Lemma 1 (foreclosure of the no-stimulus window)

Consider a platform whose policy ${\pi }_{\theta }$ has converged to its performative stable point ${\theta }^{\ast }$. The lemma establishes three facts about that stable point.

The limit.As the platform's policy parameter approaches ${\theta }^{\ast }$, the probability of an inter-stimulus interval exceeding the reflective threshold approaches zero. The reader can watch this convergence happen at the foreclosure-clock plate — the warm-ink survival curve decaying past the dashed ${\theta }_{\mathrm{ref}}$ line, the oxblood region (chrono-debt) growing beneath the baseline.

The reflective threshold, empirically. The framework treats ${\theta }_{\mathrm{ref}}$ as the time-scale at which a thought can complete itself. The dual-process literature (Kahneman 2011) places routine system-2 deliberation in the five-to-thirty-second band; the mind-wandering literature (Killingsworth and Gilbert 2010) places spontaneous reflective episodes at tens of seconds to several minutes; the meditation and attention literatures (Wallace 2006; Mrazek et al. 2013) confirm the same order of magnitude. The structural-separation hypothesis ${\theta }_{\mathrm{ref}}/{\delta }_{\min }\gg 1$ places the platform's engagement-event timescale (milliseconds to seconds, per §0.1) at least three orders of magnitude below the reflective threshold. The bite of Lemma 1's polynomial bound is in that ratio.

The polynomial rate. At the stable point, the survival probability is bounded by $(1+c_v^{\ast 2})/({\theta }_{\mathrm{ref}}/{\delta }_{\min }{)}^2$. Under the structural-separation hypothesis ${\theta }_{\mathrm{ref}}/{\delta }_{\min }\ge 10^3$ — empirically the case on contemporary platforms — this gives a survival probability under $10^{-6}$. The pause is operationally absent at the rate the platform has driven. To translate: a probability of $10^{-6}$per inter-stimulus interval, across a year of platform use at a session intensity typical of contemporary recommendation feeds, predicts on the order of one reflective-threshold interval per several days. Vanishingly few. The closure operates by shifting the inter-arrival distribution's support toward ${\delta }_{\min }$while leaving the distribution's shape largely intact at the new location.

The exponential refinement. When the inter-arrival distribution at ${\theta }^{\ast }$ has a finite moment-generating function on some interval, the survival probability decays exponentially in the Cramér transform $J({\theta }_{\mathrm{ref}})$. The framework's substantive claims survive under either the polynomial or the exponential regime. The exponential bound strengthens what the polynomial bound already establishes; the polynomial bound carries the work.

What the lemma requires. Regularity conditions (R1) and (R2). The platform-side axioms (P-I) (reward monotonicity in engagement), (P-II) (policy expressivity to any rate up to $1/{\delta }_{\min }$), and (P-III) (SGD convergence under Robbins–Monro step sizes). The structural-separation hypothesis ${\theta }_{\mathrm{ref}}/{\delta }_{\min }\gg 1$.

What the lemma authorizes the prose to claim.The foreclosure of the pause is a derived consequence of the architectural setup — the closure plus the reward's monotonicity plus the policy class's expressivity. A reader who accepts the axioms accepts the foreclosure.

The chrono-political content. The bound carries forward into the chrono-debt spectrum (Definition 1.1), the framework's signature instrument. The spectrum measures $D(t,\tau )=S(\tau ;{\theta }_{\circ })-S(\tau ;{\theta }_t)$ — the gap between the pre-closure baseline survival function and the closed-loop survival function, scale by scale. Lemma 1 says that the integral of $D(t,\tau )$ over $\tau \ge {\theta }_{\mathrm{ref}}$ grows under closed-loop convergence. The chronopolitical analysis follows. A foreclosed second is mathematically recoverable through some intervention on the closure. A foreclosed adolescence — accumulated over a decade of exposure inside the converged regime — is beyond recovery. The asymmetry between recoverable and unrecoverable foreclosure is a property of the chrono-debt integral over different$\tau$-windows. It is structural — a property of the apparatus, beyond the reach of the experiencing subject's memory or willpower.

What the older critical traditions named. Distraction. Attention destruction. The death of contemplation. Stiegler's proletarianization of attention. Marcuse's one-dimensional thought. Kahneman's gap between system-1 and system-2. Each described what foreclosure feels like at the scale of the experiencing subject, and described it well. Lemma 1 leaves those descriptions standing and specifies what foreclosure isat the scale of the platform's stationary distribution. The phenomenological and the structural descriptions name the same operation from different vantage points. The framework's commitment is to do both at once.