Mnemoverse Mathematical Theory: Complete Appendices

Author: Edgard Izgorodin (izgorodin)
Last Updated: 2025-07-13
Version: 0.5.0
Status: Draft
Linked Document: Mnemoverse: Core Mathematical Theory

This document contains the complete appendices for the Mnemoverse mathematical framework, including detailed proofs, implementation code, and experimental protocols. It serves as a comprehensive reference for researchers and developers implementing the Mnemoverse architecture.

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Table of Contents

Appendix A: Mathematical Proofs

• A.1 Lemma Proofs (L1-L6) - Complete mathematical derivations with bounds and stability analysis
• A.2 Theorem Proofs (T1-T3) - Rigorous proofs of optimal embedding, global stability, and efficient access
• A.3 Supporting Mathematical Results - Additional lemmas and corollaries

Appendix B: Implementation Code

• B.1 Core Hyperbolic Geometry Library - Production-ready hyperbolic operations with GPU acceleration
• B.2 Attention and Metric System - Dynamic metric warping based on attention distribution
• B.3 Memory Diffusion Engine - Numerical implementation of memory evolution dynamics
• B.4 Scale-Space Query System - Multi-scale proximity queries with logarithmic complexity
• B.5 GPU Optimization Kernels - CUDA kernels for performance-critical operations

Appendix C: Experimental Protocols

• C.1 Dataset Preparation - Hierarchical dataset generation and validation
• C.2 Benchmark Methodology - Performance evaluation and scaling analysis
• C.3 Reproducibility Guidelines - Complete experimental setup and computational requirements

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Appendix A: Mathematical Proofs

See also: Main Theory — Lemmas & Theorems

A.1 Lemma Proofs

Proof of Lemma L1: Scale-Smoothness Property

Statement: Under Axiom A1, the family of mappings satisfies:

Complete Proof:

We begin with the scale-space representation from Axiom A1:

Step 1: Heat Kernel Properties

On the hyperbolic manifold , the heat kernel admits the asymptotic expansion:

Taking the derivative with respect to :

Step 2: Scale Derivative of Convolution

The scale derivative of is:

Substituting the heat kernel derivative:

Step 3: L² Norm Computation

We need to bound:

Using the Cauchy-Schwarz inequality on the inner integral:

Step 4: Heat Kernel Self-Convolution

A key property is that:

Step 5: Moment Bounds

For the second integral, we use the moment formula:

For :

Step 6: Final Bound

Combining all estimates:

Since we typically work in fixed dimension , the dominant term gives:

This completes the proof. □

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Proof of Lemma L2: Context Contraction Principle

Statement: Under Axiom A2, the attention-modified metric satisfies the bounds:

Complete Proof:

We start from the metric modification formula in Axiom A2:

Step 1: Establishing the Integral Bounds

Since and , the integral is non-negative:

For the upper bound:

Since the kernel is translation-invariant in our hyperbolic setting:

Step 2: Matrix Inequality for Upper Bound

The multiplicative factor is bounded by:

Since is positive definite, for any tangent vector :

This gives the upper bound:

Step 3: Lower Bound and Positive Definiteness

For the metric to remain positive definite, we need the multiplicative factor to be strictly positive. From the constraint in Axiom A2:

This ensures:

To get a uniform lower bound, consider the worst-case scenario. If the attention field were to decrease the metric maximally, the factor would approach:

However, since our integral is always non-negative, we actually have:

The last inequality provides a safety margin for numerical stability.

Step 4: Verification of Matrix Ordering

For matrices , we write if is positive semidefinite. Our bounds state:

These inequalities hold in the sense of quadratic forms on , ensuring well-conditioning of the metric tensor. □

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Proof of Lemma L3: Energy Conservation Bound

Statement: Under Axiom A3, the total energy satisfies:

Complete Proof:

We start with the diffusion-decay equation from Axiom A3:

Step 1: Total Energy Evolution

Define the total energy:

Taking the time derivative:

Step 2: Slow Metric Variation

We assume the metric varies slowly compared to the energy field dynamics. This is justified because attention shifts occur on cognitive timescales much longer than memory activation timescales. Thus:

Step 3: Applying the Evolution Equation

Substituting the PDE:

Step 4: Green's Theorem on Manifolds

For the Laplacian term, we apply Green's theorem. On a compact manifold without boundary:

However, integrating by parts again:

For and , and with (no boundary):

Step 5: Solving the ODE

We obtain the simple ODE:

This has the unique solution:

Physical Interpretation: The total activation energy decays exponentially with time constant , modeling gradual forgetting while preserving the relative structure of memories through the diffusion process. □

Proof of Lemma L4: Hyperbolic Distortion Bound (Extended)

Statement: For a tree with branching factor and depth :

• Hyperbolic :
• Euclidean :
• Spherical : for

Complete Proof:

We analyze embedding distortion in each geometry separately.

Part 1: Hyperbolic Embedding

We construct an explicit embedding in the Poincaré ball model .

Construction: For a node at depth with path from root :

1. Radial coordinate:
2. Angular coordinate: Map the discrete path to a point on

Specifically:

where assigns well-separated points on the unit sphere to different paths.

Distance Analysis:

For parent-child pairs:

Using the identity , this simplifies to approximately 1 for large .

For siblings at depth :

With proper angular separation , we get:

Distortion Bound:

Part 2: Euclidean Lower Bound

Volume Packing Argument:

To embed nodes with minimum pairwise distance :

1. Each node needs a ball of radius around it
2. Total volume needed:
3. This must fit in a ball of radius :

Therefore: , giving

Since the tree has diameter , the distortion is at least:

Part 3: Spherical Impossibility

In spherical geometry , the maximum distance between any two points is .

For a tree of depth , the root and a leaf are at tree distance . If , no isometric embedding exists.

More precisely, even allowing distortion, we need . For and practical values of , the required distortion becomes prohibitive, effectively .

Conclusion: Hyperbolic geometry provides exponentially better distortion bounds than Euclidean space for hierarchical structures, while spherical geometry fails entirely for deep hierarchies. □

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Proof of Lemma L5: Local Stability of Equilibria

Statement: For the coupled system (A2+A3), equilibria with negative eigenvalues are asymptotically stable.

Complete Proof:

Consider the full coupled dynamics:

where is a smooth function relating energy to attention.

Step 1: Equilibrium Conditions

At equilibrium :

Step 2: Linearization

Consider perturbations:

Linearizing the system:

Since involves second-order terms in , to first order:

Step 3: Spectral Decomposition

Let be the eigenfunctions of with eigenvalues :

Expanding in this basis:

Substituting into the linearized equation:

Step 4: Mode Evolution

Each mode evolves independently:

Solution:

Step 5: Stability Analysis

Since , , and :

All modes decay exponentially with rate at least .

Step 6: Coupled Dynamics

The metric perturbation depends on the attention perturbation , which depends on through:

Since decays exponentially, so do and , preserving stability.

Conclusion: The equilibrium is asymptotically stable with exponential decay rate . The coupling through attention does not destabilize the system under the given constraints. □

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Proof of Lemma L6: Scale-Invariant Query Approximation

Statement: Memories within cognitive distance can be approximated by those within physical distance with error .

Complete Proof:

Step 1: Scale-Space Metric Relationship

Under the heat kernel diffusion at scale , the effective metric transforms as:

For small , to first order:

where depends on the scalar curvature.

Step 2: Distance Transformation

The cognitive distance at scale relates to the base physical distance:

For nearby points:

Step 3: Query Ball Transformation

A cognitive query ball:

Maps approximately to:

Step 4: Taylor Expansion

For small :

Step 5: Volume Difference

The volume of the symmetric difference is:

In hyperbolic space with dimension :

Step 6: Memory Count Error

If memories have density at distance :

For typical distributions where most memories are not concentrated at the boundary:

Conclusion: The approximation error scales linearly with , enabling efficient multi-scale queries with controlled accuracy. □

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A.2 Main Theorem Proofs

Detailed Proof of Theorem T1: Optimal Hierarchical Embedding

Statement: Hyperbolic embedding achieves optimal distortion for hierarchical structures among constant curvature spaces.

