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Finite Multiplication and Division Response Kernels

Abstract

Finite Multiplication and Division Response Kernels.

Theorem 1.1 (The exact classification at every positive horizon).

Lean statement: D5/S3/Arith/GoldenResource/SignedFiniteMultiplicationResponseKernel.eq_signed_iff

Proof. Machine-checked in Lean as D5/S3/Arith/GoldenResource/SignedFiniteMultiplicationResponseKernel.eq_signed_iff (✓ std3). ∎

Source. Repository-derived.

Commentary.

Fix a natural capacity vector on a finite coordinate set and a positive integer horizon. Each letter increases or decreases one coordinate by one, failing at capacity or at zero. A successful word returns the parity sign of its endpoint coordinate sum if every coordinate is at most one, and zero otherwise. Failure is a distinct response. The defect is the sum of the positive parts of each coordinate minus one. The guard code records each coordinate and its remaining capacity, both truncated at the horizon. Two states have equal responses on every word of length at most the horizon exactly when they are equal, or both defects exceed the horizon and their guard codes agree. Repeated letters recover both truncated boundary distances. Dividing each coordinate down to at most one separates every state whose defect does not exceed the horizon. States of larger defect retain zero readout on every successful word in the horizon, so their responses are determined by the guards. Zero capacities and an empty coordinate set are allowed.

References