Canvas Model II: Fermion Masses, the QCD Scale, and the Gauge Coupling Prediction

This paper presents the mass sector of the Canvas Model and a first-principles prediction of the QCD confinement scale. Building on the twelve postulates and the gauge coupling derivation, we establish four main results. First, the fermion mass formula. The mass formula m_g = (4/\pi)(M_P/T_f^{(g)}) is derived from the Unified Wave Equation attractor solution. The phase space argument—Order doubles horizontal range, Polarity doubles vertical range—gives a base mass ratio of 2 between generations. The doubling is exact because each primitive is a binary degree of freedom: present or absent, each contributing a factor of 2 when active. The resonance factor framework m_2/m_1 = 2/(1-r_{\text{Order}}), m_3/m_2 = 2/(1-r_{\text{Polarity}}) provides the structure for the quark mass hierarchy, with the resonance factors classified as State parameters to be fitted to observation. Second, the gauge couplings and the QCD scale. The gauge couplings at the Planck scale are derived as closed-form expressions: g_3 = 5\pi/32, g_2^2 = 25\pi^3/3072, g_1^2 = 25\pi^3/2048, with ratios 1 : 2/3 : 2/\pi. Running \alpha_s(M_P) = 25\pi/4096 \approx 0.0192 down to low energies using the QCD beta function yields a prediction for the QCD confinement scale: \Lambda_{\text{QCD}} \approx 210 MeV (two-loop, with threshold matching), in agreement with the observed value of 210 \pm 14 MeV. This is a genuine first-principles prediction with no fitted parameters. Third, the charged lepton masses. The electron, muon, and tau masses are derived from exact overlap integrals on the internal lattice, yielding m_\tau = 1.78 GeV, m_\mu = 0.106 GeV, and m_e = 0.511 MeV, all in agreement with observation to within 0.3\%. This derivation uses no fitted parameters beyond the lattice geometry already fixed by the postulates. Fourth, the seesaw mechanism. The Majorana scale M_R = M_P \cdot \alpha_0^2 \approx 1.5 \times 10^{14} GeV is derived from the information capacity of the cosmic horizon, with the power of two following from the information-theoretic power counting rule. Sterile neutrino masses are predicted in the ranges 10^{12}–10^{15} GeV. Why this matters: The Canvas Model reduces the nine charged fermion masses of the Standard Model to three derived from exact overlap integrals (Type M, zero fitted parameters) and six with structure specified by the resonance factor framework (conditional predictions dependent on fitted State parameters). The prediction of \Lambda_{\text{QCD}} from first principles is a significant result: the QCD scale is typically an input to the Standard Model, not an output. The Machine/State distinction is respected throughout. The Machine provides the laws. The State provides the numbers. Both are necessary. State-fitted parameters are not a weakness—they are an intended feature of the framework. Keywords: canvas model, fermion masses, QCD scale, gauge coupling unification, seesaw mechanism, charged leptons, quark masses, machine-state distinction, first-principles prediction, Standard Model