T6 — Closure Fixed Point (Banach Contraction)

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1. #1

   Let be the sigmoid closure operator

   $$\mathrm{Cl}_t(u)(x) = \sigma\!\big(a_{\mathrm{cl}}\,((1-\eta_{\mathrm{cl}})\,u(x) + \eta_{\mathrm{cl}}\,(P_t u)(x) - \tau_{\mathrm{cl}})\big).$$
2. #2

   T6b. Banach contraction mapping theorem. The Lipschitz constant of is

   $$L \;\leq\; (\max \sigma') \cdot a_{\mathrm{cl}} \cdot \max(1 - \eta_{\mathrm{cl}}, \eta_{\mathrm{cl}}) \;\leq\; \frac{a_{\mathrm{cl}}}{4},$$