When the Constraint Becomes a State Variable Slow-fast dynamics, a critical self-referential coupling, and a tail-index ratchet in self-referential multiplicative heavy-tailed systems

Abstract The two companion papers share one tacit premise: the allocation exponent θ is exogenous, the tail index κ★ is its passive consequence, and the lower bound on maintenance dissipation is a constraint imposed for a given κ★. This paper removes that premise. Once the allocation tilt is set by the system's internal model of itself, so that θ = θ(M), the tail index becomes a dynamical variable and the floor of the second paper continues to hold instant by instant while its value drifts with the state: the constraint turns from a boundary condition into a state variable. The situation is treated under a slow-fast decomposition, with the internal model M slow and the size distribution fast, the latter reaching instantaneously the quasi-stationary law corresponding to the current θ. Three results follow. First, the stability of the self-consistent internal model is governed by the loop gain G = ρ_θ·C′(θ), and C′(θ) has the closed form 2/[θ(κ★−1)²], so there is a critical self-referential coupling ρ_c(κ★) = √(2|m|)·(κ★−1)²/(2√κ★), the system being stable exactly when ρ_θ < ρ_c. Second, ρ_c is strictly increasing in κ★ and vanishes as κ★ → 1⁺, so along a trajectory of falling tail index the critical value itself falls: once the threshold is crossed, halting further growth of the coupling does not restore stability. This is a ratchet. Third, along the same direction the precision factor R(κ★) of the second paper falls strictly and vanishes at κ★ = 2, so thermodynamics supplies no restoring force whatever; the only restoring force is the tilting cost D(θ), which at the operating point used here is an order of magnitude weaker than the feedback. The readable conclusion is that a complex system possessing a representation of itself can drift into the region where physical constraints barely bind it, while the physical laws themselves are entirely unchanged: transfer fidelity remains one and bite goes to zero. Section 6 gives an independent amplifying mechanism, a recognition-delay spiral, whose ignition point is computed at κ★ = 2.53, that is before the floor vanishes. All results are deterministic algebra, root-finding and ODE integration, with no stochastic content.