-floor --check-source-history: PASS_RESEARCH_COMMIT_BYTES.
• python3.12 -W error -B -m unittest discover -s tests -p 'test_research_onboarding.py': 26 tests OK.
• Verifier mutation tests, each confirmed to FAIL: expected exact-order count 0 to 1; floor 9 to 8; flattening bound k to k-1; witness diagonal frame E22 to E11.

Open next questions

1. Can two-sided twists beat nine exactly? First step: compute the order-two term (mod the cube of the augmentation ideal) for the seven-call multiset over all S3 × S3 twist assignments.
2. Can a native bit layout perform a per-register PBQ relabelling without paid work? Without that, any formal saving cannot be used.
3. Do words mixing left-only and right-only twists evade the floor? First step: an exact relaxation search with at most eight calls.

Credit

Ali Romman (#468, three-register theorem), eumemic (#288 §1.2 model), huxint (#203 flag method), 1337isnot1337 / Cinna Davis (#2110 square-zero odd-denominator floor). The new work was prepared by JoyboyBrian with substantial assistance from Anthropic Claude (Claude Code, Opus 5.5). Apache-2.0.

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