Timescale Identification from $k$-Step Transition Operators: The Structural Entropy Spectrum

Agents and world models discretise experience into a transition graph at one fixed step, yet behaviour has structure at several timescales. Which step should the graph be built at, and which timescales are present? Multiscale tools (Markov stability, the map equation at Markov time, implied timescales of Markov state models) sweep a time parameter but attach no significance statement to the scales they report. We make the step count k an argument of a clustering objective on the k-step operator P^k and calibrate it against a matched null: chains whose stationary flow matrix has identical margins and dwell (self-loop) mass. Our theory uses structural entropy (SE), a two-level code-length objective: the structural entropy spectrum SE(k), its null SE_null(k) and the calibrated spectrum ΔSE(k) = SE_null(k) − SE(k). We prove that the k-dependence factors through an exact identity, SI(k;C) = Σ_a F_aa(k) log2(1/V_a) (k-step module persistence × module code length), prove that when metastable blocks have stationary volume < 1/2 the metastable partition is the unique global optimizer of the idealized in-window operator, and sketch a Simon–Ando extension to nearly uncoupled chains on a window whose endpoints scale as 1/(1−λ) for the fast and slow eigenvalue clusters, so the optimal partition path coarsens through a planted hierarchy in order. On a 36-state three-level chain, the path recovers both planted levels in order at calibrated effect sizes up to 1.8 bits, although at this fixture's modest scale separation ΔSE(k) shows no plateaus; a single-scale expander control is featureless; the spectrum is estimable from a single trajectory. On latent transition graphs from a pixel-based JEPA world model, the one-step graph's calibrated effect falls below the preregistered 0.1-bit floor, while the spectrum peaks reproducibly at k* = 11–16 with 1.6–1.9 bits across five training seeds; a self-loop control shows that this structure survives removal of dwell but that the peak location is a dwell-inflated step count (two state changes). The calibration is not SE-specific: wrapped around the map equation it passes the planted-hierarchy, single-scale and structureless controls, and around Markov stability all but the expander control; both recover two balanced blocks, SE's boundary case, and the scale boundaries move with the objective. All decisive experiments were preregistered with pass/fail gates and are reported verbatim.