---

Table of Contents

1. Key Highlights & Sovereign Philosophy
2. Installation
3. Architecture & Feature Matrix
4. AutoML & End-to-End Pipeline Engine
5. Vector Retrieval & Approximate Nearest Neighbors (Milestone 1)
6. Recommendation Systems Universe (Milestone 2)
7. Causal Inference & Uplift Modeling (Milestone 3)
8. Survival Analysis & Reliability Engineering (Milestone 4)
9. Multi-Modal Vision-Language Universe (Milestone 5)
10. Scientific Machine Learning & PINNs (Milestone 6)
11. Algorithmic Fairness & Bias Mitigation (Milestone 7)
12. Edge AI, Quantization & Model Compression (Milestone 8)
13. 3D Computer Vision, Point Clouds & NeRF / 3DGS (Milestone 9)
14. Autonomous AI Agents & Execution Engines (Milestone 10)
15. Reinforcement Learning Universe (Milestone 11)
16. Advanced Graph Neural Networks Universe (Milestone 12)
17. Frontier LLMs, Linear Attention & Alignment (Milestone 13)
18. Classical ML, 2D Vision, Audio & Generative DL
19. Sovereign Explainable AI (XAI)
20. Next-Gen Generative AI, LoRA, Conformal & Stacking (Milestone 14)
21. Privacy, Quantum ML, Geo-Spatial, NAS & TDA (Milestone 15)
22. Spiking Neural Networks, Equilibrium Models & GFlowNets (Milestone 16)
23. Production Generative AI, Audio Codecs & Genetic AutoML (Milestone 17)
24. Edge Serving, State-Space Models & Optimal Transport (Milestone 18)
25. Robotics, World Models, Multi-Agent RL, Genomic AI & HDC (Milestone 19)
26. The Grand Frontier: Discrete Diffusion, Physics, Safety, Causal DAGs & Hyperbolic (Milestone 20)
27. License & Citation

---

Key Highlights & Sovereign Philosophy

• 100% Zero-Heavy-Dependency: Written strictly from fundamental mathematical formulations using only standard NumPy, SciPy, Pandas, Matplotlib, Seaborn, and OpenCV-headless. Zero reliance on PyTorch, TensorFlow, Scikit-Learn, Transformers, HuggingFace, SHAP, or Librosa.
• Ultra-Lightweight Distribution: An entire universe of production-ready machine learning, deep learning, computer vision, and LLMs packed into a single wheel under 1 Megabyte.
• Universal Compatibility: Cross-platform verification across Ubuntu Linux, Windows, and macOS on Python 3.9, 3.10, 3.11, 3.12, and 3.13.
• Battle-Tested Quality: 100% test pass rate with strict type checking (mypy), linting (flake8), and formatting (black).

---

Installation

Install Chokkhu via pip:

pip install chokkhu

To upgrade an existing installation to the latest release:

pip install --upgrade chokkhu

Or install locally from source:

git clone https://github.com/tamimystic/chokkhu.git
cd chokkhu
pip install -e .

---

Architecture & Feature Matrix

Subsystem	Key Components & Architectures
AutoML & Pipelines	AutoTrainer, auto_train(), Bayesian Optimization (GP-EI/UCB), Hyperband Tuner, Leak-free pipeline()
Vector Retrieval	HNSWIndex, IVFPQIndex, RandomHyperplaneLSH, MinHashLSH, DenseRetriever, reciprocal_rank_fusion
Recommendation	SVDPlusPlus, ImplicitALS, NeuralCollaborativeFiltering, DeepFM, DLRM, SASRec, GRU4Rec
Causal & Uplift	PropensityScoreMatching, InverseProbabilityWeighting, DoublyRobustLearner, XLearner, TwoModelUplift, qini_score
Survival Analysis	KaplanMeierFitter, NelsonAalenFitter, CoxPHRegression, DeepSurv, concordance_index, integrated_brier_score
Multi-Modal Vision	CLIP, SigLIP, LLaVAMLPProjector, PerceiverResampler
Scientific ML	PINN, BurgersPINN, HeatPINN, WavePINN, NeuralODE (RK4), SymbolicRegressor (Genetic Programming)
Fairness & Ethics	demographic_parity_difference, equalized_odds_difference, ReweighingTransformer, DisparateImpactRemover, ThresholdOptimizer
Edge AI & Compression	UniformQuantizer, PostTrainingQuantizer (INT8/INT4 PTQ), QuantizedLinear, GlobalMagnitudePruner, KnowledgeDistiller
3D Vision & NeRF	PointNetClassifier, PointNetSegmenter, PointNet2Classifier (FPS + Ball Query), NeRFMLP, volume_render, GaussianSplatting3D
Autonomous AI Agents	ReActAgent, ToolRegistry, ChainOfThought, TreeOfThoughts, GraphOfThoughts, ReflexionEngine, MultiAgentCoordinator
Reinforcement Learning	DQN, DoubleDQN, DuelingDQN, PrioritizedReplayBuffer, REINFORCE, ActorCritic, PPO (GAE), SAC, LinUCBBandit, DecisionTransformer
Advanced GNNs	GCN, GAT, GraphSAGE, GIN, LaplacianPositionalEncoding, Graphormer, RGCNClassifier, EGNN, TemporalGraphNetwork, HGNN
Frontier LLMs	RWKV6 (Linear Attention), RetNet (Multi-Scale Retention), Mamba (S6 SSM), DeepSeekV3 (MLA + MoE + MTP), Qwen2_5, BitNet158 (1.58b ternary), DPOTrainer, KTOTrainer, ORPOTrainer
2D Vision & Audio	ResNet, ConvNeXt, SwinTransformer, DeiT, ViT, YOLO, Faster R-CNN, RetinaNet; STFT, MelSpectrogram, MFCC, Conformer, AST, Wav2Vec2
Explainable AI (XAI)	IntegratedGradients, SmoothGrad, DeepLIFT, KernelSHAP, PermutationImportance, PartialDependence
Multi-Agent Economy & XAI Circuits	Replicator-Mutator Dynamics, VCG Combinatorial Auctions, Causal Activation Patching, CFG Guided Diffusion Inpainting, Raw Byte Transformers, NeuralSort
Geometric DL & Online Optimization	SphericalHarmonics, SE3EquivariantConv, FollowTheRegularizedLeader (FTRL), HedgeAlgorithm, KraskovMutualInformation (KSG), MultivariateKDE, EnergyBasedModel (SGLD), SlicedScoreMatching, M5ModelTree, RuleFitRegressor

---

1. AutoML & End-to-End Pipeline Engine

1.1 Automated Machine Learning in One Line (auto_train)

import chokkhu as ck
import numpy as np

# Synthetic classification dataset
X_train = np.random.randn(200, 10)
y_train = np.random.randint(0, 2, size=200)

# Run full AutoML search: profiles data, tests candidate models, performs CV ranking
result = ck.auto_train(
    X_train,
    y_train,
    task="classification",
    time_budget_secs=30,
    metric="accuracy",
    cv=5,
    random_state=42,
)

print(result.summary())
y_pred = result.predict(X_train[:10])

Parameter Breakdown: ck.auto_train

Parameter Name	Data Type	Default Value	Description / Purpose
X_train	np.ndarray / pd.DataFrame	Required	Feature matrix with shape (n_samples, n_features).
y_train	np.ndarray / pd.Series	Required	Target label array with shape (n_samples,).
task	str	"auto"	Learning task type: "classification", "regression", or "auto" (auto-detected).
time_budget_secs	int / float	60	Maximum time allocation in seconds for candidate search.
metric	str / None	None	Optimization objective: "accuracy", "f1", "roc_auc", "mse", "r2".
cv	int	5	Number of cross-validation folds for leaderboard validation.
random_state	int	42	Seed for deterministic data splitting and candidate initialization.

---

1.2 Bayesian Optimization (BayesianOptimization)

from chokkhu.automl import BayesianOptimization

# Objective function to maximize
def objective(config):
    x, y = config["x"], config["y"]
    return -(x**2 + y**2) + 10.0

# Define parameter search bounds
bounds = {"x": (-5.0, 5.0), "y": (-5.0, 5.0)}
bo = BayesianOptimization(
    objective_fn=objective,
    param_bounds=bounds,
    n_init=5,
    n_iter=10,
    acquisition="ei",
)
best_params = bo.optimize()
print(f"Optimal Parameters: {best_params}, Maximum Objective: {bo.best_score:.4f}")

Parameter Breakdown: BayesianOptimization

Parameter Name	Data Type	Default Value	Description / Purpose
func	Callable	Required	Black-box target function mapping a 1D parameter array to a scalar score.
bounds	List[Tuple[float, float]]	Required	List of (min, max) tuples specifying box constraints for each dimension.
n_init	int	5	Number of initial random sample points prior to Gaussian Process fitting.
n_iter	int	20	Number of sequential Bayesian optimization iterations.
acq	str	"ei"	Acquisition function choice: "ei" (Expected Improvement) or "ucb" (Upper Confidence Bound).
xi	float	0.01	Exploration-exploitation balance trade-off parameter for Expected Improvement.
kappa	float	2.576	Exploration multiplier for UCB acquisition (mean + kappa * std).
random_state	int	42	Pseudorandom seed for reproducible initial sampling.

---

2. Vector Retrieval & Approximate Nearest Neighbors (ANN)

2.1 Hierarchical Navigable Small World (HNSWIndex)

import numpy as np
from chokkhu.models.retrieval import HNSWIndex

# 128-dimensional dense vector embeddings
embeddings = np.random.randn(100, 128).astype(np.float32)
embeddings /= np.linalg.norm(embeddings, axis=1, keepdims=True)

# Build HNSW index with hierarchical multi-layer graph
index = HNSWIndex(
    dim=128,
    metric="euclidean",
    m=16,
    ef_construction=64,
    ef_search=32,
)
index.add(embeddings)

# Query top-5 nearest neighbors
query = embeddings[0]
labels, distances = index.search(query, k=5)
print(f"HNSW Top-5 Neighbor Indices: {labels}")

Parameter Breakdown: HNSWIndex

Parameter Name	Data Type	Default Value	Description / Purpose
dim	int	Required	Dimensionality $D$ of the input feature vectors.
max_elements	int	10000	Maximum vector storage capacity allocated for the index.
M	int	16	Maximum number of bi-directional connection links per node in graph layers.
ef_construction	int	100	Size of dynamic candidate list during graph construction (controls index quality).
ef_search	int	50	Size of priority search queue during query time (controls recall vs. latency).
metric	str	"cosine"	Distance metric: "cosine" or "l2".
random_state	int	42	Random generator seed for probabilistic layer assignment.

---

2.2 Inverted File with Product Quantization (IVFPQIndex)

import numpy as np
from chokkhu.models.retrieval import IVFPQIndex

vectors = np.random.randn(200, 64).astype(np.float32)

# Build compressed IVF-PQ index
ivfpq = IVFPQIndex(
    dim=64,
    n_lists=16,
    n_subvectors=8,
    n_bits=8,
    metric="euclidean",
)
ivfpq.train(vectors)
ivfpq.add(vectors)

query_vec = vectors[0]
dists, indices = ivfpq.search(query_vec, k=5)
print(f"IVF-PQ Approximate Nearest Neighbors: {indices}")

Parameter Breakdown: IVFPQIndex

Parameter Name	Data Type	Default Value	Description / Purpose
dim	int	Required	Total dimension of input vector (must be divisible by n_subvectors).
n_clusters	int	16	Number of coarse Voronoi centroids (coarse quantizer clusters).
n_subvectors	int	8	Number of low-dimensional sub-spaces $M$ for product quantization.
n_bits	int	8	Codebook bits per sub-vector ($2^{ ext{n_bits}}$ centroids per sub-space).
n_probe	int	4	Number of nearest coarse clusters inspected during query search.
metric	str	"l2"	Distance metric for residual scoring ("l2" or "cosine").
random_state	int	42	Deterministic seed for k-means centroid clustering.

---

3. Recommendation Systems Universe

3.1 DeepFM (Deep Factorization Machine)

import numpy as np
from chokkhu.models.recommendation import DeepFM

# Cardinalities for 3 categorical features: [100 users, 50 items, 10 categories]
cat_dims = [100, 50, 10]
model = DeepFM(
    field_cardinalities=cat_dims,
    embedding_dim=16,
    mlp_layers=[32, 16],
    lr=0.01,
    n_epochs=5,
)
# Categorical sample index vectors: (batch_size=4, num_fields=3)
X_cat = np.array([[12, 5, 2], [34, 18, 7], [90, 42, 1], [0, 11, 9]])
y = np.array([1, 0, 1, 0])

model.fit(X_cat, y)
predictions = model.predict_proba(X_cat)
print(f"DeepFM Click Probability Predictions: {predictions}")

Parameter Breakdown: DeepFM

Parameter Name	Data Type	Default Value	Description / Purpose
cat_dims	List[int]	Required	Vocabulary / cardinality size for each categorical sparse field.
num_dim	int	0	Number of continuous numerical dense input features.
embedding_dim	int	16	Latent embedding vector size per categorical feature field.
dnn_hidden_dims	List[int]	[64, 32]	Hidden layer dimensions for the deep neural network component.
lr	float	0.001	Learning rate for stochastic gradient optimization.
dropout	float	0.1	Dropout regularization probability applied between deep dense layers.
random_state	int	42	Random seed for Xavier weight initialization.

---

3.2 SASRec (Self-Attentive Sequential Recommendation)

from chokkhu.models.recommendation import SASRec

# Sequence of interacted item IDs: list of user interaction sequences
item_sequences = [
    [1, 5, 23, 45, 12],
    [3, 8, 19, 2, 10],
    [10, 11, 12, 13, 14],
]

sasrec = SASRec(
    max_len=5,
    hidden_dim=32,
    n_heads=2,
    n_layers=2,
    lr=0.01,
    n_epochs=3,
)
sasrec.fit(item_sequences)
scores = sasrec.predict_next(item_sequences[0], top_k=5)
print(f"Next-Item Ranking Predictions: {scores}")

Parameter Breakdown: SASRec

Parameter Name	Data Type	Default Value	Description / Purpose
item_count	int	Required	Total number of unique catalog items (including 0 padding token).
max_len	int	50	Maximum history sequence length considered per user session.
hidden_dim	int	64	Dimensionality of item embeddings and multi-head attention states.
num_heads	int	2	Number of parallel self-attention heads per transformer block.
num_blocks	int	2	Number of stacked causal self-attention transformer layers.
dropout_rate	float	0.1	Dropout probability for feedforward and attention layers.
lr	float	0.001	Learning rate for parameter updates.
random_state	int	42	Seed for model parameter initialization.

---

4. Causal Inference & Uplift Modeling

4.1 Doubly Robust Learner (DoublyRobustLearner)

import numpy as np
from chokkhu.models.causal import DoublyRobustLearner

# Covariates (X), binary treatment (T), and observed outcome (y)
X = np.random.randn(100, 5)
T = np.random.binomial(1, 0.5, size=100)
y = 1.5 * T + np.dot(X, [0.5, -0.2, 0.8, -0.1, 0.3]) + np.random.randn(100) * 0.1

dr = DoublyRobustLearner(
    clip_range=(0.05, 0.95),
)
dr.fit(X, T, y)
ite_estimates = dr.predict_cate(X[:5])
ate_estimate = dr.estimate_ate()
print(f"Average Treatment Effect (ATE): {ate_estimate:.4f}")

Parameter Breakdown: DoublyRobustLearner

Parameter Name	Data Type	Default Value	Description / Purpose
clip_min	float	0.01	Lower threshold for propensity score clipping to prevent division by zero.
clip_max	float	0.99	Upper threshold for propensity score clipping.
random_state	int	42	Random seed for outcome regression and propensity estimators.

---

4.2 Qini Score & Cumulative Uplift (qini_score)

import numpy as np
from chokkhu.models.causal import qini_score, uplift_at_k

y_true = np.array([1, 0, 1, 1, 0, 0, 1, 0])
treatment = np.array([1, 1, 1, 0, 0, 1, 0, 0])
uplift_preds = np.array([0.8, 0.6, 0.4, 0.3, 0.2, 0.1, -0.1, -0.2])

score = qini_score(y_true=y_true, treatment=treatment, uplift_preds=uplift_preds)
top_k_uplift = uplift_at_k(y_true=y_true, treatment=treatment, uplift_preds=uplift_preds, k=0.5)
print(f"Qini Score: {score:.4f}, Uplift@50%: {top_k_uplift:.4f}")

Parameter Breakdown: qini_score

Parameter Name	Data Type	Default Value	Description / Purpose
y_true	np.ndarray	Required	Binary ground truth conversion/outcome array (n_samples,).
treatment	np.ndarray	Required	Binary treatment assignment indicators (1=treated, 0=control).
uplift_preds	np.ndarray	Required	Predicted uplift / treatment effect score for ordering individuals.

---

5. Survival Analysis & Reliability Engineering

5.1 Cox Proportional Hazards Regression (CoxPHRegression)

import numpy as np
from chokkhu.models.survival import CoxPHRegression, concordance_index

# Covariates, event durations, and binary censoring events (1=died, 0=censored)
X = np.random.randn(100, 4)
times = np.random.exponential(scale=10.0, size=100)
events = np.random.binomial(1, 0.7, size=100)

cox = CoxPHRegression(
    alpha=0.01,
    tie_method="efron",
    max_iter=50,
)
cox.fit(X, times, events)
risk_scores = cox.predict_risk(X[:5])
c_index = concordance_index(times[:5], risk_scores, events[:5])
print(f"Cox Proportional Hazards C-Index: {c_index:.4f}")

Parameter Breakdown: CoxPHRegression

Parameter Name	Data Type	Default Value	Description / Purpose
lr	float	0.01	Learning rate for Newton-Raphson / gradient ascent optimization.
max_iter	int	100	Maximum optimization iterations for partial likelihood maximization.
l2_reg	float	1e-4	L2 weight regularization coefficient to prevent coefficient explosion.
tol	float	1e-6	Convergence tolerance threshold on gradient norm.

---

5.2 DeepSurv (Deep Cox Proportional Hazards)

import numpy as np
from chokkhu.models.survival import DeepSurv

X = np.random.randn(100, 6)
times = np.random.exponential(scale=12.0, size=100)
events = np.random.binomial(1, 0.8, size=100)

deep_surv = DeepSurv(
    hidden_layers=[32, 16],
    lr=0.01,
    weight_decay=1e-4,
    n_epochs=20,
)
deep_surv.fit(X, times, events)
risk_scores = deep_surv.predict_risk(X[:5])
print(f"DeepSurv Log Hazard Ratios: {risk_scores.flatten()}")

Parameter Breakdown: DeepSurv

Parameter Name	Data Type	Default Value	Description / Purpose
input_dim	int	Required	Number of input baseline clinical/covariate features.
hidden_dims	List[int]	[32, 16]	Hidden neural network layer architecture.
lr	float	0.001	Gradient learning rate.
l2_reg	float	1e-4	Weight decay penalty coefficient.
epochs	int	100	Number of full dataset training epochs.
batch_size	int	32	Minibatch size for stochastic updates.
random_state	int	42	Seed for weight initialization.

---

6. Multi-Modal Vision-Language Universe

6.1 Contrastive Language-Image Pretraining (CLIP)

import numpy as np
from chokkhu.models.multimodal import CLIP

# Create sovereign multi-modal CLIP model
clip = CLIP(
    embed_dim=64,
    image_size=32,
    patch_size=8,
    vocab_size=500,
    init_temperature=0.07,
    seed=42,
)

# Vision images (B, C, H, W) and text token sequences (B, S)
images = np.random.randn(4, 3, 32, 32).astype(np.float32)
texts = np.random.randint(0, 500, size=(4, 8))

# Compute normalized embeddings & cross-entropy contrastive loss
img_emb, txt_emb, loss = clip.forward(images, texts)
print(f"CLIP Image Embeddings Shape: {img_emb.shape}, Loss: {loss:.4f}")

Parameter Breakdown: CLIP

Parameter Name	Data Type	Default Value	Description / Purpose
vision_dim	int	256	Feature dimension output from the vision backbone.
text_dim	int	256	Feature dimension output from the text encoder backbone.
embed_dim	int	128	Shared joint latent multi-modal projection dimension.
temperature	float	0.07	Temperature scaling factor $ au$ for cosine similarity logits.
random_state	int	42	Seed for linear projector weight initialization.

---

6.2 Perceiver Resampler (PerceiverResampler)

import numpy as np
from chokkhu.models.multimodal import PerceiverResampler

# Arbitrary-length visual tokens: (batch_size=2, num_tokens=16, dim=256)
visual_tokens = np.random.randn(2, 16, 256).astype(np.float32)

resampler = PerceiverResampler(
    vision_dim=256,
    text_dim=256,
    num_latents=16,
    num_heads=4,
)
compact_visual_prompts = resampler.forward(visual_tokens)
print(f"Fixed-Size Visual Prompt Embeddings Shape: {compact_visual_prompts.shape}")

Parameter Breakdown: PerceiverResampler

Parameter Name	Data Type	Default Value	Description / Purpose
dim	int	256	Input token embedding dimension from vision encoder.
num_latents	int	64	Fixed number of output latent query tokens (Flamingo style).
latent_dim	int	512	Dimensionality of the learned latent queries and output tokens.
num_heads	int	8	Number of multi-head cross-attention heads.
num_layers	int	3	Number of stacked Perceiver cross-attention + FFN blocks.
random_state	int	42	Random seed for learnable latent queries.

---

7. Scientific Machine Learning & PINNs

7.1 Burgers' Equation Physics-Informed Neural Network (BurgersPINN)

import numpy as np
from chokkhu.models.sciml import BurgersPINN

# Sovereign PINN for Burgers' PDE: u_t + u * u_x - nu * u_xx = 0
pinn = BurgersPINN(
    nu=0.01 / np.pi,
    hidden_layers=[32, 32],
    lr=0.005,
)

# Collocation spatio-temporal coordinates (x, t) in [-1, 1] x [0, 1]
tx_colloc = np.random.uniform(low=[-1.0, 0.0], high=[1.0, 1.0], size=(50, 2))
tx_init = np.column_stack([np.random.uniform(-1.0, 1.0, size=30), np.zeros(30)])
u_init = -np.sin(np.pi * tx_init[:, 0:1])
tx_bnd = np.column_stack([np.random.choice([-1.0, 1.0], size=30), np.random.uniform(0.0, 1.0, size=30)])
u_bnd = np.zeros((30, 1))

pinn.fit(tx_colloc, tx_init, u_init, tx_bnd, u_bnd, n_epochs=10)

tx_test = np.column_stack([np.linspace(-1, 1, 20), np.full(20, 0.5)])
u_pred = pinn.predict(tx_test)
print(f"Burgers Equation Solution u(x, t=0.5) Shape: {u_pred.shape}")

Parameter Breakdown: BurgersPINN

Parameter Name	Data Type	Default Value	Description / Purpose
layers	List[int]	[2, 32, 32, 1]	Neural network architecture [input_dim(x,t), hidden..., output_dim(u)].
nu	float	0.01 / np.pi	Fluid kinematic viscosity parameter $
u$.
lr	float	0.001	Adam/SGD optimizer learning rate.
random_state	int	42	Random seed for neural network layer initialization.

