The Boundary Conditions of the Canvas Model: Why the Dark Matter Abundance Cannot Be Derived from First Principles

The canvas model derives all Standard Model parameters from eight primitives and four pillars. The gauge couplings, Yukawa couplings, mixing angles, mass ratios, and cosmological parameters are all predictions of the theory. One quantity has resisted derivation: the dark matter abundance \Omega_{\text{DM}} \approx 0.26. This paper proves that this is not a failure of the theory but a necessary consequence of its structure. The dark matter in the canvas model consists of Planck-mass black hole remnants formed during the chain reaction of voxel nucleations at the end of inflation. Their abundance is determined by the amplitude of wave fluctuations at the Planck scale, which is set by the initial conditions of the pre-geometric canvas. The key theorem: The pre-voxel dynamics of the Unified Wave Equation are neutrally stable at small amplitudes. The oscillator amplitude is preserved, not driven to an attractor. Therefore, the fluctuation amplitude at the end of inflation depends on the initial amplitude at the beginning of inflation. The initial amplitude is not determined by the primitives—it is a boundary condition. The canvas model thus requires exactly two boundary conditions to specify the State of the Machine: 1. The age of the universe (or equivalently H_0): The laws are time-translation invariant, so any age is possible. H_0 tells us which moment we inhabit.2. The initial fluctuation amplitude (or equivalently \Omega_{\text{DM}}): The pre-voxel dynamics are neutrally stable, so any initial amplitude is preserved. \Omega_{\text{DM}} tells us which particular universe we inhabit among the ensemble of possible universes with different initial smoothness. The Machine has zero free parameters. The State has two boundary conditions. All other physical quantities—twenty-five Standard Model parameters and six cosmological parameters—are derived from the primitives. Observational consequences: The dark matter abundance is a fossil from the beginning of the universe, analogous to the cosmic microwave background anisotropies. Just as the CMB amplitude A_s \sim 10^{-9} is a boundary condition of inflation, not a prediction, the dark matter abundance \Omega_{\text{DM}} \sim 0.26 is a boundary condition of the canvas model. The canvas model predicts what dark matter IS—Planck-mass black hole remnants, absolutely stable and interacting only gravitationally. It does not predict the abundance, which encodes the initial smoothness of the universe. Both A_s and \Omega_{\text{DM}} are properties of our particular universe, not parameters of the underlying laws. Keywords: canvas model, boundary conditions, dark matter abundance, pre-voxel dynamics, neutral stability, initial conditions, Planck-mass remnants, State vs. Machine, cosmological parameters, falsifiability