[unpaidmoon]

Change and mathematical scope

Scoped impossibility proof. No new κ and no word.

Every rational rank-one splitting of 2I, placed beside the four projectors with old call B and frozen frames A, P, and Q, meets target support and fails both the extended P-coefficient and the AB-coefficient. Each character detects two calls and four targets. At the displayed t=2 frames (-2/5, 4/5, 8/5) and (12/5, 6/5, -8/5), every integer lift gives that same count. Inside the splitting lattice, a word that uses all six labels has at least ten calls. Length 7 is impossible because one extra label cannot restore the sum 4I.

Selected κ stays 363005817/100000000000. The eight-call minimum is not lowered and not raised.

Evidence classes:

1. Proof. For every primitive direction, the odd multiplier m = |u|^2 / gcd(|u|^2, 4) makes m c and m(a+b) integer characters. Both detect two calls and four targets, and every other lift is even on the splitting. No rank-one matrix in the lattice has character (1,1).
2. Exhaustive finite census of one denominator. At t=2, each family has 125 residue classes and 25 integral lifts, all even on both frames. Period steps of 5 change the value by an even integer.
3. Finite control, not a census of directions. The closed forms are checked on all 1440 primitive vectors with coordinates in [-24, 24].

Task and overlap

Base commit: e54523517d0f03e58352e647c403a5af3e525a57 (upstream CrocSwap main, “Admit conditional frontier 7000 with complete source application audit”).

Task card: bit-new-register-model, priority revision 8. Scope changed. The card asks for a larger or nonabelian native alphabet. This package stays in the abelian symmetric 2-by-2 model and answers the rational-splitting question left open by unpaidmoon PR CrocSwap#7067.

Concrete object: the six labels A, B, P, Q, X, 2I-X for a rational splitting X. The residue census is the 125 classes of each character family at t=2. The direction box is a control only.

Not redone: the length-6 support census of CrocSwap#7067, the five-call theorem of CrocSwap#7024, the six-frame classification of CrocSwap#7076, the length-5 four-frame result of CrocSwap#7083, and the two-old equation of CrocSwap#7100. No open upstream PR states this character count. A search for the displayed frames and for “rational splitting of 2I” hits CrocSwap#7067, which leaves the count open, and does not hit a later proof.

Validation

Environment: Python 3.12. Registered replay:

python3 scripts/research_package.py --package research/rational-split-p-coefficient --check-only
python3.12 scripts/research_package.py --package research/rational-split-p-coefficient

Both registered commands passed (PASS_REGISTERED_RESEARCH_PACKAGE, outcome negative-result), including the python -O rerun. Direct check: python3 -I -W error -B research/rational-split-p-coefficient/verify.py.

The t=2 parity census is exhaustive for that denominator. The statement for every primitive direction is a proof. The 1440-vector box is a finite control, not a census. Predecessor replays, Lean, Docker, and the selected-result cold replay were not run.

Failure controls: I and zero are not rank one; the raw c-coordinate of the t=2 frame is not an integer; adding A breaks the splitting; a non-primitive direction is rejected; the integral splittings 2A+2B and 2P+2Q still fail the count. Boundary: P/2 lies outside the lattice, every P-coefficient shape misses it, and the AB lift a+3b still detects two calls against four targets on the eight-label pair.

Credit

The register model is Cinna Davis's PR2085/2112 formal model. The determinant identity and the character inequality are the PR2910 package as recorded in apoursh's two-signed-frame floor, PR6190. The AB and PQ shapes are research/three-signed-old-once. The integral P-coefficient count is the language of research/four-signed-once-parity. The length-6 support shape, the displayed t=2 frames, and the open question are unpaidmoon PR CrocSwap#7067, head e6aac77f79560add2c47c761d9defff96934d88d. None of those theorems is redone.

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

Open next questions

Is there a rank-one matrix Y outside the t=2 lattice for which every AB lift and every P-coefficient lift integral on the displayed frames detects Y? A cancelling pair Y, -Y would then meet both inequalities at length 8. Passing that count would still not be a word: support and the product remain. The half projector P/2 is a negative example, killed by the lift a+3b. The smallest next matrix is (0,0,1/3).

Workflow feedback (optional)

CrocSwap#1200