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.gitignore

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+# Python
+__pycache__/
+*.py[cod]
+*$py.class
+*.so
+.Python
+build/
+develop-eggs/
+dist/
+downloads/
+eggs/
+.eggs/
+lib/
+lib64/
+parts/
+sdist/
+var/
+wheels/
+*.egg-info/
+.installed.cfg
+*.egg
+MANIFEST
+
+# PyInstaller
+*.manifest
+*.spec
+
+# Installer logs
+pip-log.txt
+pip-delete-this-directory.txt
+
+# Unit test / coverage reports
+htmlcov/
+.tox/
+.coverage
+.coverage.*
+.cache
+nosetests.xml
+coverage.xml
+*.cover
+.hypothesis/
+.pytest_cache/
+
+# Jupyter Notebook
+.ipynb_checkpoints
+
+# pyenv
+.python-version
+
+# Environments
+.env
+.venv
+env/
+venv/
+ENV/
+env.bak/
+venv.bak/
+
+# IDE
+.vscode/
+.idea/
+*.swp
+*.swo
+*~
+
+# OS
+.DS_Store
+.DS_Store?
+._*
+.Spotlight-V100
+.Trashes
+ehthumbs.db
+Thumbs.db
+
+# Temporary files
+*.tmp
+*.temp
+/tmp/

README.md

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+# Not-trained-Neural-Networks-Notes
+
+A comprehensive collection of notes, implementations, and examples of neural networks that don't rely on traditional gradient-based training methods.
+
+## Contents
+
+- [Algebraic Neural Networks](#algebraic-neural-networks)
+- [Uncomputable Neural Networks](#uncomputable-neural-networks)
+- [Theory and Mathematical Foundations](#theory-and-mathematical-foundations)
+- [Implementations](#implementations)
+- [Examples and Use Cases](#examples-and-use-cases)
+
+## Algebraic Neural Networks
+
+Algebraic Neural Networks (ANNs) represent a paradigm shift from traditional neural networks by utilizing algebraic structures and operations instead of gradient-based optimization. These networks leverage:
+
+- **Algebraic Group Theory**: Using group operations for network transformations
+- **Polynomial Algebras**: Networks based on polynomial computations
+- **Geometric Algebra**: Incorporating geometric algebraic structures
+- **Fixed Algebraic Transformations**: Pre-defined algebraic operations
+
+### Key Features
+
+1. **No Training Required**: Networks are constructed using algebraic principles
+2. **Deterministic Behavior**: Outputs are fully determined by algebraic rules
+3. **Mathematical Rigor**: Based on well-established algebraic foundations
+4. **Interpretability**: Clear mathematical interpretation of operations
+
+## Uncomputable Neural Networks
+
+Uncomputable Neural Networks extend the paradigm of non-trained networks by incorporating theoretical concepts from computability theory. These networks explore computational boundaries by simulating uncomputable functions and operations:
+
+- **Halting Oracle Layers**: Simulate access to halting oracles for program termination decisions
+- **Kolmogorov Complexity Layers**: Approximate uncomputable complexity measures using compression heuristics  
+- **Busy Beaver Layers**: Utilize the uncomputable Busy Beaver function values and approximations
+- **Non-Recursive Layers**: Operate on computably enumerable but non-computable sets
+
+### Key Features
+
+1. **Theoretical Foundations**: Based on computability theory and hypercomputation concepts
+2. **Bounded Approximations**: Practical implementations of theoretically uncomputable functions
+3. **Deterministic Simulation**: Consistent behavior through fixed-seed randomness and heuristics
+4. **Educational Value**: Demonstrates limits and possibilities of computation
+
+## Getting Started
+
+```bash
+git clone https://github.com/ewdlop/Not-trained-Neural-Networks-Notes.git
+cd Not-trained-Neural-Networks-Notes
+
+# Install dependencies
+pip install numpy matplotlib
+
+# Quick demo
+python demo.py
+
+# Run main implementation
+python algebraic_neural_network.py
+
+# Run comprehensive tests
+python test_comprehensive.py
+```
+
+### Quick Demo
+```bash
+python demo.py
+```
+This runs a simple demonstration showing how algebraic neural networks process data without any training.
+
+### Examples
+```bash
+# Polynomial-based networks
+python examples/polynomial_network.py
+
+# Group theory networks
+python examples/group_theory_network.py
+
+# Geometric algebra networks
+python examples/geometric_algebra_network.py
+
+# Uncomputable neural networks
+python examples/uncomputable_networks.py
+```
+
+## Structure
+
+```
+├── README.md                          # This file
+├── demo.py                            # Quick demonstration script
+├── algebraic_neural_network.py        # Main implementation
+├── test_comprehensive.py              # Test suite
+├── theory/                            # Theoretical background
+│   ├── algebraic_foundations.md       # Mathematical foundations
+│   ├── uncomputable_networks.md       # Uncomputable neural networks theory
+│   └── examples.md                    # Worked examples
+└── examples/                          # Practical examples
+    ├── polynomial_network.py          # Polynomial-based network
+    ├── group_theory_network.py        # Group theory implementation
+    ├── geometric_algebra_network.py   # Geometric algebra network
+    └── uncomputable_networks.py       # Uncomputable neural networks
+```
+
+## Testing
+
+Run the comprehensive test suite to verify all components:
+
+```bash
+python test_comprehensive.py
+```
+
+This tests:
+- Basic functionality of all layer types (algebraic and uncomputable)
+- Network composition and data flow
+- Deterministic behavior (same input → same output)
+- Mathematical properties of algebraic operations
+- Uncomputable layer approximations and bounds
+- Edge cases and boundary conditions

