GMRA-MATLAB Skill

Geometric Multi-Resolution Analysis with Dynamic Sufficiency

                    ┌─────────────────────────────────────┐
                    │   GMRA SHARDED RECONSTRUCTABILITY   │
                    └─────────────────────────────────────┘
                                     │
              ┌──────────────────────┼──────────────────────┐
              ▼                      ▼                      ▼
        ┌──────────┐          ┌──────────┐          ┌──────────┐
        │  MATLAB  │          │  Julia   │          │  Python  │
        │  GMRA    │◀────────▶│  Gay.jl  │◀────────▶│  PyGMRA  │
        │  MINUS   │          │  ERGODIC │          │   PLUS   │
        │   (-1)   │          │    (0)   │          │   (+1)   │
        └────┬─────┘          └────┬─────┘          └────┬─────┘
             │                     │                     │
             └─────────────────────┼─────────────────────┘
                                   ▼
                         ┌──────────────────┐
                         │  LOCAL RECON-    │
                         │  STRUCTABILITY   │
                         │  (No Trusted     │
                         │   Setup)         │
                         └──────────────────┘

Version: 1.0.0
Trit: -1 (MINUS - Validator/Reconstructor)
GF(3) Triad:

gmra-matlab (-1) ⊗ gay-julia (0) ⊗ gmra-python (+1) = 0

---

Core Insight: Deterministic Colors for Local Reconstructability

Gay.jl's SplitMix64 provides the homogeneous property required for sharding:

# Property 1: Deterministic (same seed = same color everywhere)
Gay.gay_seed(0x42D)
color1 = Gay.color_at(137)  # Always #A855F7

# Property 2: Splittable (shard computation, reconstruct locally)  
shard_a = Gay.color_at(137, seed=0x42D)  # Node A computes
shard_b = Gay.color_at(137, seed=0x42D)  # Node B computes independently
@assert shard_a == shard_b  # Local reconstructability ✓

# Property 3: No trusted setup (unlike zkSNARKs)
# SplitMix64 has no "toxic waste" - seed is public knowledge

This replaces trusted setup requirements with deterministic splittable RNG.

---

Dynamic Sufficiency Criteria

Pre-Execution Gate

Before invoking GMRA operations, verify sufficient capabilities:

class GMRASufficiencyGate:
    """
    Dynamic sufficiency for GMRA operations.
    Implements ε-machine from Computational Mechanics (SFI).
    """
    
    CAUSAL_STATES = {
        'gmra:construct': {
            'required_skills': ['linear-algebra', 'clustering', 'pca'],
            'required_tools': ['numpy', 'scipy', 'sklearn'],
            'optional': ['matlab', 'octave', 'julia'],
            'min_coverage': 0.8,
            'trit': -1
        },
        'gmra:query': {
            'required_skills': ['tree-traversal', 'nearest-neighbor'],
            'required_tools': ['numpy'],
            'optional': ['sentence-transformers'],
            'min_coverage': 0.9,
            'trit': 0
        },
        'gmra:synthesize': {
            'required_skills': ['interpolation', 'tangent-space', 'wasserstein'],
            'required_tools': ['numpy', 'scipy'],
            'optional': ['pot', 'geomloss'],
            'min_coverage': 0.85,
            'trit': +1
        },
        'gmra:visualize': {
            'required_skills': ['plotting', 'color-mapping'],
            'required_tools': ['matplotlib', 'gay-mcp'],
            'optional': ['plotly', 'graphviz'],
            'min_coverage': 0.7,
            'trit': 0
        }
    }
    
    def check_sufficiency(self, operation: str, loaded_skills: set) -> dict:
        """
        Verify sufficient skills for GMRA operation.
        
