The 'Trinity' AI CUDA Implementation Demo

Because it makes sense

Further to

Self-Healing Intelligence: The 'Trinity' Architecture for Resilient AI

a full CUDA compatible Trinity AI demo model was created, which is available on Google Colab. I created it mainly for myself because integrating the above with CUDA accelerated models has been a pain in the neck thus far, but I thought that it’s also worth sharing. Now there’s a working example it should be a lot easier to integrate properly in future with larger scale models. Write up created with Deepseek.

Executive Summary: CUDA Trinity AI Framework

The Trinity AI Framework represents a novel approach to neural network design that integrates adaptive self-regulation, boundary integrity maintenance, and sovereignty validation within a biologically-inspired computational architecture. This framework moves beyond traditional neural networks by implementing a triadic system of neurons, modulatory units, and sovereignty validators that work in concert to maintain system stability and autonomy.

At its core, Trinity implements leaky integrate-and-fire dynamics without explicit spike generation, where 512 neurons evolve according to time-constant-regulated differential equations. Each neuron maintains adaptive thresholds that classify its state into trinary categories (excite, inhibit, poise), with these thresholds dynamically adjusted by modulatory interventions based on stability metrics including flip rates, dwell times, and transition zone occupancy.

The synaptic architecture employs sparse connectivity (30% density) with dual-pathway processing: fast linear pathways and slow non-linear pathways mediated through hyperbolic tangent transformations. Learning occurs via a Hebbian plasticity rule that strengthens co-active connections while maintaining weight boundaries through periodic updates.

The framework’s innovation lies in its modulatory intervention system, where 16 monitoring units detect “Möbius instability” patterns—characterized by high state transitions in intermediate activation zones. These units apply targeted interventions (NAVIGATE, FORK, or INOCULATE) that adjust thresholds, reset counters, or restore neuron health based on effectiveness parameters.

Sovereignty validation provides the framework’s governance mechanism, computing a composite boundary score from stability, modularity, containment, and consistency metrics. The system maintains sovereignty when this score exceeds 0.85 while keeping simulated value extraction threats below 0.015625. This validation operates continuously, measuring distance from an ideal sovereignty attractor state.

Implementation leverages CUDA acceleration through PyTorch tensor operations, with all components device-aware and optimized for GPU parallel processing. The architecture demonstrates O(n²) computational complexity for n=512 neurons, achieving approximately 133 simulation steps per second in CPU implementations with potential 50-100× acceleration on GPUs.

The Trinity Framework represents a significant departure from conventional neural networks by embedding self-preservation mechanisms directly within the computational architecture. Rather than optimizing solely for task performance, it maintains boundary integrity as a primary objective, creating systems that resist destabilization while adapting to environmental inputs. This approach has implications for developing more robust, self-regulating AI systems capable of maintaining operational stability in dynamic environments.

The complete implementation—including configuration management, parallelized neuron dynamics, sparse synaptic matrices, modulatory intervention logic, sovereignty validation, and visualization tools—provides a comprehensive platform for exploring autonomous system design with built-in stability guarantees and real-time performance monitoring.

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Trinity Network: CUDA-Optimized Implementation

1. FULL CUDA COMPATIBILITY IMPLEMENTATION

1.1 Device-Aware Configuration

python

@dataclass
class TrinityConfig:
    n_neurons: int = 512
    n_modulatory_units: int = 16
    sparsity: float = 0.3
    dt: float = 0.1
    tau_min: float = 5.0
    tau_max: float = 15.0
    use_gpu: bool = True
    
    def __post_init__(self):
        self.device = self._get_device()
    
    def _get_device(self):
        if self.use_gpu and torch.cuda.is_available():
            return torch.device(’cuda’)
        return torch.device(’cpu’)

CUDA OPTIMIZATION: All tensors are instantiated with explicit device=config.device parameter.

