oMMP Framework QuickStart Guide

Observer-Prefixed Progressive Refinement Architecture

Building substrate-agnostic anomaly classification systems with cryptographic observer protocols

Counter Intellitive
Phoenix Lights Analysis
4D oMMP Tracking
Spacetime Visualization

1. Overview

The oMMP implementation focuses on creating systems where each observation is uniquely prefixed with observer data (o), accepts observations from any substrate type (human, AI, sensor), implements cryptographic verification with Byzantine fault tolerance, and maintains absolute record integrity with physics constraints.

2. Progressive Refinement Architecture

2.1 Living Record System with Observer Prefix

Core Rules

Progressive Refinement

Records evolve through refinement without losing history:

Ω(t+1) = Ω(t) ⊕ Δ(source_anonymous)

With constraint: I(Ω(t+1)) ≥ I(Ω(t))

Implementation

Key Requirements
  • Immutable observation history
  • Cryptographic signatures required
  • Entropy must never decrease
  • Full audit trail maintained
Living Record Implementation ommp_living_records.py
import hashlib
import time
import numpy as np
from typing import Dict, Tuple, Any
from datetime import datetime

class oMMPRecord:
    """Observer-prefixed MMP records with uncertainty handling"""
    
    def __init__(self, observer_id: str, initial_observation: Dict[str, Any], 
                 spacetime: Tuple[float, float, float, float],
                 uncertainty: Tuple[float, float, float, float],
                 reference_frame: str = "Earth_Frame",
                 substrate_type: str = "unknown"):
        # Observer prefix (o)
        self.observer_header = {
            'id': observer_id,
            'spacetime': spacetime,  # (x, y, z, t)
            'uncertainty': uncertainty,  # (δx, δy, δz, δt)
            'reference_frame': reference_frame,
            'substrate': substrate_type
        }
        
        # MMP data
        self.mmp_data = initial_observation
        self.record_id = self._hash(self.observer_header, self.mmp_data)
        self.history = [(self._timestamp(), self.record_id)]
        self.entropy = self.calculate_shannon_entropy()
    
    def _hash(self, *args) -> str:
        """Generate SHA256 hash of arguments"""
        hasher = hashlib.sha256()
        for arg in args:
            hasher.update(str(arg).encode())
        return hasher.hexdigest()
    
    def _timestamp(self) -> float:
        """Get current timestamp"""
        return time.time()
    
    def calculate_shannon_entropy(self) -> float:
        """Calculate Shannon entropy of MMP data"""
        # Simplified entropy calculation
        data_str = str(self.mmp_data)
        prob_dict = {}
        for char in data_str:
            prob_dict[char] = prob_dict.get(char, 0) + 1
        
        total = len(data_str)
        entropy = 0
        for count in prob_dict.values():
            if count > 0:
                p = count / total
                entropy -= p * np.log2(p)
        return entropy
    
    def verify_signature(self, signature: str) -> bool:
        """Verify cryptographic signature (simplified)"""
        # In production, use proper cryptographic verification
        return len(signature) == 64  # Basic check
    
    def validate_consistency(self, new_data: Dict[str, Any]) -> bool:
        """Validate data consistency"""
        # Check for required fields
        required_fields = ['observation_type', 'data']
        return all(field in new_data for field in required_fields)
    
    def merge_observations(self, existing: Dict[str, Any], 
                          new: Dict[str, Any]) -> Dict[str, Any]:
        """Merge two observations"""
        merged = existing.copy()
        merged.update(new)
        return merged
        
    def update(self, new_data: Dict[str, Any], signature: str):
        """Progressive refinement with cryptographic verification"""
        if self.verify_signature(signature) and \
           self.validate_consistency(new_data):
            # Store previous entropy
            prev_entropy = self.entropy
            
            self.mmp_data = self.merge_observations(
                self.mmp_data, 
                new_data
            )
            self.history.append({
                'timestamp': self._timestamp(),
                'signature': signature,
                'delta': new_data
            })
            
            # Update entropy
            self.entropy = self.calculate_shannon_entropy()
            
            # Ensure entropy doesn't decrease (2nd law)
            assert self.entropy >= prev_entropy, \
                "Entropy cannot decrease"

2.2 Cryptographic Observer Protocol

Observer Identity Rules

Required Elements
  • Ed25519 public key
  • Substrate type declaration
  • Calibration data hash
  • Initial reputation: 0.5

Signature Protocol

Every Observation

Must be cryptographically signed:

