[JosieElliston]

Result

No exact word of at most seven calls uses two added positive rank-one frames for the PR2085/2112 four-target endpoint. This holds for all directions and scales of R and S, including coincident ones, with any finite restored dirty width and free binary mixing. It answers open question 1 of JoyboyBrian's handoff #6104, the "general seven-call question" left open there. It also answers apoursh's two-frame open question in #6089. No new κ, supplier or native saving.

Package: research/bit-two-positive-frames-seven-call/ (base 721c03822).

Proof outline (necessary conditions only, no word census)

1. Determinant (PR2910). rR + sS = M = 4I - aA - bB - pP - qQ. Of the profiles with r, s >= 1 and at most 7 calls, 442 have positive semidefinite nonzero M.
2. Exact support. For each target, some subset of call labels sums to the target exponent. For a split X = rR, each support solution either holds for every X (alpha = 0, K = 0) or pins X. Exactly 7 profiles are free: (1,1,1,1,r,s) with (r,s) in {(1,1),(1,2),(2,1)}, plus the four profiles with one old frame doubled. The other profiles intersect to 32 pinned splits. All of them lie on old directions.
3. Exact-lattice characters. For each pinned split, the verifier computes an integer basis of Lambda = Z<A,B,P,R,S> and tests all eight F2 characters. Each split fails the first-order rank inequality, with 2 detected calls against 4 targets.
4. Uniform 2-local lemma for the 7 free families. R and S lie in Z_(2)<A,B,P>. Then chi_PQ (twice the off-diagonal entry) or chi_AB kills both, depending on the parity of the direction. That gives 2 detected calls against 4 targets. This extends the handoff's M = 2I, r = s = 1 argument to r or s equal to 2, and to the doubled-frame families with M = diag(1,2) and their images.

Together with #6089's single-frame theorem (at least m+7 >= 8 calls when R is used) and the eight-call floor for {A,B,P,Q}, this says no word using at most two added positive frames beats eight calls. That corollary inherits #6089's pending status.

Evidence

python3.12 -I [-O] -W error -B research/bit-two-positive-frames-seven-call/verify.py uses only the stdlib and exact Fraction arithmetic, with no repository imports. Each command takes about 30 s. Both pass, and scripts/research_package.py --package ... reports PASS_REGISTERED_RESEARCH_PACKAGE. The verifier checks:

• every finite step: 442 profiles, 7 free families, no balanced R,S support, and the 32 pinned splits each killed on the exact lattice;
• 7,784 free-family grid cases against both the lemma and the exact lattice (regression; the proof is in the README);
• an independent route that skips the support classification: 46,916 concrete splits over all 442 profiles, of which 648 pass literal subset-sum support, and none passes the exact-lattice characters;
• a positive control: at eight calls there are 77 survivors, including the duplicate-label words, and one literal eight-call word with R = A and S = B multiplied out in F2[Lambda] equals the endpoint;
• negative controls: without the support step, 28 seven-call profiles survive the characters; naive mod-2 reduction of 1/2 is rejected; a mutated target is detected.

A read-only OpenAI Codex adversarial review checked:

• the necessary conditions;
• that the support classification is complete, so no continuum can arise other than alpha = K = 0;
• the 2-adic computations and the symmetry transfer;
• the lattice routine, against covolumes on 1,000 random cases.

It found no errors. This is AI review only: no human or proof-kernel acceptance. Hosted CI was not monitored.

Not covered and open next questions

• Three or more added positive frames. Indefinite added frames.
• Realizability of the eight-call two-frame survivors, for example (1,3,0,1,1,2) with R = 2A and S = P/2.
• Native pricing, all-field claims and selected-frontier changes are out of scope.

Credit

• JoyboyBrian (HANDOFF: Joint determinant and character obstruction for two added positive rank-one BIT frames #6104): the question, the parameterization and the M = 2I character argument.
• apoursh (Classify usable positive rank-one frames and their exact call counts #6089): the one-frame classification, the paid-only reduction and the symmetries.
• Nick Petalas (PR2910): the determinant method.
• Cinna Davis (PR2085/2112): the model and endpoint.
• Philippe Van Velzen: the eight-call floor.
• The retained BIT lineage.

Prepared for JosieElliston by Claude Opus 5.5 via Claude Code, with substantial AI assistance.

🤖 Generated with Claude Code