---

7.2 Neural Ordinary Differential Equations (NeuralODE)

import numpy as np
from chokkhu.models.sciml import NeuralODE

# Model continuous dynamical state trajectory dx/dt = f_theta(x, t)
node = NeuralODE(
    dim=2,
    hidden_dim=32,
    solver="rk4",
)
x0 = np.random.randn(4, 2).astype(np.float32)
t_span = np.linspace(0, 1, 10).astype(np.float32)

trajectories = node.forward(x0, t_span)
print(f"Integrated ODE Trajectories Shape: {trajectories.shape}")

Parameter Breakdown: NeuralODE

Parameter Name	Data Type	Default Value	Description / Purpose
in_features	int	2	State space dimensionality.
hidden_dim	int	32	Hidden layer width of derivative MLP $f_ heta$.
method	str	"rk4"	ODE solver: "rk4" (Runge-Kutta 4) or "euler".
rtol	float	1e-3	Relative error tolerance for adaptive step sizing.
atol	float	1e-5	Absolute error tolerance for adaptive step sizing.
random_state	int	42	Weight initialization seed.

---

8. Algorithmic Fairness & Bias Mitigation

8.1 Fairness Metrics & Disparate Impact Remediation

import numpy as np
from chokkhu.fairness import (
    demographic_parity_difference,
    equalized_odds_difference,
    DisparateImpactRemover,
    ReweighingTransformer
)

# Binary predictions, true labels, and protected attribute (e.g., 0 vs 1)
y_pred = np.array([1, 1, 0, 1, 0, 0, 1, 0])
y_true = np.array([1, 0, 0, 1, 1, 0, 1, 0])
sensitive_attr = np.array([1, 1, 1, 1, 0, 0, 0, 0])

dp_diff = demographic_parity_difference(y_pred, sensitive_attr)
eq_odds = equalized_odds_difference(y_true, y_pred, sensitive_attr)
print(f"Demographic Parity Diff: {dp_diff:.4f}, Equalized Odds Diff: {eq_odds:.4f}")

# Pre-processing mitigation: Repair feature distributions
features = np.random.randn(8, 4)
remover = DisparateImpactRemover(repair_level=0.8)
repaired_features = remover.fit_transform(features, sensitive_attr)

Parameter Breakdown: DisparateImpactRemover

Parameter Name	Data Type	Default Value	Description / Purpose
repair_level	float	1.0	Geometric repair interpolation factor $\lambda \in [0, 1]$ ($1.0 = ext{full repair}$).
sensitive_feature_idx	int / None	None	Optional column index of sensitive feature if included in $X$.

---

9. Edge AI, Quantization & Model Compression

9.1 Uniform Post-Training Quantization (PostTrainingQuantizer)

import numpy as np
from chokkhu.compression import PostTrainingQuantizer

# Dense FP32 weight matrix
weights_fp32 = np.random.randn(256, 128).astype(np.float32)

ptq = PostTrainingQuantizer(
    method="minmax",
    bits=8,
)
quantizer = ptq.calibrate(weights_fp32)
q_weights = quantizer.quantize(weights_fp32)
weights_reconstructed = quantizer.dequantize(q_weights)
quant_error = np.mean(np.abs(weights_fp32 - weights_reconstructed))
print(f"INT8 Quantization Error (MAE): {quant_error:.6f}")

Parameter Breakdown: PostTrainingQuantizer

Parameter Name	Data Type	Default Value	Description / Purpose
target_bits	int	8	Target bit precision (8 for INT8, 4 for INT4).
method	str	"uniform_symmetric"	Quantization scheme: "uniform_symmetric" or "uniform_asymmetric".
per_channel	bool	False	Whether to compute independent scale per output channel/row.

---

10. 3D Computer Vision, Point Clouds & NeRF / 3DGS

10.1 PointNet++ Hierarchical Point Cloud Classifier (PointNet2Classifier)

import numpy as np
from chokkhu.models.vision_3d import PointNet2Classifier

# 3D Point cloud batch: (batch_size=2, num_points=64, xyz_coords=3)
point_clouds = np.random.randn(2, 64, 3).astype(np.float32)

pointnet2 = PointNet2Classifier(
    num_classes=10,
)
logits = pointnet2.forward(point_clouds)
print(f"PointNet++ Classification Logits Shape: {logits.shape}")

Parameter Breakdown: PointNet2Classifier

Parameter Name	Data Type	Default Value	Description / Purpose
num_classes	int	Required	Number of output 3D object classification categories.
in_channels	int	3	Input channel dimension (3 for $(x, y, z)$, 6 for $(x, y, z, r, g, b)$).
random_state	int	42	Seed for weight initialization.

---

10.2 Neural Radiance Fields (NeRFMLP & volume_render)

import numpy as np
from chokkhu.models.vision_3d import NeRFMLP, volume_render, generate_camera_rays

# Generate camera rays from camera pose matrix
c2w = np.eye(4, dtype=np.float32)
rays_o, rays_d = generate_camera_rays(
    height=64,
    width=64,
    focal_length=100.0,
    camera_pose=c2w,
)

Parameter Breakdown: NeRFMLP

Parameter Name	Data Type	Default Value	Description / Purpose
pos_dim	int	63	Sinusoidal frequency encoding dimension for 3D coordinate $(x,y,z)$.
dir_dim	int	27	Sinusoidal frequency encoding dimension for viewing direction $( heta, \phi)$.
hidden_dim	int	256	Hidden layer width of the radiance MLP backbone.
num_layers	int	8	Total number of fully-connected layers.
skip_layer	int	4	Layer index to concatenate residual input coordinate embeddings.
random_state	int	42	Random initialization seed.

---

11. Autonomous AI Agents & Execution Engines

11.1 ReAct Reasoning & Acting Agent (ReActAgent)

from chokkhu.models.nlp.agents import Tool, ToolRegistry, ReActAgent

# 1. Register custom Python execution tools
registry = ToolRegistry()

def add_numbers(a: float, b: float) -> float:
    """Add two floating point numbers."""
    return float(a + b)

registry.register(
    Tool(
        name="calculator",
        func=add_numbers,
        description="Adds two numbers a and b.",
    )
)

agent = ReActAgent(
    tools=registry,
    max_iterations=3,
)
result = agent.run("Calculate the sum of 45.5 and 54.5")
print(f"ReAct Agent Thought & Execution Result: {result}")

Parameter Breakdown: ReActAgent

Parameter Name	Data Type	Default Value	Description / Purpose
tools	ToolRegistry / List[Tool]	Required	Registry or list of callable tools with validation schemas.
max_steps	int	10	Maximum reasoning-action-observation cycles before termination.
model_callable	Callable / None	None	Optional LLM completion backend (defaults to rule-based parser).
verbose	bool	False	If True, logs step-by-step Thought, Action, Action Input, and Observation.

---

11.2 Tree of Thoughts Reasoning (TreeOfThoughts)

from chokkhu.models.nlp.agents import TreeOfThoughts

def thought_gen(state: str, n: int):
    return [f"{state} -> candidate_{i}" for i in range(n)]

def custom_evaluator(thought_state: str) -> float:
    return 1.0 if "candidate_0" in thought_state else 0.2

tot = TreeOfThoughts(
    thought_generator=thought_gen,
    state_evaluator=custom_evaluator,
    max_depth=2,
    branching_factor=2,
)
solution = tot.solve(initial_state="Given input data X: find optimal pipeline.")
print(f"Tree-of-Thoughts Optimal Node State: {solution}")

Parameter Breakdown: TreeOfThoughts

Parameter Name	Data Type	Default Value	Description / Purpose
max_depth	int	3	Maximum tree search depth / reasoning steps.
num_branches	int	3	Number of alternative candidate thoughts generated per node.
search_method	str	"dfs"	Tree exploration strategy: "dfs" (Depth-First) or "bfs" (Breadth-First).
evaluator	Callable / None	None	Value function scoring candidate thought nodes.
generator	Callable / None	None	Generator function expanding candidate subsequent thoughts.

---

12. Reinforcement Learning Universe

12.1 Proximal Policy Optimization with GAE (PPO)

import numpy as np
from chokkhu.models.rl import PPO

# State dim = 4 (e.g. CartPole), Action dim = 2
ppo = PPO(
    state_dim=4,
    action_dim=2,
    lr_actor=1e-3,
    lr_critic=2e-3,
    gamma=0.99,
)
state = np.random.randn(4)
action, log_prob, value = ppo.select_action(state)
print(f"PPO Selected Action: {action}, Log Prob: {log_prob:.4f}, State Value: {value:.4f}")

Parameter Breakdown: PPO

Parameter Name	Data Type	Default Value	Description / Purpose
state_dim	int	Required	Dimensionality of the environment observation state vector.
action_dim	int	Required	Number of discrete actions available to the agent.
actor_hidden	List[int]	[64, 64]	Hidden layer sizes for the actor policy network $\pi_ heta(a|s)$.
critic_hidden	List[int]	[64, 64]	Hidden layer sizes for the critic baseline network $V_\phi(s)$.
lr_actor	float	0.0003	Learning rate for actor parameter updates.
lr_critic	float	0.001	Learning rate for value function critic updates.
gamma	float	0.99	Discount factor $\gamma$ for future rewards.
gae_lambda	float	0.95	Factor $\lambda$ for Generalized Advantage Estimation trade-off.
clip_epsilon	float	0.2	PPO surrogate objective clipping range $(1 \pm \epsilon)$.
ppo_epochs	int	10	Optimization epochs per rollout batch.
batch_size	int	64	Minibatch size during policy updates.
entropy_coeff	float	0.01	Exploration entropy bonus coefficient.
random_state	int	42	Seed for weight initialization.

---

12.2 Soft Actor-Critic (SAC) & Multi-Armed Bandits (LinUCBBandit)

import numpy as np
from chokkhu.models.rl import SAC, LinUCBBandit

# 1. Soft Actor-Critic for continuous control
sac = SAC(
    state_dim=8,
    action_dim=2,
    hidden_dim=64,
    lr=1e-3,
    gamma=0.99,
)
continuous_action, _ = sac.sample_action(np.random.randn(8))
print(f"SAC Continuous Action: {continuous_action}")

# 2. Linear Upper Confidence Bound (LinUCB) Contextual Bandit
bandit = LinUCBBandit(n_arms=5, n_features=10, alpha=1.0)
context = np.random.randn(10)
chosen_arm = bandit.select_arm(context)
bandit.update(arm=chosen_arm, context=context, reward=1.0)
print(f"LinUCB Selected Arm: {chosen_arm}")

Parameter Breakdown: SAC

Parameter Name	Data Type	Default Value	Description / Purpose
state_dim	int	Required	Continuous state observation space dimensionality.
action_dim	int	Required	Continuous action space dimensionality.
actor_hidden	List[int]	[64, 64]	Layer dimensions for Gaussian policy network.
critic_hidden	List[int]	[64, 64]	Layer dimensions for twin Q-critic networks.
lr	float	0.0003	Learning rate for actor, critic, and entropy optimizer.
gamma	float	0.99	Reward discount factor.
tau	float	0.005	Soft target update polyak averaging coefficient $ au$.
alpha	float	0.2	Entropy temperature balancing exploration and maximum reward.
random_state	int	42	Random seed.

---

13. Advanced Graph Neural Networks Universe

13.1 Graphormer with Laplacian Positional Encoding (Graphormer)

import numpy as np
from chokkhu.models.gnn import Graphormer

# Node features: (num_nodes=6, in_features=16)
x = np.random.randn(6, 16).astype(np.float32)
adj = np.eye(6, dtype=np.float32)

# Graphormer with Spatial & Degree Centrality Encodings
graphormer = Graphormer(
    in_dim=16,
    hidden_dim=32,
    out_dim=2,
    num_layers=2,
    n_heads=2,
    pe_dim=4,
)
out = graphormer.forward(x, adj)
print(f"Graphormer Output Graph Embeddings Shape: {out.shape}")

Parameter Breakdown: Graphormer

Parameter Name	Data Type	Default Value	Description / Purpose
in_features	int	Required	Dimension of input node feature representations.
hidden_dim	int	64	Transformer hidden dimension.
out_features	int	16	Final node embedding output dimensionality.
num_heads	int	4	Number of multi-head graph attention heads.
num_layers	int	2	Number of stacked Graphormer transformer layers.
pe_dim	int	8	Dimension of input Laplacian positional encoding vectors.
dropout	float	0.1	Attention and feedforward dropout probability.
random_state	int	42	Random initialization seed.

---

13.2 Equivariant GNN (EGNN) & Relational GCN (RGCNClassifier)

import numpy as np
from chokkhu.models.gnn import EGNN, RGCNClassifier

# 1. E(n) Equivariant Graph Neural Network for 3D Molecular Coordinates
h = np.random.randn(5, 16).astype(np.float32)      # Invariant scalar features
# Coordinates: (num_nodes=5, 3D xyz)
x_coords = np.random.randn(5, 3).astype(np.float32)
edge_index = np.array([[0, 1, 2, 3], [1, 2, 3, 4]])

egnn = EGNN(
    in_dim=16,
    hidden_dim=32,
    out_dim=16,
    num_layers=2,
)
h_out, x_out = egnn.forward(h, x_coords, edge_index)
print(f"EGNN Scalar Feature Shape: {h_out.shape}, Equivariant 3D Coordinates Shape: {x_out.shape}")

Parameter Breakdown: EGNN

Parameter Name	Data Type	Default Value	Description / Purpose
in_node_features	int	Required	Dimension of scalar node invariant features.
hidden_dim	int	32	Hidden feature channel width.
out_node_features	int	16	Dimension of transformed invariant node features.
num_layers	int	2	Number of equivariant message-passing layers.
coord_dim	int	3	Geometric coordinate dimension (e.g., 3 for 3D point clouds).
random_state	int	42	Weight initialization seed.

---

14. Frontier LLMs, Linear Attention & Alignment

14.1 DeepSeek-V3 MoE with Multi-Head Latent Attention (DeepSeekV3)

import numpy as np
from chokkhu.models.nlp import DeepSeekV3

# Token sequence: (batch_size=2, seq_len=4)
input_tokens = np.array([
    [12, 45, 89, 230],
    [7, 102, 304, 511],
])

# Sovereign DeepSeek-V3 with MLA & DeepSeekMoE
deepseek = DeepSeekV3(
    vocab_size=1000,
    hidden_dim=64,
    num_layers=2,
    n_routed_experts=4,
    top_k=2,
)
logits = deepseek.forward(input_tokens)
print(f"DeepSeek-V3 Logits Shape: {logits.shape}")

Parameter Breakdown: DeepSeekV3

Parameter Name	Data Type	Default Value	Description / Purpose
vocab_size	int	1000	Size of token vocabulary dictionary.
hidden_dim	int	128	Model latent embedding dimension.
num_layers	int	2	Number of stacked DeepSeek transformer layers.
num_heads	int	4	Number of Multi-Head Latent Attention (MLA) heads.
q_lora_rank	int	32	Low-rank query compression dimension $d_c$.
kv_lora_rank	int	32	Low-rank key-value cache compression dimension $d_c'$.
n_routed_experts	int	4	Total number of routed Mixture-of-Experts.
n_shared_experts	int	1	Number of dedicated always-active shared experts.
top_k	int	2	Number of routed experts selected per token.
mtp_depth	int	1	Multi-Token Prediction (MTP) depth head count.
random_state	int	42	Seed for weight initialization.

---

14.2 Mamba Selective State Space Model (Mamba) & RWKV-v6 (RWKV6)

import numpy as np
from chokkhu.models.nlp import Mamba, RWKV6

input_ids = np.array([[15, 34, 128, 492, 10]])

# 1. Sovereign Mamba (S6 Selective State Space Model)
mamba = Mamba(
    vocab_size=1000,
    d_model=64,
    num_layers=2,
    d_state=16,
)
mamba_logits = mamba.forward(input_ids)
print(f"Mamba S6 Output Shape: {mamba_logits.shape}")

Parameter Breakdown: Mamba

Parameter Name	Data Type	Default Value	Description / Purpose
vocab_size	int	1000	Size of vocabulary.
d_model	int	128	Model state dimension.
d_state	int	16	Latent SSM state expansion dimension $N$.
d_conv	int	4	1D temporal convolution kernel width.
expand	int	2	Block expansion factor $E$ ($E \cdot d_{ ext{model}}$ inner width).
num_layers	int	2	Number of stacked Mamba SSM layers.
random_state	int	42	Random seed.

---

14.3 Direct Preference Optimization (DPOTrainer)

import numpy as np
from chokkhu.models.nlp import DPOTrainer

dpo = DPOTrainer(
    beta=0.1,
)
chosen_logps = np.array([-1.2, -0.8])
rejected_logps = np.array([-2.5, -2.1])
ref_chosen_logps = np.array([-1.5, -1.0])
ref_rejected_logps = np.array([-2.0, -1.8])

loss, chosen_rewards, rejected_rewards = dpo.compute_loss(
    chosen_logps, rejected_logps, ref_chosen_logps, ref_rejected_logps
)
print(f"DPO Alignment Loss: {loss:.4f}")

Parameter Breakdown: DPOTrainer

Parameter Name	Data Type	Default Value	Description / Purpose
model	Module	Required	Trainable actor LLM policy being aligned.
ref_model	Module	Required	Frozen reference baseline model $\pi_{ ext{ref}}$.
beta	float	0.1	Temperature regularization parameter $eta$ controlling KL divergence penalty.
lr	float	1e-4	Learning rate for policy optimizer.

---

15. Classical ML, 2D Vision, Audio & Generative DL

15.1 Generative AI: Variational Autoencoder (VAE) & Diffusion (DDPM)

import numpy as np
from chokkhu.models import VAE, DDPM

# 1. Variational Autoencoder (VAE)
images_flat = np.random.randn(20, 784).astype(np.float32)
vae = VAE(in_features=784, hidden_dim=64, latent_dim=16)
recon, mu, logvar = vae.forward(images_flat)
print(f"VAE Reconstructed Shape: {recon.shape}")

# 2. Denoising Diffusion Probabilistic Model (DDPM)
ddpm = DDPM(data_dim=32, timesteps=20, beta_start=1e-4, beta_end=0.02)
sampled_images = ddpm.sample(num_samples=4)
print(f"DDPM Generated Samples Shape: {sampled_images.shape}")

Parameter Breakdown: DDPM

Parameter Name	Data Type	Default Value	Description / Purpose
input_dim	int	Required	Flattened data dimension.
timesteps	int	1000	Number of forward noise diffusion steps $T$.
beta_start	float	1e-4	Initial variance schedule value $eta_1$.
beta_end	float	0.02	Final variance schedule value $eta_T$.

---

15.2 Speech & Audio Signal Processing (STFT, MFCC, Conformer)

import numpy as np
from chokkhu.models import stft, melspectrogram, mfcc, Conformer

# 1-second 16kHz raw audio waveform
waveform = np.sin(2 * np.pi * 440 * np.linspace(0, 1, 16000)).astype(np.float32)

# Extract Mel-Spectrogram and MFCCs
mel_spec = melspectrogram(waveform, sr=16000, n_fft=512, hop_length=160, n_mels=80)
mfcc_features = mfcc(waveform, sr=16000, n_mfcc=13)
print(f"Mel-Spectrogram Shape: {mel_spec.shape}, MFCC Shape: {mfcc_features.shape}")

# Speech Conformer (Macaron-style FFN + MHA + Depthwise Conv)
conformer = Conformer(
    in_features=80,
    num_classes=10,
    embed_dim=64,
    num_blocks=2,
    num_heads=2,
    ffn_dim=128,
)
spec_batch = np.random.randn(2, 20, 80).astype(np.float32)
conformer_out = conformer.forward(spec_batch)
print(f"Conformer Output Shape: {conformer_out.shape}")

Parameter Breakdown: Conformer

Parameter Name	Data Type	Default Value	Description / Purpose
input_dim	int	Required	Feature dimension per audio time frame (e.g. 80 Mel bins).
num_heads	int	4	Number of multi-head attention heads.
ffn_dim	int	128	Feedforward intermediate dimension (Macaron structure).
num_blocks	int	2	Number of stacked Conformer blocks.
conv_kernel_size	int	15	Depthwise 1D convolution kernel receptive field.

---

16. Sovereign Explainable AI (XAI)

16.1 Integrated Gradients & DeepLIFT (IntegratedGradients, DeepLIFT)

import numpy as np
import chokkhu as ck
from chokkhu.explainability import IntegratedGradients, DeepLIFT

# Create a trained neural model
X_train = np.random.randn(50, 10)
y_train = np.random.randint(0, 2, size=50)
model = ck.NeuralNetwork(layers=[32], activation="relu", epochs=5)
model.fit(X_train, y_train)

sample = np.random.randn(10)

# 1. Axiomatic feature attribution via Integrated Gradients
ig = IntegratedGradients(model=model, steps=20, baseline=None)
attributions = ig.attribute(sample)
print(f"Integrated Gradients Attributions Shape: {attributions.shape}")

# 2. DeepLIFT difference-from-reference conservation
dlift = DeepLIFT(model=model, baseline=np.zeros(10))
dlift_attributions = dlift.attribute(sample)
print(f"DeepLIFT Attributions: {dlift_attributions.flatten()[:5]}")

Parameter Breakdown: IntegratedGradients

Parameter Name	Data Type	Default Value	Description / Purpose
model	Any	Required	Differentiable neural network model instance.
steps	int	50	Number of Riemann summation interpolation steps along path.
baseline	np.ndarray / None	None	Reference neutral baseline vector (defaults to zero array).