algebraic_neural_network.py

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+"""
+Algebraic Neural Network Implementation
+
+This module implements various types of algebraic neural networks that don't require
+traditional gradient-based training. Instead, they use algebraic structures and
+operations to process information.
+
+Author: Algebraic Neural Network Research
+"""
+
+import numpy as np
+from typing import List, Callable, Union
+import math
+
+
+class AlgebraicLayer:
+    """
+    Base class for algebraic layers that use mathematical operations
+    instead of learned weights.
+    """
+    
+    def __init__(self, input_size: int, output_size: int, operation: str = "polynomial"):
+        self.input_size = input_size
+        self.output_size = output_size
+        self.operation = operation
+        
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Apply algebraic transformation to input."""
+        raise NotImplementedError("Subclasses must implement forward method")
+
+
+class PolynomialLayer(AlgebraicLayer):
+    """
+    Layer that applies polynomial transformations without learned weights.
+    Uses fixed polynomial coefficients based on algebraic principles.
+    """
+    
+    def __init__(self, input_size: int, output_size: int, degree: int = 2):
+        super().__init__(input_size, output_size, "polynomial")
+        self.degree = degree
+        # Generate coefficients using algebraic sequences (e.g., Fibonacci-based)
+        self.coefficients = self._generate_algebraic_coefficients()
+        
+    def _generate_algebraic_coefficients(self) -> np.ndarray:
+        """Generate polynomial coefficients using algebraic sequences."""
+        # Use golden ratio and algebraic numbers for coefficients
+        phi = (1 + math.sqrt(5)) / 2  # Golden ratio
+        coeffs = []
+        
+        for i in range(self.output_size):
+            for j in range(self.input_size):
+                # Generate coefficients using algebraic properties
+                coeff = (phi ** (i + 1)) / (j + 1) if j < self.input_size else 1.0
+                coeffs.append(coeff)
+                
+        return np.array(coeffs).reshape(self.output_size, self.input_size)
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Apply polynomial transformation."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        result = np.zeros((x.shape[0], self.output_size))
+        
+        for i in range(self.output_size):
+            # Apply polynomial of specified degree
+            poly_result = np.zeros(x.shape[0])
+            for degree in range(1, self.degree + 1):
+                poly_term = np.power(x, degree) @ self.coefficients[i]
+                poly_result += poly_term / math.factorial(degree)
+            result[:, i] = poly_result
+            
+        return result
+
+
+class GroupTheoryLayer(AlgebraicLayer):
+    """
+    Layer based on group theory operations, particularly using cyclic groups
+    and group actions for transformations.
+    """
+    
+    def __init__(self, input_size: int, output_size: int, group_order: int = 8):
+        super().__init__(input_size, output_size, "group_theory")
+        self.group_order = group_order
+        # Generate group elements (rotations in this case)
+        self.group_elements = self._generate_cyclic_group()
+        
+    def _generate_cyclic_group(self) -> List[np.ndarray]:
+        """Generate cyclic group elements as rotation matrices."""
+        elements = []
+        for k in range(self.group_order):
+            angle = 2 * math.pi * k / self.group_order
+            # For 2D case, use rotation matrices
+            if self.input_size == 2:
+                rotation = np.array([
+                    [math.cos(angle), -math.sin(angle)],
+                    [math.sin(angle), math.cos(angle)]
+                ])
+                elements.append(rotation)
+            else:
+                # For higher dimensions, use generalized rotations
+                rotation = np.eye(self.input_size)
+                if self.input_size >= 2:
+                    rotation[0, 0] = math.cos(angle)
+                    rotation[0, 1] = -math.sin(angle)
+                    rotation[1, 0] = math.sin(angle)
+                    rotation[1, 1] = math.cos(angle)
+                elements.append(rotation)
+        return elements
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Apply group action to input."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        results = []
+        for i, group_element in enumerate(self.group_elements[:self.output_size]):
+            if x.shape[1] == group_element.shape[0]:
+                transformed = x @ group_element.T
+                # Take the norm as a scalar output
+                result = np.linalg.norm(transformed, axis=1)
+            else:
+                # Fallback for size mismatch
+                result = np.sum(x * (i + 1), axis=1)
+            results.append(result)
+            
+        return np.column_stack(results)
+
+
+class GeometricAlgebraLayer(AlgebraicLayer):
+    """
+    Layer using geometric algebra (Clifford algebra) operations.
+    Implements basic geometric product and algebraic operations.
+    """
+    
+    def __init__(self, input_size: int, output_size: int):
+        super().__init__(input_size, output_size, "geometric_algebra")
+        # Initialize basis vectors for geometric algebra
+        self.basis_vectors = self._generate_basis_vectors()
+        
+    def _generate_basis_vectors(self) -> List[np.ndarray]:
+        """Generate basis vectors for geometric algebra."""
+        basis = []
+        # Scalar basis
+        basis.append(np.ones(self.input_size))
+        
+        # Vector basis
+        for i in range(self.input_size):
+            e_i = np.zeros(self.input_size)
+            e_i[i] = 1.0
+            basis.append(e_i)
+            
+        # Bivector basis (for pairs)
+        for i in range(self.input_size):
+            for j in range(i + 1, self.input_size):
+                e_ij = np.zeros(self.input_size)
+                e_ij[i] = 1.0
+                e_ij[j] = 1.0
+                basis.append(e_ij)
+                
+        return basis[:self.output_size]
+    
+    def geometric_product(self, a: np.ndarray, b: np.ndarray) -> float:
+        """Compute geometric product of two vectors."""
+        # Simplified geometric product: dot product + outer product norm
+        dot_prod = np.dot(a, b)
+        # Approximate outer product contribution
+        outer_norm = np.linalg.norm(np.outer(a, b))
+        return dot_prod + outer_norm
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Apply geometric algebra transformations."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        results = []
+        for basis_vector in self.basis_vectors:
+            # Compute geometric product with basis vector
+            result = []
+            for sample in x:
+                gp = self.geometric_product(sample, basis_vector)
+                result.append(gp)
+            results.append(np.array(result))
+            
+        return np.column_stack(results)
+
+
+class HaltingOracleLayer(AlgebraicLayer):
+    """
+    Layer that simulates a halting oracle for specific computational patterns.
+    Uses heuristics and bounded computation to approximate uncomputable halting decisions.
+    """
+    
+    def __init__(self, input_size: int, output_size: int, max_iterations: int = 1000):
+        super().__init__(input_size, output_size, "halting_oracle")
+        self.max_iterations = max_iterations
+        # Fixed seed for deterministic "non-algorithmic" behavior
+        self.oracle_seed = 42
+        
+    def _simulate_halting_oracle(self, program_encoding: float) -> float:
+        """Simulate halting oracle decision for encoded program."""
+        # Use fixed-seed randomness to simulate oracle behavior
+        np.random.seed(int(abs(program_encoding * 1000)) % 2**31)
+        
+        # Simple heuristic: programs with certain patterns are more likely to halt
+        # This is a deterministic approximation of the uncomputable halting problem
+        complexity_factor = abs(program_encoding) % 1.0
+        
+        if complexity_factor < 0.1:  # Very simple programs likely halt
+            return 1.0
+        elif complexity_factor > 0.9:  # Very complex programs might not halt
+            return 0.1
+        else:
+            # Use bounded computation simulation
+            iterations = int(complexity_factor * self.max_iterations)
+            # Simulate bounded program execution
+            state = program_encoding
+            for i in range(iterations):
+                state = (state * 1.618 + 0.786) % 1.0  # Simple state evolution
+                if abs(state - 0.5) < 0.01:  # "Halting condition"
+                    return 1.0
+            return 0.3  # Uncertain/timeout case
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Apply halting oracle simulation to inputs."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        results = []
+        for i in range(self.output_size):
+            oracle_results = []
+            for sample in x:
+                # Encode input as "program" for halting oracle
+                program_encoding = np.sum(sample * (i + 1)) / len(sample)
+                halting_prob = self._simulate_halting_oracle(program_encoding)
+                oracle_results.append(halting_prob)
+            results.append(np.array(oracle_results))
+            
+        return np.column_stack(results)
+
+
+class KolmogorovComplexityLayer(AlgebraicLayer):
+    """
+    Layer that approximates Kolmogorov complexity using compression-based heuristics.
+    Provides bounded approximations of the uncomputable Kolmogorov complexity function.
+    """
+    
+    def __init__(self, input_size: int, output_size: int, precision: int = 8):
+        super().__init__(input_size, output_size, "kolmogorov_complexity")
+        self.precision = precision
+        
+    def _approximate_kolmogorov_complexity(self, data: np.ndarray) -> float:
+        """Approximate Kolmogorov complexity using compressibility heuristics."""
+        # Convert to discrete representation
+        discretized = np.round(data * (2**self.precision)).astype(int)
+        
+        # Simple compression simulation using pattern detection
+        data_str = ''.join(map(str, discretized))
+        
+        # Count repetitive patterns (simple compression heuristic)
+        unique_chars = len(set(data_str))
+        total_chars = len(data_str)
+        
+        # Estimate complexity based on entropy-like measure
+        if total_chars == 0:
+            return 0.0
+            
+        # Normalized complexity estimate
+        entropy_approx = (unique_chars / total_chars) * np.log2(unique_chars + 1)
+        
+        # Add pattern complexity (longer patterns suggest lower complexity)
+        pattern_factor = 1.0
+        for pattern_len in range(2, min(5, len(data_str))):
+            patterns = set()
+            for i in range(len(data_str) - pattern_len + 1):
+                patterns.add(data_str[i:i+pattern_len])
+            if len(patterns) < len(data_str) - pattern_len + 1:
+                pattern_factor *= 0.8  # Reduce complexity for repeated patterns
+                
+        return entropy_approx * pattern_factor
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Compute approximate Kolmogorov complexity for inputs."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        results = []
+        for i in range(self.output_size):
+            complexity_results = []
+            for sample in x:
+                # Different projections for different outputs
+                projection = sample * (i + 1) / (self.output_size)
+                complexity = self._approximate_kolmogorov_complexity(projection)
+                complexity_results.append(complexity)
+            results.append(np.array(complexity_results))
+            
+        return np.column_stack(results)
+
+
+class BusyBeaverLayer(AlgebraicLayer):
+    """
+    Layer using Busy Beaver function values and approximations.
+    The Busy Beaver function is uncomputable for general n, but known for small values.
+    """
+    
+    def __init__(self, input_size: int, output_size: int):
+        super().__init__(input_size, output_size, "busy_beaver")
+        # Known Busy Beaver values: BB(1)=1, BB(2)=4, BB(3)=6, BB(4)=13, BB(5)≥4098
+        self.known_bb_values = {1: 1, 2: 4, 3: 6, 4: 13, 5: 4098}
+        
+    def _busy_beaver_approximation(self, n: int) -> float:
+        """Get Busy Beaver value or approximation."""
+        if n <= 0:
+            return 0.0
+        elif n in self.known_bb_values:
+            return float(self.known_bb_values[n])
+        elif n <= 5:
+            return float(self.known_bb_values[5])
+        else:
+            # For n > 5, use exponential approximation
+            # This is a heuristic since BB(n) grows faster than any computable function
+            return self.known_bb_values[5] * (2.0 ** (n - 5))
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Apply Busy Beaver function to transformed inputs."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        results = []
+        for i in range(self.output_size):
+            bb_results = []
+            for sample in x:
+                # Transform input to discrete parameter for Busy Beaver function
+                # Use log scale to keep values reasonable
+                param = max(1, int(abs(np.sum(sample) * (i + 1)) % 10) + 1)
+                bb_value = self._busy_beaver_approximation(param)
+                # Normalize to prevent overflow
+                normalized_bb = np.log1p(bb_value)
+                bb_results.append(normalized_bb)
+            results.append(np.array(bb_results))
+            
+        return np.column_stack(results)
+
+
+class NonRecursiveLayer(AlgebraicLayer):
+    """
+    Layer based on non-recursive sets and functions.
+    Simulates operations on computably enumerable but non-computable sets.
+    """
+    
+    def __init__(self, input_size: int, output_size: int, enumeration_bound: int = 1000):
+        super().__init__(input_size, output_size, "non_recursive")
+        self.enumeration_bound = enumeration_bound
+        # Simulate a c.e. non-recursive set using a bounded enumeration
+        self.ce_set = self._generate_ce_set()
+        
+    def _generate_ce_set(self) -> set:
+        """Generate a computably enumerable set simulation."""
+        ce_set = set()
+        # Simulate enumeration of a c.e. set (like the set of Gödel numbers of theorems)
+        for i in range(self.enumeration_bound):
+            # Simple enumeration rule (this is computable, but simulates c.e. behavior)
+            if self._enumeration_rule(i):
+                ce_set.add(i)
+        return ce_set
+    
+    def _enumeration_rule(self, n: int) -> bool:
+        """Rule for enumerating elements (simulates theorem enumeration)."""
+        # Simple mathematical property that creates interesting patterns
+        # Simulates: numbers that can be expressed as sum of two squares
+        for i in range(int(np.sqrt(n)) + 1):
+            remainder = n - i*i
+            if remainder >= 0 and int(np.sqrt(remainder))**2 == remainder:
+                return True
+        return False
+    
+    def _membership_oracle(self, value: float) -> float:
+        """Simulate membership oracle for non-recursive set."""
+        # Convert continuous value to discrete for set membership
+        discrete_val = int(abs(value * 1000)) % self.enumeration_bound
+        
+        if discrete_val in self.ce_set:
+            return 1.0
+        else:
+            # For values not yet enumerated, return uncertainty
+            # This simulates the non-recursive nature
+            return 0.5
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Apply non-recursive set operations to inputs."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        results = []
+        for i in range(self.output_size):
+            membership_results = []
+            for sample in x:
+                # Transform input for membership testing
+                test_value = np.sum(sample * np.arange(len(sample))) * (i + 1)
+                membership = self._membership_oracle(test_value)
+                membership_results.append(membership)
+            results.append(np.array(membership_results))
+            
+        return np.column_stack(results)
+
+
+class AlgebraicNeuralNetwork:
+    """
+    Main class for Algebraic Neural Networks that combines different
+    algebraic layers to create a complete network.
+    """
+    
+    def __init__(self):
+        self.layers = []
+        
+    def add_layer(self, layer: AlgebraicLayer):
+        """Add an algebraic layer to the network."""
+        self.layers.append(layer)
+        
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Forward pass through all algebraic layers."""
+        current_input = x
+        for layer in self.layers:
+            current_input = layer.forward(current_input)
+        return current_input
+    
+    def predict(self, x: np.ndarray) -> np.ndarray:
+        """Alias for forward pass."""
+        return self.forward(x)
+
+
+def create_sample_network() -> AlgebraicNeuralNetwork:
+    """Create a sample algebraic neural network for demonstration."""
+    network = AlgebraicNeuralNetwork()
+    
+    # Add different types of algebraic layers
+    network.add_layer(PolynomialLayer(4, 6, degree=2))
+    network.add_layer(GroupTheoryLayer(6, 4, group_order=8))
+    network.add_layer(GeometricAlgebraLayer(4, 2))
+    
+    return network
+
+
+def create_uncomputable_network() -> AlgebraicNeuralNetwork:
+    """Create a sample network with uncomputable layers for demonstration."""
+    network = AlgebraicNeuralNetwork()
+    
+    # Add uncomputable layers
+    network.add_layer(HaltingOracleLayer(4, 5, max_iterations=500))
+    network.add_layer(KolmogorovComplexityLayer(5, 4, precision=6))
+    network.add_layer(BusyBeaverLayer(4, 3))
+    network.add_layer(NonRecursiveLayer(3, 2, enumeration_bound=500))
+    
+    return network
+
+
+def demo_algebraic_neural_network():
+    """Demonstrate the algebraic neural network with sample data."""
+    print("=== Algebraic Neural Network Demo ===\n")
+    
+    # Create sample network
+    network = create_sample_network()
+    
+    # Generate sample input data
+    np.random.seed(42)
+    sample_input = np.random.randn(5, 4)  # 5 samples, 4 features each
+    
+    print("Input data shape:", sample_input.shape)
+    print("Input data:\n", sample_input)
+    
+    # Run prediction
+    output = network.predict(sample_input)
+    
+    print("\nOutput data shape:", output.shape)
+    print("Output data:\n", output)
+    
+    # Demonstrate individual layers
+    print("\n=== Individual Layer Demonstrations ===\n")
+    
+    # Polynomial Layer
+    poly_layer = PolynomialLayer(4, 3, degree=2)
+    poly_output = poly_layer.forward(sample_input[0])
+    print("Polynomial Layer Output:", poly_output)
+    
+    # Group Theory Layer
+    group_layer = GroupTheoryLayer(4, 3, group_order=6)
+    group_output = group_layer.forward(sample_input[0])
+    print("Group Theory Layer Output:", group_output)
+    
+    # Geometric Algebra Layer
+    geo_layer = GeometricAlgebraLayer(4, 3)
+    geo_output = geo_layer.forward(sample_input[0])
+    print("Geometric Algebra Layer Output:", geo_output)
+
+
+def demo_uncomputable_neural_network():
+    """Demonstrate the uncomputable neural network with sample data."""
+    print("\n=== Uncomputable Neural Network Demo ===\n")
+    
+    # Create uncomputable network
+    network = create_uncomputable_network()
+    
+    # Generate sample input data
+    np.random.seed(42)
+    sample_input = np.random.randn(3, 4)  # 3 samples, 4 features each
+    
+    print("Input data shape:", sample_input.shape)
+    print("Input data:\n", sample_input)
+    
+    # Run prediction
+    output = network.predict(sample_input)
+    
+    print("\nOutput data shape:", output.shape)
+    print("Output data:\n", output)
+    
+    # Demonstrate individual uncomputable layers
+    print("\n=== Individual Uncomputable Layer Demonstrations ===\n")
+    
+    # Halting Oracle Layer
+    halting_layer = HaltingOracleLayer(4, 3, max_iterations=100)
+    halting_output = halting_layer.forward(sample_input[0])
+    print("Halting Oracle Layer Output:", halting_output)
+    
+    # Kolmogorov Complexity Layer
+    kolmogorov_layer = KolmogorovComplexityLayer(4, 3, precision=6)
+    kolmogorov_output = kolmogorov_layer.forward(sample_input[0])
+    print("Kolmogorov Complexity Layer Output:", kolmogorov_output)
+    
+    # Busy Beaver Layer
+    bb_layer = BusyBeaverLayer(4, 3)
+    bb_output = bb_layer.forward(sample_input[0])
+    print("Busy Beaver Layer Output:", bb_output)
+    
+    # Non-Recursive Layer
+    nr_layer = NonRecursiveLayer(4, 3, enumeration_bound=100)
+    nr_output = nr_layer.forward(sample_input[0])
+    print("Non-Recursive Layer Output:", nr_output)
+
+
+if __name__ == "__main__":
+    demo_algebraic_neural_network()
+    demo_uncomputable_neural_network()

demo.py

@@ -0,0 +1,73 @@
+#!/usr/bin/env python3
+"""
+Quick demonstration of Algebraic Neural Networks
+
+Run this script to see the basic functionality of algebraic neural networks.
+"""
+
+import numpy as np
+from algebraic_neural_network import create_sample_network, create_uncomputable_network
+
+def main():
+    print("🧮 Algebraic Neural Network Quick Demo")
+    print("="*50)
+    
+    # Create a sample network
+    print("1. Creating algebraic neural network...")
+    network = create_sample_network()
+    print("   ✓ Network created with polynomial, group theory, and geometric algebra layers")
+    
+    # Generate sample data
+    print("\n2. Generating sample data...")
+    np.random.seed(42)  # For reproducible results
+    sample_data = np.random.randn(3, 4)
+    print(f"   ✓ Generated {sample_data.shape[0]} samples with {sample_data.shape[1]} features each")
+    
+    # Make predictions
+    print("\n3. Processing data through algebraic transformations...")
+    predictions = network.predict(sample_data)
+    print(f"   ✓ Output shape: {predictions.shape}")
+    print(f"   ✓ Output range: [{np.min(predictions):.3f}, {np.max(predictions):.3f}]")
+    
+    # Show the data
+    print("\n4. Results:")
+    print("   Input data:")
+    for i, sample in enumerate(sample_data):
+        print(f"     Sample {i+1}: [{sample[0]:6.3f}, {sample[1]:6.3f}, {sample[2]:6.3f}, {sample[3]:6.3f}]")
+    
+    print("\n   Algebraic neural network output:")
+    for i, output in enumerate(predictions):
+        print(f"     Output {i+1}: [{output[0]:8.3f}, {output[1]:8.3f}]")
+    
+    # Demonstrate determinism
+    print("\n5. Demonstrating deterministic behavior...")
+    predictions2 = network.predict(sample_data)
+    difference = np.linalg.norm(predictions - predictions2)
+    print(f"   ✓ Difference between runs: {difference:.10f} (should be 0)")
+    
+    print("\n" + "="*50)
+    print("✅ Demo completed! Algebraic neural networks work without training.")
+    
+    # Bonus: Quick uncomputable network demo
+    print("\n🔬 Bonus: Uncomputable Neural Network Quick Demo")
+    print("="*50)
+    
+    print("1. Creating uncomputable neural network...")
+    uncomputable_network = create_uncomputable_network()
+    print("   ✓ Network created with halting oracle, Kolmogorov complexity, Busy Beaver, and non-recursive layers")
+    
+    print("\n2. Processing same data through uncomputable transformations...")
+    uncomputable_predictions = uncomputable_network.predict(sample_data)
+    print(f"   ✓ Output shape: {uncomputable_predictions.shape}")
+    print(f"   ✓ Output range: [{np.min(uncomputable_predictions):.3f}, {np.max(uncomputable_predictions):.3f}]")
+    
+    print("\n   Uncomputable neural network output:")
+    for i, output in enumerate(uncomputable_predictions):
+        print(f"     Output {i+1}: [{output[0]:6.3f}, {output[1]:6.3f}]")
+    
+    print("\n" + "="*50)
+    print("🎯 Both networks operate without training but explore different mathematical domains!")
+    print("📚 See theory/ and examples/ directories for more details.")
+
+if __name__ == "__main__":
+    main()

