⚙️ CHAMBER XIV: Φ-SCALE v0.8.0 + τ-VALIDATOR

Operator XIV — Self-Contained with τ-Invariant Analysis
Configuration
Visualization
τ-Field Evolution
Δ_scale(μ) & Π(μ)
Standard Metrics
μ★
—
φ Error
—
Current μ
—
Δ_scale
—
Π
—
Status
Ready
🔬 τ-Invariant Validation Analysis
Closure Residual (Δτ)
—
Closure Signature
—
Classification
—
N★ (Transcendence)
—
Ψ-Drift (final)
—
τ-Closure Status
—

📘 Note on Derived τ-Structures: Derived τ-structures may exhibit high stability and reproducibility without satisfying τ-closure. Certification requires Δτ → 0 under refinement. Stable structures with Δτ > 10⁻⁶ indicate composition of primary mechanisms, not primariness itself.

C⊗ Extended Validation

C⊗₃ (Reproducibility)
—
C⊗₄ (τ-Closure)
—
C⊗₅ (Stability)
—
Certification
—
📚 Laboratory Guide

v0.8.0 Updates: τ-Invariant Integration

New in v0.8.0:

  • Δτ Calculator: Real-time closure residual computation
  • Φ→Ψ→τ Mapping: Full regime progression with N★ threshold detection
  • Signature Classification: Automatic detection of closure mechanisms (O/R/P)
  • Structural Half-Life: Operator XII collapse dynamics
  • Extended C⊗ Validation: τ-informed certification criteria

Operator XIV: Φ-Scale Hypothesis

Theoretical Foundation: The recursive scaling operator XIV implements:

τn+1(x) = τn(x) + λ sin(τn(Sμx) - τn(x)) + σ ξ

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

τ-Invariant Analysis

Closure Signature of φ: OR

  • O (Orthogonal): √5 component in φ = (1+√5)/2
  • R (Relaxation): Fibonacci ratio equilibration dynamics
  • Classification: DERIVED τ-structure (not primary)

Expected Behavior:

  • Δτ(φ) decreases exponentially with depth (R dominates)
  • t₁/₂ = finite but large (derived stability)
  • N★ ≈ 15-25 μ-steps for Φ→Ψ→τ transcendence
  • Survives noise σ ≤ 0.05, collapses for σ > 0.1

Validation Workflow

  1. Standard Run: Click "▶ Run Sweep" for basic μ★ detection
  2. Reproducibility Test: Click "🔬 Multi-Seed" for C⊗₃ (CV ≤ 1%)
  3. Stability Test: Click "📊 Depth Sweep" for C⊗₅ plateau
  4. Review τ-Analysis: Check Δτ, signature, and certification status
  5. Export Bundle: Save complete JSON with τ-validation data

Version: 0.8.0 | Engine: TauFieldEngineN v0.7.0 | Validator: TauInvariantValidator v1.0.0