Navigator Codex & Mathematical Manual

Reference manual for mathematical formulas, optimization algorithms, non-linear hidden layers, transformer attention mechanisms, and generative diffusion noise schedules.

ARCHIPELAGO SECTOR CONCEPT CARDS

1. Single-Layer Perceptron & Boolean Gates

The Logic Shores (Sector Alpha-1)

y = step(w₁·x₁ + w₂·x₂ + b)

A Perceptron is the foundational artificial neuron. It computes a weighted linear sum of input signals (x₁, x₂) and applies a step function threshold to output 0 (OFF) or 1 (ON). Adjusting weights rotates the decision boundary line, while bias translates it up or down.

KEY TAKEAWAY: Can solve any linearly separable logic gate (AND, OR, NOT), but requires multiple layers (MLP) for non-linear XOR problems.

2. Linear Regression & Gradient Descent

Predictive Plains (Sector Beta-2)

w := w - α · (∂J / ∂w)

Gradient Descent is an iterative optimization algorithm used to minimize Mean Squared Error loss J(w). By calculating the derivative slope (gradient) at the current parameter position, the model steps downhill along the loss curve. Learning rate α controls step magnitude.

KEY TAKEAWAY: Too small an α results in slow convergence; too large an α causes step overshooting and instability.

3. 2D Spatial Convolution & Feature Map Pooling

Convolutional Coast (Sector Gamma-3)

S(i,j) = (I * K)(i,j) = Σ_m Σ_n I(i+m, j+n) · K(m,n)

Convolutional Neural Networks (CNNs) slide a small weight matrix (Kernel) across spatial image grids to compute dot products. This extracts spatial feature contours such as vertical edges, gradients, and textures.

KEY TAKEAWAY: Max Pooling downsamples feature map resolution while preserving peak spatial feature activations.

4. Multi-Layer Perceptrons & Non-Linear Activation

Neural Nebula (Sector Delta-4)

y = step( W⁽²⁾ · ReLU(W⁽¹⁾x + b⁽¹⁾) + b⁽²⁾ )

Multi-layer neural networks introduce hidden layers and non-linear activation functions (ReLU, Sigmoid). By combining multiple linear decision half-spaces, MLPs solve non-linearly separable problems like XOR.

KEY TAKEAWAY: Non-linear activations are essential; without them, stacking multiple layers collapses back into a single linear transformation.

5. Transformers & Scaled Dot-Product Self-Attention

Mount Generative (Sector Omega-5)

Attention(Q, K, V) = softmax(Q·Kᵀ / √dₖ) · V

Self-Attention allows neural models to dynamically weigh relationships between tokens in a sequence regardless of distance. Queries (Q) represent current token requests, Keys (K) represent context tags, and Values (V) contain representation vectors.

KEY TAKEAWAY: Softmax converts dot-product similarity scores into normalized attention probabilities summing to 1.0.

6. Generative Diffusion Models & Latent Samplers

Diffusion Deep (Sector Abyss-6)

x_{t-1} = 1/√α_t · ( x_t - (β_t / √(1-ᾱ_t)) · ε_θ(x_t, t) )

Diffusion models reverse a forward noise process that degrades images into Gaussian noise. By learning to predict and subtract noise at each timestep t, diffusion samplers construct crisp latent images out of chaos.

KEY TAKEAWAY: Classifier-Free Guidance (w) steers the noise prediction toward conditional target prompts.

NAVIGATOR RANK PROGRESSION ROADMAP

L10 - 249 XP

Cadet Navigator

UNLOCKED BY DEFAULT

L2250 - 499 XP

Signal Apprentice

REQUIRES LOGIC SHORES CALIBRATION

L3500 - 749 XP

Neural Voyager

REQUIRES PREDICTIVE PLAINS CONVERGENCE

L4750 - 999 XP

Archipelago Architect

REQUIRES CONVOLUTIONAL COAST KERNEL

L51000 - 1249 XP

Neural Master Architect

REQUIRES NEURAL NEBULA MLP

L61250 - 1499 XP

Archipelago Visionary

REQUIRES MOUNT GENERATIVE ATTENTION

L7+1500+ XP

Archipelago Master Creator

APEX PROMOTION (DIFFUSION DEEP DENOISE)