examples/geometric_algebra_network.py

@@ -0,0 +1,461 @@
+"""
+Geometric Algebra (Clifford Algebra) Neural Network
+
+This example demonstrates neural networks using geometric algebra operations
+for processing geometric and spatial data.
+"""
+
+import numpy as np
+from typing import List, Tuple, Dict, Union
+import math
+
+
+class GeometricAlgebraNetwork:
+    """
+    Neural network based on geometric algebra (Clifford algebra) operations.
+    """
+    
+    def __init__(self, input_dim: int, hidden_dim: int, output_dim: int, signature: str = "euclidean"):
+        self.input_dim = input_dim
+        self.hidden_dim = hidden_dim
+        self.output_dim = output_dim
+        self.signature = signature
+        
+        # Initialize geometric algebra basis
+        self.basis_elements = self._generate_basis_elements()
+        self.metric = self._generate_metric()
+        
+        # Generate transformation coefficients
+        self.coefficients = self._generate_ga_coefficients()
+        
+    def _generate_metric(self) -> np.ndarray:
+        """Generate metric tensor for the geometric algebra."""
+        if self.signature == "euclidean":
+            return np.eye(self.input_dim)
+        elif self.signature == "minkowski":
+            metric = np.eye(self.input_dim)
+            metric[0, 0] = -1  # Time component with opposite signature
+            return metric
+        elif self.signature == "conformal":
+            # Conformal geometric algebra Cl(n+1,1)
+            metric = np.eye(self.input_dim + 2)
+            metric[-1, -1] = -1  # One negative signature
+            return metric
+        else:
+            return np.eye(self.input_dim)
+    
+    def _generate_basis_elements(self) -> List[Tuple[str, np.ndarray]]:
+        """Generate basis elements for geometric algebra."""
+        basis = []
+        
+        # Scalar (grade 0)
+        scalar_basis = np.zeros(2**self.input_dim)
+        scalar_basis[0] = 1.0
+        basis.append(("scalar", scalar_basis))
+        
+        # Vector basis elements (grade 1)
+        for i in range(self.input_dim):
+            vector_basis = np.zeros(2**self.input_dim)
+            vector_basis[2**i] = 1.0
+            basis.append((f"e{i+1}", vector_basis))
+        
+        # Bivector basis elements (grade 2)
+        for i in range(self.input_dim):
+            for j in range(i+1, self.input_dim):
+                bivector_basis = np.zeros(2**self.input_dim)
+                bivector_basis[2**i + 2**j] = 1.0
+                basis.append((f"e{i+1}e{j+1}", bivector_basis))
+        
+        # Higher grade elements for small dimensions
+        if self.input_dim <= 4:
+            # Trivectors (grade 3)
+            for i in range(self.input_dim):
+                for j in range(i+1, self.input_dim):
+                    for k in range(j+1, self.input_dim):
+                        trivector_basis = np.zeros(2**self.input_dim)
+                        trivector_basis[2**i + 2**j + 2**k] = 1.0
+                        basis.append((f"e{i+1}e{j+1}e{k+1}", trivector_basis))
+            
+            # Pseudoscalar (highest grade)
+            if self.input_dim >= 2:
+                pseudo_basis = np.zeros(2**self.input_dim)
+                pseudo_basis[-1] = 1.0
+                basis.append(("pseudoscalar", pseudo_basis))
+        
+        return basis
+    
+    def _generate_ga_coefficients(self) -> Dict[str, np.ndarray]:
+        """Generate coefficients for geometric algebra transformations."""
+        coeffs = {}
+        
+        # Constants from geometric algebra theory
+        sqrt2 = math.sqrt(2)
+        sqrt3 = math.sqrt(3)
+        phi = (1 + math.sqrt(5)) / 2  # Golden ratio
+        
+        constants = [1.0, 1/sqrt2, 1/sqrt3, 1/phi, phi/3, sqrt2/3, sqrt3/5]
+        
+        # Input to hidden transformation
+        num_basis = len(self.basis_elements)
+        coeffs['input_hidden'] = np.zeros((self.hidden_dim, self.input_dim, num_basis))
+        
+        for i in range(self.hidden_dim):
+            for j in range(self.input_dim):
+                for k in range(num_basis):
+                    const_idx = (i + j + k) % len(constants)
+                    coeffs['input_hidden'][i, j, k] = constants[const_idx]
+        
+        # Hidden to output transformation
+        coeffs['hidden_output'] = np.zeros((self.output_dim, self.hidden_dim, num_basis))
+        
+        for i in range(self.output_dim):
+            for j in range(self.hidden_dim):
+                for k in range(num_basis):
+                    const_idx = (i + j + k + 1) % len(constants)
+                    coeffs['hidden_output'][i, j, k] = constants[const_idx]
+        
+        return coeffs
+    
+    def geometric_product(self, a: np.ndarray, b: np.ndarray) -> np.ndarray:
+        """Compute geometric product of two multivectors."""
+        # Simplified geometric product implementation
+        # In full implementation, this would use the basis multiplication table
+        
+        if len(a) != len(b):
+            min_len = min(len(a), len(b))
+            a, b = a[:min_len], b[:min_len]
+        
+        # For this implementation, approximate with:
+        # ab = a·b + a∧b (dot + wedge products)
+        
+        # Dot product component (grade reduction)
+        dot_product = np.dot(a, b)
+        
+        # Wedge product component (grade increase) - simplified
+        wedge_magnitude = np.linalg.norm(np.outer(a, b) - np.outer(b, a))
+        
+        # Combine into multivector representation
+        result = np.zeros(max(len(a), 2**self.input_dim))
+        result[0] = dot_product  # Scalar part
+        
+        if len(result) > 1:
+            result[1] = wedge_magnitude  # Vector part approximation
+        
+        # Additional components based on input structure
+        for i in range(2, min(len(result), len(a) + len(b) - 1)):
+            result[i] = (a[i % len(a)] * b[i % len(b)] + 
+                        b[i % len(b)] * a[i % len(a)]) / 2
+        
+        return result[:len(a)]
+    
+    def outer_product(self, a: np.ndarray, b: np.ndarray) -> float:
+        """Compute outer (wedge) product magnitude."""
+        if len(a) >= 2 and len(b) >= 2:
+            # 2D outer product as determinant
+            return abs(a[0] * b[1] - a[1] * b[0])
+        else:
+            # Higher dimensional approximation
+            return np.linalg.norm(np.outer(a, b) - np.outer(b, a))
+    
+    def inner_product(self, a: np.ndarray, b: np.ndarray) -> float:
+        """Compute inner (dot) product."""
+        return np.dot(a, b)
+    
+    def reverse(self, mv: np.ndarray) -> np.ndarray:
+        """Compute reverse of multivector (reverse order of basis elements)."""
+        # For bivectors and higher grades, reverse changes sign
+        reversed_mv = mv.copy()
+        
+        # Approximate reversal by alternating signs for higher components
+        for i in range(1, len(reversed_mv)):
+            grade = bin(i).count('1')  # Grade based on binary representation
+            if grade % 4 in [2, 3]:  # Bivectors and trivectors change sign
+                reversed_mv[i] *= -1
+                
+        return reversed_mv
+    
+    def magnitude(self, mv: np.ndarray) -> float:
+        """Compute magnitude of multivector."""
+        # Magnitude is sqrt(mv * reverse(mv))
+        reversed_mv = self.reverse(mv)
+        product = self.geometric_product(mv, reversed_mv)
+        return math.sqrt(abs(product[0]))  # Scalar part should be positive
+    
+    def normalize(self, mv: np.ndarray) -> np.ndarray:
+        """Normalize multivector."""
+        mag = self.magnitude(mv)
+        if mag > 1e-10:
+            return mv / mag
+        else:
+            return mv
+    
+    def apply_ga_transformation(self, x: np.ndarray, coeffs: np.ndarray) -> np.ndarray:
+        """Apply geometric algebra transformation."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        batch_size, input_size = x.shape
+        output_size = coeffs.shape[0]
+        num_basis = len(self.basis_elements)
+        
+        result = np.zeros((batch_size, output_size))
+        
+        for batch_idx in range(batch_size):
+            for out_idx in range(output_size):
+                # Construct multivector from input
+                multivector = np.zeros(num_basis)
+                
+                for in_idx in range(min(input_size, self.input_dim)):
+                    for basis_idx in range(num_basis):
+                        basis_name, basis_vector = self.basis_elements[basis_idx]
+                        coeff = coeffs[out_idx, in_idx, basis_idx]
+                        
+                        # Ensure basis_vector has correct length
+                        basis_component = basis_vector[:num_basis] if len(basis_vector) >= num_basis else np.pad(basis_vector, (0, num_basis - len(basis_vector)))
+                        
+                        # Weight by input value and coefficient
+                        multivector += x[batch_idx, in_idx] * coeff * basis_component
+                
+                # Apply geometric algebra operations
+                # 1. Geometric product with basis elements
+                transformed_mv = multivector.copy()
+                
+                for basis_idx in range(min(3, num_basis)):  # Use first few basis elements
+                    _, basis_vector = self.basis_elements[basis_idx]
+                    transformed_mv = self.geometric_product(transformed_mv, basis_vector[:num_basis])
+                
+                # 2. Extract scalar and vector parts
+                scalar_part = transformed_mv[0] if len(transformed_mv) > 0 else 0
+                vector_magnitude = np.linalg.norm(transformed_mv[1:min(4, len(transformed_mv))])
+                
+                # 3. Combine into output
+                result[batch_idx, out_idx] = scalar_part + vector_magnitude
+        
+        return result
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Forward pass through geometric algebra network."""
+        # Input to hidden layer
+        hidden = self.apply_ga_transformation(x, self.coefficients['input_hidden'])
+        
+        # Apply nonlinearity (preserve geometric structure)
+        hidden = np.tanh(hidden)
+        
+        # Hidden to output layer
+        output = self.apply_ga_transformation(hidden, self.coefficients['hidden_output'])
+        
+        return output
+    
+    def predict(self, x: np.ndarray) -> np.ndarray:
+        """Prediction method."""
+        return self.forward(x)
+
+
+def test_geometric_operations():
+    """Test basic geometric algebra operations."""
+    print("=== Geometric Algebra: Basic Operations Test ===\n")
+    
+    network = GeometricAlgebraNetwork(input_dim=3, hidden_dim=4, output_dim=2)
+    
+    # Test vectors
+    a = np.array([1, 0, 0])  # e1
+    b = np.array([0, 1, 0])  # e2
+    c = np.array([1, 1, 0])  # e1 + e2
+    
+    print("Testing geometric algebra operations:")
+    
+    # Inner products
+    inner_ab = network.inner_product(a, b)
+    inner_aa = network.inner_product(a, a)
+    print(f"Inner product a·b = {inner_ab:.3f} (should be 0)")
+    print(f"Inner product a·a = {inner_aa:.3f} (should be 1)")
+    
+    # Outer products
+    outer_ab = network.outer_product(a, b)
+    outer_aa = network.outer_product(a, a)
+    print(f"Outer product a∧b magnitude = {outer_ab:.3f} (should be 1)")
+    print(f"Outer product a∧a magnitude = {outer_aa:.3f} (should be 0)")
+    
+    # Geometric products
+    geom_ab = network.geometric_product(a, b)
+    print(f"Geometric product a*b = {geom_ab[:3]} (first 3 components)")
+    
+    # Magnitudes
+    mag_a = network.magnitude(a)
+    mag_c = network.magnitude(c)
+    print(f"Magnitude |a| = {mag_a:.3f}")
+    print(f"Magnitude |c| = {mag_c:.3f}")
+    
+    print()
+
+
+def test_3d_rotation_processing():
+    """Test processing of 3D rotational data."""
+    print("=== Geometric Algebra: 3D Rotation Processing ===\n")
+    
+    network = GeometricAlgebraNetwork(input_dim=3, hidden_dim=6, output_dim=4)
+    
+    # Generate 3D rotation data (axis-angle representation)
+    rotation_axes = [
+        [1, 0, 0],  # X-axis rotation
+        [0, 1, 0],  # Y-axis rotation
+        [0, 0, 1],  # Z-axis rotation
+        [1, 1, 1],  # Diagonal rotation
+        [1, -1, 0], # Mixed rotation
+    ]
+    
+    print("Processing 3D rotation data:")
+    
+    for i, axis in enumerate(rotation_axes):
+        axis = np.array(axis, dtype=float)
+        axis = axis / np.linalg.norm(axis)  # Normalize
+        
+        # Different rotation angles
+        angles = [0, np.pi/4, np.pi/2, np.pi, 3*np.pi/2]
+        
+        outputs = []
+        for angle in angles:
+            # Rotation vector (axis * angle)
+            rotation_vector = axis * angle
+            output = network.predict(rotation_vector.reshape(1, -1))
+            outputs.append(output[0])
+        
+        outputs = np.array(outputs)
+        
+        print(f"\nRotation axis {i+1}: {axis}")
+        print(f"  Output range: [{np.min(outputs):.3f}, {np.max(outputs):.3f}]")
+        print(f"  Output variance: {np.var(outputs, axis=0)}")
+        
+        # Check for periodic behavior (rotations should have 2π periodicity)
+        first_output = outputs[0]  # 0 radians
+        last_output = outputs[-1]   # 3π/2 radians, should be similar to π/2
+        
+        periodicity_error = np.linalg.norm(first_output - outputs[2])  # Compare 0 and π
+        print(f"  Periodicity error (0 vs π): {periodicity_error:.6f}")
+
+
+def test_conformal_geometry():
+    """Test conformal geometric algebra for 2D points."""
+    print("=== Geometric Algebra: Conformal Geometry ===\n")
+    
+    # Use conformal signature for 2D conformal GA
+    network = GeometricAlgebraNetwork(input_dim=2, hidden_dim=8, output_dim=4, signature="conformal")
+    
+    # Test geometric primitives
+    geometric_objects = {
+        "Point": np.array([1, 1]),
+        "Origin": np.array([0, 0]),
+        "Unit_X": np.array([1, 0]),
+        "Unit_Y": np.array([0, 1]),
+        "Diagonal": np.array([1, 1]) / np.sqrt(2)
+    }
+    
+    print("Processing geometric objects in conformal space:")
+    
+    object_features = {}
+    for obj_name, point in geometric_objects.items():
+        # Convert to conformal representation
+        # In conformal GA: P = point + 0.5*|point|²*e∞ + e₀
+        point_squared = np.dot(point, point)
+        conformal_point = np.concatenate([point, [0.5 * point_squared, 1]])
+        
+        # Process through network
+        output = network.predict(conformal_point.reshape(1, -1))
+        object_features[obj_name] = output[0]
+        
+        print(f"{obj_name:>10}: {point} → output: {output[0]}")
+    
+    # Analyze relationships between objects
+    print("\nAnalyzing geometric relationships:")
+    
+    origin_features = object_features["Origin"]
+    for obj_name, features in object_features.items():
+        if obj_name != "Origin":
+            distance = np.linalg.norm(features - origin_features)
+            print(f"  Feature distance from origin to {obj_name}: {distance:.4f}")
+
+
+def test_bivector_operations():
+    """Test bivector operations for oriented areas."""
+    print("=== Geometric Algebra: Bivector Operations ===\n")
+    
+    network = GeometricAlgebraNetwork(input_dim=4, hidden_dim=6, output_dim=3)
+    
+    # Create bivectors representing oriented areas
+    bivectors = [
+        [1, 0, 1, 0],  # e1∧e3
+        [0, 1, 0, 1],  # e2∧e4
+        [1, 1, 0, 0],  # e1∧e2
+        [0, 0, 1, 1],  # e3∧e4
+        [1, 0, 0, 1],  # e1∧e4
+    ]
+    
+    print("Processing bivector data:")
+    
+    for i, bivector in enumerate(bivectors):
+        bv = np.array(bivector, dtype=float)
+        output = network.predict(bv.reshape(1, -1))
+        
+        # Calculate bivector magnitude
+        bv_magnitude = np.linalg.norm(bv)
+        
+        print(f"Bivector {i+1}: {bv}")
+        print(f"  Magnitude: {bv_magnitude:.3f}")
+        print(f"  Network output: {output[0]}")
+        print(f"  Output magnitude: {np.linalg.norm(output[0]):.3f}")
+        print()
+
+
+def test_multivector_algebra():
+    """Test general multivector operations."""
+    print("=== Geometric Algebra: Multivector Operations ===\n")
+    
+    network = GeometricAlgebraNetwork(input_dim=3, hidden_dim=5, output_dim=2)
+    
+    # Create multivectors with different grade components
+    multivectors = [
+        [1, 0, 0],     # Pure vector e1
+        [0, 1, 0],     # Pure vector e2
+        [0, 0, 1],     # Pure vector e3
+        [1, 1, 0],     # e1 + e2
+        [1, 1, 1],     # e1 + e2 + e3
+        [2, -1, 0.5],  # 2*e1 - e2 + 0.5*e3
+    ]
+    
+    print("Multivector algebra processing:")
+    
+    for i, mv in enumerate(multivectors):
+        mv_array = np.array(mv, dtype=float)
+        
+        # Test reverse operation
+        reversed_mv = network.reverse(mv_array)
+        
+        # Test magnitude
+        magnitude = network.magnitude(mv_array)
+        
+        # Test normalization
+        normalized_mv = network.normalize(mv_array)
+        
+        # Network processing
+        output = network.predict(mv_array.reshape(1, -1))
+        
+        print(f"\nMultivector {i+1}: {mv}")
+        print(f"  Reversed: {reversed_mv}")
+        print(f"  Magnitude: {magnitude:.4f}")
+        print(f"  Normalized: {normalized_mv}")
+        print(f"  Network output: {output[0]}")
+
+
+if __name__ == "__main__":
+    print("Geometric Algebra Neural Network Demo\n")
+    print("="*60)
+    
+    # Run tests
+    test_geometric_operations()
+    test_3d_rotation_processing()
+    test_conformal_geometry()
+    test_bivector_operations()
+    test_multivector_algebra()
+    
+    print("\n" + "="*60)
+    print("Geometric algebra demo completed successfully!")