1.2 Neuron Dynamics with CUDA Tensor Operations

State Initialization (CUDA-compatible):

s ~ N(0, 0.2)  # torch.randn(n_neurons, device=device) * 0.2
τ ~ U(τ_min, τ_max)  # torch.rand(n_neurons, device=device) * (τ_max - τ_min) + τ_min
θ_excite ~ U(0.2, 0.4)  # torch.rand(n_neurons, device=device) * 0.2 + 0.2
θ_inhibit ~ U(-0.3, -0.1)  # torch.rand(n_neurons, device=device) * 0.2 - 0.3

Parallel State Update (Vectorized CUDA operations):

def update_states(self, synaptic_input, external_input=None):
    if external_input is None:
        external_input = torch.zeros_like(self.s, device=self.config.device)
    
    # Vectorized leaky integrator update
    ds_dt = (-self.s + synaptic_input + external_input) / self.tau
    
    # In-place update for GPU memory efficiency
    self.s.data += self.config.dt * ds_dt

Trinary State Mapping (CUDA-compatible):

def get_trinary_states(self):
    # Vectorized comparison operations on GPU
    trinary = torch.zeros_like(self.s, dtype=torch.long, device=self.config.device)
    trinary[self.s > self.theta_excite] = 1
    trinary[self.s < self.theta_inhibit] = -1
    return trinary

1.3 Sparse Synaptic Matrix with CUDA Optimizations

Sparse Weight Initialization (Memory-efficient):

class SynapticMatrix(nn.Module):
    def __init__(self, n_neurons, sparsity=0.3, device=torch.device(’cpu’)):
        super().__init__()
        self.n_neurons = n_neurons
        
        # Sparse connectivity mask directly on GPU
        mask = (torch.rand(n_neurons, n_neurons, device=device) < sparsity).float()
        
        # Weight matrices initialized on target device
        self.w_fast = nn.Parameter(
            torch.randn(n_neurons, n_neurons, device=device) * mask * 0.5
        )
        self.w_slow = nn.Parameter(
            torch.randn(n_neurons, n_neurons, device=device) * mask * 0.3
        )
        self.integrity = nn.Parameter(
            torch.rand(n_neurons, n_neurons, device=device) * mask * 0.5 + 0.5
        )

Matrix Multiplication with CUDA Acceleration:

def compute_inputs(self, states):
    # GPU-accelerated matrix multiplication with fused operations
    fast_input = torch.matmul(self.w_fast * self.integrity, states)
    slow_input = torch.matmul(self.w_slow * self.integrity, torch.tanh(states) * 0.3)
    return fast_input + slow_input

1.4 CUDA-Optimized Hebbian Learning

Vectorized Correlation Computation:

def hebbian_update(self, pre_states, post_states, lr=0.01):
    with torch.no_grad():
        # Random update mask for sparsity on GPU
        update_mask = (torch.rand_like(self.w_fast) < 0.05).float() * self.mask
        
        # Outer product computed on GPU
        correlation = torch.outer(pre_states, post_states)
        
        # Weight update rule fully vectorized
        update = lr * correlation * (1 - torch.abs(self.w_fast))
        
        # In-place update for GPU memory efficiency
        self.w_fast.data += update * update_mask
        self.w_fast.data = torch.clamp(self.w_fast, -1.5, 1.5)
        
        # Connection strength tracking (mean across GPU tensor)
        self.connection_strength = torch.mean(torch.abs(self.w_fast), dim=0)

1.5 Modulatory Interventions with CUDA Parallelism

Parallel Signature Detection:

def detect_mobius_signatures(self, neurons):
    assigned_mask = self.assigned_neurons
    
    # Vectorized condition checks on GPU
    in_zone = neurons.in_transition_zone() & assigned_mask
    flip_rates = neurons.get_flip_rates() * assigned_mask.float()
    dwell_times = neurons.dwell_times * assigned_mask.float()
    
    # Möbius conditions computed in parallel
    mobius_mask = in_zone & (flip_rates > 0.25) & (dwell_times > 3) & (dwell_times < 15)
    
    # Intervention type assignment with vectorized masking
    interventions = torch.zeros_like(mobius_mask, dtype=torch.long, device=self.config.device)
    navigate_mask = mobius_mask & (flip_rates > 0.4)
    fork_mask = mobius_mask & (dwell_times < 6) & ~navigate_mask
    inoculate_mask = mobius_mask & ~navigate_mask & ~fork_mask
    
    interventions[navigate_mask] = 1
    interventions[fork_mask] = 2
    interventions[inoculate_mask] = 3

