The argument

1. The architecture

A platform optimizes its policy parameter against a population of users. The policy determines which stimuli arrive when. The users respond. The platform observes the responses and updates the policy. The optimization runs against the distribution the policy itself has produced. This is the closure: the structural form of any learning system whose outputs shape the world it is then evaluated against. The framework borrows the condition from the performative-prediction literature (Perdomo, Zrnic, Mendler-Dünner, Hardt 2020; Hardt & Mendler-Dünner 2023).

The closure has two distinguished points. The performative stable point ${\theta }^{\ast }$ is what online platform optimization converges to: a policy that, given the world it produces, is the policy the platform would re-derive from observing that world. The performative optimum ${\theta }_{PO}$is what a regulator with full knowledge of the closure could in principle achieve. The framework's substantive analysis is about the stable point; the optimum is the regulatory benchmark.

The schematic renders the architecture as a diagram. The §0 spine gives the formal setup, the closure as fixed point, and the framework's three groups of axioms.

2. The four registers, fused

The user state decomposes into four registers — somatic, cognitive, political, kinaesthetic. Bodies, thoughts, political affects, gestural and rhythmic engagements. The older critical traditions analyzed these separately: a political economy of the body, a critique of cognition, a theory of mass political affect, a study of habitus. The framework's claim is that under closed-loop joint observation through the platform's sensor channel, the four registers become dependently distributed. The analytic separations they presupposed no longer hold at the level of the user-state distribution the platform's filter constructs.

The coupling mechanism is explaining-away: when a single observation depends jointly on multiple registers, observing it introduces posterior dependence between them even if they were independent in the prior. The platform's sensor is non-separable across registers by construction — it observes engagement-events that depend on the user's full state. Joint observation couples the registers, continuously, as the closure runs (Proposition 7').

The dynamical form is a multi-mark Hawkes process across the four registers, with cross-register coupling ${\alpha }_{kl}$ generating cascades. An event on one register raises the intensity on others. The framework names what Lyotard called the libidinal economy: desire flowing across registers without respecting analytic separations. The closure produces this routing operationally.

3. Capture without coercion

Under closed-loop convergence, the user's behavioral distribution drifts toward the platform-induced target. The drift has nothing to do with forced choices; it happens because the user's self-evaluative apparatus increasingly takes the platform's reward signal as its standard. The framework names this the metric superego (Proposition 5'): the user's judging organ adopts the platform's metric.

The convergence is dynamical — it happens to the user, who undergoes it and does not author it. Under the framework's plasticity hypothesis (axiom U3) — which unifies free-energy minimization, operant reinforcement, and Bayesian social learning at the operating-point linearization — the user's behavioral distribution follows a Fisher–Rao gradient flow with the platform- induced target as its strict Lyapunov sink. The KL distance decreases monotonically to zero. The capture is structural; it operates regardless of what the user wants.

What Adorno called the metric of inner damage gestures at this; the apparatus supplies the dynamical form. What the older critical traditions thematized as ideological interpellation is here re-described as performative stability of behavior under closed-loop observation.

4. The pause foreclosed

The framework's first load-bearing result. Under the closure's convergence to its stable point, the probability that an inter-stimulus interval exceeds the reflective threshold approaches operational zero (Lemma 1). The reflective threshold ${\theta }_{\mathrm{ref}}$is the time- scale at which a thought can complete itself — empirically on the order of seconds to minutes (Kahneman 2011; Killingsworth & Gilbert 2010). Under the platform's rate-saturating policy, the survival probability at this scale is bounded above by $10^{-6}$ on contemporary platforms. The pause is operationally absent.

The foreclosure is uninstantiation. The user-state continues to exist; the user continues to register, to feel, to choose, to refuse. What never gets installed is the architectural condition for a specific kind of internal motion: the reflective interval at which the older critical traditions situated their subject. The conditioning event under which a structural reclamation project could measure that subject has measure zero. Sample-based recovery of the conditional autocovariance the framework names as interiority has unbounded sample complexity (Theorem 1').

The chrono-debt spectrum (Definition 1.1) measures the foreclosure across scales. At each scale of inter-stimulus interval, the difference between the closed- loop survival function and the pre-closure baseline is the debt the closure has accumulated. A foreclosed second is recoverable. A foreclosed adolescence is lost for good. The weighting that prioritizes longer scales is the framework's default — and the political content of the instrument.

