Numerical Simulations of the Canvas Model: Beta Finder, Threshold Search, and Waveform Verification

This paper presents three numerical simulation programs that verify key predictions of the canvas model — a unified framework where spacetime, quantum fields, and gravity emerge from wave intersections on a pre-geometric canvas. The canvas model makes specific numerical predictions at three levels: the internal structure of particles (masses and couplings), the conditions for particle formation (thresholds), and the dynamics of bound states (waveform shapes). Each level is tested through simulation. --- What this paper provides: · The Beta Finder (Section 2): A self-iterating optimizer that determines the internal lattice parameter \beta = 1.868164 by matching the predicted charm quark mass to observation. With this single parameter fixed, the fermion mass hierarchy m_t : m_c : m_u \approx 1 : 0.0073 : 0.000052 emerges, spanning five orders of magnitude consistent with observation. Higher harmonics (n \geq 8) are exponentially suppressed (e^{-55\beta} \sim 10^{-45}), explaining why no fourth generation is observed.· The 1+1D Threshold Search (Section 3): A binary search that maps the boundary for bound state formation. The minimum threshold R_{\text{min}} \approx 0.537 is identified. The predicted operating point R = 4.0 (from the spacetime threshold R_{ST} = d+1 = 4) lies well above this minimum. The search demonstrates that bound state formation is not trivial — there exists a genuine threshold below which no structure forms.· The 3+1D Waveform Verification (Section 4): A GPU-accelerated simulation (Taichi on Radeon RX 560, 48^3 lattice, 1500 time steps) that tests the central falsifiable prediction of the framework. At the predicted spacetime threshold R = 4.0, colliding Gaussian wave packets form stable bound states in 5 out of 5 independent runs (100% success).· Waveform asymmetry measurement: The predicted ratio T_{\text{rise}}/T_{\text{fall}} = \pi/2 \approx 1.5708 is measured at 1.568 \pm 0.012 — agreement within 0.2\%. Convergence testing at three lattice resolutions (24^3, 48^3, 96^3) shows the ratio approaching \pi/2 as resolution increases.· Harmonic analysis (independent confirmation): The asymmetry parameter \alpha = (\pi-2)/(\pi+2) \approx 0.222 predicts 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 measures \phi_1 = 0.698 \pm 0.015 rad and |\hat{\psi}_2|/|\hat{\psi}_1| = 0.253 \pm 0.008 — both within measurement uncertainty. These are not fitted parameters; they are derived directly from \alpha.· Pillar IV dynamics (work in progress): A self-adjusting simulation in which the weights evolve from generic initial conditions toward the attractor fixed point has been attempted but does not yet achieve stable convergence. Weight explosion and bound state loss are the primary difficulties. This remains an open computational challenge. --- Key results: Quantity Predicted Measured Agreement\beta (internal lattice parameter) — 1.868164 Determined from charm massm_t : m_c : m_u 1 : 0.0073 : 0.000052 Matches observation ✓R_{\text{min}} (minimum threshold) — \approx 0.537 Threshold existsT_{\text{rise}}/T_{\text{fall}} \pi/2 = 1.5708 1.568 \pm 0.012 Within 0.2\%\phi_1 (first harmonic phase) 0.697 rad 0.698 \pm 0.015 rad ✓( \hat{\psi}_2 / \hat{\psi}_1 --- What this paper does NOT provide: · A full dynamical simulation where the weights evolve from generic initial conditions under Pillar IV (work in progress)· Experimental confirmation of the \pi/2 asymmetry (the simulation is numerical; laboratory test is pending) --- All simulation code is publicly available at:https://github.com/eolvvin --- Keywords: canvas model, numerical simulation, Beta Finder, fermion mass hierarchy, threshold search, bound state formation, waveform asymmetry, \pi/2 prediction, 3+1D simulation, Taichi GPU, harmonic analysis, Pillar IV, Feed Equation