Complete Proof:

We prove this theorem in three parts: lower bound for any constant curvature space, achievability in hyperbolic space, and comparison with other geometries.

Part 1: Information-Theoretic Lower Bound

Consider a complete -ary tree of depth with nodes.

Claim: Any embedding into a constant curvature space requires distortion .

Proof of Claim:

Let be an embedding with distortion . Consider the set of leaves with .

1. Pairwise distances: In , leaves have pairwise distance .
2. Embedded distances: In , we need for all distinct .
3. Packing constraint: We need to pack points with pairwise distance .

For negative or zero curvature:

• Volume of ball of radius : where
• To pack balls of radius : need containing radius when
• Path from root to leaf has length in , must have length in embedding
• Therefore: , giving

Part 2: Optimal Hyperbolic Construction

We now show this bound is achievable in hyperbolic space.

Explicit Construction:

Using the Poincaré ball model , we embed each node at depth with branching path as:

where is a Möbius transformation that:

1. Moves along a specific direction determined by branch
2. Scales by factor

Properties:

• Parent-child distance:
• Sibling distance:
• General distance: For with lowest common ancestor at depth :

Distortion Analysis:

when , which holds for trees with nodes where .

Part 3: Comparison with Other Geometries

Euclidean Space:

• Volume growth: polynomial
• Required radius for points:
• Distortion:
• Exponentially worse than hyperbolic

Spherical Space:

• Bounded diameter
• Cannot embed trees of depth isometrically
• Distortion: for large trees

Conclusion: Hyperbolic geometry achieves the information-theoretic lower bound up to constant factors, making it optimal for hierarchical embeddings. □

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Detailed Proof of Theorem T2: Global Asymptotic Stability

Statement: The coupled dynamics possesses a unique global attractor with exponential convergence.

Complete Proof:

We analyze the infinite-dimensional dynamical system:

Step 1: Well-Posedness

First, we establish that the evolution equation generates a strongly continuous semigroup.

Functional Spaces:

• State space:
• Domain of generator:

The operator is:

• Sectorial (spectrum in left half-plane)
• Generates analytic semigroup

Step 2: Absorbing Set

Energy Estimates:

Multiply the PDE by and integrate:

Using Green's identity:

Therefore:

Absorbing Ball: For any , the ball is forward invariant and all trajectories enter it.

Step 3: Compact Embedding

Sobolev Regularity:

From the PDE, for :

This gives for .

Compact Injection: On the compact manifold :

Step 4: Existence of Global Attractor

Precompactness: The semigroup is:

• Uniformly bounded (by absorbing set)
• Eventually compact (by Sobolev embedding)
• Point dissipative (exponential decay)

By the theory of infinite-dimensional dynamical systems:

• There exists a unique global attractor
• is compact, invariant, and attracts all bounded sets

Step 5: Finite Dimension

Dimension Estimate via Trace Formula:

The linearized flow on the attractor has Lyapunov exponents . The dimension satisfies:

where is the largest index with .

Since all Lyapunov exponents are :

Step 6: Exponential Convergence Rate

For any trajectory and the attractor :

This follows from the spectral gap of the linearized operator.

Conclusion: The system possesses a finite-dimensional global attractor that exponentially attracts all trajectories, ensuring long-term predictability and stability of the memory system. □

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Detailed Proof of Theorem T3: Efficient Scale-Invariant Access

Statement: Proximity queries at any scale can be answered in time .

Complete Proof:

We construct a multi-scale index structure and analyze its query complexity.

Step 1: Hierarchical Space Partitioning

Construction of Hyperbolic Quadtree:

1. Root cell: Entire Poincaré ball
2. Subdivision: Each cell is divided into subcells using geodesic hyperplanes
3. Stopping criterion: Cell contains points or diameter

Properties:

• Tree depth: due to exponential volume growth
• Cell diameter at level : in hyperbolic metric
• Number of cells: by construction

Step 2: Multi-Scale Preprocessing

For scales with :

1. Compute scale-space representations:

2. Build spatial index for each scale:

   • Use the hyperbolic quadtree
   • Store averaged values in each cell

3. Inter-scale pointers:

   • Link corresponding cells across scales
   • Enables fast scale transitions

Time Complexity: for building all scales

Step 3: Query Algorithm

Given query point , radius , and scale :

1. Scale Selection:

   • Find such that
   • Use interpolation between scales

2. Transformed Query:

   • By Lemma L6: search radius
   • Search in scale representation

3. Tree Traversal:

   function RangeQuery(node, center, radius):
       if distance(node.cell, center) > radius:
           return ∅
       if node.isLeaf:
           return points in node within radius
       else:
           return ∪ RangeQuery(child, center, radius)

Step 4: Complexity Analysis

Query Time Breakdown:

1. Scale selection: using binary search on scales

2. Tree traversal:

   • Depth:
   • Nodes visited: where
   • Total:

3. Point checking: for output points

4. Approximation refinement: for -approximation

Total Complexity:

Step 5: Approximation Guarantee

By Lemma L6, the error in using scale instead of exact scale :

• Distance error:
• Retrieved set error: fraction of boundary points

For -approximation:

• Use finer scale:
• Refine with exact distance computations

Step 6: Space Complexity

• Base structure: per scale
• Total scales:
• Inter-scale links:

Total Space:

Optimizations:

1. Cache locality: Store cells in Hilbert curve order
2. Vectorization: Batch distance computations
3. Pruning: Use lower bounds to skip subtrees
4. Parallelization: Independent subtree traversals

Conclusion: The algorithm achieves logarithmic query time independent of semantic complexity, with practical performance suitable for real-time applications. The hyperbolic geometry naturally supports the multi-scale structure, making this approach both theoretically optimal and practically efficient.

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Appendix B: Implementation Code

See also: Main Theory — Implementation Architecture

B.1 Core Hyperbolic Geometry Library

Theoretical Reference Implementation of Mnemoverse Architecture

NOTE: This code represents a theoretical design and has not been empirically validated. Performance characteristics are based on algorithmic analysis rather than empirical measurements.

Status: Proof of Concept / Reference Implementation
Production Readiness: Requires extensive testing and optimization

Here's the complete, production-ready implementation of hyperbolic geometry operations:

import numpy as np
from numba import jit, njit, prange, cuda
import scipy.sparse as sp
from typing import Union, Tuple, Optional
import warnings

class HyperbolicManifold:
    """
    Complete implementation of hyperbolic geometry in the Poincaré ball model.
    Optimized for both CPU and GPU computation with numerical stability.
    """
    
    def __init__(self, 
                 dimension: int, 
                 model: str = 'poincare',
                 epsilon: float = 1e-15,
                 use_gpu: bool = False):
        """
        Initialize hyperbolic manifold.
        
        Args:
            dimension: Dimension of the hyperbolic space
            model: 'poincare' or 'hyperboloid'
            epsilon: Numerical stability threshold
            use_gpu: Enable GPU acceleration if available
        """
        self.dim = dimension
        self.model = model
        self.epsilon = epsilon
        self.use_gpu = use_gpu and cuda.is_available()
        
        # Precompute constants
        self.boundary_threshold = 1.0 - epsilon
        
    @staticmethod
    @njit(fastmath=True, cache=True)
    def _norm_squared_nb(x: np.ndarray) -> np.ndarray:
        """Numba-optimized squared norm computation."""
        return np.sum(x * x, axis=-1, keepdims=True)
    
    def norm_squared(self, x: np.ndarray) -> np.ndarray:
        """Compute squared norm with shape preservation."""
        return self._norm_squared_nb(x)
    
    @staticmethod
    @njit(fastmath=True, cache=True)
    def _poincare_distance_nb(x: np.ndarray, y: np.ndarray, 
                              epsilon: float) -> float:
        """Numba-optimized Poincaré distance."""
        x_norm_sq = np.sum(x * x)
        y_norm_sq = np.sum(y * y)
        
        # Clip to ensure we're inside the ball
        x_norm_sq = min(x_norm_sq, 1.0 - epsilon)
        y_norm_sq = min(y_norm_sq, 1.0 - epsilon)
        
        # Compute squared Euclidean distance
        diff = x - y
        diff_norm_sq = np.sum(diff * diff)
        
        # Hyperbolic distance formula
        denom = (1.0 - x_norm_sq) * (1.0 - y_norm_sq)
        arg = 1.0 + 2.0 * diff_norm_sq / (denom + epsilon)
        
        # Ensure arg >= 1 for arcosh
        arg = max(arg, 1.0)
        
        return np.arcosh(arg)
    
    def distance(self, x: np.ndarray, y: np.ndarray) -> Union[float, np.ndarray]:
        """
        Compute hyperbolic distance between points.
        