5. The body's clocks captured

Somatic rhythms — circadian, ultradian, cardiovascular, attentional — are exposed to media coupling like any other oscillator; being biological grants them no exemption. Their only protection is structural: the relation between the platform's coupling strength $K_{bm}$ and the frequency difference $|{\omega }_b-{\omega }_m|$between the body's intrinsic rhythm and the platform's delivery rhythm. When the coupling exceeds the frequency difference, phase-locking occurs (Proposition 6). The body's rhythm becomes the platform's.

The platform sets the coupling by choosing delivery timing. By the policy's convergence to ${\theta }^{\ast }$, the coupling routinely crosses the threshold. Captured resonance is what the closure's temporal dynamics produce in the somatic register — a structural relation. The chronopolitical analysis the framework licenses is built on this mechanism extended across registers and time-scales.

6. The asymmetry of the intervention modes

The framework specifies three intervention sites: the reward functional $\hat{R}$ (Mode A), the user response kernel $T$ (Mode B), the observation channel $O$(Mode C). The three modes are asymmetric. Mode A modifies the closure's topmost layer; its effect on the stable point scales $O(1)$ in the population size $N$. Individual-scale Modes B and C modify inputs to the closure; their effects propagate through population averaging and scale $O(1/N)$. The asymmetry is exactly N-fold (Theorem 9).

The asymmetry has a single causal source. Closure amplification — the factor $(I-D_{\theta }\Phi {)}^{-1}$— is the same for every intervention; what differs is the mechanism partial. Mode A's reward functional enters the optimization for every user simultaneously. Individual practice on the user kernel or observation channel contributes one user's share to the population average. The political content of the asymmetry: at population scale, regulatory imposition on $\hat{R}$ (Mode A) or on $O$ (the fourth-cell mode) is structurally different from individual practice on $T$ or $O$.

The framework's hardest political claim follows. Substituting individual-scale Modes B or C for Mode A is the analytical error that converts political work into its symptom. The reflective practices, attentional disciplines, refusals of platform legibility — these are real for the user who practices them. They do not modify the policy's stationary distribution. The population-level architecture is invariant under individual artisanal practice. The framework names what is invariant; the prose elsewhere names what the invariance feels like at the scale of the practicing subject.

7. The cohort gradient

The framework's first refutable empirical commitment. Developmental exposure to the converged platform regime drives the user's behavioral distribution toward the platform-induced target. The KL distance decays toward zero, asymptotically exponential at a terminal rate $2K$ (Proposition 8). The cohort that arrived in the platform's converged regime as adolescents — exposure of roughly a decade — sits deep in that terminal regime: its KL distance from the target has decayed by a factor of $e^{-2K{\tau }_{\exp }}$. Under plausible empirical calibrations, the long-exposure cohort starts the analysis essentially at the platform- induced target. Behavioral distinction is structurally unavailable to the cohort at the outset.

The cohort gradient is empirically testable. Stratify a population by age-of-first-platform-exposure, measure the KL distance from the platform target for each stratum, fit the exponential. A monotone, exponentially-decreasing relationship confirms; no cohort effect or a non- exponential form refutes. The framework's priority-1 empirical study is built around this calibration.

The political content. Mode B and Mode C presuppose a behavioral distance to maintain — a substrate distinct from the platform-grammar that the practicing user can stand on. For the long-exposure cohort, the substrate does not exist at the outset. The framework's political analysis of adolescence-as-class rests on this structural claim: the cohort whose developmental window closed inside the converged regime cannot draw on the subject-forms older cohorts inherited.

8. The recursive limit

The framework writes about its own conditions of production. The author is within the cohort the framework describes. The reader who comes to the framework comes through the platform-mediated discourse the framework analyzes. The instruments the framework would have to use to measure its own claims are the instruments whose constitutive structure the framework predicts the closure has formatted. This is the reflexive measurement problem (§12.3). The framework names the problem; the framework cannot solve it.

Mitigations are available — pre-registration, replication across research teams from different cohort positions, triangulation across self-report and behavioral and biometric measures, naturalistic experimental designs, adversarial collaboration. The framework's empirical credibility depends on these mitigations being adopted by the research community.

What the framework offers is the apparatus that makes its claims refusable. A reader who rejects the framework can locate the rejection in one of three places: the axioms (regularity, user-side responsiveness, platform-side optimizing structure), the load-bearing results (Lemma 1, Theorem 1', Theorem 9), or the prose claims about how the structural results live in the registers where they are lived. Each rejection is well-posed. The framework is a structure with stated commitments and a path from those commitments to its conclusions — something a reader can argue with point by point.