        Returns:
            {
                'sufficient': bool,
                'coverage': float,
                'missing': list,
                'fallback': str or None
            }
        """
        if operation not in self.CAUSAL_STATES:
            return {'sufficient': False, 'coverage': 0.0, 
                    'missing': ['unknown-operation'], 'fallback': None}
        
        state = self.CAUSAL_STATES[operation]
        required = set(state['required_skills'] + state['required_tools'])
        
        covered = required & loaded_skills
        missing = required - loaded_skills
        coverage = len(covered) / len(required) if required else 1.0
        
        # Check optional tools for fallback
        fallback = None
        if coverage < state['min_coverage']:
            for opt in state['optional']:
                if opt in loaded_skills:
                    fallback = opt
                    break
        
        return {
            'sufficient': coverage >= state['min_coverage'],
            'coverage': coverage,
            'missing': list(missing),
            'fallback': fallback,
            'trit': state['trit']
        }

MATLAB Availability Detection

import subprocess
import shutil

def detect_matlab_environment() -> dict:
    """
    Detect MATLAB/Octave availability for GMRA.
    Follows dynamic sufficiency: prefer native, fallback gracefully.
    """
    result = {
        'matlab': False,
        'octave': False,
        'julia': False,
        'python_fallback': True,
        'recommended': 'python'
    }
    
    # Check MATLAB
    matlab_path = shutil.which('matlab')
    if matlab_path:
        try:
            proc = subprocess.run(['matlab', '-batch', 'disp("ok")'], 
                                  capture_output=True, timeout=30)
            result['matlab'] = proc.returncode == 0
        except:
            pass
    
    # Check Octave (FOSS MATLAB alternative)
    octave_path = shutil.which('octave')
    if octave_path:
        try:
            proc = subprocess.run(['octave', '--eval', 'disp("ok")'],
                                  capture_output=True, timeout=10)
            result['octave'] = proc.returncode == 0
        except:
            pass
    
    # Check Julia (for Gay.jl integration)
    julia_path = shutil.which('julia')
    if julia_path:
        try:
            proc = subprocess.run(['julia', '-e', 'println("ok")'],
                                  capture_output=True, timeout=10)
            result['julia'] = proc.returncode == 0
        except:
            pass
    
    # Determine recommendation
    if result['matlab']:
        result['recommended'] = 'matlab'  # Original Maggioni toolbox
    elif result['octave']:
        result['recommended'] = 'octave'  # FOSS alternative
    elif result['julia']:
        result['recommended'] = 'julia'   # Gay.jl + ACSets integration
    else:
        result['recommended'] = 'python'  # Pure Python fallback
    
    return result

---

GMRA Sources (Multi-Language)

1. MATLAB (Original Maggioni Toolbox)

Source:

https://mauromaggioni.duckdns.org/?page_id=105

Status: ⚠️ Primary site unreliable, use mirrors

% Maggioni GMRA Toolbox usage (if available)
% Download from: https://github.com/samuelgerber/gmra (C++ core)

% Load data
X = randn(1000, 50);  % 1000 points in 50D

% Build GMRA tree
opts = struct();
opts.MaxLevels = 4;
opts.IntrinsicDim = 8;
gmra = GMRA(X, opts);

% Query at level
[approx, cell_idx] = gmra.project(query, level);

2. R Package (C++ Backend)

Source:

https://github.com/samuelgerber/gmra

# Install
library(devtools)
devtools::install_github("samuelgerber/gmra")

# Usage
library(gmra)
tree <- gmra_create(X, max_scale = 4)
result <- gmra_query(tree, query_point)

3. Python (Pure Implementation)

Source: Local implementation with Gay.jl bridge

from sklearn.decomposition import PCA
from sklearn.cluster import KMeans
import numpy as np

class SimpleGMRA:
    """
    GMRA with Gay.jl deterministic coloring.
    Enables local reconstructability for sharded queries.
    """
    
    def __init__(self, X, max_levels=4, intrinsic_dim=8, gay_seed=0x42D):
        self.X = X
        self.max_levels = max_levels
        self.d = intrinsic_dim
        self.gay_seed = gay_seed
        self.tree = []
        self._build_tree()
    
    def _gay_color(self, level: int, cell_idx: int) -> str:
        """
        Deterministic color for (level, cell).
        Enables local reconstructability across shards.
        """
        # SplitMix64-style hash
        seed = (self.gay_seed ^ (level * 0x9E3779B9) ^ (cell_idx * 0x85EBCA6B))
        seed = ((seed >> 16) ^ seed) * 0x45D9F3B
        seed = ((seed >> 16) ^ seed) * 0x45D9F3B
        seed = (seed >> 16) ^ seed
        
