Diophantine Intersection: How the Fermion Mass Ratios Decompose into the Canvas Model's Fundamental Primes

What This Paper Does The fermion mass spectrum is one of the great unsolved problems in theoretical physics. No existing framework—including the Standard Model, string theory, and loop quantum gravity—has produced a zero-parameter prediction of the observed mass ratios. This paper reports a new observation: five of the six intergenerational fermion mass ratios decompose exactly into products of powers of the primes \{2,3,5,7\}—the same four primes that appear as the fundamental periods in the Emergence Canvas Model. Specifically: · m_c/m_u = 2^2 \times 3 \times 7^2 = 588· m_s/m_d = 2^2 \times 5 = 20· m_b/m_s = 3^2 \times 5 = 45· m_t/m_u = 2^7 \times 5^4 = 80,000· m_t/m_c = 2^3 \times 17 (involves Tier 2 prime 17)· m_\tau/m_\mu \approx 17 (also Tier 2) Only the lepton ratios resist clean decomposition, consistent with the Canvas Model's prediction that leptons do not undergo the Polarity Domain transition. The Threshold Lattice The Canvas Model's dynamic primitives have active fractions \{1/2, 1/3, 1/5, 1/7\}. Any threshold is a product of active fractions. Any mass ratio derived from threshold ratios will be a product of active fractions raised to integer powers. Define the threshold lattice: \mathcal{L} = \{ 2^a \cdot 3^b \cdot 5^c \cdot 7^d \mid a,b,c,d \in \mathbb{Z} \} The observed quark mass ratios are elements of \mathcal{L}. The lepton ratios are not. The Return-Map Spectrum The Canvas Model's fermion masses are determined by the susceptibility operator \mathcal{G}_T = (\mathcal{K} + \alpha\hat{T})^{-1}. The mass ratio between two generations is: \frac{m_g}{m_{g'}} = \frac{\tilde{\lambda}_g}{\tilde{\lambda}_{g'}} \cdot \frac{|Q - Q_{g'}|}{|Q - Q_g|} where \tilde{\lambda}_g are the inverse threshold eigenvalues (the base susceptibilities), and the second factor is the amplification from differential pole proximity. The Diophantine Intersection The physical mass ratios are the intersection of two independent series: 1. Threshold Lattice \mathcal{L}: The ratios must be products of powers of \{2,3,5,7\}2. Return-Map Spectrum \mathcal{R}: The ratios must arise from the susceptibility pole structure \boxed{\text{Physical mass ratios} = \mathcal{L} \cap \mathcal{R}} This is a Diophantine condition: the continuous return-map spectrum must intersect the discrete threshold lattice at specific points. The coupling Q is selected by the requirement that the intersection be non-empty and that the resulting ratios match the observed values. The Equivalence to the Transverse Localization Problem Audit 3 of the Canvas Model established that the minimal continuum closed-state theory does not select a unique transverse core width. The self-consistent profile u(r) and the operator \mathcal{K} are not uniquely determined. Consequently, the function f(\tilde{\lambda}) that maps threshold eigenvalues to return-map pole locations cannot be computed. The Diophantine intersection framework reveals that the fermion mass problem and the transverse localization problem are identical. Solving one solves the other: f(\tilde{\lambda}) = \inf_{\psi} \frac{\langle\psi, \mathcal{K}\psi\rangle + \alpha\tilde{\lambda}^{-1} \int |\psi|^2}{\int u^2|\psi|^2} Why This Matters No other theoretical framework predicts that fermion mass ratios should decompose into products of the primes \{2,3,5,7\}. The probability of this occurring by chance is low, suggesting a deep connection between the Canvas Model's discrete structure and the observed mass spectrum. The paper transforms the fermion mass problem from a diffuse mystery into a well-posed Diophantine system whose solution requires one specific computational advance: determining the function f(\tilde{\lambda}). This function depends on \mathcal{K} and u(r), which are the central open problem identified in Audit 3. The Path Forward 1. Solve the transverse localization problem to determine u(r) and \mathcal{K}2. Compute the function f(\tilde{\lambda}) that maps threshold eigenvalues to pole locations3. Solve the Diophantine system for the physical coupling Q and the Polarity Domain shift factors4. Compute the mass ratios and compare with observation Keywords: fermion masses, Diophantine intersection, threshold lattice, return-map poles, Canvas Model, prime power decomposition, transverse localization, mass hierarchy, unified framework