Metric superego

What the plate operates. Proposition 5' establishes that under axiom U3 and Lemma 1's convergence ${\theta }_t\to {\theta }^{\ast }$, the user's behavioral distribution converges to the platform-induced target ${\pi }_{\hat{R}}^{\ast }$. The plate visualizes this on the 3-choice simplex.

The simplex. Each user-action distribution ${\nu }_u$ over three choices $(a_1,a_2,a_3)$ is a point in the triangle. Corners are pure-strategy distributions; the center is uniform. The platform-induced target ${\pi }_{\hat{R}}^{\ast }$is another point somewhere in the triangle — wherever the platform's reward functional has placed the maximum over user actions under $\mathcal{D}({\theta }^{\ast })$.

The dynamics. Fisher–Rao gradient flow of $D_{KL}(\nu \Vert {\pi }_{\hat{R}}^{\ast })$ moves ${\nu }_u(t)$in the “straight” direction toward ${\pi }_{\hat{R}}^{\ast }$on the simplex (geodesic in the Fisher–Rao metric). At each step the KL distance strictly decreases (LaSalle's invariance principle applied to gradient flow on the compact simplex).

What the Lyapunov readout shows. The right panel plots $D_{KL}({\nu }_u(t)\Vert {\pi }_{\hat{R}}^{\ast })$ on log scale. The curve is strictly decreasing — the Lyapunov function certificate. Convergence rate scales with the plasticity parameter ${\beta }_u$; the half-life of the KL distance is roughly $\log 2/(2{\beta }_u)$.

What the proposition authorizes the prose to claim. The user's self-evaluative distribution drifts toward the platform's ledger. The judging organ is the platform's metric, internalized as the user's standard. The convergence is mechanism-agnostic — free- energy minimization, operant reinforcement, and Bayesian social learning all produce the same Fisher–Rao gradient flow at the operating-point linearization. The plate renders the structural form; specific user-side mechanisms vary the rate but not the destination.