        # Map to LCH color space
        hue = (seed % 360)
        chroma = 50 + (seed >> 8) % 50
        lightness = 45 + (seed >> 16) % 20
        
        return f"lch({lightness}% {chroma} {hue})"
    
    def _build_tree(self):
        """Build hierarchical tree with deterministic colors."""
        n_clusters_per_level = [3, 6, 18, len(self.X)]
        
        for level, n_clusters in enumerate(n_clusters_per_level[:self.max_levels]):
            if n_clusters >= len(self.X):
                # Finest level: individual points
                cells = [{
                    'mean': self.X[i],
                    'basis': np.eye(self.d)[:, :min(self.d, self.X.shape[1])],
                    'indices': [i],
                    'color': self._gay_color(level, i),
                    'level': level
                } for i in range(len(self.X))]
            else:
                cells = self._build_level(n_clusters, level)
            
            self.tree.append(cells)
    
    def _build_level(self, n_clusters: int, level: int) -> list:
        """Build one GMRA level with local PCA."""
        kmeans = KMeans(n_clusters=n_clusters, random_state=42, n_init=10)
        labels = kmeans.fit_predict(self.X)
        
        cells = []
        for k in range(n_clusters):
            mask = (labels == k)
            indices = np.where(mask)[0]
            X_cell = self.X[mask]
            
            if len(X_cell) == 0:
                continue
            
            mean = X_cell.mean(axis=0)
            
            # Local PCA for tangent space
            if len(X_cell) > self.d:
                pca = PCA(n_components=min(self.d, len(X_cell), X_cell.shape[1]))
                pca.fit(X_cell - mean)
                basis = pca.components_.T
            else:
                basis = np.eye(X_cell.shape[1])[:, :self.d]
            
            cells.append({
                'mean': mean,
                'basis': basis,
                'indices': indices.tolist(),
                'color': self._gay_color(level, k),
                'level': level
            })
        
        return cells
    
    def project(self, x: np.ndarray, level: int) -> tuple:
        """Project to GMRA approximation at level."""
        cells = self.tree[level]
        
        dists = [np.linalg.norm(x - cell['mean']) for cell in cells]
        nearest_idx = np.argmin(dists)
        cell = cells[nearest_idx]
        
        # Project to tangent space
        x_centered = x - cell['mean']
        if cell['basis'].shape[0] == len(x_centered):
            coords = cell['basis'].T @ x_centered
            x_approx = cell['mean'] + cell['basis'] @ coords
        else:
            x_approx = cell['mean']
        
        return x_approx, nearest_idx, cell
    
    def local_reconstruct(self, level: int, cell_idx: int) -> dict:
        """
        Local reconstructability: given (level, cell_idx),
        reconstruct full cell information without global state.
        
        This is the key property enabled by Gay.jl determinism.
        """
        cell = self.tree[level][cell_idx]
        
        # Color is reconstructable from (level, cell_idx) alone
        expected_color = self._gay_color(level, cell_idx)
        assert cell['color'] == expected_color, "Color mismatch - reconstruction failed"
        
        return {
            'level': level,
            'cell_idx': cell_idx,
            'color': cell['color'],
            'n_points': len(cell['indices']),
            'reconstructable': True
        }

4. Julia (Gay.jl + ACSets Integration)

Source:

/Users/bob/ies/GMRA_ACSET_GOKO_MORPHISMS.jl

using Gay, ACSets, Catlab

"""
GMRA with Gay.jl coloring and ACSet morphism structure.
"""
struct ColoredGMRA
    levels::Vector{Vector{Dict}}
    gay_seed::UInt64
end

function ColoredGMRA(X::Matrix, max_levels::Int=4; seed::UInt64=0x42D)
    Gay.gay_seed(seed)
    
    levels = Vector{Vector{Dict}}()
    n_clusters = [3, 6, 18, size(X, 1)]
    
    for (level, k) in enumerate(n_clusters[1:max_levels])
        cells = build_level(X, k, level, seed)
        push!(levels, cells)
    end
    
    ColoredGMRA(levels, seed)
end

function local_reconstruct(gmra::ColoredGMRA, level::Int, cell_idx::Int)
    """
    Reconstruct cell from (level, cell_idx) alone.
    No global state needed - Gay.jl provides determinism.
    """
    Gay.gay_seed(gmra.gay_seed)
    expected_color = Gay.color_at(level * 1000 + cell_idx)
    
    cell = gmra.levels[level][cell_idx]
    @assert cell[:color] == expected_color "Local reconstruction verified"
    
    cell
end

---

GF(3) Conservation in GMRA Hierarchy

def verify_gf3_across_levels(gmra: SimpleGMRA) -> dict:
    """
    Verify GF(3) trit conservation across GMRA levels.
    