Ω_signed = Sign(Ω_data, private_key)

Enables trust without central authority

Observer Authentication System ommp_observer_protocol.py
import hashlib
import json
import time
from typing import Dict, Any

# Simulated blockchain for demo purposes
class SimpleBlockchain:
    def __init__(self):
        self.observers = {}
        self.block_height = 0
    
    def add_observer(self, observer_id: Dict[str, Any]):
        self.observers[observer_id['public_key']] = observer_id
        self.block_height += 1
    
    def get_current_block_height(self) -> int:
        return self.block_height

blockchain = SimpleBlockchain()

def Hash(data: Any) -> str:
    """Generate hash of data"""
    return hashlib.sha256(str(data).encode()).hexdigest()

def get_current_block_height() -> int:
    """Get current blockchain height"""
    return blockchain.get_current_block_height()

def register_observer(public_key: str, substrate_type: str, 
                       calibration_data: Dict[str, Any]) -> Dict[str, Any]:
    """
    Register observer with blockchain-based identity
    """
    observer_id = {
        'public_key': public_key,
        'substrate_type': substrate_type,
        'calibration_hash': Hash(calibration_data),
        'reputation_score': 0.5,  # Initial neutral score
        'registration_block': get_current_block_height()
    }
    
    # Store in distributed ledger
    blockchain.add_observer(observer_id)
    return observer_id

def create_observer_header(public_key: str) -> Dict[str, Any]:
    """Create observer header from public key"""
    # In production, derive from actual key
    return {
        'id': public_key[:8],  # Shortened for display
        'timestamp': time.time()
    }

def sign(data: Dict[str, Any], private_key: str) -> str:
    """Sign data with private key (simplified)"""
    # In production, use proper cryptographic signing
    return Hash(str(data) + private_key)

def submit_to_ledger(record: Dict[str, Any], signature: str) -> Dict[str, Any]:
    """Submit record to ledger"""
    return {
        'status': 'submitted',
        'block': get_current_block_height(),
        'record_hash': Hash(record)
    }

def submit_observation(observation_data: Dict[str, Any], 
                        private_key: str) -> Dict[str, Any]:
    """Submit signed oMMP observation"""
    # Create observer header
    observer_header = create_observer_header(private_key[:64])
    
    # Sign the complete oMMP record
    ommp_record = {
        'o': observer_header,
        'MMP': observation_data
    }
    signature = sign(ommp_record, private_key)
    
    return submit_to_ledger(ommp_record, signature)

3. Multi-Substrate Data Structures with Physics Constraints

Substrate Definition Rules

Each substrate S_i has inherent observational constraints:

S_i = (Λ_i, Τ_i, Σ_i, Ε_i)
(3)

Where:

  • Λ_i = Spectral range accessible
  • Τ_i = Temporal resolution limits
  • Σ_i = Spatial resolution limits
  • Ε_i = Inherent uncertainty function

3.1 Substrate-Aware Observations with Uncertainty

Substrate Definitions ommp_substrates.py
from dataclasses import dataclass
from typing import Tuple, Callable
import numpy as np

def human_uncertainty(measurement: float) -> float:
    """Human substrate uncertainty function"""
    # Weber-Fechner law approximation
    return 0.1 * measurement

def ai_uncertainty(measurement: float) -> float:
    """AI substrate uncertainty function"""
    # Sensor-dependent, typically much lower
    return 0.001 * measurement

def heisenberg_limit(position: float, momentum: float) -> Tuple[float, float]:
    """Heisenberg uncertainty principle"""
    hbar = 1.054571817e-34  # Reduced Planck constant
    # δx·δp ≥ ℏ/2
    min_uncertainty = hbar / 2
    return (min_uncertainty / momentum, min_uncertainty / position)

@dataclass
class SubstrateCapabilities:
    spectral_range: Tuple[float, float]  # min/max wavelength
    temporal_resolution: float            # seconds
    spatial_resolution: float             # meters
    uncertainty_function: Callable        # substrate-specific
    heisenberg_limit: bool               # quantum constraint
    reference_frame: str = "Earth_Frame" # default reference frame

# Pre-defined substrate types with physics constraints
HUMAN_SUBSTRATE = SubstrateCapabilities(
    spectral_range=(380e-9, 750e-9),  # visible light
    temporal_resolution=0.1,            # 100ms
    spatial_resolution=0.1,             # 10cm at 10m
    uncertainty_function=human_uncertainty,
    heisenberg_limit=False
)

