[Dugongue]

Adds a sufficient family of paid power-moment improvements that the existing capped-mass order cannot accept. An arbitrary-order binomial rank stencil strictly lowers Σ r^(1−a) for every real 0 < a < 1, while preserving child count and rank mass. The full proof uses the iterated finite-difference integral and alternating derivative signs; the underlying classical identity is explicitly credited.

The two concrete moves are 3×rank 2 + rank 4 → rank 1 + 3×rank 3 and 4×rank 2 + 4×rank 4 → rank 1 + 6×rank 3 + rank 5. Their capped sums have both signs; an explicit concave test cost actually increases. Thus this extends sufficient certificate coverage rather than repricing a previously accepted concave-order move.

A uniform rational cell-integral certificate prices positive added costs. On the actual current PR3815/3857 histogram, across saving [0.0017, 0.0018], 3,502 copies of the first move or 7,099 of the second pay one extra rank-1 child; both pass the aggregate inventory check. These are sufficient implementation targets, not circuit constructions. Operand compatibility, stock, chronology, dirty cleanup, compiler and full physical costs still require an actual program. No new κ or selected-frontier change is claimed.

Registered Python 3.12 replay passes: 42 exact radical cases, 168 grid bounds, six independently enclosed full-profile power differences, inventory/overhead/domain rejects, actual Git-free CLI and source/result corruptions, repeated with assertions disabled. Includes complete informal proof, exact receipts and bounded reproduction commands; no Lean formalization or independent human review.

Credit PR3863 for the paid concave-profile framework and retained distinct power oracle; the ranked-pair verification and its PR57/60/62/63 lineage; MathProof's sufficient replacement-with-reserve precedent; ChaloupeMusic/PR3815, hexapode/PR3802, dataisfire/PR3734 and PR3857 for the unchanged modern full histogram. All predecessor notices remain. Substantial OpenAI Codex assistance.

Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90, priority revision 6. Fresh overlap scans cover 571 open PR bodies and updated all-state records through #3866, with related actual source comparisons and partial Discussion HTML. No exhaustive novelty claim.