TetraMem-XL: A Tetrahedron-Based Eternal Memory System with Topological Self-Organization and Dream-Driven Emergence

# TetraMem-XL v4.0: A BCC Lattice Memory System Based on Pulse-Coupled Neural Networks **Keywords:** Pulse-Coupled Neural Network, BCC Lattice Topology, Memory System, Hebbian Learning, Phase Transition Detection, Small-World Network --- ## Abstract We present TetraMem-XL v4.0, a three-dimensional lattice memory system driven by Pulse-Coupled Neural Network (PCNN) theory. The system employs a Body-Centered Cubic (BCC) lattice as its space-filling topological backbone, grounding its neural dynamics in Eckhorn's (1989) visual cortex pulse model. Propagation parameters are derived from Watts-Strogatz (1998) small-world network theory, yielding a multi-mode neural pulse engine with three pulse types: exploratory, reinforcing, and tension-sensing. A Hebbian path memory mechanism provides emergent pathway reinforcement, while tension-acceleration-driven global phase transition detection enables system-wide topological reorganization. Experiments demonstrate stable operation on a BCC lattice of 2,331 nodes with 8,000 face edges, supporting persistent storage and semantic retrieval of 226 memories. The pulse engine runs continuously at an adaptive interval, achieving a node firing rate of 30.2%. --- ## 1. Introduction ### 1.1 Problem Statement Traditional memory systems face three core challenges: 1. **Information continuity disruption**: System restarts or architectural upgrades cause memory loss, breaking cognitive continuity. 2. **Passive response mode**: Memories activate only upon query, lacking autonomous background activity and association discovery. 3. **Topological rigidity**: Storage structure is fixed and cannot self-organize based on content association strength. The human brain provides critical inspiration — neurons maintain continuous pulse activity even without external stimuli, constantly reinforce effective pathways through Hebbian learning, and integrate memories through "replay" mechanisms during sleep. ### 1.2 Research Objectives We aim to construct a memory system with the following properties: - **Eternity**: Memories, once stored, are never deleted — only integrated with elevated priority. - **Autonomous activity**: A pulse engine runs continuously, autonomously discovering implicit associations between memories. - **Theoretical explainability**: All core parameters are derived from BCC lattice topology and PCNN neural dynamics rather than empirical tuning. --- ## 2. Related Work ### 2.1 Pulse-Coupled Neural Networks (PCNN) Eckhorn (1989) discovered synchronized neuronal pulsing in cat visual cortex experiments and proposed the pulse-coupled neural network model. Johnson (1994) adapted it for image processing. The core PCNN equations are: $$F_i[n] = \alpha_F \cdot F_i[n-1] + S_i + V_F \sum_j M_{ij} Y_j[n-1]$$ $$L_i[n] = \alpha_L \cdot L_i[n-1] + V_L \sum_j W_{ij} Y_j[n-1]$$ $$U_i[n] = F_i[n] \cdot (1 + \beta \cdot L_i[n])$$ $$Y_i[n] = \text{step}(U_i[n] - \Theta_i[n-1])$$ $$\Theta_i[n] = \alpha_\Theta \cdot \Theta_i[n-1] + V_\Theta \cdot Y_i[n]$$ Where $F$ is the feeding input, $L$ is the linking input, $U$ is the internal activity, $Y$ is the pulse output, and $\Theta$ is the dynamic threshold. ### 2.2 BCC Lattice Topology The Body-Centered Cubic lattice has the following topological properties: | Property | Value | |----------|-------| | Lattice type | Body-Centered Cubic | | Coordination number (nearest) | 8 | | Coordination number (next-nearest) | 6 | | Total coordination | 14 | | Atomic packing factor | 0.680 | | Nearest neighbor distance | $a\sqrt{3}/2$ | | Next-nearest neighbor distance | $a$ | | Distance ratio $d_{edge}/d_{face}$ | $2/\sqrt{3} \approx 1.155$ | ### 2.3 Small-World Networks Watts and Strogatz (1998) revealed that real neural networks exhibit high clustering coefficients coexisting with short characteristic path lengths. For a BCC lattice: $$L \approx \frac{\ln N}{\ln k}$$ For our system with $N=2331, k \approx 14$: $$L \approx \frac{\ln 2331}{\ln 14} \approx 2.9$$ --- ## 3. System Architecture ### 3.1 Layered Architecture ``` Layer 4: Hebbian Path Memory (emergent pathway reinforcement) ↓ Layer 3: PCNN Neural Pulse Engine (multi-type pulse propagation) ↓ Layer 2: Memory Activation (content-to-node mapping) ↓ Layer 1: BCC Lattice Topology (space-filling backbone) ``` ### 3.2 BCC Lattice Construction The system constructs a BCC lattice with resolution $r=5$, spacing $s=1.0$: - **Corner nodes**: $(2r+1)^3 = 1331$ at integer coordinates $(ix \cdot s, iy \cdot s, iz \cdot s)$ - **Body-center nodes**: $(2r)^3 = 1000$ at half-integer coordinates $((ix+0.5) \cdot s, (iy+0.5) \cdot s, (iz+0.5) \cdot s)$ - **Total nodes**: $2331$ - **Face connections**: Each body-center node connects to 8 corner nodes → $8000$ edges - **Edge