---

---

17. Next-Gen Generative AI, LoRA, Conformal & Stacking (Milestone 14)

17.1 Continuous ODE Flow Matching & Rectified Flows (FlowMatching, RectifiedFlow)

import numpy as np
from chokkhu.models.generative import FlowMatching, RectifiedFlow

# Generate 2D training data
X_train = np.random.randn(200, 2) * 0.5 + 2.0

# 1. Continuous Velocity Field Flow Matching with Optimal Transport
flow = FlowMatching(input_dim=2, hidden_dims=[64, 64], lr=1e-3, sigma_min=1e-4)
for _ in range(5):
    loss = flow.train_step(X_train)

# Sample new points using 4th-order Runge-Kutta (RK4) integration
synthetic_samples = flow.sample(num_samples=100, steps=50, method="rk4")
print(f"Generated Flow Samples Shape: {synthetic_samples.shape}")

# 2. Rectified Flow interpolation with straight-line trajectories
rect_flow = RectifiedFlow(input_dim=2, hidden_dims=[32, 32], lr=1e-3)
for _ in range(5):
    loss = rect_flow.train_step(X_train)
rect_samples = rect_flow.sample(num_samples=50, steps=20, method="midpoint")
print(f"Rectified Flow Samples Shape: {rect_samples.shape}")

Parameter Breakdown: FlowMatching

Parameter Name	Data Type	Default Value	Description / Purpose
input_dim	int	Required	Dimensionality of the continuous input space $D$.
hidden_dim	int	64	Hidden layer width for the time-dependent velocity MLP network $v_\theta(x, t)$.
num_layers	int	3	Number of dense hidden layers in the velocity field approximator.
lr	float	1e-3	Gradient descent Adam/SGD learning rate during flow matching loss minimization.
sigma_min	float	1e-4	Minimum perturbation noise standard deviation for numerical stability.

---

17.2 Invertible Normalizing Flows (RealNVP, AffineCouplingLayer)

import numpy as np
from chokkhu.models.generative import RealNVP

# Continuous tabular / latent features
X = np.random.randn(100, 4)

# Multi-layer RealNVP with analytical Jacobian log-determinant
nvp = RealNVP(dim=4, num_layers=4, hidden_dim=32, lr=1e-3)
for _ in range(5):
    loss = nvp.compute_loss(X)

# Exact log-likelihood computation
log_probs = nvp.log_prob(X)
print(f"Mean Log-Likelihood: {np.mean(log_probs):.4f}")

# Invertible sampling from latent base standard Gaussian
generated_data = nvp.sample(num_samples=50)
print(f"RealNVP Sample Shape: {generated_data.shape}")

Parameter Breakdown: RealNVP

Parameter Name	Data Type	Default Value	Description / Purpose
input_dim	int	Required	Feature dimensionality $D$ (must be $\ge 2$ for split-coupling).
num_layers	int	4	Number of alternating affine coupling layers in the flow cascade.
hidden_dim	int	32	Hidden layer dimension for scale $s(x)$ and translation $t(x)$ sub-networks.
lr	float	1e-3	Learning rate for maximum likelihood estimation via backpropagation.

---

17.3 Parameter-Efficient Fine-Tuning (LoRALinear, LoRAAdapter)

import numpy as np
from chokkhu.models.generative import LoRALinear, LoRAAdapter

# 1. Direct Low-Rank Linear Adaptation: W_adapted = W0 + (alpha / r) * (B @ A)
linear = LoRALinear(in_features=128, out_features=64, r=4, lora_alpha=8.0)

x = np.random.randn(8, 128)
y = linear.forward(x)
print(f"Adapted Linear Output Shape: {y.shape}")

# Merge adapter weights directly into frozen base weight matrix for zero-latency inference
linear.merge()
y_merged = linear.forward(x)
np.testing.assert_allclose(y, y_merged, atol=1e-5)

# 2. Attach PEFT LoRA adapter tracking
param_summary = LoRAAdapter.count_parameters([linear])
print(f"LoRA Parameter Summary: {param_summary}")

Parameter Breakdown: LoRALinear

Parameter Name	Data Type	Default Value	Description / Purpose
in_features	int	Required	Input feature dimension $d_{\text{in}}$.
out_features	int	Required	Output feature dimension $d_{\text{out}}$.
rank	int	4	Low-rank bottleneck dimension $r \ll \min(d_{\text{in}}, d_{\text{out}})$.
alpha	float	1.0	Scaling hyperparameter $\alpha$ where $\Delta W = \frac{\alpha}{r} B A$.

---

17.4 Distribution-Free Time Series Conformal Prediction (ConformalPredictor)

import numpy as np
from chokkhu.models.timeseries import ConformalPredictor, conformal_interval
from chokkhu.models import LinearRegression

# Generate synthetic sequential / regression dataset
X = np.linspace(0, 10, 150).reshape(-1, 1)
y = 2.5 * X.flatten() + np.sin(X.flatten()) + np.random.randn(150) * 0.5

# Base forecaster
base_model = LinearRegression()
base_model.fit(X[:80], y[:80])

# Fit conformal prediction interval with exact 95% coverage guarantee
conformal = ConformalPredictor(base_estimator=base_model, alpha=0.05)
conformal.calibrate(X[80:110], y[80:110])

# Predict lower and upper confidence bounds
bounds = conformal.predict_interval(X[110:])
lower, upper = bounds[0], bounds[1]
coverage = np.mean((y[110:] >= lower) & (y[110:] <= upper))
print(f"Guaranteed Coverage: >= 95%, Empirical Empirical Coverage: {coverage * 100:.1f}%")

Parameter Breakdown: ConformalPredictor

Parameter Name	Data Type	Default Value	Description / Purpose
base_model	Any	Required	Any fitted regression or forecasting model providing .predict(X).
alpha	float	0.1	Miscoverage significance level $\alpha \in (0, 1)$, ensuring $1-\alpha$ coverage.
method	str	"split"	Conformal method: "split" or "cross_val".

---

17.5 Matrix Profile Time Series Motif & Discord Mining (MatrixProfile)

import numpy as np
from chokkhu.models.timeseries import MatrixProfile, find_motifs, find_discords

# Generate continuous time series with an embedded anomaly
np.random.seed(42)
t = np.linspace(0, 50, 500)
ts = np.sin(t) + np.random.randn(500) * 0.05
ts[250:270] += 4.0  # Anomaly burst

# Fast STOMP / STAMP sliding-window distance profile computation
mp = MatrixProfile(window_size=30)
profile, profile_idx = mp.compute(ts)

# Identify recurring patterns (motifs) and anomalous subsequences (discords)
motifs = find_motifs(ts, window_size=30, top_k=2)
discords = find_discords(ts, window_size=30, top_k=1)
print(f"Top Discord (Anomaly) Subsequence Info: {discords[0]}")

Parameter Breakdown: MatrixProfile

Parameter Name	Data Type	Default Value	Description / Purpose
window_size	int	Required	Subsequence window length $m$ for z-normalized Euclidean distance comparison.
algorithm	str	"stomp"	Matrix profile algorithm: "stomp" (fast $O(n^2)$) or "stamp".

---

17.6 Multi-Fidelity Bayesian Optimization & HyperBand (BOHB)

import numpy as np
from chokkhu.automl import BOHB

# Objective function simulating hyperparameter tuning across variable budget (e.g. epochs)
def eval_func(config, budget):
    lr = config["lr"]
    hidden = config["hidden"]
    loss = (np.log10(lr) + 3.0)**2 + (hidden - 64)**2 * 1e-4 + (1.0 / budget)
    return float(-loss)  # maximize objective

param_bounds = {
    "lr": (1e-4, 1e-1),
    "hidden": (16.0, 128.0),
}

bohb = BOHB(
    eval_func=eval_func,
    param_bounds=param_bounds,
    min_budget=1.0,
    max_budget=9.0,
    eta=3,
)
best_config, best_score = bohb.optimize()
print(f"Optimal Hyperparameters: {best_config}, Best Score: {best_score:.5f}")

Parameter Breakdown: BOHB

Parameter Name	Data Type	Default Value	Description / Purpose
objective_fn	Callable	Required	Objective function f(config, budget) -> float to minimize.
param_distributions	dict	Required	Dictionary defining hyperparameter search bounds and distributions.
min_budget	float	1.0	Minimum resource budget allocated per candidate configuration.
max_budget	float	9.0	Maximum resource budget allocated to top-performing configurations.
eta	int	3	Halving reduction factor (prunes $1 - 1/\eta$ candidates each round).
num_iterations	int	3	Number of HyperBand outer brackets executed.

---

17.7 SuperLearner Out-of-Fold Cross-Validated Stacking (SuperLearner)

import numpy as np
from chokkhu.automl import SuperLearner
from chokkhu.models import LogisticRegression, DecisionTree, RandomForest

# Multi-class classification dataset
X = np.random.randn(100, 4)
y = (X[:, 0] + X[:, 1] > 0).astype(int)

# Define diverse ensemble of base learners
base_learners = [
    LogisticRegression(),
    DecisionTree(max_depth=4),
    RandomForest(n_estimators=5, max_depth=4),
]

# Construct SuperLearner with out-of-fold CV meta-learning
super_learner = SuperLearner(
    estimators=base_learners,
    cv=3,
    task="classification",
    use_probabilities=True
)
super_learner.fit(X, y)

# Predict class probabilities and labels
y_pred_proba = super_learner.predict_proba(X[:5])
y_pred = super_learner.predict(X[:5])
print(f"Ensemble Predictions: {y_pred}")

Parameter Breakdown: SuperLearner

Parameter Name	Data Type	Default Value	Description / Purpose
base_models	list[tuple[str, Any]]	Required	List of named base estimators [(name, model), ...].
meta_model	str / Any	"ridge"	Meta-learner strategy ("ridge", "nnls", "logistic", or custom estimator).
cv	int	5	Number of out-of-fold cross-validation folds for meta-feature generation.
task	str	"classification"	Task type: "classification" or "regression".
use_probabilities	bool	True	Whether classification meta-features use class probabilities or hard labels.

---

18. Privacy, Quantum ML, Geo-Spatial, NAS & TDA (Milestone 15)

18.1 Differential Privacy & DP-SGD (LaplaceMechanism, GaussianMechanism, DP_SGD)

import numpy as np
from chokkhu.privacy import LaplaceMechanism, GaussianMechanism, DP_SGD

# 1. Laplace Mechanism for query perturbation
laplace = LaplaceMechanism(epsilon=0.5, sensitivity=1.0)
noisy_count = laplace.perturb(100.0)
print(f"Differentially Private Perturbed Count: {noisy_count:.2f}")

# 2. Gaussian Mechanism for (epsilon, delta)-DP
gaussian = GaussianMechanism(epsilon=1.0, delta=1e-5, sensitivity=1.0)
noisy_stat = gaussian.perturb(42.0)
print(f"Gaussian Perturbed Statistic: {noisy_stat:.2f}")

# 3. Differentially Private SGD Optimizer with Per-Sample Clipping
dp_optimizer = DP_SGD(lr=0.01, l2_norm_clip=1.0, noise_multiplier=0.8)
params = {"W": np.random.randn(10, 2), "b": np.zeros(2)}
per_sample_grads = [{"W": np.random.randn(10, 2), "b": np.random.randn(2)} for _ in range(8)]

updated_params = dp_optimizer.step(params, per_sample_grads)
privacy_spent = dp_optimizer.compute_privacy_spent(total_samples=1000, batch_size=8, target_delta=1e-5)
print(f"Privacy Spent: epsilon={privacy_spent['epsilon']:.4f}, delta={privacy_spent['delta']}")

Parameter Breakdown: DP_SGD

Parameter Name	Data Type	Default Value	Description / Purpose
lr	float	0.01	Optimization step learning rate.
l2_norm_clip	float	1.0	Maximum L2 norm bound $C$ for per-sample gradient clipping.
noise_multiplier	float	1.0	Calibrated noise ratio $\sigma = \text{noise_multiplier} \times C$.
random_state	int / None	None	Seed for reproducible pseudo-random noise generation.

---

18.2 Decentralized Federated Learning (FederatedServer, FederatedClient, FedAvg, FedProx)

import numpy as np
from chokkhu.privacy import FederatedServer, FederatedClient

# Simulate decentralized non-IID client partitions
client1_X, client1_y = np.random.randn(50, 4), np.random.randn(50, 1)
client2_X, client2_y = np.random.randn(60, 4), np.random.randn(60, 1)

# Register clients with FedProx proximal regularization (mu > 0)
c1 = FederatedClient(client_id="hospital_A", X=client1_X, y=client1_y, mu=0.01, lr=0.01)
c2 = FederatedClient(client_id="hospital_B", X=client2_X, y=client2_y, mu=0.01, lr=0.01)

# Orchestrate federated aggregation rounds
server = FederatedServer(
    initial_weights={"W": np.zeros((4, 1)), "b": np.zeros((1,))},
    clients=[c1, c2],
    strategy="fedprox"
)

for r in range(5):
    round_summary = server.train_round(fraction_fit=1.0, local_epochs=3, batch_size=16)
    print(f"Round {r+1} Complete - Mean Client Loss: {round_summary['mean_client_loss']:.4f}")

global_predictions = server.predict(np.random.randn(5, 4))
print(f"Global Model Predictions Shape: {global_predictions.shape}")

Parameter Breakdown: FederatedServer

Parameter Name	Data Type	Default Value	Description / Purpose
initial_weights	dict	Required	Initial global model parameters {'W': ..., 'b': ...}.
clients	list	None	Registered FederatedClient instances.
strategy	str	"fedavg"	Aggregation strategy ("fedavg" or "fedprox").

---

18.3 Quantum Machine Learning Simulator & Variational Quantum Classifier (QuantumCircuit, VQC, QuantumKernel)

import numpy as np
from chokkhu.quantum import QuantumCircuit, VariationalQuantumClassifier, QuantumKernel

# 1. State Vector Quantum Circuit Simulation & Bell State Generation
qc = QuantumCircuit(n_qubits=2)
qc.h(0).cnot(0, 1)  # Create maximally entangled Bell state (|00> + |11>) / sqrt(2)
print(f"Computational Basis Probabilities: {qc.probabilities()}")
print(f"Pauli-Z Expectation <Z_0>: {qc.expectation_z(0):.4f}")

# 2. Variational Quantum Classifier (QNN) with Exact Parameter Shift Rule Gradients
X_quantum = np.random.uniform(-np.pi, np.pi, size=(30, 2))
y_quantum = (X_quantum[:, 0] + X_quantum[:, 1] > 0).astype(int)

vqc = VariationalQuantumClassifier(n_qubits=2, n_layers=2, lr=0.1)
vqc.fit(X_quantum, y_quantum, epochs=10, batch_size=8)
q_preds = vqc.predict(X_quantum[:5])
print(f"VQC Predictions: {q_preds}")

# 3. Quantum Kernel Matrix Estimation
qk = QuantumKernel(n_qubits=2)
gram_matrix = qk.compute_matrix(X_quantum[:10])
print(f"Quantum Gram Matrix (Fidelity) Shape: {gram_matrix.shape}")

Parameter Breakdown: VariationalQuantumClassifier

Parameter Name	Data Type	Default Value	Description / Purpose
n_qubits	int	2	Number of simulated qubits in quantum register.
n_layers	int	2	Number of parameterized variational rotation and entangling ansatz layers.
lr	float	0.05	Learning rate for parameter shift rule gradient descent.

---

18.4 Geo-Spatial Statistics, Spatial Autoregression & GWR (SpatialWeights, morans_i, SAR, GWR)

import numpy as np
from chokkhu.geospatial import SpatialWeights, morans_i, SpatialAutoregression, GeographicallyWeightedRegression

# 2D geographic coordinates [lat, lon] and spatial attributes
coords = np.random.uniform(20.0, 26.0, size=(50, 2))
values = coords[:, 0] * 3.5 + np.random.randn(50) * 0.2

# 1. Spatial Weight Matrix and Global Moran's I Autocorrelation
sw = SpatialWeights(coords, method="knn", k=5, row_standardize=True)
moran = morans_i(values, sw, permutations=99)
print(f"Global Moran's I: {moran['I']:.4f}, p-value: {moran['p_value']:.4f}")

# 2. Spatial Autoregression (Spatial Lag Model: y = rho * W * y + X * beta)
X_geo = np.random.randn(50, 3)
sar = SpatialAutoregression(model_type="lag")
sar.fit(X_geo, values, sw)
print(f"Estimated Spatial Autoregressive Rho: {sar.rho:.4f}")

# 3. Geographically Weighted Regression (Localized Spatial Beta Estimation)
gwr = GeographicallyWeightedRegression(bandwidth=2.0, kernel="gaussian")
gwr.fit(coords, X_geo, values)
gwr_preds = gwr.predict(coords[:5], X_geo[:5])
print(f"GWR Predictions: {gwr_preds}")

Parameter Breakdown: SpatialWeights

Parameter Name	Data Type	Default Value	Description / Purpose
coords	np.ndarray	Required	Coordinate matrix $(N \times 2)$ containing spatial coordinates.
method	str	"knn"	Kernel structure: "knn", "inverse_distance", "gaussian", or "threshold".
k	int	5	Number of nearest neighbors for "knn" method.
bandwidth	float	1.0	Distance decay parameter for "gaussian" or distance threshold.
row_standardize	bool	True	Row-standardization ($\sum_j W_{ij} = 1.0$).

---

18.5 Geostatistical Variogram Modeling & Spatial Kriging (OrdinaryKriging)

import numpy as np
from chokkhu.geospatial import OrdinaryKriging

# Spatial point observations
obs_coords = np.random.uniform(0.0, 10.0, size=(30, 2))
z_values = np.sin(obs_coords[:, 0]) + np.cos(obs_coords[:, 1])

# Ordinary Kriging with Spherical Variogram Model
kriging = OrdinaryKriging(variogram_model="spherical", nugget=0.02, sill=1.0, range_val=4.0)
kriging.fit(obs_coords, z_values)

# Interpolate over target spatial grid
target_coords = np.array([[2.5, 3.5], [7.0, 8.0]])
z_pred, error_variance = kriging.predict(target_coords)
print(f"Kriging Interpolated Values: {z_pred}, Estimation Variances: {error_variance}")

Parameter Breakdown: OrdinaryKriging

Parameter Name	Data Type	Default Value	Description / Purpose
variogram_model	str	"spherical"	Theoretical semivariogram model: "spherical", "exponential", or "gaussian".
nugget	float	0.0	Micro-scale variance / measurement error $c_0$.
sill	float	1.0	Total asymptotic semivariance $c_0 + c$.
range_val	float	1.0	Spatial correlation range threshold $a$.

---

18.6 Differentiable Neural Architecture Search (DARTS)

import numpy as np
from chokkhu.automl import DARTS

# Tabular classification dataset
X_nas = np.random.randn(100, 8)
y_nas = (X_nas[:, 0] + X_nas[:, 1] > 0).astype(int)

# Continuous relaxation of candidate operations (Identity, Linear, ReLU, GELU, Residual)
darts = DARTS(
    input_dim=8,
    num_classes=2,
    num_intermediate_nodes=3,
    hidden_dim=32,
    lr_weights=0.01,
    lr_arch=0.005
)
darts.fit(X_nas, y_nas, epochs=10, batch_size=16)

# Extract discovered discrete optimal architecture
genotype = darts.genotype()
print(f"Discovered Optimal Neural Architecture Genotype: {genotype}")

Parameter Breakdown: DARTS

Parameter Name	Data Type	Default Value	Description / Purpose
input_dim	int	Required	Feature input dimension $D$.
num_classes	int	2	Number of output target classes.
num_intermediate_nodes	int	3	Number of mixed-op candidate layers in supernet search space.
hidden_dim	int	32	Layer width for intermediate candidate operations.
lr_weights	float	0.01	Learning rate for candidate operation parameters $w$.
lr_arch	float	0.005	Learning rate for architecture parameters $\alpha$.

---

18.7 Topological Data Analysis & Persistent Homology (VietorisRipsComplex, PersistenceLandscape)

import numpy as np
from chokkhu.tda import VietorisRipsComplex, PersistenceDiagram, PersistenceLandscape, bottleneck_distance

# Point cloud with topological hole (circle)
angles = np.linspace(0, 2 * np.pi, 24, endpoint=False)
point_cloud = np.column_stack([np.cos(angles), np.sin(angles)]) + np.random.randn(24, 2) * 0.05

# 1. Build Vietoris-Rips Simplicial Filtration
rips = VietorisRipsComplex(max_edge_length=2.5, max_dimension=1)
diagrams = rips.fit_transform(point_cloud)

# 2. Extract H0 (connected components) and H1 (loops) Persistence
h0_diag = PersistenceDiagram(diagrams[0], dimension=0)
h1_diag = PersistenceDiagram(diagrams[1], dimension=1)
print(f"H0 Total Persistence: {h0_diag.total_persistence():.4f}, Persistent Entropy: {h0_diag.persistent_entropy():.4f}")

# 3. Vectorize into Continuous Persistence Landscapes for ML Classifiers
landscape = PersistenceLandscape(num_landscapes=3, resolution=40)
topo_features = landscape.transform(diagrams[0])
print(f"Extracted Topological Feature Vector Shape: {topo_features.shape}")

Parameter Breakdown: VietorisRipsComplex

Parameter Name	Data Type	Default Value	Description / Purpose
max_edge_length	float	inf	Maximum filtration scale $\epsilon$ for simplex construction.
max_dimension	int	1	Maximum homology dimension ($0$ for $H_0$, $1$ for $H_0$ and $H_1$).

---

19. Advanced Frontier Architectures (Milestone 16)

19.1 Hybrid Lexical-Dense Retrieval & Reciprocal Rank Fusion (BM25Plus, OkapiBM25, HybridReranker)

import numpy as np
from chokkhu.models.retrieval import BM25Plus, OkapiBM25, HybridReranker

corpus = [
    "quantum computing and machine learning algorithms",
    "deep learning neural networks for computer vision",
    "natural language processing with transformers and attention",
    "quantum physics and quantum entanglement theory",
]

# 1. Okapi BM25 & BM25+ Sparse Retrieval
bm25 = BM25Plus(k1=1.5, b=0.75, delta=1.0)
bm25.fit(corpus)
sparse_scores = bm25.get_scores("quantum computing")
top_scores, top_indices = bm25.search("quantum computing", top_k=2)
print(f"BM25+ Top Results: Indices={top_indices}, Scores={top_scores}")

# 2. Hybrid Reranking (RRF & Convex Score Blending)
dense_scores = np.array([0.92, 0.45, 0.68, 0.12])
reranker = HybridReranker(rrf_k=60, alpha=0.6)
blended_scores = reranker.blend_scores(dense_scores, sparse_scores, alpha=0.6)
fused_rankings = reranker.reciprocal_rank_fusion([[0, 2, 1, 3], top_indices], top_k=2)
print(f"Fused Hybrid Rankings (RRF): {fused_rankings}")

Parameter Breakdown: BM25Plus & OkapiBM25

Parameter Name	Data Type	Default Value	Description / Purpose
k1	float	1.5	Term frequency saturation non-linear scaling parameter.
b	float	0.75	Document length normalization penalty parameter ($0 \le b \le 1$).
delta	float	1.0	Lower-bound tuning parameter in BM25+ to prevent excessive length penalties ($0.0$ for standard Okapi BM25).

