AIM Framework glossary
Critical Loop Gain
Canonical definition
Critical Loop Gain (Gcrit). The Critical Loop Gain Gcrit is the threshold value of the internal loop gain G = κS · β · γ · σ'(V*) at which the fixed point of the fast valuation dynamics loses linearised stability. For the minimal three-node fast-valuation system specified in Appendix A.4, Gcrit = 1 exactly, recoverable by Routh-Hurwitz analysis of the characteristic polynomial of the Jacobian J, and independent of the relaxation times τM, τV, τS. Below threshold (G < 1) the fixed point is stable, perturbations decay, and the system behaves as a transient social-learning mechanism. Above threshold (G > 1) the fixed point is unstable, perturbations compound, and the system exhibits Mimetic Premium compounding and rivalry escalation. The bifurcation at G = 1 is necessarily a Saddle-Node Fold in the three-node skeleton with positive κS, β, γ, σ' (the Hopf locus is structurally unreachable; see Appendix A.5.3). For the joint fast-plus-slow coupled system the threshold is denoted Gcrit^joint, and for the full network-extended system Gcrit^total; both are governed by spectral conditions on the corresponding block-Jacobians, with the additive decompositions of A.5.6 and A.5.7.4 understood as small-cross-coupling expansions rather than exact stability conditions.
Explanation
The Internal Loop Gain is the dimensionless number that decides whether mimetic activation in an Individual or population decays harmlessly or compounds into runaway, and the Critical Loop Gain is the threshold it is measured against. Four quantities set the Internal Loop Gain, and each is something the architecture or its institutional environment can in principle modify: how strongly environmental Salience drives mimetic signal; the weight the Decision Hub assigns to Mimetic Desire in common-currency integration; the gain on the feedback from realised valuation to environmental Salience; and the local slope of the valuation-to-salience mapping at the operating point, the level of realised valuation at which that slope reaches zero being the Salience Ceiling. Their product crossing the threshold is what flips the system from sub-critical to supercritical. The relaxation times do not move the threshold but they govern how fast the system reaches instability once it crosses, how strongly it amplifies pre-bifurcation noise (critical slowing down), and how long a supercritical loop can compound before external reality intrudes (Decoherence Time). Subcriticality Maintenance is the architectural goal of keeping the Internal Loop Gain below the threshold across the populations and institutions in which M-dynamics operate, by combined Class 1 interventions, on the Ignition Input, and Class 2 interventions, on the three couplings inside the loop.
Epistemic status
Framework-Specific Result. Derived from Appendix A.4 by Routh-Hurwitz analysis of the linearised three-node Jacobian; numerically confirmed across four orders of magnitude in each τ.
Derivation
Appendix A, Section A.5; Appendix A, Section A.5.3; Chapter 4, Section 4.4.13.2.