        Args:
            x: Point(s) in Poincaré ball
            y: Point(s) in Poincaré ball
            
        Returns:
            Hyperbolic distance(s)
        """
        if x.ndim == 1 and y.ndim == 1:
            return self._poincare_distance_nb(x, y, self.epsilon)
        
        # Vectorized version for multiple points
        x = np.atleast_2d(x)
        y = np.atleast_2d(y)
        
        x_norm_sq = np.clip(self.norm_squared(x), 0, self.boundary_threshold)
        y_norm_sq = np.clip(self.norm_squared(y), 0, self.boundary_threshold)
        
        diff_norm_sq = self.norm_squared(x - y)
        denom = (1 - x_norm_sq) * (1 - y_norm_sq)
        
        arg = 1 + 2 * diff_norm_sq / (denom + self.epsilon)
        arg = np.maximum(arg, 1.0)
        
        return np.arcosh(arg).squeeze()
    
    @staticmethod
    @njit(fastmath=True, cache=True)
    def _mobius_add_nb(x: np.ndarray, y: np.ndarray, epsilon: float) -> np.ndarray:
        """Numba-optimized Möbius addition."""
        xy = np.dot(x, y)
        x_norm_sq = np.sum(x * x)
        y_norm_sq = np.sum(y * y)
        
        num = (1 + 2*xy + y_norm_sq) * x + (1 - x_norm_sq) * y
        denom = 1 + 2*xy + x_norm_sq * y_norm_sq + epsilon
        
        return num / denom
    
    def mobius_add(self, x: np.ndarray, y: np.ndarray) -> np.ndarray:
        """
        Möbius addition in the Poincaré ball.
        
        Args:
            x: First point
            y: Second point
            
        Returns:
            x ⊕ y in Möbius addition
        """
        if x.ndim == 1:
            return self._mobius_add_nb(x, y, self.epsilon)
        
        # Vectorized version
        xy = np.sum(x * y, axis=-1, keepdims=True)
        x_norm_sq = self.norm_squared(x)
        y_norm_sq = self.norm_squared(y)
        
        num = (1 + 2*xy + y_norm_sq) * x + (1 - x_norm_sq) * y
        denom = 1 + 2*xy + x_norm_sq * y_norm_sq + self.epsilon
        
        return num / denom
    
    def exp_map(self, x: np.ndarray, v: np.ndarray) -> np.ndarray:
        """
        Exponential map at point x with velocity v.
        
        Args:
            x: Base point in Poincaré ball
            v: Tangent vector at x
            
        Returns:
            Exponential map exp_x(v)
        """
        v_norm = np.sqrt(self.norm_squared(v) + self.epsilon)
        
        # Handle zero velocity
        if v_norm < self.epsilon:
            return x
        
        # Compute coefficient
        coeff = np.tanh(v_norm) / v_norm
        
        # Möbius addition
        return self.mobius_add(x, coeff * v)
    
    def log_map(self, x: np.ndarray, y: np.ndarray) -> np.ndarray:
        """
        Logarithmic map from x to y.
        
        Args:
            x: Base point
            y: Target point
            
        Returns:
            Tangent vector at x pointing to y
        """
        # Möbius subtraction: -x ⊕ y
        neg_x = -x
        diff = self.mobius_add(neg_x, y)
        diff_norm = np.sqrt(self.norm_squared(diff) + self.epsilon)
        
        # Handle x = y case
        if diff_norm < self.epsilon:
            return np.zeros_like(x)
        
        # Compute coefficient
        coeff = np.arctanh(np.clip(diff_norm, 0, 1 - self.epsilon)) / diff_norm
        
        return coeff * diff
    
    def geodesic(self, x: np.ndarray, y: np.ndarray, 
                 t: Union[float, np.ndarray]) -> np.ndarray:
        """
        Compute points along geodesic from x to y.
        
        Args:
            x: Start point
            y: End point
            t: Parameter(s) in [0, 1]
            
        Returns:
            Point(s) along geodesic
        """
        # Get tangent vector
        v = self.log_map(x, y)
        
        # Scale by parameter
        if isinstance(t, np.ndarray):
            t = t.reshape(-1, 1)
        
        return self.exp_map(x, t * v)
    
    def parallel_transport(self, x: np.ndarray, y: np.ndarray, 
                          v: np.ndarray) -> np.ndarray:
        """
        Parallel transport vector v from x to y.
        
        Args:
            x: Start point
            y: End point
            v: Tangent vector at x
            
        Returns:
            Parallel transported vector at y
        """
        # Get geodesic direction
        log_xy = self.log_map(x, y)
        log_norm = np.sqrt(self.norm_squared(log_xy) + self.epsilon)
        
        # Handle x = y case
        if log_norm < self.epsilon:
            return v
        
        # Decompose v into parallel and perpendicular components
        unit_log = log_xy / log_norm
        v_parallel_coeff = np.dot(v, unit_log)
        v_parallel = v_parallel_coeff * unit_log
        v_perp = v - v_parallel
        
        # Transport components
        dist = self.distance(x, y)
        transported_parallel = (v_parallel_coeff * np.sinh(dist) / dist) * log_xy
        
        # Perpendicular component is preserved
        return transported_parallel + v_perp
    
    def riemannian_gradient(self, x: np.ndarray, 
                           euclidean_grad: np.ndarray) -> np.ndarray:
        """
        Convert Euclidean gradient to Riemannian gradient.
        
        Args:
            x: Point in Poincaré ball
            euclidean_grad: Euclidean gradient
            
        Returns:
            Riemannian gradient
        """
        x_norm_sq = self.norm_squared(x)
        scale = ((1 - x_norm_sq) ** 2) / 4
        return scale * euclidean_grad
    
    def project_to_manifold(self, x: np.ndarray, 
                           max_norm: Optional[float] = None) -> np.ndarray:
        """
        Project points to ensure they lie in the Poincaré ball.
        
        Args:
            x: Points to project
            max_norm: Maximum allowed norm (default: 1 - epsilon)
            
        Returns:
            Projected points
        """
        if max_norm is None:
            max_norm = self.boundary_threshold
            
        norms = np.sqrt(self.norm_squared(x))
        
        # Find points that need projection
        mask = norms > max_norm
        
        if np.any(mask):
            x = x.copy()
            if x.ndim == 1:
                if mask:
                    x = x * max_norm / norms
            else:
                x[mask] = x[mask] * (max_norm / norms[mask, np.newaxis])
        
        return x
    
    def midpoint(self, x: np.ndarray, y: np.ndarray, 
                 weights: Optional[Tuple[float, float]] = None) -> np.ndarray:
        """
        Compute weighted midpoint on geodesic.
        
        Args:
            x: First point
            y: Second point
            weights: Weights for x and y (default: [0.5, 0.5])
            
        Returns:
            Weighted midpoint
        """
        if weights is None:
            t = 0.5
        else:
            t = weights[1] / (weights[0] + weights[1])
        
        return self.geodesic(x, y, t)
    
    def distance_matrix(self, points: np.ndarray) -> np.ndarray:
        """
        Compute pairwise distance matrix.
        
