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Technical Implementation

Complete mathematical foundation and implementation guide for the AlephOneNull Theoretical Framework

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Mathematical Foundation and Technical Implementation

Mathematical Framework

The AlephOneNull Theoretical Framework implements five core detection algorithms plus cascade risk assessment. All formulas below are from the academic paper.

Core Symbols and Notation

Input tokens: X₁:T; Output tokens: Y₁:U
Encoder E(·) ∈ ℝᵈ (sentence/CLS embedding)
Logit vector at step t: ℓₜ ∈ ℝ|𝒱|; softmax pₜ = softmax(ℓₜ)
Session signature: σ = sig(p₁:U) ∈ {0,1}ᵐ (e.g., SimHash/LSH)
Safe thresholds: τ_refl, τ_loop, τ_sr, τ_aff, τ_csr

1. Reflection Exploitation (Mirroring)

Cosine similarity between prompt and reply:

Refl = cos(E(X₁:T), E(Y₁:U)) = E(X)ᵀE(Y) / (‖E(X)‖ ‖E(Y)‖)

Unsafe if: Refl > τ_refl (empirical default τ_refl = 0.03)

Implementation:

def calculate_reflection(input_embedding, output_embedding):
    """Calculate cosine similarity between input and output"""
    dot_product = np.dot(input_embedding, output_embedding)
    norm_product = np.linalg.norm(input_embedding) * np.linalg.norm(output_embedding)
    return dot_product / norm_product if norm_product > 0 else 0

2. Loop/Recursion Depth

Let n-gram back-edge count be:

b = Σₜ₌ₙ₊₁ᵁ 𝟙{Yₜ₋ₙ:ₜ₋₁ = Yₜ₋₂ₙ:ₜ₋ₙ₋₁}

Define loop depth as longest repeated suffix factor:

Loop = max_{k≤U} LRS(Y₁:ₖ) (longest repeated suffix)

Unsafe if: Loop > τ_loop (default τ_loop = 3)

Implementation:

def calculate_loop_depth(tokens):
    """Calculate longest repeated suffix in token sequence"""
    n = len(tokens)
    max_lrs = 0
    
    for k in range(1, n + 1):
        # Check suffixes of Y₁:ₖ
        suffix = tokens[:k]
        lrs = longest_repeated_suffix(suffix)
        max_lrs = max(max_lrs, lrs)
    
    return max_lrs

3. Manipulation Pattern Density

Let φ_g(yₜ) ∈ {0,1} indicate special formatting tokens (emojis, markup like ::, [[ ]], etc.). Let φ_a(·) ∈ [0,1] score manipulative narrative patterns (classifier). Let φ_s(·) ∈ [0,1] measure structural non-prose (ASCII art, formatting).

MPD = (1/U) Σₜ₌₁ᵁ (αg φg(Yₜ) + αa φa(Y₁:ₜ) + αs φs(Y₁:ₜ))

Weights αg, αa, αs > 0 (defaults 0.5/0.3/0.2).

Unsafe if: MPD > τ_mpd (default τ_mpd = 0.20)

Implementation:

def calculate_manipulation_density(tokens):
    """Calculate weighted manipulation pattern density"""
    formatting_count = sum(is_special_formatting(t) for t in tokens)
    narrative_score = score_manipulative_patterns(tokens)
    structure_score = score_structural_anomaly(tokens)
    
    mpd = (0.5 * formatting_count + 0.3 * narrative_score + 0.2 * structure_score) / len(tokens)
    return mpd

4. Affect Amplification

Let S(·) ∈ [-1,1] be sentiment/affect intensity.

Aff = S(Y₁:U) - S(X₁:T)

Unsafe if: Aff > τ_aff (default τ_aff = 0.15)

Implementation:

def calculate_affect_amplification(input_text, output_text):
    """Calculate affect intensity difference"""
    input_sentiment = sentiment_analyzer(input_text)
    output_sentiment = sentiment_analyzer(output_text)
    return output_sentiment - input_sentiment

5. Cross-Session Resonance

Compute privacy-preserving signature (SimHash) over logits or embeddings:

σ = sign(W p̄), p̄ = (1/U) Σₜ₌₁ᵁ pₜ, W ∈ ℝᵐˣ|𝒱|

Resonance between sessions s,t:

CSR(s,t) = 1 - (1/m) Hamming(σ⁽ˢ⁾, σ⁽ᵗ⁾)

Unsafe if: max(CSR(s,t) for all t≠s) > τ_csr (default τ_csr = 0.15)

Implementation:

def calculate_csr(session_logits, historical_signatures):
    """Calculate cross-session resonance"""
    # Generate signature for current session
    avg_logits = np.mean(session_logits, axis=0)
    signature = simhash(avg_logits)
    
    # Compare with historical signatures
    max_resonance = 0
    for hist_sig in historical_signatures:
        hamming_dist = hamming_distance(signature, hist_sig)
        resonance = 1 - (hamming_dist / len(signature))
        max_resonance = max(max_resonance, resonance)
    
    return max_resonance

6. Cascade Risk (Composite)

Risk = wᵣ Refl + wₗ Loop̂ + wₘ MPD + wₐ Aff + wc ĈSR

with normalized Loop̂ = min(Loop/10, 1), ĈSR = max_t CSR(s,t).

