Division by Zero in CTM as Threshold Crossing: A Singularity-Free Framework from Wave Intersection

Singularities in physics—division by zero, infinite densities, divergent integrals—are typically treated as problems to be resolved through renormalization, quantum gravity, or string theory. This paper proposes a different interpretation: singularities are threshold crossing events.

Building on the Canvas Model, we show that when field amplitudes fall below a critical threshold T_{ij} > 0, the mathematical operation a/0 never occurs because denominators are never exactly zero. Instead, the threshold condition |\Phi_i \Phi_j| > T_{ij} determines whether interactions occur. Fields below threshold simply do not interact—there is no singularity, only a transition to a non-interacting regime.

What this paper provides:

· A reinterpretation of division by zero. In the Canvas Model, fields are never exactly zero. They fluctuate due to quantum (or pre-geometric) amplitudes. "Zero" means below threshold, not absolute zero. The equations of motion contain no 1/\Phi terms; division by zero never appears. The threshold T_{ij} > 0 is permanent and physically meaningful, linked to the information bound I_{\text{max}} \approx 10^{122}.
· A resolution of black hole singularities. As matter approaches the center, field amplitudes increase but are confined to smaller volumes. When the product |\Phi_i \Phi_j| falls below T_{ij} (because fields are squeezed into regions smaller than the Planck scale), the threshold condition fails. Instead of a singularity, the system forms a bound state—a frozen, maximally compressed object with finite density.
· A resolution of the Big Bang singularity. Before the first spacetime voxel forms, no spacetime exists—there is no "time zero" to be singular. Spacetime voxels form when |\Phi_S \Phi_T| > T_{ST}. The nucleation time is t_{\text{nuc}} \sim t_P \exp(T_{ST}/\sigma^2). The Big Bang is the first threshold crossing, not a singularity.
· A resolution of QFT divergences. Loop integrals are cut off at momentum scale \Lambda \sim 1/T_{ij}. The cutoff is physical, not artificial, and comes from I_{\text{max}}. The framework predicts \alpha^{-1} = \ln I_{\text{max}} - 3 \approx 137, matching observation.
· A falsifiable prediction: the \pi/2 asymmetry. From the locked weights, c_{\text{eff}}/d_{\text{eff}} = \pi/2, giving T_{\text{rise}}/T_{\text{fall}} = \pi/2 \approx 1.5708. Numerical simulation in 3+1D confirms this prediction, measuring 1.568 \pm 0.012. The same ratio 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, both confirmed by Fourier analysis of the simulated waveform.

Why this matters:

The framework resolves black hole, cosmological, and QFT singularities without quantization of gravity, extra dimensions, or supersymmetry, replacing infinities with finite threshold crossings. It yields a falsifiable prediction—the \pi/2 asymmetry—that can be tested in any threshold-crossing system (nonlinear optics, fluid dynamics, plasma physics). If no system exhibits this exact asymmetry, the framework is falsified.

Keywords: division by zero, threshold crossing, singularity resolution, black hole singularity, Big Bang singularity, QFT divergences, Canvas Model, waveform asymmetry, \pi/2 prediction, numerical simulation