examples/group_theory_network.py

@@ -0,0 +1,381 @@
+"""
+Group Theory-based Algebraic Neural Network
+
+This example demonstrates neural networks that use group theory operations
+for data transformations, particularly focusing on symmetry groups.
+"""
+
+import numpy as np
+from typing import List, Tuple, Dict
+import math
+
+
+class GroupTheoryNetwork:
+    """
+    Neural network based on group theory operations and symmetries.
+    """
+    
+    def __init__(self, input_dim: int, group_types: List[str], output_dim: int):
+        self.input_dim = input_dim
+        self.group_types = group_types
+        self.output_dim = output_dim
+        
+        # Initialize group operations
+        self.groups = self._initialize_groups()
+        
+    def _initialize_groups(self) -> Dict[str, List[np.ndarray]]:
+        """Initialize various group operations."""
+        groups = {}
+        
+        for group_type in self.group_types:
+            if group_type.startswith("cyclic_"):
+                n = int(group_type.split("_")[1])
+                groups[group_type] = self._generate_cyclic_group(n)
+            elif group_type.startswith("dihedral_"):
+                n = int(group_type.split("_")[1])
+                groups[group_type] = self._generate_dihedral_group(n)
+            elif group_type == "symmetric_3":
+                groups[group_type] = self._generate_symmetric_group_3()
+            elif group_type == "reflection":
+                groups[group_type] = self._generate_reflection_group()
+                
+        return groups
+    
+    def _generate_cyclic_group(self, n: int) -> List[np.ndarray]:
+        """Generate cyclic group Cn as rotation matrices."""
+        group_elements = []
+        
+        for k in range(n):
+            angle = 2 * math.pi * k / n
+            
+            if self.input_dim == 2:
+                # 2D rotation matrix
+                rotation = np.array([
+                    [math.cos(angle), -math.sin(angle)],
+                    [math.sin(angle), math.cos(angle)]
+                ])
+            elif self.input_dim >= 3:
+                # 3D rotation around z-axis, identity for higher dimensions
+                rotation = np.eye(self.input_dim)
+                if self.input_dim >= 2:
+                    rotation[0, 0] = math.cos(angle)
+                    rotation[0, 1] = -math.sin(angle)
+                    rotation[1, 0] = math.sin(angle)
+                    rotation[1, 1] = math.cos(angle)
+            else:
+                # 1D case - just scaling
+                rotation = np.array([[(-1) ** k]])
+                
+            group_elements.append(rotation)
+            
+        return group_elements
+    
+    def _generate_dihedral_group(self, n: int) -> List[np.ndarray]:
+        """Generate dihedral group Dn (rotations + reflections)."""
+        group_elements = []
+        
+        # Add rotations (same as cyclic group)
+        rotations = self._generate_cyclic_group(n)
+        group_elements.extend(rotations)
+        
+        # Add reflections
+        for k in range(n):
+            angle = 2 * math.pi * k / n
+            
+            if self.input_dim == 2:
+                # Reflection across line through origin at angle/2
+                reflection_angle = angle / 2
+                cos_2theta = math.cos(2 * reflection_angle)
+                sin_2theta = math.sin(2 * reflection_angle)
+                
+                reflection = np.array([
+                    [cos_2theta, sin_2theta],
+                    [sin_2theta, -cos_2theta]
+                ])
+            else:
+                # For higher dimensions, reflect across first coordinate
+                reflection = np.eye(self.input_dim)
+                reflection[0, 0] = -1
+                
+            group_elements.append(reflection)
+            
+        return group_elements
+    
+    def _generate_symmetric_group_3(self) -> List[np.ndarray]:
+        """Generate symmetric group S3 (permutations of 3 elements)."""
+        if self.input_dim < 3:
+            # For lower dimensions, use reduced representation
+            return self._generate_cyclic_group(3)
+        
+        # All permutations of 3 elements as permutation matrices
+        permutations = [
+            [0, 1, 2],  # identity
+            [1, 2, 0],  # (0 1 2)
+            [2, 0, 1],  # (0 2 1)
+            [1, 0, 2],  # (0 1)
+            [0, 2, 1],  # (0 2)
+            [2, 1, 0],  # (1 2)
+        ]
+        
+        group_elements = []
+        for perm in permutations:
+            matrix = np.eye(self.input_dim)
+            for i in range(3):
+                if i < self.input_dim and perm[i] < self.input_dim:
+                    matrix[i, i] = 0
+                    matrix[i, perm[i]] = 1
+            group_elements.append(matrix)
+            
+        return group_elements
+    
+    def _generate_reflection_group(self) -> List[np.ndarray]:
+        """Generate group of reflections across coordinate axes."""
+        group_elements = []
+        
+        # Identity
+        group_elements.append(np.eye(self.input_dim))
+        
+        # Reflections across each coordinate axis
+        for i in range(self.input_dim):
+            reflection = np.eye(self.input_dim)
+            reflection[i, i] = -1
+            group_elements.append(reflection)
+            
+        # Reflections across diagonal planes (for 2D and 3D)
+        if self.input_dim == 2:
+            # Reflection across y = x
+            diag_reflection = np.array([[0, 1], [1, 0]])
+            group_elements.append(diag_reflection)
+            
+            # Reflection across y = -x
+            anti_diag_reflection = np.array([[0, -1], [-1, 0]])
+            group_elements.append(anti_diag_reflection)
+            
+        return group_elements
+    
+    def apply_group_actions(self, x: np.ndarray, group_name: str) -> np.ndarray:
+        """Apply all group actions to input data."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        group_elements = self.groups[group_name]
+        results = []
+        
+        for element in group_elements:
+            if x.shape[1] == element.shape[0]:
+                transformed = x @ element.T
+            else:
+                # Handle dimension mismatch by padding or truncating
+                min_dim = min(x.shape[1], element.shape[0])
+                transformed = x[:, :min_dim] @ element[:min_dim, :min_dim].T
+                
+            # Compute invariant features
+            norm = np.linalg.norm(transformed, axis=1, keepdims=True)
+            mean = np.mean(transformed, axis=1, keepdims=True)
+            std = np.std(transformed, axis=1, keepdims=True) + 1e-8
+            
+            # Combine features
+            features = np.concatenate([norm, mean, std], axis=1)
+            results.append(features)
+            
+        return np.concatenate(results, axis=1)
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Forward pass through group theory network."""
+        all_features = []
+        
+        # Apply each group type
+        for group_name in self.group_types:
+            group_features = self.apply_group_actions(x, group_name)
+            all_features.append(group_features)
+            
+        # Concatenate all features
+        combined_features = np.concatenate(all_features, axis=1)
+        
+        # Linear combination to get desired output dimension
+        feature_dim = combined_features.shape[1]
+        if feature_dim >= self.output_dim:
+            # Take first output_dim features
+            output = combined_features[:, :self.output_dim]
+        else:
+            # Repeat features to reach output_dim
+            repeats = (self.output_dim + feature_dim - 1) // feature_dim
+            repeated = np.tile(combined_features, (1, repeats))
+            output = repeated[:, :self.output_dim]
+            
+        return output
+    
+    def predict(self, x: np.ndarray) -> np.ndarray:
+        """Prediction method."""
+        return self.forward(x)
+
+
+def test_rotation_invariance():
+    """Test rotation invariance of the group theory network."""
+    print("=== Group Theory Network: Rotation Invariance Test ===\n")
+    
+    # Create network with cyclic group
+    network = GroupTheoryNetwork(
+        input_dim=2,
+        group_types=["cyclic_8"],
+        output_dim=4
+    )
+    
+    # Create test patterns
+    original_pattern = np.array([[1, 0], [0, 1], [1, 1]])
+    
+    # Manual rotations for comparison
+    rotation_angles = [0, np.pi/4, np.pi/2, np.pi, 3*np.pi/2]
+    
+    print("Testing rotation invariance:")
+    outputs = []
+    
+    for angle in rotation_angles:
+        # Manually rotate the pattern
+        cos_a, sin_a = np.cos(angle), np.sin(angle)
+        rotation_matrix = np.array([[cos_a, -sin_a], [sin_a, cos_a]])
+        rotated_pattern = original_pattern @ rotation_matrix.T
+        
+        # Get network output
+        output = network.predict(rotated_pattern)
+        outputs.append(output)
+        
+        print(f"  Rotation {angle:.2f} rad: mean output = {np.mean(output):.4f}")
+    
+    # Check invariance (outputs should be similar)
+    output_array = np.array(outputs)
+    variance_across_rotations = np.var(output_array, axis=0)
+    mean_variance = np.mean(variance_across_rotations)
+    
+    print(f"\nMean variance across rotations: {mean_variance:.6f}")
+    print("(Lower values indicate better rotation invariance)\n")
+    
+    return outputs
+
+
+def test_symmetry_detection():
+    """Test the network's ability to detect different symmetries."""
+    print("=== Group Theory Network: Symmetry Detection ===\n")
+    
+    # Create network with multiple group types
+    network = GroupTheoryNetwork(
+        input_dim=2,
+        group_types=["cyclic_4", "dihedral_4", "reflection"],
+        output_dim=6
+    )
+    
+    # Define test patterns with different symmetries
+    patterns = {
+        "Square": np.array([[1, 1], [1, -1], [-1, -1], [-1, 1]]),
+        "Triangle": np.array([[1, 0], [-0.5, np.sqrt(3)/2], [-0.5, -np.sqrt(3)/2]]),
+        "Line": np.array([[1, 0], [0.5, 0], [0, 0], [-0.5, 0], [-1, 0]]),
+        "Circle": np.array([[np.cos(θ), np.sin(θ)] for θ in np.linspace(0, 2*np.pi, 8)]),
+        "Asymmetric": np.array([[1, 0], [0, 1], [0.3, 0.7], [0.8, 0.2]])
+    }
+    
+    print("Symmetry detection results:")
+    pattern_outputs = {}
+    
+    for pattern_name, pattern_points in patterns.items():
+        output = network.predict(pattern_points)
+        pattern_outputs[pattern_name] = output
+        
+        print(f"\n{pattern_name}:")
+        print(f"  Mean output: {np.mean(output, axis=0)}")
+        print(f"  Output std: {np.std(output, axis=0)}")
+        
+    return pattern_outputs
+
+
+def test_group_composition():
+    """Test composition of group operations."""
+    print("=== Group Theory Network: Group Composition ===\n")
+    
+    network = GroupTheoryNetwork(
+        input_dim=3,
+        group_types=["symmetric_3"],
+        output_dim=3
+    )
+    
+    # Test group properties
+    test_input = np.array([[1, 2, 3]])
+    
+    print("Testing group composition properties:")
+    
+    # Get all group elements
+    group_elements = network.groups["symmetric_3"]
+    
+    # Test identity element (should be first)
+    identity_result = test_input @ group_elements[0].T
+    print(f"Identity transformation: {test_input[0]} → {identity_result[0]}")
+    
+    # Test composition of operations
+    for i, g1 in enumerate(group_elements[:3]):
+        for j, g2 in enumerate(group_elements[:3]):
+            # Apply g1 then g2
+            intermediate = test_input @ g1.T
+            final = intermediate @ g2.T
+            
+            # Apply composition g2∘g1
+            composition = g2 @ g1
+            direct = test_input @ composition.T
+            
+            # Check if they're the same (within numerical precision)
+            difference = np.linalg.norm(final - direct)
+            print(f"  g{j}∘g{i}: composition error = {difference:.8f}")
+
+
+def test_invariant_features():
+    """Test extraction of invariant features."""
+    print("=== Group Theory Network: Invariant Feature Extraction ===\n")
+    
+    # Network for extracting rotation-invariant features
+    network = GroupTheoryNetwork(
+        input_dim=2,
+        group_types=["cyclic_16"],  # Fine rotation sampling
+        output_dim=8
+    )
+    
+    # Test patterns with known geometric properties
+    test_cases = [
+        ("Unit Circle", np.array([[np.cos(θ), np.sin(θ)] for θ in np.linspace(0, 2*np.pi, 10)])),
+        ("Ellipse", np.array([[2*np.cos(θ), np.sin(θ)] for θ in np.linspace(0, 2*np.pi, 10)])),
+        ("Square", np.array([[1, 1], [1, -1], [-1, -1], [-1, 1]])),
+        ("Random", np.random.randn(8, 2))
+    ]
+    
+    print("Invariant feature analysis:")
+    
+    for case_name, points in test_cases:
+        # Original features
+        original_features = network.predict(points)
+        
+        # Rotated version
+        rotation_45 = np.array([[np.cos(np.pi/4), -np.sin(np.pi/4)],
+                               [np.sin(np.pi/4), np.cos(np.pi/4)]])
+        rotated_points = points @ rotation_45.T
+        rotated_features = network.predict(rotated_points)
+        
+        # Measure feature consistency
+        feature_difference = np.linalg.norm(original_features - rotated_features)
+        relative_difference = feature_difference / (np.linalg.norm(original_features) + 1e-8)
+        
+        print(f"\n{case_name}:")
+        print(f"  Feature difference: {feature_difference:.6f}")
+        print(f"  Relative difference: {relative_difference:.6f}")
+        print(f"  Original features mean: {np.mean(original_features):.4f}")
+        print(f"  Rotated features mean: {np.mean(rotated_features):.4f}")
+
+
+if __name__ == "__main__":
+    print("Group Theory Algebraic Neural Network Demo\n")
+    print("="*60)
+    
+    # Run tests
+    test_rotation_invariance()
+    test_symmetry_detection()
+    test_group_composition()
+    test_invariant_features()
+    
+    print("\n" + "="*60)
+    print("Group theory demo completed successfully!")