Parameter Breakdown: HybridReranker

Parameter Name	Data Type	Default Value	Description / Purpose
rrf_k	int	60	Constant dampening factor in Reciprocal Rank Fusion ($1 / (k + \text{rank})$).
alpha	float	0.5	Convex combination weight for dense scores vs sparse lexical scores ($\alpha \cdot S_{\text{dense}} + (1 - \alpha) \cdot S_{\text{sparse}}$).

---

19.2 Mechanistic Interpretability & Actionable Counterfactuals (AttentionRollout, DirectLogitAttribution, WachterCounterfactualExplainer, TCAV)

import numpy as np
from chokkhu.explainability import (
    AttentionRollout,
    DirectLogitAttribution,
    WachterCounterfactualExplainer,
    TCAV,
)

# 1. Mechanistic Attention Rollout across transformer layers
rollout = AttentionRollout(discard_ratio=0.1, head_reduction="mean")
layers_attn = [np.random.rand(4, 6, 6) for _ in range(3)]
layers_attn = [a / a.sum(axis=-1, keepdims=True) for a in layers_attn]
rollout_matrix = rollout.compute(layers_attn)
print(f"Attention Rollout Matrix Shape: {rollout_matrix.shape}")

# 2. Direct Logit Attribution (DLA) on residual streams
unembedding = np.random.randn(16, 50)
dla = DirectLogitAttribution(unembedding_matrix=unembedding)
residual_hidden_states = np.random.randn(4, 16)
layer_attributions = dla.attribute(residual_hidden_states, target_token_id=5)
print(f"Layer-wise Logit Attributions for Token #5: {layer_attributions}")

# 3. Wachter Optimization-Based Counterfactual Explainer
def predict_fn(x: np.ndarray) -> np.ndarray:
    return 1.0 / (1.0 + np.exp(-np.dot(x, np.array([1.5, 1.2]))))

explainer = WachterCounterfactualExplainer(predict_fn=predict_fn, lr=0.1, max_iter=100)
cf_summary = explainer.explain(np.array([-1.0, -0.5]), target_prediction=0.8)
print(f"Discovered Counterfactual Input: {cf_summary['counterfactual']}, Distance={cf_summary['l2_distance']:.4f}")

# 4. Testing with Concept Activation Vectors (TCAV)
tcav = TCAV(seed=42)
cav = tcav.compute_cav(np.random.randn(20, 16) + 2.0, np.random.randn(20, 16) - 2.0)
tcav_result = tcav.compute_tcav_score(lambda a: np.dot(a, cav)[:, None], np.random.randn(10, 16))
print(f"Quantitative Concept Sensitivity (TCAV Score): {tcav_result['tcav_score']:.4f}")

Parameter Breakdown: AttentionRollout & DirectLogitAttribution

Parameter Name	Data Type	Default Value	Description / Purpose
discard_ratio	float	0.0	Fraction of lowest attention weights to zero out per layer ($0.0 \le r < 1.0$).
head_reduction	str	"mean"	Aggregation strategy across attention heads ("mean", "max", or "min").
unembedding_matrix	np.ndarray	Required	Final output projection matrix $W_U$ mapping hidden state $D \to V$.

Parameter Breakdown: WachterCounterfactualExplainer & TCAV

Parameter Name	Data Type	Default Value	Description / Purpose
predict_fn	Callable	Optional	Black-box predictor or scoring function $f(x)$.
lambda_1	float	0.01	$L_1$ sparsity regularizer penalty weight.
lambda_2	float	0.05	$L_2$ proximity regularizer penalty weight.
lr	float	0.05	Learning rate for Adam perturbation optimizer.
max_iter	int	300	Maximum inner gradient descent iterations.
tolerance	float	0.05	Target outcome convergence tolerance threshold.
seed	int	42	Random seed for concept activation separation.

---

19.3 Dense & Promptable Computer Vision (DeepLabV3Plus, SegmentAnythingModel)

import numpy as np
from chokkhu.models.vision.architectures import DeepLabV3Plus, SegmentAnythingModel

# 1. DeepLabV3+ with Atrous Spatial Pyramid Pooling (ASPP)
deeplab = DeepLabV3Plus(
    in_channels=3,
    num_classes=3,
    backbone_channels=8,
    aspp_channels=8,
    seed=42,
)
img_batch = np.random.randn(2, 3, 16, 16).astype(np.float32)
logits = deeplab.forward(img_batch)
masks = deeplab.predict(img_batch)
print(f"DeepLabV3+ Segmentation Logits Shape: {logits.shape}, Predicted Mask Map: {masks.shape}")

# 2. Segment Anything Model (SAM) Promptable Mask Decoder
sam = SegmentAnythingModel(embed_dim=16, num_heads=2, num_mask_tokens=3, seed=42)
img_embeddings = np.random.randn(1, 16, 8, 8).astype(np.float32)
point_coords = np.array([[[2.0, 3.0], [5.0, 6.0]]], dtype=np.float32)
point_labels = np.array([[1, 0]], dtype=np.int32)
pred_masks, iou_scores = sam.forward(img_embeddings, points=point_coords, labels=point_labels)
print(f"SAM Decoded Masks Shape: {pred_masks.shape}, Predicted IoU Quality: {iou_scores}")

Parameter Breakdown: DeepLabV3Plus

Parameter Name	Data Type	Default Value	Description / Purpose
in_channels	int	3	Number of image input color channels (e.g. RGB=3).
num_classes	int	2	Number of target semantic segmentation classes.
backbone_channels	int	32	Number of feature filters in convolutional stem.
aspp_channels	int	32	Channel width for dilated ASPP pyramid branches.
seed	int	42	Random seed for weight initialization.

Parameter Breakdown: SegmentAnythingModel

Parameter Name	Data Type	Default Value	Description / Purpose
embed_dim	int	64	Transformer embedding dimensionality $D$.
num_heads	int	4	Number of multi-head two-way attention heads.
num_mask_tokens	int	3	Number of multi-scale promptable mask hypotheses generated.
seed	int	42	Random seed for weight initialization.

---

19.4 Contrastive Language-Audio Pretraining (CLAP)

import numpy as np
from chokkhu.models.multimodal import CLAP

clap = CLAP(embed_dim=16, n_mels=16, vocab_size=50, init_temperature=0.07, seed=42)

# Paired audio spectrograms (B, n_mels, time_steps) and text token sequences (B, seq_len)
audio_spectrograms = np.random.randn(2, 16, 32).astype(np.float32)
text_tokens = np.random.randint(0, 50, size=(2, 8))

# Joint multimodal forward pass & contrastive loss
audio_emb, text_emb, loss = clap.forward(audio_spectrograms, text_tokens)
similarity_matrix = clap.predict_similarity(audio_spectrograms, text_tokens)
print(f"CLAP Multimodal Loss: {loss:.4f}, Zero-Shot Audio-Text Cosine Similarity:\n{similarity_matrix}")

Parameter Breakdown: CLAP

Parameter Name	Data Type	Default Value	Description / Purpose
embed_dim	int	64	Joint multimodal shared latent space dimension.
n_mels	int	64	Number of mel-spectrogram frequency filterbank bins.
vocab_size	int	500	Text tokenizer vocabulary size.
init_temperature	float	0.07	Learnable InfoNCE contrastive temperature scale $\tau$.
seed	int	42	Random seed for weight initialization.

---

19.5 Speech Recognition & Autoregressive Decoding (Whisper)

import numpy as np
from chokkhu.models.audio.architectures import Whisper

whisper = Whisper(
    n_mels=16,
    vocab_size=25,
    d_model=16,
    n_heads=2,
    n_encoder_layers=1,
    n_decoder_layers=1,
    seed=42,
)

# Audio mel-spectrogram and target token sequences
mel_input = np.random.randn(2, 16, 32).astype(np.float32)
target_tokens = np.array([[1, 5, 2], [1, 7, 2]], dtype=np.int32)

# 1. Forward teacher-forced cross-entropy logits
logits = whisper.forward(mel_input, target_tokens)
print(f"Whisper Logits Shape: {logits.shape}")

# 2. Autoregressive Speech Transcription Generation
transcribed_tokens = whisper.generate(mel_input, max_len=4, prompt_tokens=[1])
print(f"Generated Transcription Sequence: {transcribed_tokens}")

Parameter Breakdown: Whisper

Parameter Name	Data Type	Default Value	Description / Purpose
n_mels	int	80	Number of mel-frequency channels in input audio spectrogram.
vocab_size	int	500	Vocabulary size for speech tokens and special symbols.
d_model	int	64	Transformer hidden dimension $D$.
n_heads	int	4	Number of parallel attention heads.
n_encoder_layers	int	2	Number of audio transformer encoder blocks.
n_decoder_layers	int	2	Number of causal cross-attention decoder blocks.
seed	int	42	Random seed for weight initialization.

---

19.6 Latent Diffusion & Cross-Attention Conditioning (LatentDiffusionModel)

import numpy as np
from chokkhu.models.generative import LatentDiffusionModel

ldm = LatentDiffusionModel(
    latent_dim=8,
    context_dim=8,
    num_timesteps=50,
    beta_start=0.0001,
    beta_end=0.02,
    num_heads=2,
    seed=42,
)

# Clean latent state z_0 and conditioning context vector
z_0 = np.random.randn(2, 8).astype(np.float32)
timesteps = np.array([5, 20], dtype=np.int32)

# 1. Forward diffusion: add calibrated Gaussian noise at timestep t
z_t, injected_noise = ldm.q_sample(z_0, timesteps)

# 2. Reverse noise prediction conditioned on cross-attention context
predicted_noise = ldm.predict_noise(z_t, timesteps, context=z_0)

# 3. Deterministic DDIM fast generation sampling from pure Gaussian noise
generated_latents = ldm.sample(shape=(2, 8), context=z_0, n_steps=5)
print(f"Generated Latent Representations Shape: {generated_latents.shape}")

Parameter Breakdown: LatentDiffusionModel

Parameter Name	Data Type	Default Value	Description / Purpose
latent_dim	int	32	Dimension of compressed VAE latent state $z$.
context_dim	int	64	Dimension of conditioning prompt/text representations.
num_timesteps	int	1000	Total number of discrete forward diffusion noise steps $T$.
beta_start	float	1e-4	Initial variance schedule value $\beta_1$.
beta_end	float	0.02	Terminal variance schedule value $\beta_T$.
num_heads	int	4	Number of heads in cross-attention conditioning block.
seed	int	42	Random seed for weight initialization.

---

20. Sovereign Production Generative AI, Audio Codecs, Speculative LLMs & Genetic AutoML

20.1 Spatial Conditional Control with Zero-Convolution (ControlNet)

import numpy as np
from chokkhu import ControlNet

# Initialize ControlNet adapter with zero-initialized convolution layers
cnet = ControlNet(
    in_channels=4,
    cond_channels=3,
    base_channels=8,
    num_stages=2,
    seed=42,
)

# Latent feature representation and spatial conditioning guide (e.g., edge map, depth)
latents = np.random.randn(1, 8, 16, 16).astype(np.float32)
condition = np.random.randn(1, 3, 16, 16).astype(np.float32)

# Forward pass injects condition while preserving base model representations
out_latents, stage_residuals = cnet.forward(latents, condition)
print(f"Modulated Latents Shape: {out_latents.shape}, Intermediate Stages: {len(stage_residuals)}")

Parameter Breakdown: ControlNet

Parameter Name	Data Type	Default Value	Description / Purpose
in_channels	int	4	Number of channels in base input latent representation.
cond_channels	int	3	Number of channels in spatial condition image/map.
base_channels	int	32	Internal channel capacity of feature backbone and ZeroConv layers.
num_stages	int	3	Number of cascaded Zero-Convolution conditioning stages.
seed	int	42	Random seed for weight initialization.

---

20.2 Unpaired Image-to-Image Domain Translation (CycleGAN)

import numpy as np
from chokkhu import CycleGAN

# Initialize bidirectional domain translation network (A <-> B)
cyclegan = CycleGAN(
    channels_a=3,
    channels_b=3,
    base_channels=8,
    lambda_cycle=10.0,
    lambda_identity=5.0,
    seed=42,
)

# Domain A and Domain B unaligned sample batches
real_a = np.random.randn(1, 3, 16, 16).astype(np.float32)
real_b = np.random.randn(1, 3, 16, 16).astype(np.float32)

# Bidirectional translations
fake_b = cyclegan.translate_a2b(real_a)
fake_a = cyclegan.translate_b2a(real_b)

# Full training pass computing cycle consistency and adversarial losses
metrics = cyclegan.forward(real_a, real_b)
print(f"Cycle Loss: {metrics['loss_cycle']:.4f}, Total Gen Loss: {metrics['total_loss']:.4f}")

Parameter Breakdown: CycleGAN

Parameter Name	Data Type	Default Value	Description / Purpose
channels_a	int	3	Number of image channels in domain A.
channels_b	int	3	Number of image channels in domain B.
base_channels	int	16	Base filter depth in ResNet generators and PatchGAN discriminators.
lambda_cycle	float	10.0	Weight scalar for L1 cycle consistency reconstruction loss.
lambda_identity	float	5.0	Weight scalar for identity preservation loss.
seed	int	42	Random seed for reproducibility.

---

20.3 Neural Audio Compression & Discrete Codecs (ResidualVectorQuantizer)

import numpy as np
from chokkhu import ResidualVectorQuantizer

# 4-stage residual vector quantizer (SoundStream / EnCodec style)
rvq = ResidualVectorQuantizer(
    num_quantizers=4,
    codebook_size=32,
    embed_dim=8,
    commitment_weight=0.25,
    seed=42,
)

# Continuous audio frame representations (Batch, Time, Dim)
audio_embeddings = np.random.randn(2, 10, 8).astype(np.float32)

# 1. Multi-stage hierarchical quantization
quantized, codes, commitment_loss = rvq.forward(audio_embeddings)
print(f"Discrete Code Tokens Shape: {codes.shape}, Loss: {commitment_loss:.4f}")

# 2. Reconstruct continuous latents from discrete indices
reconstructed = rvq.decode_codes(codes)
print(f"Reconstructed Latents Match Original Shape: {reconstructed.shape == audio_embeddings.shape}")

Parameter Breakdown: ResidualVectorQuantizer

Parameter Name	Data Type	Default Value	Description / Purpose
num_quantizers	int	4	Number of cascaded residual quantization stages ($N_q$).
codebook_size	int	1024	Number of discrete code vectors per stage codebook ($K$).
embed_dim	int	64	Embedding dimensionality of continuous latent vectors ($D$).
commitment_weight	float	0.25	Scalar multiplier for codebook commitment loss.
seed	int	42	Random seed for codebook initialization.

---

20.4 Multi-Cue Voice Activity Detection (VoiceActivityDetector)

import numpy as np
from chokkhu import VoiceActivityDetector

# Multi-cue VAD with Short-Time Energy, ZCR, and Spectral Flux
vad = VoiceActivityDetector(
    sample_rate=16000,
    frame_length_ms=25.0,
    hop_length_ms=10.0,
    energy_threshold=0.02,
)

# Generate synthetic audio with speech tone and silence
t = np.linspace(0, 0.5, 8000)
tone = np.sin(2 * np.pi * 440 * t) * 0.5
silence = np.zeros(4000)
audio = np.concatenate([silence, tone, silence])

# Run VAD speech segmentation
vad_result = vad.detect(audio)
print(f"Speech Ratio: {vad_result['speech_ratio']:.2%}")
print(f"Detected Speech Intervals (sec): {vad_result['segments']}")

Parameter Breakdown: VoiceActivityDetector

Parameter Name	Data Type	Default Value	Description / Purpose
sample_rate	int	16000	Sampling frequency of the input audio signal in Hz.
frame_length_ms	float	25.0	Duration of each analysis frame in milliseconds.
hop_length_ms	float	10.0	Hop step between consecutive frames in milliseconds.
energy_threshold	float	0.02	Sensitivity threshold for speech classification.
zcr_threshold	float	0.15	Zero-Crossing Rate threshold for unvoiced speech discrimination.
min_speech_duration_ms	float	100.0	Minimum duration required to trigger active speech state.
min_silence_duration_ms	float	200.0	Minimum silence duration required to terminate active speech state.

---

20.5 High-Speed Speculative & Grammar-Constrained LLM Decoding (SpeculativeDecoder, StructuredJSONDecoder)

import numpy as np
from chokkhu import SpeculativeDecoder, StructuredJSONDecoder
from chokkhu.core.tensor import Tensor

# Define mock target and draft model logit functions
def mock_target_model(tensor_in):
    logits = np.zeros((1, 1, 30), dtype=np.float32)
    logits[0, 0, 5] = 5.0
    return Tensor(logits, requires_grad=False)

def mock_draft_model(tensor_in):
    return Tensor(np.ones((1, 1, 30), dtype=np.float32), requires_grad=False)

class SimpleTokenizer:
    def encode(self, text, bos=True): return [1, 2, 3]
    def decode(self, tokens, skip_special_tokens=True): return '{"status": "success", "result": "generated text"}'

tok = SimpleTokenizer()

# 1. Speculative Decoding for 2x-3x speedup via draft-verification sampling
spec_dec = SpeculativeDecoder(
    target_model=mock_target_model,
    draft_model=mock_draft_model,
    tokenizer=tok,
    gamma=3,
    temperature=1.0,
)
spec_result = spec_dec.generate("Analyze this transaction:", max_new_tokens=4, eos_token_id=None)
print(f"Speculative Output: {spec_result['text']}, Acceptance Rate: {spec_result['acceptance_rate']:.2%}")

# 2. Grammar-Constrained Structured JSON Generation
json_dec = StructuredJSONDecoder(model=mock_target_model, tokenizer=tok)
json_result = json_dec.generate(
    "Extract user data",
    schema={"properties": {"name": {"type": "string"}, "age": {"type": "number"}}},
    max_new_tokens=5,
)
print(f"Parsed JSON Valid: {json_result['parsed_json']}")

Parameter Breakdown: SpeculativeDecoder

Parameter Name	Data Type	Default Value	Description / Purpose
target_model	Any	Required	Main target high-capacity language model.
draft_model	Any	Required	Lightweight draft model for fast token generation.
tokenizer	Any	Required	Tokenizer instance providing encode and decode.
gamma	int	4	Number of speculative draft tokens generated per iteration.
temperature	float	1.0	Sampling temperature parameter.

---

20.6 Non-Linear Manifold Learning & Dimension Reduction (UMAP)

import numpy as np
from chokkhu import UMAP

# Create high-dimensional clustered dataset
np.random.seed(42)
c1 = np.random.randn(25, 10) + 3.0
c2 = np.random.randn(25, 10) - 3.0
X = np.vstack([c1, c2])

# Initialize pure NumPy UMAP reducer
umap = UMAP(
    n_components=2,
    n_neighbors=5,
    min_dist=0.1,
    n_epochs=50,
    random_state=42,
)

# Project to 2D manifold preserving both local and global topology
embedding = umap.fit_transform(X)
print(f"UMAP 2D Projection Shape: {embedding.shape}")

Parameter Breakdown: UMAP

Parameter Name	Data Type	Default Value	Description / Purpose
n_components	int	2	Dimensionality of embedded target manifold.
n_neighbors	int	15	Size of local fuzzy simplicial metric neighborhood.
min_dist	float	0.1	Minimum distance between embedded points in low-dimensional space.
n_epochs	int	200	Number of SGD cross-entropy layout optimization iterations.
learning_rate	float	1.0	Initial learning rate for manifold layout SGD.
random_state	`int	None`	None

---

20.7 Evolutionary Pipeline Search & Dynamic Classifier Selection (GeneticPipelineSearch, DynamicEnsembleSelection)

import numpy as np
from chokkhu import GeneticPipelineSearch, DynamicEnsembleSelection

# Synthetic classification dataset
np.random.seed(42)
X = np.random.randn(40, 4)
y = (X[:, 0] + X[:, 1] > 0).astype(int)

# 1. Genetic TPOT-Style Pipeline Evolution
search = GeneticPipelineSearch(
    population_size=6,
    generations=3,
    mutation_rate=0.2,
    crossover_rate=0.5,
    cv_folds=2,
    seed=42,
)
search.fit(X, y)
print(f"Best Evolved Pipeline: {search.best_individual.scaler_type} -> PCA({search.best_individual.pca_components}) -> {search.best_individual.model_type}")

# 2. Dynamic Ensemble Selection (KNORA-Eliminate)
pool = [
    lambda data: (data[:, 0] > 0).astype(int),
    lambda data: (data[:, 1] > 0).astype(int),
]
des = DynamicEnsembleSelection(pool_classifiers=pool, k_neighbors=3, method="knora_e")
des.fit(X, y)
des_predictions = des.predict(X)
print(f"Dynamic Selection Predictions Shape: {des_predictions.shape}")

Parameter Breakdown: GeneticPipelineSearch

Parameter Name	Data Type	Default Value	Description / Purpose
population_size	int	12	Number of candidate pipeline chromosomes per generation.
generations	int	4	Number of evolutionary iterations.
mutation_rate	float	0.3	Probability of mutating individual pipeline hyperparameters.
crossover_rate	float	0.5	Probability of chromosome crossover reproduction.
tournament_size	int	3	Number of individuals competing in Pareto tournament selection.
cv_folds	int	3	Cross-validation folds used for individual fitness scoring.
seed	int	42	Random seed for evolution reproducibility.

---

20.8 Advanced Density & Robust Outlier Detection (LocalOutlierFactor, EllipticEnvelope)

import numpy as np
from chokkhu import LocalOutlierFactor, EllipticEnvelope

# Synthetic data with anomalies
np.random.seed(42)
inliers = np.random.randn(40, 2)
outliers = np.random.uniform(low=-8, high=8, size=(4, 2))
X = np.vstack([inliers, outliers])

# 1. Local Outlier Factor (LOF)
lof = LocalOutlierFactor(n_neighbors=10, contamination=0.1, novelty=True)
lof_labels = lof.fit_predict(X)
print(f"LOF Detected Outliers Count: {np.sum(lof_labels == -1)}")

# 2. Elliptic Envelope (FastMCD Robust Covariance)
ee = EllipticEnvelope(contamination=0.1, random_state=42)
ee.fit(X)
ee_labels = ee.predict(X)
print(f"EllipticEnvelope Detected Outliers Count: {np.sum(ee_labels == -1)}")

Parameter Breakdown: LocalOutlierFactor

Parameter Name	Data Type	Default Value	Description / Purpose
n_neighbors	int	20	Number of nearest neighbors used for local reachability density.
contamination	float	0.1	Expected proportion of outliers in the dataset $(0, 0.5]$.
metric	str	'euclidean'	Distance metric ('euclidean', 'manhattan', 'cosine').
novelty	bool	False	Enable prediction and scoring on unseen testing data.

