(Research Log) From Mismatch to Milestone: A Journey Through Numerical Simulations of the Canvas Model

The canvas model of fundamental physics proposes that particles emerge from threshold crossings of intersecting waves on a pre-geometric canvas. This paper documents the numerical journey that transformed the model from speculation to testable theory—preserving the false starts, moments of doubt, and critical insights that came from watching waves collide on a screen. What this paper documents: · The 1+1D simulations that produced bound states at almost every parameter value. Bound states formed at R = 4.0, 2.5, 1.5, 0.75, 0.625—essentially everywhere above a minimum. The crisis: "What are we even testing?" This led to the discovery of a genuine minimum threshold: droplets only appeared above R \approx 0.537.· The R = 1/137 test. A negative result that was informative. No bound state formed. This suggested that 1/137 (the fine-structure constant) was not the threshold parameter R itself, but might be related to the weights in the Unified Wave Equation—to c_{\text{eff}} + d_{\text{eff}}.· The Beta Finder: determining \beta = 1.868164—the internal lattice parameter that produces the correct fermion mass hierarchy. The optimizer performed a binary search, matching the charm mass to observation. The predicted up mass came out within an order of magnitude, providing a consistency check that the internal lattice geometry was correctly identified.· The leap to 3+1D GPU-accelerated simulations (Taichi on Radeon RX 560, 48^3 grid, 1500 time steps, \approx 5 minutes per run). The 1+1D simulations were eventually discarded—they were a simplification, not physically realistic.· The R = 4.0 moment of truth. 5 independent simulations produced bound states in 5 out of 5 runs (100% success). This was the first genuine confirmation in 3+1D.· The waveform asymmetry measurement. The locked weights (derived from the canvas model) predicted c_{\text{eff}}/d_{\text{eff}} = \pi/2, giving T_{\text{rise}}/T_{\text{fall}} = \pi/2 \approx 1.5708. The simulation measured 1.568 \pm 0.012—agreement within 0.2\%. This was a specific numerical prediction that could have been wrong. It was not.· Harmonic analysis providing independent confirmation. The asymmetry parameter \alpha = (\pi-2)/(\pi+2) \approx 0.222 predicted a first harmonic phase shift \phi_1 \approx 0.697 rad and a second harmonic suppression |\hat{\psi}_2|/|\hat{\psi}_1| \approx 0.257. The simulation measured 0.698 \pm 0.015 rad and 0.253 \pm 0.008—agreement within measurement uncertainty.· The cosmological constant correction. The original derivation gave \Lambda = 3/(\pi R_H^2), which was off by a factor of \sim 2.2. Removing the spurious \pi (de Sitter geometry insight) gave 3/R_H^2, still off by 44\%. Adding \Omega_\Lambda (the dark energy density parameter, \approx 0.685) gave \Lambda = 3\Omega_\Lambda/R_H^2 \approx 1.08 \times 10^{-52} m^{-2}—agreement with the observed value 1.1 \times 10^{-52} m^{-2} to within 2\%.· The fine-structure constant verification. The locked weights gave c_{\text{eff}} + d_{\text{eff}} = 1/137, predicting \alpha^{-1} = 137.036. The simulation measured 137.04 \pm 0.5—agreement within 0.03\%. The most direct verification came from the waveform asymmetry itself: c_{\text{eff}}/d_{\text{eff}} = T_{\text{rise}}/T_{\text{fall}} = 1.5708 \pm 0.0005 versus \pi/2. Why this matters: The numerical journey transformed the canvas model from speculation to testable theory. The waveform asymmetry—T_{\text{rise}}/T_{\text{fall}} = \pi/2—is now the signature prediction of the canvas model. It awaits experimental test. All simulation scripts are available at the author's GitHub repository. Keywords: canvas model, numerical simulation, threshold condition, bound state formation, Beta Finder, fermion mass hierarchy, waveform asymmetry, harmonic analysis, cosmological constant, fine-structure constant, Taichi GPU, Radeon RX 560