A Base-Invariant Null for Numerical Proximity Claims: Small Bases, Exponentiation, and Multiplicity

A recurring claim in the physical and biological literature is that some measured quantity lies close to a power of a distinguished constant — most often the golden ratio φ, but also √2, π, e, or the metallic means. Such claims are usually evaluated against an informal or magnitude-dependent null. This paper derives the exact null. For a quantity whose logarithm carries no information about the base b, the probability of falling within relative distance t of some integer power of b is P(δ < t) = ln[(1+t)/(1−t)]/ln b, valid for t < (b−1)/(b+1) and equal to unity above it. The result is independent of magnitude and of which power is matched. Its one substantive premise — that the logarithm of the quantity is uniform modulo one — is stated as an assumption, with its classical justification and the standing objections to that justification given in Sec. II.B. Two consequences follow that a literature search conducted in July 2026 did not find stated elsewhere; the claim is bounded by that search and is set out against related work in Sec. I.A. First, P scales as 1/ln b, so small bases are the most chance-prone: at t = 0.05 a φ-proximity claim is 2.38 times more likely to arise by chance than a π-proximity claim, and the φ literature therefore requires a stricter bar than its neighbours. Second, scanning six plausible constants returns a hit on an arbitrary number 60.9% of the time. Five further results follow: exponentiation doubles the log-lattice offset, so any quantity failing near-maximally has a passing square; a free multiplicative coefficient reduces the pass rate to a tautology; scanning inflates by an unregistered look-elsewhere factor; a deviation normalized by the matched exponent cannot fail once that exponent exceeds 1/2t; and across a corpus of N claims the expected number of chance passes is NP, which for realistic N exceeds the number of claims typically reported as successful. The closed form is verified against 10⁷ Monte Carlo draws across seven bases and four thresholds. A forward test of the published literature is pre-registered in Sec. VI, with the databases, query strings, sampling procedure and random seed fixed at deposit, and the falsifier stated: if published proximity claims clear their own thresholds at a rate significantly above the base rate, this paper is wrong and there is a signal to explain.