Emergence XXIX: The Three Simplified Equations of the Canvas Model

The Emergence series derives all of physics and mathematics from eight primitives. Paper I introduced six core equations. This paper shows that the entire framework reduces to three equations. The three equations are: 1. The Unified Wave Equation: \Phi(v) = a v + b \Phi_0 + c \ddot{\Phi} + d \pi(v). Generates all dynamics from the four dynamic primitives (Order, Amplitude, Acceleration, Polarity). Every equation of motion in physics is a special case of this single equation with appropriate constant weights.2. The Threshold Condition: |\Phi_i \Phi_j| > T_{ij}. Selects stable structures from the four property primitives (Dimension, Angle, Chirality, Charge). Converts continuous wave dynamics into discrete particles, spacetime voxels, and physical properties. Subsumes quantization, back-reaction, and the emergent spacetime lattice.3. The Eigenvalue Equation: \hat{T}_{ij} c^j = \lambda c_i. Organizes the selected structures into spectral families. Determines gauge charges (eigenvectors), masses and couplings (eigenvalues), and mixing angles (projections). Subsumes the gauge coupling and the covariant derivative. The six core equations of Paper I are the expanded form: the emergent spacetime lattice is a specific instance of the threshold condition; quantization and back-reaction are its dynamical consequences; the gauge coupling is the eigenvalue equation applied to the internal space. The three equations are the minimal set. The eight primitives are the axioms. The universe is the spectrum. This paper is a synthesis. It does not contain new derivations; it organizes the results of the entire Emergence series into a compact, elegant architecture. Readers new to the canvas model should start with the Position Paper or Paper I. Keywords: canvas model, unified wave equation, threshold condition, eigenvalue equation, emergence, primitives, synthesis, three equations