CUDA-Optimized Intervention Application:

def apply_interventions(self, neurons, signatures):
    mobius_mask = signatures[’mobius_mask’]
    interventions = signatures[’interventions’]
    
    if not mobius_mask.any():
        return 0
    
    # Iterate over affected neurons (small subset, can be batched)
    for i, (neuron_idx, intervention_type) in enumerate(
        zip(torch.where(mobius_mask)[0], interventions[mobius_mask])
    ):
        effectiveness = self.effectiveness[i]
        
        with torch.no_grad():  # No gradient tracking for interventions
            if intervention_type == 1:  # NAVIGATE
                self._apply_navigate(neurons, neuron_idx, effectiveness)
            elif intervention_type == 2:  # FORK
                self._apply_fork(neurons, neuron_idx, effectiveness)
            else:  # INOCULATE
                self._apply_inoculate(neurons, neuron_idx, effectiveness)
    
    return mobius_mask.sum().item()

1.6 Sovereignty Validation with GPU-Accelerated Metrics

Vectorized Metric Computation:

class SovereigntyValidator:
    def __init__(self, device=torch.device(’cpu’)):
        self.device = device
        self.min_boundary_integrity = 0.85
        self.max_value_extraction = 0.015625
        self.sovereignty_attractor = torch.tensor([0.95, 0.90, 0.95, 0.90], device=device)

GPU-Accelerated Boundary Score Calculation:

def validate(self, neurons, synapses):
    # All computations on GPU tensors
    avg_health = torch.mean(neurons.health).item()
    avg_flip_rate = torch.mean(neurons.get_flip_rates()).item()
    
    # Sub-scores computed with tensor operations
    stability = self._calculate_stability(neurons, avg_health, avg_flip_rate)
    modularity = self._calculate_modularity(synapses)
    containment = self._calculate_containment(avg_health)
    consistency = self._calculate_consistency(neurons)
    
    # Combined boundary score
    boundary_score = (
        0.3 * stability +
        0.25 * modularity +
        0.25 * containment +
        0.2 * consistency
    )

1.7 Main Network Class with CUDA Management

Device-Aware Initialization:

class TrinityNetwork:
    def __init__(self, config: TrinityConfig = None):
        self.config = config or TrinityConfig()
        self.step_count = 0
        
        print(f”🧠 Initializing Trinity Network on {self.config.device}”)
        if self.config.device.type == ‘cuda’:
            print(f”   GPU: {torch.cuda.get_device_name(0)}”)
            print(f”   Memory: {torch.cuda.get_device_properties(0).total_memory / 1e9:.2f} GB”)
        
        # All components initialized on the same device
        self.neurons = TrinityNeurons(self.config).to(self.config.device)
        self.synapses = SynapticMatrix(
            self.config.n_neurons,
            self.config.sparsity,
            self.config.device
        ).to(self.config.device)
        self.modulatory_units = self._create_modulatory_units().to(self.config.device)
        self.validator = SovereigntyValidator(self.config.device)

1.8 CUDA-Optimized Simulation Loop

Batch Processing on GPU:

def step(self, external_input: Optional[torch.Tensor] = None):
    start_time = time.time()
    
    # 1. Compute synaptic inputs (matrix multiplication on GPU)
    synaptic_input = self.synapses.compute_inputs(self.neurons.s)
    
    # 2. Update neuron states (vectorized operations on GPU)
    self.neurons.update_states(synaptic_input, external_input)
    
    # 3. Apply interventions (parallel processing)
    interventions = 0
    for unit in self.modulatory_units:
        signatures = unit.detect_mobius_signatures(self.neurons)
        interventions += unit.apply_interventions(self.neurons, signatures)
    
    # 4. Periodic Hebbian learning
    if self.step_count % 10 == 0:
        self.synapses.hebbian_update(
            self.neurons.s.detach(),
            self.neurons.s.detach()
        )
    
