Research: exhaust rational rank cuts before bit scalar-word search

[petalas]

Summary

The four-role example in Discussion #2200 had only a handwritten projection rejection before its planned scalar search. Add a reusable exact screen for every rational linear projection/rank cut of a bounded paid-frame multiset.

paid-frame subset spans, at most 2^n
  compute the exact minimum cut margin
  emit an integer separating functional when the margin is negative

The proof and checker show why paid spans cover all rational functionals. The CLI accepts address dimension at most four and at most twelve paid frames/target roles. This is a recorded changed scope from bit-new-register-model, priority revision 5. Canonical claim.

Evidence

• Before: all nine coordinate cuts pass the discarded eight-frame example; its combined projection obstruction was checked by hand.
  After: all 16 paid spans are checked and an exact witness gives two surviving paid factors against three target roles, excluding every scalar choice/order and restored dirty workspace in the stated model.
• All 525 independent integer-functional oracle cases agree. The nine-call direct word passes, its missing-call mutation rejects, and the capacity control reaches all 4,096 spans. Seventeen invalid-input controls and actual CLI/receipt mutations pass.
• Registered replay passed on CPython 3.12.15; optimized standalone replay and CLI controls passed on CPython 3.14.8, macOS arm64. Source pins, warning-as-error compilation, Ruff checks and committed-byte preflight pass.

python3.12 -B scripts/research_package.py --package research/bit-rational-rank-cut-screen

The projection inequality belongs to Ali Romman's PR #468 and its retained eumemic/huxint lineage. The example and original rejecting functional belong to @1337isnot1337 in Discussion #2200. The contribution is the complete finite screen and rerunnable evidence. Substantial AI assistance was used for derivation, implementation, internal checks and writing; no independent human review is claimed.

Merge Danger

Door: two-way.

Blast Radius: research.

Only the new opt-in package is added. Passing establishes necessary cuts only; determinant equality, full formal identities, native realization, complete supplier costs and the selected cold replay are outside scope. No supplier admission, selected-frontier change or new kappa is claimed. Hosted CI is separate from the local evidence above.