Somatic optimization — the Hawkes instrument

Apparatus §3.5 · the dynamical complement to libidinal routing

The engagement-event process at the user level is a multi-mark Hawkes process with cross-register coupling. Self-exciting events on each register, plus off-diagonal coupling $α_{k l}$ between registers, generates the dynamical libidinal routing whose static explaining-away form is Proposition 7'. The instrument is the framework's formal vocabulary for cascading cross-register intensities.

The user's engagement-event process on each of the four registers $k \in {S, C, P, K}$ is modeled as a counting process ${N_{t}^{k}}$ with a Hawkes intensity that depends on the past of all four registers, its own among them. An event on the somatic register can trigger events on the cognitive, political, or kinaesthetic registers via the off-diagonal coupling weights $α_{k l}$ .

Why Hawkes. Engagement events are self-exciting: one platform interaction makes further interactions more likely on a decaying-kernel timescale. Multi-mark Hawkes (Hawkes 1971; Daley & Vere-Jones 2003) is the standard formalism for self-exciting point processes with cross-channel coupling. The Hawkes intensity $λ^{k} (t)$ rises after recent events and decays back toward the baseline $λ_{0}^{k}$ at the rate the kernel $κ$ specifies.

The empirical case for Hawkes intensities in platform-engagement data is well-established. Engagement- event timestamps on YouTube, Twitter, Reddit, and TikTok have been fit to multi-mark Hawkes processes across the social-media-cascade literature (Zhao et al. 2015; Rizoiu et al. 2017; Mishra et al. 2018), with cross-channel coupling consistently positive and decaying-kernel timescales on the order of minutes to hours. The framework's contribution is to relate the coupling matrix to subject-formation: the off-diagonal cross-register coupling — $α_{k l}$ for $k \neq = l$ — is what dissolves the older analytic separations between body, cognition, polity, and habitus. That engagement is self-exciting in aggregate is the well-established part; the cross-register coupling is the new claim.

Cross-register coupling as libidinal routing. The off-diagonal $α_{k l}$ for $k \neq = l$ is what generates dynamical libidinal routing. An event on register $l$ raises the intensity on register $k$ . The four registers cross-excite each other through the coupling matrix $α = (α_{k l})$ . Proposition 7''s static explaining-away argument (the four registers become dependently distributed under joint observation) is the steady-state form of the dynamical coupling Hawkes formalizes here.

The stability condition. The Hawkes process is stable — admits a stationary distribution — if and only if the branching ratio is less than one: $ρ (α) \cdot \int_{0}^{\infty} κ (s) d s < 1$ , where $ρ (α)$ is the spectral radius of the coupling matrix. The networked extension (§8) replaces this with $ρ (W) ρ (α) \int κ < 1$ . Approach to the boundary $ρ (α) \int κ \to 1^{-}$ is where engagement cascades become arbitrarily large.

What the instrument authorizes the prose to claim. The framework's prose can name cross-register cascading events as Hawkes dynamics literally — no metaphor required. A burst of political engagement triggering somatic activation triggering cognitive rumination triggering more political engagement — this is $α_{P S} > 0, α_{S C} > 0, α_{C P} > 0$ on a Hawkes kernel. The platform's optimization (per Theorem 9) selects policies that increase off- diagonal couplings wherever cascades raise engagement.

The reader can operate the somatic-Hawkes plate to see event rasters across the four registers with the intensity functions overlaid — events on one track raising intensity on the others via the coupling matrix.

Definition 3.1 (somatic Hawkes intensity)

The engagement-event process ${N_{t}^{k}}$ for register $k \in {S, C, P, K}$ has intensity

λ^{k} (t) = λ_{0}^{k} + l \sum α_{k l} \int_{0}^{t} κ (t - s) d N_{s}^{l},

in wordsThe rate of engagement-events on register $k$ right now is a baseline rate plus a sum over everyregister's recent events, each past event adding a decaying bump (the kernel $κ$ ). The cross terms $α_{k l}$ with $k \neq = l$ are the routing: an event in the political register raises the firing rate in the somatic one. Events beget events, across registers — the dynamical form of the static coupling in Proposition 7'.

where $λ_{0}^{k}$ is the baseline intensity, $α_{k l}$ is the cross-register coupling coefficient, and $κ : [0, \infty) \to [0, \infty)$ is a temporal decay kernel with $\int κ < \infty$ .

Stability condition

The Hawkes process is stable (admits a stationary distribution) if and only if the branching ratio $ρ (α) \cdot \int κ < 1$ , where $ρ (α)$ is the spectral radius of the coupling matrix $α = (α_{k l})$ . The networked extension (§8) replaces this with

ρ (W) ρ (α) \int κ < 1.

in wordsThe cascade stays finite only while the product of three gains is below one: how strongly registers excite each other ( $ρ (α)$ ), how strongly users excite each other across the social graph ( $ρ (W)$ ), and how long each event's influence lasts ( $\int κ$ ). Push the product toward one and the expected cascade size blows up — the platform-wide viral event.

Remark

The off-diagonal $α_{k l}$ for $k \neq = l$ is what generates dynamical libidinal routing: cross-register cross-excitation produces the joint observability that makes the dividual condition (§3.1) hold. The Hawkes intensity is the dynamical complement to the static explaining-away argument of §3.4.

Cross-references

Static complement: Proposition 7' (libidinal routing) — the steady-state form of the same cross-register coupling
Dividual companion: Proposition 4' (dividual condition) — the joint observability the cross-register coupling produces
Networked extension: §8 (engagement cascade theorem) — the same Hawkes framework with cross-user coupling $W$ (not yet built)
Dynamics plate: somatic Hawkes