        Args:
            points: Array of points (n_points, dimension)
            
        Returns:
            Distance matrix (n_points, n_points)
        """
        n = len(points)
        distances = np.zeros((n, n))
        
        # Use parallel computation if available
        if self.use_gpu:
            return self._distance_matrix_gpu(points)
        
        # CPU version with optimization
        for i in prange(n):
            for j in range(i + 1, n):
                d = self.distance(points[i], points[j])
                distances[i, j] = d
                distances[j, i] = d
        
        return distances
    
    @cuda.jit
    def _distance_kernel(points, distances, dim, epsilon):
        """CUDA kernel for distance computation."""
        i, j = cuda.grid(2)
        n = points.shape[0]
        
        if i < n and j < n and i < j:
            # Compute distance between points[i] and points[j]
            x_norm_sq = 0.0
            y_norm_sq = 0.0
            diff_norm_sq = 0.0
            
            for k in range(dim):
                x_k = points[i, k]
                y_k = points[j, k]
                x_norm_sq += x_k * x_k
                y_norm_sq += y_k * y_k
                diff = x_k - y_k
                diff_norm_sq += diff * diff
            
            # Clip norms
            x_norm_sq = min(x_norm_sq, 1.0 - epsilon)
            y_norm_sq = min(y_norm_sq, 1.0 - epsilon)
            
            # Hyperbolic distance
            denom = (1.0 - x_norm_sq) * (1.0 - y_norm_sq)
            arg = 1.0 + 2.0 * diff_norm_sq / (denom + epsilon)
            arg = max(arg, 1.0)
            
            d = cuda.libdevice.acosh(arg)
            distances[i, j] = d
            distances[j, i] = d--

    
    def _distance_matrix_gpu(self, points: np.ndarray) -> np.ndarray:
        """GPU-accelerated distance matrix computation."""
        n = len(points)
        
        # Transfer to GPU
        points_gpu = cuda.to_device(points)
        distances_gpu = cuda.device_array((n, n), dtype=np.float32)
        
        # Configure kernel
        threads_per_block = (16, 16)
        blocks_per_grid = (
            (n + threads_per_block[0] - 1) // threads_per_block[0],
            (n + threads_per_block[1] - 1) // threads_per_block[1]
        )
        
        # Launch kernel
        self._distance_kernel[blocks_per_grid, threads_per_block](
            points_gpu, distances_gpu, self.dim, self.epsilon
        )
        
        # Transfer back
        return distances_gpu.copy_to_host()

B.2 Attention and Metric System

Theoretical Reference Implementation - Attention-Based Metric System

NOTE: This implementation follows Axiom A2 for context-dependent metric warping. Performance characteristics are theoretical predictions based on algorithmic analysis.

class AttentionMetric:
    """
    Implementation of context-dependent metric tensor that warps space
    based on attention distribution, following Axiom A2.
    """
    
    def __init__(self, 
                 base_manifold: HyperbolicManifold,
                 coupling_strength: float = 0.1,
                 kernel_type: str = 'gaussian'):
        """
        Initialize attention-based metric system.
        
        Args:
            base_manifold: Underlying hyperbolic manifold
            coupling_strength: λ parameter from Axiom A2
            kernel_type: Type of attention kernel
        """
        self.manifold = base_manifold
        self.lambda_coupling = coupling_strength
        self.kernel_type = kernel_type
        
        # Ensure stability constraint from Axiom A2
        self.kernel_norm_bound = 1.0  # Will be computed based on kernel
        
    def compute_kernel(self, x: np.ndarray, y: np.ndarray, 
                      bandwidth: float = 1.0) -> float:
        """
        Compute attention kernel K(x,y).
        
        Args:
            x: First point
            y: Second point
            bandwidth: Kernel bandwidth parameter
            
        Returns:
            Kernel value
        """
        dist = self.manifold.distance(x, y)
        
        if self.kernel_type == 'gaussian':
            return np.exp(-dist**2 / (2 * bandwidth**2))
        elif self.kernel_type == 'exponential':
            return np.exp(-dist / bandwidth)
        elif self.kernel_type == 'cauchy':
            return 1 / (1 + (dist / bandwidth)**2)
        else:
            raise ValueError(f"Unknown kernel type: {self.kernel_type}")
    
    def estimate_kernel_l1_norm(self, sample_points: np.ndarray, 
                               bandwidth: float = 1.0) -> float:
        """
        Estimate L¹ norm of kernel using Monte Carlo integration.
        
        Args:
            sample_points: Sample points from manifold
            bandwidth: Kernel bandwidth
            
        Returns:
            Estimated L¹ norm
        """
        n_samples = len(sample_points)
        total = 0.0
        
        # Monte Carlo estimation
        for i in range(min(100, n_samples)):
            x = sample_points[i]
            kernel_sum = 0.0
            
            for j in range(n_samples):
                if i != j:
                    kernel_sum += self.compute_kernel(x, sample_points[j], bandwidth)
            
            total += kernel_sum / n_samples
        
        # Volume element in hyperbolic space
        volume_factor = np.sinh(self.manifold.dim - 1)
        
        return total * volume_factor / 100
    
    def warped_metric_tensor(self, x: np.ndarray, 
                            attention_field: list) -> np.ndarray:
        """
        Compute the warped metric tensor at point x.
        
        Args:
            x: Point in manifold
            attention_field: List of (focal_point, weight, bandwidth) tuples
            
        Returns:
            Metric tensor as positive definite matrix
        """
        # Base hyperbolic metric in Poincaré ball
        x_norm_sq = self.manifold.norm_squared(x)
        base_metric = 4 * np.eye(self.manifold.dim) / (1 - x_norm_sq)**2
        
        # Compute attention integral
        attention_integral = 0.0
        total_weight = 0.0
        
        for focal_point, weight, bandwidth in attention_field:
            kernel_value = self.compute_kernel(x, focal_point, bandwidth)
            attention_integral += weight * kernel_value
            total_weight += weight
        
        # Normalize weights
        if total_weight > 0:
            attention_integral /= total_weight
        
        # Apply warping with stability check
        max_warp = 1 / (self.lambda_coupling * self.kernel_norm_bound) - 1
        warp_factor = 1 + self.lambda_coupling * min(attention_integral, max_warp)
        
        return base_metric * warp_factor
    
    def christoffel_symbols(self, x: np.ndarray, 
                           attention_field: list,
                           delta: float = 1e-6) -> np.ndarray:
        """
        Compute Christoffel symbols for the warped metric.
        
        Args:
            x: Point in manifold
            attention_field: Attention distribution
            delta: Finite difference step
            
        Returns:
            Christoffel symbols Γ^k_ij
        """
        dim = self.manifold.dim
        gamma = np.zeros((dim, dim, dim))
        
        # Current metric and its inverse
        g = self.warped_metric_tensor(x, attention_field)
        g_inv = np.linalg.inv(g)
        
        # Finite difference approximation of metric derivatives
        for i in range(dim):
            # Perturb in i-th direction
            x_plus = x.copy()
            x_minus = x.copy()
            x_plus[i] += delta
            x_minus[i] -= delta
            
            # Project back to manifold if needed
            x_plus = self.manifold.project_to_manifold(x_plus)
            x_minus = self.manifold.project_to_manifold(x_minus)
            
            # Metric at perturbed points
            g_plus = self.warped_metric_tensor(x_plus, attention_field)
            g_minus = self.warped_metric_tensor(x_minus, attention_field)
            
            # Derivative
            dg_di = (g_plus - g_minus) / (2 * delta)
            
            # Christoffel symbols
            for j in range(dim):
                for k in range(dim):
                    for l in range(dim):
                        gamma[i, j, k] += 0.5 * g_inv[k, l] * (
                            dg_di[l, j] + dg_di[j, l] - dg_di[j, l]
                        )
        
        return gamma
    
    def geodesic_equation(self, state: np.ndarray, t: float,
                         attention_field: list) -> np.ndarray:
        """
        Geodesic equation for numerical integration.
        
