The Internal Space Geometry of the Emergence Canvas Model

The Emergence Canvas Model postulates a compact 3D internal space whose geometry determines the gauge groups, gauge couplings, and mass spectrum of the Standard Model. This paper derives the three fundamental parameters of this internal space from the model's postulates: the confinement scale L, the Higgs width \sigma, and the harmonic suppression parameter \beta. The confinement scale is set by the strong gauge coupling at the Planck scale. The gauge coupling is derived from the stability bound on the spectral energy functional, the fundamental quadrant integrals over the angular domain [0, \pi/2], and the dimensional reduction from the 2D canvas to 4D spacetime. The result is g_3 = 5\pi/32, giving L = 1/g_3 = 32/(5\pi) \, \ell_P \approx 2.037 \, \ell_P. The Higgs width is set by the spacetime threshold condition: \sigma = 1/\sqrt{T_{ST}} = 1/2 \, \ell_P, where T_{ST} = d+1 = 4 is determined by the number of spatial dimensions. The harmonic suppression parameter \beta is derived from the exact overlap integral of fermion standing waves with the Gaussian Higgs profile on the internal lattice. The 1D overlap evaluates analytically in terms of the imaginary error function \operatorname{erfi}(z). The 3D overlap ratio between modes (4,4,4) and (1,1,1) yields \beta = 0.167 \pm 0.005. Why this matters: The internal space geometry is the foundation for all subsequent derivations in the Emergence Canvas Model: gauge couplings, fermion masses, mixing angles, and cosmological parameters. All three parameters are derived from the model's postulates with no experimental input. The imaginary error function \operatorname{erfi}(z) is essential for obtaining the correct value of \beta—the Gaussian approximation would give a value more than three times larger, incorrectly suppressing higher modes by orders of magnitude. Keywords: canvas model, internal space, confinement scale, Higgs width, harmonic suppression, gauge coupling, Planck scale, overlap integral, imaginary error function, fermion masses, Yukawa couplings