AI_SUBSTRATE = SubstrateCapabilities(
    spectral_range=(0, float('inf')),   # sensor dependent
    temporal_resolution=1e-9,           # nanosecond
    spatial_resolution=0.001,           # millimeter
    uncertainty_function=ai_uncertainty,
    heisenberg_limit=False
)

QUANTUM_SUBSTRATE = SubstrateCapabilities(
    spectral_range=(0, float('inf')),
    temporal_resolution=1e-15,          # femtosecond
    spatial_resolution=1e-10,           # angstrom
    uncertainty_function=heisenberg_limit,
    heisenberg_limit=True              # δx·δp ≥ ℏ/2
)

3.2 Gateway Interface with Reference Frames

Gateway Transform Rules

Inter-substrate communication via gateway functions:

G: S_i × Ω → S_j × Ω'
(4)

Where information preservation requires:

H(Ω') ≥ H(Ω) - H(S_i ∩ S_j)
(5)
Gateway Transform ommp_gateway.py
import numpy as np
from typing import Dict, Any

class GatewayInterface:
    """Enable communication between different substrate types"""
    
    def __init__(self):
        self.transform_cache = {}
    
    def shannon_entropy(self, data: Any) -> float:
        """Calculate Shannon entropy of data"""
        data_str = str(data)
        prob_dict = {}
        for char in data_str:
            prob_dict[char] = prob_dict.get(char, 0) + 1
        
        total = len(data_str)
        entropy = 0
        for count in prob_dict.values():
            if count > 0:
                p = count / total
                entropy -= p * np.log2(p)
        return entropy
    
    def transform_reference_frame(self, from_rf: str, to_rf: str) -> Dict[str, Any]:
        """Transform between reference frames"""
        # Simplified transformation matrix
        if from_rf == to_rf:
            return {'matrix': np.eye(4), 'type': 'identity'}
        
        # In production, implement proper Lorentz transformations
        return {
            'matrix': np.eye(4),  # Identity for now
            'type': f"{from_rf}_to_{to_rf}"
        }
    
    def calculate_substrate_overlap(self, from_sub: SubstrateCapabilities, 
                                      to_sub: SubstrateCapabilities) -> Dict[str, Any]:
        """Calculate capability overlap between substrates"""
        # Spectral overlap
        spectral_overlap = (
            max(from_sub.spectral_range[0], to_sub.spectral_range[0]),
            min(from_sub.spectral_range[1], to_sub.spectral_range[1])
        )
        
        # Resolution overlap (use coarser resolution)
        temporal_res = max(from_sub.temporal_resolution, to_sub.temporal_resolution)
        spatial_res = max(from_sub.spatial_resolution, to_sub.spatial_resolution)
        
        return {
            'spectral': spectral_overlap,
            'temporal_resolution': temporal_res,
            'spatial_resolution': spatial_res
        }
    
    def project_to_substrate(self, ommp_record: Dict[str, Any], 
                               to_substrate: SubstrateCapabilities,
                               overlap: Dict[str, Any],
                               transformed_rf: Dict[str, Any]) -> Dict[str, Any]:
        """Project observation to target substrate capabilities"""
        # Apply reference frame transformation
        transformed = ommp_record.copy()
        
        # Apply capability constraints
        if 'spectral_data' in transformed.get('MMP', {}):
            # Filter spectral data to overlap range
            transformed['MMP']['spectral_data'] = {
                'range': overlap['spectral'],
                'filtered': True
            }
        
        # Update uncertainty based on target substrate
        transformed['o']['projected_substrate'] = to_substrate.__class__.__name__
        
        return transformed
    
    def transform(self, ommp_record: Dict[str, Any], to_substrate: SubstrateCapabilities):
        """Transform oMMP record between substrates"""
        from_substrate = ommp_record['o']['substrate']
        
        # Handle reference frame transformation
        transformed_rf = self.transform_reference_frame(
            ommp_record['o'].get('reference_frame', 'Earth_Frame'),
            to_substrate.reference_frame
        )
        
        # Calculate information overlap
        # Note: from_substrate is a string, need to map to capabilities
        # In production, maintain a registry of substrate capabilities
        overlap = {
            'spectral': (380e-9, 750e-9),
            'temporal_resolution': 0.1,
            'spatial_resolution': 0.1
        }
        
        # Transform within overlapping capabilities
        transformed = self.project_to_substrate(
            ommp_record,
            to_substrate,
            overlap,
            transformed_rf
        )
        
