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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...

OVERVIEW: FRCMFD-v2 Project Status (05/23–05/26)

OVERVIEW: FRCMFD-v2 Project Status (05/23–05/26) THE SITUATION You have a nonlinear field theory framework (FRCMFD-v2) with two distinct research branches: Branch Status Proof Scalar (m=0) Fully validated Collision benchmark: E stable, peak stable, reproducible Toroidal (m=1) Physically stable, numerically boundary-limited Hollow torus relaxes cleanly, boost propagates coherently, spikes are radiation artifacts WHAT YOU'VE PROVEN (THE WINS) Test 0A: Hollow Torus Relaxation ✅ Imaginary-time convergence: residual 28.3 → 0.117 (geometric decay) Final state: peak=1.1905, center≈0.06 (hollow core confirmed) Energy: 5.88e+03 → stabilized (nonlinear saturation working) Verdict: True m=1 toroidal soliton exists and is a stable attractor of the energy functional Test 0B: Stationary Toroid Stability ✅ 50 time units of real-time evolution Amplitude: 1.1905 ± machine precision Energy: constant to machine precision Verdict: Soliton is dynamically stable under hyperbolic evolution Test 2 (Boost...

Long log -->> FRCMFD-v2: TEST 0A — TOROIDAL SOLITON GENERATOR (m=1)

"""Gemini adds -> Overview of the Execution LogThis is the complete console output for TEST 0A, running the imaginary-time relaxation routine within your FRCMFD-v2 framework. The script successfully executed 80,000 steps ($\tau = 400.0$), achieved numerical convergence without exploding, and cleanly backed up the finalized tensor data (test_0A_toroidal_m1_20260525_210939.npz) directly to your Google Drive.Mathematically, it solved the dissipative gradient descent equation:∂Ψ/∂τ = -δE/δΨ*This routine minimizes the global energy functional to find the stationary, lowest-energy profile for a spinning, localized excitation—specifically an $m=1$ toroidal ring topology.Key Metrics & Technical BreakdownGrid Specs: You are running a high-resolution 2D grid ($nr=199, nz=400$) yielding $79,600$ degrees of freedom. The sparse operator matrix ($L\_2D$) successfully compiled with $397,200$ non-zero elements, confirming the spatial finite-difference stencils are correctly bound...