# Theorem — Ω–Recursive Field System: Error-Corrected Algebraic Closure

**Certificate:** `MC-OMEGA-RECURSIVE-CLOSURE-20260927-001`  
**Status:** `PASS`  
**Evidence:** E0 exact symbolic identities inside the corrected system; E1 Python numerical validation; E1 quantum small-oscillation simulation.

## I. Continuity and covariant field
\[
\partial_t\psi+\nabla\cdot J=0,\quad J=-D\nabla\psi
\Longrightarrow
\boxed{\partial_t\psi=D\nabla^2\psi}
\]
\[
J^\mu=-D\nabla^\mu\psi,\quad \nabla_\mu J^\mu=0
\Longrightarrow
\boxed{\Box\psi=0}
\]
\[
T_{\mu\nu}=\nabla_\mu\psi\nabla_\nu\psi-\frac12g_{\mu\nu}(\nabla\psi)^2,\quad
\Box\psi=0
\Longrightarrow
\boxed{\nabla^\mu T_{\mu\nu}=0}
\]

## II. Conserved geometric coupling
\[
\nabla^\mu G_{\mu\nu}=0,\qquad \nabla^\mu g_{\mu\nu}=0
\]
\[
\mathcal H_{\mu\nu}:=aG_{\mu\nu}+bg_{\mu\nu},\qquad \nabla^\mu\mathcal H_{\mu\nu}=0
\]
\[
\mathcal H_{\mu\nu}=\kappa_0T_{\mu\nu},\quad
\Lambda:=\frac ba,\quad \kappa:=\frac{\kappa_0}{a}
\]
\[
\boxed{G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}}
\]

## III. Newtonian limit
\[
G_{00}\simeq\nabla^2\Phi,\qquad T_{00}\simeq\frac12(\nabla\psi)^2
\]
\[
\boxed{\nabla^2\Phi=\frac\kappa2(\nabla\psi)^2}
\]
\[
\rho:=\alpha(\nabla\psi)^2,\qquad \boxed{\kappa=8\pi G\alpha}
\]
\[
\boxed{\nabla^2\Phi=4\pi G\rho}
\]
\[
\boxed{\Phi=\frac\kappa2\nabla^{-2}[(\nabla\psi)^2]+\Phi_h},\qquad \nabla^2\Phi_h=0
\]
\[
r>R_{\rm src}\Longrightarrow \Phi(r)=-\frac{GM}{r}
\Longrightarrow
\boxed{\mathbf F=-\frac{GMm}{r^2}\hat{\mathbf r}}
\]

## IV. Recursive fixed point
\[
\psi_{n+1}=\frac12\left(\psi_n+\frac{\phi^{-5}}{\psi_n}\right),\qquad
\phi=\frac{1+\sqrt5}{2}
\]
\[
\psi_{n+1}=\psi_n=\psi_*
\Longrightarrow
\boxed{\psi_*=\phi^{-5/2}}
\]

## V. Exact discrete-scale potential
\[
\boxed{
V(\chi)=\lambda\chi^4
\sin^2\left(\pi\frac{\ln(\chi/\chi_0)}{\ln\phi}\right)
}
\]
\[
\boxed{V(\phi\chi)=\phi^4V(\chi)}
\]
\[
\chi_n=\chi_0\phi^n,\qquad V(\chi_n)=V'(\chi_n)=0
\]
\[
\boxed{V''(\chi_n)=\frac{2\lambda\pi^2}{(\ln\phi)^2}\chi_n^2}
\]
\[
m_n^2:=V''(\chi_n)
\Longrightarrow
\boxed{m_n=m_0\phi^n}
\]

## VI. Decay kernel
\[
\boxed{
P_{n\to n-k}=
\frac{e^{-\lambda_dk}}{\sum_{j=1}^n e^{-\lambda_dj}}
},\qquad
\sum_kP_{n\to n-k}=1
\]
\[
\boxed{1\ge\phi^{-a}+\phi^{-b}}
\]
\[
(1,1):\times,\quad
(1,2):\checkmark,\quad
(2,2):\checkmark,\quad
(2,3):\checkmark,\quad
(3,3):\checkmark,\quad
(2,4):\checkmark
\]
\[
\prod_i e^{-\lambda_dk_i}=e^{-\lambda_d(n-m)}
\Longrightarrow
\boxed{P(n\to m)\propto e^{-\lambda_d(n-m)}}
\]

## VII. Numeric notation
\[
\phi=1.618033988749895,\qquad
\ln\phi=0.481211825059603,\qquad
\psi_*=0.300283106000778
\]
\[
m_0=0.511\ {\rm MeV},\qquad
\lambda=3.063265509773806e-03
\]

\[
\begin{array}{c|c}
n&m_n\ ({\rm MeV})\\
\hline
0&0.511000000\\
11&101.691567774\\
17&1824.781143097\\
19&4777.339054658\\
20&7729.896966219\\
21&12507.236020878\\
22&20237.132987097\\
23&32744.369007975\\
24&52981.501995072\\
25&85725.871003046\\
26&138707.372998118\\
\end{array}
\]

\[
\max|V(\chi_n)|=1.582e-26,\qquad
\max\left|\frac{m_{n+1}}{m_n}-\phi\right|=2.220e-16
\]

## VIII. Quantum small-oscillation simulation
\[
V(\chi_n+q)=V(\chi_n)+\frac12V''(\chi_n)q^2+O(q^3)
\]
\[
\omega_n:=\sqrt{V''(\chi_n)}=m_n
\]
\[
\hat H_n=\frac12\hat p_n^2+\frac12\omega_n^2\hat q_n^2
\]
\[
\boxed{E_{n,k}=\left(k+\frac12\right)\omega_n}
\]
\[
\boxed{
\frac{\Delta E_{n+1}}{\Delta E_n}
=
\frac{\omega_{n+1}}{\omega_n}
=
\phi
}
\]
\[
\dim\mathcal H_{\rm sim}=48
\]
\[
\max|E_k^{\rm num}-(k+\tfrac12)\omega_n|=4.263e-14
\]
\[
\max\|U_n^\dagger U_n-I\|_2=3.754e-15
\]
\[
\max\left|\frac{\Delta E_{n+1}}{\Delta E_n}-\phi\right|=1.554e-15
\]
\[
\boxed{\text{SYMBOLIC + NUMERIC + QUANTUM: 19/19 PASS}}
\]