Bridge and size: the Kodaira–Spencer morphism, arithmetic derivatives and Step (xi) of IUT as lax 2-cells

This note is an essay that proposes a viewpoint; it contains almost no new mathematical results. In function fields, the height inequality (the analogue of Szpiro's inequality) follows from the Kodaira–Spencer morphism. The note places this classical argument, the "bridges" that are built in number fields from an additive relation a + b = c (the p-adic logarithms of Baker's method and the arithmetic derivatives of Pasten), and Step (xi) of the proof of Corollary 3.12 of inter-universal Teichmüller theory (IUT) III side by side in the form of the lax 2-cells of the earlier note Segment or loop? Mochizuki himself describes the Kodaira–Spencer morphism as analogous to the "license to confuse" of IUT. The difficulty splits into a bridge (that a 2-cell exists) and a size (the blurring that must be allowed). This restates two requirements stated by Mochizuki himself: a common container that is large enough to house both sides, and small enough to yield a nontrivial estimate. In function fields both come automatically from the structure. A bridge from an additive relation exists unconditionally, but bounding its size is equivalent to the abc conjecture (Pasten). In IUT, conversely, the blurring is small by the upper bound of IUT IV, while the bridge (the inclusion in Step (xi)) is not derived in the written argument. A bridge from an additive relation does not cross the Θ-link: the Θ-link carries the unit portions by full poly-isomorphisms, so that specific global elements such as 1 − λ of the Legendre form are not carried. Forcing them across gives a contradiction at the level of units already on actual initial Θ-data, and absorbing it by blurring needs exactly ℓ times the wall (k − 1)Q of Corollary 3.12, which grows with q. The curves on which isogenies give a bridge are excluded from the initial Θ-data by Definition 3.1 (c) of IUT I. The reading of the note is that the ring structure (in particular, the addition) that IUT gives up in order to circumvent the absence of a global multiplicative subspace is precisely the material of a Kodaira–Spencer-type bridge. To complete IUT as an argument of Kodaira–Spencer type, one would have to build, inside the multiradial representation, a bridge whose blurring does not grow with q; for a single datum this claim is equivalent to the height inequality, and if it holds for all data the abc conjecture follows. Homotopy type theory can state this precisely, but cannot construct it; what it contributed here is that, by requiring one to say along which path one identifies, it made visible that an additive bridge is information on the side of structure. No claim is made about whether IUT is correct. This is a preprint. Use of generative AI. The question of this note — whether the number-field version of the Kodaira–Spencer argument, which gives the height inequality for function fields, can be realized as a quantity on the construction of IUT — and the decision to write it up as a separate note arose in the author's discussions with generative AI tools; the author judged them important and decided to make them the subject of this note. Generative AI tools (large language models) were used, under the author's direction, in the computations, in checking the literature and in the writing. The author takes full responsibility for the contents of this note. The deposit contains the note and an archive of the folder bridge-and-size/ (the LaTeX source, the script that computes the two tables, the local data of the seven initial Θ-data and the output), made from tag v0.19.0 of github.com/turahigashi/iut-explicit-initial-theta. The files are licensed under CC BY 4.0.