The Mathematics of Representation: Quantitative Metamathematics under Coordinate Changes
Wenjie Yang · Zenodo (CERN European Organization for Nuclear Research) · 2026
Quantitative metamathematics depends on costs, machine coordinates, proof codings,and filtrations. We distinguish the effects of these choices. Under computable costs affinelyequivalent to raw length, the convergent normalized projection spectra have exact range(0, 1] ∩ LCE(0′) on every theory interval [Q, S] with a c.e. Σ1-sound upper theory S ⊇IΣ1 + exp; one cost realizes each value uniformly throughout the interval. For an arbitraryconsistent c.e. extension of Q, we also realize prescribed lower and upper limits withthe corresponding effective cut complexities. Cellwise provable equivalence preserves theentire survival class. Complete-window refutation budgets eventually dominate every totalcomputable function, even after replacing length windows by any computable increasing exhaustive sequence. By contrast, effective catalogues of individually provable sentences have computable budgets; classical finite consistency supplies a restricted-family speed separation without entropy separation. For probabilities on Cantor space, we give a pathwise Hellinger decomposition and construct an exhaustive clopen filtration whose equal-splitting measure is continuously equivalent to any given full-support atomless probability, with density arbitrarily close to one. An explicit local splitting bound identifies the conditional intersection masses needed for cost control. A final appendix separates a proved finite filtering lemma from the still conditional arithmetic compiler bridge. Noconclusion about saturation of the fixed raw PA spectrum is inferred from a change of representation.Keywords: incompleteness; syntactic spectra; representation dependence; proof length; Stone spaces; Hellinger process; equal-splitting measures.