Derivation of the Unified Wave Equation Weights from Physical Constraints

The Unified Wave Equation (UWE) of the canvas model contains four dimensionless weights (a, b, c, d) corresponding to the four dynamic primitives: Order, Amplitude, Acceleration, and Polarity. These weights are not free parameters. This paper derives their values from five independent physical constraints, each arising from a different sector of the framework. What this paper provides: · Five physical constraints that lock all four weights: 1. Dimension constraint: The effective Order weight couples to the number of spatial dimensions n = 3, giving a_{\text{eff}} = a_{\text{base}} \times 3 = 3. 2. Strong force constraint: The effective Amplitude weight couples to the strong charge q_s = 3, giving b_{\text{eff}} = b_{\text{base}} \times 3 = 1. 3. Electromagnetic constraint: The sum of the Acceleration and Polarity effective weights equals the fine-structure constant: c_{\text{eff}} + d_{\text{eff}} = 1/137. 4. Attractor symmetry constraint: The Feed Equation (Pillar IV) drives the system to c_{\text{base}} = d_{\text{base}} at the fixed point. 5. Spacetime threshold constraint: The sum of the effective Order and Amplitude weights equals the spacetime threshold R_{ST} = 4.· Solution of the system: a_{\text{base}} = 1, b_{\text{base}} = 1/3, c_{\text{base}} = d_{\text{base}} = 1/[137(1+\pi/2)] \approx 0.002839.· Geometric modulation by property primitives: The effective weights are modulated by the angle between spatial axes (\theta = \pi/2, P7) and the chirality (h = +1, P5): a_{\text{eff}} = 3, b_{\text{eff}} = 1, c_{\text{eff}} = (\pi/2)/[137(1+\pi/2)] \approx 0.00446, d_{\text{eff}} = 1/[137(1+\pi/2)] \approx 0.00284.· The \pi/2 ratio: c_{\text{eff}}/d_{\text{eff}} = (\pi/2)/1 = \pi/2. This is a pure geometric prediction, independent of the numerical value of the fine-structure constant. It follows from the orthogonality of the spatial axes (\theta = \pi/2) and the positive chirality (h = +1).· Waveform asymmetry and harmonic signatures: The ratio \pi/2 produces T_{\text{rise}}/T_{\text{fall}} = \pi/2 \approx 1.5708, asymmetry parameter \alpha = (\pi-2)/(\pi+2) \approx 0.222, first harmonic phase shift \phi_1 = \pi\alpha \approx 0.697 rad, and second harmonic suppression |\hat{\psi}_2|/|\hat{\psi}_1| = (\pi-2)/(\sqrt{2}\pi) \approx 0.257. All have been verified in 3+1D numerical simulation.· The complete UWE with locked weights: Base form: \Phi(v) = v + \frac{1}{3}\Phi_0 + \frac{1}{137(1+\pi/2)}\frac{d^2\Phi}{dv^2} + \frac{1}{137(1+\pi/2)}\operatorname{sgn}(\Phi(v)) Effective form: \Phi(v) = 3v + \Phi_0 + \frac{\pi/2}{137(1+\pi/2)}\frac{d^2\Phi}{dv^2} + \frac{1}{137(1+\pi/2)}\operatorname{sgn}(\Phi(v)) Why this matters: The derivation reduces four apparently free parameters to a small set of physical inputs: two discrete geometric integers (n=3, q_s=3), one dynamical attractor condition (c_{\text{base}} = d_{\text{base}}), and one empirical scale (\alpha \approx 1/137). The ratio \pi/2 requires no empirical input—it is a pure geometric prediction of the framework. Every weight traces back to a physical principle. No free parameters remain. Keywords: Unified Wave Equation, UWE weights, canvas model, dynamic primitives, Order, Amplitude, Acceleration, Polarity, fine-structure constant, \pi/2 ratio, waveform asymmetry, attractor dynamics, Pillar IV, geometric modulation, property primitives