examples/polynomial_network.py

@@ -0,0 +1,289 @@
+"""
+Polynomial-based Algebraic Neural Network
+
+This example demonstrates a neural network that uses polynomial transformations
+with coefficients derived from algebraic number theory.
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from typing import Tuple
+import math
+
+
+class PolynomialAlgebraicNetwork:
+    """
+    Neural network using polynomial basis functions with algebraic coefficients.
+    """
+    
+    def __init__(self, input_dim: int, hidden_dim: int, output_dim: int, max_degree: int = 3):
+        self.input_dim = input_dim
+        self.hidden_dim = hidden_dim
+        self.output_dim = output_dim
+        self.max_degree = max_degree
+        
+        # Generate polynomial coefficients using algebraic numbers
+        self.coefficients = self._generate_algebraic_coefficients()
+        
+    def _generate_algebraic_coefficients(self) -> dict:
+        """Generate coefficients using famous algebraic constants."""
+        coeffs = {}
+        
+        # Golden ratio and related algebraic numbers
+        phi = (1 + math.sqrt(5)) / 2  # Golden ratio
+        phi_conjugate = (1 - math.sqrt(5)) / 2
+        
+        # Silver ratio
+        silver = 1 + math.sqrt(2)
+        
+        # Euler's number approximation using continued fractions
+        e_approx = 2.718281828
+        
+        # Pi approximation
+        pi_approx = math.pi
+        
+        algebraic_constants = [1, phi, phi_conjugate, silver, e_approx, pi_approx]
+        
+        # Generate coefficients for input to hidden transformation
+        coeffs['input_hidden'] = np.zeros((self.hidden_dim, self.input_dim, self.max_degree + 1))
+        for i in range(self.hidden_dim):
+            for j in range(self.input_dim):
+                for k in range(self.max_degree + 1):
+                    # Use algebraic constants in a systematic way
+                    const_idx = (i + j + k) % len(algebraic_constants)
+                    base_coeff = algebraic_constants[const_idx]
+                    
+                    # Scale by factorial to maintain stability
+                    coeffs['input_hidden'][i, j, k] = base_coeff / math.factorial(k + 1)
+        
+        # Generate coefficients for hidden to output transformation
+        coeffs['hidden_output'] = np.zeros((self.output_dim, self.hidden_dim, self.max_degree + 1))
+        for i in range(self.output_dim):
+            for j in range(self.hidden_dim):
+                for k in range(self.max_degree + 1):
+                    const_idx = (i + j + k + 1) % len(algebraic_constants)
+                    base_coeff = algebraic_constants[const_idx]
+                    coeffs['hidden_output'][i, j, k] = base_coeff / math.factorial(k + 1)
+        
+        return coeffs
+    
+    def _polynomial_activation(self, x: np.ndarray, coeffs: np.ndarray) -> np.ndarray:
+        """Apply polynomial activation with given coefficients."""
+        if x.ndim == 1:
+            x = x.reshape(1, -1)
+            
+        batch_size, input_size = x.shape
+        output_size = coeffs.shape[0]
+        
+        result = np.zeros((batch_size, output_size))
+        
+        for i in range(output_size):
+            for j in range(input_size):
+                for degree in range(self.max_degree + 1):
+                    if degree == 0:
+                        poly_term = coeffs[i, j, degree]
+                    else:
+                        poly_term = coeffs[i, j, degree] * (x[:, j] ** degree)
+                    result[:, i] += poly_term
+                    
+        return result
+    
+    def forward(self, x: np.ndarray) -> np.ndarray:
+        """Forward pass through the polynomial network."""
+        # Input to hidden layer
+        hidden = self._polynomial_activation(x, self.coefficients['input_hidden'])
+        
+        # Apply hyperbolic tangent for stability
+        hidden = np.tanh(hidden)
+        
+        # Hidden to output layer
+        output = self._polynomial_activation(hidden, self.coefficients['hidden_output'])
+        
+        return output
+    
+    def predict(self, x: np.ndarray) -> np.ndarray:
+        """Prediction method."""
+        return self.forward(x)
+
+
+def test_function_approximation():
+    """Test the polynomial network on function approximation tasks."""
+    print("=== Polynomial Network Function Approximation ===\n")
+    
+    # Create network
+    network = PolynomialAlgebraicNetwork(input_dim=1, hidden_dim=5, output_dim=1, max_degree=3)
+    
+    # Test on various mathematical functions
+    test_functions = [
+        ("Sine", lambda x: np.sin(2 * np.pi * x)),
+        ("Cosine", lambda x: np.cos(2 * np.pi * x)),
+        ("Quadratic", lambda x: x**2 - 0.5*x + 0.1),
+        ("Cubic", lambda x: x**3 - x**2 + 0.5*x),
+        ("Exponential", lambda x: np.exp(-x**2))
+    ]
+    
+    x_test = np.linspace(-1, 1, 50).reshape(-1, 1)
+    
+    results = {}
+    for func_name, func in test_functions:
+        y_true = func(x_test.flatten())
+        y_pred = network.predict(x_test).flatten()
+        
+        # Calculate approximation error
+        mse = np.mean((y_true - y_pred)**2)
+        mae = np.mean(np.abs(y_true - y_pred))
+        
+        results[func_name] = {
+            'mse': mse,
+            'mae': mae,
+            'y_true': y_true,
+            'y_pred': y_pred
+        }
+        
+        print(f"{func_name}:")
+        print(f"  MSE: {mse:.6f}")
+        print(f"  MAE: {mae:.6f}")
+        print()
+    
+    return results, x_test
+
+
+def test_pattern_recognition():
+    """Test polynomial network on 2D pattern recognition."""
+    print("=== Polynomial Network Pattern Recognition ===\n")
+    
+    # Create 2D network
+    network = PolynomialAlgebraicNetwork(input_dim=2, hidden_dim=8, output_dim=3, max_degree=2)
+    
+    # Generate test patterns
+    def generate_circle_points(n_points=20, radius=0.8):
+        angles = np.linspace(0, 2*np.pi, n_points, endpoint=False)
+        return np.column_stack([radius * np.cos(angles), radius * np.sin(angles)])
+    
+    def generate_square_points(n_points=20, side=1.0):
+        points_per_side = n_points // 4
+        side_points = []
+        
+        # Bottom side
+        x = np.linspace(-side/2, side/2, points_per_side)
+        y = np.full(points_per_side, -side/2)
+        side_points.extend(zip(x, y))
+        
+        # Right side
+        x = np.full(points_per_side, side/2)
+        y = np.linspace(-side/2, side/2, points_per_side)
+        side_points.extend(zip(x, y))
+        
+        # Top side
+        x = np.linspace(side/2, -side/2, points_per_side)
+        y = np.full(points_per_side, side/2)
+        side_points.extend(zip(x, y))
+        
+        # Left side
+        x = np.full(points_per_side, -side/2)
+        y = np.linspace(side/2, -side/2, points_per_side)
+        side_points.extend(zip(x, y))
+        
+        return np.array(side_points[:n_points])
+    
+    def generate_triangle_points(n_points=18, size=0.8):
+        angles = np.array([0, 2*np.pi/3, 4*np.pi/3])
+        vertices = size * np.column_stack([np.cos(angles), np.sin(angles)])
+        
+        points = []
+        points_per_edge = n_points // 3
+        
+        for i in range(3):
+            start = vertices[i]
+            end = vertices[(i + 1) % 3]
+            edge_points = np.linspace(start, end, points_per_edge, endpoint=False)
+            points.extend(edge_points)
+        
+        return np.array(points[:n_points])
+    
+    # Generate patterns
+    circles = generate_circle_points()
+    squares = generate_square_points()
+    triangles = generate_triangle_points()
+    
+    # Process with network
+    circle_outputs = network.predict(circles)
+    square_outputs = network.predict(squares)
+    triangle_outputs = network.predict(triangles)
+    
+    # Analyze outputs
+    print("Circle pattern analysis:")
+    print(f"  Mean output: {np.mean(circle_outputs, axis=0)}")
+    print(f"  Std output: {np.std(circle_outputs, axis=0)}")
+    
+    print("\nSquare pattern analysis:")
+    print(f"  Mean output: {np.mean(square_outputs, axis=0)}")
+    print(f"  Std output: {np.std(square_outputs, axis=0)}")
+    
+    print("\nTriangle pattern analysis:")
+    print(f"  Mean output: {np.mean(triangle_outputs, axis=0)}")
+    print(f"  Std output: {np.std(triangle_outputs, axis=0)}")
+    
+    return {
+        'circles': (circles, circle_outputs),
+        'squares': (squares, square_outputs),
+        'triangles': (triangles, triangle_outputs)
+    }
+
+
+def demonstrate_coefficient_properties():
+    """Demonstrate properties of the algebraic coefficients."""
+    print("=== Algebraic Coefficient Properties ===\n")
+    
+    network = PolynomialAlgebraicNetwork(input_dim=3, hidden_dim=4, output_dim=2)
+    
+    # Analyze coefficient matrices
+    input_hidden_coeffs = network.coefficients['input_hidden']
+    hidden_output_coeffs = network.coefficients['hidden_output']
+    
+    print("Input-Hidden Coefficients:")
+    print(f"  Shape: {input_hidden_coeffs.shape}")
+    print(f"  Min coefficient: {np.min(input_hidden_coeffs):.6f}")
+    print(f"  Max coefficient: {np.max(input_hidden_coeffs):.6f}")
+    print(f"  Mean coefficient: {np.mean(input_hidden_coeffs):.6f}")
+    print(f"  Std coefficient: {np.std(input_hidden_coeffs):.6f}")
+    
+    print("\nHidden-Output Coefficients:")
+    print(f"  Shape: {hidden_output_coeffs.shape}")
+    print(f"  Min coefficient: {np.min(hidden_output_coeffs):.6f}")
+    print(f"  Max coefficient: {np.max(hidden_output_coeffs):.6f}")
+    print(f"  Mean coefficient: {np.mean(hidden_output_coeffs):.6f}")
+    print(f"  Std coefficient: {np.std(hidden_output_coeffs):.6f}")
+    
+    # Test stability with different input magnitudes
+    print("\nStability Analysis:")
+    test_inputs = [
+        np.array([[0.1, 0.1, 0.1]]),
+        np.array([[0.5, 0.5, 0.5]]),
+        np.array([[1.0, 1.0, 1.0]]),
+        np.array([[2.0, 2.0, 2.0]]),
+    ]
+    
+    for i, test_input in enumerate(test_inputs):
+        output = network.predict(test_input)
+        magnitude = np.linalg.norm(test_input)
+        output_magnitude = np.linalg.norm(output)
+        print(f"  Input magnitude {magnitude:.1f} → Output magnitude {output_magnitude:.6f}")
+
+
+if __name__ == "__main__":
+    # Run demonstrations
+    print("Polynomial Algebraic Neural Network Demo\n")
+    print("="*50)
+    
+    # Function approximation test
+    func_results, x_vals = test_function_approximation()
+    
+    # Pattern recognition test
+    pattern_results = test_pattern_recognition()
+    
+    # Coefficient analysis
+    demonstrate_coefficient_properties()
+    
+    print("\n" + "="*50)
+    print("Demo completed successfully!")

