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Change and mathematical scope
On the exact selected #7001 paired BIT word, the nine owned rank-seven descent rows project to a binary [111,9] code of minimum weight 48. Under the retained monotone source flow and rational chart catalogue, changing any one through four of the nine descents cannot lower the paid moment. The proof cancels the largest recovery transports and charges the residual source-line cover. It allows arbitrary timing, XOR recovery and reuse of earlier descended values.
At κ = 0.0040075385047236, any five-summary change saving at least 5/3 paid units per native copy would require eight rank-four transports and at most one rank-one transport. The necessary source-cover screen exhausted 59 states with no surviving set. Smaller savings from five summaries and joint changes to six through nine remain open, as do new chart spaces and nonmonotone routes.
NO VERIFIED GAIN. This changes neither the selected ordinary BIT saving nor the complete conditional κ. The benchmark reproduced in the campaign is #7145 at κ = 0.003643438506419.
Task, inputs and replay
Refines
bit-helper-contracton canonical basee54523517d0f03e58352e647c403a5af3e525a57. Exact live BIT word blobb718d6b2848215ad785022106dfa9f871a3d1c0c, SHA-25651203e7fc75ba10bd3ac3f8e675759f47e13d262bea66b6600d1e515095cad78.Registered CPython 3.12.14, standalone, metadata and committed-byte checks passed. The replay verifies the rational chart annihilators, all 2,862 public MOVEs, 511 nonzero summary combinations, the joint cost inequalities and the capped cover. Duplicate-summary and corrupted-capacity controls reject. A separate bitset implementation matched the code distance.
Credit and next test
Ali Romman’s #5421 supplied the owned relation mechanism. DreamingOfClouds, zxv, CubeJerry and retained authors supplied the selected native and paired controller. The archived G20 nine-owned package applies the mechanism to another word. This proof adds the source-specific code audit and joint lower bound. Codex and subagents materially assisted. The next open test is exact recovery of six selected summaries from a charged quotient cover.