Definability of quantization inputs: a Padoa-style classification and its boundary

Abstract

Quantum theory carries several discrete or numerical inputs that are routinely described as "not derived" from an underlying substrate. This paper asks a sharper question: for which of these inputs can "not derived" be upgraded to a proof of undefinability, and for which is this structurally impossible? Padoa's classical method of primitive notions answers the first kind and, as shown here, cleanly refuses the second. The inputs are classified by their logical position relative to a similarity type: an input is amenable to Padoa's method exactly when it is a predicate on a fixed type, and is not amenable when it is a parameter of the type itself or a weight on an ensemble. Both poles are exhibited. At one pole, circulation quantization: reformulating a companion result, the predicate is shown Padoa-undefinable over the structure of realized jets, the nontrivial part being the existence of a model pair; here the method also computes the exact reduct threshold at which definitions begin to exist, direction by direction. At the other pole, the spinor sign: the model pair is nearly definitional and the entire mathematical weight sits in an explicit locality hypothesis about the reduct, stated rather than assumed. Finally, the amplitude parameter (the reduced Planck constant) and Born-type typicality weights are shown to fall outside the method. For the amplitude parameter a dichotomy is proved for the single-theory formulation; for typicality weights a semantic obstruction is documented with a counterexample. In both cases the two available formulations give, respectively, an answer without blindness and blindness without content — the reason these inputs are excluded structurally. The boundary is offered as a result, not an apology: the classification tells one in advance which foundational "inputs" can be argued about by this method at all. Companion to "Locality, cycles, and the Wallstrom condition: an exact dichotomy and an undefinability theorem" (Zenodo, DOI 10.5281/zenodo.21727985). A self-contained SymPy script verifying the paper's symbolic claims is provided as supplementary material.