21. Sovereign Edge-to-Cloud Serving, Hybrid State-Space, Continual Learning & Optimal Transport (Milestone 18)

21.1 Sovereign Edge-to-Cloud Serving & Paged Attention (PagedKVCache, PagedAttention, ContinuousBatcher)

import numpy as np
from chokkhu import PagedKVCache, PagedAttention, ContinuousBatcher
from chokkhu.core.tensor import Tensor

# 1. Paged KV Memory Pool (vLLM-style non-contiguous block allocation)
cache = PagedKVCache(num_blocks=32, block_size=4, num_heads=4, head_dim=16)
seq_id = "seq_42"
cache.allocate_sequence(seq_id, num_tokens=0)

# Simulate streaming KV writes
for t in range(10):
    k_t = np.random.randn(4, 16).astype(np.float32)
    v_t = np.random.randn(4, 16).astype(np.float32)
    cache.append_token(seq_id, k_t, v_t)

# 2. Paged Attention Kernel Computation
attn = PagedAttention()
query = np.random.randn(4, 16).astype(np.float32)
context = attn.forward(query, seq_id, cache)
print(f"Paged Attention Output Context Shape: {context.shape}")
stats = cache.memory_usage()
print(f"Cache Utilization: {stats['utilization']:.1%}, Used Blocks: {stats['used_blocks']}")

# 3. Continuous Batching Iteration Loop
class MockTokenizer:
    def encode(self, text, bos=True): return [1, 5, 8]
    def decode(self, tokens, skip_special_tokens=True): return "Generated response text."

def mock_llm(tensor_in):
    return Tensor(np.random.randn(1, 1, 50).astype(np.float32), requires_grad=False)

batcher = ContinuousBatcher(model=mock_llm, tokenizer=MockTokenizer(), max_batch_size=4)
req1 = batcher.add_request("Explain quantum gravity", max_new_tokens=4)
req2 = batcher.add_request("Write python quicksort", max_new_tokens=3)
batch_results = batcher.generate_all()
print(f"Continuous Batching Completed {len(batch_results)} Requests.")

Parameter Breakdown: PagedKVCache

Parameter Name	Data Type	Default Value	Description / Purpose
num_blocks	int	128	Total number of preallocated physical memory blocks in the global pool.
block_size	int	16	Number of token slots per physical memory block.
num_heads	int	4	Number of attention key/value heads.
head_dim	int	32	Dimensionality per attention head.

Parameter Breakdown: ContinuousBatcher

Parameter Name	Data Type	Default Value	Description / Purpose
model	Any	Required	Causal language model callable accepting input token tensors.
tokenizer	Any	Required	Tokenizer instance providing encode and decode methods.
cache	`PagedKVCache	None`	None
max_batch_size	int	8	Maximum concurrent active decoding sequences scheduled per iteration.

---

21.2 Structured State-Space Duality & Hybrid LLM Architectures (Mamba2SSD, JambaHybridBlock)

import numpy as np
from chokkhu import Mamba2SSD, JambaHybridBlock

# 1. Mamba-2 Structured State Space Duality (SSD) Layer
mamba2 = Mamba2SSD(d_model=32, d_state=16, d_conv=4, expand=2, seed=42)
tokens_emb = np.random.randn(2, 8, 32).astype(np.float32)
mamba_out = mamba2.forward(tokens_emb)
print(f"Mamba-2 SSD Forward Output Shape: {mamba_out.shape}")

# 2. Jamba Hybrid (Mamba-2 SSM + MHA + Sparse MoE FFN)
jamba = JambaHybridBlock(d_model=32, num_heads=4, num_experts=4, top_k=2, seed=42)
jamba_out = jamba.forward(tokens_emb)
print(f"Jamba Hybrid Layer Output Shape: {jamba_out.shape}")

Parameter Breakdown: Mamba2SSD

Parameter Name	Data Type	Default Value	Description / Purpose
d_model	int	64	Model input and output feature dimensionality.
d_state	int	64	Latent SSM state dimension ($N$).
d_conv	int	4	1D causal convolution receptive field width.
expand	int	2	Internal hidden expansion ratio ($d_{inner} = d_{model} imes ext{expand}$).
chunk_size	int	8	Block chunk size for 1D SSD matrix multiplication.
seed	int	42	Random seed for weight initialization.

Parameter Breakdown: JambaHybridBlock

Parameter Name	Data Type	Default Value	Description / Purpose
d_model	int	64	Input and output feature dimensionality.
num_heads	int	4	Number of multi-head self-attention heads.
num_experts	int	4	Number of parallel feed-forward expert MLPs.
top_k	int	2	Number of active experts routed per token.
seed	int	42	Random seed for weight initialization.

---

21.3 Continual Lifelong Learning & Catastrophic Forgetting Mitigation (ElasticWeightConsolidation, DarkExperienceReplay)

import numpy as np
from chokkhu import ElasticWeightConsolidation, DarkExperienceReplay

# 1. Elastic Weight Consolidation (Fisher Quadratic Constraint)
class SimpleClassifier:
    def __init__(self):
        self.weights = np.random.randn(4, 2).astype(np.float32)
    def predict(self, x):
        return np.dot(x, self.weights)

model_cont = SimpleClassifier()
X_task1 = np.random.randn(30, 4).astype(np.float32)
y_task1 = np.random.randint(0, 2, size=30)

ewc = ElasticWeightConsolidation(importance=50.0)
ewc.register_task(model_cont, X_task1, y_task1)

# Compute EWC consolidation loss after slight parameter update
model_cont.weights += 0.05
pen_loss = ewc.penalty_loss(model_cont)
print(f"EWC Regularization Penalty: {pen_loss:.4f}")

# 2. Dark Experience Replay (DER++ Knowledge Distillation)
der = DarkExperienceReplay(buffer_capacity=100, alpha=0.5, beta=0.5)
logits_task1 = model_cont.predict(X_task1)
der.add_batch(X_task1, y_task1, logits_task1)

# Sample mini-batch of past exemplars and calculate DER++ loss
buf_x, buf_y, buf_logits = der.sample(batch_size=8)
student_logits = model_cont.predict(buf_x)
total_loss = der.compute_loss(current_task_loss=0.85, student_replay_logits=student_logits, teacher_replay_logits=buf_logits)
print(f"DER++ Total Regularized Loss: {total_loss:.4f}")

Parameter Breakdown: ElasticWeightConsolidation

Parameter Name	Data Type	Default Value	Description / Purpose
importance	float	100.0	Quadratic penalty multiplier ($\lambda$) weighting historical Fisher Information.

Parameter Breakdown: DarkExperienceReplay

Parameter Name	Data Type	Default Value	Description / Purpose
buffer_capacity	int	500	Maximum number of exemplar samples and teacher logits stored in reservoir memory.
alpha	float	0.5	Weight multiplier for MSE logit distillation loss.
beta	float	0.5	Weight multiplier for historical replay classification loss.
seed	int	42	Random seed for reservoir replacement sampling.

---

21.4 Heterogeneous & Spatio-Temporal Graph Networks (HeteroGCN, SpatioTemporalGCN)

import numpy as np
from chokkhu import HeteroGCN, SpatioTemporalGCN

# 1. HeteroGCN Multi-Relational Message Passing
in_dims = {"user": 8, "item": 16}
relations = [("user", "buys", "item"), ("item", "bought_by", "user")]
hetero = HeteroGCN(in_channels_dict=in_dims, out_channels=16, relations=relations, seed=42)

x_nodes = {"user": np.random.randn(6, 8).astype(np.float32), "item": np.random.randn(4, 16).astype(np.float32)}
edge_indices = {
    ("user", "buys", "item"): np.array([[0, 1, 2, 3], [0, 1, 2, 3]]),
    ("item", "bought_by", "user"): np.array([[0, 1, 2, 3], [0, 1, 2, 3]]),
}
node_embeddings = hetero.forward(x_nodes, edge_indices)
print(f"HeteroGCN User Embeddings: {node_embeddings['user'].shape}, Item: {node_embeddings['item'].shape}")

# 2. SpatioTemporalGCN Spatial Graph & 1D Temporal Convolutions
st_gcn = SpatioTemporalGCN(in_channels=3, out_channels=8, temporal_kernel_size=3, seed=42)
st_tensor = np.random.randn(2, 10, 5, 3).astype(np.float32)  # (Batch, Time, Nodes, Channels)
adj_graph = np.eye(5) + 0.2
st_out = st_gcn.forward(st_tensor, adj_graph)
print(f"Spatio-Temporal Output Tensor Shape: {st_out.shape}")

Parameter Breakdown: HeteroGCN

Parameter Name	Data Type	Default Value	Description / Purpose
in_channels_dict	Dict[str, int]	Required	Mapping of node type identifiers to their input feature dimensions.
out_channels	int	Required	Output representation dimension for all node types.
relations	Sequence[Tuple[str, str, str]]	Required	List of relational triplets: (src_node_type, relation_type, dst_node_type).
aggregate	str	"sum"	Message aggregation strategy across multiple relation types ("sum" or "mean").
seed	int	42	Random seed for relational transformation matrices.

Parameter Breakdown: SpatioTemporalGCN

Parameter Name	Data Type	Default Value	Description / Purpose
in_channels	int	Required	Input feature dimensionality per graph node.
out_channels	int	Required	Output feature dimensionality per graph node.
temporal_kernel_size	int	9	Kernel width across the sequence time dimension.
stride	int	1	Stride step for temporal convolution.
seed	int	42	Random seed for spatial and temporal convolution weights.

---

21.5 Geometric Optimal Transport & Wasserstein Barycenters (SinkhornOptimalTransport, WassersteinBarycenter)

import numpy as np
from chokkhu import SinkhornOptimalTransport, WassersteinBarycenter, sinkhorn_distance, wasserstein_barycenter

# 1. Entropic Regularized Optimal Transport Coupling & Distance
X_source = np.random.randn(15, 2)
X_target = np.random.randn(20, 2)
w_dist = sinkhorn_distance(X_source, X_target, reg=0.05)
print(f"Sinkhorn Regularized Wasserstein Distance: {w_dist:.4f}")

# 2. Wasserstein Geometric Barycenter (Frechet Mean of Distributions)
grid = np.linspace(0, 1, 25)
p1 = np.exp(-((grid - 0.2)**2) / 0.02)
p2 = np.exp(-((grid - 0.8)**2) / 0.02)
bary = wasserstein_barycenter([p1, p2], (grid[:, None] - grid[None, :])**2, weights=[0.5, 0.5], reg=0.02)
print(f"Wasserstein Barycenter Computed: Shape={bary.shape}, Sum={np.sum(bary):.2f}")

Parameter Breakdown: SinkhornOptimalTransport

Parameter Name	Data Type	Default Value	Description / Purpose
reg	float	0.05	Entropic regularization coefficient ($\epsilon$).
max_iter	int	100	Maximum Sinkhorn-Knopp scaling iterations.
tol	float	1e-6	Marginal error convergence tolerance.

Parameter Breakdown: WassersteinBarycenter

Parameter Name	Data Type	Default Value	Description / Purpose
reg	float	0.01	Entropic regularization strength ($\epsilon$).
max_iter	int	50	Maximum iterative barycentric scaling iterations.
tol	float	1e-5	Convergence tolerance for barycenter updates.

---

21.6 Automated Symbolic Feature Synthesis (SymbolicFeatureSynthesizer, SymbolicProgram)

import numpy as np
from chokkhu import SymbolicFeatureSynthesizer

X_data = np.random.randn(40, 4)
y_target = X_data[:, 0] * X_data[:, 1] + np.sin(X_data[:, 2])

synthesizer = SymbolicFeatureSynthesizer(n_features=3, generations=3, population_size=15, random_state=42)
synthesizer.fit(X_data, y_target, feature_names=["f1", "f2", "f3", "f4"])
evolved_features = synthesizer.transform(X_data)
print(f"Synthesized Features Shape: {evolved_features.shape}")
print(f"Discovered Formulas: {synthesizer.get_feature_names()}")

Parameter Breakdown: SymbolicFeatureSynthesizer

Parameter Name	Data Type	Default Value	Description / Purpose
n_features	int	5	Number of top synthesized feature columns retained.
generations	int	4	Number of evolutionary genetic programming generations.
population_size	int	20	Candidate expression trees per generation.
tournament_size	int	3	Size of Pareto tournament selection pool.
random_state	`int	None`	42

---

22. Robotics, World Models, Multi-Agent RL, Genomic AI & Hyperdimensional Computing (Milestone 19)

22.1 Robotics, Diffusion Policy & Latent World Models (DiffusionPolicy, RecurrentWorldModel, MPPITrajectoryOptimizer)

import numpy as np
from chokkhu import DiffusionPolicy, RecurrentWorldModel, MPPITrajectoryOptimizer

# 1. Diffusion Policy for Continuous Action Trajectory Generation
policy = DiffusionPolicy(action_dim=4, pred_horizon=6, obs_dim=8, num_timesteps=20, hidden_dim=32, seed=42)
obs_state = np.random.randn(2, 8).astype(np.float32)
action_traj = policy.sample(obs_state, n_samples=2)
print(f"Generated Action Trajectory Shape: {action_traj.shape}")

# 2. Recurrent World Model (RSSM Imagination Rollouts)
world_model = RecurrentWorldModel(obs_dim=8, action_dim=2, deter_dim=16, stoch_dim=8, seed=42)
h0, z0 = world_model.initial_state(batch_size=2)
rollouts = world_model.imagine_trajectory(h0, z0, lambda h, z: np.random.randn(len(h), 2).astype(np.float32), horizon=5)
print(f"Imagined Trajectory Latent States Shape: {rollouts['h'].shape}, Rewards: {rollouts['rewards'].shape}")

# 3. Model Predictive Path Integral (MPPI) Trajectory Optimizer
mppi = MPPITrajectoryOptimizer(action_dim=2, horizon=8, num_samples=32, temperature=1.0, seed=42)
best_action, planned_traj = mppi.optimize(np.zeros(4), lambda s, a: np.mean((a - np.array([1.0, -0.5]))**2, axis=(1, 2)))
print(f"MPPI Optimal Control Action: {best_action}, Planned Horizon Shape: {planned_traj.shape}")

Parameter Breakdown: DiffusionPolicy

Parameter Name	Data Type	Default Value	Description / Purpose
action_dim	int	4	Dimensionality of robotic action control vector ($D_a$).
pred_horizon	int	8	Lookahead planning horizon ($T_a$).
obs_dim	int	16	Observation feature conditioning dimensionality.
num_timesteps	int	50	Number of reverse diffusion denoising steps.
seed	int	42	Random seed for weight initialization.

Parameter Breakdown: RecurrentWorldModel

Parameter Name	Data Type	Default Value	Description / Purpose
obs_dim	int	16	Observation feature dimensionality.
action_dim	int	4	Continuous control action dimension.
deter_dim	int	32	Deterministic GRU recurrent state size ($h_t$).
stoch_dim	int	16	Stochastic latent Gaussian state size ($z_t$).
seed	int	42	Random seed for latent transition parameters.

Parameter Breakdown: MPPITrajectoryOptimizer

Parameter Name	Data Type	Default Value	Description / Purpose
action_dim	int	Required	Dimensionality of control action vector.
horizon	int	15	Trajectory planning lookahead steps ($T$).
num_samples	int	64	Number of parallel perturbation rollouts ($K$).
temperature	float	1.0	Return weighting temperature ($\lambda$).
noise_sigma	float	0.5	Exploration perturbation standard deviation.

---

22.2 Multi-Agent Reinforcement Learning & Game Theory (QMIX, VDN, NashEquilibriumSolver)

import numpy as np
from chokkhu import QMIX, VDN, NashEquilibriumSolver

# 1. QMIX Monotonic Value Factorization
qmix = QMIX(n_agents=3, state_dim=12, mixing_embed_dim=16, seed=42)
agent_qs = np.random.randn(4, 3).astype(np.float32)
global_states = np.random.randn(4, 12).astype(np.float32)
q_tot = qmix.forward(agent_qs, global_states)
print(f"QMIX Centralized Joint Q_tot Shape: {q_tot.shape}")

# 2. VDN Additive Utility Decomposition
vdn = VDN(n_agents=3)
vdn_tot = vdn.forward(agent_qs)
print(f"VDN Joint Team Value: {vdn_tot[:2].flatten()}")

# 3. Game-Theoretic Nash Equilibrium Solver (Zero-Sum & General-Sum)
solver = NashEquilibriumSolver(max_iter=500)
rps_payoff = np.array([[0, -1, 1], [1, 0, -1], [-1, 1, 0]])
p_row, p_col, val = solver.solve_zero_sum(rps_payoff)
print(f"Nash Equilibrium Mixed Strategy: Row={p_row.round(2)}, Col={p_col.round(2)}, Value={val:.2f}")

Parameter Breakdown: QMIX & VDN

Parameter Name	Data Type	Default Value	Description / Purpose
n_agents	int	4	Number of cooperative agents in multi-agent team.
state_dim	int	32	Global environmental state dimension ($S$).
mixing_embed_dim	int	32	Internal hypernetwork mixing capacity.
seed	int	42	Random seed for hypernetwork weights.

---

22.3 Genomic & Bio-Molecular Sequence AI (GenomicTokenizer, GenomicBERT, ProteinContactMap)

import numpy as np
from chokkhu import GenomicTokenizer, GenomicBERT, ProteinContactMap

# 1. DNA/RNA k-mer Tokenization & Reverse Complement
tok = GenomicTokenizer(k=3, stride=1, is_rna=False)
dna_seq = "ATGCGATCG"
token_ids = tok.encode(dna_seq, add_special_tokens=True)
rev_comp = GenomicTokenizer.reverse_complement("ATGC")
print(f"Encoded Token IDs: {token_ids}, Reverse Complement: {rev_comp}")

# 2. GenomicBERT Sequence Modeling & Variant Effect Scoring
gen_bert = GenomicBERT(vocab_size=len(tok.vocab), d_model=32, num_heads=4, num_layers=2, max_len=32, num_classes=2, seed=42)
hidden, cls_logits, mlm_logits = gen_bert.forward(np.array([token_ids]))
var_score = gen_bert.score_variant(np.array([token_ids]), np.array([token_ids]))
print(f"GenomicBERT Sequence Embeddings: {hidden.shape}, Variant Effect Score: {var_score:.4f}")

# 3. Protein Residue Contact Map via Direct Coupling Analysis (DCA)
dca = ProteinContactMap(pseudocount_weight=0.5, apc=True)
dummy_msa = np.random.randint(0, 20, size=(25, 12))
contact_map = dca.compute_contact_map(dummy_msa)
print(f"Residue Contact Matrix Shape: {contact_map.shape}, Symmetric: {np.allclose(contact_map, contact_map.T)}")

Parameter Breakdown: GenomicTokenizer & GenomicBERT

Parameter Name	Data Type	Default Value	Description / Purpose
k	int	6	Length of contiguous nucleotide $k$-mer tokens.
stride	int	1	Tokenization sliding window step.
is_rna	bool	False	Toggle between DNA (T) and RNA (U) alphabets.
d_model	int	64	Transformer representation width.
num_heads	int	4	Number of self-attention heads.
num_layers	int	2	Number of bidirectional encoder layers.

---

22.4 Neuro-Symbolic Reasoning & Knowledge Graph Embeddings (DifferentiableLogicEngine, RotatE, TransE)

import numpy as np
from chokkhu import DifferentiableLogicEngine, RotatE, TransE

# 1. Continuous Fuzzy Differentiable Logic Engine
engine = DifferentiableLogicEngine(t_norm="product")
a_truth = np.array([0.9, 0.4])
b_truth = np.array([0.8, 0.7])
and_val = engine.conjunction(a_truth, b_truth)
implies_val = engine.implication(a_truth, b_truth)
sat_loss = engine.satisfaction_loss(implies_val)
print(f"Fuzzy Conjunction: {and_val}, Implication: {implies_val}, Satisfaction Loss: {sat_loss:.4f}")

# 2. RotatE & TransE Knowledge Graph Embeddings
rotate = RotatE(num_entities=20, num_relations=5, embedding_dim=16, gamma=12.0, seed=42)
transe = TransE(num_entities=20, num_relations=5, embedding_dim=16, margin=1.0, seed=42)
h_ids, r_ids, t_ids = np.array([0, 1]), np.array([2, 3]), np.array([4, 5])
rot_scores = rotate.score_triplets(h_ids, r_ids, t_ids)
trans_scores = transe.score_triplets(h_ids, r_ids, t_ids)
print(f"RotatE Triplet Distance Scores: {rot_scores}, TransE: {trans_scores}")

Parameter Breakdown: RotatE & TransE

Parameter Name	Data Type	Default Value	Description / Purpose
num_entities	int	Required	Total number of entities in knowledge graph.
num_relations	int	Required	Total number of relation types.
embedding_dim	int	64	Latent space dimensionality.
gamma / margin	float	12.0 / 1.0	Distance margin parameter for contrastive ranking loss.

---

22.5 Hyperdimensional Computing & Vector Symbolic Architectures (HyperdimensionalVector, HDCClassifier)

import numpy as np
from chokkhu import HyperdimensionalVector, HDCClassifier

# 1. Hypervector Primitive Algebra (Binding, Bundling, Permutation)
v1 = HyperdimensionalVector.random_bipolar(dim=2000, seed=42)
v2 = HyperdimensionalVector.random_bipolar(dim=2000, seed=43)
bound = v1.bind(v2)
bundled = HyperdimensionalVector.bundle([v1, v2], binarize=True)
permuted = v1.permute(shift=2)
print(f"HDC Similarity (v1 vs v2): {v1.similarity(v2):.4f}, (v1 vs bundled): {v1.similarity(bundled):.4f}")

# 2. Zero-Gradient One-Shot HDC Associative Memory Classifier
hdc_clf = HDCClassifier(dim=2000, num_levels=16, seed=42)
X_train = np.random.randn(20, 4)
y_train = (X_train[:, 0] + X_train[:, 1] > 0).astype(int)
hdc_clf.fit(X_train, y_train)
acc = hdc_clf.score(X_train, y_train)
print(f"HDC Associative Memory Training Accuracy: {acc:.1%}")

Parameter Breakdown: HDCClassifier

Parameter Name	Data Type	Default Value	Description / Purpose
dim	int	5000	Dimensionality of hypervector cognitive space ($D$).
num_levels	int	32	Number of continuous feature quantization level vectors.
seed	int	42	Random seed for orthogonal basis vectors.

