visualization II.

reclamation trajectories

the conditional autocovariance as a set that fails to grow

Theorem 1 (the reclamation theorem) says that under (1)–(5) and (A1)–(A5) the conditional autocovariance $I (τ) := Cov (u_{t}, u_{t + τ} ∣ E_{τ})$ is undefined-on-trajectory in the limit $t \to \infty$ for $τ > θ_{ref}$ — the conditioning event $E_{τ} := {s_{t^{'}} = 0 for t < t^{'} \leq t + τ}$ has measure zero (Lemma 1). Run the simulation. Each line is a user; trajectories that experience an $E_{τ}$ event turn oxblood. Under the closed-loop policy, the oxblood set does not grow. The visualization is the theorem rendered as a set that fails to appear.

θ_{ref}

=users =policy =

users simulated: 50

simulated time: 0.0s / 300s

window threshold $θ_{ref}$ : 30s

trajectories with a no-input window ≥ $θ_{ref}$ : 0 / 50 (0.0%)

Each ink-50 line is a single user under the closed loop. Trajectories that experience a no-input window of duration ≥

θ_{ref}

are rendered in oxblood. Under the converged closed-loop policy (β = 0), the oxblood set does not grow — the reclamation theorem rendered as a set that fails to appear. Switch the policy to

with-bonus (β = 10)

— Mode A — and oxblood trajectories emerge: the architectural intervention is the only one that produces them.

what to look for

At default parameters with the closed-loop policy, the counter stays at 0 as the simulation runs to its horizon. No trajectory experiences a reflective-length no-input window. This is Lemma 1 made visible as the absence of an event: the failure of the highlighted set to populate.

Switch the policy to $with-bonus (β = 10)$ — Mode A (Proposition 1), architectural — and re-run. Now oxblood trajectories emerge. The regularized objective $\hat{R}^{'} (θ; β) = E [\hat{R}] + β \cdot E [\sum b (τ^{(k)})]$ rewards inter-stimulus intervals $τ > τ^{*}$ ; the operating point along the path $\tilde{θ} (β)$ moves to one with positive probability on long windows. The contrast is the framework's central political claim: only the architectural mode breaks Lemma 1 and recovers what Theorem 1 forecloses.

Implementation note. 50 users at default. The simulation engine implements equations (1)–(5) of the formal exposition with simplified response dynamics; each user's state evolves independently. The stimulus raster at the bottom shows tick-level events for the representative user.