[unpaidmoon]

Change and mathematical scope

Scoped character proof. No new κ and no word.

In the PR2085/2112 formal register model, the seven-call profile (a,b,p,q,r,s,t) = (1,1,0,2,1,1,1) admits no word. The determinant equation is R+S+T = I+2P, with matrix [[2,1],[1,2]] and coefficient vector (1,1,2). Every decomposition into three nonzero symmetric rational rank-one matrices admits an AB or PQ character that detects two calls and four targets. The selected record stays κ = 781451/250000000.

This is a proof for this one named census profile. It is not a multiplication theorem, not an exhaustive census of the other profiles, and not a heuristic search. Rational solutions of the matrix equation exist; A+B+2P is one, and the same character excludes it. The other seventeen three-frame census profiles, including the sign-flipped (1,1,2,0)/(1,1,1), and every decomposition with four or more added frames, remain open. The settled profiles (1,1,1,1)/(1,1,1), (0,2,1,1)/(1,1,1), and (2,0,1,1)/(1,1,1) are not redone.

The bilinear form is 2 x1 y1 - x1 y2 - x2 y1 + 2 x2 y2. The forms 3 x1 y1 + x2 y2 and x1 y1 + 3 x2 y2 are special to diag(1,3) and diag(3,1) and do not transfer. The axis equation here is v1^2 (4l-α)(αβ-γ^2) + 4 α k (α Q - 3 l v1^2) = 0, with Q = v1^2 - v1 v2 + v2^2.

Task and overlap

Session r206. The assigned question is one three-frame count profile already printed in discussion comment 19377345, other than (1,1,1,1), (0,2,1,1), and (2,0,1,1). The tuple read from that comment is (1,1,0,2)/(1,1,1), one of the twelve profiles listed there with 24 positive-dimensional spaces.

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840 (upstream CrocSwap main, re-fetched 2026-11-10 and unchanged). Priority revision 8. The priority file still quotes an older selected decimal; the frontier value used here is the selected record.

Task card: bit-new-register-model, scope_status: changed. The card asks for a larger or nonabelian native alphabet. This package stays in the abelian symmetric 2-by-2 model and answers one profile left open by PR #7035.

Overlap recheck on 2026-11-10, with scripts/find_overlap.py '1,1,0,2' --refresh against open upstream PRs and discussion #1166 (339 open-PR records and 7013 discussion records, complete pagination; local note inventory complete at the base commit): the string occurs in census comment 19377345, which lists the profile as still open, and as an unrelated digit substring in research/p104-coupled-lifting-residues. No open PR settles (1,1,0,2). Open #7013 and #7035 are the excluded siblings (0,2,1,1) and (2,0,1,1). No fork branch was already named for this profile.

Validation

Python 3.12. From the repository root:

python3 scripts/research_package.py --package research/three-signed-q-twice --check-only
python3 scripts/research_package.py --package research/three-signed-q-twice
python3 scripts/check_research_commit.py --package research/three-signed-q-twice --ref HEAD --check-modes

Observed: PASS_PACKAGE_METADATA_AND_PINS, PASS_REGISTERED_RESEARCH_PACKAGE with outcome=completed, and PASS_RESEARCH_COMMIT_BYTES at 7af3a03dade6bd166ad08d7afc6879b0726fe100, including the executable-mode check. The direct check is python3 -I -W error -B research/three-signed-q-twice/verify.py, also run with -O. Both printed PASS three-signed-q-twice character checks.

verify.py expands the scale-bracket, axis-numerator, axis-rearrangement, and Cramer-determinant polynomials and checks that each identity is exact. It exhausts the 32 odd residue classes of the mod-8 discriminant lemma and the 64 axis residue classes of the mod-4 obstruction. It checks that every primitive vector has q ≡ 2 (mod 4), with the three mod-2 classes exhaustive for that parity. It then checks explicit characters on five declared frames: the rational solution A+B+2P, a negative frame, a fractional frame, a parallel opposite-scale pair, and a frame equal to the old matrix Q. Each witness detects two calls and four targets. Failure controls reject a tampered sum, reject I as a rank-one summand, reject an even-even-even form, and reject the diag(1,3) and diag(3,1) bilinear forms as solutions of this residual. The single-line box and the axis-reduction box are finite corroborations. They are not the proof; Sections 4 through 6 of PROOF.md are.

The source-history flag of scripts/check_research_commit.py was not run: this checkout is shallow. The pinned base 17469fff78c22941280f04d39dc5a1faee09c840 is the parent of the package commit, and git merge-base --is-ancestor confirms that relationship. Not run: Lean, Docker, the selected-result cold replay, and scripts/research_selected_baseline.py.

Credit

The register model is Cinna Davis's PR2085/2112 formal model. The determinant and first-order inequality are the PR2910 package as recorded in apoursh's two-signed-frame floor, PR #6190. The census is Nick Petalas's discussion comment 19377345. The released all-old-once handoff is comment 19377832. The (0,2,1,1) and (2,0,1,1) exclusions are unpaidmoon PRs #7013 and #7035. Those three profiles are not redone here.

Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent human or formal-kernel review. New work is Apache-2.0.

Open next questions

Which of the other seventeen three-frame count profiles survive the same test? A useful first step is to read one tuple that discussion comment 19377345 already prints, other than (1,1,1,1), (0,2,1,1), (2,0,1,1), and (1,1,0,2), write that tuple's own determinant residual, and test one rational rank-one solution with the bilinear form of that residual. The sign-flipped tuple (1,1,2,0)/(1,1,1) has residual [[2,-1],[-1,2]] and is not settled here. The form 2 x1 y1 - x1 y2 - x2 y1 + 2 x2 y2 belongs to [[2,1],[1,2]] and does not transfer.

Four or more added signed rank-one frames are outside this three-summand character and are not a next question of this package.

Workflow feedback (optional)

The census comment prints the tuple, and the predecessor states that the diagonal bilinear forms do not transfer. Recomputing the bracket, the axis numerator, and the Cramer matrix from [[2,1],[1,2]], and checking those polynomials in the verifier, is enough to keep this profile apart from diag(1,3) and diag(3,1). Linking this once in discussion #1200 is enough.