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

What This Paper Does 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. Diophantine Structure of Quark Mass Ratios 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 and solving the susceptibility operator \mathcal{G}_T = (\mathcal{K} + \alpha\hat{T})^{-1}, 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 = 3^{-3} \cdot 5^{-1}· s_1^{(d)} = 7/1350 = 2^{-1} \cdot 3^{-3} \cdot 5^{-2} \cdot 7· s_1^{(u)} = 1/5670 = 2^{-1} \cdot 3^{-4} \cdot 5^{-1} \cdot 7^{-1} All shift factors are elements of the threshold lattice \mathcal{L} = \{ 2^a \cdot 3^b \cdot 5^c \cdot 7^d \}. This emerged from the solution and was not imposed as a constraint. Reproduced and Predicted Ratios The solution exactly reproduces the observed quark mass ratios: · m_c/m_u = 588 (input)· m_s/m_d = 20 (input)· m_b/m_s = 45 (input) And predicts: · m_t/m_u = 79,\!968 (observed \approx 80,\!000, 0.04\% difference)· m_b/m_d = 900 (observed \approx 895, 0.6\% difference) 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. 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 most promising new candidate is P_1P_2P_3P_5 at Q = 5.0, exactly matching the physical coupling, with a mass proxy of approximately 1.2 TeV. Limitations All mass values presented are post-dictions obtained by fitting the coupling and shift parameters to observed quark ratios. First-principles derivation of these parameters from the axioms remains the central open problem of the programme. The numerical data confirming Q_g \propto \tilde{\lambda}_g provides a constraint: any viable derivation must produce this linear relationship with the specific proportionality constant \alpha = 1/6. Data Availability The complete numerical data supporting the pole structure analysis and the 255-combination combinatorial audit are provided in the supplementary material. Keywords: Canvas Model, fermion masses, quark mass ratios, threshold lattice, return-map poles, Polarity Domain shifts, new particle candidates, Diophantine intersection, stability constraints, unified framework