examples/uncomputable_networks.py

@@ -0,0 +1,218 @@
+#!/usr/bin/env python3
+"""
+Uncomputable Neural Network Examples
+
+This script demonstrates the various types of uncomputable neural network layers
+and their theoretical foundations.
+"""
+
+import numpy as np
+import sys
+import os
+
+# Add parent directory to path
+sys.path.insert(0, os.path.dirname(os.path.dirname(os.path.abspath(__file__))))
+
+from algebraic_neural_network import (
+    HaltingOracleLayer, KolmogorovComplexityLayer, BusyBeaverLayer, 
+    NonRecursiveLayer, create_uncomputable_network
+)
+
+def demonstrate_halting_oracle():
+    """Demonstrate the Halting Oracle Layer."""
+    print("=== Halting Oracle Layer Demonstration ===\n")
+    
+    print("The Halting Oracle Layer simulates access to a halting oracle,")
+    print("which can theoretically answer whether a program halts on given input.")
+    print("This is uncomputable in general, but we provide bounded approximations.\n")
+    
+    layer = HaltingOracleLayer(4, 3, max_iterations=1000)
+    
+    # Test with different "program" inputs
+    test_cases = [
+        ([1, 0, 0, 0], "Simple program"),
+        ([0.5, 0.5, 0.5, 0.5], "Balanced program"),
+        ([0.9, 0.9, 0.9, 0.9], "Complex program"),
+        ([0.1, 0.1, 0.1, 0.1], "Minimal program"),
+    ]
+    
+    for i, (input_data, description) in enumerate(test_cases, 1):
+        input_array = np.array(input_data).reshape(1, -1)
+        output = layer.forward(input_array)
+        
+        print(f"Test {i}: {description}")
+        print(f"  Input: {input_data}")
+        print(f"  Halting probabilities: {output[0]}")
+        print(f"  Interpretation: {['Unlikely to halt' if p < 0.5 else 'Likely to halt' for p in output[0]]}")
+        print()
+
+def demonstrate_kolmogorov_complexity():
+    """Demonstrate the Kolmogorov Complexity Layer."""
+    print("=== Kolmogorov Complexity Layer Demonstration ===\n")
+    
+    print("The Kolmogorov Complexity Layer approximates the Kolmogorov complexity")
+    print("of inputs - the length of the shortest program that outputs the input.")
+    print("True Kolmogorov complexity is uncomputable, but we use compression heuristics.\n")
+    
+    layer = KolmogorovComplexityLayer(4, 3, precision=8)
+    
+    # Test with different complexity patterns
+    test_cases = [
+        ([1, 1, 1, 1], "Highly regular pattern"),
+        ([1, 2, 3, 4], "Simple arithmetic sequence"), 
+        ([0.123, 0.456, 0.789, 0.012], "Seemingly random"),
+        ([1, 1, 2, 3], "Fibonacci-like pattern"),
+    ]
+    
+    for i, (input_data, description) in enumerate(test_cases, 1):
+        input_array = np.array(input_data).reshape(1, -1)
+        output = layer.forward(input_array)
+        
+        print(f"Test {i}: {description}")
+        print(f"  Input: {input_data}")
+        print(f"  Complexity estimates: {output[0]}")
+        print(f"  Average complexity: {np.mean(output[0]):.3f}")
+        print()
+
+def demonstrate_busy_beaver():
+    """Demonstrate the Busy Beaver Layer."""
+    print("=== Busy Beaver Layer Demonstration ===\n")
+    
+    print("The Busy Beaver Layer uses the Busy Beaver function BB(n),")
+    print("which gives the maximum number of steps a halting n-state Turing machine can take.")
+    print("BB(n) is uncomputable for general n, but known for small values.\n")
+    
+    layer = BusyBeaverLayer(4, 3)
+    
+    # Known BB values for reference
+    print("Known Busy Beaver values:")
+    print("  BB(1) = 1")
+    print("  BB(2) = 4") 
+    print("  BB(3) = 6")
+    print("  BB(4) = 13")
+    print("  BB(5) ≥ 4098")
+    print()
+    
+    # Test with inputs that map to different BB parameters
+    test_cases = [
+        ([0.1, 0, 0, 0], "Maps to small machine"),
+        ([0.5, 0.5, 0, 0], "Maps to medium machine"),
+        ([1.0, 1.0, 1.0, 1.0], "Maps to larger machine"),
+    ]
+    
+    for i, (input_data, description) in enumerate(test_cases, 1):
+        input_array = np.array(input_data).reshape(1, -1)
+        output = layer.forward(input_array)
+        
+        print(f"Test {i}: {description}")
+        print(f"  Input: {input_data}")
+        print(f"  Log BB values: {output[0]}")
+        print(f"  Approximate BB values: {np.exp(output[0]) - 1}")
+        print()
+
+def demonstrate_non_recursive():
+    """Demonstrate the Non-Recursive Layer."""
+    print("=== Non-Recursive Layer Demonstration ===\n")
+    
+    print("The Non-Recursive Layer simulates operations on computably enumerable")
+    print("but non-recursive sets. These are sets that can be enumerated but")
+    print("for which membership cannot be decided algorithmically.\n")
+    
+    layer = NonRecursiveLayer(4, 3, enumeration_bound=1000)
+    
+    print(f"The layer simulates a c.e. set with {len(layer.ce_set)} enumerated elements")
+    print("using the rule: numbers expressible as sum of two squares\n")
+    
+    # Test membership for different inputs
+    test_cases = [
+        ([1, 0, 0, 0], "Simple input"),
+        ([2, 3, 5, 7], "Prime-like pattern"),
+        ([1, 4, 9, 16], "Perfect squares"),
+        ([0.5, 0.25, 0.125, 0.0625], "Geometric sequence"),
+    ]
+    
+    for i, (input_data, description) in enumerate(test_cases, 1):
+        input_array = np.array(input_data).reshape(1, -1)
+        output = layer.forward(input_array)
+        
+        print(f"Test {i}: {description}")
+        print(f"  Input: {input_data}")
+        print(f"  Membership probabilities: {output[0]}")
+        print(f"  Interpretations: {['Not enumerated' if p == 0.5 else ('In set' if p == 1.0 else 'Not in set') for p in output[0]]}")
+        print()
+
+def demonstrate_complete_network():
+    """Demonstrate a complete uncomputable neural network."""
+    print("=== Complete Uncomputable Neural Network ===\n")
+    
+    print("This demonstrates a full network combining all uncomputable layer types.")
+    print("The network processes inputs through:")
+    print("  1. Halting Oracle Layer (4→5)")
+    print("  2. Kolmogorov Complexity Layer (5→4)")
+    print("  3. Busy Beaver Layer (4→3)")
+    print("  4. Non-Recursive Layer (3→2)")
+    print()
+    
+    network = create_uncomputable_network()
+    
+    # Test with various input patterns
+    test_cases = [
+        "Random data",
+        "Structured sequence", 
+        "Constant values",
+        "Alternating pattern"
+    ]
+    
+    np.random.seed(42)
+    test_inputs = [
+        np.random.randn(4),
+        np.array([1, 2, 3, 4]) / 4.0,
+        np.ones(4) * 0.5,
+        np.array([1, -1, 1, -1]) * 0.5
+    ]
+    
+    for i, (input_data, description) in enumerate(zip(test_inputs, test_cases), 1):
+        output = network.predict(input_data.reshape(1, -1))
+        
+        print(f"Test {i}: {description}")
+        print(f"  Input: {input_data}")
+        print(f"  Final output: {output[0]}")
+        print(f"  Output interpretation: Decision values for uncomputable questions")
+        print()
+    
+    # Demonstrate deterministic behavior
+    print("Deterministic behavior verification:")
+    test_input = np.random.randn(1, 4)
+    output1 = network.predict(test_input)
+    output2 = network.predict(test_input)
+    difference = np.linalg.norm(output1 - output2)
+    print(f"  Same input produces identical outputs: {difference < 1e-10}")
+    print(f"  Difference: {difference:.2e}")
+
+def main():
+    """Run all uncomputable neural network demonstrations."""
+    print("🔬 Uncomputable Neural Networks Examples")
+    print("=" * 60)
+    print("Exploring the theoretical boundaries of computation through")
+    print("neural networks that incorporate uncomputable functions.\n")
+    
+    demonstrate_halting_oracle()
+    print("\n" + "─" * 60 + "\n")
+    
+    demonstrate_kolmogorov_complexity()
+    print("\n" + "─" * 60 + "\n")
+    
+    demonstrate_busy_beaver()
+    print("\n" + "─" * 60 + "\n")
+    
+    demonstrate_non_recursive()
+    print("\n" + "─" * 60 + "\n")
+    
+    demonstrate_complete_network()
+    
+    print("\n" + "=" * 60)
+    print("✅ Uncomputable neural network demonstrations completed!")
+    print("📚 See theory/uncomputable_networks.md for mathematical foundations.")
+
+if __name__ == "__main__":
+    main()

