Description This paper develops a non-degenerate k-ary extension of a binary Bellman-type model for operational signalling under measurement dependence. In earlier work, operational signalling was shown to emerge under repeated interaction even when no single round exhibits identifiable control. However, extending this framework beyond the binary setting introduces structural degeneracies: naive k-ary generalisations collapse or overestimate the signalling coefficient. This paper resolves these issues by introducing a block-overlap principle, which leads to a corrected one-step signalling coefficient and a complete structural characterisation of the model. The coefficient admits a max-linear formulation over block scales, revealing an underlying piecewise-linear geometry and enabling exact analysis. Main results Exact one-step coefficientA corrected k-ary signalling coefficient Γₖ (p) defined via block overlap, admitting a max-linear representation. Sharp bounds 1Hk−1≤Γk (p) ≤1, 1H₊-₁ ₖ (p) 1, Hk−11≤Γk (p) ≤1, where Hk−1H₊-₁Hk−1 is the harmonic number. Harmonic optimalityThe harmonic profile pr∝1/rpᵣ 1/rpr∝1/r uniquely minimises the signalling coefficient. Geometric structureΓₖ (p) is piecewise-linear and determined by block-scale extrema. Multi-round amplification (conditional) Under structural assumptions, signalling amplifies according to an (p) =1− (1−Γk (p) ) n. aₙ (p) = 1 - (1 - ₖ (p) ) ⁿ. an (p) =1− (1−Γk (p) ) n. Significance This work establishes the correct k-ary formulation of the operational signalling model and identifies the harmonic profile as the critical extremal structure. It provides the structural foundation for subsequent developments in the series, including adaptive dynamics, persistence of multiscale structure, and global control laws. Keywords operational signalling; measurement dependence; Bellman recursion; k-ary models; harmonic distribution; max-linear structure; information amplification; multiscale systems Notes This paper forms part of a series on operational signalling under measurement dependence. See accompanying repositories for related works in the series.
Bob Jefferson (Sat,) studied this question.