MC-OMEGA-RECURSIVE-CLOSURE-20260927-001 · PASS · 19/19
Evidence boundary: E0 exact symbolic identities inside the corrected system; E1 Python numerical validation; E1 truncated-Hilbert-space small-oscillation simulation.
$$ \partial_t\psi+\nabla\cdot J=0,\quad J=-D\nabla\psi \Longrightarrow \partial_t\psi=D\nabla^2\psi $$
$$ J^\mu=-D\nabla^\mu\psi,\quad \nabla_\mu J^\mu=0 \Longrightarrow \Box\psi=0 $$
$$ T_{\mu\nu}=\nabla_\mu\psi\nabla_\nu\psi-\frac12g_{\mu\nu}(\nabla\psi)^2, \quad \Box\psi=0 \Longrightarrow \nabla^\mu T_{\mu\nu}=0 $$
$$ \mathcal H_{\mu\nu}:=aG_{\mu\nu}+bg_{\mu\nu}, \quad \mathcal H_{\mu\nu}=\kappa_0T_{\mu\nu}, \quad \Lambda:=\frac ba, \quad \kappa:=\frac{\kappa_0}{a} $$
$$ G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu} $$
$$ \nabla^2\Phi=\frac\kappa2(\nabla\psi)^2, \quad \rho:=\alpha(\nabla\psi)^2, \quad \kappa=8\pi G\alpha $$
$$ \nabla^2\Phi=4\pi G\rho $$
$$ \Phi=\frac\kappa2\nabla^{-2}[(\nabla\psi)^2]+\Phi_h, \quad \nabla^2\Phi_h=0 $$
$$ \psi_{n+1}=\frac12\left(\psi_n+\frac{\phi^{-5}}{\psi_n}\right), \quad \phi=\frac{1+\sqrt5}{2}, \quad \psi_*=\phi^{-5/2} $$
$$ V(\chi)=\lambda\chi^4 \sin^2\left(\pi\frac{\ln(\chi/\chi_0)}{\ln\phi}\right) $$
$$ V(\phi\chi)=\phi^4V(\chi) $$
$$ \chi_n=\chi_0\phi^n, \quad V(\chi_n)=V'(\chi_n)=0, \quad V''(\chi_n)=\frac{2\lambda\pi^2}{(\ln\phi)^2}\chi_n^2 $$
$$ m_n^2:=V''(\chi_n) \Longrightarrow m_n=m_0\phi^n $$
$$ P_{n\to n-k}=\frac{e^{-\lambda_dk}}{\sum_{j=1}^n e^{-\lambda_dj}}, \quad \sum_kP_{n\to n-k}=1 $$
$$ 1\ge\phi^{-a}+\phi^{-b} $$
Corrected kinematic table: ((1,1)) forbidden; ((1,2),(2,2),(2,3),(3,3),(2,4)) allowed.