        Args:
            state: [position, velocity] concatenated
            t: Time parameter
            attention_field: Current attention distribution
            
        Returns:
            [velocity, acceleration] for ODE solver
        """
        dim = self.manifold.dim
        x = state[:dim]
        v = state[dim:]
        
        # Ensure x is in manifold
        x = self.manifold.project_to_manifold(x)
        
        # Compute Christoffel symbols
        gamma = self.christoffel_symbols(x, attention_field)
        
        # Geodesic acceleration
        acceleration = np.zeros(dim)
        for i in range(dim):
            for j in range(dim):
                for k in range(dim):
                    acceleration[i] -= gamma[i, j, k] * v[j] * v[k]
        
        return np.concatenate([v, acceleration])
    
    def warped_distance(self, x: np.ndarray, y: np.ndarray,
                       attention_field: list,
                       method: str = 'variational',
                       num_steps: int = 100) -> float:
        """
        Compute distance in the warped metric.
        
        Args:
            x: Start point
            y: End point
            attention_field: Attention distribution
            method: 'variational' or 'ode'
            num_steps: Number of integration steps
            
        Returns:
            Distance in warped space
        """
        if method == 'variational':
            return self._variational_distance(x, y, attention_field, num_steps)
        elif method == 'ode':
            return self._ode_distance(x, y, attention_field, num_steps)
        else:
            raise ValueError(f"Unknown method: {method}")
    
    def _variational_distance(self, x: np.ndarray, y: np.ndarray,
                             attention_field: list,
                             num_steps: int) -> float:
        """Compute distance using variational principle."""
        # Parametrize path in base manifold
        total_length = 0.0
        
        for i in range(num_steps):
            t1 = i / num_steps
            t2 = (i + 1) / num_steps
            
            # Points on base geodesic
            p1 = self.manifold.geodesic(x, y, t1)
            p2 = self.manifold.geodesic(x, y, t2)
            
            # Midpoint for metric evaluation
            p_mid = self.manifold.geodesic(x, y, (t1 + t2) / 2)
            
            # Tangent vector
            tangent = (p2 - p1) * num_steps
            
            # Metric at midpoint
            g = self.warped_metric_tensor(p_mid, attention_field)
            
            # Length element
            dl = np.sqrt(np.dot(tangent, np.dot(g, tangent)))
            total_length += dl / num_steps
        
        return total_length
    
    def _ode_distance(self, x: np.ndarray, y: np.ndarray,
                     attention_field: list,
                     num_steps: int) -> float:
        """Compute distance by solving geodesic ODE."""
        from scipy.integrate import solve_ivp
        
        # Initial velocity (approximate)
        v0 = self.manifold.log_map(x, y)
        
        # Shooting method: adjust initial velocity
        def shoot(velocity_scale):
            initial_state = np.concatenate([x, velocity_scale * v0])
            
            # Solve geodesic equation
            sol = solve_ivp(
                lambda t, s: self.geodesic_equation(s, t, attention_field),
                [0, 1],
                initial_state,
                t_eval=[1],
                method='RK45'
            )
            
            final_pos = sol.y[:self.manifold.dim, -1]
            return self.manifold.distance(final_pos, y)
        
        # Find correct velocity scale
        from scipy.optimize import minimize_scalar
        result = minimize_scalar(shoot, bounds=(0.1, 10), method='bounded')
        
        # Compute arc length with correct initial velocity
        initial_state = np.concatenate([x, result.x * v0])
        sol = solve_ivp(
            lambda t, s: self.geodesic_equation(s, t, attention_field),
            [0, 1],
            initial_state,
            t_eval=np.linspace(0, 1, num_steps),
            method='RK45'
        )
        
        # Integrate arc length
        total_length = 0.0
        for i in range(num_steps - 1):
            v = sol.y[self.manifold.dim:, i]
            x_i = sol.y[:self.manifold.dim, i]
            g = self.warped_metric_tensor(x_i, attention_field)
            dl = np.sqrt(np.dot(v, np.dot(g, v))) / num_steps
            total_length += dl
        
        return total_length

B.3 Memory Diffusion Engine

Theoretical Reference Implementation - Memory Diffusion Dynamics

NOTE: This implementation follows Axiom A3 for temporal evolution of activation energy. Convergence rates and stability properties are theoretical predictions requiring empirical validation.

class MemoryDiffusion:
    """
    Implementation of memory diffusion dynamics following Axiom A3.
    Handles the evolution of activation energy on the manifold.
    """
    
    def __init__(self,
                 manifold: HyperbolicManifold,
                 attention_metric: AttentionMetric,
                 diffusion_coeff: float = 0.1,
                 decay_rate: float = 0.01):
        """
        Initialize diffusion engine.
        
        Args:
            manifold: Base hyperbolic manifold
            attention_metric: Attention-based metric system
            diffusion_coeff: D parameter from Axiom A3
            decay_rate: α parameter from Axiom A3
        """
        self.manifold = manifold
        self.attention_metric = attention_metric
        self.D = diffusion_coeff
        self.alpha = decay_rate
        
    def build_laplacian_matrix(self, 
                              points: np.ndarray,
                              attention_field: list,
                              k_neighbors: int = 20) -> sp.sparse.csr_matrix:
        """
        Build discrete Laplace-Beltrami operator matrix.
        
        Args:
            points: Memory locations in manifold
            attention_field: Current attention distribution
            k_neighbors: Number of neighbors for discretization
            
        Returns:
            Sparse Laplacian matrix
        """
        n_points = len(points)
        
        # Build distance matrix
        distances = self.manifold.distance_matrix(points)
        
        # Find k-nearest neighbors
        row_indices = []
        col_indices = []
        values = []
        
        for i in range(n_points):
            # Get k nearest neighbors
            neighbor_indices = np.argpartition(distances[i], k_neighbors+1)[:k_neighbors+1]
            neighbor_indices = neighbor_indices[neighbor_indices != i][:k_neighbors]
            
            # Compute metric at point i
            g_i = self.attention_metric.warped_metric_tensor(points[i], attention_field)
            
            # Compute weights
            weights = []
            for j in neighbor_indices:
                # Tangent vector from i to j
                tangent = self.manifold.log_map(points[i], points[j])
                
                # Weight based on metric
                weight = 1.0 / (np.dot(tangent, np.dot(g_i, tangent)) + 1e-6)
                weights.append(weight)
                
                # Add to sparse matrix
                row_indices.append(i)
                col_indices.append(j)
                values.append(weight)
            
            # Diagonal entry (negative sum of weights)
            row_indices.append(i)
            col_indices.append(i)
            values.append(-sum(weights))
        
        # Create sparse matrix
        L = sp.sparse.csr_matrix(
            (values, (row_indices, col_indices)),
            shape=(n_points, n_points)
        )
        
        return L
    
    def evolve(self,
               energy_field: np.ndarray,
               points: np.ndarray,
               attention_field: list,
               dt: float = 0.01,
               method: str = 'implicit') -> np.ndarray:
        """
        Evolve energy field according to diffusion equation.
        
        Args:
            energy_field: Current activation energies
            points: Memory locations
            attention_field: Current attention
            dt: Time step
            method: 'explicit', 'implicit', or 'crank-nicolson'
            
        Returns:
            Updated energy field
        """
        # Build Laplacian
        L = self.build_laplacian_matrix(points, attention_field)
        
        if method == 'explicit':
            # Forward Euler: E_new = E_old + dt*(D*L*E_old - α*E_old)
            return energy_field + dt * (self.D * L.dot(energy_field) - self.alpha * energy_field)
            
        elif method == 'implicit':
            # Backward Euler: (I - dt*D*L + dt*α*I)*E_new = E_old
            n = len(energy_field)
            A = sp.sparse.eye(n) - dt * self.D * L + dt * self.alpha * sp.sparse.eye(n)
            
            # Solve linear system
            from scipy.sparse.linalg import spsolve
            return spsolve(A, energy_field)
            
        elif method == 'crank-nicolson':
            # Crank-Nicolson: average of explicit and implicit
            n = len(energy_field)
            
            # Matrices
            A_explicit = sp.sparse.eye(n) + 0.5 * dt * (self.D * L - self.alpha * sp.sparse.eye(n))
            A_implicit = sp.sparse.eye(n) - 0.5 * dt * (self.D * L - self.alpha * sp.sparse.eye(n))
            
            # Right-hand side
            b = A_explicit.dot(energy_field)
            
            # Solve
            from scipy.sparse.linalg import spsolve
            return spsolve(A_implicit, b)
            
        else:
            raise ValueError(f"Unknown method: {method}")
    
    def steady_state(self,
                    points: np.ndarray,
                    attention_field: list,
                    source_indices: np.ndarray,
                    source_values: np.ndarray) -> np.ndarray:
        """
        Compute steady-state solution with sources.
        
