mechanism · §4

the causal chain

the proof of Theorem 1 against the intervention modes

The structural argument of §4 as a directed graph. The six assumptions (A1)–(A6) feed three intermediate results: the policy reaches $θ^{*}$ at the engagement-maximizer, the Asmussen renewal bound holds on the survival function, and the no-input event probability collapses (Lemma 1). From Lemma 1, the estimator variance for $I (τ)$ diverges, and Theorem 1 follows: $I (τ)$ is undefined-on-trajectory.

The framework's structural claim about the three modes of intervention is visible directly on this chain. Mode A intercepts at the $θ^{*}$ node — by adding $β \cdot b (τ^{(k)})$ to the reward, Assumption (A2)'s monotonicity fails. Modes B and C do not intercept this chain at all: they hold structures the chain doesn't depend on.

policy =Mode B active (

L_{m} > 0

)Mode C active (

σ_{C} > 0

)

§4. The proof chain from the assumptions to Theorem 1, with each node firing when its premise holds. Under closed-loop, the chain fires to completion and reclamation is foreclosed. Switch to with-bonus and Mode A cuts the chain at the

θ^{*}

node — Assumption (A2) reward-monotonicity fails under

β \hat{R}^{'}

. Modes B and C light their footers without modifying the chain: they hold parallel structures (

D_{KL}

Σ

) that the proof of Theorem 1 does not depend on.

what to look for

At closed-loop, every node fires — the chain completes and Theorem 1's terminal node lights in oxblood: RECLAMATION ⌀. Switch to with-bonus and the red Mode A cut appears between the (A2) node and the $θ^{*}$ node; every downstream node dims; the terminal node goes dark. Theorem 1 does not fire.

Toggle the Mode B and Mode C indicators. Their footers brighten when active. The chain itself does not respond — Mode B and Mode C operate on parallel structures (the user's self-evaluative divergence $D_{KL}$ and the platform's surplus $Σ$ respectively). They produce real effects, but not the effect the chain produces. The framework's central political claim is this asymmetry: substituting one mode for another is substituting an intervention on one chain for an intervention on a different chain. The chains do not exchange.