The central advance of this version: the golden ratio is not postulated. It emerges as the first stable attractor reached by a primordial evaporation dynamics starting from a fully symmetric state R = 1.
(a) Geometric. In 3D, ordered structures with 5-fold rotation axes require the icosahedral group I ≅ A₅. The golden ratio is inscribed in the icosahedron vertices: (0, ±1, ±τ).
(b) Spectral. A stationary interference pattern with long-range order requires an irrational frequency ratio. τ = [1;1,1,1,...] is the most irrational number, yielding maximal robustness against rational approximants.
(c) Thermodynamic. Partial crystallisation at R ≈ τ² induces density feedback via Gρ_τ = c²/[16π(τ²+4/3)], creating a negative restoring force that locks the ratio.
The lock-in timescale is a pure geometric constant derived from the surface-to-volume ratio of the acceptance window W (rhombic triacontahedron):
The lock-in occurs at t ~ 10⁻⁴² s — firmly pre-inflationary. The present universe is the permanent post-lock-in phase.
The quantum vacuum is modeled as a two-mode elastic medium — a contracting tension T_grav and a dilating tension T_exp. The vacuum is not a quasi-crystal. It is an elastic continuum whose tension-wave interference pattern has quasi-crystalline geometry.
After lock-in at R = τ², the two modes propagate with frequency ratio ω_G/ω_e = 1/τ. Because τ is irrational, the pattern is non-periodic; combined with the projection geometry ℤ⁶→ℝ³, this yields quasi-crystalline order — the AKN tiling — with long-range correlations.
Liquid time (σ ≈ 0): tension stored internally. The quantum vacuum. Gravity is diffuse.
Flowing time (σ oscillating): tension propagates as waves. Radiation.
Solid time (σ = α, locked): tension crystallised at a stable node. Matter.
All from the single dynamic mechanism. Zero adjustable free parameters at the core level.
| Quantity | Nexum | Experiment | Residual |
|---|---|---|---|
| 1/α | 137.0360 | 137.03600 | 0.001% |
| m_μ/m_e | τ¹¹⁽¹⁺α⁾ = 206.768 | 206.768 | 0.036% |
| Koide Q | 2/3 (exact) | 0.666661 | 5×10⁻⁶ |
| Quark masses (6) | <2.2% all | PDG 2024 | 0 param |
| Ω_Λ | (τ+1)/(τ+2) = 0.7236 | 0.7244 | 0.11% |
| G (m³/kg/s²) | 6.6754×10⁻¹¹ | 6.6743×10⁻¹¹ | 0.016% |
| η_B | 6.1005×10⁻¹⁰ | 6.1000×10⁻¹⁰ | 0.01% |
| Hubble tension | +8.2% | +8.3±0.3% | 0.1% |
| sin²θ_W (unif.) | 3/8 (exact) | 3/8 (GUT) | exact |
| κτ (lock-in scale) | ≈ 2.00 (geometric) | pre-inflation | closed S87 |
| ε² (phasonic noise) | 225α³/[72π(τ+2)τ⁴ ln τ] | — | closed S87 |
New results (S86–87) highlighted in green.