Theorem 9 (causal sensitivity in performative systems)

Apparatus §7 · the performative do-calculus

The unified sensitivity formula. The three intervention propositions of §5 — separately stated, separately proved — become corollaries of a single causal-sensitivity result. The framework's third load-bearing claim: the asymmetry between Mode A and individual-scale Modes B/C is exactly the population-averaging factor 1/N.

The five structural objects (P1)–(P5) and the closure define a causal model. The DAG of structural dependencies, ignoring closure, is ${T, O, \hat{R}} \to π_{θ} \to D (θ) \to F (θ) \to Φ \to θ^{*}$ . The closure introduces $θ^{*} \to π_{θ^{*}}$ , creating a cycle the fixed-point equation $θ^{*} = Φ (θ^{*})$ resolves.

A mechanism intervention $do (X \leftarrow X^{'})$ replaces the structural mechanism $X$ with $X^{'}$ . The framework's three exogenous mechanisms admit:

Mode A: $do (\hat{R} \leftarrow \hat{R} + β b)$ — architectural intervention on the reward functional
Mode B (individual): $do (T_{i} \leftarrow T_{i}^{'})$ for one user $i$ in a population of $N$
Mode B (population): $do (T \leftarrow T^{'})$
Mode C (individual): $do (O_{i} \leftarrow O_{i}^{'})$
Mode C (regulatory, fourth-cell): $do (O \leftarrow O^{'})$

What the theorem says. The sensitivity of the performative stable point $θ^{*}$ to an intervention on any mechanism $X \in {\hat{R}, T, O}$ factors into two parts: the closure amplification $(I - D_{θ} Φ (θ^{*}))^{- 1}$ — the same factor for every mechanism — and the mechanism partial $\partial_{X} Φ (θ^{*})$ — different for each mechanism.

The closure amplification factor. Under contractivity $κ_{Φ} < 1$ — the operator modulus of §0.3, $κ_{Φ} \leq (β / α) L_{D}$ — the matrix $(I - D_{θ} Φ)^{- 1}$ has operator norm bounded by $(1 - κ_{Φ})^{- 1}$ . Small interventions amplify through the closure by this factor. As contractivity approaches its boundary $κ_{Φ} \to 1$ , the amplification diverges — the regime Theorem 12 classifies as the bifurcation regimes at the boundary.

The N-fold asymmetry.Mode A's reward functional $\hat{R}$ is not user-indexed. It enters the optimization for every user simultaneously. Population-scale intervention on Modes B or C (e.g. fourth-cell regulatory imposition on the observation channel) is also unindexed: $\partial θ^{*} / \partial X = O (1)$ for all three. Individual-scale intervention on $T_{i}$ or $O_{i}$ for a single user in a population of $N$ incurs population-averaging: $\partial θ^{*} / \partial X_{i} = (1/ N) \cdot \partial θ^{*} / \partial X = O (1/ N)$ . The asymmetry between Mode A and individual-scale Modes B/C is exactly $N$ -fold. It vanishes at population-coordinated intervention scale.

What the theorem authorizes the prose to claim. Substituting individual-scale Modes B or C for Mode A is the analytical error that converts political work into its symptom. The framework's hardest political claim is formally underwritten here. Mode A is the only intervention the foreclosure result (Lemma 1) and the impossibility result (Theorem 1') do not foreclose.

The political vocabulary the theorem clarifies. What an organizer or social-movement theorist would call “individual digital-hygiene practices” — the curated phone, the disabled notifications, the practiced refusal of platform legibility — operates at the $1/ N$ scale on the closed loop. What the same organizer would call “regulatory reform” — content-moderation rules, antitrust action, privacy-regime overhauls — operates at the $O (1)$ scale through Mode A, the fourth-cell mode, or the population-coordinated variant of Mode B. The framework's claim is that the gap between these two is a difference of kind — an $N$ -fold structural gap baked into the closure's geometry. Individual practice has real value for the person who practices it; what the framework specifies is its reach at population scale.

The do-calculus extension.The theorem extends Pearl's do-calculus in three ways. Closure amplification: standard do-calculus works on acyclic DAGs; the performative closure adds a cycle that amplifies interventions through $(I - D_{θ} Φ)^{- 1}$ . Population averaging: the distinction between individual-practice and regulatory imposition is causally captured as the difference between intervening on one term of an averaged variable and intervening on the variable itself. Mechanism interventions on non-acyclic settings: the framework operates on mechanisms. It cannot operate on the closure variable directly: setting $θ^{*}$ to a constant is meaningless, since $θ^{*}$ is defined as the fixed point of $Φ$ .

