⚛️ CHAMBER XL: PHASE-EXPOSURE DIAGNOSTICS

UNNS Entanglement Resolution v1.3 (Ω̂ϕ-Enabled) Ω̂ϕ ACTIVE PUBLICATION READY
Configuration
⚠️ τ-field modes: Hard limit N≤300 (auto-clamped)
Creates κ-stable amplitude channels from phase
▶ Ω̂ϕ Phase-Lift Configuration
Ω̂ϕ Status: ENABLED (Pre-κ Operator)
Phase-lift transducer: τ → (τ, ρ cos φ, ρ sin φ)
Creates κ-stable channels for nonseparable correlations
Note: Synthetic coupled/independent modes → ρ=1.0 (no τ-field)
▶ Robustness Configuration
Phase Exposure Results (Ω̂ϕ-Enhanced)
Omega-phi: rho_A
—
Omega-phi: rho_B
—
I₁ (Var ω)
—
I₂ (⟨cos ωΔt⟩)
—
σ_τ (decorr)
—
CHSH |S| (Omega-phi)
—
CHSH |S|
—
|S| Erased
—
D_KL (EM)
—
p-value
—
Weinberg
—
Visualization
Phase Tracks φ_t
I₃(τ) Autocorrelation
Stride Sweep |S|
Time-Shift Null
Robustness Matrix
Test Status Key Metrics Details
Stride Sweep — — —
Time-Shift Null — — —
Surrogate Test — — —
Σ-Gating — — —
📖 Ω̂ϕ Phase-Lift Transducer (Pre-κ Operator)

Core Definition

The Ω̂ϕ operator transforms τ-fields or phase trajectories into κ-stable channels:

Ω̂ϕ(τ_t) := (τ_t, ρ_t cos φ_t, ρ_t sin φ_t)

where:

  • φ_t = phase extracted from τ (gradient-angle or spectral)
  • ρ_t = phase confidence envelope (gradient magnitude or spectral power)
  • (ρ cos φ, ρ sin φ) = "phase tag channels" (κ-stable geometric projections)

Why Ω̂ϕ Matters

Problem: Phase correlations are destroyed by common κ-operators (windowing, thresholding, binning)

Symptom: |S_erased| ≤ 2 even when genuine nonseparability exists

Solution: Ω̂ϕ embeds phase as amplitude-like channels before κ is applied

κ-Stability Guarantee

For any admissible κ (linear averaging, windowing, thresholding):

E[(ρ_A cos φ_A)(ρ_B cos φ_B) + (ρ_A sin φ_A)(ρ_B sin φ_B)] = E[ρ_A ρ_B cos(φ_A - φ_B)]

This means:

  • ✅ Phase correlations survive windowing
  • ✅ CHSH violations remain detectable under coarse observables
  • ✅ Observability erasure is structurally prevented

Observability Trichotomy

With Ω̂ϕ, Chamber XL can distinguish three cases:

  1. Absence: |S_lifted| ≤ 2 → No nonseparability exists
  2. Erasure Artifact: |S_lifted| > 2 but |S_erased| ≤ 2 → Nonseparability exists but is κ-erased
  3. κ-Stable: |S_lifted| > 2 and |S_erased| > 2 → Nonseparability survives coarse projections

Implementation Details

  • Position: Ω̂ϕ MUST be applied before any κ or Σ operator
  • ρ Methods: gradient_magnitude (2D fields), spectral_power (FFT), uniform (ρ=1)
  • ρ Clipping: Prevents numerical instability (default ρ_max = 10)
  • Output: 3-channel field or augmented trajectory
⚠️ Critical:
  • Applying Ω̂ϕ AFTER κ is a category error
  • Single-mode Fourier phase is explicitly forbidden (use multi-mode aggregation)
  • Phase-erased proxies (|cos φ|) invalidate Ω̂ϕ

Version: 1.0 | Status: Production | Paper: "A Pre-κ Phase-Exposure Operator and a Rigorous UNNS Resolution of the Entanglement Question"

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