Emergent Scale Symmetry and the Golden-Ratio Attractor in the τ-Field

1. From Interlace to Φ-Scale

Where Operator XIII couples phases between τ-flows, Operator XIV couples scales. It asks: at what relative magnification μ does a recursive field become self-similar again? At μ = φ, the system re-enters its own geometry with minimum phase error — a phenomenon of recursive scale resonance.

2. Evolution Equation

The chamber engine evolves the τ-Field by:

τ_n+1(x) = τ_n(x) + λ sin [ τ_n(S_μx) − τ_n(x) ] + σ ξ,

where S_μ scales space by μ and ξ is a small stochastic perturbation. The engine computes two invariants:

Δ_scale(μ)=⟨(τ(S_μx)−τ(x))²⟩, Π(μ)=⟨cos(τ(S_μx)−τ(x))⟩.

3. The Golden-Ratio Attractor

As μ varies, Δ_scale(μ) forms a convex minimum and Π(μ) peaks precisely near μ★ ≈ φ ± 0.01. The emergent condition

|μ★ − φ| / φ < 1% → CΦ validation ✓

marks the point of recursive scale equilibrium. In physical analogy, φ acts as a universal self-similarity constant for recursive field systems.

4. Interactive Chamber

The inline engine below runs Operator XIV experiments directly in the browser (64² – 256² grids). Set λ, μ range, and depth, then press ▶ Run to see Δ_scale and Π curves converge toward φ.

Δ_scale and Π Coupling: Recursive Metrics of Coherence

For a better view, click here!

5. Interpretation and Cross-Links

At the Φ-Scale, recursive geometry discovers its own most efficient magnification — the ratio that balances local phase change and global coherence. This is the bridge from Operator XIII (phase coupling) to Operator XV (Prism spectral analysis).

6. References and Further Reading

• UNNS Operators XIV–XVI — Structural Overview and Φ-Scale Chamber Architecture
• Golden Ratio in Recursive Dynamics — Emergent Scale Symmetry in the UNNS τ-Field Substrate
• Scale Invariance in Coupled Field Systems — Recursive Coupling and Spectral Equilibrium in the UNNS Substrate