Intensional semantics for comparison problems in arithmetic geometry

Abstract

This work proposes a semantic environment for arithmetic geometry in which passage between presentations carries an explicit weight. Morphism costs induce a Lawvere generalized distance by taking infima over transports, and heights satisfy the corresponding transport bounds. A labeled quantitative $(2,1)$-category retains invertible $2$-morphisms as coherent comparisons between transports. Its vertical groupoids admit a transport interpretation inspired by the identity types of Voevodsky's C-systems. If a height factors through the underlying object, then a vertical comparison has equal endpoints, whatever its transport cost. We formulate an intensional Szpiro inequality for elliptic packages, where the discriminant height transports along a morphism with a controlled model defect, and we relate a bound of this form to the $abc$ conjecture. The defect remains visible between non-minimal and normalized presentations, while conductor complexity stays attached to the underlying curve. We then treat the Birch and Swinnerton-Dyer conjecture in its pre-modularity form, using strong point-count asymptotics with an explicit convergence requirement. The refined constant retains separately typed arithmetic factors. A finite differential correction makes the period independent of the chosen differential and links its normalization to model transport. Within the declared finite-descent theory, Tate--Shafarevich finiteness reduces to finitely many closed prime columns of Selmer shadows and a uniform cutoff. For $y^2=x^3-q^{2}x$ with $q\equiv 3(\mathrm{mod}8)$ prime, two-isogeny descent closes the two-primary column uniformly. The statements admit expression in ${\mathsf{ACA}}_0$ relative to certified descent and Mordell--Weil data, with certified Cauchy names for the period and regulator.