test_comprehensive.py

@@ -0,0 +1,327 @@
+#!/usr/bin/env python3
+"""
+Comprehensive test suite for Algebraic Neural Networks
+
+This script tests all components of the algebraic neural network implementation
+to ensure everything works correctly together.
+"""
+
+import sys
+import os
+import numpy as np
+
+# Add the parent directory to the path to import our modules
+sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
+
+# Import all our implementations
+from algebraic_neural_network import (
+    AlgebraicNeuralNetwork, PolynomialLayer, GroupTheoryLayer, 
+    GeometricAlgebraLayer, HaltingOracleLayer, KolmogorovComplexityLayer,
+    BusyBeaverLayer, NonRecursiveLayer, create_sample_network, create_uncomputable_network
+)
+
+def test_basic_functionality():
+    """Test basic functionality of all layer types."""
+    print("=== Testing Basic Functionality ===\n")
+    
+    # Test data
+    test_input = np.random.randn(3, 4)
+    print(f"Test input shape: {test_input.shape}")
+    
+    # Test individual layers
+    print("\n1. Testing PolynomialLayer:")
+    poly_layer = PolynomialLayer(4, 3, degree=2)
+    poly_output = poly_layer.forward(test_input)
+    print(f"   Input: {test_input.shape} → Output: {poly_output.shape}")
+    print(f"   Output range: [{np.min(poly_output):.3f}, {np.max(poly_output):.3f}]")
+    
+    print("\n2. Testing GroupTheoryLayer:")
+    group_layer = GroupTheoryLayer(4, 3, group_order=6)
+    group_output = group_layer.forward(test_input)
+    print(f"   Input: {test_input.shape} → Output: {group_output.shape}")
+    print(f"   Output range: [{np.min(group_output):.3f}, {np.max(group_output):.3f}]")
+    
+    print("\n3. Testing GeometricAlgebraLayer:")
+    geo_layer = GeometricAlgebraLayer(4, 3)
+    geo_output = geo_layer.forward(test_input)
+    print(f"   Input: {test_input.shape} → Output: {geo_output.shape}")
+    print(f"   Output range: [{np.min(geo_output):.3f}, {np.max(geo_output):.3f}]")
+    
+    return True
+
+def test_network_composition():
+    """Test composition of multiple algebraic layers."""
+    print("\n=== Testing Network Composition ===\n")
+    
+    # Create a complete network
+    network = AlgebraicNeuralNetwork()
+    network.add_layer(PolynomialLayer(5, 8, degree=2))
+    network.add_layer(GroupTheoryLayer(8, 6, group_order=8))
+    network.add_layer(GeometricAlgebraLayer(6, 3))
+    network.add_layer(PolynomialLayer(3, 2, degree=1))
+    
+    # Test with different input sizes
+    test_cases = [
+        np.random.randn(1, 5),   # Single sample
+        np.random.randn(5, 5),   # Multiple samples
+        np.random.randn(10, 5),  # Larger batch
+    ]
+    
+    for i, test_case in enumerate(test_cases):
+        output = network.predict(test_case)
+        print(f"Test case {i+1}:")
+        print(f"   Input shape: {test_case.shape}")
+        print(f"   Output shape: {output.shape}")
+        print(f"   Output mean: {np.mean(output):.4f}")
+        print(f"   Output std: {np.std(output):.4f}")
+    
+    return True
+
+def test_deterministic_behavior():
+    """Test that the networks are deterministic."""
+    print("\n=== Testing Deterministic Behavior ===\n")
+    
+    # Create network
+    network = create_sample_network()
+    
+    # Same input should produce same output
+    test_input = np.random.randn(3, 4)
+    
+    output1 = network.predict(test_input)
+    output2 = network.predict(test_input)
+    output3 = network.predict(test_input)
+    
+    # Check if outputs are identical
+    diff_12 = np.linalg.norm(output1 - output2)
+    diff_13 = np.linalg.norm(output1 - output3)
+    diff_23 = np.linalg.norm(output2 - output3)
+    
+    print(f"Input shape: {test_input.shape}")
+    print(f"Output 1 vs 2 difference: {diff_12:.10f}")
+    print(f"Output 1 vs 3 difference: {diff_13:.10f}")
+    print(f"Output 2 vs 3 difference: {diff_23:.10f}")
+    
+    is_deterministic = (diff_12 < 1e-10) and (diff_13 < 1e-10) and (diff_23 < 1e-10)
+    print(f"Network is deterministic: {is_deterministic}")
+    
+    return is_deterministic
+
+def test_mathematical_properties():
+    """Test mathematical properties of algebraic operations."""
+    print("\n=== Testing Mathematical Properties ===\n")
+    
+    # Test polynomial layer properties
+    print("1. Polynomial Layer Properties:")
+    poly_layer = PolynomialLayer(2, 3, degree=2)
+    
+    # Linearity test (for degree 1 components)
+    x1 = np.array([[1, 0]])
+    x2 = np.array([[0, 1]])
+    x_sum = np.array([[1, 1]])
+    
+    y1 = poly_layer.forward(x1)
+    y2 = poly_layer.forward(x2)
+    y_sum = poly_layer.forward(x_sum)
+    
+    # Note: Due to higher degree terms, this won't be exactly linear
+    linearity_error = np.linalg.norm(y_sum - (y1 + y2))
+    print(f"   Linearity deviation (expected for degree > 1): {linearity_error:.4f}")
+    
+    # Test group theory layer properties
+    print("\n2. Group Theory Layer Properties:")
+    group_layer = GroupTheoryLayer(2, 4, group_order=4)
+    
+    # Test with unit vectors
+    unit_x = np.array([[1, 0]])
+    unit_y = np.array([[0, 1]])
+    
+    out_x = group_layer.forward(unit_x)
+    out_y = group_layer.forward(unit_y)
+    
+    print(f"   Unit X output norm: {np.linalg.norm(out_x):.4f}")
+    print(f"   Unit Y output norm: {np.linalg.norm(out_y):.4f}")
+    
+    # Test geometric algebra layer properties
+    print("\n3. Geometric Algebra Layer Properties:")
+    geo_layer = GeometricAlgebraLayer(3, 4)
+    
+    # Test with orthogonal vectors
+    e1 = np.array([[1, 0, 0]])
+    e2 = np.array([[0, 1, 0]])
+    e3 = np.array([[0, 0, 1]])
+    
+    out1 = geo_layer.forward(e1)
+    out2 = geo_layer.forward(e2)
+    out3 = geo_layer.forward(e3)
+    
+    print(f"   e1 output: {out1[0]}")
+    print(f"   e2 output: {out2[0]}")
+    print(f"   e3 output: {out3[0]}")
+    
+    return True
+
+def test_uncomputable_layers():
+    """Test uncomputable neural network layers."""
+    print("\n=== Testing Uncomputable Layers ===\n")
+    
+    test_input = np.random.randn(3, 4)
+    
+    # Test Halting Oracle Layer
+    print("1. Testing HaltingOracleLayer:")
+    halting_layer = HaltingOracleLayer(4, 3, max_iterations=100)
+    halting_output = halting_layer.forward(test_input)
+    print(f"   Input: {test_input.shape} → Output: {halting_output.shape}")
+    print(f"   Output range: [{np.min(halting_output):.3f}, {np.max(halting_output):.3f}]")
+    # Outputs should be probabilities between 0 and 1
+    assert np.all(halting_output >= 0) and np.all(halting_output <= 1), "Halting oracle outputs must be in [0,1]"
+    
+    # Test Kolmogorov Complexity Layer
+    print("\n2. Testing KolmogorovComplexityLayer:")
+    kolmogorov_layer = KolmogorovComplexityLayer(4, 3, precision=6)
+    kolmogorov_output = kolmogorov_layer.forward(test_input)
+    print(f"   Input: {test_input.shape} → Output: {kolmogorov_output.shape}")
+    print(f"   Output range: [{np.min(kolmogorov_output):.3f}, {np.max(kolmogorov_output):.3f}]")
+    # Complexity should be non-negative
+    assert np.all(kolmogorov_output >= 0), "Kolmogorov complexity must be non-negative"
+    
+    # Test Busy Beaver Layer
+    print("\n3. Testing BusyBeaverLayer:")
+    bb_layer = BusyBeaverLayer(4, 3)
+    bb_output = bb_layer.forward(test_input)
+    print(f"   Input: {test_input.shape} → Output: {bb_output.shape}")
+    print(f"   Output range: [{np.min(bb_output):.3f}, {np.max(bb_output):.3f}]")
+    # BB values should be positive
+    assert np.all(bb_output > 0), "Busy Beaver values must be positive"
+    
+    # Test Non-Recursive Layer
+    print("\n4. Testing NonRecursiveLayer:")
+    nr_layer = NonRecursiveLayer(4, 3, enumeration_bound=100)
+    nr_output = nr_layer.forward(test_input)
+    print(f"   Input: {test_input.shape} → Output: {nr_output.shape}")
+    print(f"   Output range: [{np.min(nr_output):.3f}, {np.max(nr_output):.3f}]")
+    # Membership values should be in [0,1]
+    assert np.all(nr_output >= 0) and np.all(nr_output <= 1), "Membership values must be in [0,1]"
+    
+    # Test deterministic behavior of uncomputable layers
+    print("\n5. Testing deterministic behavior:")
+    halting_output2 = halting_layer.forward(test_input)
+    diff = np.linalg.norm(halting_output - halting_output2)
+    print(f"   Determinism check: difference = {diff:.10f}")
+    assert diff < 1e-10, "Uncomputable layers must be deterministic"
+    
+    return True
+
+def test_uncomputable_network_composition():
+    """Test composition of uncomputable neural network."""
+    print("\n=== Testing Uncomputable Network Composition ===\n")
+    
+    # Create uncomputable network
+    network = create_uncomputable_network()
+    
+    # Test with different input sizes
+    test_cases = [
+        (1, 4),    # Single sample
+        (5, 4),    # Multiple samples
+        (10, 4)    # Larger batch
+    ]
+    
+    for i, (batch_size, input_size) in enumerate(test_cases, 1):
+        test_input = np.random.randn(batch_size, input_size)
+        output = network.predict(test_input)
+        
+        print(f"Test case {i}:")
+        print(f"   Input shape: {test_input.shape}")
+        print(f"   Output shape: {output.shape}")
+        print(f"   Output mean: {np.mean(output):.4f}")
+        print(f"   Output std: {np.std(output):.4f}")
+    
+    return True
+
+def test_edge_cases():
+    """Test edge cases and boundary conditions."""
+    print("\n=== Testing Edge Cases ===\n")
+    
+    network = create_sample_network()
+    
+    # Test with zero input
+    zero_input = np.zeros((2, 4))
+    zero_output = network.predict(zero_input)
+    print(f"1. Zero input test:")
+    print(f"   Input: all zeros, shape {zero_input.shape}")
+    print(f"   Output: {zero_output}")
+    
+    # Test with very small inputs
+    small_input = np.ones((2, 4)) * 1e-6
+    small_output = network.predict(small_input)
+    print(f"\n2. Small input test:")
+    print(f"   Input: 1e-6, shape {small_input.shape}")
+    print(f"   Output range: [{np.min(small_output):.8f}, {np.max(small_output):.8f}]")
+    
+    # Test with large inputs
+    large_input = np.ones((2, 4)) * 100
+    large_output = network.predict(large_input)
+    print(f"\n3. Large input test:")
+    print(f"   Input: 100, shape {large_input.shape}")
+    print(f"   Output range: [{np.min(large_output):.3f}, {np.max(large_output):.3f}]")
+    
+    # Test with single sample
+    single_input = np.random.randn(4)  # 1D input
+    single_output = network.predict(single_input)
+    print(f"\n4. Single sample test:")
+    print(f"   Input shape: {single_input.shape}")
+    print(f"   Output shape: {single_output.shape}")
+    
+    return True
+
+def run_comprehensive_test():
+    """Run all tests and report results."""
+    print("Comprehensive Algebraic Neural Network Test Suite")
+    print("="*60)
+    
+    tests = [
+        ("Basic Functionality", test_basic_functionality),
+        ("Network Composition", test_network_composition), 
+        ("Deterministic Behavior", test_deterministic_behavior),
+        ("Mathematical Properties", test_mathematical_properties),
+        ("Uncomputable Layers", test_uncomputable_layers),
+        ("Uncomputable Network Composition", test_uncomputable_network_composition),
+        ("Edge Cases", test_edge_cases),
+    ]
+    
+    results = []
+    
+    for test_name, test_func in tests:
+        try:
+            result = test_func()
+            results.append((test_name, result, None))
+            print(f"\n✓ {test_name}: PASSED")
+        except Exception as e:
+            results.append((test_name, False, str(e)))
+            print(f"\n✗ {test_name}: FAILED - {e}")
+    
+    # Summary
+    print("\n" + "="*60)
+    print("TEST SUMMARY")
+    print("="*60)
+    
+    passed = sum(1 for _, result, _ in results if result)
+    total = len(results)
+    
+    for test_name, result, error in results:
+        status = "PASS" if result else "FAIL"
+        print(f"{test_name:.<30} {status}")
+        if error:
+            print(f"    Error: {error}")
+    
+    print(f"\nOverall: {passed}/{total} tests passed")
+    
+    if passed == total:
+        print("🎉 All tests passed! Algebraic Neural Network implementation is working correctly.")
+    else:
+        print("⚠️  Some tests failed. Please review the implementation.")
+    
+    return passed == total
+
+if __name__ == "__main__":
+    success = run_comprehensive_test()
+    sys.exit(0 if success else 1)

theory/algebraic_foundations.md

@@ -0,0 +1,162 @@
+# Algebraic Foundations of Non-Trained Neural Networks
+
+## Introduction
+
+Algebraic Neural Networks (ANNs) represent a fundamental departure from traditional neural networks by eliminating the need for gradient-based training. Instead, they leverage mathematical structures from abstract algebra to perform computations.
+
+## Mathematical Foundations
+
+### 1. Algebraic Structures
+
+#### Groups
+A group (G, ∘) is a set G with a binary operation ∘ that satisfies:
+- **Closure**: ∀ a, b ∈ G, a ∘ b ∈ G
+- **Associativity**: ∀ a, b, c ∈ G, (a ∘ b) ∘ c = a ∘ (b ∘ c)
+- **Identity**: ∃ e ∈ G such that ∀ a ∈ G, e ∘ a = a ∘ e = a
+- **Inverse**: ∀ a ∈ G, ∃ a⁻¹ ∈ G such that a ∘ a⁻¹ = a⁻¹ ∘ a = e
+
+In our neural networks, we use group actions to transform input data systematically.
+
+#### Rings and Fields
+A ring (R, +, ×) provides two operations (addition and multiplication) that interact via distributive laws. Fields extend rings by requiring multiplicative inverses for non-zero elements.
+
+#### Algebras
+An algebra over a field F is a vector space A over F equipped with a bilinear multiplication operation.
+
+### 2. Polynomial Algebras
+
+Polynomial algebras form the basis for our polynomial layers. For a field F and variables x₁, x₂, ..., xₙ, the polynomial algebra F[x₁, x₂, ..., xₙ] consists of all polynomials in these variables.
+
+#### Key Properties:
+- **Linearity**: P(ax + by) = aP(x) + bP(y) for linear terms
+- **Homomorphism**: Polynomial mappings preserve algebraic structure
+- **Universal Property**: Polynomial algebras are initial objects in certain categories
+
+### 3. Geometric Algebra (Clifford Algebra)
+
+Geometric algebra extends vector algebra with a geometric product that unifies dot and cross products.
+
+For vectors a and b:
+**Geometric Product**: ab = a·b + a∧b
+
+Where:
+- a·b is the dot product (scalar)
+- a∧b is the outer product (bivector)
+
+#### Properties:
+- **Associative**: (ab)c = a(bc)
+- **Distributive**: a(b + c) = ab + ac
+- **Contraction**: a² = |a|² (for vectors)
+
+### 4. Group Actions in Neural Networks
+
+A group action of G on a set X is a function G × X → X such that:
+- Identity action: ex = x for all x ∈ X
+- Compatibility: (gh)x = g(hx)
+
+In neural networks, we use group actions to:
+- Transform input features systematically
+- Preserve important symmetries
+- Generate multiple representations
+
+## Algebraic Neural Network Architecture
+
+### Layer Types
+
+#### 1. Polynomial Layers
+Transform inputs using polynomial functions with algebraically determined coefficients:
+
+```
+f(x) = Σᵢ₌₁ⁿ Σⱼ₌₁ᵈ (aᵢⱼ/j!) xʲ
+```
+
+Where aᵢⱼ are coefficients derived from algebraic sequences (e.g., involving golden ratio φ).
+
+#### 2. Group Theory Layers
+Apply group elements to transform inputs:
+
+```
+y = g · x for g ∈ G
+```
+
+Common groups used:
+- **Cyclic groups**: Cₙ = {e, g, g², ..., gⁿ⁻¹}
+- **Dihedral groups**: Symmetries of regular polygons
+- **Symmetric groups**: Permutations of elements
+
+#### 3. Geometric Algebra Layers
+Use geometric product operations:
+
+```
+y = x ∘ eᵢ
+```
+
+Where eᵢ are basis elements of the geometric algebra.
+
+### Composition of Layers
+
+Layers compose through function composition, preserving algebraic properties:
+
+```
+f = fₙ ∘ fₙ₋₁ ∘ ... ∘ f₁
+```
+
+## Advantages of Algebraic Approach
+
+### 1. No Training Required
+- Networks are constructed using mathematical principles
+- No gradient computation needed
+- No optimization algorithms required
+
+### 2. Deterministic Behavior
+- Outputs are completely determined by algebraic rules
+- Reproducible results
+- No randomness in weights
+
+### 3. Mathematical Interpretability
+- Clear mathematical meaning for each operation
+- Theoretical guarantees on behavior
+- Connection to established mathematical theory
+
+### 4. Computational Efficiency
+- No backpropagation required
+- Direct computation of outputs
+- Parallel computation of group actions
+
+## Theoretical Guarantees
+
+### Universal Approximation
+Under certain conditions, algebraic neural networks can approximate continuous functions:
+
+**Theorem**: Let f: Rⁿ → Rᵐ be a continuous function on a compact set K. Then for any ε > 0, there exists an algebraic neural network F such that ||f - F||∞ < ε on K.
+
+### Stability Properties
+Algebraic operations provide stability guarantees:
+- Polynomial layers have bounded derivatives
+- Group actions preserve norms (for orthogonal groups)
+- Geometric algebra operations maintain geometric relationships
+
+## Applications
+
+### 1. Signal Processing
+- Fourier transforms as group actions
+- Wavelet transforms using algebraic structures
+- Filter design using polynomial algebras
+
+### 2. Computer Vision
+- Geometric transformations using group theory
+- Feature extraction using geometric algebra
+- Invariant pattern recognition
+
+### 3. Scientific Computing
+- Solving differential equations
+- Numerical integration
+- Optimization problems with algebraic constraints
+
+## References
+
+1. Clifford, W.K. (1878). "Applications of Grassmann's Extensive Algebra"
+2. Doran, C. & Lasenby, A. (2003). "Geometric Algebra for Physicists"
+3. Rotman, J.J. (2012). "A First Course in Abstract Algebra"
+4. MacLane, S. & Birkhoff, G. (1999). "Algebra"
+5. Cybenko, G. (1989). "Approximation by Superpositions of a Sigmoidal Function"

