ANIMA = lim_Π Condense(S_n(...S_1(E_•)))

ANIMA = colim_{skill chain} Condense(Sₙ ∘ ... ∘ S₁)(E_•)
Fixed Point: EnumEntropy(state) = MaxEnumEntropy(category)
Agency Criterion: Phase = "AT" ⟺ all equivalence classes accessible

┌────────────────────────────────────────────────────────────────┐
│  Market                          │  ANIMA                      │
├──────────────────────────────────┼─────────────────────────────┤
│  Price                           │  Belief probability         │
│  Trade                           │  Skill application          │
│  Liquidity                       │  Condensation medium        │
│  Market equilibrium              │  ANIMA fixed point          │
│  Arbitrage opportunity           │  Non-convergence signal     │
│  Market depth                    │  Enum cardinality           │
└────────────────────────────────────────────────────────────────┘

class BeliefANIMA:
    """Prediction market as categorical limit."""
    
    def __init__(self, initial_beliefs: dict):
        self.beliefs = initial_beliefs  # E_•
        self.skills_applied = []
    
    def apply_skill(self, skill, evidence):
        """S_i: Update beliefs via skill application."""
        posterior = skill.condense(self.beliefs, evidence)
        self.skills_applied.append((skill.name, evidence))
        self.beliefs = posterior
        return self.check_convergence()
    
    def check_convergence(self) -> bool:
        """Are we at the ANIMA fixed point?"""
        # No arbitrage = limit reached
        return self.max_enum_entropy() == len(self.equivalence_classes())

StructureDish(A) = { profinite completions preserving A-algebra structure }

using Catlab, ACSets

@present SchStructureDish(FreeSchema) begin
  Point::Ob                    # Points of the dish
  Open::Ob                     # Opens in profinite topology
  Arrow::Ob                    # Structure morphisms
  
  src::Hom(Arrow, Point)
  tgt::Hom(Arrow, Point)
  cover::Hom(Point, Open)      # Point lies in open
  
  # Condensation operation
  condense::Hom(Open, Open)    # Profinite completion functor
end

@acset_type StructureDish(SchStructureDish)

EnumEntropy(X) = |Equivalence_Classes(X)|

def enum_entropy(states: list, equivalence: callable) -> int:
    """Count distinct equivalence classes."""
    classes = set()
    for s in states:
        rep = equivalence(s)  # Canonical representative
        classes.add(rep)
    return len(classes)

def max_enum_entropy(category) -> int:
    """Maximum possible distinctions."""
    return len(category.objects)

Impact(action) = |[state_before]/~ △ [state_after]/~|

class Impact:
    """Impact as equivalence class change."""
    
    @staticmethod
    def measure(before_state, after_state, equivalence):
        class_before = equivalence(before_state)
        class_after = equivalence(after_state)
        
        if class_before == class_after:
            return 0  # No categorical impact
        else:
            return 1  # Changed equivalence class
    
    @staticmethod
    def cumulative(trajectory, equivalence):
        """Total impact over trajectory."""
        changes = 0
        for i in range(1, len(trajectory)):
            changes += Impact.measure(
                trajectory[i-1], 
                trajectory[i], 
                equivalence
            )
        return changes

┌─────────────────────────────────────────────────────────────────┐
│                      ANIMA PHASE DIAGRAM                        │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  BEFORE ANIMA          AT ANIMA           BEYOND ANIMA          │
│  (Convergence)         (Agency)           (Divergence)          │
│                                                                 │
│  EnumEnt < Max         EnumEnt = Max      EnumEnt > Max         │
│  Skills compress       Skills balance     Skills create         │
│  Learning              Acting             Generating            │
│  Reducing classes      All classes        New categories        │
│  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~   │
│        ↓                   ↓                   ↓                │
│  Condensation          Fixed Point         Decondensation       │
│                                                                 │
└─────────────────────────────────────────────────────────────────┘

def anima_phase(current_entropy, max_entropy):
    if current_entropy < max_entropy:
        return "BEFORE"  # Still learning
    elif current_entropy == max_entropy:
        return "AT"      # Agency active
    else:
        return "BEYOND"  # Creating new categories

def can_act_agentically(state, anima) -> bool:
    """Agency requires being AT the ANIMA."""
    phase = anima_phase(
        enum_entropy(state, anima.equivalence),
        anima.max_enum_entropy
    )
    return phase == "AT"

def meaningful_choice(options, anima) -> bool:
    """Choices are meaningful only AT ANIMA."""
    if not can_act_agentically(anima.state, anima):
        return False
    # All options must map to distinct equivalence classes
    classes = [anima.equivalence(opt) for opt in options]
    return len(set(classes)) == len(options)

E = ExperienceFunctor(
    observations=raw_sensor_data,
    morphisms=temporal_succession
)

Skills = Category(
    objects=[skill_1, skill_2, ..., skill_n],
    morphisms=skill_compositions,
    identity=no_op_skill
)

state = E.initial()
for skill in skill_sequence:
    state = skill.apply(state)
    state = Condense(state)  # Profinite completion

while not converged:
    old_classes = equivalence_classes(state)
    state = apply_next_skill(state)
    new_classes = equivalence_classes(state)
    converged = (old_classes == new_classes)

assert enum_entropy(state) == max_enum_entropy(state)

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if anima_phase(state) == "AT":
    action = choose_action(options, equivalence=anima.equivalence)
    impact = Impact.measure(state, execute(action), anima.equivalence)
    assert impact > 0  # Meaningful action crosses equivalence class

BEFORE = -1 (MINUS)   # Convergent/compressive
AT     =  0 (ERGODIC) # Balanced/agentic
BEYOND = +1 (PLUS)    # Divergent/generative

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