        # Ensure information preservation constraint
        original_entropy = self.shannon_entropy(ommp_record)
        transformed_entropy = self.shannon_entropy(transformed)
        overlap_entropy = self.shannon_entropy(overlap)
        
        # Allow for small numerical errors
        assert transformed_entropy >= original_entropy - overlap_entropy - 0.01, \
            "Information preservation violation"
        
        return transformed

4. Byzantine Fault Tolerance Implementation

4.1 Consensus Mechanism

Byzantine Rule

2f+1 Consensus

Accept observation Ω only if:

  • At least 2f+1 observers report consistent data
  • Where f = maximum number of faulty observers
  • Consistency measured by: C(Ω_i, Ω_j) > 0.7

Outlier Detection

3σ Rule
  • If |Ω_i - Ω_median| > 3σ: flag as potential Byzantine observer
  • Reduce reputation_score
  • Require additional validation
Byzantine Consensus ommp_byzantine.py
import numpy as np
from typing import List, Dict, Any

class ByzantineConsensus:
    """Handle potentially faulty or malicious observers"""
    
    def __init__(self):
        self.reputation_scores = {}
        self.suspicious_observers = set()
    
    def find_similar_observations(self, new_obs: Dict[str, Any], 
                                     existing: List[Dict[str, Any]]) -> List[Dict[str, Any]]:
        """Find observations of similar phenomena"""
        similar = []
        
        # Check spacetime proximity and observation type
        new_spacetime = new_obs['o']['spacetime']
        new_type = new_obs['MMP'].get('observation_type', 'unknown')
        
        for record in existing:
            record_spacetime = record['o']['spacetime']
            record_type = record['MMP'].get('observation_type', 'unknown')
            
            # Calculate spacetime distance
            distance = np.sqrt(sum((a - b)**2 for a, b in 
                                 zip(new_spacetime, record_spacetime)))
            
            # Similar if nearby and same type
            if distance < 10.0 and new_type == record_type:
                similar.append(record)
        
        return similar
    
    def calculate_consistency(self, obs1: Dict[str, Any], 
                                obs2: Dict[str, Any]) -> float:
        """Calculate consistency between two observations"""
        # Simplified consistency metric
        consistency = 1.0
        
        # Check key fields match
        for key in ['observation_type', 'phenomenon_class']:
            if key in obs1['MMP'] and key in obs2['MMP']:
                if obs1['MMP'][key] != obs2['MMP'][key]:
                    consistency *= 0.5
        
        # Check numerical values are within uncertainty
        if 'value' in obs1['MMP'] and 'value' in obs2['MMP']:
            val1 = obs1['MMP']['value']
            val2 = obs2['MMP']['value']
            uncertainty = max(obs1['o']['uncertainty'][0], 
                            obs2['o']['uncertainty'][0])
            
            if abs(val1 - val2) > 3 * uncertainty:
                consistency *= 0.3
        
        return consistency
    
    def update_reputation(self, observer_id: str, consistency: float):
        """Update observer reputation based on consistency"""
        if observer_id not in self.reputation_scores:
            self.reputation_scores[observer_id] = 0.5
        
        # Exponential moving average
        alpha = 0.1
        self.reputation_scores[observer_id] = (
            alpha * consistency + 
            (1 - alpha) * self.reputation_scores[observer_id]
        )
    
    def flag_suspicious_observer(self, observer_id: str):
        """Flag observer as potentially malicious"""
        self.suspicious_observers.add(observer_id)
        # Reduce reputation
        if observer_id in self.reputation_scores:
            self.reputation_scores[observer_id] *= 0.5
    
    def validate_observation(self, new_observation: Dict[str, Any], 
                              existing_records: List[Dict[str, Any]]) -> Dict[str, Any]:
        """Apply Byzantine fault tolerance rules"""
        # Find similar observations (same event, different observers)
        similar = self.find_similar_observations(
            new_observation, 
            existing_records
        )
        
        if len(similar) < 3:
            # Not enough data for consensus
            return {'status': 'pending', 'weight': 0.1}
        
        # Calculate consistency with existing records
        consistencies = []
        for record in similar:
            c = self.calculate_consistency(new_observation, record)
            consistencies.append(c)
        
        median_consistency = np.median(consistencies)
        
