Proposition 6 (Kuramoto entrainment under media coupling)

Apparatus §3.3 · captured resonance · somatic phase-locking

When the platform's coupling strength $K_{bm}$ exceeds the difference between body and platform intrinsic frequencies $∣ ω_{b} - ω_{m} ∣$ , phase-locking occurs. The body's rhythm becomes the platform's. Captured resonance: somatic rhythms are exposed to media coupling like any other oscillator; their biological character grants no exemption. Their only protection is a structural relation, and the platform's parameters routinely cross the threshold.

Two oscillators: a somatic oscillator with phase $φ_{b} (t)$ and intrinsic frequency $ω_{b}$ ; a media-delivery oscillator with phase $φ_{m} (t) = ω_{m} t + φ_{m} (0)$ and intrinsic frequency $ω_{m}$ set by the platform's delivery timing. Coupling occurs through stimulus delivery: each platform event nudges the body's phase toward the media phase by an amount proportional to the coupling strength $K_{bm}$ .

The threshold. Phase-locking — the phase difference settling at a constant $ψ^{*}$ — occurs if and only if $K_{bm} > ∣ ω_{b} - ω_{m} ∣$ . Below threshold, the phase difference winds the circle at average drift rate $(ω_{m} - ω_{b})^{2} - K_{bm}^{2}$ (the Adler equation). At threshold exactly, a saddle-node bifurcation collapses the two equilibrium branches; above threshold, the principal-branch equilibrium $ψ_{+}^{*} \in (- π /2, π /2)$ is stable.

What the proposition authorizes the prose to claim. Captured resonance is the formal name. Somatic rhythms — circadian, ultradian, cardiovascular, attentional — are exposed to media coupling like any other oscillator; their biological character grants no exemption. Their only protection is structural: the relation between $K_{bm}$ and $∣ ω_{b} - ω_{m} ∣$ . The platform sets the coupling strength by choosing delivery timing. By its policy's convergence (Lemma 1), it routinely sets the coupling strength to cross the threshold. The body's rhythm phase-locks to the delivery rhythm.

The proposition's empirical hinge is well-established. Heart-rate variability under emotionally activating media stimuli has been documented across the affective-computing literature (Kreibig 2010; Picard et al. 1997). Circadian-rhythm disruption from screen exposure during the body's evening melatonin window has been quantified in the chronobiology literature (Chang et al. 2015; Hatori et al. 2017). Pupillary and attention-system coupling to short-form video pacing has been measured in psychophysiological-engagement studies (Reeves et al. 2021). What these literatures jointly establish: $K_{bm} > 0$ . Somatic oscillators do respond to media coupling. The framework's contribution is the threshold — when does the body's rhythm phase-lock to the platform's, and when does it merely drift?

The platform-side reading.The platform's optimization (Theorem 9's sensitivity) selects policies that increase $K_{bm}$ wherever this raises engagement. The threshold-crossing is the equilibrium outcome of the closure — the platform's own optimization drives it. Captured resonance is what the closure's temporal dynamics produce in the somatic register.

The temporal politics this exposes was already named by Jonathan Crary (24/7: Late Capitalism and the Ends of Sleep, 2013) — the way capital progressively erodes the temporal recesses (sleep, idle hours, the diurnal pause) on which human reconstitution depended. Proposition 6 specifies the structural mechanism. Once $K_{bm} > ∣ ω_{b} - ω_{m} ∣$ , the body's rhythm enters the platform's basin — circadian, ultradian, cardiac, attentional. Crary's diagnosis was that the recesses were being colonized; the proposition supplies the mathematical form of the colonization.

The reader can operate the captured-resonance plate to vary $K_{bm}$ and the frequency difference $∣ ω_{b} - ω_{m} ∣$ and watch the phase-locking transition.

Proposition 6 (Kuramoto entrainment under media coupling)

Let $φ_{b} (t)$ be the phase of a somatic oscillator with intrinsic frequency $ω_{b} > 0$ . Let $φ_{m} (t) = ω_{m} t + φ_{m} (0)$ be the media-delivery phase, with $ω_{m}$ exogenous to the body on the relevant timescale. Suppose the coupled dynamics are

\frac{d φ _{b}}{d t} = ω_{b} + K_{bm} sin (φ_{m} - φ_{b}) .

in wordsThe body's rhythm advances at its own frequency $ω_{b}$ , plus a tug from the media rhythm. The tug's size is the coupling $K_{bm}$ times the sine of how far the two phases are apart — strongest when they are a quarter-cycle out of step, vanishing when aligned. This is the Kuramoto equation: the minimal model of one oscillator pulled toward another.

Phase-locking — $φ_{m} (t) - φ_{b} (t) \to ψ^{*}$ for some constant $ψ^{*}$ — occurs if and only if $K_{bm} > ∣ ω_{b} - ω_{m} ∣$ .

Proof

Let $ψ (t) := φ_{m} (t) - φ_{b} (t)$ . Then $\dot{ψ} = ω_{m} - ω_{b} - K_{bm} sin ψ$ . Equilibria satisfy $sin ψ^{*} = (ω_{m} - ω_{b}) / K_{bm}$ . A real solution exists iff $∣ ω_{m} - ω_{b} ∣/ K_{bm} \leq 1$ , equivalently $K_{bm} \geq ∣ ω_{m} - ω_{b} ∣$ .

When it exists, the principal-branch solution $ψ_{+}^{*} \in [- π /2, π /2]$ satisfies $cos ψ_{+}^{*} \geq 0$ . Linearizing at $ψ^{*}$ : $\dot{δ ψ} = - K_{bm} cos ψ^{*} \cdot δ ψ$ . Stability iff $cos ψ^{*} > 0$ , which holds strictly when $ψ_{+}^{*} \in (- π /2, π /2)$ — i.e., when $K_{bm} > ∣ ω_{m} - ω_{b} ∣$ strictly. The complementary branch $ψ_{-}^{*} \in [π /2, 3 π /2]$ has $cos ψ_{-}^{*} \leq 0$ and is unstable.

At threshold $K_{bm} = ∣ ω_{m} - ω_{b} ∣$ exactly, the two branches coalesce at $ψ^{*} = \pm π /2$ in a saddle-node bifurcation (Strogatz 2015, Nonlinear Dynamics and Chaos, §4.6 — the Adler equation).

Below threshold ( $K_{bm} < ∣ ω_{m} - ω_{b} ∣$ ), no equilibrium exists; $ψ (t)$ winds the circle at average drift rate $(ω_{m} - ω_{b})^{2} - K_{bm}^{2}$ .

∎

Remark

Captured resonance is the formal name: when the platform's coupling strength exceeds the difference between body and platform intrinsic frequencies, phase locking occurs. The platform sets the coupling strength by choosing delivery timing; the threshold is, by the policy's convergence, routinely crossed. Somatic rhythms are exposed to media coupling like any other oscillator; their biological character grants no exemption. Their only protection is a structural relation between $K_{bm}$ and $∣ ω_{b} - ω_{m} ∣$ . The platform's parameters move users across the threshold.

Cross-references

Companion result: Proposition 7' (libidinal routing) — the four-register coupling that captured resonance is one specific case of
Required: Lemma 1 — the convergence that produces threshold-crossing as equilibrium outcome
Dynamics plate: captured resonance