Temporal Coalgebra

Coalgebra: (X, γ: X → F(X))   # State → Observable structure
Final:     νF = lim F^n(1)     # Greatest fixpoint

Bisimulation R ⊆ X × Y:
  (x, y) ∈ R ⟹ F(R)(γ_X(x), γ_Y(y))

For derivation observation:

# Observe derivation stream
observe(derivation) = { head: current_step, tail: rest_of_derivation }

# Two derivations are equivalent iff:
bisimilar?(d1, d2) == (observe(d1).head == observe(d2).head &&
                       bisimilar?(observe(d1).tail, observe(d2).tail))

Why Coalgebra for Verification?

1. Behavioral equivalence: Same observations = same system
2. Infinite structures: Streams, trees, processes
3. Game semantics: Attacker/defender games are coalgebraic
4. Lazy evaluation: Observe only what's needed

Gadgets

1. ObservationFunctor

Transform derivations into observations:

functor = TemporalCoalgebra::ObservationFunctor.new(
  source: :derivation_chain,
  target: :observation_stream
)
observation = functor.apply(derivation)
observation.head      # => current observable state
observation.tail      # => remaining stream (lazy)
observation.finite?   # => false (potentially infinite)

2. FinalCoalgebra

Construct the final coalgebra for type F:

final = TemporalCoalgebra::FinalCoalgebra.new(
  functor: stream_functor,
  approximation_depth: 100
)
final.carrier           # => νF (greatest fixpoint)
final.universal?(coal)  # => check if coal maps uniquely
final.unfold(seed)      # => generate infinite structure

3. BisimulationChecker

Verify behavioral equivalence:

checker = TemporalCoalgebra::BisimulationChecker.new
checker.add_system(:system_a, coalgebra_a)
checker.add_system(:system_b, coalgebra_b)

result = checker.check_bisimilar!
result[:bisimilar]         # => true/false
result[:distinguishing_trace]  # => if false, witness
result[:depth_checked]     # => how deep we verified

4. StreamCoalgebra

Work with infinite streams:

stream = TemporalCoalgebra::StreamCoalgebra.new(seed: 0x42D)
stream.head              # => first element
stream.tail              # => rest of stream
stream.take(10)          # => first 10 elements
stream.drop(5).head      # => 6th element
stream.map { |x| x * 2 } # => transformed stream
stream.zip(other_stream) # => paired stream

5. ThreeMatchBisimulation

Integration with three-match for game verification:

game = TemporalCoalgebra::ThreeMatchBisimulation.new(
  attacker: player_a,
  defender: player_b,
  three_match_gadget: gadget
)
game.play_round!
game.defender_wins?      # => strategies are bisimilar
game.attacker_wins?      # => found distinguishing move
game.gf3_conserved?      # => trit sum = 0

Commands

# Observe derivation chain
just coalgebra-observe

# Check bisimulation
just coalgebra-bisim system_a system_b

# Generate stream from seed
just coalgebra-stream 0x42D 20

# Verify game equivalence
just coalgebra-game

API

require 'temporal_coalgebra'

# Create observation system
obs = TemporalCoalgebra::Observer.new(
  trit: -1,
  functor: :stream
)

# Observe derivation
stream = obs.observe(derivation_chain)

# Check equivalence
bisim = obs.bisimilar?(stream_a, stream_b)

# Integrate with three-match
game_result = obs.verify_game(three_match_gadget)

Integration with GF(3) Triads

Forms valid triads with ERGODIC (0) and PLUS (+1) skills:

temporal-coalgebra (-1) ⊗ unworld (0) ⊗ gay-mcp (+1) = 0 ✓
temporal-coalgebra (-1) ⊗ glass-bead-game (0) ⊗ cider-clojure (+1) = 0 ✓
temporal-coalgebra (-1) ⊗ acsets (0) ⊗ rubato-composer (+1) = 0 ✓

Mathematical Foundation

Coalgebra Definition

Coalgebra for F: (X, γ: X → F(X))
  - X is carrier (state space)
  - γ is structure map (observation)
  - F is endofunctor (observation type)

Stream Functor

F(X) = A × X    (head × tail)
νF ≅ A^ω       (infinite sequences)

Bisimulation

R is bisimulation ⟺
  ∀(x,y) ∈ R: (γ_X(x), γ_Y(y)) ∈ F(R)

Behavioral equivalence: x ∼ y ⟺ ∃R bisimulation. (x,y) ∈ R

Coinduction Principle

To prove P(ν F), show:
  P is F-consistent: P(x) ⟹ P(tail(x))
  P(seed) holds

Example Output

─── Temporal Coalgebra Observation ───
Source: Derivation chain (length ∞)
Functor: Stream (head × tail)

Observation:
  head: { seed: 0x42D, color: #2626D8, trit: -1 }
  tail: <lazy stream>

Bisimulation Check:
  System A: derivation_chain_1
  System B: derivation_chain_2
  
  Depth 0: heads match ✓
  Depth 1: tails match ✓
  Depth 2: tails match ✓
  ...
  Depth 100: tails match ✓

Result: BISIMILAR (observationally equivalent)
GF(3) Trit: -1 (MINUS/Observer)

─── Three-Match Integration ───
Game: Attacker vs Defender
Rounds: 12
Winner: Defender (strategies bisimilar)
GF(3) conserved: true

Skill Name: temporal-coalgebra Type: Observation / Bisimulation Verification Trit: -1 (MINUS) Color: #2626D8 (Blue) GF(3): Forms valid triads with ERGODIC + PLUS skills Dual: Algebra (construction) ↔ Coalgebra (observation)