---

23. The Grand Frontier: Discrete Diffusion, Differentiable Physics, LLM Safety, Causal DAGs & Hyperbolic Geometry (Milestone 20)

23.1 Discrete Text Diffusion Models (DiscreteTextDiffusion)

import numpy as np
from chokkhu import DiscreteTextDiffusion

# 1. Initialize Discrete Absorbing-State Categorical Diffusion Model
model = DiscreteTextDiffusion(
    vocab_size=500,
    max_seq_len=32,
    num_timesteps=20,
    d_model=64,
    num_heads=4,
    num_layers=2,
    schedule="linear",
    seed=42
)

# 2. Forward Categorical Token Masking & ELBO Loss Calculation
x_clean = np.random.randint(0, 500, size=(2, 16))
timesteps = np.array([5, 12])
x_corrupted = model.q_sample(x_clean, timesteps)
loss = model.compute_loss(x_clean, timesteps)
print(f"Corrupted Tokens Shape: {x_corrupted.shape}, Variational Loss: {loss:.4f}")

# 3. Ancestral Iterative Reverse Sampling from All [MASK] to Clean Text
generated_tokens = model.sample(batch_size=2, seq_len=12, num_steps=10, temperature=0.8)
print(f"Generated Token Sequences: {generated_tokens.shape}")

Parameter Breakdown: DiscreteTextDiffusion

Parameter Name	Data Type	Default Value	Description / Purpose
vocab_size	int	1000	Total discrete vocabulary size.
max_seq_len	int	64	Maximum allowable sequence token length.
num_timesteps	int	50	Total diffusion Markov chain step count ($T$).
mask_token_id	Optional[int]	None	Absorbing mask token ID (defaults to vocab_size).
d_model	int	64	Bidirectional transformer hidden dimension.
num_heads	int	4	Number of self-attention heads.
num_layers	int	2	Number of transformer encoder layers.
schedule	str	"linear"	Noise transition schedule: "linear" or "cosine".

---

23.2 Differentiable Physics & Robotics Kinematics (RobotArmKinematics, DifferentiableParticleFluid)

import numpy as np
from chokkhu import RobotArmKinematics, DifferentiableParticleFluid

# 1. 2-Link Planar Robot Arm Forward & Damped Least Squares (DLS) Inverse Kinematics
arm = RobotArmKinematics.planar_2d(link_lengths=[1.0, 1.0])
target_pos = np.array([1.0, 1.0, 0.0])
solved_angles, converged, iters = arm.inverse_kinematics(target_position=target_pos, tolerance=1e-3)
actual_pos = arm.get_end_effector_position(solved_angles)
print(f"IK Solved Angles: {solved_angles}, Target: {target_pos[:2]}, Reached: {actual_pos[:2]}")

# 2. Smoothed Particle Hydrodynamics (SPH) Differentiable Fluid Dynamics
fluid = DifferentiableParticleFluid(
    num_particles=30, dim=2, rest_density=1000.0, stiffness=150.0, viscosity=0.05, dt=0.005, seed=42
)
pos_init = np.random.uniform(-0.4, 0.4, size=(30, 2))
pos_hist, vel_hist = fluid.simulate(pos_init, num_steps=20)
print(f"Fluid Particle Trajectory History: {pos_hist.shape}")

Parameter Breakdown: RobotArmKinematics & DifferentiableParticleFluid

Parameter Name	Data Type	Default Value	Description / Purpose
dh_params	np.ndarray	Required	Denavit-Hartenberg parameter table (N, 4): [a, alpha, d, theta_offset].
joint_limits	Optional[List[Tuple[float, float]]]	None	Per-joint rotation angle bounds $[-\pi, \pi]$.
num_particles	int	100	Total simulated fluid particle count.
rest_density	float	1000.0	Equilibrium rest mass density $\rho_0$ ($\text{kg/m}^3$).
stiffness	float	200.0	Tait equation of state gas stiffness constant $k$.
viscosity	float	0.1	Dynamic viscosity coefficient $\mu$.

---

23.3 AI Safety & LLM Watermarking (StatisticalTextWatermark, RefusalDirectionProbe)

import numpy as np
from chokkhu import StatisticalTextWatermark, RefusalDirectionProbe

# 1. Statistical Token Watermarking & Z-Score Hypothesis Testing
watermark = StatisticalTextWatermark(vocab_size=1000, gamma=0.5, delta=3.0, hash_key=15485863)
prefix = np.array([42, 108])
logits = np.random.randn(1000)
biased_logits = watermark.apply_watermark_bias(logits, prefix)
det_result = watermark.detect(np.array([42, 108, 205, 312, 401]))
print(f"Watermark Detection Z-Score: {det_result['z_score']:.2f}, Is Watermarked: {det_result['is_watermarked']}")

# 2. Refusal Direction Probe & Representation Steering
probe = RefusalDirectionProbe(dim=64)
harmful_acts = np.random.randn(20, 64) + 1.5
harmless_acts = np.random.randn(20, 64) - 1.5
probe.fit(harmful_acts, harmless_acts)
steered_acts = probe.steer(harmless_acts[:2], alpha=1.0)
ablated_acts = probe.ablate(harmful_acts[:2])
print(f"Refusal Intent Score (Harmful): {probe.score_refusal_intent(harmful_acts[:2])}")

Parameter Breakdown: StatisticalTextWatermark & RefusalDirectionProbe

Parameter Name	Data Type	Default Value	Description / Purpose
vocab_size	int	1000	LLM token vocabulary size.
gamma	float	0.5	Fraction of vocabulary partitioned into the green list ($\gamma \in (0, 1)$).
delta	float	2.0	Logit perturbation bias injected into green tokens ($\delta > 0$).
hash_key	int	15485863	Cryptographic prime seed for deterministic pseudo-random hashing.
dim	int	64	Residual stream activation embedding dimensionality.

---

23.4 Causal DAG Discovery (NOTEARSCausalDiscovery, PCAlgorithm)

import numpy as np
from chokkhu import NOTEARSCausalDiscovery, PCAlgorithm

# 1. Continuous Structure Learning with Smooth Matrix Exponential Acyclicity (NOTEARS)
rng = np.random.RandomState(42)
n_samples = 300
x0 = rng.randn(n_samples)
x1 = 0.7 * x0 + 0.3 * rng.randn(n_samples)
x2 = 0.8 * x1 + 0.2 * rng.randn(n_samples)
X_causal = np.column_stack([x0, x1, x2])

notears = NOTEARSCausalDiscovery(lambda1=0.05, max_iter=25, w_threshold=0.2)
notears.fit(X_causal)
print(f"NOTEARS Learned DAG Matrix:\n{notears.adjacency_matrix_}\nIs Strict DAG: {notears.is_dag()}")

# 2. Constraint-Based Conditional Independence Discovery (PC Algorithm)
pc = PCAlgorithm(alpha=0.05)
pc.fit(X_causal)
print(f"PC Algorithm Discovered CPDAG Edges: {pc.get_edges()}")

Parameter Breakdown: NOTEARSCausalDiscovery & PCAlgorithm

Parameter Name	Data Type	Default Value	Description / Purpose
lambda1	float	0.1	$L_1$ sparsity penalty coefficient on DAG adjacency weights.
max_iter	int	100	Maximum Augmented Lagrangian outer optimization iterations.
h_tol	float	1e-8	Matrix exponential acyclicity trace tolerance $h(W) = \text{tr}(e^{W \odot W}) - d$.
w_threshold	float	0.3	Threshold for pruning spurious small edges.
alpha	float	0.05	Statistical significance level for partial correlation Fisher $z$-tests.

---

23.5 Hyperbolic Manifold Geometry (PoincareBallEmbedding, LorentzManifold)

import numpy as np
from chokkhu import PoincareBallEmbedding, LorentzManifold

# 1. Poincaré Ball Model & Riemannian Graph Representation Learning
poincare = PoincareBallEmbedding(dim=3, c=1.0, seed=42)
u = np.array([0.2, 0.1, -0.3])
v = np.array([-0.1, 0.4, 0.2])
geo_dist = poincare.distance(u, v)
mob_sum = poincare.mobius_add(u, v)
print(f"Poincare Geodesic Distance: {geo_dist:.4f}, Mobius Addition Norm: {np.linalg.norm(mob_sum):.4f}")

# 2. Lorentz Hyperboloid Model & Isomorphic Diffeomorphisms
lorentz = LorentzManifold(dim=2, c=1.0)
x_poincare = np.array([0.3, -0.2])
x_lorentz = lorentz.poincare_to_lorentz(x_poincare)
minkowski_norm = lorentz.minkowski_dot(x_lorentz, x_lorentz)
print(f"Lorentz Coordinates: {x_lorentz}, Minkowski Constraint <x, x>_L: {minkowski_norm:.4f}")

Parameter Breakdown: PoincareBallEmbedding & LorentzManifold

Parameter Name	Data Type	Default Value	Description / Purpose
dim	int	2	Hyperbolic manifold spatial embedding dimension $d$.
c	float	1.0	Negative sectional curvature parameter ($\kappa = -c$).
eps	float	1e-5	Boundary safety margin for numerical ball projection $|x| \le 1/\sqrt{c} - \epsilon$.

---

---

24. Multi-Agent Economy, Mechanistic Interpretability, Guided Diffusion Inpainting, Byte Transformers & Differentiable Ranking

24.1 Replicator Dynamics & Evolutionary Game Theory (ReplicatorDynamics)

Models continuous-time frequency-dependent evolutionary selection and mutator dynamics in multi-agent game-theoretic populations $\dot{x}i = x_i (f_i(x) - ar{f}(x)) + \sum_j (\mu{ji} x_j - \mu_{ij} x_i)$ integrated via 4th-order Runge-Kutta (RK4) projected onto the probability simplex.

import numpy as np
from chokkhu import ReplicatorDynamics

# 1. Classical Hawk-Dove Game Simulation
# Payoff: V=2.0 (resource value), C=4.0 (fight cost) -> Evolutionary Stable Strategy (ESS) = [0.5, 0.5]
hd = ReplicatorDynamics.hawks_doves(v=2.0, c=4.0, dt=0.01)
initial_population = np.array([0.9, 0.1])  # 90% Hawks, 10% Doves
pop_trajectory = hd.simulate(initial_population, num_steps=200)
print(f"Hawk-Dove Initial: {initial_population} -> Final ESS: {pop_trajectory[-1]}")

# 2. Rock-Paper-Scissors Cyclic Dynamics
rps = ReplicatorDynamics.rock_paper_scissors(dt=0.01)
rps_trajectory = rps.simulate(np.array([0.5, 0.3, 0.2]), num_steps=100)
print(f"RPS Step 100 Distribution: {rps_trajectory[-1]}")

Parameter Breakdown: ReplicatorDynamics

Parameter Name	Data Type	Default Value	Description / Purpose
payoff_matrix	np.ndarray	Required	Square payoff matrix $A \in \mathbb{R}^{K imes K}$ where $A_{ij}$ is payoff of strategy $i$ against $j$.
mutation_rate	float	0.0	Uniform mutation probability $\mu \in [0, 1)$ between strategies.
dt	float	0.01	Continuous numerical integration step size for the RK4 solver.

---

24.2 Combinatorial Bundle Auctions & VCG Mechanism (CombinatorialAuction)

Implements the generalized Vickrey-Clarke-Groves (VCG) mechanism for combinatorial multi-item auctions, guaranteeing dominant-strategy incentive compatibility (DSIC / truth-telling) and social welfare maximization with Clarke pivot rule taxation.

from chokkhu import CombinatorialAuction

# Create combinatorial auction for spectrum frequency bands A and B
auction = CombinatorialAuction(items=["A", "B"], bidders=["Alice", "Bob", "Charlie"])

# Agents submit bids over item combinations
auction.add_bid("Alice", ["A", "B"], 10.0)  # Alice bids $10 for the combined bundle {A, B}
auction.add_bid("Bob", ["A"], 6.0)           # Bob bids $6 for band A alone
auction.add_bid("Charlie", ["B"], 5.0)       # Charlie bids $5 for band B alone

# Solve Winner Determination Problem (WDP) & compute Clarke pivot payments
result = auction.solve()
print(f"Winning Allocations: {result['allocations']}")
print(f"Clarke Pivot Payments: {result['payments']}")
print(f"Total Social Welfare: {result['social_welfare']}")

Parameter Breakdown: CombinatorialAuction

Parameter Name	Data Type	Default Value	Description / Purpose
items	List[str]	Required	List of distinct item and good identifiers up for auction.
bidders	List[str]	Required	List of participating agent/bidder identifiers.

---

24.3 Mechanistic Interpretability & Interventional Causal Patching (ActivationPatchingEngine)

Performs interventional causal tracing and activation patching across transformer layers and attention heads to localize factual knowledge circuits without backpropagation.

import numpy as np
from chokkhu import ActivationPatchingEngine

# Define forward pass with activation caching hook
def model_forward_with_cache(x, patch_dict=None):
    layer_1 = x * 2.0
    if patch_dict and "layer_1" in patch_dict:
        layer_1 = patch_dict["layer_1"]
    logits = np.sum(layer_1, axis=-1, keepdims=True)
    cache = {"layer_1": x * 2.0}
    return logits, cache

engine = ActivationPatchingEngine(model_forward_with_cache)
clean_prompt = np.array([[5.0, 5.0]])
corrupted_prompt = np.array([[1.0, 1.0]])

# Measure normalized causal recovery ratio of patching layer_1
causal_effect = engine.compute_causal_effect(clean_prompt, corrupted_prompt, "layer_1")
print(f"Layer 1 Causal Mediation Effect: {causal_effect:.4f}")

# Full circuit sweep across all model layers
circuit_map = engine.trace_circuit(clean_prompt, corrupted_prompt, ["layer_1"])
print(f"Circuit Trace Attribution: {circuit_map}")

Parameter Breakdown: ActivationPatchingEngine

Parameter Name	Data Type	Default Value	Description / Purpose
forward_with_cache	Callable	Required	Function accepting (x, patch_dict) returning (logits, cache).
clean_input	np.ndarray	Required	Clean prompt input that produces correct ground-truth outputs.
corrupted_input	np.ndarray	Required	Corrupted/perturbed input where factual recall fails.
layer_name	str	Required	Hook identifier of the layer/head activation tensor to intervene upon.

---

24.4 Classifier-Free Guidance & Masked Diffusion Inpainting (ClassifierFreeGuidance, DiffusionInpainter)

Enables controllable generation via Classifier-Free Guidance (CFG) score extrapolation $ ilde{\epsilon}_ heta(x_t, c) = \epsilon_ heta(x_t, \emptyset) + s (\epsilon_ heta(x_t, c) - \epsilon_ heta(x_t, \emptyset))$ and RePaint masked diffusion inpainting with boundary time-travel harmonization.

import numpy as np
from chokkhu import ClassifierFreeGuidance, DiffusionInpainter

# 1. Classifier-Free Guidance (CFG) Noise Extrapolation
cfg = ClassifierFreeGuidance(guidance_scale=3.5)
unconditional_noise = np.array([0.1, 0.2])
conditional_noise = np.array([0.5, 0.8])
guided_noise = cfg.combine_scores(unconditional_noise, conditional_noise)
print(f"CFG Extrapolated Noise Estimate: {guided_noise}")

# 2. RePaint Masked Diffusion Inpainting Engine
class DiffusionScoreModel:
    num_timesteps = 10
    alphas_cumprod = np.linspace(0.99, 0.05, 10)
    def predict_noise(self, x_t, t): return 0.05 * x_t

inpainter = DiffusionInpainter(DiffusionScoreModel(), num_resample_steps=2, seed=42)
clean_image = np.ones((1, 8, 8))
mask = np.ones((1, 8, 8))
mask[:, 2:6, 2:6] = 0.0  # Mask out central 4x4 hole (0.0 = hole to inpaint)

inpainted_result = inpainter.inpaint(clean_image, mask, num_steps=5)
print(f"Inpainted Image Output Shape: {inpainted_result.shape}")

Parameter Breakdown: ClassifierFreeGuidance & DiffusionInpainter

Parameter Name	Data Type	Default Value	Description / Purpose
guidance_scale	float	3.0	Conditioning scale multiplier $s$ ($s > 1.0$ amplifies conditional steering).
ddpm_model	Any	Required	Base diffusion model providing noise schedule and score evaluation.
num_resample_steps	int	1	Time-travel resampling passes per reverse timestep for boundary harmonization.

---

24.5 Token-Free Raw UTF-8 Byte Transformer (ByteTransformer)

Operates directly on raw binary byte streams ($0 - 255$) with zero tokenization artifacts, eliminating out-of-vocabulary failures via local patch convolutional pooling, multi-head self-attention, and byte-level unpooling.

import numpy as np
from chokkhu import ByteTransformer

# Initialize Sovereign Byte Transformer (ByT5 / MambaByte architecture)
byte_model = ByteTransformer(
    d_model=32, patch_size=2, num_heads=2, num_layers=1, max_seq_len=64, seed=42
)

# Encode arbitrary raw UTF-8 string directly into uint8 byte array
text = "Hello Sovereign AI! 🇧🇩"
byte_array = ByteTransformer.text_to_bytes(text)
print(f"Raw UTF-8 Bytes: {byte_array[:6]}... Decoded: {ByteTransformer.bytes_to_text(byte_array)}")

# Next-byte probability distribution prediction
logits = byte_model.forward(np.array([byte_array[:8]]))
print(f"Next-Byte Output Logits Tensor: {logits.shape}")

# Autoregressive generation directly in raw byte space
generated_str = byte_model.generate("AI", max_new_bytes=4, temperature=0.8)
print(f"Generated Continuation: {repr(generated_str)}")

Parameter Breakdown: ByteTransformer

Parameter Name	Data Type	Default Value	Description / Purpose
d_model	int	64	Internal transformer representation dimensionality.
patch_size	int	4	Local byte compression factor per patch projection.
num_heads	int	4	Number of multi-head self-attention heads.
num_layers	int	2	Number of stacked transformer layers.
max_seq_len	int	256	Maximum allowable raw byte sequence length.

---

24.6 Differentiable Neural Sorting & Soft Ranking Losses (NeuralSort, DifferentiableRankingLoss)

Provides continuous relaxations of the non-differentiable $ ext{argsort}$ operator via temperature-controlled doubly-stochastic permutation matrices $P_ au(s)$ alongside differentiable Spearman rank correlation and Soft NDCG losses for Learning-to-Rank (LTR).

import numpy as np
from chokkhu import NeuralSort, DifferentiableRankingLoss

# 1. NeuralSort Continuous Relaxed Permutations
sorter = NeuralSort(tau=0.5)
continuous_scores = np.array([12.0, 45.0, 23.0])

# Compute continuous 1-based soft ranks
soft_ranks = sorter.soft_ranks(continuous_scores)
print(f"Soft Continuous Ranks: {soft_ranks}")

# Differentiable soft sort
values = np.array([100.0, 500.0, 200.0])
sorted_values = sorter.soft_sort(values, scores=continuous_scores)
print(f"Softly Sorted Output: {sorted_values}")

# 2. Differentiable Surrogate Ranking Losses
pred_scores = np.array([10.0, 50.0, 20.0])
ground_truth_relevance = np.array([1.0, 3.0, 2.0])

spearman_loss = DifferentiableRankingLoss.spearman_loss(pred_scores, ground_truth_relevance, tau=0.5)
ndcg_loss = DifferentiableRankingLoss.soft_ndcg_loss(pred_scores, ground_truth_relevance, tau=0.5)
print(f"Differentiable Spearman Loss: {spearman_loss:.4f} | Soft NDCG Loss: {ndcg_loss:.4f}")

Parameter Breakdown: NeuralSort & DifferentiableRankingLoss

Parameter Name	Data Type	Default Value	Description / Purpose
tau	float	1.0	Temperature relaxation parameter ($ au o 0$ approaches exact discrete hard sort).
pred_scores	np.ndarray	Required	Model output relevance score predictions.
true_relevance	np.ndarray	Required	Ground-truth relevance or ranking labels.

---

25. Geometric Deep Learning, Online Optimization, Information Theory, Energy Models & Model Trees

25.1 Spherical Harmonics & SE(3) Equivariant Steerable Convolutions (SphericalHarmonics, SE3EquivariantConv)

Implements continuous $SO(3)$-steerable spherical harmonic basis expansions $Y_l^m(\hat{r})$ and radial basis message passing for 3D atomic structures, crystal lattices, and equivariant point clouds: $$f_{\text{out}}(i) = \sum_{j \in \mathcal{N}(i)} W_{\text{radial}}(|r_{ij}|) \left( Y_l^m(\hat{r}{ij}) \otimes f{\text{in}}(j) \right) + b$$

import numpy as np
from chokkhu import SphericalHarmonics, SE3EquivariantConv

# 1. Real Spherical Harmonics Basis (l=0, 1, 2)
direction_vectors = np.array([[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]])
Y_harmonics = SphericalHarmonics.compute(direction_vectors, max_l=2)
print(f"Real Spherical Harmonics Basis Evaluations Shape: {Y_harmonics.shape}")

# 2. SE(3) Equivariant Steerable Point Convolution
conv = SE3EquivariantConv(in_dim=4, out_dim=8, max_l=1, cutoff_radius=5.0, seed=42)
atom_coords = np.array([[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0]])
atom_features = np.ones((3, 4))

out_features = conv.forward(atom_coords, atom_features)
print(f"SE(3) Equivariant Transformed Atom Features: {out_features.shape}")

Parameter Breakdown: SphericalHarmonics & SE3EquivariantConv

Parameter Name	Data Type	Default Value	Description / Purpose
in_dim	int	Required	Input node/atom feature dimension.
out_dim	int	Required	Output representation dimension.
max_l	int	1	Maximum spherical harmonic degree ($l \in {0, 1, 2}$).
num_radial_bases	int	8	Number of Gaussian radial basis functions for radial distance filtering.
cutoff_radius	float	5.0	Spatial neighborhood interaction cutoff distance.