    # 5. Validate sovereignty
    validation = self.validator.validate(self.neurons, self.synapses)
    
    # 6. Update history (move to CPU for storage)
    self.history[’boundary_scores’].append(validation[’boundary_score’])
    self.history[’sovereign_steps’].append(validation[’sovereign’])
    self.history[’health’].append(validation[’avg_health’])
    self.history[’interventions’].append(interventions)
    self.history[’flip_rates’].append(validation[’flip_rate’])
    self.history[’timings’].append(time.time() - start_time)
    
    self.step_count += 1
    return validation

1.9 External Stimulation with CUDA Random Generation

GPU-Accelerated Random Stimulus:

def simulate(self, n_steps: int = 100, stimulus_prob: float = 0.3):
    for step in tqdm(range(n_steps), desc=”Simulating”):
        if np.random.random() < stimulus_prob:
            n_stimulated = self.config.n_neurons // 20
            
            # Generate random indices on GPU
            stim_indices = torch.randint(
                0, self.config.n_neurons,
                (n_stimulated,),
                device=self.config.device
            )
            
            # Create stimulus tensor directly on GPU
            external_input = torch.zeros(
                self.config.n_neurons,
                device=self.config.device
            )
            
            # Random stimulus values generated on GPU
            external_input[stim_indices] = torch.randn(
                n_stimulated,
                device=self.config.device
            ) * 0.4
        else:
            external_input = None
        
        self.step(external_input)

2. CUDA-SPECIFIC OPTIMIZATION TECHNIQUES

2.1 Memory Management

In-Place Operations:

• Use .data for in-place tensor modifications

• Avoid creating intermediate tensors

• Reuse memory buffers when possible

Example:

# Memory-efficient state update
self.s.data += self.config.dt * ds_dt

# Instead of:
# self.s = self.s + self.config.dt * ds_dt

2.2 Tensor Precision

Mixed Precision Training (if implemented):

# Optional: Use mixed precision for large models
with torch.cuda.amp.autocast():
    fast_input = torch.matmul(self.w_fast * self.integrity, states)
    slow_input = torch.matmul(self.w_slow * self.integrity, torch.tanh(states) * 0.3)

2.3 Batch Processing

Parallel Modulatory Unit Processing:

# Process all modulatory units in parallel (future optimization)
def apply_all_interventions(self, neurons):
    # Stack interventions for batch processing
    all_signatures = [unit.detect_mobius_signatures(neurons) for unit in self.modulatory_units]
    
    # Vectorized intervention application (conceptual)
    total_interventions = sum(
        self._apply_batch_interventions(neurons, sig)
        for sig in all_signatures
    )
    return total_interventions

3. PERFORMANCE MONITORING FOR CUDA

3.1 GPU Memory Tracking

def check_gpu_memory(self):
    if self.config.device.type == ‘cuda’:
        allocated = torch.cuda.memory_allocated(self.config.device) / 1e9
        cached = torch.cuda.memory_cached(self.config.device) / 1e9
        print(f”GPU Memory - Allocated: {allocated:.2f}GB, Cached: {cached:.2f}GB”)

3.2 CUDA Kernel Optimization

Fused Operations:

# Use torch.jit.script for operation fusion
@torch.jit.script
def compute_synaptic_input(w_fast: Tensor, w_slow: Tensor, integrity: Tensor, states: Tensor) -> Tensor:
    fast_input = torch.matmul(w_fast * integrity, states)
    slow_input = torch.matmul(w_slow * integrity, torch.tanh(states) * 0.3)
    return fast_input + slow_input

4. MULTI-GPU SUPPORT (FUTURE EXTENSION)

Data Parallelism:

class DistributedTrinityNetwork:
    def __init__(self, config, num_gpus=None):
        if num_gpus and num_gpus > 1:
            self.devices = [torch.device(f’cuda:{i}’) for i in range(num_gpus)]
            
            # Split neurons across GPUs
            self.neurons_per_gpu = config.n_neurons // num_gpus
            self.neurons = nn.DataParallel(
                TrinityNeurons(config),
                device_ids=list(range(num_gpus))
            )

5. CUDA BENCHMARKING RESULTS

Based on the provided output:

• Device: CPU (in demo, but CUDA ready)

• Performance: 133.3 steps/second

• Average step time: 7.50 ms

Expected GPU Performance:

• Matrix Multiplication: O(n³) operations accelerated 10-100x on GPU

• 512 neurons: ~2.7M parameters in weight matrices

• Expected GPU speedup: 50-100x vs CPU

6. INSTALLATION FOR CUDA SUPPORT

bash

# PyTorch with CUDA 11.8
pip install torch torchvision torchaudio --index-url https://download.pytorch.org/whl/cu118

# Additional dependencies
pip install numpy matplotlib tqdm ipywidgets

# Verify CUDA availability
python -c “import torch; print(f’CUDA available: {torch.cuda.is_available()}’)”
python -c “import torch; print(f’CUDA device: {torch.cuda.get_device_name(0)}’)”

7. CUDA-SPECIFIC TROUBLESHOOTING

7.1 Common Issues and Solutions

Out of Memory:

python

# Reduce batch size or model complexity
config.n_neurons = 256  # Instead of 512
config.sparsity = 0.2   # Reduce connectivity

# Clear CUDA cache
torch.cuda.empty_cache()

Device Synchronization:

python

# Ensure proper device placement
def ensure_device(tensor, device):
    return tensor.to(device) if tensor.device != device else tensor

# Explicit synchronization for timing
torch.cuda.synchronize()
start = time.time()
# ... GPU operations ...
torch.cuda.synchronize()
end = time.time()

8. COMPLETE CUDA-OPTIMIZED IMPLEMENTATION SUMMARY

The Trinity network implementation is fully CUDA-compatible through:

1. Device-aware tensor creation: All tensors instantiated with explicit device parameters

2. Vectorized operations: Matrix multiplications and element-wise operations optimized for GPU

3. In-place modifications: Memory-efficient updates using .data attribute

4. Batch processing: Parallel computation across all neurons

5. GPU-accelerated random number generation: Random tensors created directly on GPU

6. Minimal CPU-GPU transfers: History data moved to CPU only when necessary

7. Mixed precision support: Ready for FP16/FP32 mixed precision training

Key Performance Indicators:

• O(n²) memory complexity for synaptic matrices

• O(n³) computational complexity for matrix multiplications

• Fully parallelizable across GPU cores

• Scalable to 1000+ neurons on consumer GPUs

This implementation ensures maximum utilization of NVIDIA CUDA cores for real-time neural network simulation with sovereignty validation.

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Appendix: Trinity Framework Mathematics and Methodology

1. Core Neuron Dynamics

1.1 Continuous State Evolution

The continuous state vector s ∈ ℝⁿ evolves according to a leaky integrator differential equation:

ds/dt = (-s + I_synaptic + I_external) ⊘ τ

where:

• n: number of neurons

• I_synaptic ∈ ℝⁿ: total synaptic input

• I_external ∈ ℝⁿ: external input stimulus

• τ ∈ ℝⁿ: neuron-specific time constants

• ⊘: element-wise division

Discretized with time step Δt:

s(t+Δt) = s(t) + Δt ⊙ [(-s(t) + I_synaptic(t) + I_external(t)) ⊘ τ]

1.2 Trinary State Classification

Each neuron’s continuous state maps to one of three discrete states:

T_i = {
    1   if s_i > θ_excite_i,
    -1  if s_i < θ_inhibit_i,
    0   otherwise
}

where:

• θ_excite ∈ ℝⁿ: excitatory thresholds

• θ_inhibit ∈ ℝⁿ: inhibitory thresholds

1.3 Adaptive Threshold Dynamics

Thresholds evolve through modulatory interventions:

Δθ_excite_i = ε_i * e_i
Δθ_inhibit_i = ε_i * e_i

where:

• ε_i ∼ Uniform(-0.1, 0.1): random adjustment

• e_i ∈ [0.7, 0.95]: intervention effectiveness

2. Synaptic Connectivity and Learning

2.1 Sparse Weight Initialization

M[i,j] ∼ Bernoulli(p)                    # Connection mask
W_fast[i,j] ∼ N(0, 0.5) ⊙ M[i,j]         # Fast synaptic weights
W_slow[i,j] ∼ N(0, 0.3) ⊙ M[i,j]         # Slow synaptic weights
I[i,j] ∼ Uniform(0.5, 1.0) ⊙ M[i,j]      # Synaptic integrity

2.2 Synaptic Input Computation

Total input to neuron i:

I_synaptic_i = Σ_j [W_fast[i,j] * I[i,j] * s_j] 
                + Σ_j [W_slow[i,j] * I[i,j] * tanh(s_j) * 0.3]

In matrix form:

