⚖️ CHAMBER κ₁

Purpose

Chamber κ₁ operates on τ-stable ensembles from Chamber κ₀, selecting states based on intrinsic symmetry measures. It does not modify τ-dynamics or generate new states—it purely selects among existing τ-stable outcomes.

Symmetry Measures

• Σ₁ (Mean Bias): |⟨x⟩| — translation symmetry breaking
• Σ₂ (Reflection): Σ|x[i] − x[N−1−i]| — spatial inversion asymmetry
• Σ₃ (Spectral Odd-Mode): Fraction of power in odd Fourier modes
• Σ₄ (Chirality): |Σ(x[i] − x[i+1])| — rotational orientation bias

Selection Policies

• Pure: argmin Σᵢ — select minimum of single measure
• Lexicographic: Σᵢ → Σⱼ → Σₖ — hierarchical tie-breaking
• Negative Control: argmax Σᵢ — anti-selection for comparison

κ₁ vs κ₁′ Modes

κ₁′ Null-Axis Criteria

An axis Σᵢ is classified as null if any of:

• Zero variance: Var(Σᵢ) < 10⁻⁶
• High degeneracy: >50% of states share minimum value
• Vanishing gap: Gap to 2nd minimum < 10⁻³

Example: In your run, Σ₄ (chirality) showed Var ≈ 0, degeneracy = 100%, gap = 0 → classified as null

Validation Criteria

κ₁ Standard Criteria:

• Cκ1-1: At least 2 states differ in each Σᵢ
• Cκ1-2: ≥95% of policies resolve deterministically
• Cκ1-3: κ₁ does not alter U or W distributions
• Cκ1-4: Different policies select different states
• Cκ1-5: Report degeneracy metrics for all policies

κ₁′ Additional Criteria:

• Cκ1′-1: At least one Σᵢ classified as null and excluded
• Cκ1′-2: Determinism improves relative to κ₁ standard
• Cκ1′-3: Selections are subset of κ₁ selections (consistency)

Required κ₀ Data Format

{
  "ensemble": [
    {
      "state_id": "s_0042",
      "config": [x₀, x₁, ..., x₆₃],  ← Required for full κ₁
      "energy": 1.234,
      "wall_count": 8,
      "mean": 0.023
    },
    ...
  ]
}