Locality by Representation: Certified Learned Modular Multiplication at 2048 Bits from a 37,767-Parameter Carry-Save Montgomery Automaton

Exact arithmetic at unseen widths can be obtained by learning a finite transition rule and certifying every input to that rule. In a carry-save Montgomery representation, a modular-multiplication tick uses a radius-one bit window and two broadcast bits. Four learned cells, containing 37,767 parameters, implement the tick, carry resolution, comparison, and conditional subtraction. Exhaustive certificates cover their complete local alphabets. An explicit induction then proves exact modular multiplication for every odd modulus and every operand width satisfying the stated storage bounds. The same cells reach 2048-bit prime moduli in two recorded runs of the SAIR evaluator inside its published sandbox: all 1,000 scored rows correct in each run, in 71.2 and 60.6 seconds. A boundary example explains why small residues can require a full-width carry chain, and why empirical accuracy alone cannot replace certification.