Foreclosure clock

What the plate operates. Lemma 1 (foreclosure) establishes that under (R1), (R2), (P-I)–(P-III), the survival probability $S({\theta }_{\mathrm{ref}};{\theta }_t)$ approaches operational zero as ${\theta }_t\to {\theta }^{\ast }$. The plate renders this convergence: as the closure's optimization runs, the warm-ink survival curve decays past the dashed ${\theta }_{\mathrm{ref}}$ line; the oxblood-shaded region grows beneath the dashed cream baseline.

The controls.

• β, α, L_D → κ_Φ (contractivity). The plate exposes the operator modulus through its three constituents: reward smoothness $\beta$, strong-concavity modulus $\alpha$, and distribution sensitivity $L_{\mathcal{D}}$. It computes ${\kappa }_{\Phi }=(\beta /\alpha )L_{\mathcal{D}}$; axiom R1 (under R2's strong concavity) requires ${\kappa }_{\Phi }<1$. Driving ${\kappa }_{\Phi }$ toward 1 slows convergence — the closure operator $\Phi$becomes “barely a contraction.” The closure-amplification factor $(I-D_{\theta }\Phi {)}^{-1}$ (Theorem 9) diverges as ${\kappa }_{\Phi }\to 1^{-}$ — the crisis boundary (Theorem 12).
• θ_ref / δ_min (structural separation). Empirically $10^3$ to $10^5$on contemporary platforms. Higher ratios deepen the foreclosure; Lemma 1's polynomial bound gives $S\le (1+c_v^{\ast 2})/({\theta }_{\mathrm{ref}}/{\delta }_{\min }{)}^2$, so doubling the ratio quarters the upper bound.

The sim engine.A 1-D scalar reduction of the apparatus's setup (P1)-(P5). User state collapses to scalar $u$; policy $\theta$ ∈ [0, 1] indexes the firing-rate (with ${\theta }^{\ast }=1$ corresponding to the rate ceiling $1/{\delta }_{\min }$ per Lemma 1 Step 2); ISIs are exponential at rate $\theta /{\delta }_{\min }$; the policy update is geometric convergence ${\theta }_{t+1}-{\theta }^{\ast }={\kappa }_{\Phi }({\theta }_t-{\theta }^{\ast })$ at modulus ${\kappa }_{\Phi }=(\beta /\alpha )L_{\mathcal{D}}$. The simplifications preserve Lemma 1's qualitative behavior; the full multi-register sim is Phase 3's primary engine.

What the reader produces. Press play. Watch the survival curve decay past ${\theta }_{\mathrm{ref}}$. The S(θ_ref; θ_t) readout drops from $\sim e^{-0.01}$ at $t=0$ to $\sim e^{-1000}$ at ${\theta }_t\to 1$. The regime you produced — where $S({\theta }_{\mathrm{ref}};{\theta }_t)$ is operationally zero — is what Lemma 1 names.