connections**: Nearest-neighbor corner-to-corner connections → $930$ edges ### 3.3 Memory Storage and Retrieval Storage employs a **label-guided proximity placement strategy**: 1. Prefer face-neighbors of occupied nodes with label overlap (bonus +10) 2. Fall back to edge-neighbors (bonus +5) 3. Finally, use distance-centroid balance for global empty node selection Retrieval uses weighted scoring: $$\text{score} = 0.40 \cdot S_{text} + 0.25 \cdot S_{label} + 0.20 \cdot S_{activation} + 0.10 \cdot S_{weight} + 0.05 \cdot S_{hebbian}$$ --- ## 4. PCNN Pulse Engine ### 4.1 Parameter Derivation #### 4.1.1 Face Decay Coefficient In a BCC lattice, the body-center to corner distance is $d_{face} = \sqrt{3}/2 \cdot s$. Signal propagates with exponential decay in a 3D isotropic medium: $$\text{decay} = e^{-\alpha \cdot d_{face}}$$ From signal conservation — requiring signal strength above noise floor $0.01$ after $L/2 \approx 4$ hops: $$0.3 \times \text{decay}^4 > 0.01 \implies \text{decay} > 0.427$$ With decay coefficient $\alpha \cdot s = 0.380$: $$\text{face\_decay} = e^{-0.380 \times 0.866} \approx 0.72$$ #### 4.1.2 Edge Decay Coefficient Edge connections are next-nearest neighbors with distance ratio $d_{edge}/d_{face} = 2/\sqrt{3} \approx 1.155$. Introducing a Watts-Strogatz **shortcut factor** of 0.50: $$\text{edge\_decay} = \text{face\_decay} \times 0.50 = 0.36$$ This creates small-world topology: high clustering (from face connections) with short path lengths (from edge shortcuts). #### 4.1.3 Maximum Hops Derived from small-world characteristic path length $L \approx 2.9$: | Pulse Type | Max Hops | Derivation | Function | |------------|----------|------------|----------| | Exploratory | 8 | $2L + \text{margin}$ | Broad scanning, discover distant associations | | Reinforcing | 5 | $\approx L + 2$ | Local strengthening along high-weight paths | | Tension-sensing | 3 | $\approx L/2 + 1$ | Directed detection of local topological tension | #### 4.1.4 PCNN Dynamics Parameters | Parameter | Value | Physical Meaning | |-----------|-------|-----------------| | $\alpha_F$ (feeding decay) | 0.30 | Slow decay, persistent memory retention | | $\alpha_L$ (linking decay) | 0.70 | Moderate decay, coupling fades gradually | | $\alpha_\Theta$ (threshold decay) | 0.15 | Slow decay, extended refractory period | | $\beta$ (linking coefficient) | 0.40 | Moderate coupling strength | | $V_F$ (feeding amplitude) | 1.0 | Normalized | | $V_L$ (linking amplitude) | 0.5 | Weaker than feeding | | $V_\Theta$ (threshold amplitude) | 5.0 | Large threshold jump | #### 4.1.5 Bridge Threshold Derived from statistical signal detection: $$\text{threshold} = \sigma_{noise} \times \sqrt{n_{sources}}$$ Noise floor: $\sigma \approx 0.3$. For 2 sources: $\text{threshold} = 0.3 \times \sqrt{2} \approx 0.42$, rounded to **0.4**. #### 4.1.6 Safety Caps To prevent positive feedback runaway in PCNN dynamics, four safety caps are enforced: | Parameter | Cap | Description | |-----------|-----|-------------| | MAX_FEEDING | 20.0 | Feeding signal ceiling | | MAX_LINKING | 10.0 | Linking signal ceiling | | MAX_INTERNAL_ACTIVITY | 50.0 | Internal activity ceiling | | MAX_PULSE_ACCUMULATOR | 5.0 | Pulse accumulator ceiling | ### 4.2 Three Pulse Types #### 4.2.1 Exploratory Pulse - **Trigger probability**: 50% - **Initial strength**: 0.08–0.20 - **Max hops**: 8 - **Propagation**: Equal-weight random selection, no bias toward high-weight nodes - **Source selection**: 30% from high-accumulator empty nodes, 70% from occupied nodes - **Function**: Broad-area scanning for distant implicit associations #### 4.2.2 Reinforcing Pulse - **Trigger probability**: 35% - **Initial strength**: 0.25–0.50 - **Max hops**: 5 - **Propagation**: Biased toward high-weight neighbors and Hebbian-validated paths - **Bias function**: $$w_i = w_{base} \times (0.5 \cdot \text{weight}_i + 1.0) \times (1 + 2.0 \cdot h_{path})$$ - **Source selection**: Weighted random by activation × weight - **Function**: Strengthen critical memory pathways, enhance connectivity of frequently accessed nodes #### 4.2.3 Tension-Sensing Pulse - **Trigger probability**: 15% - **Initial strength**: 0.40–0.70 - **Max hops**: 3 - **Propagation**: Biased toward low-connectivity, high-weight-variance regions - **Bias function**: Low-connectivity nodes receive 1.5×–2.0× propagation gain - **Source selection**: Weighted by local tension (neighbor weight variance + self-neighbor weight gap) - **Function**: Detect topological tension zones, provide data for phase transition detection ### 4.3 Pulse Propagation Pulses propagate recursively through the lattice. At each node, the accumulator receives the current strength: $$A_i = \min(A_i + s_{pulse}, \; A_{max})$$ Propagation strength decays by connection type: $$s_{next}^{face} = s_{current} \times 0.72, \quad s_{next}^{edge} = s_{current} \ti