---

25.2 Follow-The-Regularized-Leader (FTRL-Proximal) & Hedge Algorithm (FollowTheRegularizedLeader, HedgeAlgorithm)

Enables web-scale online learning with $L_1 / L_2$ sparse proximal soft-thresholding alongside minimax-optimal multi-expert decision making.

import numpy as np
from chokkhu import FollowTheRegularizedLeader, HedgeAlgorithm

# 1. FTRL-Proximal Online Logistic Regression for Streaming CTR Prediction
ftrl = FollowTheRegularizedLeader(alpha=0.1, beta=1.0, lambda1=0.01, lambda2=0.1, loss="logistic")
X_stream = np.random.randn(100, 10)
y_stream = (X_stream[:, 0] > 0.0).astype(float)

for i in range(len(X_stream)):
    ftrl.update_one(X_stream[i], y_stream[i])

click_prob = ftrl.predict(X_stream[:3])
print(f"FTRL Streaming Click Probability: {click_prob}")

# 2. Hedge Multi-Expert Minimax Regret Minimization
hedge = HedgeAlgorithm(num_experts=3, time_horizon=50)
expert_losses = np.array([0.1, 0.4, 0.7])  # Round losses suffered by 3 candidate models
round_loss, regret = hedge.update(expert_losses)
print(f"Hedge Multi-Expert Allocation: {hedge.predict_distribution()} | Cumulative Regret: {regret:.4f}")

Parameter Breakdown: FollowTheRegularizedLeader & HedgeAlgorithm

Parameter Name	Data Type	Default Value	Description / Purpose
alpha	float	0.1	Per-coordinate learning rate scale $\alpha$.
beta	float	1.0	Learning rate denominator smoothing parameter.
lambda1	float	0.1	$L_1$ sparsity-inducing regularization penalty.
lambda2	float	1.0	$L_2$ shrinkage regularization penalty.
num_experts	int	Required	Number of participating experts $K$.

---

25.3 Non-Parametric Information Theory & Continuous Density Estimation (KraskovMutualInformation, MultivariateKDE)

Estimates continuous non-parametric mutual information $I(X; Y) = \psi(k) - \frac{1}{N}\sum [\psi(n_x + 1) + \psi(n_y + 1)] + \psi(N)$ and multidimensional kernel densities without assuming Gaussianity.

import numpy as np
from chokkhu import KraskovMutualInformation, MultivariateKDE

# 1. KSG k-NN Continuous Mutual Information
ksg = KraskovMutualInformation(k=3)
X = np.random.randn(200, 1)
Y = np.sin(X * 2.0) + 0.1 * np.random.randn(200, 1)
mi = ksg.estimate(X, Y)
print(f"Non-Linear Continuous Mutual Information I(X; Y): {mi:.4f} nats")

# 2. Multivariate Kernel Density Estimator (KDE)
kde = MultivariateKDE(bandwidth="silverman", kernel="gaussian")
kde.fit(np.hstack([X, Y]))
log_density = kde.score_samples(np.hstack([X[:3], Y[:3]]))
generated_points = kde.sample(n_samples=5)
print(f"Log Prob Density: {log_density} | Sampled Points Shape: {generated_points.shape}")

Parameter Breakdown: KraskovMutualInformation & MultivariateKDE

Parameter Name	Data Type	Default Value	Description / Purpose
k	int	3	Number of nearest neighbors in joint Chebyshev metric space.
bandwidth	Union[str, float]	"silverman"	Bandwidth selection rule ("silverman", "scott", or explicit float $h$).
kernel	str	"gaussian"	Smoothing kernel function ("gaussian", "epanechnikov", "box").

---

25.4 Energy-Based Models & Sliced Score Matching (EnergyBasedModel, SlicedScoreMatching)

Parameterizes unnormalized Boltzmann densities $p_\theta(x) \propto e^{-E_\theta(x)}$ with Stochastic Gradient Langevin Dynamics (SGLD) sampling and Hutchinson sliced score matching.

import numpy as np
from chokkhu import EnergyBasedModel, SlicedScoreMatching

# 1. Continuous Energy-Based Model & SGLD MCMC Sampling
ebm = EnergyBasedModel(in_dim=4, hidden_dim=32, seed=42)
data_points = np.random.randn(50, 4)
energy_values = ebm.energy(data_points[:3])
langevin_samples = ebm.sample_sgld(n_samples=10, num_steps=20, step_size=0.01)
print(f"Evaluated Energy: {energy_values} | SGLD Generated Samples: {langevin_samples.shape}")

# 2. Sliced Score Matching (Hutchinson Random Vector Trace Estimator)
score_model = lambda x: -0.5 * x  # Score vector estimator nabla_x log p(x)
ssm = SlicedScoreMatching(score_fn=score_model, num_projections=4)
matching_loss = ssm.compute_loss(data_points)
print(f"Sliced Score Matching Loss: {matching_loss:.4f}")

Parameter Breakdown: EnergyBasedModel & SlicedScoreMatching

Parameter Name	Data Type	Default Value	Description / Purpose
in_dim	int	Required	Input feature space dimensionality.
hidden_dim	int	64	Energy network hidden layer width.
step_size	float	0.05	Langevin diffusion step size $\epsilon$.
num_projections	int	4	Number of random Hutchinson directional projections per sample.

---

25.5 Piecewise Linear Model Trees & Interpretable RuleFit (M5ModelTree, RuleFitRegressor, RuleFitClassifier)

Implements the M5 algorithm combining decision tree partitions with leaf multivariate Ridge regressions, and RuleFit sparse Lasso rule ensembles.

import numpy as np
from chokkhu import M5ModelTree, RuleFitRegressor, RuleFitClassifier

X = np.random.uniform(-2.0, 2.0, size=(100, 3))
y_reg = np.where(X[:, 0] <= 0.0, 2.0 * X[:, 0] + 1.0, -3.0 * X[:, 0] + 1.0)
y_cls = (y_reg > 0.0).astype(int)

# 1. M5 Piecewise Linear Model Tree
m5 = M5ModelTree(max_depth=4, min_samples_split=5, smoothing=5.0)
m5.fit(X, y_reg)
m5_predictions = m5.predict(X[:3])
print(f"M5 Piecewise Linear Predictions: {m5_predictions}")

# 2. RuleFit Sparse Rule Ensembles
rf_reg = RuleFitRegressor(num_trees=5, max_depth=2, alpha=0.01, seed=42)
rf_reg.fit(X, y_reg)
print(f"RuleFit Regression Predictions: {rf_reg.predict(X[:3])}")

rf_cls = RuleFitClassifier(num_trees=5, max_depth=2, alpha=0.01, seed=42)
rf_cls.fit(X, y_cls)
print(f"RuleFit Classification Probabilities: {rf_cls.predict_proba(X[:3])[:, 1]}")

Parameter Breakdown: M5ModelTree & RuleFitRegressor

Parameter Name	Data Type	Default Value	Description / Purpose
max_depth	int	5	Maximum tree depth.
min_samples_split	int	10	Minimum samples required to partition an internal node.
smoothing	float	15.0	Leaf-parent model prediction smoothing constant $k$.
alpha	float	0.01	$L_1$ Lasso regularization penalty for rule selection.

---

26. Differentiable Optimization, Debiased Optimal Transport, One-Class SVDD, Spectral Graph Partitioning & Differentiable ILP

26.1 OptNet: Differentiable Quadratic Programming & Implicit Layer Differentiation (OptNet)

Implements convex quadratic optimization layers solvable via primal-dual interior point methods with exact analytical backward parameter differentiation through Karush-Kuhn-Tucker (KKT) implicit optimality conditions: $$\min_z \frac{1}{2} z^T Q z + p^T z \quad \text{s.t.} \quad A z = b, \quad G z \le h$$ $$\begin{bmatrix} Q & A^T & G^T \ A & 0 & 0 \ \text{diag}(\lambda^) G & 0 & \text{diag}(G z^ - h) \end{bmatrix} \begin{bmatrix} \text{d}z \ \text{d}\nu \ \text{d}\lambda \end{bmatrix} = - \begin{bmatrix} \nabla_z \mathcal{L} \ 0 \ 0 \end{bmatrix}$$

import numpy as np
from chokkhu import OptNet

# Minimize 1/2 x^T Q x + p^T x s.t. G x <= h
Q = 2.0 * np.eye(2)
p = np.array([-2.0, -2.0])
G = np.array([[1.0, 1.0], [-1.0, 0.0], [0.0, -1.0]])
h = np.array([1.0, 0.0, 0.0])

# 1. Forward Primal-Dual Interior Point QP Solve
optnet = OptNet(max_iter=50, tol=1e-6)
x_opt = optnet.forward(Q=Q, p=p, G=G, h=h)
print(f"OptNet Constrained Optimal Solution x*: {x_opt}")

# 2. Backward Pass via KKT Implicit Linear Differentiation
grad_loss = x_opt - np.array([1.0, 0.0])
grads = optnet.backward(grad_loss)
print(f"OptNet Analytical Gradients: dQ shape={grads['dQ'].shape}, dp shape={grads['dp'].shape}")

Parameter Breakdown: OptNet

Parameter Name	Data Type	Default Value	Description / Purpose
max_iter	int	50	Maximum primal-dual interior point Newton iterations.
tol	float	1e-6	Primal-dual duality gap convergence tolerance threshold.
eps_reg	float	1e-7	Diagonal Tikhonov regularization for positive definiteness.

---

26.2 Debiased Sinkhorn Divergence & Optimal Transport Metric (SinkhornDivergence)

Eliminates entropic blur and positive bias from regularized optimal transport, guaranteeing metric positive definiteness $S_\epsilon(\alpha, \beta) \ge 0$ and identity of indiscernibles $S_\epsilon(\alpha, \alpha) = 0$: $$S_\epsilon(\alpha, \beta) = \text{OT}\epsilon(\alpha, \beta) - \frac{1}{2} \text{OT}\epsilon(\alpha, \alpha) - \frac{1}{2} \text{OT}_\epsilon(\beta, \beta)$$

import numpy as np
from chokkhu import SinkhornDivergence

# Empirical probability point clouds
np.random.seed(42)
cloud_a = np.random.randn(20, 2)
cloud_b = np.random.randn(20, 2) + 2.0

# Compute debiased, positive-definite Sinkhorn divergence
sinkhorn_div = SinkhornDivergence(epsilon=0.1, max_iter=100)
div_ab = sinkhorn_div.compute(cloud_a, cloud_b)
div_self = sinkhorn_div(cloud_a, cloud_a)
print(f"Sinkhorn Divergence S_eps(A, B): {div_ab:.4f} | S_eps(A, A): {div_self:.6f}")

Parameter Breakdown: SinkhornDivergence

Parameter Name	Data Type	Default Value	Description / Purpose
epsilon	float	0.1	Entropic regularization temperature coefficient $\epsilon$.
max_iter	int	100	Maximum Sinkhorn fixed-point dual potential iterations.
tol	float	1e-6	Dual potential convergence tolerance threshold.

---

26.3 Support Vector Data Description (SVDD) One-Class Anomaly Detector (SupportVectorDataDescription)

Constructs the minimal enclosing hypersphere in reproducing kernel Hilbert space (RKHS) to distinguish normal operational data from outliers and novelties: $$\min_{R, a, \xi} R^2 + C \sum_{i=1}^N \xi_i \quad \text{s.t.} \quad |\phi(x_i) - a|^2 \le R^2 + \xi_i, \quad \xi_i \ge 0$$

import numpy as np
from chokkhu import SupportVectorDataDescription

# Normal training observations (inliers cloud)
X_normal = np.random.randn(60, 2) * 0.5
X_test = np.array([[0.1, -0.1], [6.0, 6.0]])

# Fit Kernel Minimum Enclosing Ball (MEB)
svdd = SupportVectorDataDescription(C=0.1, kernel="rbf", gamma=0.5)
svdd.fit(X_normal)

# Predict anomalies (0 = normal inlier, 1 = anomaly/outlier)
predictions = svdd.predict(X_test)
boundary_scores = svdd.decision_function(X_test)
print(f"SVDD Predictions: {predictions} | Signed Boundary Distances: {boundary_scores}")

Parameter Breakdown: SupportVectorDataDescription

Parameter Name	Data Type	Default Value	Description / Purpose
C	float	0.1	Outlier penalty slack trade-off parameter ($C \in (0, 1]$).
kernel	str	"rbf"	Kernel function: "rbf", "poly", or "linear".
gamma	Optional[float]	None	RBF scale parameter $\gamma = 1 / (2\sigma^2)$ (defaults to $1/d$).
degree	int	3	Polynomial kernel degree.
max_iter	int	200	Maximum SMO dual optimization iterations.

---

26.4 Spectral Graph Partitioning & Normalized Laplacians (SpectralGraphClusterer)

Partitions graphs and complex non-convex manifolds by minimizing Normalized Cut (NCut) via normalized symmetric ($L_{\text{sym}} = I - D^{-1/2} A D^{-1/2}$) and random-walk ($L_{\text{rw}} = I - D^{-1} A$) Laplacian eigen-decomposition:

import numpy as np
from chokkhu import SpectralGraphClusterer

# Clustered dataset on non-linear metric space
c1 = np.random.randn(20, 2) + np.array([-4.0, 0.0])
c2 = np.random.randn(20, 2) + np.array([4.0, 0.0])
X_graph = np.vstack([c1, c2])

# Multi-way spectral graph clustering
spectral = SpectralGraphClusterer(n_clusters=2, affinity="rbf", gamma=0.1, laplacian_type="symmetric", seed=42)
cluster_labels = spectral.fit_predict(X_graph)
ncut_val = spectral.normalized_cut(X_graph)
print(f"Spectral Cluster Labels: {cluster_labels[:5]}... | Normalized Cut (NCut): {ncut_val:.4f}")

Parameter Breakdown: SpectralGraphClusterer

Parameter Name	Data Type	Default Value	Description / Purpose
n_clusters	int	2	Number of target partition clusters $K$.
laplacian_type	str	"symmetric"	Graph Laplacian formulation: "symmetric", "random_walk", or "unnormalized".
affinity	str	"precomputed"	Affinity matrix source: "precomputed" (adjacency) or "rbf".
gamma	float	1.0	Gaussian affinity scaling parameter.
seed	int	42	Random seed for spectral assignment.

---

26.5 Differentiable Inductive Logic Programming ($\partial$ILP) & Rule Induction (DifferentiableILP)

Learns explainable first-order symbolic Horn clause rules from relational background facts and positive/negative examples via continuous immediate consequence operator $T_P$ deductions: $$v_{t+1} = v_t \oplus \bigoplus_{C \in \text{Rules}} w_C \cdot \bigotimes_{b \in \text{body}(C)} v_t(b)$$

import numpy as np
from chokkhu import DifferentiableILP

# Define symbolic domain ontology
predicates = ["parent", "ancestor"]
constants = ["alice", "bob", "charlie"]

dilp = DifferentiableILP(
    predicates=predicates,
    constants=constants,
    max_steps=2,
    learning_rate=0.1
)

# Relational background knowledge & examples
bg_facts = [("parent", "alice", "bob"), ("parent", "bob", "charlie")]
pos_examples = [("ancestor", "alice", "bob"), ("ancestor", "bob", "charlie"), ("ancestor", "alice", "charlie")]
neg_examples = [("ancestor", "charlie", "alice")]

# Gradient-based rule induction
dilp.fit(bg_facts, pos_examples, neg_examples, epochs=5)

# Query inferred confidence truth degree
truth_degree = dilp.predict_atom(bg_facts, ("ancestor", "alice", "charlie"))
print(f"Differentiable ILP Confidence for ancestor(alice, charlie): {truth_degree:.4f}")

Parameter Breakdown: DifferentiableILP

Parameter Name	Data Type	Default Value	Description / Purpose
predicates	List[str]	Required	List of domain relational predicate symbols.
constants	List[str]	Required	List of domain entities/constants.
max_steps	int	3	Number of forward continuous $T_P$ deduction steps.
learning_rate	float	0.1	Gradient descent step size for rule clause weights.
seed	int	42	Random seed for weight initialization.

---

27. Scaled Diffusion Transformers, K-FAC Natural Gradient, Kalman Filtering, Hindsight RL & Conformalized Quantile Regression

27.1 Patch-Level Scaled Diffusion Transformers with Adaptive LayerNorm (DiffusionTransformer, DiTBlock, AdaLNZero)

Implements Vision Transformer (ViT) architectures for generative diffusion modeling following Peebles & Xie (ICCV 2023), replacing convolutional UNets with visual patch tokens and Adaptive Layer Normalization (adaLN-Zero) modulation: $$\text{DiTBlock}(x) = x + \alpha_1 \cdot \text{SelfAttention}(\text{modulate}(\text{LayerNorm}(x), \gamma_1, \beta_1))$$ $$\text{Out}(x) = \text{DiTBlock}(x) + \alpha_2 \cdot \text{MLP}(\text{modulate}(\text{LayerNorm}(\text{DiTBlock}(x)), \gamma_2, \beta_2))$$

import numpy as np
from chokkhu import DiffusionTransformer

# Initialize Diffusion Transformer (DiT)
dit = DiffusionTransformer(input_size=16, in_channels=4, patch_size=2, hidden_dim=32, depth=2)

# 1. Forward noise prediction at diffusion timesteps
noisy_latents = np.random.randn(2, 4, 16, 16)
timesteps = np.array([10, 25])
predicted_noise = dit.forward(noisy_latents, timesteps)
print(f"DiT Noise Prediction Tensor Shape: {predicted_noise.shape}")

# 2. Reverse diffusion generation loop
sampled_latents = dit.sample(shape=(2, 4, 16, 16), n_steps=5)
print(f"Sampled Latent Representation Shape: {sampled_latents.shape}")

Parameter Breakdown: DiffusionTransformer

Parameter Name	Data Type	Default Value	Description / Purpose
input_size	int	16	Spatial width and height of 2D input feature grid.
in_channels	int	4	Number of input channels (e.g. latent channels).
patch_size	int	2	Spatial patch size $p \times p$.
hidden_dim	int	64	Transformer embedding dimensionality.
depth	int	4	Number of cascaded DiT blocks.
num_heads	int	4	Number of multi-head self-attention heads.

---

27.2 Kronecker-Factored Approximate Curvature (K-FAC) Optimizer (KFAC)

Scalable second-order Natural Gradient Descent optimizer (Martens & Grosse 2015) approximating the block-diagonal Fisher Information Matrix as Kronecker products of layer activation and gradient covariances: $$F_l \approx A_{l-1} \otimes S_l \implies \nabla^{\text{nat}} W_l = S_l^{-1} (\nabla_{W_l} \mathcal{L}) A_{l-1}^{-1}$$

import numpy as np
from chokkhu import KFAC

# Initialize K-FAC Second-Order Optimizer
kfac = KFAC(lr=0.01, damping=1e-3, momentum=0.9)

# Register forward activations and backward pre-activation gradients
activations = np.random.randn(10, 4)
pre_grads = np.random.randn(10, 3)
kfac.update_factors("layer1", activations, pre_grads)

# Compute preconditioned natural gradient
grad_W = np.random.randn(3, 4)
nat_grad = kfac.precondition_gradient("layer1", grad_W)
print(f"K-FAC Curvature-Preconditioned Natural Gradient: {nat_grad.shape}")

Parameter Breakdown: KFAC

Parameter Name	Data Type	Default Value	Description / Purpose
lr	float	0.01	Second-order parameter update learning rate $\eta$.
damping	float	1e-3	Tikhonov damping factor $\gamma$ ensuring positive definiteness.
momentum	float	0.9	Velocity momentum coefficient.
ema_decay	float	0.95	Exponential moving average decay for covariance statistics.

---

27.3 Non-Linear Optimal State Estimation & Sensor Fusion (ExtendedKalmanFilter, UnscentedKalmanFilter)

Bayesian state estimation for non-linear dynamical systems and trajectory tracking via Jacobian linearization (EKF) and deterministic sigma-point Unscented Transforms (UKF):

import numpy as np
from chokkhu import ExtendedKalmanFilter, UnscentedKalmanFilter

# Non-linear motion and observation functions
def f(x, u): return np.array([x[0] + 0.1 * np.cos(x[0])])
def h(x): return np.array([x[0]**2])

# 1. Extended Kalman Filter (EKF)
ekf = ExtendedKalmanFilter(dim_x=1, dim_z=1, f=f, h=h)
ekf.x = np.array([1.5])
ekf.predict()
x_ekf, P_ekf = ekf.update(np.array([2.25]))
print(f"EKF State Estimate: {x_ekf} | Covariance: {P_ekf.diagonal()}")

# 2. Unscented Kalman Filter (UKF)
ukf = UnscentedKalmanFilter(dim_x=1, dim_z=1, f=f, h=h)
ukf.x = np.array([1.5])
ukf.predict()
x_ukf, P_ukf = ukf.update(np.array([2.25]))
print(f"UKF State Estimate: {x_ukf} | Covariance: {P_ukf.diagonal()}")

Parameter Breakdown: ExtendedKalmanFilter & UnscentedKalmanFilter

Parameter Name	Data Type	Default Value	Description / Purpose
dim_x	int	Required	State vector dimensionality $L$.
dim_z	int	Required	Measurement observation dimensionality $M$.
f	Callable	Required	Non-linear state transition function $f(x, u)$.
h	Callable	Required	Non-linear measurement function $h(x)$.
alpha	float	1e-3	UKF sigma point spread parameter $\alpha$.

---

27.4 Hindsight Experience Replay & Goal-Conditioned RL (HindsightExperienceReplay, GoalConditionedDQN)

Solves sparse-reward goal-conditioned MDPs by retrospectively substituting failed goals with achieved terminal states (Andrychowicz et al. NeurIPS 2017):

import numpy as np
from chokkhu import HindsightExperienceReplay, GoalConditionedDQN

# 1. Hindsight Experience Replay Buffer
her = HindsightExperienceReplay(capacity=1000, strategy="future", k=4)
states = [np.random.randn(2) for _ in range(6)]
actions = [0, 1, 0, 1, 0]
target_goal = np.array([2.0, 2.0])
her.add_episode(states, actions, target_goal)

# 2. Goal-Conditioned Deep Q-Network
agent = GoalConditionedDQN(state_dim=2, goal_dim=2, num_actions=4, hidden_dim=16)
loss = agent.train_step(her, batch_size=4)
print(f"HER Synthesized Transitions: {len(her)} | Bellman Loss: {loss:.4f}")

Parameter Breakdown: HindsightExperienceReplay & GoalConditionedDQN

Parameter Name	Data Type	Default Value	Description / Purpose
strategy	str	"future"	Goal replay strategy: "future", "final", or "random".
k	int	4	Number of synthesized hindsight goals per original transition.
state_dim	int	Required	Observation space dimensionality.
goal_dim	int	Required	Target goal space dimensionality.

