Create README.md

README.md

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+---
+tags:
+- markov-chains
+- complex-analysis
+- non-euclidean-geometry
+- classics-revival
+- experimental
+license: apache-2.0
+library_name: pytorch
+---
+
+# Möbius Markov Chain - The Classics Revival
+
+**Non-Euclidean Probabilistic Systems with Dynamic Geometry**
+
+**Experimental Research Code** - Functional but unoptimized, expect rough edges
+
+## What Is This?
+
+Möbius Markov Chain operates Markov processes in complex space with dynamically warped geometry via Möbius transformations. The state space itself evolves based on current states, creating probabilistic dynamics that adapt their geometric structure during evolution.
+
+**Core Innovation**: Markov transition probabilities computed in continuously warped complex space, where the geometry itself learns to optimize transition dynamics.
+
+## Architecture Highlights
+
+- **Complex State Space**: States positioned in the complex plane
+- **Dynamic Möbius Transformations**: Learnable conformal mappings f(z) = (az+b)/(cz+d)
+- **Geometric Transition Matrices**: Distances computed in transformed space
+- **State-Dependent Warping**: Geometry evolves based on current state distribution
+- **Conformal Invariance**: Preserves angles while warping distances
+- **Distance Kernels**: Multiple kernel options for probability computation
+
+## Quick Start
+```python
+from mobius_markov import MobiusMarkovSystem
+
+# Create non-Euclidean Markov system
+system = MobiusMarkovSystem(
+    num_states=8,
+    state_embedding_dim=32,
+    evolution_steps=5
+)
+
+# Initialize state distribution
+initial_state = torch.zeros(batch_size, num_states)
+initial_state[:, 0] = 1.0  # Start in state 0
+
+# Evolve through warped space
+output = system(initial_state, return_full_trajectory=True)
+
+# Generate sequence predictions
+sequence = system.predict_sequence(initial_state, sequence_length=10)
+```
+
+## Current Status
+- **Working**: Möbius transformations, complex state dynamics, geometric evolution, sequence prediction
+- **Rough Edges**: No validation on standard Markov tasks, numerical stability needs improvement
+- **Still Missing**: Advanced conformal mappings, multi-scale geometry, distributed state spaces
+- **Performance**: Novel behavior on toy problems, needs real-world validation
+- **Memory Usage**: Moderate, dominated by complex number operations
+- **Speed**: Reasonable for small state spaces, optimization needed for large systems
+
+## Mathematical Foundation
+Möbius transformations are defined as:
+```
+f(z) = (az + b)/(cz + d)
+```
+where a,b,c,d are learnable complex parameters with ad - bc ≠ 0.
+
+State positions in complex space determine transition probabilities:
+```
+P_ij = kernel(d_transformed(z_i, z_j))
+```
+
+Distance kernels include:
+- **Gaussian**: `P ∝ exp(-d²/2σ²)`
+- **Inverse**: `P ∝ 1/d^α`
+- **Linear**: `P ∝ max(0, 1-d)`
+
+The geometry evolves according to:
+```
+∂θ/∂t = η × state_embedding_evolution(current_distribution)
+```
+
+where θ represents the Möbius transformation parameters.
+
+## Research Applications
+- **Non-Euclidean machine learning**
+- **Adaptive probabilistic models**
+- **Complex systems with geometric constraints**
+- **Hyperbolic neural networks**
+- **Conformal prediction systems**
+
+## Installation
+```bash
+pip install torch numpy matplotlib
+# Download mobius_markov.py from this repo
+```
+
+## The Classics Revival Collection
+
+Möbius Markov Chain is part of a larger exploration of foundational algorithms enhanced with modern neural techniques:
+
+-  Evolutionary Turing Machine
+-  Hebbian Bloom Filter
+-  Hopfield Decision Graph
+-  Liquid Bayes Chain
+-  Liquid State Space Model
+- **Möbius Markov Chain** ← You are here
+-  Memory Forest
+
+## Citation
+```bibtex
+@misc{mobiusmarkov2025,
+  title={Möbius Markov Chain: Non-Euclidean Probabilistic Systems},
+  author={Jae Parker 𓅸 1990two},
+  year={2025},
+  note={Part of The Classics Revival Collection}
+}
+```