Realizable, not derivable: what a spectral certificate can and cannot do for a structural mass ansatz

A family of structural mass ansätze fits the fermion mass spectrum of the Standard Model with a small number of locked constants and small integer exponents. Whether such a fit is evidence of structure or of reverse engineering has been debated under the heading of numerology. I argue that this debate lacks a category, and supply it with a case study. For one such ansatz we exhibit → I exhibit a pair of certificates: an existence theorem, showing that the spectrum is realized by an explicit finite spectral triple satisfying the KO‑dimension, order‑zero and first‑order axioms of noncommutative geometry; and a non‑derivability theorem, showing that no horizontal extension of the algebra can select the spectrum within the order‑zero window. The two results delimit a precise epistemic status—realizable, not derivable—which is neither derivation nor numerology. We propose → I propose a three‑place taxonomy (derived / realizable‑not‑derived / not realizable) for mass ansätze in axiomatically constrained frameworks, locate standard examples within it, and identify the remaining arbiter of such claims: a formal description‑length audit.