Proposition 7' (libidinal routing via explaining-away)

Apparatus §3.4 · the four registers coupled

The four registers — somatic, cognitive, political, kinaesthetic — start factorially independent in the pre-closure prior. Under closed-loop joint observation through the platform's observation channel $O$ , they become dependently distributed. Explaining-away does the coupling. The four-register subject is libidinally one under the closure. This is the formal content of Lyotard's libidinal economy claim.

The pre-closure baseline assumes a factorizing prior: $p_{0} (u^{S}, u^{C}, u^{P}, u^{K}) = \prod_{a} p_{0}^{a} (u^{a})$ . The four registers are independent under the prior. The framework's claim is that closed-loop joint observation through the platform's observation channel $O$ couples the registers.

The coupling mechanism: explaining-away. A textbook property of Bayesian networks (Pearl 1988, Probabilistic Reasoning in Intelligent Systems, Ch. 2). When an observation $y_{s}$ depends jointly on two register variables $u^{i}$ and $u^{j}$ in a non-separable way, observing $y_{s}$ introduces posterior dependence between $u^{i}$ and $u^{j}$ even if they were independent in the prior. The platform's $O$ is non-separable across registers (by construction — it observes engagement-events that depend on the user's full state). So joint observation couples the registers via explaining-away.

What the proposition says. Two claims that turn out to be equivalent: (i) the posterior joint distribution over $(u^{i}, u^{j})$ does not factor; (ii) the posterior mutual information $I_{Π_{t}} (u^{i}; u^{j}) > 0$ strictly. The equivalence is the definitional content of mutual information as the KL divergence from joint to product-of- marginals.

What the proposition authorizes the prose to claim. The four registers are libidinally routed under the closure. Lyotard's libidinal economy — desire flowing across registers without respect for analytic separations — is what the closed loop produces operationally. Subject, Citizen, Person, Consumer, once dynamically decoupled, are now dependently distributed via the platform's observation channel. The pre-closure baseline (factorizing prior) is a stipulated modeling choice: the framework takes it as given and specifies what libidinal routing is, relative to it.

Each of the four registers had its own critical tradition. The body had phenomenology, somatic theory, affect studies. The mind had cognitive science and the discourse on attention. The political had ideology critique and the literatures on the public sphere. The kinaesthetic had Bourdieu's work on habitus and the broader sociology of bodily practice. Each tradition presupposed that its register could be analyzed at least relatively independently of the others — that the body did not need to be modeled jointly with the political imaginary, that cognition did not need to be modeled jointly with gestural memory. Proposition 7' specifies the structural conditions under which that presupposition breaks. Under closed-loop joint observation through $O$ , the four registers become posterior-dependent; the analytic separations the older traditions assumed available no longer hold at the level of the platform's filter.

The dynamical complement. §3.5 (the somatic Hawkes instrument) supplies the dynamical form of the same coupling: the cross-register coupling matrix $α = (α_{k l})$ with non-zero off-diagonals is the dynamical machinery that produces the joint observability the explaining-away argument here uses statically.

The reader can operate the libidinal-routing plate to watch the 4×4 mutual- information matrix of the four registers form across time — off-diagonals climbing from cream toward oxblood as the closure converges.

Proposition 7' (libidinal routing via explaining-away)

Let the prior factor: $p_{0} (u^{S}, u^{C}, u^{P}, u^{K}) = \prod_{a} p_{0}^{a} (u^{a})$ . Let $O (\cdot ∣ u)$ be non-separable in $(i, j)$ for some $i \neq = j$ . Then for any $t \geq 1$ and any $y_{1 : t}$ in a positive-measure subset of trajectories under $D (θ^{*})$ :

(i) Non-factorization.

Π_{t} (u^{i}, u^{j} ∣ y_{1 : t}) \neq = Π_{t} (u^{i} ∣ y_{1 : t}) Π_{t} (u^{j} ∣ y_{1 : t}) .

in wordsThe platform's joint belief about two registers does not split into a product of separate beliefs. Knowing what the platform infers about the user's somatic state changes what it infers about their political state — however independent the two were in the prior, observing the loop has tied them together.

