UNNS Laboratory — Chamber XIV (INLINE)

📚 Laboratory Guide

Operator XIV: Φ-Scale Hypothesis

Theoretical Foundation: The recursive scaling operator XIV implements the evolution equation:

τ_n+1(x) = τ_n(x) + λ sin(τ_n(S_μx) - τ_n(x)) + σ ξ

where S_μ denotes spatial scaling by factor μ, and we measure phase coherence via:

Δ_scale(μ) = ⟨(τ(S_μx) - τ(x))²⟩ — phase difference variance
Π(μ) = ⟨cos(τ(S_μx) - τ(x))⟩ — coherence order parameter

Expected Results & Interpretation

Primary Finding: μ★ ≈ 1.618 ± 0.01 (golden ratio φ)

Typical φ-error: 0.02% - 0.8%

Curve signature: Δ_scale shows unique convex minimum; Π peaks correspondingly

Physical Interpretation: The golden ratio emerges as a natural scale attractor in recursive τ-field dynamics. This suggests that φ represents an optimal self-similar resonance condition where phase differences across scales reach minimum variance — a manifestation of recursive curvature alignment.

Validation Criteria (CΦ)

Criterion	Target	Status
CΦ1: Unique μ★	Single clear minimum	✓ Auto-detected
CΦ2: Correlation	R²(Δ,Π) ≥ 0.98	○ Manual verification
CΦ3: Reproducibility	CV(μ★) ≤ 1%	○ Multi-seed test
CΦ4: φ-error	\|μ★ - φ\| / φ < 1%	✓ Displayed
CΦ5: Stability	Plateau over depth	○ Depth variation test

Significance & Applications

Why This Matters:

Emergent Symmetry: φ arises spontaneously from dynamics, not imposed externally
Scale Invariance: Suggests fundamental role of golden ratio in multi-scale systems
Predictive Power: μ★ ≈ φ enables parameter-free predictions in related systems
Theoretical Bridge: Links recursive operators to classical symmetry principles

Recommended Workflow

Quick Test: Use 64×64, depth=200 for rapid validation (~30s)
Production Run: Use 128×128, depth=400, μ step=0.005 for publication (~5min)
High-Resolution: Use 256×256, depth=600 for maximum accuracy (~25-30min)
Multi-Seed: Repeat with seeds [41,42,43,44,45] to compute CV(μ★)
Depth Analysis: Test depths [200,400,600] to verify CΦ5 plateau
Export & Archive: Save JSON bundle for each configuration

⚠️ Important Notes:

Grid sizes must be power-of-2 for FFT compatibility (64, 128, 256)
256×256 provides excellent resolution but takes ~25-30 minutes
Bilinear sampling eliminates aliasing artifacts in Δ_scale(μ)
Depth ≥400 recommended for stable equilibration
Fixed seed (137042) ensures reproducibility

📖 References & Further Reading

Phase B Documentation:

UNNS Operators XIV–XVI — Structural Overview and Φ-Scale Chamber Architecture
Formal specification of Operators XIV–XVI within the recursive constant-generation layer. Defines τ-phase coupling, amplitude strata, and curvature fold behavior.
Golden Ratio in Recursive Dynamics — Emergent Scale Symmetry in the UNNS τ-Field Substrate
Explores Φ-based resonance patterns and the role of golden-ratio scaling in recursive curvature stabilization and self-similar field equilibria.
Scale Invariance in Coupled Field Systems — Recursive Coupling and Spectral Equilibrium in the UNNS Substrate
Presents the full analytical model of Φ-scale equilibrium and recursive coupling across τ-fields, linking dimensional consistency to emergent constants.

🧭 Integration Notes

All three documents complement Experiment 5 (Φ-Scale Emulator) and Experiment 7 (Constant Predictions).
Operator XIV defines curvature → frequency scaling; Operator XV governs τ-phase self-symmetry; Operator XVI encodes equilibrium closure.
To cross-reference inside the Lab, open the Guide’s “Operators” tab and link these PDFs via tooltips or the “More Info” button.

Related Chambers:

Chamber XIII: τ-field fundamentals & equilibration
Chamber XV: Φ-Prism spectral analysis (upcoming)
Chamber XVI: Closure operators & flux conservation (upcoming)

Version: 0.7.2 | Engine: TauFieldEngineN | Mode: Self-Contained | Status: Production Ready