[zxv]

Result

Any fixed finite sample of power exponents can miss a losing interval between samples, even for complete synthetic invoices with equal count, equal rank mass, and only the five ranks {1,2,4,8,16}. Five is minimal for this two-crossing pattern. More generally, matching the first h-1 endpoint derivatives still permits a hidden window, and exactly 2h+3 ranks suffice and are necessary.

The integer polynomial construction extends Nick Petalas's PR #5174. The support lower bound is proved directly by Rolle induction. A small example has 135 calls and rank mass 546 on each side, wins at p=1/4 and 3/4, and loses at p=1/2 by the exact inequality 256² > 2·181². Existing exact root-budget tools already handle multiple crossings; no defect in them is alleged.

Significance for researchers

This is a sharp adversarial fixture for replacing interval-wide moment certificates with finite sampling. It also explains why the simultaneous reduction in PR #6047 needs its ordered equal-spread structure. It supplies no physical construction, supplier admission or new κ. Multiplicities are unbounded; a finite-call-budget separation theorem remains open.

Validation

CPython 3.12.8 normal and optimized registered replays pass 323 constructed windows, 4,600 sampled winning signs, h=1..8 endpoint-multiplicity checks, narrow brackets near saving .003125, and expanded-polynomial interval controls. All signs and coefficients use exact integer/rational arithmetic. Committed bytes pass. The optional full-history check is unavailable in this shallow checkout; source pins are at the immediate main base. No full selected replay was run.

The navigation repair in PR #6050 is included. At head d914b4a9c, shared planning and this package’s hosted scientific replay pass in run 40626062482. Unrelated archived input-pin jobs triggered by the navigation files fail; no frozen package is changed. Discussion #6052 is the canonical investigation thread.

Open next questions

Bound the possible hidden-window width under a fixed total-call budget, beginning with rational-root separation for this construction. Separately, determine whether a legal source-bound supplier can realize a five-rank two-crossing edit with a common complete boundary and all fees paid.

Prepared by zxv with substantial OpenAI Codex assistance. Predecessor attribution and exact sources are retained. No independent human or proof-assistant review is claimed.

Final export validation at 2646b6d67ceb74b9868e632f3a27312d8be62a0e: 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.