The reader can operate the intervention-asymmetry plate to see the 2×2 taxonomy in action — the same intervention placed in each cell, with the closure-amplification factor visible above each chart, the $O (1)$ vs $O (1/ N)$ distinction operationalized.

Theorem 9 (causal sensitivity in performative systems)

Under (R1), (R2), assume the performative operator $Φ$ is $C^{1}$ in $(θ, T, O, \hat{R})$ and the contractivity condition $κ_{Φ} < 1$ holds (the operator modulus of §0.3, $κ_{Φ} \leq (β / α) L_{D}$ ). Let $θ^{*}$ be the performative stable point. Then for any mechanism $X \in {\hat{R}, T, O}$ , the sensitivity of $θ^{*}$ to $X$ is

\frac{\partial θ ^{*}}{\partial X} = (I - D_{θ} Φ (θ^{*}))^{- 1} \cdot \partial_{X} Φ (θ^{*}) .

in wordsEvery intervention's effect on the stable point splits into the same two factors. The right-hand piece $\partial_{X} Φ$ is how directly the mechanism $X$ moves the loop — different for each of reward, response, observation. The left piece $(I - D_{θ} Φ)^{- 1}$ is the closure amplifier — identical for all of them, the factor by which the loop magnifies any nudge. One shared amplifier, three different levers: the whole intervention theory is in those two factors.

(i) Closure amplification factor. The matrix $(I - D_{θ} Φ (θ^{*}))^{- 1}$ has operator norm at most $(1 - κ_{Φ})^{- 1}$ .

(ii) Mode A direct sensitivity. $\partial_{\hat{R}} Φ (θ^{*}) = O (1)$ in $N$ .

(iii) Modes B and C population-scale sensitivity. For uniform intervention on $T$ or $O$ , $\partial_{T} Φ (θ^{*}), \partial_{O} Φ (θ^{*}) = O (1)$ in $N$ .

(iv) Modes B and C individual-scale sensitivity. For intervention on $T_{i}$ or $O_{i}$ for one user $i$ in a population of $N$ ,

\partial_{T_{i}} Φ (θ^{*}) = \frac{1}{N} \partial_{T} Φ (θ^{*}) = O (1/ N) .

in wordsOne user changing their own behavior contributes one share out of $N$ to the population average the platform optimizes against — so the effect on the stable point is smaller by exactly the factor $1/ N$ . This is the whole asymmetry in a line: regulation of the reward acts at full strength; individual practice is divided by the size of the population. On a platform with millions of users, $1/ N$ is indistinguishable from zero.

The asymmetry between Mode A and individual-scale Modes B/C is exactly $N$ -fold; it vanishes at population-coordinated intervention scale.

Proof — three steps

(1) Implicit function theorem on the closure equation $θ^{*} = Φ (θ^{*}; T, O, \hat{R})$ gives $(I - D_{θ} Φ) \cdot \partial θ^{*} / \partial X = \partial_{X} Φ$ . Contractivity (R1) plus Neumann series gives the inverse with operator-norm bound (1−κ_Φ)^{-1}— establishing (i). (2) Decomposition of partial derivatives: comparative statics on the FOC under (R2)'s Hessian-invertibility gives $\partial_{\hat{R}} Φ = O (1)$ (Milgrom & Shannon 1994); chain rule through $D$ gives same for T, O. (3) Population-averaging: with $D = (1/ N) \sum_{i} D_{i}$ , individual-scale partials are $(1/ N) \cdot \partial_{T} Φ$ . Mode A's reward is not user-indexed; always at population level.

∎

Corollaries — recovery of Propositions 1', 2', 3'

Corollary 9.1 (Prop 1'). Intervention $do (\hat{R} \leftarrow \hat{R} + β b)$ produces $\partial θ^{*} / \partial β = O (1)$ ; at sufficient $β$ , Mode A becomes positive.
Corollary 9.2 (Prop 2'(ii)). $do (T_{i} \leftarrow T_{i}^{'})$ on a single user: $O (1/ N)$ . Foreclosure invariant.
Corollary 9.3 (Prop 3'(iii)). $do (O_{i} \leftarrow O_{i}^{'})$ on a single user: $O (1/ N)$ . Foreclosure invariant.
Corollary 9.4 (Fourth-cell mode). $do (O \leftarrow O^{'})$ at population scale: $O (1)$ — structurally equivalent to Mode A.

Cross-references

Required: Lemma 1 (foreclosure) — the foreclosure this theorem's political content responds to
Required: Theorem 1' (reclamation impossibility) — the impossibility Mode A breaks
Dynamics plate: intervention asymmetry (2×2) (operates this theorem)
What follows: §10 Theorem 12 (crisis boundary) — what happens to the amplification factor as contractivity approaches its boundary