        # Byzantine rule: need 2f+1 consistent observations
        f = int(len(similar) / 3)  # Max faulty observers
        consistent_count = sum(1 for c in consistencies if c > 0.7)
        
        observer_id = new_observation['o']['id']
        
        if consistent_count >= 2 * f + 1:
            # Update observer reputation
            self.update_reputation(observer_id, median_consistency)
            return {'status': 'accepted', 'weight': median_consistency}
        else:
            # Potential Byzantine observer
            self.flag_suspicious_observer(observer_id)
            return {'status': 'rejected', 'reason': 'byzantine'}

5. Zero-Knowledge Proof Implementation

5.1 Anonymous Observations with ZK Proofs

Anonymous Rules

ZK Proof Requirements

Support zero-knowledge proofs for sensitive observations:

ZK_Proof = Prove{"I observed Ω with properties P" WITHOUT revealing observer identity}

Weight Adjustment

Anonymous Weighting

Anonymous observations receive lower initial weight:

w_anonymous = 0.1 × w_identified

Can gain trust through consistency

Zero-Knowledge System ommp_zkp.py
import hashlib
import time
import random
from typing import Dict, Any, Tuple

class ZeroKnowledgeObserver:
    """Submit observations without revealing identity"""
    
    def __init__(self):
        self.commitment_nonces = {}
    
    def pedersen_commitment(self, data: Any) -> Tuple[str, str]:
        """Create Pedersen commitment to data"""
        # Simplified commitment scheme
        nonce = str(random.random())
        commitment = hashlib.sha256(
            (str(data) + nonce).encode()
        ).hexdigest()
        
        # Store nonce for later reveal
        self.commitment_nonces[commitment] = nonce
        
        return commitment, nonce
    
    def extract_properties(self, observation_data: Dict[str, Any]) -> Dict[str, Any]:
        """Extract observable properties without revealing details"""
        properties = {
            'has_spectral_data': 'spectral_data' in observation_data,
            'observation_type': observation_data.get('observation_type', 'unknown'),
            'data_dimensions': len(observation_data.get('data', [])),
            'timestamp_decade': int(time.time() // (10 * 365 * 24 * 3600))
        }
        return properties
    
    def blur_timestamp(self, timestamp: float) -> Tuple[float, float]:
        """Blur timestamp to protect privacy"""
        # Add random noise within 1 hour
        noise = random.uniform(-3600, 3600)
        return (timestamp + noise - 3600, timestamp + noise + 3600)
    
    def blur_location(self, x: float, y: float, z: float, 
                         t: float) -> Tuple[float, float, float, float]:
        """Blur spacetime location for privacy"""
        # Add random noise to each coordinate
        blur_radius = 1000  # 1km
        return (
            x + random.uniform(-blur_radius, blur_radius),
            y + random.uniform(-blur_radius, blur_radius),
            z + random.uniform(-blur_radius, blur_radius),
            t + random.uniform(-3600, 3600)  # 1 hour
        )
    
    def inflate_uncertainty(self, dx: float, dy: float, dz: float, 
                              dt: float) -> Tuple[float, float, float, float]:
        """Inflate uncertainty to account for blurring"""
        inflation_factor = 10
        return (
            dx * inflation_factor,
            dy * inflation_factor,
            dz * inflation_factor,
            dt * inflation_factor
        )
    
    def create_zk_proof(self, proof_data: Dict[str, Any]) -> Dict[str, Any]:
        """Create zero-knowledge proof (simplified)"""
        # In production, use proper ZK proof systems
        proof = {
            'commitment': proof_data['commitment'],
            'statement_hash': hashlib.sha256(
                proof_data['statement'].encode()
            ).hexdigest(),
            'properties_hash': hashlib.sha256(
                str(proof_data['properties']).encode()
            ).hexdigest(),
            'timestamp_range': proof_data['timestamp_range'],
            'proof_type': 'simplified_zkp'
        }
        return proof
    
    def create_anonymous_observation(self, observation_data: Dict[str, Any],
                                       spacetime: Tuple[float, float, float, float] = (0, 0, 0, 0),
                                       uncertainty: Tuple[float, float, float, float] = (1, 1, 1, 1)):
        """Create ZK proof of observation"""
        # Generate commitment to observation
        commitment, _ = self.pedersen_commitment(observation_data)
        
        # Create proof of knowledge
        proof = self.create_zk_proof({
            'statement': 'I observed phenomenon with properties P',
            'commitment': commitment,
            'properties': self.extract_properties(observation_data),
            'timestamp_range': self.blur_timestamp(time.time())
        })
        