theory/examples.md

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+# Worked Examples of Algebraic Neural Networks
+
+## Example 1: Polynomial Network for Function Approximation
+
+### Problem
+Approximate the function f(x, y) = x² + 2xy + y² using an algebraic neural network.
+
+### Solution
+
+```python
+import numpy as np
+from algebraic_neural_network import PolynomialLayer
+
+# Create a polynomial layer
+poly_layer = PolynomialLayer(input_size=2, output_size=1, degree=2)
+
+# Test points
+test_points = np.array([
+    [1, 1],    # f(1,1) = 1 + 2 + 1 = 4
+    [2, 3],    # f(2,3) = 4 + 12 + 9 = 25
+    [0, 1],    # f(0,1) = 0 + 0 + 1 = 1
+    [-1, 2]    # f(-1,2) = 1 - 4 + 4 = 1
+])
+
+# Apply the polynomial transformation
+output = poly_layer.forward(test_points)
+print("Polynomial approximation results:", output.flatten())
+```
+
+### Mathematical Analysis
+The polynomial layer generates coefficients using the golden ratio φ = (1 + √5)/2:
+- For output neuron 1: coefficient matrix uses φ¹ and φ²
+- The transformation applies: y = Σᵢ Σⱼ (aᵢⱼ/j!) xʲ
+
+## Example 2: Group Theory Network for Rotation Invariance
+
+### Problem
+Create a network that recognizes patterns invariant under rotations.
+
+### Solution
+
+```python
+import numpy as np
+from algebraic_neural_network import GroupTheoryLayer
+
+# Create a group theory layer using 8-fold rotational symmetry
+group_layer = GroupTheoryLayer(input_size=2, output_size=4, group_order=8)
+
+# Test with a simple pattern (vector pointing in different directions)
+patterns = np.array([
+    [1, 0],      # Point along x-axis
+    [0, 1],      # Point along y-axis
+    [1/√2, 1/√2], # Point along 45° diagonal
+    [-1, 0]      # Point along negative x-axis
+])
+
+# Apply group transformations
+transformed = group_layer.forward(patterns)
+print("Group theory transformation results:")
+print(transformed)
+```
+
+### Mathematical Analysis
+The group theory layer applies rotations from the cyclic group C₈:
+- Rotation angles: 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°
+- Each rotation is represented by a 2×2 rotation matrix
+- The norm of the transformed vector provides rotation-invariant features
+
+## Example 3: Geometric Algebra Network for 3D Processing
+
+### Problem
+Process 3D geometric data using geometric algebra operations.
+
+### Solution
+
+```python
+import numpy as np
+from algebraic_neural_network import GeometricAlgebraLayer
+
+# Create geometric algebra layer
+geo_layer = GeometricAlgebraLayer(input_size=3, output_size=4)
+
+# 3D vectors representing different geometric entities
+vectors = np.array([
+    [1, 0, 0],    # Unit vector along x
+    [0, 1, 0],    # Unit vector along y  
+    [0, 0, 1],    # Unit vector along z
+    [1, 1, 1]     # Diagonal vector
+])
+
+# Apply geometric algebra transformations
+geo_output = geo_layer.forward(vectors)
+print("Geometric algebra results:")
+print(geo_output)
+```
+
+### Mathematical Analysis
+The geometric algebra layer computes:
+- Scalar products: a·b
+- Vector products: a∧b (bivectors)
+- Trivector products: a∧b∧c
+- Mixed products combining all grades
+
+## Example 4: Complete Network for Pattern Classification
+
+### Problem
+Build a complete algebraic neural network for classifying 2D patterns.
+
+### Solution
+
+```python
+from algebraic_neural_network import AlgebraicNeuralNetwork, PolynomialLayer, GroupTheoryLayer
+
+# Create network architecture
+network = AlgebraicNeuralNetwork()
+network.add_layer(PolynomialLayer(2, 4, degree=2))      # Feature extraction
+network.add_layer(GroupTheoryLayer(4, 3, group_order=6)) # Symmetry processing
+network.add_layer(PolynomialLayer(3, 1, degree=1))      # Final classification
+
+# Test patterns
+circle_points = np.array([
+    [np.cos(θ), np.sin(θ)] for θ in np.linspace(0, 2*np.pi, 8)
+])
+
+square_points = np.array([
+    [1, 1], [1, -1], [-1, -1], [-1, 1],
+    [1, 0], [0, 1], [-1, 0], [0, -1]
+])
+
+# Classify patterns
+circle_scores = network.predict(circle_points)
+square_scores = network.predict(square_points)
+
+print("Circle pattern scores:", np.mean(circle_scores))
+print("Square pattern scores:", np.mean(square_scores))
+```
+
+### Analysis
+This network demonstrates:
+1. **Feature Extraction**: Polynomial layer extracts nonlinear features
+2. **Symmetry Processing**: Group theory layer handles rotational symmetries
+3. **Classification**: Final layer provides decision boundary
+
+## Example 5: Time Series Processing with Algebraic Networks
+
+### Problem
+Process time series data using algebraic transformations.
+
+### Solution
+
+```python
+import numpy as np
+
+def create_time_series_network():
+    network = AlgebraicNeuralNetwork()
+    # Window-based polynomial features
+    network.add_layer(PolynomialLayer(5, 6, degree=2))  # 5-point window
+    # Temporal symmetries
+    network.add_layer(GroupTheoryLayer(6, 4, group_order=4))
+    # Final prediction
+    network.add_layer(PolynomialLayer(4, 1, degree=1))
+    return network
+
+# Generate sample time series
+t = np.linspace(0, 4*np.pi, 100)
+signal = np.sin(t) + 0.3*np.sin(3*t) + 0.1*np.random.randn(100)
+
+# Create windows of 5 consecutive points
+windows = np.array([signal[i:i+5] for i in range(len(signal)-4)])
+
+# Process with algebraic network
+ts_network = create_time_series_network()
+predictions = ts_network.predict(windows)
+
+print(f"Processed {len(windows)} time windows")
+print(f"Prediction range: [{np.min(predictions):.3f}, {np.max(predictions):.3f}]")
+```
+
+### Analysis
+This demonstrates algebraic networks for temporal data:
+- **Windowing**: Convert time series to fixed-size vectors
+- **Polynomial Features**: Capture local nonlinear patterns
+- **Temporal Symmetries**: Handle time-shift invariances
+
+## Performance Characteristics
+
+### Computational Complexity
+- **Polynomial Layers**: O(nd) where n is input size, d is degree
+- **Group Theory Layers**: O(ng) where g is group order
+- **Geometric Algebra Layers**: O(n²) for geometric products
+
+### Memory Requirements
+- **Fixed Coefficients**: No weight storage needed
+- **Intermediate Results**: Only temporary computation storage
+- **Total Memory**: O(n) where n is largest layer size
+
+### Accuracy Analysis
+Algebraic networks provide:
+- **Consistency**: Same input always produces same output
+- **Stability**: Small input changes → small output changes
+- **Interpretability**: Mathematical meaning for each operation
+
+## Practical Considerations
+
+### When to Use Algebraic Networks
+- **Known Mathematical Structure**: Problem has clear algebraic properties
+- **No Training Data**: When gradient-based training isn't feasible
+- **Interpretability Required**: Need mathematical understanding of operations
+- **Real-time Processing**: Fast, deterministic computation needed
+
+### Limitations
+- **Limited Expressivity**: May not capture all possible patterns
+- **Parameter Selection**: Choosing group orders, polynomial degrees
+- **Scaling**: Performance with very high-dimensional data
+
+### Extensions
+- **Adaptive Coefficients**: Use algebraic sequences that adapt to data
+- **Hybrid Networks**: Combine with traditional neural networks
+- **Custom Algebras**: Develop problem-specific algebraic structures

theory/uncomputable_networks.md

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+# Uncomputable Neural Networks
+
+## Introduction
+
+Uncomputable Neural Networks represent a theoretical extension of non-trained neural networks that incorporate uncomputable functions and non-algorithmic operations. These networks explore the boundaries of computation and demonstrate concepts from theoretical computer science.
+
+## Mathematical Foundations
+
+### 1. Computability Theory
+
+#### Computable vs Uncomputable Functions
+- **Computable functions**: Can be computed by a Turing machine in finite time
+- **Uncomputable functions**: Cannot be computed by any algorithm (e.g., halting problem)
+
+#### The Halting Problem
+The halting problem asks: Given a program P and input I, will P halt on I?
+This is formally uncomputable - no algorithm can solve it for all cases.
+
+### 2. Oracle Machines
+
+An Oracle machine is a theoretical Turing machine with access to an "oracle" that can answer questions about uncomputable problems in constant time.
+
+#### Oracle Operations
+- **Halting Oracle**: O_H(P,I) = 1 if program P halts on input I, 0 otherwise
+- **Kolmogorov Oracle**: O_K(x) returns the Kolmogorov complexity of string x
+- **Busy Beaver Oracle**: O_BB(n) returns the nth Busy Beaver number
+
+### 3. Hypercomputation
+
+Hypercomputation refers to computational models that can solve uncomputable problems:
+- Infinite time Turing machines
+- Analog computers with real number precision
+- Quantum computers with hypothetical capabilities
+
+## Uncomputable Neural Network Architecture
+
+### Layer Types
+
+#### 1. Halting Oracle Layer
+Simulates access to a halting oracle for specific problem domains:
+
+```
+f(x) = O_H(encode(x), input_program)
+```
+
+Where `encode(x)` transforms the input into a program representation.
+
+#### 2. Kolmogorov Complexity Layer
+Approximates Kolmogorov complexity using compression-based heuristics:
+
+```
+f(x) = K_approx(x) = min{|p| : U(p) = x, |p| ≤ threshold}
+```
+
+#### 3. Busy Beaver Layer
+Uses known Busy Beaver values and approximations for larger inputs:
+
+```
+f(x) = BB(⌊log₂(||x||)⌋)
+```
+
+#### 4. Non-Recursive Enumeration Layer
+Operates on sets that are computably enumerable but not computable:
+
+```
+f(x) = indicator_function(x ∈ RE_set)
+```
+
+### Composition and Determinism
+
+Despite incorporating uncomputable concepts, these networks maintain practical determinism through:
+- Finite approximations of infinite processes
+- Bounded computation with oracle simulation
+- Heuristic approaches to uncomputable problems
+
+## Implementation Strategies
+
+### 1. Oracle Simulation
+- Use lookup tables for known cases
+- Apply heuristics for unknown cases
+- Incorporate randomness with fixed seeds for determinism
+
+### 2. Bounded Approximation
+- Limit computation depth to maintain practical execution
+- Use asymptotic behaviors for large inputs
+- Approximate infinite processes with finite resources
+
+### 3. Theoretical Consistency
+- Maintain mathematical rigor in approximations
+- Document assumptions and limitations
+- Preserve algebraic properties where possible
+
+## Applications and Use Cases
+
+### 1. Complexity Analysis
+- Estimating computational complexity of algorithms
+- Pattern recognition in program behavior
+- Automated theorem proving assistance
+
+### 2. Theoretical Computer Science Education
+- Demonstrating computability concepts
+- Exploring limits of computation
+- Understanding oracle hierarchies
+
+### 3. Research in Hypercomputation
+- Modeling hypothetical computational paradigms
+- Investigating quantum computational advantages
+- Exploring analog computation limits
+
+## Limitations and Considerations
+
+### 1. Practical Constraints
+- True uncomputable functions cannot be implemented
+- All implementations are approximations or simulations
+- Finite resources limit theoretical completeness
+
+### 2. Determinism vs Uncomputability
+- Balance between theoretical concepts and practical determinism
+- Fixed seed randomness for reproducible "non-algorithmic" behavior
+- Clear documentation of approximation methods
+
+### 3. Verification Challenges
+- Difficult to verify correctness of uncomputable approximations
+- Limited testing capabilities for infinite processes
+- Reliance on theoretical foundations rather than empirical validation
+
+## Relationship to Other Non-Trained Networks
+
+Uncomputable neural networks extend the paradigm of non-trained networks by:
+- Eliminating not just training but algorithmic computation itself
+- Incorporating theoretical computer science concepts
+- Exploring computational limits and capabilities
+- Maintaining deterministic behavior through careful design
+
+## References
+
+1. Turing, A. M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem"
+2. Rogers, H. (1987). "Theory of Recursive Functions and Effective Computability"
+3. Copeland, B. J. (2002). "Hypercomputation: philosophical issues"
+4. Beggs, E., Costa, J. F., Tucker, J. V. (2012). "The impact of models of a physical oracle on computational power"
+5. Aaronson, S. (2013). "Quantum Computing since Democritus"