Ward Identity Checker

Ward Identity Checker

Verify GF(3) conservation as Ward identities across RG flow with Markov blanket separation.

Seed

741086072858456200

Core Principle

Ward identities express symmetry conservation: Σ trit = 0 (mod 3) at every renormalization group (RG) level. Violations indicate "relevant operators" that break the symmetry.

MCP Calibration Data

markov_blanket:
  internal_states: ["#3FF1A7", "#10B99D", "#DF9811"]
  sensory_indices: [1, 2, 3]
  
active_inference:
  prediction_error: 0.5692
  free_energy: 0.6692
  recommendation: perceptual_inference
  
reafference:
  prediction: "#F7E182"
  sensation: "#3FF1A7"
  result: identity_mismatch

Predicates

GF3Conserved(level)

GF3Conserved(L) := Σᵢ trit(cᵢ) ≡ 0 (mod 3)
  where cᵢ ∈ colors_at_level(L)

BlanketIntact(state)

BlanketIntact(s) := ∀ internal ∈ s.internal_states,
  ∃ blanket ∈ s.sensory_states ∪ s.active_states
  such that internal ⊥ external | blanket

NoLeakage(flow)

NoLeakage(f) := GF3Conserved(f.source) ∧ GF3Conserved(f.target)
  ∧ Σ trit(f.source) = Σ trit(f.target)

Ward Identity Check Protocol

def check_ward_identity(colors: list[str], level: int) -> dict:
    """Verify Σ trit = 0 at RG level."""
    trits = [hex_to_trit(c) for c in colors]
    total = sum(trits) % 3
    
    return {
        "level": level,
        "trit_sum": total,
        "conserved": total == 0,
        "violation_type": None if total == 0 else "relevant_operator",
        "correction_needed": (3 - total) % 3
    }

def hex_to_trit(hex_color: str) -> int:
    """Map hex to GF(3) via hue angle."""
    r, g, b = int(hex_color[1:3], 16), int(hex_color[3:5], 16), int(hex_color[5:7], 16)
    hue = compute_hue(r, g, b)
    return int(hue / 120) % 3  # 0-119 → 0, 120-239 → 1, 240-359 → 2

Markov Blanket Separation

┌─────────────────────────────────────────────┐
│                 EXTERNAL                     │
│   (exafference: world-caused sensations)    │
├─────────────────────────────────────────────┤
│              BLANKET STATES                  │
│  Sensory: idx [1,2,3] → colors observed     │
│  Active:  predictions emitted               │
├─────────────────────────────────────────────┤
│                 INTERNAL                     │
│  #3FF1A7 (trit 1) ─┐                        │
│  #10B99D (trit 1) ─┼─ Σ = 3 ≡ 0 (mod 3) ✓  │
│  #DF9811 (trit 1) ─┘                        │
└─────────────────────────────────────────────┘

Violation Detection

When reafference check shows identity mismatch (prediction ≠ sensation):

Usage

# Via Gay.jl MCP
gay_seed 741086072858456200
gay_markov_blanket --internal-seed 741086072858456200 --sensory-indices "1,2,3"

# Check conservation
just ward-check --level 0 --colors "#3FF1A7,#10B99D,#DF9811"

Integration

• cybernetic-immune: Use Ward violations as non-self markers
• active-inference: Free energy ≈ Ward violation magnitude
• unworld: Color chain derivations must preserve Ward identity

References

• Ward-Takahashi identities in QFT
• Friston's Markov blanket formalism
• GF(3) trit arithmetic for color conservation

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/​claude-scientific-skills ecosystem:

Graph Theory

• networkx [○] via bicomodule
  • Universal graph hub

Bibliography References

• general: 734 citations in bib.duckdb

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.