(ii) Strict mutual information.

I_{Π_{t}} (u^{i}; u^{j} ∣ y_{1 : t}) > 0.

in wordsThe same fact stated as a number: the mutual information between the two registers is strictly above zero — real, measurable shared information. This is the exact sense in which the registers are libidinally routed; the plate visualizes it as the off-diagonal cells of the 4×4 information matrix lighting up.

Proof

By Bayes,

Π_{t} (u^{i}, u^{j} ∣ y_{1 : t}) \propto p_{0}^{i} (u^{i}) p_{0}^{j} (u^{j}) \cdot L_{t} (u^{i}, u^{j}),

in wordsBy Bayes' rule the joint posterior is the two independent priors multiplied by a likelihood term $L_{t}$ that ties the registers together through the shared observations. The whole question of coupling reduces to one thing: does $L_{t}$ itself factor into separate pieces for each register, or not?

where $L_{t} (u^{i}, u^{j}) = \int [\prod_{s} O (y_{s} ∣ u)] \prod_{a \neq = i, j} p_{0}^{a} (u^{a}) d u^{a}$ .

The posterior factors iff $L_{t}$ factors as $h_{i} (u^{i}) h_{j} (u^{j})$ , equivalently iff the mixed partial $\partial^{2} lo g L_{t} / \partial u^{i} \partial u^{j}$ vanishes identically. By non-separability of $O$ in $(u^{i}, u^{j})$ , the single-step mixed partial $\partial^{2} lo g O (y_{s} ∣ u) / \partial u^{i} \partial u^{j}$ is non-zero on a positive-measure set of $(y_{s}, u)$ for each $s$ . The marginal-likelihood mixed partial is

\frac{\partial ^{2} lo g L _{t}}{\partial u ^{i} \partial u ^{j}} = \int_{u^{- ij}} [s \sum \frac{\partial ^{2} lo g O ( y _{s} ∣ u )}{\partial u ^{i} \partial u ^{j}}] \cdot \frac{p _{0}^{- ij} ( u ^{- ij} ) \prod _{s} O ( y _{s} ∣ u )}{L _{t} ( u ^{i} , u ^{j} )} d u^{- ij} .

in wordsThis is the coupling test. A likelihood factors exactly when its log has zero cross-derivative in the two registers, so the left side is the quantity that has to vanish for the registers to stay independent. It is an average of the platform sensor's own cross-derivative — how jointly the sensor responds to the two registers at once. When the sensor is non-separable, that inner term is non-zero, the whole integral is non-zero, and the posterior cannot factor. The coupling is inherited directly from the sensor.

This is a continuous function of the observed trajectory $y_{1 : t}$ that depends non-trivially on $y_{1 : t}$ . Its vanishing set is a closed measure-zero subset of trajectory space under $D (θ^{*})$ . For $y_{1 : t}$ in the complementary positive- measure set, $L_{t}$ does not factor. Hence (i). Equivalent to (ii) since $I_{Π_{t}} (u^{i}; u^{j}) = D_{K L} (Π_{t} (u^{i}, u^{j}) ∥ Π_{t} (u^{i}) Π_{t} (u^{j})) > 0$ .

∎

Remark

Proposition 7' supplies the formal content of Lyotard's libidinal economyclaim. The four registers — Subject, Citizen, Person, Consumer — once dynamically decoupled, are dependently distributed under closed-loop, with the dependence routed through the platform's joint observation channel. The pre-closure baseline is a stipulated modeling choice (factorizing prior $p_{0}$ ). Closed-loop joint observation couples the registers via explaining-away — a textbook property of Bayesian networks (Pearl 1988, Ch. 2).

Cross-references

Companion result: Proposition 6 (Kuramoto phase-locking) — captured resonance is one specific case of the cross-register coupling
Required: Proposition 4' (dividual condition) — the platform's posterior is what carries the coupling formally
Dynamical complement: §3.5 somatic Hawkes (the dynamical form of the same cross-register coupling)
Dynamics plate: libidinal routing