From attitude to instrument: coverage theorems for integer-decomposition claims in fundamental physics

For a century, claims that a measured constant equals a combination of small integers have recurred in fundamental physics, and the word used to reject them—numerology—has remained an attitude rather than an instrument. This paper supplies the instrument for one precisely delimited claim class, the integer-decomposition claim. We prove two coverage the orems, a universal deficit lower bound, and a degeneracy asymptotic: for the pool {3, . . . , 8} every integer in [−20, 0] is representable with coefficients in {−1, 0, 1}; for two coprime bases the non-representable targets above an explicit threshold are exactly the Sylvester gap set union its reflection, so the coverage deficit is Θ(1/B) with explicit constants; every pool obeys δB ≥ g/(B P X); and dB(n) grows as Bk−1 at every fixed target. The weight scale is axiomatized and embedded in Mayo’s severity framework. A three-place taxonomy separates decomposition, functional-form, and principle-level claims, with the Koide relation and the history of fine-structure numerology as controls. The instrument is then turned on the au thor’s own framework. The construction is built for working use: every verdict it returns is a finite computation a referee can repeat. The criterion is time-indexed: saturation forbids claiming in advance that a coincidence is the shadow of deep structure; it does not forbid that outcome.