[WIP] Investigate predictive power of phase gradient |∇φ|

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<issue_title>Research: Phase Gradient |∇φ| Predictive Power Validation</issue_title>
<issue_description># Phase Gradient |∇φ| Predictive Power Validation

Context

The telemetry module (src/tnfr/physics/fields.py) provides phase gradient measurement |∇φ| as a research-phase metric. Current evidence shows weak correlation with coherence (corr ≈ -0.13), qualifying it as "EM-like, long-range" interaction regime.

Current Status: RESEARCH (NON-CANONICAL)

From §10-11 evidence:

• Correlation: |∇φ| vs ΔC ≈ -0.13 (weak)
• Range: Long (network-wide effects)
• Mechanism: Directional phase tension

Objective

Investigate whether phase gradient |∇φ| can achieve predictive power comparable to Φ_s (|corr| > 0.5) and establish unique safety criteria not captured by structural potential alone.

Research Tasks

1. Extended Correlation Analysis

Goal: Test if |∇φ| correlation improves under specific conditions

Experiments:

• Test correlation at different intensity ranges (I ∈ [1.5, 2.5])
• Separate analysis by topology (ring, scale_free, ws, tree, grid)
• Track time-resolved evolution: |∇φ|(t) vs C(t) trajectories
• Test RA-dominated sequences (resonance amplification) vs OZ-heavy

Hypothesis: |∇φ| may show stronger correlation in:

• Resonance propagation scenarios (RA/UM operators)
• Long-range topologies (ring, grid)
• Pre-fragmentation windows (high stress before collapse)

Acceptance Criteria:

• Document corr(|∇φ|, ΔC) across conditions
• Identify regime where |corr| > 0.3 (if any)
• Compare with Φ_s baseline (corr = -0.822)

2. Path-Integrated Gradient Analysis

Goal: Test if cumulative phase gradient predicts coupling effectiveness

Theory: From §3, UM/RA effectiveness should correlate with path-integrated |∇φ| along coupling edges.

Implementation:

def path_integrated_gradient(G, path):
    """Sum |∇φ| along shortest path between nodes."""
    total = 0
    for i, j in zip(path[:-1], path[1:]):
        phi_i = G.nodes[i]['phase']
        phi_j = G.nodes[j]['phase']
        total += abs(wrap_angle(phi_j - phi_i))
    return total

Experiments:

• Measure path gradient before/after UM/RA application
• Correlate with coupling strength (effective information transfer)
• Compare with coherence length ξ_C predictions

Acceptance Criteria:

• Establish if high path gradient impedes or facilitates coupling
• Document threshold values for effective RA propagation

3. Unique Safety Criterion Development

Goal: Identify if |∇φ| provides safety information not captured by Φ_s

Current Gap:

• Φ_s predicts global stability (Δ Φ_s < 2.0 escape threshold)
• |∇φ| may capture local directional stress before global collapse

Tests:

• Track |∇φ|_max evolution in fragmentation events
• Test if |∇φ| spike precedes Φ_s escape (early warning)
• Measure |∇φ| variance distribution (hotspot detection)

Proposed Criteria:

Safety_gradient = max(|∇φ|) < threshold_gradient

Experiments: Calibrate threshold_gradient via:

• False positive rate (stable systems flagged)
• False negative rate (fragmentation missed)
• Lead time (steps before Φ_s escape)

Acceptance Criteria:

• threshold_gradient value with <5% false positive rate
• Lead time ≥ 2 steps before fragmentation
• Demonstrate cases where |∇φ| alerts but Φ_s doesn't (or vice versa)

4. Cross-Domain Validation

Goal: Test |∇φ| predictions in biological/social/AI applications (domain neutrality)

Proposed Domains:

• Biological: Neural synchronization patterns (phase coherence in oscillator networks)
• Social: Opinion dynamics (phase = position on issue spectrum)
• AI: Distributed learning (phase = gradient alignment)

Implementation:

• Adapt |∇φ| computation to domain-specific phase definitions
• Run simulations with domain-appropriate topologies
• Compare |∇φ| predictive power across domains

Acceptance Criteria:

• |∇φ| correlation consistent across ≥2 domains (CV < 20%)
• Domain-specific interpretation documented
• Evidence of universal mechanism

Promotion Criteria to Canonical

From fields.py documentation:

1. Predictive Power: |corr| > 0.5 comparable to Φ_s (currently ~0.13)
2. Unique Safety Criteria: Not redundant with Φ_s < 2.0 threshold
3. Cross-Domain Validation: Biological, social, AI applications

Constraints

Preserve Invariants:

• Invariant fermga/TNFR-Python-Engine#1: No direct EPI mutation (|∇φ| is read-only telemetry)
• Invariant Permite personalizar ventana de estabilidad para RE’MESH #5: Phase verification still required for UM/RA (|∇φ| doesn't replace |φᵢ - φⱼ| ≤ Δφ_max check)
• Invariant Revert "Clarify EPI history comment" #10: Domain neutrality in core; physics analogies isolated to docs

Classification: RESEARCH until promotion criteria satisfied

Tools & References

Existing Tools:

• `src/tnfr/physics...

Custom agent used: TNFR
TNFR Agent

• Fixes Research: Phase Gradient |∇φ| Predictive Power Validation #2933

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