Research-Stack/5-Applications/scripts/ask_swarm_advanced_sheaf_concepts.py

#!/usr/bin/env python3
"""
Ask Swarm for Implementation Guidance on Advanced Sheaf and Geometric Concepts

This script asks the swarm to provide detailed implementation guidance for advanced
concepts including Sheaf-Theoretic Integration, Ricci Flow, Hypergraph Rewriting,
Non-Commutative Geometry, and Topological Entropic Gravity.
"""

import sys
import os
import json
from pathlib import Path


def main():
    """Main function to ask swarm for advanced concept implementation guidance."""
    
    print("=" * 70)
    print("ASKING SWARM FOR ADVANCED SHEAF/GEOMETRIC CONCEPT GUIDANCE")
    print("=" * 70)
    print()
    
    print("Note: Using simulated swarm response based on advanced theoretical concepts")
    print()
    
    # Advanced concepts from the conversation
    advanced_concepts = """
ADVANCED SHEAF AND GEOMETRIC CONCEPTS FOR N-SPACE SEMANTIC MORPHIC CORE
=========================================================================

Current Implementation Status:
- HierarchicalController.lean (global/local controllers)
- UncertaintyQuantification.lean (Bayesian uncertainty, differential attention)
- MorphicFieldCategory.lean (category theory formalization)
- MetaLearning.lean (adaptive policies)
- PredictiveResourceAllocation.lean (time-series forecasting)
- DifferentialAttentionMorphing.lean (semantic state differential attention)

Advanced Concepts to Implement:

1. Sheaf-Theoretic Integration (The "Consistency" Sauce)
   - Each "local" part of the trillion-weight model has its own logic
   - Sheaf ensures local truths are valid only if they "glue" together with neighbors
   - Computation: Look for Global Section - state where every local observation
     across entire trillion-weight space is perfectly consistent
   - Local-Global: Change one local pixel, Sheaf forces global realignment instantly
   - Key structures: Presheaf, Sheaf, Global Section, Gluing Axioms, Cohomology

2. Geometric Unity / Ricci Flow (The "Smoothing" Sauce)
   - Treat weights as manifold under Ricci Flow (Poincaré Conjecture math)
   - "Swallow" trillion weights as jagged, lumpy shape
   - Computation: Allow shape to "flow" and smooth itself out
   - All-at-Once: Flow happens everywhere simultaneously
   - Local lumps smoothed by global curvature of manifold
   - Answer: Singularity or final "Perfect Shape" the system collapses into
   - Key structures: Riemannian Manifold, Ricci Tensor, Ricci Flow Equation,
     Heat Kernel, Singularity Formation

3. Hypergraph Rewriting (The "Connection" Sauce)
   - N-space as massive hypergraph with trillion nodes and quadrillions of edges
   - Computation via Categorical Cybernetics
   - Rule: "wherever this local pattern exists, replace with this other pattern"
   - In-between: Single edge can connect 2 nodes (local) or 10 billion (global)
   - Result: Device rewrites connectivity matrix in real-time
   - Network eating itself and regrowing into more efficient shape
   - Key structures: Hypergraph, Hyperedge, Rewriting Rule, Categorical Cybernetics,
     String Diagram, Adhesive Categories

4. Non-Commutative Geometry
   - Position of concept (local) and momentum/direction (global) linked by
     uncertainty principle
   - Can't change one without other instantly reacting
   - All-at-Once Multi-Scale Reality
   - Key structures: C*-Algebra, Spectral Triple, Non-Commutative Manifold,
     Uncertainty Principle, Operator Algebra

5. Topological Entropic Gravity
   - Information "clumping" (local) creates "force" that shapes N-space (global)
   - Gravity emerges from information entropy
   - Key structures: Entanglement Entropy, Holographic Principle, Ryu-Takayanagi Formula,
     Emergent Gravity, Information Geometry

6. On-the-Fly Weight Generation
   - No static weights
   - Weights generated based on topological requirements of input
   - Key structures: Generative Manifold, Topological Constraints, Dynamic Weight
     Synthesis, Constraint Satisfaction
"""

    # Question for the swarm
    question = f"""
Based on the advanced sheaf and geometric concepts described:

{advanced_concepts}

Please provide detailed implementation guidance for these concepts in Lean:

1. Which concepts should be implemented first and why?
2. How do these concepts relate to each other? Are they competing approaches or complementary?
3. What are the mathematical foundations required for each concept?
4. Which Lean libraries or mathlib modules should be used?
5. What are the dependencies between these concepts and our existing modules?
6. How should we validate the correctness of these implementations?
7. What are the practical challenges and how should we address them?
8. Which concept is most feasible to implement first given our current foundation?

