The Intersection Programme: A Computational Roadmap for the Emergence Canvas Model

What This Paper Does The Emergence Canvas Model has reached an inflection point. The analytical phase—deriving gauge structures, mixing angles, cosmological parameters, and the generation eigenvectors from the eight primitives and four pillars—is substantially complete. The remaining open problems reduce to a single bottleneck: the function f that maps threshold eigenvalues to return-map pole locations, which depends on the self-consistent transverse profile u(r) and the operator \mathcal{K}. This paper proposes a new computational paradigm for completing the model. Instead of attempting further analytical derivations, we identify ten distinct infinite series that the Canvas Model's axioms generate exactly. Physical predictions are the numbers that belong to the intersection of all relevant series simultaneously. The Ten Infinite Series Series Origin Contains\mathcal{L} Active fractions 1/2, 1/3, 1/5, 1/7 2^a 3^b 5^c 7^d\mathcal{R} Dynamic subset periods (lcm values) Multiples of lcm(S)\mathcal{P} Return-map susceptibility poles Mass ratios from Q and shift factors\mathcal{L}_2 Tier 2 primes 11, 13, 17, 19 2^a 3^b 5^c 7^d 11^e 13^f 17^g 19^h\mathcal{H} Harmonic modes on internal lattice P_{\text{mode}} e^{-\beta\Sigma^2}\mathcal{G} Geometric overlaps with Higgs (\mathcal{A} Gauge coupling attractor 1 : 2/3 : 2/\pi, 5\pi/32\mathcal{C} Cosmological parameters \Omega_\Lambda, n_s, \Omega_k, r\mathcal{V} Pre-voxel nucleation P_{\text{nuc}} \approx 0.01299\mathcal{D} Detachment mode rates Resonance, beat, cusp rates The Intersection Principle A physical quantity must belong to ALL relevant series simultaneously. The fermion mass ratios, for example, must be elements of the threshold lattice \mathcal{L}, must arise from the return-map susceptibility spectrum \mathcal{P}, must include harmonic suppression from the internal lattice \mathcal{H}, and must be consistent with the geometric overlap structure \mathcal{G}. The intersection of these four series constrains the unknown parameters sufficiently to determine them. The Computational Strategy 1. Generate each series numerically by varying the free parameters over their allowed ranges2. Find pairwise intersections between series for each observable3. Propagate constraints across sectors (the same parameters must work for all sectors)4. Determine the unknown function f from the intersection conditions5. Make blind predictions for quantities not used in the fitting The Simplest Starting Computation The simplest intersection to compute first is the mass ratios. Parameterize the function f(\tilde{\lambda}) (e.g., as C \tilde{\lambda}^p), vary C, p, Q, and the shift factors, compute the mass ratios from the pole formula, and check whether they belong to \mathcal{L} and match observation. Success and Failure Criteria The intersection programme succeeds if there exists a unique parameter set such that: · All mass ratios are elements of \mathcal{L} (for quarks) or explained as non-\mathcal{L} (for leptons)· All mass ratios satisfy the return-map pole formula with the same Q and f· The harmonic mode numbers and geometric overlaps are consistent with the effective Higgs angles· The gauge couplings at laboratory scales match observation after RG evolution· The cosmological parameters match observation· Blind predictions match observation The programme fails if: · No parameter set produces non-empty intersections for all sectors· A parameter set produces intersections but predicted values do not match observation· Multiple disconnected parameter sets produce equally good matches with no selection principle· The required function f is inconsistent with the return-map dynamics from the axioms Why This Matters This transforms the Canvas Model from a collection of individual derivations into a unified computational search for the unique parameter set that satisfies all axiomatic constraints simultaneously. If the intersection is non-empty and matches observation, the model is validated. If the intersection is empty, the model is falsified. Either outcome is scientifically decisive. Keywords: Canvas Model, intersection programme, infinite series, threshold lattice, return-map poles, harmonic suppression, geometric overlaps, computational roadmap, fermion masses, unified framework, falsifiability