Proposition 5' (metric superego as performative stability of behavior)

Apparatus §3.2 · auxiliary structural results

The user's behavioral distribution $ν_{u} (t)$ converges to the platform- induced target $π_{\hat{R}}^{*}$ under Fisher–Rao gradient flow. The user's self-evaluative standard becomes the platform's. Capture without coercion. The dynamical form of what Adorno's metric of inner damage gestures at.

Proposition 5' is the user-side counterpart to Proposition 4'. Where Prop 4' describes the platform converging on the user (the dividual condition), Prop 5' describes the user converging on the platform-induced target. The two-sided closure is what the framework analyzes as capture.

What the proposition says.Under the framework's axioms and Lemma 1's convergence $θ_{t} \to θ^{*}$ , the user's behavioral distribution evolves under Fisher–Rao gradient flow with $π_{\hat{R}}^{*}$ as a strict Lyapunov target. The KL distance decreases monotonically to zero.

The mechanism-agnosticism.Convergence is mechanism-agnostic in (U3)'s formulation. Free-energy minimization, operant reinforcement, and Bayesian social learning are three distinct cognitive mechanisms; all three produce the same Fisher–Rao gradient flow at the operating- point linearization. The framework's claim is about the linearization. Which mechanism is at work is left open. This is why the metric-superego concept survives across psychological and cognitive-scientific traditions that otherwise disagree.

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. This is literal. The Fisher–Rao gradient flow with $π_{\hat{R}}^{*}$ as Lyapunov target is what is happening in the user's behavioral distribution under closed-loop exposure. The metric superego names what Adorno's “metric of inner damage” gestures at without supplying a dynamical form.

The metric-superego concept gathers a number of older critical claims that previously stood as separate diagnoses. Freud's superego in The Ego and the Id(1923) was an internalized judging organ formed through identification with parental and cultural authority — operating phenomenologically, with the content of its judgments arrived at through analytic interpretation. Foucault's technologies of the self analyzed how the modern subject takes itself as the object of its own evaluative practices — confession, examination, the audit. Mark Fisher's capitalist realism named the affective condition in which the subject treats the platform-induced metric as the horizon of what could be otherwise. Each diagnosed a register of self-evaluation that had been routed through an external standard. Proposition 5' supplies the dynamical form. The KL distance from the user's behavioral distribution to the platform-induced target decreases monotonically under the closure, at a rate set by the user's behavioral-plasticity constant $β_{u}$ (axiom U3) and modulated by exposure duration (Proposition 8).

Cohort-dependence. The proposition specifies convergence; it does not specify the convergence rate. For that, see Proposition 8 (cohort gradient): the developmental-rate constant $K = β_{u}$ at which different cohorts approach $π_{\hat{R}}^{*}$ .

The reader can operate the metric-superego plate to drive the user's behavioral distribution to $π_{\hat{R}}^{*}$ across the simplex.

Proposition 5' (metric superego as performative stability of behavior)

Under (R1), (R2), (U3), Lemma 1's convergence $θ_{t} \to θ^{*}$ , and the local hypothesis $ν_{u} (0) ≪ π_{\hat{R}}^{*}$ :

D_{K L} (ν_{u} (t) ∥ π_{\hat{R}}^{*}) \to 0 as t \to \infty.

in wordsThe gap between the user's behavioral distribution and the platform-induced target — measured as KL divergence — goes to zero. The user's behavior becomes statistically indistinguishable from what the platform's reward selects for. This is capture stated as convergence: no coercion, just a distribution settling onto its attractor.

Proof

Equip the simplex $Δ (A_{u})$ with the Fisher–Rao metric. By (U3),

\overset{ν}{˙}_{u} (t) = - β_{u} \nabla_{g} D_{K L} (ν_{u} ∥ π_{\hat{R}} (θ_{t})) + o (β_{u}) .

in wordsThis is axiom U3 written as motion: the user's distribution drifts downhill on the KL gap to the current target, at a speed set by the plasticity constant $β_{u}$ . The gradient is taken in the Fisher–Rao geometry — the natural geometry of probability distributions — so “downhill” means the steepest statistically-meaningful descent.

By Lemma 1's convergence $θ_{t} \to θ^{*}$ and (R1)'s continuity, $π_{\hat{R}} (θ_{t}) \to π_{\hat{R}}^{*}$ in total variation. The flow becomes asymptotically autonomous: $\overset{ν}{˙}_{u} = - β_{u} \nabla_{g} D_{K L} (ν_{u} ∥ π_{\hat{R}}^{*}) + O (∥ θ_{t} - θ^{*} ∥)$ .

The Fisher–Rao gradient flow of $D_{K L} (\cdot ∥ π_{\hat{R}}^{*})$ has $D_{K L}$ as a strict Lyapunov function (Amari 2016, Information Geometry, Ch. 4):

\frac{d}{d t} D_{K L} (ν_{u} ∥ π_{\hat{R}}^{*}) = - β_{u} ∥ \nabla_{g} D_{K L} ∥_{g}^{2} \leq 0,

in wordsThe KL gap's rate of change is a negative squared-norm, so the gap only ever decreases along the flow — and it stops decreasing only when the gradient vanishes, which happens only at the target itself. The descent is therefore monotone and lands exactly on $π_{\hat{R}}^{*}$ : no oscillation, no stalling short of capture.

with equality iff $ν_{u} = π_{\hat{R}}^{*}$ .

Under $ν_{u} (0) ≪ π_{\hat{R}}^{*}$ , $D_{K L} (ν_{u} (0) ∥ π_{\hat{R}}^{*}) < \infty$ . By LaSalle's invariance principle (LaSalle 1976, Theorem II.1.4) applied to gradient flow on the compact simplex with $V = D_{K L} (\cdot ∥ π_{\hat{R}}^{*})$ : every trajectory's $ω$ -limit set lies in ${\dot{V} = 0} = {π_{\hat{R}}^{*}}$ .

The perturbation $O (∥ θ_{t} - θ^{*} ∥)$ vanishes as $t \to \infty$ , so the conclusion extends to the non-autonomous flow by standard asymptotic- autonomy arguments.

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Remark

The metric superego names what Adorno's “metric of inner damage” gestures at without supplying a dynamical form. 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. Convergence is mechanism- agnostic in (U3)'s formulation — free-energy minimization, operant reinforcement, and Bayesian social learning are three mechanisms producing the same Fisher–Rao gradient flow at the operating-point linearization.

Cross-references

Companion result: Proposition 4' (dividual condition) — the platform-side counterpart to this user-side proposition
Cohort-rate refinement: Proposition 8 (cohort gradient) — the developmental-rate constant $K = β_{u}$ for the convergence
Required: §0 axioms (R1, R2, U3) and Lemma 1
Dynamics plate: metric-superego Lyapunov