Reading the IUT dispute through indexed identifications: an educational model in undergraduate mathematics

The dispute between Scholze–Stix and Mochizuki over Corollary 3.12 of inter-universal Teichmüller theory (IUT) III has been described in words: are isomorphic copies to be identified or not? This note gives an educational model that states the shape of the dispute precisely using only undergraduate mathematics: sets and maps, multiplication of positive real numbers, graphs and inequalities. Instead of the equality sign it uses a symbol x ≈f y that records which identification is used, with four rules declared at the outset: identity and inverse, composition, action on numbers, and sets of indices. In this notation an identification that does not return to 1 around a loop (a monodromy) becomes a formula. Both Mochizuki's own elementary numerical model and the three options described by Scholze–Stix then sit on one line, the inequality (k−1)Q ≤ E. What is agreed is that an identification which does not return to 1 around a loop leads to a contradiction if it is used consistently. What is disputed is whether the license, in Mochizuki's word, to identify up to relatively small error terms can be obtained independently of the conclusion. The model does not decide this question. Nothing in the mathematics of the note is new; every proposition is at the level of an undergraduate exercise, and exercises with answers are included. Nothing is claimed about the truth of Corollary 3.12, Theorem 1.10 of IUT IV, or the abc conjecture. A companion paper states the same model in homotopy type theory and checks its quantitative core in Lean: doi:10.5281/zenodo.23931344. This is a preprint. Use of generative AI. The idea of this note (to write the shape of the dispute in undergraduate mathematics, with symbols for indexed identifications in place of the equality sign) is the author's. Generative AI tools (large language models) were used extensively, under the author's direction, in working out the propositions and their proofs 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 indexed-identifications/ (the LaTeX source), made from tag v0.13.0 of github.com/turahigashi/iut-explicit-initial-theta. The files are licensed under CC BY 4.0.