The Emergence Canvas Unified Framework: A Finite Generative Architecture for Quantum Structure, Chiral Matter, Relativistic Particles, Spacetime, Gravity, Effective Field Theory, and Cosmology

This monograph presents the Emergence Canvas Unified Framework (ECUF) as a single finite generative architecture rather than a collection of domain-specific constructions. The paper is deliberately status-explicit: every substantive bridge is marked as derived, computed, conditional, calibrated, open, or ruled out. It does not claim a completed theory of everything. Its purpose is to state, in one place, the strongest mathematics that survives the programme audits and the exact remaining closure problem. What the Framework Does The microscopic starting point is a finite coprime recurrence structure with primitive periods 2, 3, 5, 7, whose least common multiple is L = 210. The Chinese remainder theorem gives Z_210 ≅ Z_2 × Z_3 × Z_5 × Z_7, from which a finite tensor-product Hilbert space, Weyl operator algebra, discrete first-order calculus, curvature, and a Dirichlet pullback metric can be constructed. A separate five-dimensional internal module admits Λ^even C^5 ≅ 1 ⊕ 10 ⊕ \bar{5}, providing the representation capacity of one chiral Standard Model family; the Z_6 center condition and anomaly equations reproduce the Standard Model hypercharge pattern in the audited minimal-faithful domain. The framework then bifurcates into two physical bridges. The spacetime bridge requires an event network with unbounded order, a low-energy quadratic spatial symbol in three effective dimensions, hyperbolicity, and a spin/Dirac structure. Under these conditions a Lorentzian continuum can emerge, but finite recurrence alone does not derive physical time, dimension three, or the Lorentz cone. The matter bridge requires nonlinear localization. The frozen relativistic Dirac–scalar completion supports finite-energy localized branches; Lorentz invariance implies the collective-coordinate action S_eff = -E_0 ∫ dt √(1 - v²/c²), and therefore M_i = E_0/c². With universal metric coupling, M_g = M_i. Representative numerical branches are bound and pass substantial spectral-stability tests, but their absolute mass scale is not yet predicted. The cosmological sector is built from one frozen scalar–tensor action with F = M_*² + (ξ/2)φ² and V = (A/2)φ² - (B/4)φ⁴ + (C/8)φ⁶. Its constant-field accelerated solutions obey an exact cubic. The exact covariant theory has been implemented through an Einstein–Boltzmann pipeline and confronted with Planck, DESI DR2, and Pantheon+. The best restricted audited point is essentially tied with the matched GR control in raw likelihood, Δχ² ≈ -0.077, while complexity penalties favor GR/ΛCDM. What the Paper Contains · A complete finite mathematical spine: CRT decomposition, Weyl algebra on Z_210, discrete calculus, Dirichlet pullback metric· Chiral matter capacity from Λ^even C^5, with SU(5) decomposition and hypercharge constraints from anomaly equations· A conditional bridge to Lorentzian spacetime from event propagation and spin structure· A relativistic localization mechanism that derives M_i = E_0/c² from Lorentz invariance· A covariant scalar–tensor cosmology with exact field equations, perturbation stability, and full likelihood comparison with Planck, DESI DR2, and Pantheon+· A rigorous no-go ledger: anomaly cancellation cannot derive three generations; finite square operators cannot give nonzero chiral index; massless exterior Dirac localization fails for real-frequency bound states; ordinary gradients cannot cure UV recurrence runaway; voxels alone do not imply quantum gravity· A deletion-and-regeneration test for genuine predictions: a quantity qualifies only if its experimental value is deleted from the construction and the frozen engine still returns a unique value Why This Matters The synthesis exposes the decisive scientific status. Canvas now possesses nontrivial mathematical bridges across recurrence geometry, chiral representation capacity, conditional emergent spacetime, stable relativistic composite matter, gauge constraints, and viable effective cosmology. However, several arrows remain conditional: the microscopic selection of three spatial dimensions and Lorentzian propagation, three physical generations, the minimal-faithful representation principle, ultraviolet stabilization, the absolute carrier scale, and primitive derivation of successful cosmological parameters. The framework is therefore best characterized as an incomplete candidate for fundamental unification with an operational many-output forward (one-button) engine and a finite, explicit first-principles closure ledger. The remaining frontier is not construction of the button, but removal of residual calibrated or constitutive selectors from inside it. Keywords: emergence canvas, unified framework, finite recurrence, chiral matter, relativistic localization, scalar–tensor cosmology, gauge coupling unification, fermion masses, CKM matrix, PMNS matrix, neutrino masses, cosmological constant, dark matter, inflation, strong CP problem, measurement problem, one-button engine