% Fitness-proportionate selection
P(i) = \frac{f_i}{\sum_{j=1}^{N} f_j}

% Replicator dynamics
\dot{x}_i = x_i \left[ f_i(x) - \bar{f}(x) \right]

         Cooperate    Defect
Cooperate   R,R        S,T
Defect      T,S        P,P

where T > R > P > S (temptation > reward > punishment > sucker)

Rule 110: [111→0] [110→1] [101→1] [100→0] [011→1] [010→1] [001→1] [000→0]

A^{t+\Delta t} = \left[ A^t + \Delta t \cdot G(K * A^t) \right]_0^1

G_{\mu,\sigma}(x) = 2e^{-\frac{(x-\mu)^2}{2\sigma^2}} - 1

% Velocity field from kernel convolution
\vec{v}(x) = \nabla G(K * A^t)

% Mass-conserving update via continuity equation
A^{t+1} = A^t - \nabla \cdot (A^t \cdot \vec{v})

% With multispecies extension
A_i^{t+1} = A_i^t - \nabla \cdot \left(A_i^t \cdot \sum_j w_{ij} \vec{v}_j\right)

\left[\left[A_i^t + \Delta t G(K * A_i^t)\right]_0^1 + \sum_{j \in N(i)} k_{ji} \cdot E_{ji}^t\right]_0^1

def nca_step(grid, model):
    # Perceive: Sobel filters for gradients
    perception = perceive(grid)  # [identity, sobel_x, sobel_y, ...]
    
    # Update: Neural network
    delta = model(perception)
    
    # Apply with stochastic mask
    mask = torch.rand_like(delta) < 0.5
    return grid + delta * mask

class Agent:
    def __init__(self):
        self.sugar = initial_sugar
        self.metabolism = random.randint(1, 4)
        self.vision = random.randint(1, 6)
    
    def move(self, landscape):
        # Look in cardinal directions up to vision
        best = max(visible_sites, key=lambda s: s.sugar)
        self.position = best
        self.sugar += best.sugar - self.metabolism

\vec{v}_{new} = w_s \cdot \text{separation} + w_a \cdot \text{alignment} + w_c \cdot \text{cohesion}

\mathcal{F} = \underbrace{D_{KL}[q(\theta)||p(\theta)]}_{\text{complexity}} + \underbrace{\mathbb{E}_q[-\log p(y|\theta)]}_{\text{accuracy}}

/Users/bob/ies/hatchery_repos/bmorphism__alien/
├── source/       # CUDA kernels
├── resources/    # Simulation configs
└── GAY.md        # Gay.jl integration

undefined

undefined

% Mutation-selection balance
\hat{p} = \frac{\mu}{s}

% Wright-Fisher drift
\text{Var}(\Delta p) = \frac{p(1-p)}{2N}

% Gray-Scott
\frac{\partial u}{\partial t} = D_u \nabla^2 u - uv^2 + f(1-u)
\frac{\partial v}{\partial t} = D_v \nabla^2 v + uv^2 - (f+k)v

% Information synergy
I_{\text{syn}}(X \rightarrow Y) = I_{\text{tot}} - \sum_{i=1}^{n} I_{\text{ind}}(X_i)

\frac{dx_i}{dt} = x_i\left(r_i + \sum_{j=1}^{n} A_{ij} x_j\right)

/Users/bob/ies/paper_extracts/alife2025/
├── ALIFE2025_full.md          # 925KB markdown
├── ALIFE2025_tex.zip          # 11MB LaTeX
├── tex_extracted/
│   └── fed660c6-.../
│       ├── *.tex              # 7283 lines
│       └── images/            # 153 figures
└── conversion_status.json

/Users/bob/ies/
├── axelrod-evolution-of-cooperation.md
├── epstein-axtell-growing-artificial-societies.txt
├── wooldridge-multiagent-systems.txt
└── hatchery_repos/bmorphism__alien/

using Gay

undefined

just alife-toc                    # Full table of contents
just alife-paper 42               # Get paper at page 42
just alife-equation "lenia"       # Find Lenia equations
just alife-axelrod                # Axelrod summary
just alife-sugarscape             # Sugarscape patterns
just alife-alien                  # ALIEN simulation info
just alife-lenia "orbium"         # Lenia creature lookup

undefined

undefined

graph TB
    subgraph Evolution
        GA[Genetic Algorithms]
        OEE[Open-Ended Evolution]
        NS[Natural Selection]
    end
    
    subgraph Emergence
        CA[Cellular Automata]
        NCA[Neural CA]
        Lenia[Lenia]
    end
    
    subgraph Agents
        ABM[Agent-Based Models]
        Swarm[Swarm Intelligence]
        GABM[Generative ABM]
    end
    
    subgraph Chemistry
        BZ[BZ Reaction]
        Auto[Autopoiesis]
        Chem[Artificial Chemistry]
    end
    
    GA --> OEE
    CA --> NCA --> Lenia
    ABM --> Swarm --> GABM
    BZ --> Auto --> Chem
    
    OEE --> Emergence
    Lenia --> Agents
    GABM --> Chemistry

@proceedings{alife2025,
  title     = {ALIFE 25: Ciphers of Life},
  editor    = {Witkowski, O. and Adams, A.M. and Sinapayen, L.},
  year      = {2025},
  pages     = {337}
}

@book{axelrod1984,
  title     = {The Evolution of Cooperation},
  author    = {Axelrod, Robert},
  year      = {1984},
  publisher = {Basic Books}
}

@book{epstein1996,
  title     = {Growing Artificial Societies},
  author    = {Epstein, Joshua M. and Axtell, Robert},
  year      = {1996},
  publisher = {MIT Press}
}

% Flow-Lenia mass conservation
A^{t+1} = A^t + \nabla \cdot (A^t \cdot \vec{v}(K * A^t))

% EngramNCA hidden memory
h^{t+1} = \sigma(W_h \cdot [v^t, h^t] + b_h)

% DiffLogic gate probability
p(g) = \text{softmax}(\theta_g) \quad g \in \{\text{AND}, \text{OR}, \text{XOR}, ...\}

% Monoid operational closure
\text{Aut}(S) \cong \text{Mon}(\mathcal{C}), \quad |\text{Ob}| = 1