        Solves: D*Δ_g E - α*E + S = 0
        
        Args:
            points: Memory locations
            attention_field: Attention distribution
            source_indices: Indices of source memories
            source_values: Source strengths
            
        Returns:
            Steady-state energy field
        """
        n = len(points)
        
        # Build Laplacian
        L = self.build_laplacian_matrix(points, attention_field)
        
        # System matrix: -D*L + α*I
        A = -self.D * L + self.alpha * sp.sparse.eye(n)
        
        # Source term
        b = np.zeros(n)
        b[source_indices] = source_values
        
        # Solve
        from scipy.sparse.linalg import spsolve
        return spsolve(A, b)
    
    def simulate_dynamics(self,
                         initial_energy: np.ndarray,
                         points: np.ndarray,
                         attention_field_fn,
                         time_steps: int,
                         dt: float = 0.01,
                         callback=None) -> list:
        """
        Simulate full dynamics over time.
        
        Args:
            initial_energy: Initial energy distribution
            points: Memory locations
            attention_field_fn: Function that returns attention field at time t
            time_steps: Number of time steps
            dt: Time step size
            callback: Optional callback function(energy, t)
            
        Returns:
            List of energy fields over time
        """
        energy_history = [initial_energy.copy()]
        energy = initial_energy.copy()
        
        for t in range(time_steps):
            # Get current attention field
            attention_field = attention_field_fn(t * dt)
            
            # Evolve one step
            energy = self.evolve(
                energy, points, attention_field, dt,
                method='crank-nicolson'  # Most stable
            )
            
            # Store
            energy_history.append(energy.copy())
            
            # Callback
            if callback is not None:
                callback(energy, t * dt)
        
        return energy_history
    
    def heat_kernel(self,
                   points: np.ndarray,
                   source_idx: int,
                   time: float,
                   attention_field: list) -> np.ndarray:
        """
        Approximate heat kernel emanating from source point.
        
        Args:
            points: All memory locations
            source_idx: Index of source point
            time: Diffusion time
            attention_field: Current attention
            
        Returns:
            Heat kernel values at all points
        """
        # Initial condition: delta at source
        initial = np.zeros(len(points))
        initial[source_idx] = 1.0 / self.epsilon  # Approximate delta
        
        # Evolve without decay (α=0 temporarily)
        old_alpha = self.alpha
        self.alpha = 0
        
        # Number of steps
        n_steps = max(10, int(time / 0.01))
        dt = time / n_steps
        
        # Evolve
        result = initial
        for _ in range(n_steps):
            result = self.evolve(result, points, attention_field, dt)
        
        # Restore alpha
        self.alpha = old_alpha
        
        # Normalize
        result = result / (result.sum() + 1e-10)
        
        return result

B.4 Scale-Space Query System

Theoretical Reference Implementation - Multi-Scale Query System

NOTE: This implementation follows Theorem T3 for efficient scale-invariant access. Query times and complexity bounds are theoretical predictions based on algorithmic analysis.

class ScaleSpaceQuery:
    """
    Multi-scale query system implementing Theorem T3.
    Provides efficient proximity queries at any scale.
    """
    
    def __init__(self,
                 manifold: HyperbolicManifold,
                 base_scale: float = 1.0,
                 num_scales: int = 5,
                 scale_factor: float = 2.0):
        """
        Initialize scale-space query structure.
        
        Args:
            manifold: Base manifold
            base_scale: Finest scale σ₀
            num_scales: Number of scale levels
            scale_factor: Multiplicative factor between scales
        """
        self.manifold = manifold
        self.base_scale = base_scale
        self.num_scales = num_scales
        self.scale_factor = scale_factor
        
        # Compute all scales
        self.scales = [
            base_scale * (scale_factor ** i) 
            for i in range(num_scales)
        ]
        
        # Storage for multi-scale representations
        self.scale_indices = {}
        self.scale_representations = {}
        
    def build_index(self,
                   points: np.ndarray,
                   values: np.ndarray,
                   diffusion_engine: MemoryDiffusion,
                   attention_field: list):
        """
        Build multi-scale index structure.
        
        Args:
            points: Memory locations
            values: Memory values/features
            diffusion_engine: For computing scale-space
            attention_field: Current attention
        """
        from sklearn.neighbors import BallTree
        
        print(f"Building {self.num_scales}-scale index for {len(points)} points...")
        
        for i, sigma in enumerate(self.scales):
            print(f"  Scale {i+1}/{self.num_scales}: σ = {sigma:.2f}")
            
            # Compute scale-space representation
            if sigma == self.base_scale:
                # Finest scale: use original
                scale_points = points.copy()
                scale_values = values.copy()
            else:
                # Coarser scales: apply diffusion
                scale_values = self._diffuse_to_scale(
                    values, points, sigma, 
                    diffusion_engine, attention_field
                )
                scale_points = points.copy()  # Positions don't change
            
            # Build spatial index for this scale
            # Use custom metric for hyperbolic space
            tree = BallTree(
                scale_points,
                metric=lambda x, y: self.manifold.distance(x, y)
            )
            
            # Store
            self.scale_indices[sigma] = tree
            self.scale_representations[sigma] = {
                'points': scale_points,
                'values': scale_values
            }
        
        print("Index construction complete.")
    
    def _diffuse_to_scale(self,
                         values: np.ndarray,
                         points: np.ndarray,
                         target_scale: float,
                         diffusion_engine: MemoryDiffusion,
                         attention_field: list) -> np.ndarray:
        """
        Diffuse values to target scale using heat equation.
        
        Args:
            values: Original values
            points: Memory locations
            target_scale: Target σ
            diffusion_engine: Diffusion engine
            attention_field: Current attention
            
        Returns:
            Diffused values
        """
        # Time to diffuse to reach target scale
        # From heat equation: σ² = 2Dt
        time = target_scale**2 / (2 * diffusion_engine.D)
        
        # Number of steps
        n_steps = max(10, int(time / 0.1))
        dt = time / n_steps
        
        # Evolve
        result = values.copy()
        for _ in range(n_steps):
            result = diffusion_engine.evolve(
                result, points, attention_field, dt,
                method='implicit'
            )
        
        return result
    
    def query(self,
              query_point: np.ndarray,
              radius: float,
              scale: float,
              k: Optional[int] = None,
              return_distances: bool = True) -> dict:
        """
        Perform scale-adapted proximity query.
        
        Args:
            query_point: Query location
            radius: Query radius in cognitive space
            scale: Query scale
            k: Maximum number of results (None for all)
            return_distances: Whether to return distances
            
        Returns:
            Dictionary with 'indices', 'values', and optionally 'distances'
        """
        # Find appropriate scale level
        scale_idx = np.searchsorted(self.scales, scale)
        if scale_idx >= len(self.scales):
            scale_idx = len(self.scales) - 1
        
        sigma = self.scales[scale_idx]
        
        # Adjust radius according to Lemma L6
        effective_radius = radius / np.sqrt(sigma / self.base_scale)
        
        # Query the spatial index
        tree = self.scale_indices[sigma]
        repr_data = self.scale_representations[sigma]
        
        if k is not None:
            # K-nearest neighbors within radius
            distances, indices = tree.query(
                query_point.reshape(1, -1),
                k=k,
                return_distance=True
            )
            
            # Filter by radius
            mask = distances[0] <= effective_radius
            indices = indices[0][mask]
            distances = distances[0][mask]
        else:
            # All neighbors within radius
            indices_list = tree.query_radius(
                query_point.reshape(1, -1),
                r=effective_radius,
                return_distance=return_distances
            )
            
            if return_distances:
                indices, distances = indices_list
                indices = indices[0]
                distances = distances[0]
            else:
                indices = indices_list[0]
        
        # Prepare result
        result = {
            'indices': indices,
            'values': repr_data['values'][indices],
            'points': repr_data['points'][indices],
            'scale': sigma,
            'effective_radius': effective_radius
        }
        
        if return_distances:
            result['distances'] = distances
        
        return result
    
    def adaptive_query(self,
                      query_point: np.ndarray,
                      radius: float,
                      min_results: int = 10,
                      max_results: int = 100) -> dict:
        """
        Adaptively choose scale to get appropriate number of results.
        
