Pattern Agency on Active Media: A Spectral Criterion for Self-Maintaining Patterns, with a Cross-Substrate Measurement Interface

Abstract

An agent is taken to be a pattern on an active medium: a configuration that persists by consuming the medium's free-energy throughput. Two matrices are attached to such a pattern — a state matrix C of non-negative capacities, and the transverse monodromy A, the rank-2 reduction of the linearised Poincaré return map of the pattern's cycle. Keeping them apart is load-bearing: the entries of A carry no sign restriction, while the chart and the state dynamics require non-negative capacities. Independent rescaling of the two coordinates acts on A by conjugation, and on the coupling plane as an SO(1,1) boost whose invariant interval is the loop gain L = s² − q². The complete operator invariants are therefore (Tr A, det A); individual gates are not observables, though a canonical gauge — boost to the coupling's rest frame — fixes them without any measurement, and a vanishing gate is gauge-invariant exactly on the null cone L = 0. Orbital stability is the Jury region |det A| < 1 ∧ |Tr A| < 1 + det A, whose three edges are three distinct ways of leaving homeostasis: a complex pair crossing the unit circle, an expanding direction (growth or replication), and period doubling. Rotation per return θ = maxᵢ|arg λᵢ| orders the subjects, with zone boundaries fixed by ω = μ and Tr A = 0; the discriminant factors as Δ = 4(h² + s² − q²), so an agent is a pattern whose antisymmetric coupling dominates its symmetric part — the matrix counterpart of solenoidal flow dominating dissipative flow at a non-equilibrium steady state. Because det, Tr and the spectrum are transpose-invariant, the algebra cannot say which of two coupled patterns is the agent. Orientation comes either from a thermodynamic functional, whose role is therefore symmetry breaking rather than discrimination, or from commensurability of periods — a driver dominates when τ_reflex < τ_A ≤ τ_volition/2, which is measurable with a stopwatch. Target space is admitted without teleology as the multiplicity of coexisting attracting cycles, c₂₂ = log₂|T|, so reprogramming is a basin switch driven by the informational layer alone; and an agent with L < 0 provably cannot leave the stable region by self-action at constant trace. Morphogenetic transients are handled by the mode-resolved family A(k) = J − k²D, in which Turing's criterion is a sign change of the determinant — the same edge — indexed by wavenumber rather than by time. The multi-agent step is a block construction with an anonymizing projection onto the symmetric mode, which discards the contributing index and is the existence condition for the next rung; matrix multiplication is period matching across a rung; and the coarse-graining depth available to a collective is bounded by its coupling strength, so tight coupling limits collective size. A measurement interface reduces heterogeneous systems to one ruler (kBT per characteristic time, with Kolmogorov–Sinai entropy and statistical complexity on the informational axis) and a registry of thirty-three subjects spans cellular automata, chemical oscillators, a laser soliton, self-organising dusty plasma, bacterial chemotaxis, planarian regeneration, synthetic biobots, insects, collectives, a 1948 electromechanical homeostat and language-model inference, together with analytic anchors whose matrices are known in closed form. Every grounded value carries its source, its venue type and its status (measured, counted, model assumption, derived, illustrative), and only the first constitutes external support: no metabolic rate of a synthetic biobot has been measured, and no diffusion tensor has been measured for any subject, so the gauge is fixed nowhere by experiment. Seven falsifiable predictions and twenty-five open problems are stated, the largest being that the rank-2 projection is asserted rather than constructed