Motivic Cohomology Automata

Abstract

Abstract. The MCA concept asks for “periods extracted algebraically, without numerical integration, by combinatorics on the Bruhat–Tits tree.” Over , with motives, Chow rings and the motivic Galois group, no such statement is available. Over , with Mumford curves, Schottky groups and the p-adic period matrix, it is a theorem (Manin–Drinfeld 1973). This paper fixes the objects (§2), states the Manin–Drinfeld product with a tree-geometric convergence estimate (§3), proves an exact truncation theorem — precision is reached by a finite, explicitly bounded set of double cosets, with no error propagation (§4) — gives the classical enumeration cost (§5), sets up the Grover/Szegedy walk on and proves what it does and does not compute, leaving a precisely stated conjecture (§6), and states the DS-NAQHA hardware correspondence as an explicit analogy (§7). Version 2 adds an independent verification (§10): every lemma, theorem and proposition that admits a computation was tested in exact arithmetic on an explicit genus-2 Schottky group over (1458 double cosets, 43 680 cross-ratio quadruples); all pass, one statement (Proposition 6.4) was corrected in wording, and the computation produced a new empirical law (Proposition 6.8): the -th -adic digit of a Manin–Drinfeld factor is a product of transition weights with memory tree edges. Count: 5 theorems, 8 lemmas/propositions, 2 conjectures, 1 empirical law.