    Property: Σ trits ≡ 0 (mod 3) at each level
    """
    results = {}
    
    for level, cells in enumerate(gmra.tree):
        # Assign trits based on cell position modulo 3
        trits = [(i % 3) - 1 for i in range(len(cells))]  # {-1, 0, +1}
        total = sum(trits)
        
        results[level] = {
            'n_cells': len(cells),
            'trit_sum': total,
            'conserved': total % 3 == 0,
            'colors': [cell['color'] for cell in cells[:3]]
        }
    
    return results

---

Dynamic Sufficiency Workflow

┌─────────────────────────────────────────────────────────────────────┐
│                    GMRA SUFFICIENCY WORKFLOW                        │
├─────────────────────────────────────────────────────────────────────┤
│                                                                     │
│  1. DETECT ENVIRONMENT                                              │
│     ├─ MATLAB available? → Use Maggioni toolbox                     │
│     ├─ Octave available? → Use FOSS fallback                        │
│     ├─ Julia available? → Use Gay.jl + ACSets                       │
│     └─ Python only? → Use SimpleGMRA                                │
│                                                                     │
│  2. CHECK SUFFICIENCY                                               │
│     ├─ Required skills loaded?                                      │
│     ├─ Coverage ≥ threshold?                                        │
│     └─ GF(3) triad balanced?                                        │
│                                                                     │
│  3. EXECUTE WITH LOCAL RECONSTRUCTABILITY                           │
│     ├─ Build tree with Gay.jl colors                                │
│     ├─ Shard across nodes if distributed                            │
│     └─ Verify: any shard can reconstruct cell from (level, idx)     │
│                                                                     │
│  4. VALIDATE                                                        │
│     ├─ GF(3) conservation verified?                                 │
│     ├─ Color determinism verified?                                  │
│     └─ No trusted setup required? ✓                                 │
│                                                                     │
└─────────────────────────────────────────────────────────────────────┘

---

Installation

# Python (always available)
pip install numpy scipy scikit-learn sentence-transformers

# Optional: R package (C++ backend)
R -e 'devtools::install_github("samuelgerber/gmra")'

# Optional: Julia (Gay.jl integration)
julia -e 'using Pkg; Pkg.add(["Gay", "ACSets", "Catlab"])'

# Optional: Octave (FOSS MATLAB alternative)
brew install octave  # macOS
apt install octave   # Debian/Ubuntu

---

Key Properties

Property	Mechanism	Benefit
Deterministic	SplitMix64 seed	Same color everywhere
Splittable	Independent streams	Parallel/distributed
No Trusted Setup	Public seed	Unlike zkSNARKs
Local Reconstructability	(level, idx) → cell	Sharding-friendly
GF(3) Conservation	Σ trits ≡ 0 (mod 3)	Algebraic invariant

---

---

End-of-Skill Interface

References

1. Allard, Chen & Maggioni (2012) - GMRA foundation (ACHA)
2. Crutchfield & Young - Computational Mechanics (SFI)
3. samuelgerber/gmra - C++/R implementation
4. Gay.jl - Deterministic coloring with SPI
5. GMRA_ACSET_GOKO_MORPHISMS.jl - Local Julia implementation

---

Skill Name: gmra-matlab
Type: Multi-Resolution Analysis with Dynamic Sufficiency
Trit: -1 (MINUS - Validator/Reconstructor)
Sufficiency Triad:

gmra-matlab (-1) ⊗ gay-julia (0) ⊗ gmra-python (+1) = 0

---

Autopoietic Marginalia

The interaction IS the skill improving itself.

Every use of this skill is an opportunity for worlding:

• MEMORY (-1): Record what was learned
• REMEMBERING (0): Connect patterns to other skills
• WORLDING (+1): Evolve the skill based on use

Add Interaction Exemplars here as the skill is used.