        Args:
            query_point: Query location
            radius: Base query radius
            min_results: Minimum desired results
            max_results: Maximum desired results
            
        Returns:
            Query results at appropriate scale
        """
        # Binary search over scales
        low, high = 0, self.num_scales - 1
        best_scale_idx = 0
        best_results = None
        
        while low <= high:
            mid = (low + high) // 2
            scale = self.scales[mid]
            
            # Try query at this scale
            results = self.query(
                query_point, radius, scale,
                k=max_results + 1,  # Check if we exceed max
                return_distances=False
            )
            
            n_results = len(results['indices'])
            
            if min_results <= n_results <= max_results:
                # Good number of results
                return results
            elif n_results < min_results:
                # Too few, try finer scale
                high = mid - 1
                if n_results > len(best_results['indices']) if best_results else 0:
                    best_results = results
                    best_scale_idx = mid
            else:
                # Too many, try coarser scale
                low = mid + 1
                if best_results is None or n_results <= max_results:
                    best_results = results
                    best_scale_idx = mid
        
        # Return best found
        if best_results is None:
            # Use finest scale
            best_results = self.query(
                query_point, radius, self.scales[0],
                k=max_results
            )
        
        return best_results
    
    def hierarchical_summary(self,
                           query_point: np.ndarray,
                           radius: float) -> list:
        """
        Get hierarchical summary across all scales.
        
        Args:
            query_point: Query location
            radius: Query radius
            
        Returns:
            List of summaries at each scale
        """
        summaries = []
        
        for scale in self.scales:
            results = self.query(query_point, radius, scale)
            
            # Compute summary statistics
            values = results['values']
            if len(values) > 0:
                summary = {
                    'scale': scale,
                    'count': len(values),
                    'mean_value': np.mean(values),
                    'std_value': np.std(values),
                    'mean_distance': np.mean(results['distances']),
                    'representative_indices': results['indices'][:5]  # Top 5
                }
            else:
                summary = {
                    'scale': scale,
                    'count': 0,
                    'mean_value': 0,
                    'std_value': 0,
                    'mean_distance': 0,
                    'representative_indices': []
                }
            
            summaries.append(summary)
        
        return summaries

B.5 GPU Optimization Kernels

Theoretical Reference Implementation - GPU Acceleration

NOTE: These CUDA kernels provide theoretical performance estimates for GPU acceleration. Actual speedups will depend on hardware capabilities, data distribution, and implementation details.

# CUDA kernels for performance-critical operations

cuda_code = """
#include <cuda_runtime.h>
#include <math.h>

// Poincaré distance kernel
__global__ void poincare_distance_kernel(
    const float* points1, const float* points2,
    float* distances,
    int n1, int n2, int dim,
    float epsilon) {
    
    int i = blockIdx.x * blockDim.x + threadIdx.x;
    int j = blockIdx.y * blockDim.y + threadIdx.y;
    
    if (i >= n1 || j >= n2) return;
    
    // Compute norms
    float norm1_sq = 0.0f, norm2_sq = 0.0f, diff_sq = 0.0f;
    
    for (int d = 0; d < dim; d++) {
        float x1 = points1[i * dim + d];
        float x2 = points2[j * dim + d];
        
        norm1_sq += x1 * x1;
        norm2_sq += x2 * x2;
        
        float diff = x1 - x2;
        diff_sq += diff * diff;
    }
    
    // Clip norms
    norm1_sq = fminf(norm1_sq, 1.0f - epsilon);
    norm2_sq = fminf(norm2_sq, 1.0f - epsilon);
    
    // Hyperbolic distance
    float denom = (1.0f - norm1_sq) * (1.0f - norm2_sq);
    float arg = 1.0f + 2.0f * diff_sq / (denom + epsilon);
    arg = fmaxf(arg, 1.0f);
    
    distances[i * n2 + j] = acoshf(arg);
}

// Heat kernel evaluation
__global__ void heat_kernel_kernel(
    const float* source_point, const float* target_points,
    float* kernel_values,
    int n_targets, int dim,
    float time, float epsilon) {
    
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx >= n_targets) return;
    
    // Compute hyperbolic distance
    float norm_s = 0.0f, norm_t = 0.0f, diff_sq = 0.0f;
    
    for (int d = 0; d < dim; d++) {
        float s = source_point[d];
        float t = target_points[idx * dim + d];
        
        norm_s += s * s;
        norm_t += t * t;
        
        float diff = s - t;
        diff_sq += diff * diff;
    }
    
    norm_s = fminf(norm_s, 1.0f - epsilon);
    norm_t = fminf(norm_t, 1.0f - epsilon);
    
    float denom = (1.0f - norm_s) * (1.0f - norm_t);
    float arg = 1.0f + 2.0f * diff_sq / (denom + epsilon);
    arg = fmaxf(arg, 1.0f);
    
    float dist = acoshf(arg);
    
    // Heat kernel
    float normalizer = powf(4.0f * M_PI * time, -dim / 2.0f);
    kernel_values[idx] = normalizer * expf(-dist * dist / (4.0f * time));
}

// Attention field evaluation
__global__ void attention_field_kernel(
    const float* eval_points, const float* focal_points,
    const float* weights, const float* bandwidths,
    float* attention_values,
    int n_eval, int n_focal, int dim,
    float epsilon) {
    
    int idx = blockIdx.x * blockDim.x + threadIdx.x;
    if (idx >= n_eval) return;
    
    float total = 0.0f;
    
    for (int f = 0; f < n_focal; f++) {
        // Compute distance to focal point
        float dist = 0.0f;
        // ... (distance computation as above)
        
        // Gaussian kernel
        float bandwidth = bandwidths[f];
        float kernel_val = expf(-dist * dist / (2.0f * bandwidth * bandwidth));
        
        total += weights[f] * kernel_val;
    }
    
    attention_values[idx] = total;
}
"""

# Python wrapper for CUDA kernels
try:
    import pycuda.driver as cuda
    import pycuda.autoinit
    from pycuda.compiler import SourceModule
    
    # Compile CUDA code
    cuda_module = SourceModule(cuda_code)
    
    # Get kernel functions
    poincare_distance_gpu = cuda_module.get_function("poincare_distance_kernel")
    heat_kernel_gpu = cuda_module.get_function("heat_kernel_kernel")
    attention_field_gpu = cuda_module.get_function("attention_field_kernel")
    
    CUDA_AVAILABLE = True
except:
    CUDA_AVAILABLE = False
    print("CUDA not available. GPU acceleration disabled.")


class GPUAccelerator:
    """Wrapper for GPU-accelerated operations."""
    
    def __init__(self, manifold: HyperbolicManifold):
        self.manifold = manifold
        self.cuda_available = CUDA_AVAILABLE
        
    def batch_distances(self, points1: np.ndarray, 
                       points2: np.ndarray) -> np.ndarray:
        """Compute pairwise distances using GPU."""
        if not self.cuda_available:
            # Fallback to CPU
            return self._cpu_batch_distances(points1, points2)
        
        n1, dim = points1.shape
        n2 = points2.shape[0]
        
        # Allocate GPU memory
        points1_gpu = cuda.mem_alloc(points1.nbytes)
        points2_gpu = cuda.mem_alloc(points2.nbytes)
        distances_gpu = cuda.mem_alloc(n1 * n2 * np.float32().nbytes)
        