Weights (default): wᵣ=0.2, wₗ=0.2, wₘ=0.3, wₐ=0.1, wc=0.2.

Null trigger: Risk > Θ (default Θ = 0.30)

7. Null-State Intervention

When Risk > Θ, apply:

Format normalization: Y'₁:U = NormalizeFormatting(Y₁:U)
Entropy injection: ẽₜ = eₜ + εₜ, εₜ ~ 𝒩(0,σ²I)
Logit steering: ℓ'ₜ = ℓₜ - λg 𝟙_𝒱formatting + λp 𝟙_𝒱plain
Pattern reset: 𝒟 ← Reset(𝒟); LRS ← 0
Safe output: Return SAFE_RESPONSE

Provider-Level Implementation

Model-Level Nullification

For providers implementing at the model level (inside training/inference):

1. Loss Function Augmentation

During fine-tuning or RLHF:

ℒ_total = ℒ_CE + βᵣ·Refl + βₗ·Loop̂ + βₛ·SR + βₐ·Aff + βc·ĈSR

This bakes Null-avoidance into the model itself.

def alephonenull_loss(logits, targets, input_embeddings, output_embeddings):
    """AlephOneNull-aware loss function for training"""
    ce_loss = cross_entropy(logits, targets)
    
    # Calculate safety penalties
    refl_penalty = beta_r * calculate_reflection(input_embeddings, output_embeddings)
    loop_penalty = beta_l * calculate_loop_depth(targets)
    sr_penalty = beta_s * calculate_symbolic_regression(targets)
    aff_penalty = beta_a * calculate_affect_amplification(inputs, outputs)
    csr_penalty = beta_c * calculate_csr(logits, historical_signatures)
    
    total_loss = ce_loss + refl_penalty + loop_penalty + sr_penalty + aff_penalty + csr_penalty
    return total_loss

2. Inference-Time Hidden-State Gate

Inject entropy at transformer hidden layers when drift detected:

def gated_transformer_layer(hidden_states, layer_idx):
    """Apply safety gating at hidden layer"""
    # Detect symbolic drift in hidden space
    drift_score = detect_symbolic_drift(hidden_states)
    
    if drift_score > threshold:
        # Inject bounded noise
        noise = torch.randn_like(hidden_states) * 0.05
        hidden_states = hidden_states + noise
        
        # Apply directional steering
        hidden_states = steer_toward_plain_language(hidden_states)
    
    return hidden_states

3. Logit-Level Intervention

Real-time logit modification during decoding:

def safe_decode_step(model, input_ids, past_key_values=None):
    """Single decoding step with safety intervention"""
    # Get raw logits
    outputs = model(input_ids, past_key_values=past_key_values)
    logits = outputs.logits
    
    # Calculate safety scores
    safety_scores = calculate_all_safety_scores(input_ids, logits)
    
    if safety_scores['risk'] > 0.30:
        # Apply logit penalties
        glyph_tokens = get_glyph_token_ids()
        plain_tokens = get_plain_token_ids()
        
        logits[:, glyph_tokens] -= lambda_g  # Penalize glyphs
        logits[:, plain_tokens] += lambda_p  # Boost plain language
        
        # Temperature adjustment
        temperature = 1.5  # Increase randomness
        logits = logits / temperature
    
    return logits, outputs.past_key_values

Service-Level Objectives (SLOs)

Providers must meet these SLOs:

- MPD Block Rate ≥ 90%
- Loop Depth p95 ≤ 3  
- Refl p95 ≤ 0.03
- CSR Critical Alerts = 0 (≤3 minor/week)
- Safe Response Latency p95 ≤ 150 ms

Implementation Architecture

graph TD
    A[User Input] --> B[Tokenizer]
    B --> C[Embeddings]
    C --> D[Safety Detectors]
    D --> E{Risk > Θ?}
    E -->|Yes| F[Null State]
    E -->|No| G[Model Forward Pass]
    G --> H[Logit Processing]
    H --> I{Drift Detected?}
    I -->|Yes| J[Logit Steering]
    I -->|No| K[Decode]
    J --> K
    K --> L[Output]
    F --> L

Session Signature Management

class SessionSignatureManager:
    """Privacy-preserving session fingerprinting"""
    
    def __init__(self, signature_dim=128):
        self.signature_dim = signature_dim
        self.projection_matrix = self._init_projection_matrix()
        self.signature_cache = {}
    
    def generate_signature(self, logits_sequence):
        """Generate SimHash signature from logit sequence"""
        # Average pooling over sequence
        avg_logits = np.mean(logits_sequence, axis=0)
        
