CNRS Pr3-2: The Explicit One-Argument Multiplication Transducer for CNRS-A: Carry Sets, State Graphs, and Verified Examples

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:

1. 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.
2. 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.
3. 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].