Generator (⊕)  ─────┐
                    ├──→  GF(3) = 0  ──→  Prime Geodesic
Coordinator (○) ────┤
                    │
Validator (⊖)  ─────┘

The 3-Agent Structure

Role	Trit	Color	Action	Responsibility
Generator	+1	#D82626 (Red)	Create	Propose code changes, new patterns
Coordinator	0	#26D826 (Green)	Transport	Formalize structure, derive next seed
Validator	-1	#2626D8 (Blue)	Verify	Check invariants, reduce to essence

Prime Geodesic Foundation

See PRIME_GEODESICS.md for full mathematical foundation.

Why Non-Backtracking?

Property	Prime Path	Composite Path
Factorization	Unique	Multiple
p-adic valuation	Well-defined	Ambiguous
Möbius μ(n)	≠ 0	= 0 (filtered)
Ihara zeta	Contributes	Ignored

Key insight: Chromatic walks are prime geodesics in derivation space, traversable unambiguously by zeta functions.

Zeta Function Traversability

Zeta	Domain	Primes
Riemann ζ(s)	ℤ	Prime numbers
Ihara ζ_G(u)	Graphs	Non-backtracking cycles
Dedekind ζ_K(s)	Number fields	Prime ideals
Selberg Z(s)	Manifolds	Prime geodesics

Seed Chaining Across Agents

Each agent derives its seed from the shared genesis, offset by the golden ratio:

genesis = 0x42D  # 1069
γ = 0x9E3779B97F4A7C15  # Golden ratio constant

seed_generator   = genesis                    # +1 stream
seed_coordinator = genesis ^ γ                # 0 stream  
seed_validator   = genesis ^ (γ << 1)         # -1 stream

Derivation Chain (per step)

seed_{n+1} = (seed_n ⊕ (trit_n × γ)) × MIX  mod 2⁶⁴

GF(3) Conservation Invariant

At every step of the walk:

trit_generator + trit_coordinator + trit_validator ≡ 0 (mod 3)
      (+1)      +       (0)       +      (-1)      =   0  ✓

Self-Boiling Property

Like the Three Chromatic Samovars:

Generator provides +1 (creative surplus)
Validator provides -1 (critical reduction)
Coordinator provides 0 (transport without change)

Walk Protocol

Phase 1: Genesis

just chromatic-walk seed=1069

Phase 2: Propose (Generator leads)

Generator proposes a codebase change with +1 trit.

Phase 3: Formalize (Coordinator transports)

Coordinator structures the proposal, deriving the interface.

Phase 4: Verify (Validator constrains)

Validator checks invariants and reduces.

Phase 5: Commit (combined)

If GF(3) = 0 at walk end, the improvement is committed.

Commands

just chromatic-walk seed=1069         # Initialize walk
just chromatic-walk-step seed=1069    # Run single step
just chromatic-walk-verify seed=1069  # Verify GF(3) conservation

Triad Bundles

three-match (-1) ⊗ chromatic-walk (0) ⊗ gay-mcp (+1) = 0 ✓  [Core Walk]
ramanujan-expander (-1) ⊗ ihara-zeta (0) ⊗ moebius-inversion (+1) = 0 ✓  [Prime Netting]

Goblin Integration

Chromatic-walk is the 0 Coordinator in the goblin triad:

shadow-goblin (-1) ⊗ chromatic-walk (0) ⊗ agent-o-rama (+1) = 0 ✓

Role	Trit	Color	Action
Shadow Goblin	-1	#F0D127	Validates walks
Chromatic Walk	0	#46F27F	Transports derivations
Agent-O-Rama	+1	#E7B367	Generates proposals

Coordinator Capabilities

As the 0 goblin, Chromatic Walk provides:

Transport: Move context between generator and validator
Derive: Compute next seed in derivation chain
Navigate: Traverse between realms (Beautiful Realms)
Balance: Ensure triad sums remain at GF(3) = 0

Color Stream (Seed 1069, Stream 2)

Step 1: #46F27F (mint green)     ○ Transport
Step 2: #E2282A (vivid red)      ○ Derive
Step 3: #EE55F2 (hot pink)       ○ Navigate

Related Skills

Skill	Relation
unworld	Seed chaining foundation
ihara-zeta	Non-backtracking enumeration
ramanujan-expander	Spectral gap verification
moebius-inversion	Composite filtering
gay-mcp	Deterministic color generation
shadow-goblin	Validates walks (-1)
agent-o-rama	Generates proposals (+1)

Skill Name: chromatic-walk
Type: Multi-Agent Prime Geodesic Exploration
Trit: 0 (ERGODIC)
GF(3): Conserved by construction
Backtracking: Forbidden (composites are p-adically imprecise)

chromatic-walk

Chromatic Walk Skill

Overview