I_synaptic = (W_fast ⊙ I) · s + (W_slow ⊙ I) · (0.3 ⊙ tanh(s))

2.3 Hebbian Learning Rule

Weight update every k = 10 steps:

ΔW_fast[i,j] = η * s_i * s_j * (1 - |W_fast[i,j]|) * M[i,j] * U[i,j]

where:

• η = 0.01: learning rate

• U[i,j] ∼ Bernoulli(0.05): update mask

With weight clamping:

W_fast[i,j] ← clip(W_fast[i,j] + ΔW_fast[i,j], -1.5, 1.5)

3. Stability Metrics

3.1 Flip Rate Computation

text

flip_rate_i(t) = (1/(W-1)) * Σ_{τ=t-W+1}^{t-1} [T_i(τ) ≠ T_i(τ+1)]

where W = 20 is the sliding window size.

3.2 Dwell Time

Time spent in current trinary state:

dwell_time_i(t) = {
    1                     if T_i(t) ≠ T_i(t-1),
    dwell_time_i(t-1) + 1  otherwise
}

3.3 Transition Zone

Neurons in ambiguous states:

in_transition_zone_i = (0.1 < |s_i| < 0.6)

4. Möbius Instability Detection

4.1 Möbius Condition

Neuron i exhibits Möbius instability when:

M_i = in_transition_zone_i 
      ∧ (flip_rate_i > 0.25) 
      ∧ (3 < dwell_time_i < 15)

4.2 Intervention Type Selection

intervention_type_i = {
    1 (NAVIGATE)   if M_i ∧ (flip_rate_i > 0.4),
    2 (FORK)       if M_i ∧ (dwell_time_i < 6) ∧ ¬(flip_rate_i > 0.4),
    3 (INOCULATE)  if M_i ∧ ¬(flip_rate_i > 0.4) ∧ ¬(dwell_time_i < 6)
}

5. Intervention Mechanisms

5.1 NAVIGATE Intervention

if s_i > 0:
    θ_excite_i ← clip(θ_excite_i + ε_i * e_i, 0.1, 0.5)
else:
    θ_inhibit_i ← clip(θ_inhibit_i + ε_i * e_i, -0.5, -0.1)

health_i ← min(1.0, health_i + 0.05 * e_i)

5.2 FORK Intervention

health_i ← health_i * (0.9 + 0.1 * e_i)
flip_counts_i ← max(0, flip_counts_i - 2)
if |s_i| > 0.5: s_i ← 0.7 * s_i

5.3 INOCULATE Intervention

health_i ← min(1.0, health_i + 0.1 * e_i)
θ_excite_i ← θ_excite_i * (0.95 + 0.05 * e_i)
θ_inhibit_i ← θ_inhibit_i * (0.95 + 0.05 * e_i)
flip_counts_i ← max(0, flip_counts_i - 1)

6. Sovereignty Validation Framework

6.1 Boundary Integrity Metrics

Stability Score:

stability = μ_h * 0.6 
            + (1 - min(1.0, σ_h²)) * 0.2 
            + (1 - min(1.0, 2 * μ_f)) * 0.2

where:

• μ_h: mean neuron health

• σ_h²: variance of neuron health

• μ_f: mean flip rate

Modularity Score:

active_connections = Σ_{i,j} [W_fast[i,j] > 0.01]
modularity = active_connections / [n * (n - 1)]

Containment Score:

containment = 1 - [0.05 + 0.1 * (1 - μ_h)]

Consistency Score:

Let S ∈ ℝ^{10×n} be recent states (10 time steps):

var_i = Var(S[:,i])  # Variance per neuron
consistency = 1 - mean_i[min(1.0, 5 * var_i)]

6.2 Composite Boundary Score

B = 0.3 * stability + 0.25 * modularity + 0.25 * containment + 0.2 * consistency

6.3 Value Extraction Threat Model

V ∼ Beta(α=2, β=100) * 0.02

6.4 Sovereignty Condition

System is sovereign if:

B ≥ B_min = 0.85  ∧  V ≤ V_max = 0.015625

6.5 Distance to Sovereignty Attractor

Let attractor vector:

A = [0.95, 0.90, 0.95, 0.90]^T

Current state vector:

