Intensional semantics for comparison problems in arithmetic geometry

Abstract

This work proposes a semantic environment for arithmetic geometry in which the passage between two presentations of one object carries a measurable weight, rather than dissolving into a transparent identification. Following Lawvere, a category equipped with a cost on its morphisms is enriched over the monoidal poset $([0,\infty],\ge,+)$, so that a height becomes a map controlled by a generalized distance. Following Voevodsky, the identity type of a C-system is a locus of structure open to transport. A labeled quantitative $(2,1)$-category reconciles the two readings, with invertible $2$-morphisms witnessing that two transports differ only by a coherent change of presentation. A comparison between two labeled realizations of one object becomes a transport inequality. If it is asked to descend functorially to an inequality between extensional invariants, then it collapses to a tautology, unless a non-zero load is assigned to the transport itself, at which point the inequality regains its content. We formulate an intensional Szpiro inequality for elliptic packages, where the discriminant height transports along a morphism with a controlled model defect, and relate a bound of this shape to the $abc$ conjecture. We then treat the Birch and Swinnerton-Dyer conjecture in its pre-modularity form, so that the rank and refined forms descend to $\mathrm{ACA}_{0}$ without analytic continuation. The refined constant reads as a transport cost in which the regulator, the period, the local Tamagawa data, and the order of the Tate--Shafarevich group $\mathrm{Sha}(E/K)$ are held apart by the descent obstruction. We reduce the finiteness of $\mathrm{Sha}(E/K)$ to one hypothesis.