[zxv]

Result

The complete fixed-cardinality layers in Nick Petalas's PR #5901 have one uniquely cheapest invoice simultaneously for every power tariff r^p, 0 < p < 1: spread the lowest-index blocks. The 252- and 462-member examples therefore each reduce to one candidate for power pricing, while their published all-concave antichain result remains valid.

The proof uses an exact triangular-kernel identity for second differences and a strict exchange argument. It also covers arbitrary increasing block centers with one common spread width. This answers the predecessor's second open question for complete layers; partial pools, varying widths, changed fees and physical realizability remain separate.

Significance for researchers

An exponential all-concave frontier need not obstruct the narrower power moments used for κ. This provides an exact reduction on a named predecessor family and identifies the structure that a legal supplier would need to inherit. It supplies no supplier, native implementation, selected-frontier update or new κ.

Validation

CPython 3.12.8 registered normal and optimized replays pass 84,357 strict complete-invoice comparisons at 21 rational exponents, using integer nth-root interval bounds. Hinge, endpoint, reversed-exponent, changed-target and added-fee controls pass. Committed-byte preflight passes. The optional source-history check refuses this shallow repository; all declared pins resolve at the reachable main-branch ancestor. Selected supplier replay was not run. The sparse worktree occupies about 5.4 MB.

Open next questions

Can two independent equal-width rank exchanges be realized by an actual supplier with a common boundary and complete target/restoration/native fee? Extract two source-bound exchanges before applying this reduction. For a partial layer lacking the canonical optimum, the remaining power frontier is also open.

Prepared by zxv with substantial OpenAI Codex assistance. The predecessor family and its attribution are retained and pinned; no independent human or proof-assistant review is claimed.

The navigation repair in PR #6050 is included. At head 8235b482d, shared planning and this package’s hosted scientific replay pass in run 40626058649. Unrelated archived input-pin jobs triggered by the navigation files fail; no frozen package is changed. Discussion #6052 is the canonical investigation thread.

Final export validation at 841acb94e5d59b750ab15c2483389d30e5674186: official committed export, integrity and declared-input coverage pass. A fresh optimized Git-free replay with empty PATH and read-only exported files passes; every exported byte remains unchanged.