The Fermion Mass Ratios and New Particle Candidates in the Emergence Canvas Model: Diophantine Structure, Return-Map Poles, and Stability Constraints

This paper reports two advances in the Emergence Canvas Model programme: a structural solution to the quark mass ratios and a stability analysis of new-particle candidates. What This Paper Does Part 1: The Diophantine Mass Ratio Solution. Five of the six intergenerational quark mass ratios decompose exactly into products of powers of the primes \{2,3,5,7\}—the same primes that appear as the fundamental periods of the model's dynamic primitives. The base mass ratios 1:7:35 are derived from the primitive threshold fractions without any mass input. Numerical computation of the return-map pole structure, performed by scanning the coupling Q from 0.8 to 8.9, reveals that the pole locations are exactly proportional to the inverse threshold eigenvalues, with ratios 6:42:210 for all Q > 0.85. A Diophantine intersection framework selects a consistent solution: the physical coupling Q = 5, with Polarity Domain shift factors s_2 = 1/135, s_1^{(d)} = 7/1350, and s_1^{(u)} = 1/5670, all of which are elements of the threshold lattice. This solution exactly reproduces the observed ratios m_c/m_u = 588, m_s/m_d = 20, and m_b/m_s = 45, and yields m_t/m_u = 79,968 (observed \approx 80,000) and m_b/m_d = 900 (observed \approx 895). The lepton ratios are shown not to be elements of the threshold lattice, consistent with the model's prediction that leptons lack the Polarity Domain transition. Part 2: New Particle Candidates and Stability Analysis. The original predictions of scalar bosons at 519 GeV and 346 GeV are shown to have pole locations Q = 11.67 and Q = 17.5, exceeding the Feed stability bound Q < 9. They are therefore excluded as stable return-map fixed points, though they may persist as unstable resonances. Fifteen alternative connected primitive combinations survive the stability cut, with mass proxies in the range 1–7 TeV. The state P_1P_2P_3P_5 at Q = 5.0 exactly matches the physical coupling and is identified as the most promising new-particle candidate at approximately 1.2 TeV. Why This Matters The quark mass hierarchy has been one of the deepest puzzles in particle physics: why do masses span six orders of magnitude with a specific pattern? The Canvas Model's architecture—eight primitives, a threshold lattice of \{2,3,5,7\}, and a return-map pole structure—provides a coherent explanation for this pattern. The same primes that synchronize the canvas dynamics also determine the mass ratios. The Diophantine solution demonstrates that a single consistent parameter set reproduces observation, and that the model's internal consistency constraints eliminate certain new-particle candidates while pointing to others. What This Paper Does Not Claim The mass values presented are post-dictions, not first-principles predictions. The parameters Q, s_2, s_1^{(d)}, and s_1^{(u)} are fitted to observed quark ratios. The central open problem is the first-principles derivation of these parameters from the axioms, which requires solving the transverse localization problem. The new-particle candidates are conditional on assumptions that have not been derived from first principles. Keywords: Emergence Canvas Model, quark mass ratios, Diophantine structure, threshold lattice, return-map poles, stability constraints, new particle candidates, Polarity Domain, threshold eigenvalues, post-dictions