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

We present a Space–Time Membrane (STM) model in which an eight-parameter, Planck-anchored elasticity equation is explored as a single deterministic framework for emergent quantum-like, gauge-like and gravitational phenomena. The master PDE ρ∂²ₜu + T∇²u − (E_STM(μ) + ΔE)∇⁴u + η∇⁶u − ργ∂ₜu − λu³ − guΨΨ = 0 is fixed by a 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 an effective spinor Ψ; local internal rephasings then support U(1) × SU(2) × SU(3)–type gauge structures as zero-energy wave/anti-wave cycles. Coarse-graining rapid sub-Planck oscillations yields Schrödinger-like envelope dynamics, and we use non-Markovian GKSL master equations to model deterministic decoherence and obtain Born-rule–compatible statistics in representative measurement-style scenarios. In the gravitational sector, the transverse–traceless tensor modes reduce uniquely to an Einstein–Λ limit in the infrared under standard spin-2 bootstrap assumptions and acquire k⁴–k⁶ corrections in the ultraviolet; enhanced short-range stiffness replaces singularities with solitonic cores whose microcanonical mode counting, within this STM solitonic-core model, yields a leading-order area-law black-hole entropy with Hawking-like temperature and grey-body factors, and suggests specific deviations in strong-field and ringdown observables.Flavour is treated as a consistency check rather than a full derivation. A calibrated z = 3 scalar Functional Renormalisation Group (FRG) analysis (Appendix Y.10), anchored to the STM elastic coefficients, shows that an open set of ultraviolet couplings flows to an effective triple-well scalar potential at a finite infrared scale; the three resulting elastic basins are used as generation-labelled mass scales in the Yukawa sector. A flat-prior scan over STM elastic bands, with no flavour-specific tuning, then reproduces all nine CKM moduli at PDG-2024 precision (primary band) and yields a compact PMNS parameter-space fit at few-unit χ², supported by robustness tests (seed sweeps, ablations, down-scaled draws and a PMNS-target bootstrap). A minimal SMEFT bridge reproduces the tree-level e⁺e⁻ → μ⁺μ⁻ line shape, including γ–Z interference and leptonic running of α(s). The STM construction realises a z=3 Lifshitz-type scaling and, within this class, is super-renormalisable and ghost-free in the UV, admits an Osterwalder–Schrader reconstruction with a spin–statistics theorem on globally hyperbolic backgrounds and anomaly cancellation for the bimodal spinor, and remains stable under small BRST-compatible GKSL deformations. It is directly testable in principle via membrane/metamaterial interferometry, gravitational-wave dispersion and selected collider channels. Overall, in its present calibrated form STM should be viewed as a deterministic elasticity candidate for emergent quantum and gravitational phenomena, offering concrete numerical recipes and experimental benchmarks, and providing a framework that can be systematically tested, extended and, if necessary, falsified.