⚡ CHAMBER XXXIX: κ-LIMITED PROPAGATION PROTOCOL

⚙️ Regime Presets:

κ₂ Stress-Test Controls

⚠️ Experiment Protocol:

1. Initialization: Prepare dual τ-field with k-filtered excitation
2. Evolution: Run dispersive dynamics (β-coupled Laplacian)
3. Phase Velocity: Measure v_phase from spectral ω(k) relation
4. κ₁ Test: Check local registration continuity
5. κ₂ Test: Verify cross-observer consistency
6. κ₃ Test: Test re-entry persistence

Purpose & Theoretical Foundation

Chamber XXXIX implements the κ-Limited Propagation Principle (KLP) test protocol. Rather than asserting "nothing can exceed c," KLP states that propagation admissibility depends on what kind of structure is propagating:

Key Prediction: You can engineer v_phase > c, but it will necessarily exhibit κ₁, κ₂, or κ₃ collapse—it cannot carry persistent causal structure.

Experimental Design

We use a dual τ-field system with dispersive coupling:

∂τ₁/∂t = λ sin(Δτ₁) - β ∇²τ₁
∂τ₂/∂t = λ sin(Δτ₂) - β ∇²τ₂

The dispersive term β∇²τ creates a frequency-dependent phase velocity via the dispersion relation ω(k). By filtering to specific k-modes, we engineer regimes where v_phase = ω/k > c_eff.

Test Protocols

κ₁ Test: Local Registrability

Question: Can you track discrete propagation steps locally?

Method: Measure spatial registration continuity via gradient correlation:

C_κ₁ = ⟨∇τ(t) · ∇τ(t+Δt)⟩ / (|∇τ(t)| |∇τ(t+Δt)|)

Collapse Criterion: C_κ₁ < 0.3 indicates failure to maintain local tracking

κ₂ Test: Cross-Observer Consistency

Question: Do boosted observers agree on propagation invariants?

Method: Apply Lorentz-like transformation and check invariant preservation:

τ'(x', t') = τ(γ(x' + vt'), γ(t' + vx'/c²))

Collapse Criterion: Observer-dependent metrics indicate κ₂ failure

κ₃ Test: Re-Entry Persistence

Question: Does the propagation leave a stable, replayable record?

Method: Extract "propagated structure," reset field, replay, and compare:

R_κ₃ = ⟨(τ_replay - τ_original)²⟩^(1/2) / ⟨τ²⟩^(1/2)

Collapse Criterion: R_κ₃ > 0.5 indicates non-persistent structure

Expected Results

Critical Insight: Sub-ceiling velocity (v < c_eff) does NOT guarantee κ-admissibility. Clean carrier requires σ=0 AND low dispersion. Regime III demonstrates that speed limits are insufficient - structural gates matter.

Interpretation Guidelines

• κ₁ Collapse: Pattern is not a locally registrable transport event (no step-by-step carrier)
• κ₂ Collapse: Observers disagree on what propagated → frame-dependent artifact
• κ₃ Collapse: Cannot extract stable causal record → ephemeral phase pattern
• SUBCRITICAL_NONREGISTRABLE: Valid regime - v < c but noise/dispersion destroys κ₁ carrier (NOT an anomaly)
• NO_KAPPA_ADMISSIBLE_PROPAGATION: κ₁ collapse with null velocity - no measurable carrier exists (valid result, not data failure)

KLP Verdict: If v > c_eff but structure persists through all three gates, KLP is falsified. Otherwise, KLP is supported with classification of propagation regime.

Recommended Workflow

1. Regime I (Admissible): σ=0, β=0.01, k_filter=0.8, duration=600 → Clean carrier, C_κ₁>0.5, all gates PASS
2. Regime III (Nonregistrable): σ=0.01, β=0.12, k_filter=2.5, duration=400 → Diffusive, κ₁ COLLAPSE despite v
3. Regime IV (Phase Collapse): β=0.25, k_filter=5.0, duration=400 → Superluminal phase, all gates COLLAPSE
4. Analysis: Compare κ-gate results across regimes to validate structural vs kinematic admissibility
5. Validation: Verify Regime I requires BOTH v

Theoretical Significance

Chamber XXXIX provides positive evidence for KLP by demonstrating:

• Structural Dominance: κ-admissibility depends on coherence (σ, β), not just velocity
• Regime Separation: Four experimentally distinguishable propagation classes
• Falsifiability: KLP makes specific, testable predictions about collapse modes
• Operational Clarity: "Faster than light" requires specifying what moves and how it's registered
• Substrate Generality: Results should hold in any κ-admissible substrate, not just this τ-field implementation

Connection to Lorentz Symmetry: If KLP holds, Lorentz invariance emerges as the unique symmetry preserving κ-admissibility across inertial observers—exactly the reinterpretation proposed in the companion theoretical document.

Version: 1.0.0 | Engine: TauFieldEngineN (N=2, FFT) | Status: Experimental Protocol