mechanism · §1

the closed loop

equations (1)–(6) wired as a directed graph

The framework's foundational structure. Four nodes — user state $u_{t}$ , engagement signal $y_{t}$ , policy $π$ , stimulus $s_{t}$ — connected by four edges corresponding to equations (1)–(4) of the formal exposition. The contour is closed by construction: every stimulus the user receives is shaped, recursively, by the policy's prior shaping of the user. The exogenous channel $e_{t} \equiv 0$ from equation (6) stands outside the loop — definitional, not propositional. The propositional consequence (that no-input windows of reflective length vanish) is Lemma 1, derived from this architecture.

Equations (1)–(6). The four edges step in sequence: the user's state

u_{t}

produces an engagement signal

y_{t}

by the observation function (2), the platform updates its policy parameters

θ_{t}

by gradient ascent (4), the policy selects the next stimulus

s_{t}

(3), and the stimulus reshapes the user's state (1). The exogenous channel

e_{t} \equiv 0

stands outside the contour — the loop has no input it did not, in some prior step, contribute to shaping (6).

what to look for

The pulse cycles through the four loop equations in order. The edge in oxblood is the one currently firing. There is no inlet to the contour; the only marker outside is the foreclosed exogenous channel $e_{t} \equiv 0$ .

This is the architecture from which everything else descends. The foreclosure lemma, the reclamation theorem, the three modes of intervention, the six instruments — all are properties of this wiring or interventions on it.