<!-- Propagated to cursor | Trit: -1 | Source: .ruler/skills/three-match -->

Three-Match Skill: 3-SAT via Colored Subgraph Isomorphism

Status: ✅ Production Ready Trit: -1 (MINUS - conservative/geodesic) Principle: Local constraints → Global correctness Frame: Non-backtracking geodesics with Möbius filtering

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Overview

Three-Match reduces 3-SAT to 3-coloring which reduces to colored subgraph isomorphism. The 3-MATCH gadget enforces constraints LOCALLY via:

1. Non-backtracking geodesics (prime paths, μ(n) ≠ 0)
2. Möbius inversion filtering (back-and-forth cancellation)
3. GF(3) conservation (sum ≡ 0 mod 3)

Correct by construction: If local geodesic constraints are satisfied, global 3-SAT solution is guaranteed.

Core Formula

# Three colors match at depth d iff:
# - Pairwise differences have 3-adic valuation ≥ d
# - No backtracking (each color unique in path)
# - GF(3) sum ≡ 0 (mod 3)

v₃(|a - b|) ≥ d  ∧  v₃(|b - c|) ≥ d  ∧  v₃(|c - a|) ≥ d

Why Non-Backtracking?

1. Prime paths: μ(n) ≠ 0 ⟺ n is squarefree
2. No revisiting: Each state appears once in geodesic
3. Möbius filtering: Composites (backtracking) cancel out
4. Spectral gap: Ramanujan property (λ₂ ≤ 2√(k-1))

Gadgets

1. ThreeMatch Gadget

Three colors forming a valid local constraint:

match = ThreeMatchGeodesicGadget::ThreeMatch.new(seed: 0x42D, depth: 1)
match.color_a  # => { trit: -1, hex: "#2626D8", polarity: :minus }
match.color_b  # => { trit: 0, hex: "#26D826", polarity: :ergodic }
match.color_c  # => { trit: 1, hex: "#D82626", polarity: :plus }
match.gf3_conserved?  # => true

2. NonBacktrackingGeodesic

Prime path through color space:

geo = NonBacktrackingGeodesic.new(seed: seed, length: 8).generate!
geo.prime?           # => true (no backtracking)
geo.moebius_product  # => ±1 (non-zero for primes)
geo.moebius_filter   # => filtered path (only primes kept)

3. ColoredSubgraphGadget

3-SAT clause reduction:

gadget = ColoredSubgraphGadget.new(seed: seed)
gadget.add_clause(1, -2, 3)   # (x₁ ∨ ¬x₂ ∨ x₃)
gadget.add_clause(-1, 2, 4)   # (¬x₁ ∨ x₂ ∨ x₄)
gadget.build_gadgets!
gadget.correct_by_construction?  # => true

4. BackAndForthFilter

Möbius inversion bidirectionally:

filter = BackAndForthFilter.new(seed: seed)
result = filter.full_cycle(sequence)
# Primes kept, composites filtered

Commands

# Run 3-MATCH demo
just three-match

# Test gadget correctness
just test-three-match

# Combine with unworld
just unworld-match

API

require 'three_match_geodesic_gadget'

# Create gadget
match = ThreeMatchGeodesicGadget::ThreeMatch.new(seed: seed)

# Verify constraints
match.gf3_conserved?      # GF(3) sum = 0
match.matches_at_depth?(1) # 3-adic valuation ≥ 1

# Build geodesic
geo = ThreeMatchGeodesicGadget::NonBacktrackingGeodesic.new(
  seed: seed, length: 12
).generate!

# Check primality
geo.prime?  # No backtracking?

Integration with Unworld

The 3-MATCH chain uses seed-chaining for gadget sequence:

chain = Unworld::ThreeMatchChain.new(genesis_seed: seed, length: 4)
chain.unworld[:matches].each do |m|
  puts "#{m[:colors]} | GF(3): #{m[:gf3]}"
end

Mathematical Foundation

Möbius Function

μ(n) = { 1     if n = 1
       { (-1)^k if n = p₁p₂...pₖ (distinct primes)
       { 0     if n has squared prime factor

Möbius Inversion

f(n) = Σ_{d|n} g(d)  ⟹  g(n) = Σ_{d|n} μ(n/d) f(d)

3-adic Valuation

v₃(n) = max { k : 3^k | n }

Example Output

─── 3-MATCH Gadget ───
3-MATCH(d=1): #D8267F #2CD826 #4FD826
  GF(3) conserved: true
  Matches at depth 1: true

─── Non-Backtracking Geodesic ───
Geodesic(PRIME, μ=1): #D8267F → #2CD826 → #4FD826 → ...
  Prime path: true
  Möbius product: 1

─── Colored Subgraph Gadget (3-SAT) ───
  Clauses: 3
  GF(3) all conserved: true
  Prime geodesics: 3
  Correct by construction: true

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Skill Name: three-match Type: 3-SAT Reduction / Colored Subgraph Isomorphism Trit: -1 (MINUS) GF(3): Conserved by construction Geodesics: Non-backtracking (prime paths only)