visualization I.

foreclosure clock

the survival probability of the reflective window, made physically legible

Lemma 1 (foreclosure of the no-input window) establishes that the survival function of the inter-stimulus interval at the converged policy $θ^{*}$ satisfies $S (θ_{ref}; θ^{*}) \leq C (c_{v}) exp (- θ_{ref} / δ_{m i n})$ (Asmussen, Ch. V). The exponent $θ_{ref} / δ_{m i n}$ is on the order of a thousand to a hundred thousand for realistic platforms. The bound is therefore mathematically positive but operationally indistinguishable from zero. The visualization renders this fact as a length comparison: the bound is smaller than the smallest physical scale.

θ_{ref}

δ_{m i n}

$θ_{ref} / δ_{m i n}$ = 3,000

$S (θ_{ref}; θ^{*}) \leq C (c_{v}) e^{- 3, 000} \approx 1 0^{- 1, 303}$

the foreclosure bound

1 0^{- 1, 303}

the Planck length

1 0^{- 35}

compared against:

Lemma 1. Under Assumptions (A1)–(A6), the survival function of the inter-stimulus interval at the converged policy

θ^{*}

satisfies

S (θ_{ref}; θ^{*}) \leq C (c_{v}) exp (- θ_{ref} / δ_{m i n})

. At realistic parameter values — a reflective threshold of seconds to minutes, a tick rate of milliseconds to sub-second — the exponent

θ_{ref} / δ_{m i n}

is on the order of

1 0^{3}

1 0^{5}

, so the bound is on the order of

e^{- 1000}

e^{- 100, 000}

: mathematically positive, operationally indistinguishable from zero.

what to look for

Switch $δ_{m i n}$ between 1ms and 1s. At 1s, the bound has visible length — the "closure" doesn't yet hold. At 10ms (the realistic tick rate for engagement platforms), the bound collapses to invisibility against every physical reference. There is no physical scale at which the bound is meaningfully non-zero under the converged policy.

Switch the reference to the Planck length — the smallest physically meaningful quantity in the universe. The foreclosure bound is still many orders of magnitude smaller. This is what the proof means when it says the bound is operationally indistinguishable from zero. The mathematics has no operational referent below the Planck scale; the bound lives below.