Prove one-donor T740 operator and residual-rank rigidity

[thakrarsagar]

PR #3263 found that rank-valid T740 axis allocations can fail the full eight-stream operator. This package resolves its proposed four-idle-site plus one-partial-donor boundary: at owner 0 / port 0, with sites 8, 10, 12, 13 and 14 variable and all other phases, hosted prefixes, translations and the schedule fixed, the complete operator is the identity exactly when every residual phase ratio is 1. Consequently independent residual bases must retain ranks 48, 48, 48, 48, 1.

The proof replays all 73 character bits symbolically as exact quadratic phase polynomials over Z/4, rather than sampling characters. Its final 16 terms contain no ambient-character dependence. A second checker independently reconstructs the operator from pinned raw JSON, without importing either the primary checker or predecessor executable. The rank conclusion follows from the exact character average; it does not imply basis uniqueness.

Validation: registered CPython 3.12.15 replay passed; the independent checker agrees with the retained certificate; six control groups cover hidden quadratic errors, omission of the actual donor axis, missing hosted/bridge operations, source corruption, explicit Gaussian arithmetic, and distinct bases with the same phase. Source pins, metadata, whitespace and committed publication bytes pass (23 files).

Scope: this fixes child 15 and the other hosted sites. It does not classify allocations changing those sites, change the phase interface, provide a native/full supplier, or improve the selected bound. Selected-frontier reproduction was not rerun because no selected artifact changes. Open next questions and exact hypotheses are in the package.

Coordination and attribution: canonical claim. Builds on Gonzalo Mier's #3199/#3263 operator work, Tekalign's #3204 four-site rigidity, and Rohan Arun's #3200/#3251 rank allocations; full pinned heads and source credits are retained. The adjacent character-dependent chart investigation in comment 19003688 remains separate. Developed with substantial OpenAI Codex assistance.