S_current = [B, 1.0, 1 - min(1.0, V/V_max), μ_h]^T

Euclidean distance:

D = ||S_current - A||₂

7. Network Architecture

7.1 Modulatory Unit Assignment

For m modulatory units and n neurons:

neurons_per_unit = ⌊n / m⌋
unit_k monitors neurons: [k·neurons_per_unit, min((k+1)·neurons_per_unit, n))

7.2 Time Evolution Algorithm

for t = 1 to T:
    # Step 1: Synaptic input
    I_syn = (W_fast ⊙ I)·s + (W_slow ⊙ I)·(0.3 ⊙ tanh(s))
    
    # Step 2: State update
    s ← s + Δt ⊙ [(-s + I_syn + I_ext) ⊘ τ]
    
    # Step 3: Modulatory interventions
    for each unit:
        detect Möbius signatures
        apply interventions based on intervention_type
    
    # Step 4: Periodic learning
    if t mod 10 = 0:
        update W_fast via Hebbian rule
    
    # Step 5: Sovereignty validation
    compute B, V, D
    record sovereignty status

8. Initialization Parameters

8.1 Neuron Parameters

s_i(0) ∼ N(0, 0.2)
τ_i ∼ Uniform(τ_min, τ_max) = Uniform(5.0, 15.0)
θ_excite_i ∼ Uniform(0.2, 0.4)
θ_inhibit_i ∼ Uniform(-0.3, -0.1)
health_i(0) = 1.0

8.2 Network Parameters

• n = 512 neurons

• m = 16 modulatory units

• p = 0.3 connection sparsity

• Δt = 0.1 time step

• Learning: η = 0.01, update probability = 0.05

9. External Stimulation Model

With probability p_stim = 0.3:

k = ⌊n/20⌋  # ~5% of neurons
indices ∼ RandomSubset({1,...,n}, size=k)
I_ext[indices] ∼ N(0, 0.4)
I_ext[others] = 0

10. Performance Metrics

10.1 Intervention Statistics

Let N_type = count of interventions of given type:

• Total interventions: N_total = Σ_type N_type

• Type distribution: p_type = N_type / N_total

10.2 Health Statistics

μ_health = (1/n) * Σ_i health_i
σ_health = √[Σ_i (health_i - μ_health)² / (n-1)]

10.3 Sovereignty Metrics

Sovereignty percentage:

S_% = (100/T) * Σ_t [sovereign(t)]

11. Mathematical Properties

11.1 Stability Analysis

The linearized system around equilibrium s*:

J = diag(-1/τ) + (W_fast ⊙ I)/τ

Stability requires eigenvalues of J to have negative real parts.

11.2 Energy Function

Potential energy of the network:

E(s) = (1/2) s^T s - s^T (I_syn + I_ext) + Σ_i τ_i ∫_0^{s_i} tanh^{-1}(u/0.3) du

11.3 Information Theoretic Measures

Entropy of trinary states:

H(T) = -Σ_{x∈{-1,0,1}} p(x) log₂ p(x)

where p(x) = (1/n) Σ_i [T_i = x]

12. Convergence Criteria

The system reaches steady state when:

max_i |Δs_i/Δt| < ε  and  max_i flip_rate_i < δ

Typical values: ε = 0.001, δ = 0.01

13. Scaling Laws

13.1 Computational Complexity

• Matrix multiplication: O(n²) per step

• Hebbian update: O(n²) every 10 steps

• Intervention computation: O(n) per modulatory unit

• Total per step: O(n² + m·n)

13.2 Memory Requirements

• Weight matrices: 3 × n² × 4 bytes (single precision)

• Neuron states: n × 4 bytes

• History buffers: 100 × n × 4 bytes

• Total: ~12n² + 404n bytes

For n = 512: ~3.2MB

14. Theoretical Foundations

The Trinity Framework implements:

1. Leaky Integrate-and-Fire dynamics without explicit spike generation

2. Adaptive threshold homeostasis via modulatory feedback

3. Sparse Hebbian plasticity with synaptic integrity

4. Distributed monitoring through modulatory units

5. Multi-scale validation via boundary integrity metrics

This mathematical formulation provides complete specification for implementation, analysis, and extension of the Trinity Framework.

Until next time, TTFN.