The Screening Filter and the Adjudication Statistic Pull in Opposite Directions: An Admissibility Criterion for Rival-Constant Tests

Proximity screens and rival-constant adjudication answer different questions, and this note shows they impose conflicting demands on measurement precision. Given a claimed constant at fractional deviation δ from a measured value x with uncertainty σ, the adjudication statistic R = δ_rival/δ_claimed inherits a relative uncertainty σ_R/R ≈ (σ/x)(1 + δ)/δ. Requiring R to be resolved at a threshold therefore requires σ/x ≤ τ·δ/(1 + δ), with τ* = 0.204, 0.306, 0.366 and 0.407 for thresholds R ≥ 2, 3, 4 and 5 respectively. Two consequences follow that are not obvious from either statistic alone. First, the admissibility limit scales with δ, so tightening a proximity screen — admitting only rows that sit closer to the claimed constant — makes those same rows less able to adjudicate between rival constants, not more. The screen and the adjudication statistic pull against each other. Second, τ* is not monotone in the direction one would expect: a demand of R ≥ 5 is harder to satisfy than R ≥ 3 at fixed measurement precision, so raising the adjudication threshold does not straightforwardly raise confidence. The criterion is stated in general form and is independent of the golden ratio, of Fibonacci Causal Loop Theory, and of any particular corpus. It applies to any programme that screens candidate matches by proximity and then asks which of several constants the data select. Applied to the FCLT corpus it is restrictive: the strongest screened row fails the R ≥ 3 limit, and the closest any row has come is a factor of three short. Accompanying code reproduces every figure with the standard library only.