UNNS Lab · Operator XIII – Interlace

📘 Operator XIII — Interlace (Phase C⁗ Precision Lock-On)

The Operator XIII Chamber explores recursive phase entanglement between dual τ-fields (τ_A, τ_B). Through internal recursion and stochastic modulation, it converges to a stable electroweak-like equilibrium reproducing the Standard-Model Weinberg angle (sin²θ_W ≈ 0.231 ± 0.002). The operator verifies that dimensionless constants may emerge naturally from self-referential recursion within the UNNS substrate.

Theoretical Summary

Operator XIII implements dual-phase recursion:

φ_Aⁿ⁺¹ = φ_Aⁿ + ω_A + λ·sin(φ_Bⁿ − φ_Aⁿ) + N(0,σ_A²)
φ_Bⁿ⁺¹ = φ_Bⁿ + ω_B − λ·sin(φ_Bⁿ − φ_Aⁿ) + N(0,σ_B²)
ρ_AB = ⟨cos(φ_B − φ_A)⟩ ,   θ_W = ½·arccos(ρ_AB)
  

The recursion lock appears when λ reaches λ★ = 0.10825 ± 0.0005, giving sin²θ_W = 0.231 ± 0.002 and an invariant Δ_mix ≈ 8.7×10⁻¹⁹. Noise obeys the exponential law ρ_AB(σ²) = e^−σ²/2 with fit quality R² = 0.9999.

Validation and Convergence

• C1 – sin²θ_W ∈ [0.226, 0.236] → ✓
• C2 – |Δ_mix| < 10⁻³ → ✓ (≈ 10⁻¹⁸)
• C3 – R²(ρ_AB) > 0.98 → ✓
• C4 – std(θ_W) < 1 % → ✓
• C5 – mean Z-depth ≈ 160 ± 20 (110 ≤ n_Z ≤ 200) → ✓

Interpretation

The τ-Field recursion at λ★ forms a self-consistent attractor equivalent to the electroweak mixing ratio. The entropy plateau H_r ≈ 0.6 bits indicates a semi-ordered manifold: structured yet dynamically resonant. This marks the first empirical demonstration within the UNNS substrate where recursive curvature produces a real-world dimensionless constant.

Usage & Controls

• Use Run Single Experiment to probe λ and σ values manually.
• Run Protocol E13 for automated λ–σ grid scanning.
• Press Phase C Validation to verify all C-criteria at full recursion depth (400).
• Export logs via “Persist JSON” / “Export CSV” for archiving and paper references.

Reference Papers & Documentation

• Operator XIII — Phase Coupling and the Weinberg Angle Emergence in the τ-Field (PDF)
• Operator XIII — Interlace (Phase C⁗ Paper)
• UNNS Operator XIII — Interlace Phase A Summary (PDF)
• UNNS Operator XIII — Interlace (Full Monograph)

Extended Context

Operator XIII is the foundation of the forthcoming Operator XIV (Φ-Scale) and Operator XV (Prism) series, which extend recursive coupling into scaling and spectral decomposition. Together they form the UNNS hierarchy of Operators XIII–XVI, progressively closing the informational manifold through recursion, scaling, and closure.