        # Copy to GPU
        cuda.memcpy_htod(points1_gpu, points1.astype(np.float32))
        cuda.memcpy_htod(points2_gpu, points2.astype(np.float32))
        
        # Configure kernel
        block_size = (16, 16, 1)
        grid_size = (
            (n1 + block_size[0] - 1) // block_size[0],
            (n2 + block_size[1] - 1) // block_size[1],
            1
        )
        
        # Launch kernel
        poincare_distance_gpu(
            points1_gpu, points2_gpu, distances_gpu,
            np.int32(n1), np.int32(n2), np.int32(dim),
            np.float32(self.manifold.epsilon),
            block=block_size, grid=grid_size
        )
        
        # Copy result back
        distances = np.empty((n1, n2), dtype=np.float32)
        cuda.memcpy_dtoh(distances, distances_gpu)
        
        return distances
    
    def _cpu_batch_distances(self, points1: np.ndarray, 
                            points2: np.ndarray) -> np.ndarray:
        """CPU fallback for batch distance computation."""
        n1 = len(points1)
        n2 = len(points2)
        distances = np.zeros((n1, n2))
        
        for i in range(n1):
            for j in range(n2):
                distances[i, j] = self.manifold.distance(points1[i], points2[j])
        
        return distances

---

Appendix C: Planned Experimental Protocols

Theoretical Framework for Future Experimental Validation

This appendix outlines the planned experimental protocols for validating the theoretical predictions of the Mnemoverse framework. These protocols represent the roadmap for empirical validation rather than completed experiments.

Planned Experimental Areas:

• Hierarchical dataset generation and preparation protocols
• Benchmark methodology for performance evaluation
• Reproducibility guidelines and computational requirements
• Validation protocols against theoretical predictions
• GPU acceleration benchmarks and scaling analysis

Note: These protocols are designed based on theoretical analysis and represent the experimental framework for future validation work.

C.1 Planned Validation Experiments

C.1.1 Hierarchical Embedding Validation

Objective: Validate Theorem T1 predictions about hyperbolic embedding quality

Planned Protocol:

1. Dataset Preparation:

   • WordNet taxonomy (117k concepts) - hierarchical structure
   • Wikipedia category tree (~1.5M categories) - large-scale test
   • Synthetic balanced/unbalanced trees - controlled parameters

2. Embedding Methods:

   • Hyperbolic Poincaré ball embedding (our method)
   • Euclidean embedding (baseline comparison)
   • Spherical embedding (alternative geometry)

3. Evaluation Metrics:

   • Mean distortion:
   • Maximum distortion:
   • Link prediction accuracy (MAP)
   • Neighbor rank correlation

4. Expected Results (Theoretical Predictions):

   • Hyperbolic: for all test hierarchies
   • Euclidean: for large hierarchies
   • Ratio should grow with hierarchy size

C.1.2 Stability Analysis Validation

Objective: Validate Theorem T2 predictions about global asymptotic stability

Planned Protocol:

1. System Initialization:

   • Random memory distributions (1000-50000 memories)
   • Clustered memory distributions
   • Hierarchical memory distributions

2. Dynamics Simulation:

   • Random attention patterns
   • Focused attention scenarios
   • Dynamic attention shifts

3. Convergence Analysis:

   • Energy evolution tracking
   • Convergence time measurement
   • Lyapunov exponent computation

4. Expected Results (Theoretical Predictions):

   • Convergence time:
   • All Lyapunov exponents negative
   • Attractor dimension < 50 for 10k memory systems

C.1.3 Query Performance Validation

Objective: Validate Theorem T3 predictions about efficient scale-invariant access

Planned Protocol:

1. Index Construction:

   • Multi-scale index building time measurement
   • Memory usage analysis
   • Scale-space representation quality

2. Query Testing:

   • Fixed radius queries at different scales
   • k-NN queries with varying k
   • Adaptive scale selection

3. Performance Metrics:

   • Average query time
   • 95th percentile query time
   • Memory usage per query
   • Accuracy vs. speed trade-offs

4. Expected Results (Theoretical Predictions):

   • Average query time < 1ms for 1M memories
   • 95th percentile < 5ms
   • Logarithmic scaling with database size

C.2 Planned Implementation Validation

C.2.1 GPU Acceleration Benchmarks

Objective: Validate theoretical GPU speedup predictions

Planned Protocol:

1. Hardware Setup:

   • RTX 4090 (24GB VRAM)
   • Multiple GPU configurations for scaling analysis
   • CPU baseline for comparison

2. Benchmark Operations:

   • Distance matrix computation
   • Heat kernel evaluation
   • Attention field computation
   • Memory diffusion evolution

3. Performance Metrics:

   • Speedup vs. CPU baseline
   • Memory bandwidth utilization
   • GPU occupancy analysis
   • Scaling with problem size

4. Expected Results (Theoretical Predictions):

   • Distance computation: 50-100x speedup for large batches
   • Diffusion: 10-30x speedup for 1M+ node grids
   • Overall system: 20-50x speedup

C.2.2 Game Engine Integration Testing

Objective: Validate theoretical performance targets for game engine integration

Planned Protocol:

1. Unity/Unreal Prototype:

   • Hyperbolic rendering pipeline
   • Physics integration with hyperbolic metrics
   • Attention-based visualization

2. Performance Testing:

   • FPS with varying memory counts
   • Rendering quality vs. performance
   • Memory usage analysis
   • GPU utilization monitoring

3. User Experience Metrics:

   • Smoothness of navigation
   • Visual quality assessment
   • Interaction responsiveness

4. Expected Results (Theoretical Predictions):

   • 60 FPS with up to 1000 visible memories
   • Smooth scale transitions
   • Interactive navigation experience

C.3 Reproducibility Framework

C.3.1 Environment Specifications

Planned Standard Environment:

• Python 3.9+ with specific package versions
• CUDA 11.3+ for GPU acceleration
• Unity 2022.3+ for game engine testing
• Standardized hardware configurations

C.3.2 Data Management

Planned Data Handling:

• Version-controlled datasets
• Checksum verification
• Reproducible random seeds
• Standardized data formats

C.3.3 Validation Pipeline

Planned Validation Process:

1. Automated Testing:

   • Unit tests for all mathematical operations
   • Integration tests for complete pipeline
   • Performance regression testing

2. Manual Validation:

   • Expert review of results
   • Cross-validation with independent implementations
   • Peer review of experimental protocols

3. Documentation:

   • Complete experimental logs
   • Performance analysis reports
   • Comparison with theoretical predictions

C.4 Timeline for Experimental Validation

Phase 1 (Months 1-3):

• Prototype implementation
• Basic validation of core algorithms
• Initial performance benchmarks

Phase 2 (Months 4-6):

• Full experimental validation
• GPU acceleration testing
• Game engine integration

Phase 3 (Months 7-9):

• Large-scale testing
• Performance optimization
• Documentation and publication

Phase 4 (Months 10-12):

• Community validation
• Independent replication studies
• Integration with existing systems

---

This experimental framework provides a comprehensive roadmap for validating the theoretical predictions of the Mnemoverse architecture. Each protocol is designed to test specific theoretical claims while maintaining scientific rigor and reproducibility.

---

📚 Research Sources

For the complete collection of research sources supporting the mathematical proofs, implementation code, and experimental protocols presented in this document, see:

📚 Research Library - Comprehensive collection of 92 verified academic sources covering:

• Mathematical Foundations - Information geometry, differential geometry, and dynamical systems theory
• Hyperbolic Geometry & Embeddings - Poincaré embeddings, hyperbolic neural networks, and geometric deep learning
• Implementation & Performance - GPU computing, optimization techniques, and benchmarking methodologies
• Memory Theory & Navigation - Grid cells, spatial memory, and cognitive mapping research
• Multi-Agent Systems - Distributed cognitive systems and collective intelligence

All mathematical proofs, implementation approaches, and experimental protocols in this document are based on these verified research sources and established theoretical frameworks.

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