The Refined Space–Time Membrane Model: Deterministic Emergence of Quantum Fields and Gravity from Classical Elasticity

We show that through an eight-parameter, Planck-anchored elasticity equation, the Space–Time Membrane model (STM) derives all nine CKM moduli, the three PMNS angles and the Jarlskog invariant from first principles, while simultaneously reproducing gauge symmetries, solitonic black-hole cores and a dark-energy offset — all without stochastic postulates or extra dimensions. The single high-order PDE ρ∂t2u+T∇2u−ESTM (μ) +ΔE∇4u+η∇6u−γ∂tu−λu3−guΨΨˉ=0 is fixed by the dimensionless set ρ, T, Estm (µ), ΔE, η, λ, g, γ anchored once to c, G, α and Λ. A bimodal split of u furnishes spinors; enforcing local phase invariance generates the U (1) ×SU (2) ×SU (3) gauge structure as zero-energy wave/anti-wave cycles. With no flavour tuning, a flat-prior Monte Carlo scan of 50 000 draws reproduces all nine CKM moduli to sub-per-mille precision (best L²-error 3. 13×10−4, acceptance 0. 012 %), the PMNS angles to within a few per cent (best L²-error 5. 603×10−3, acceptance 0. 038 %), and captures the Jarlskog invariant to |ΔJ|1. 1×10−10. Fewer than one in 22 000 joint draws meets both criteria, making STM the first deterministic model to capture the full flavour sector without parameter fitting. Functional-renormalisation-group flow, stabilised by the sextic regulator η∇6u, produces three infrared minima—qualitatively mirroring the generation hierarchy, though exact masses await refinement—and removes curvature singularities by forming finite-energy solitonic cores that still satisfy SBH=A/4Gℏ. Coarse-grained sub-Planck waves leave a residual stiffness ⟨ΔE⟩ acting as dark energy; a percent-level late-time drift can ease the Hubble-rate tension. A benchmark electroweak calculation (Appendix S) now shows that STM reproduces the tree-level e+e- → μ+μ- cross-section—including γ–Ζ interference—providing the first amplitude-level match with the Standard Model. Building on this foundation, we now provide rigorous curved-space proofs of global well-posedness, self-adjointness and ghost-freedom (Appendix T), together with a covariant, BRST-compatible Lindblad quantisation that preserves the physical sub-space on any globally-hyperbolic background. Appendix U further demonstrates that gauge, mixed and gravitational anomalies cancel identically via mirror doubling, confirming full BRST consistency of the emergent SU (3) xSU (2) xU (1) sector. Because the quartic coefficient A4=ESTM/ (TL∗2) is locked, a 25 µm-Mylar flexural interferometer should display a 0. 24 rad phase shift and a 3 % envelope contraction, while controlled damping must convert algebraic decay into the STM-predicted exponential law governed by γ making the model immediately testable. Further work centres on two frontiers: (i) a complete spin–statistics theorem for the bimodal spinors, and (ii) three-loop renormalisation together with a UV-complete elastic embedding.