Order-Dependent Update Dynamics in a Gated State Space: A Minimal Model of Path Dependence

Many sequential processes involve operations whose effects depend on ordering. This paper introduces a minimal computational toy model of order-dependent dynamics using a gated state space and two stochastic update rules. The model is not intended as an empirical claim but as a formal abstraction. We define a three-component state (S₁, S₂, S₃) with a gating variable ⊥ ∈ [0,1]. Two operators are introduced: V (perturbation) and A (audit/compression). Their sequential composition is evaluated under matched randomness. An order-dependent divergence measure is derived: Δ = |δ| (1 − λ) ⊥ where δ is a random perturbation and λ a decay rate. Simulations show bounded, monotonic divergence that persists under weak recovery and is eliminated under strong recovery. In the irreversible regime (ρ = 0), the residual after first-order divergence is bounded (|Rₜ| < 0.02); under weak recovery the residual remains small but is not formally bounded at this value. The model provides a formal mechanism for studying order-sensitive sequential updates. Illustrative analogies to relational domains are discussed separately as non-validated interpretations.