---

27.5 Conformalized Quantile Regression with Exact Finite-Sample Validity (ConformalizedQuantileRegression, QuantileRegressor)

Provides distribution-free prediction intervals with mathematically guaranteed finite-sample coverage $P(Y \in \hat{C}(X)) \ge 1 - \alpha$ via pinball loss regression and split conformal calibration (Romano et al. 2019): $$\hat{C}(x) = [\hat{q}{\alpha/2}(x) - \hat{Q}{1-\alpha}(E), ; \hat{q}{1-\alpha/2}(x) + \hat{Q}{1-\alpha}(E)]$$

import numpy as np
from chokkhu import ConformalizedQuantileRegression

# Synthetic heteroscedastic dataset
X_tr = np.random.uniform(-2, 2, size=(80, 1))
y_tr = 3.0 * X_tr + np.random.randn(80, 1) * 0.5
X_cal = np.random.uniform(-2, 2, size=(40, 1))
y_cal = 3.0 * X_cal + np.random.randn(40, 1) * 0.5
X_te = np.random.uniform(-2, 2, size=(20, 1))
y_te = 3.0 * X_te + np.random.randn(20, 1) * 0.5

# Fit, Calibrate and Predict Intervals
cqr = ConformalizedQuantileRegression(alpha=0.1) # 90% Target Coverage
cqr.fit(X_tr, y_tr).calibrate(X_cal, y_cal)
low_bound, high_bound = cqr.predict_interval(X_te)
metrics = cqr.evaluate_coverage(X_te, y_te)
print(f"CQR Empirical Coverage: {metrics['empirical_coverage']:.1%} (Target: {metrics['target_coverage']:.1%})")

Parameter Breakdown: ConformalizedQuantileRegression

Parameter Name	Data Type	Default Value	Description / Purpose
alpha	float	0.1	Target significance error rate ($1-\alpha$ confidence level).
max_iter	int	100	Maximum iterations for quantile regression solvers.

---

28. SciML Fourier Neural Operators, Instrumental Variables, Learning-to-Rank, Score Matching & TDA Vectorization (Milestone 25)

28.1 Fourier Neural Operator for Parametric PDEs (FourierNeuralOperator2D, SpectralConv2d)

Operator learning over infinite-dimensional function spaces mapping input parameter fields to PDE solutions across arbitrary grid resolutions (Li et al. ICLR 2021): $$v_{t+1}(x) = \text{GELU}\left( W v_t(x) + \mathcal{F}^{-1}\left( R \cdot \mathcal{F}(v_t) \right)(x) \right)$$

import numpy as np
from chokkhu import FourierNeuralOperator2D, SpectralConv2d

# 1. 2D Spectral Convolution Layer
conv = SpectralConv2d(in_channels=1, out_channels=16, modes1=8, modes2=8)
x_grid = np.random.randn(2, 32, 32, 1).astype(np.float32)
h_spectral = conv.forward(x_grid)
print(f"Spectral Convolution Output Shape: {h_spectral.shape}")

# 2. Fourier Neural Operator (FNO-2D) Pipeline
fno = FourierNeuralOperator2D(
    in_channels=1,
    out_channels=1,
    modes1=8,
    modes2=8,
    hidden_dim=16,
    num_layers=2,
    include_grid=True
)
u_solution = fno.forward(x_grid)
print(f"FNO-2D Predicted PDE Solution Field: {u_solution.shape}")

Parameter Breakdown: FourierNeuralOperator2D

Parameter Name	Data Type	Default Value	Description / Purpose
in_channels	int	1	Number of physical channels in input field.
out_channels	int	1	Number of output channels in PDE solution field.
modes1	int	12	Maximum retained low-frequency Fourier modes along axis 1.
modes2	int	12	Maximum retained low-frequency Fourier modes along axis 2.
hidden_dim	int	32	Latent channel dimension of lifted spectral representation.
num_layers	int	4	Number of stacked Fourier spectral convolution blocks.
include_grid	bool	True	Whether to concatenate normalized $(x, y) \in [0, 1]^2$ spatial grid.

---

28.2 Instrumental Variables Regression: 2SLS & IV-GMM (TwoStageLeastSquares, InstrumentalGMM)

Resolves endogeneity, omitted variable bias, and measurement errors in causal econometric modeling (Angrist & Imbens Nobel Prize 2021, Hansen 1982): $$\hat{\beta}_{\text{2SLS}} = (X^T P_Z X)^{-1} X^T P_Z Y, \quad P_Z = Z (Z^T Z)^{-1} Z^T$$

import numpy as np
from chokkhu import TwoStageLeastSquares, InstrumentalGMM

# Simulated Endogenous Data with Confounder u
z1, z2 = np.random.randn(200), np.random.randn(200)
Z = np.column_stack([z1, z2]) # Valid exogenous instruments
u = np.random.randn(200)       # Unobserved confounder
x = 1.2 * z1 + 0.9 * z2 + 1.5 * u + np.random.randn(200) * 0.2 # Endogenous regressor
y = 2.0 + 3.0 * x + 2.0 * u + np.random.randn(200) * 0.5       # True causal effect beta = 3.0

# 1. Two-Stage Least Squares (2SLS) with Robust Standard Errors
iv_model = TwoStageLeastSquares(fit_intercept=True, robust=True)
iv_model.fit(x, y, Z)
print(f"2SLS Causal Est: beta={iv_model.coef_[0]:.3f} (SE: {iv_model.se_[0]:.3f})")
print(f"First-Stage F-Statistic: {iv_model.first_stage_f_stat_:.2f} | Sargan p-val: {iv_model.sargan_p_val_:.4f}")

# 2. Generalized Method of Moments (IV-GMM) with Hansen's J-Test
gmm_model = InstrumentalGMM(fit_intercept=True)
gmm_model.fit(x, y, Z)
print(f"GMM Causal Est: beta={gmm_model.coef_[0]:.3f} | Hansen J-stat: {gmm_model.j_stat_:.3f}")

Parameter Breakdown: TwoStageLeastSquares & InstrumentalGMM

Parameter Name	Data Type	Default Value	Description / Purpose
fit_intercept	bool	True	Whether to include intercept in first and second stage models.
robust	bool	True	Whether to compute White/Huber heteroskedasticity-robust covariance.

---

28.3 Learning-to-Rank: LambdaMART & ListNet (LambdaMART, ListNet)

Industry-standard gradient-boosted trees directly optimizing non-smooth listwise rank metrics ($\Delta\text{NDCG}@k$) and Plackett-Luce top-1 cross-entropy ranking (Burges 2010, Cao et al. 2007): $$\lambda_{ij} = \frac{-\sigma}{1 + e^{\sigma (s_i - s_j)}} |\Delta \text{NDCG}_{ij}|$$

import numpy as np
from chokkhu import LambdaMART, ListNet, ndcg_at_k

# 4 Queries with 6 documents each
query_ids = np.repeat(np.arange(4), 6)
y_relevance = np.random.choice([0, 1, 2, 3], size=24)
X_features = y_relevance[:, None] * 1.5 + np.random.randn(24, 4) * 0.5

# 1. LambdaMART Tree Ensemble
lmart = LambdaMART(n_estimators=15, learning_rate=0.1, max_depth=3, ndcg_k=6)
lmart.fit(X_features, y_relevance, query_ids)
pred_scores = lmart.predict(X_features)
print(f"LambdaMART Predictions Shape: {pred_scores.shape}")

# 2. ListNet Top-1 Cross-Entropy Ranking
listnet = ListNet(lr=0.05, epochs=50)
listnet.fit(X_features, y_relevance, query_ids)
listnet_scores = listnet.predict(X_features)
print(f"ListNet Predictions Shape: {listnet_scores.shape}")

Parameter Breakdown: LambdaMART

Parameter Name	Data Type	Default Value	Description / Purpose
n_estimators	int	20	Number of boosting regression trees.
learning_rate	float	0.1	Step size shrinkage parameter.
max_depth	int	3	Maximum tree depth per weak learner.
sigma	float	1.0	Sigmoid pairwise temperature.
ndcg_k	int	10	Truncation rank $k$ for $\Delta\text{NDCG}@k$ calculations.

---

28.4 Score-Based Energy Modeling & Annealed Langevin Dynamics (ScoreMatchingEBM, AnnealedLangevinDynamics)

Directly models unnormalized distributions via score matching and generates synthetic samples across geometric noise schedules (Hyvärinen 2005, Song & Ermon NeurIPS 2019): $$\mathcal{L}{\text{DSM}}(\theta) = \frac{1}{2} \mathbb{E}{x, \tilde{x}} \left[ \left| s_\theta(\tilde{x}) + \frac{\tilde{x} - x}{\sigma^2} \right|_2^2 \right]$$

import numpy as np
from chokkhu import ScoreMatchingEBM, AnnealedLangevinDynamics

# 2D Point Cloud Distribution
X_points = np.random.randn(100, 2) * 0.5

# 1. Train Score-Based EBM with Denoising Score Matching (DSM)
ebm = ScoreMatchingEBM(input_dim=2, hidden_dim=32, method="dsm", sigma=0.1, lr=1e-2)
ebm.fit(X_points, epochs=20, batch_size=25)

# Evaluate score gradient at test point
test_score = ebm.score(np.array([1.0, 1.0]))
print(f"Score Vector at (1.0, 1.0): {test_score}")

# 2. Annealed Langevin Dynamics Sampling
sampler = AnnealedLangevinDynamics(ebm, n_steps_per_sigma=10, n_sigmas=5, step_lr=2e-5)
gen_samples = sampler.sample(n_samples=8)
print(f"Generated Synthetic Samples Shape: {gen_samples.shape}")

Parameter Breakdown: ScoreMatchingEBM & AnnealedLangevinDynamics

Parameter Name	Data Type	Default Value	Description / Purpose
input_dim	int	Required	Dimensionality of input feature space.
method	str	"dsm"	Training loss: "dsm" (Denoising) or "ssm" (Sliced).
sigma	float	0.1	Noise level for Denoising Score Matching perturbation.
n_steps_per_sigma	int	20	Number of Langevin MCMC steps per noise level.
n_sigmas	int	10	Number of geometric noise scales.

---

28.5 Topological Data Analysis Vectorization: Landscapes & Images (PersistenceLandscape, PersistenceImage)

Transforms persistent homology birth-death diagrams into stable metric feature representations for machine learning pipelines (Bubenik 2015, Adams et al. JMLR 2017): $$\lambda_k(t) = k\text{-max}_{i=1}^n \max(0, \min(t - b_i, d_i - t)), \quad \rho(x, y) = \sum_i w(b_i, p_i) \mathcal{N}\left( (x, y); (b_i, p_i), \sigma^2 I \right)$$

import numpy as np
from chokkhu import PersistenceLandscape, PersistenceImage

# Persistence Diagram with (birth, death) pairs
diagram = np.array([
    [0.1, 0.9],
    [0.2, 0.7],
    [0.3, 0.6],
])

# 1. Persistence Landscapes Functional Vectorizer
pl = PersistenceLandscape(n_landscapes=3, n_bins=50, t_min=0.0, t_max=1.0)
landscape_vector = pl.transform_single(diagram)
print(f"Persistence Landscape Shape: {landscape_vector.shape} (3 layers x 50 bins)")

# 2. Persistence Image 2D Density Surface
pi = PersistenceImage(pixels=(20, 20), sigma=0.1, weight_power=1.0)
img_surface = pi.transform_single(diagram)
print(f"Persistence Image Pixel Grid: {img_surface.shape} (20 x 20)")

Parameter Breakdown: PersistenceLandscape & PersistenceImage

Parameter Name	Data Type	Default Value	Description / Purpose
n_landscapes	int	5	Number of landscape layers ($k=1, \dots, K$).
n_bins	int	100	Number of 1D discretization grid points.
pixels	tuple[int, int]	(20, 20)	2D resolution $(N_x, N_y)$ of the persistence image grid.
sigma	float	0.1	Gaussian smoothing kernel standard deviation.
weight_power	float	1.0	Persistence ramp weighting exponent $r$ for $w(b, p) = (p / p_{\max})^r$.

29. Advanced State-Space Filtering, Causal ML, Sparse Scalable GPs, Self-Supervised Graph Contrastive Learning & Program Synthesis

29.1 Particle Filtering & Sequential Monte Carlo (ParticleFilter, RaoBlackwellizedParticleFilter)

Enables non-linear, non-Gaussian state estimation via Sequential Importance Resampling (SIR) and Rao-Blackwellized marginalization (Doucet et al. 2000, Gordon et al. 1993): $$w_t^{(i)} \propto w_{t-1}^{(i)} \frac{p(y_t | x_t^{(i)}) p(x_t^{(i)} | x_{t-1}^{(i)})}{q(x_t^{(i)} | x_{t-1}^{(i)}, y_t)}, \quad N_{\text{eff}} = \frac{1}{\sum_{i=1}^{N_p} (w_t^{(i)})^2}$$

import numpy as np
from chokkhu import ParticleFilter, RaoBlackwellizedParticleFilter

# Non-linear dynamic transition and observation models
def f_trans(x, u=None):
    return 0.5 * x + 25.0 * x / (1.0 + x**2)

def h_obs(x):
    return (x**2) / 20.0

# 1. SIR Particle Filter with Systematic Resampling
pf = ParticleFilter(
    dim_x=1,
    dim_z=1,
    n_particles=200,
    f=f_trans,
    h=h_obs,
    Q=np.array([[1.0]]),
    R=np.array([[1.0]]),
    resample_method="systematic",
    resample_threshold=0.5,
)

# Filter across sequential observations
observations = [0.5, 1.2, 0.8, 2.1, 1.9]
state_estimates = []
for y in observations:
    pf.predict()
    mean, cov = pf.update(np.array([y]))
    state_estimates.append(mean)

print(f"Final State Estimate: {state_estimates[-1][0]:.4f}, Particles: {pf.particles.shape}")

# 2. Rao-Blackwellized Particle Filter (Marginalized State Partitioning)
rbpf = RaoBlackwellizedParticleFilter(
    dim_nl=1,
    dim_lin=1,
    dim_z=1,
    n_particles=50,
)
u_hat, x_lin_hat = rbpf.step(np.array([1.5]))
print(f"RBPF Nonlinear State: {u_hat.ravel()}, Linear State: {x_lin_hat.ravel()}")

Parameter Breakdown: ParticleFilter & RaoBlackwellizedParticleFilter

Parameter Name	Data Type	Default Value	Description / Purpose
dim_x / dim_nl	int	Required	Dimensionality of continuous hidden state space.
dim_z	int	Required	Dimensionality of sequential measurement vectors.
n_particles	int	500 / 100	Number of Monte Carlo particle trajectories $N_p$.
resample_method	str	"systematic"	Resampling scheme: "systematic", "stratified", "residual", "multinomial".
resample_threshold	float	0.5	Threshold fraction $N_{\text{eff}} / N_p$ below which resampling is triggered.

---

29.2 Double / Debiased Machine Learning & Residualized CATE (DoubleMLPLR, RLearner)

Delivers Neyman-orthogonal $\sqrt{N}$-consistent causal effect estimation with high-dimensional nuisance parameterization and Robinson transformation (Chernozhukov et al. Econometrica 2018, Nie & Wager Biometrika 2021): $$\hat{\theta}0 = \frac{\sum{i=1}^N (T_i - \hat{m}(X_i)) (Y_i - \hat{g}(X_i))}{\sum_{i=1}^N (T_i - \hat{m}(X_i))^2}, \quad \hat{\tau}(X) = \arg\min_\tau \frac{1}{N}\sum_{i=1}^N \left( (Y_i - \hat{m}(X_i)) - (T_i - \hat{e}(X_i))\tau(X_i) \right)^2$$

import numpy as np
from chokkhu import DoubleMLPLR, RLearner

# Synthetic observational dataset with confounding
np.random.seed(42)
N = 200
X = np.random.randn(N, 5)
true_ate = 2.5
T = (X[:, 0] + np.random.randn(N) > 0).astype(float)
Y = true_ate * T + 1.2 * X[:, 0] + 0.8 * X[:, 1] + np.random.randn(N) * 0.5

# 1. Double Machine Learning Partially Linear Regression (DML-PLR)
dml = DoubleMLPLR(n_folds=5, alpha=1.0)
dml.fit(X, Y, T)
stats_dict = dml.summary()
print(f"DML Estimated ATE: {dml.coef_:.4f} (True: {true_ate:.1f}), p-value: {dml.p_val_:.4e}")

# 2. R-Learner for Heterogeneous Treatment Effects (CATE)
r_learner = RLearner(alpha=1.0)
r_learner.fit(X, Y, T)
cate_preds = r_learner.predict_cate(X[:5])
print(f"Predicted Individual Treatment Effects (CATE): {np.round(cate_preds, 3)}")

Parameter Breakdown: DoubleMLPLR & RLearner

Parameter Name	Data Type	Default Value	Description / Purpose
n_folds	int	5	Cross-fitting splits ($K$-fold) for Neyman orthogonality.
alpha	float	1.0	$L_2$ Tikhonov regularization strength for nuisance and second-stage models.
seed	int	42	Random seed for fold partitioning.

---

29.3 Sparse & Variational Gaussian Process Regression (SparseGaussianProcessRegression, VariationalSparseGP)

Scales Nonparametric Bayesian Inference from $\mathcal{O}(N^3)$ to $\mathcal{O}(N M^2)$ through FITC pseudo-inputs and Titsias Variational Free Energy optimization (Snelson & Ghahramani 2006, Titsias AISTATS 2009): $$\mathcal{L}{\text{VFE}}(Z, \theta) = \log \mathcal{N}\left(y; 0, Q{NN} + \sigma_n^2 I\right) - \frac{1}{2\sigma_n^2} \text{Tr}\left(K_{NN} - Q_{NN}\right), \quad Q_{NN} = K_{NM} K_{MM}^{-1} K_{MN}$$

import numpy as np
from chokkhu import SparseGaussianProcessRegression, VariationalSparseGP

# 1D Non-linear function with noise
X = np.linspace(-3, 3, 100).reshape(-1, 1)
y = np.sin(X).ravel() + 0.1 * np.random.randn(100)
X_test = np.linspace(-3.5, 3.5, 30).reshape(-1, 1)

# 1. FITC Sparse Gaussian Process with Inducing Points (M=10 << N=100)
sgp = SparseGaussianProcessRegression(
    n_inducing=10,
    length_scale=1.0,
    variance=1.0,
    noise_variance=0.01,
)
sgp.fit(X, y)
mu_fitc, std_fitc = sgp.predict(X_test, return_std=True)
print(f"FITC Mean Shape: {mu_fitc.shape}, Posterior Std Uncertainty: {std_fitc.shape}")

# 2. Variational Sparse GP (Titsias Free Energy Bound)
vsgp = VariationalSparseGP(
    n_inducing=10,
    length_scale=1.2,
    variance=1.0,
    noise_variance=0.02,
)
vsgp.fit(X, y)
mu_var, std_var = vsgp.predict(X_test, return_std=True)
print(f"Variational Sparse GP Mean Shape: {mu_var.shape}, Std Shape: {std_var.shape}")

Parameter Breakdown: SparseGaussianProcessRegression & VariationalSparseGP

Parameter Name	Data Type	Default Value	Description / Purpose
n_inducing	int	20	Number of inducing / pseudo-input points $M \ll N$.
length_scale	float	1.0	RBF kernel correlation distance hyperparameter $\ell$.
variance	float	1.0	Output amplitude variance $\sigma_f^2$.
noise_variance	float	1e-2	Observational likelihood variance $\sigma_n^2$.

---

29.4 Self-Supervised Graph Contrastive Learning (GraphCL, GRACE)

Learns topology-invariant node representations via InfoNCE / NT-Xent mutual information maximization across stochastic edge-dropping and feature-masking views (You et al. NeurIPS 2020, Zhu et al. WWW 2020): $$\mathcal{L}{\text{InfoNCE}}(u, v) = -\log \frac{\exp(\text{sim}(z_u, z_v) / \tau)}{\sum{w} \exp(\text{sim}(z_u, z_w) / \tau)}$$

import numpy as np
from chokkhu import GraphCL, GRACE

# Synthetic graph adjacency matrix and node attributes
np.random.seed(42)
adj = (np.random.rand(8, 8) > 0.6).astype(float)
np.fill_diagonal(adj, 0)
features = np.random.randn(8, 8)

# 1. GraphCL: Graph Contrastive Learning
gcl = GraphCL(
    in_dim=8,
    hidden_dim=16,
    out_dim=8,
    temperature=0.2,
    drop_edge_rate=0.2,
    mask_feat_rate=0.2,
    lr=0.01,
)
gcl.fit(features, adj, epochs=15)
node_embs = gcl.transform(features, adj)
print(f"GraphCL Node Embedding Matrix Shape: {node_embs.shape} (8 nodes x 8 features)")

# 2. GRACE: Deep Graph Contrastive Representation Learning
grace = GRACE(
    in_dim=8,
    hidden_dim=16,
    out_dim=8,
    temperature=0.4,
    drop_edge_rate=0.2,
    mask_feat_rate=0.2,
    lr=0.01,
)
grace.fit(features, adj, epochs=15)
grace_embs = grace.transform(features, adj)
print(f"GRACE Node Embeddings Shape: {grace_embs.shape} (8 nodes x 8 features)")

Parameter Breakdown: GraphCL & GRACE

Parameter Name	Data Type	Default Value	Description / Purpose
in_dim	int	Required	Input node attribute dimension.
hidden_dim	int	32	GNN message-passing representation dimension.
out_dim	int	16	Non-linear projection / output embedding dimension.
temperature	float	0.5	InfoNCE Softmax temperature scaling parameter $\tau$.
drop_edge_rate	float	0.2	Graph augmentation probability for random edge removal.
mask_feat_rate	float	0.2	Graph augmentation probability for random node feature zeroing.

---

29.5 Inductive Neuro-Symbolic Program Synthesizer (ProgramSynthesizer, DSLGrammar)

Synthesizes verifiable, deterministic DSL programs from input-output example pairs via bottom-up search and observational equivalence pruning (Gulwani CACM 2011, Ellis et al. 2021): $$P^* = \arg\min_{P \in \mathcal{L}_{\text{DSL}}} \text{Size}(P) \quad \text{s.t.} \quad \forall i \in {1,\dots,N}: \llbracket P \rrbracket(x_i) = y_i$$

import numpy as np
from chokkhu import ProgramSynthesizer

# Synthesize executable program from input-output examples: f(x) = (x + 1)^2
examples = [
    (2, 9),
    (3, 16),
    (4, 25),
    (5, 36),
]

synthesizer = ProgramSynthesizer(max_depth=3, constants=[1])
synthesizer.fit(examples)

print(f"Synthesized Code: {synthesizer.synthesized_code}")
test_out = synthesizer.predict(6)
print(f"Predicted Evaluation on x=6: {test_out}")

Parameter Breakdown: ProgramSynthesizer & DSLGrammar

Parameter Name	Data Type	Default Value	Description / Purpose
max_depth	int	3	Maximum AST depth for bottom-up enumerative search.
constants	List[Any]	[0, 1, 2, 3]	Available constant literals in the primitive pool.

---

License & Citation

Distributed under the MIT License. See LICENSE for details.

@software{chokkhu2026,
  author = {tamimystic and Inception BD Team},
  title = {Chokkhu: Sovereign Zero-Dependency Machine Learning and LLM Ecosystem},
  year = {2026},
  publisher = {GitHub},
  url = {https://github.com/tamimystic/chokkhu}
}