[ToikY]

Change and mathematical scope

For three mandatory proper-rank BIT children, an invariant covector subspace gives one physical grade per cover orbit even when no normalized middle grade is fixed by every right route. If both outer children preserve the grade and the target negates it, a selected-rank cut forces asymptotic middle volume at least two original fields. With the two extremal cuts, the best total optional paid saving is zero on the upper stock of 1,776 qualifying equal-mask pairs. The complete #7001 raw deficit remains about 60.10389921914735 at target κ = 0.0040075385047236. NO VERIFIED GAIN in κ.

Task and overlap

Canonical base e545235. Changes the bit-new-register-model task to a three-child #7001 route. #7201 bounds two children and supplies the extremal cuts. #7190 uses fixed normalized grades. The new criterion allows a right route to negate or mix the normalized grade space. On the explicit port-1 rank-(1,54,1) table the invariant space has dimension 54 and its pointwise fixed normalized subspace has dimension zero. The separate CI navigation commit reuses the #7161 prerequisite.

Validation

From a clean checkout, use CPython 3.12 with assertions:

python3.12 -B scripts/research_package.py --package research/bit-cover-orbit-grade

The checker authenticates three exact in-tree #7001 inputs, computes the invariant subspace, checks two cover orbits at digit cardinalities 3, 5 and 9, and encloses the whole-stock price. The active-cycle control returns dimension zero. Independent proof audit and clean replay passed, including source and receipt mutation rejection. Local registered replay, 10 byte/mode/source-history checks and six navigation tests passed. Hosted checks pending at submission.

Credit

CubeJerry #7001 supplies the caller and tariff. Credit #6726/#6834/#6980 and delayed-cover authors for native and whole-fibre routes, #7163/#7190/#7201 for the rank arguments, Round 17–18 of #7111 for the port-1 route and pricing, and Clear1nsight #7161 for the complete comparator. OpenAI Codex and subagents materially assisted.

Open next questions

Find one width-independent route table with zero invariant grade space and an exact independent signed output and return. The full paid volume and route, compiler, precision and tape bills remain necessary. Cross-pair sharing and additional paid positions are outside the proved scope.