Core FRCMFD-v2 Evolution Equation
[
\Phi(\mathbf{x},t)=A(\mathbf{x},t)e^{iS(\mathbf{x},t)}
]
Complex monad field decomposed into amplitude and phase.
---
[
\rho_M(\mathbf{x},t)=|\Phi(\mathbf{x},t)|^2
]
Monad-field configuration density.
---
[
\mathbf{J}_M=\rho_M\nabla S
]
Configuration-flow / phase-current relation.
---
[
\partial_t\rho_M+\nabla\cdot\mathbf{J}_M=0
]
Continuity equation for monad-field conservation.
---
# Core FRCMFD-v2 Evolution Equation
[
i\partial_t\Phi
===============
-\alpha\nabla^2\Phi
+
\beta|\Phi|^2\Phi
+
V_{\mathrm{env}}(\mathbf{x},t)\Phi
+
V_{\mathrm{top}}(\mathbf{x},t)\Phi
]
Nonlinear monad-field evolution including environmental and topological coupling sectors.
---
# Cylindrical Toroidal Form
For axisymmetric winding sectors:
[
\Phi(r,\theta,z,t)
==================
f(r,z,t)e^{im\theta}
]
Topological winding decomposition.
---
[
\nabla^2
========
\partial_r^2
+
\frac{1}{r}\partial_r
+
\partial_z^2
------------
\frac{m^2}{r^2}
]
Cylindrical Laplacian with centrifugal winding term.
---
# Toroidal FRCMFD Equation
[
i\partial_t f
=============
-\alpha
\left(
\partial_r^2
+
\frac{1}{r}\partial_r
+
\partial_z^2
------------
\frac{m^2}{r^2}
\right)f
+
\beta|f|^2f
+
V_{\mathrm{env}}f
]
Current working toroidal evolution equation.
---
# Environmental Coupling Sector
[
V_{\mathrm{env}}
================
V_{\rho}
+
V_{\mathrm{anis}}
+
V_{\mathrm{shear}}
+
V_{\mathrm{vort}}
]
Decomposition into scalar and geometric coupling sectors.
---
## Scalar Density Sector
[
V_{\rho}\propto\delta\rho
]
Original scalar-density coupling hypothesis (currently unsupported empirically).
---
## Directional / Bulk-Flow Sector
[
V_{\mathrm{anis}}
\propto
\hat{n}\cdot\mathbf{v}_{\mathrm{bulk}}
]
Directional anisotropy coupling.
---
## Shear Coupling Sector
[
V_{\mathrm{shear}}
\propto
\sigma_{ij}\hat n_i\hat n_j
]
Velocity-shear tensor coupling.
---
## Vorticity / Spin Coupling
[
V_{\mathrm{vort}}
\propto
\boldsymbol{\omega}\cdot\mathbf{L}
]
Spin–vorticity alignment sector.
---
# Residual Observable Pipeline
[
\Delta\gamma_{\mathrm{resid}}
=============================
## \gamma_{\mathrm{obs}}
## \gamma_{\Lambda\mathrm{CDM}}
\gamma_{\mathrm{baryonic}}
]
Residualized observable used in environmental coupling tests.
---
# Dominant Dataset Confound
[
\rho(\mathrm{distance},V_{\mathrm{pec}})
\approx
-0.73
]
Current dominant geometric/environmental correlation structure.
---
# Self-Adjoint Cylindrical Operator Construction
[
L_r
===
W_r^{-1/2}MW_r^{-1/2}
]
Weighted self-adjoint radial operator.
---
[
L_{2D}
======
I_z\otimes L_r
+
L_z\otimes I_r
]
Full cylindrical 2D operator.
---
# Self-Adjointness Constraint
[
\langle v,Lu\rangle
===================
\langle Lv,u\rangle
]
Discrete weighted Hermiticity condition.
---
# Hamiltonian Functional
[
H[\Phi]
=======
\int
\left[
\alpha|\nabla\Phi|^2
+
\frac{\beta}{2}|\Phi|^4
+
V_{\mathrm{env}}|\Phi|^2
\right]
dV
]
Current effective energy functional.
---
# Hamiltonian Evolution Form
[
\partial_t^2\Phi
================
-\frac{\delta H}{\delta\Phi^\ast}
]
Variational Hamiltonian field evolution structure.
---
# Toroidal Boost Initialization
[
\Phi_{\mathrm{boost}}
=====================
\Phi_0(r,z)e^{ik_{\mathrm{boost}}z}
]
Phase-gradient boosted toroidal state.
---
# Boundary-Recollision Scaling Hypothesis
[
t_{\mathrm{collision}}
\sim
\frac{L_z}{v_{\mathrm{wake}}}
]
Current interpretation of late-time toroidal spike behavior under periodic boundaries.
