Dialectica Skill (ERGODIC 0)
Proof-as-game interpretation via Gödel's Dialectica
Trit: 0 (ERGODIC)
Color: #26D826 (Green)
Role: Coordinator/Transporter
Dialectica transforms proofs into games:
A ⊢ B becomes ∃x. ∀y. R(x, y)
Where:
D(A ∧ B) = ∃(x,x').∀(y,y'). D(A)[x,y] ∧ D(B)[x',y']
D(A → B) = ∃f,F. ∀x,y. D(A)[x, F(x,y)] → D(B)[f(x), y]
D(∀z.A) = ∃f. ∀z,y. D(A)[f(z), y]
D(∃z.A) = ∃(z,x). ∀y. D(A)[x, y]
# World hop via Dialectica
def dialectica_hop(proposition, world_state)
# Transform proposition to game
game = {
proponent_moves: extract_witnesses(proposition),
opponent_moves: extract_challenges(proposition),
winning: atomic_condition(proposition)
}
# Play generates new world
new_world = play_game(game, world_state)
# GF(3) conservation check
verify_gf3(world_state, new_world)
end
Proponent (∃)
↓ witness x
Opponent (∀)
↓ challenge y
Proponent
↓ response (via f, F)
...
Atomic check R(x,y)
Dialectica splits into multiplicative/additive:
A ⊸ B = (A⊥ ⅋ B) # Linear implication
A ⊗ B # Tensor (both needed)
A & B # With (choice)
A ⊕ B # Plus (given)
!A # Of course (reusable)
?A # Why not (garbage)
Chu(Set, ⊥) ≃ *-autonomous category
Objects: (A⁺, A⁻, ⟨-,-⟩: A⁺ × A⁻ → ⊥)
three-match (-1) ⊗ dialectica (0) ⊗ gay-mcp (+1) = 0 ✓
proofgeneral-narya (-1) ⊗ dialectica (0) ⊗ rubato-composer (+1) = 0 ✓
clj-kondo-3color (-1) ⊗ dialectica (0) ⊗ cider-clojure (+1) = 0 ✓
# Transform proof to game
just dialectica-game "A → B"
# Play one round
just dialectica-play witness challenge
# Check linear decomposition
just dialectica-linear prop
Dialectica produces:
Hom_Dial((A,X,α), (B,Y,β)) =
{ (f,F) : A×Y → B, A×Y → X |
α(a, F(a,y)) ≤ β(f(a,y), y) }