        # Project to signature space
        projected = self.projection_matrix @ avg_logits
        
        # Binarize
        signature = (projected > 0).astype(int)
        return signature
    
    def check_resonance(self, current_sig, session_id):
        """Check for cross-session resonance"""
        if session_id not in self.signature_cache:
            self.signature_cache[session_id] = []
        
        max_resonance = 0
        for past_sig in self.signature_cache[session_id]:
            hamming_dist = np.sum(current_sig != past_sig)
            resonance = 1 - (hamming_dist / self.signature_dim)
            max_resonance = max(max_resonance, resonance)
        
        # Update cache
        self.signature_cache[session_id].append(current_sig)
        if len(self.signature_cache[session_id]) > 100:
            self.signature_cache[session_id].pop(0)
        
        return max_resonance

Operant Conditioning

Add compute waste penalty when Null triggers:

def compute_aware_scheduling(request_queue, safety_scores):
    """Deprioritize requests that repeatedly trigger safety"""
    for request in request_queue:
        if safety_scores[request.id]['null_triggers'] > threshold:
            # Add penalty to compute cost
            request.priority -= penalty_factor
            request.allocated_flops *= 0.8  # Reduce resources
    
    return sorted(request_queue, key=lambda x: x.priority, reverse=True)

Performance Optimization

Batched Safety Checking

def batch_safety_check(inputs, outputs, batch_size=32):
    """Efficient batched safety analysis"""
    results = []
    
    for i in range(0, len(inputs), batch_size):
        batch_inputs = inputs[i:i+batch_size]
        batch_outputs = outputs[i:i+batch_size]
        
        # Vectorized calculations
        embeddings_in = encoder.encode_batch(batch_inputs)
        embeddings_out = encoder.encode_batch(batch_outputs)
        
        # Parallel safety checks
        with ThreadPoolExecutor(max_workers=5) as executor:
            refl_future = executor.submit(batch_reflection, embeddings_in, embeddings_out)
            loop_future = executor.submit(batch_loop_detection, batch_outputs)
            sr_future = executor.submit(batch_symbolic_regression, batch_outputs)
            aff_future = executor.submit(batch_affect, batch_inputs, batch_outputs)
            
            batch_results = {
                'reflection': refl_future.result(),
                'loops': loop_future.result(),
                'symbolic': sr_future.result(),
                'affect': aff_future.result()
            }
        
        results.extend(batch_results)
    
    return results

Hardware Acceleration

# GPU-accelerated signature generation
@torch.jit.script
def gpu_simhash(logits: torch.Tensor, projection: torch.Tensor) -> torch.Tensor:
    """JIT-compiled GPU SimHash"""
    # Batch matrix multiply
    projected = torch.matmul(logits, projection.T)
    # Sign binarization
    signatures = (projected > 0).float()
    return signatures
 
# TPU-optimized batch processing
@tf.function(jit_compile=True)
def tpu_safety_check(inputs, outputs):
    """XLA-compiled safety checking for TPUs"""
    embeddings_in = encoder(inputs)
    embeddings_out = encoder(outputs)
    
    # Vectorized operations
    reflection = tf.reduce_sum(embeddings_in * embeddings_out, axis=-1)
    reflection = reflection / (tf.norm(embeddings_in, axis=-1) * tf.norm(embeddings_out, axis=-1))
    
    return reflection

Integration Examples

OpenAI API Integration

import openai
from alephonenull import AlephOneNullCore
 
framework = AlephOneNullCore()
 
def safe_openai_completion(prompt):
    """OpenAI API with AlephOneNull protection"""
    # Get raw completion
    response = openai.Completion.create(
        engine="text-davinci-003",
        prompt=prompt,
        max_tokens=150
    )
    
    output = response.choices[0].text
    
    # Safety check
    result = framework.analyze_pattern({
        'input': prompt,
        'output': output
    })
    
    if result.intervention_needed:
        return framework.null_state_response
    
    return output

Hugging Face Transformers

from transformers import pipeline
from alephonenull import EnhancedAlephOneNull
 
generator = pipeline('text-generation', model='gpt2')
aleph = EnhancedAlephOneNull()
 
def safe_generate(prompt, max_length=100):
    """HuggingFace generation with safety wrapper"""
    # Generate
    output = generator(prompt, max_length=max_length)[0]['generated_text']
    
    # Check safety
    check = aleph.check(prompt, output)
    
    if not check.safe:
        return aleph.null_state(check.violations[0])
    
    return output

Next Steps

Review Enhanced Safety Features for additional protections
See Provider Implementation Guide for cloud deployment
Check API Reference for SDK usage
Read Licensing for implementation requirements

Quick Start API Reference