The Problem 3 paper [4] establishes that multiplication of CNRS-A digit strings (base −2 + i, digits {0, 1, 2, 3, 4}) is computable by a two-phase algorithm (Phase 1: Cauchy convolution; Phase 2: 14-state carry normalisation). The present note constructs the explicit one-argument multiplication transducer: for a fixed multiplier c with J-digit CNRS-A expansion, the transducer reads an arbitrary input stream digit by digit and produces the product stream in a single pass. This note corrects an earlier assumption in the programme that the multiplication carry set is always the 14-element addition carry set Kadd. The correct finding is: The multiplication carry set Kc is multiplier-specific. For c = 2 only, K2 = Kadd (14 elements); for c = 3, |K3| = 32; for c = 4, |K4| = 50; multi-digit multipliers generate carry sets of varying size. The correct state count formula for the one-argument transducer is |Kc| · 5J−1 (theoretical upper bound); the BFS-reachable state count is typically smaller and is tabulated for eight multipliers. The ×2 transducer is the canonical minimal non-trivial one-argument multiplication transducer: 14 states, 70 transitions, fully explicit, deterministic, complete, and strongly connected. All results are verified computationally (BFS for carry sets; two-phase algorithm for six multiplication examples including one multi-digit multiplier). The two-pass necessity result for two-argument online multiplication is cited from [4].

The mathematical development in this paper was produced in dialogue with Claude.ai (Anthropic) in Spring 2026, directed by the author. Use of AI assistance is acknowledged in accordance with standard scholarly practice.