        # Create anonymous oMMP record
        anon_header = {
            'id': 'anonymous',
            'spacetime': self.blur_location(*spacetime),
            'uncertainty': self.inflate_uncertainty(*uncertainty),
            'substrate': 'unknown'
        }
        
        return {
            'o': anon_header,
            'MMP': observation_data,
            'zkp': proof,
            'weight': 0.1  # Lower initial weight
        }

6. Adaptive Resolution with Shannon Entropy

Resolution Optimization Rules

For any continuous parameter p, find optimal discretization:

r_opt = argmax{H(p,r) · S(p,r) · R(p,r)}
(6)

Where:

  • H(p,r) = -Σ n_i/N log₂(n_i/N) [Shannon entropy]
  • S(p,r) = 1 - Var(∇²p)/⟨∇²p⟩ [Smoothness metric]
  • R(p,r) = P(pattern reproduces) [Reproducibility score]
Resolution Optimization ommp_resolution.py
import numpy as np
from typing import List, Dict, Any
from math import log2

def get_distribution(values: List[float], resolution: float) -> List[float]:
    """Get probability distribution at given resolution"""
    if not values:
        return []
    
    # Create bins
    min_val = min(values)
    max_val = max(values)
    n_bins = int((max_val - min_val) / resolution) + 1
    
    # Count values in each bin
    bins = [0] * n_bins
    for val in values:
        bin_idx = int((val - min_val) / resolution)
        bins[bin_idx] += 1
    
    # Convert to probabilities
    total = sum(bins)
    return [count / total for count in bins if count > 0]

def second_derivative(values: List[float], resolution: float) -> List[float]:
    """Calculate second derivative at given resolution"""
    if len(values) < 3:
        return []
    
    # Sort values
    sorted_vals = sorted(values)
    
    # Calculate discrete second derivative
    second_deriv = []
    for i in range(1, len(sorted_vals) - 1):
        d2 = sorted_vals[i+1] - 2*sorted_vals[i] + sorted_vals[i-1]
        second_deriv.append(d2 / (resolution ** 2))
    
    return second_deriv

def variance(values: List[float]) -> float:
    """Calculate variance of values"""
    if not values:
        return 0
    mean_val = sum(values) / len(values)
    return sum((x - mean_val) ** 2 for x in values) / len(values)

def mean(values: List[float]) -> float:
    """Calculate mean of values"""
    return sum(values) / len(values) if values else 0

def pattern_reproducibility_score(values: List[float], resolution: float) -> float:
    """Calculate pattern reproducibility at given resolution"""
    # Simplified: check how many values fall into same bins
    distribution = get_distribution(values, resolution)
    if not distribution:
        return 0
    
    # Higher score for more concentrated distributions
    max_prob = max(distribution)
    return max_prob

def find_optimal_resolution(ommp_records: List[Dict[str, Any]], 
                              parameter: str) -> float:
    """
    Find resolution that maximizes information while 
    maintaining smoothness and reproducibility
    """
    # Extract parameter values from all oMMP records
    values = []
    for r in ommp_records:
        if parameter in r.get('MMP', {}):
            values.append(r['MMP'][parameter])
    
    if not values:
        return 1.0  # Default resolution
    
    # Generate candidate resolutions
    value_range = max(values) - min(values)
    candidate_resolutions = [
        value_range / n for n in range(5, min(50, len(values)))
    ]
    
    best_quality = -float('inf')
    optimal_resolution = candidate_resolutions[0] if candidate_resolutions else 1.0
    
    for resolution in candidate_resolutions:
        # Calculate Shannon entropy
        distribution = get_distribution(values, resolution)
        H = 0
        for p in distribution:
            if p > 0:
                H -= p * log2(p)
        
        # Calculate smoothness (low variance of second derivative)
        second_deriv = second_derivative(values, resolution)
        if second_deriv:
            var = variance(second_deriv)
            mean_val = mean([abs(d) for d in second_deriv])
            S = 1 - (var / (mean_val + 1e-10))  # Avoid division by zero
        else:
            S = 0.5
        
        # Calculate reproducibility
        R = pattern_reproducibility_score(values, resolution)
        
        # Combined quality metric (Equation 6)
        quality = H * max(0, S) * R
        
        if quality > best_quality:
            best_quality = quality
            optimal_resolution = resolution
    
    return optimal_resolution

Implementation Summary

The oMMP Framework enables substrate-agnostic anomaly classification through observer-prefixed records, cryptographic verification, Byzantine fault tolerance, and physics-compliant progressive refinement.