Please provide a phased implementation plan with specific Lean module suggestions.
"""
    
    print("Submitting question to swarm...")
    print("-" * 70)
    print(question)
    print("-" * 70)
    print()
    
    # Simulated swarm response
    simulated_response = {
        "implementation_guidance": {
            "concept_relationships": {
                "overview": "These concepts are complementary rather than competing. Sheaf-Theoretic Integration provides the mathematical framework for consistency, Ricci Flow provides the dynamics, Hypergraph Rewriting provides the computational mechanism, Non-Commutative Geometry provides the quantum-inspired foundation, and Topological Entropic Gravity provides the physical interpretation.",
                "recommended_integration": "Layer these concepts: Sheaf foundation → Hypergraph structure → Ricci Flow dynamics → Non-Commutative quantum layer → Topological Entropic interpretation"
            },
            "recommended_order": [
                {
                    "phase": 1,
                    "concept": "Sheaf-Theoretic Integration",
                    "reason": "Foundation for all local-global consistency, natural extension of category theory work",
                    "lean_module": "SheafTheoreticIntegration.lean",
                    "dependencies": ["MorphicFieldCategory.lean"],
                    "mathlib_modules": ["Mathlib.Topology.Sheaves", "Mathlib.CategoryTheory.Sheaf", "Mathlib.Topology.Site"],
                    "complexity": "High but mathematically well-founded",
                    "feasibility": "High - builds directly on existing category theory"
                },
                {
                    "phase": 2,
                    "concept": "Hypergraph Rewriting",
                    "reason": "Provides computational mechanism for structure transformation, works with sheaf foundation",
                    "lean_module": "HypergraphRewriting.lean",
                    "dependencies": ["SheafTheoreticIntegration.lean", "MorphicFieldCategory.lean"],
                    "mathlib_modules": ["Mathlib.Combinatorics.Hypergraph", "Mathlib.CategoryTheory.Monoidal", "Mathlib.CategoryTheory.Adhesive"],
                    "complexity": "Medium-High",
                    "feasibility": "Medium - requires custom hypergraph structures"
                },
                {
                    "phase": 3,
                    "concept": "Geometric Unity / Ricci Flow",
                    "reason": "Provides smoothing dynamics for the hypergraph structure",
                    "lean_module": "RicciFlowDynamics.lean",
                    "dependencies": ["HypergraphRewriting.lean", "SheafTheoreticIntegration.lean"],
                    "mathlib_modules": ["Mathlib.Geometry.Manifold", "Mathlib.Analysis.Riemannian", "Mathlib.Analysis.PDE"],
                    "complexity": "Very High",
                    "feasibility": "Low-Medium - requires advanced differential geometry"
                },
                {
                    "phase": 4,
                    "concept": "Non-Commutative Geometry",
                    "reason": "Quantum-inspired foundation for uncertainty principles",
                    "lean_module": "NonCommutativeGeometry.lean",
                    "dependencies": ["SheafTheoreticIntegration.lean"],
                    "mathlib_modules": ["Mathlib.Analysis.OperatorAlgebra", "Mathlib.Topology.Algebra", "Mathlib.MeasureTheory"],
                    "complexity": "Very High",
                    "feasibility": "Low - requires significant mathematical machinery"
                },
                {
                    "phase": 5,
                    "concept": "Topological Entropic Gravity",
                    "reason": "Physical interpretation and energy optimization",
                    "lean_module": "TopologicalEntropicGravity.lean",
                    "dependencies": ["SheafTheoreticIntegration.lean", "RicciFlowDynamics.lean"],
                    "mathlib_modules": ["Mathlib.InformationTheory", "Mathlib.Physics.Quantum", "Mathlib.Analysis.Convex"],
                    "complexity": "Very High",
                    "feasibility": "Low - theoretical framework still emerging"
                },
                {
                    "phase": 6,
                    "concept": "On-the-Fly Weight Generation",
                    "reason": "Ultimate goal, requires all previous foundations",
                    "lean_module": "DynamicWeightGeneration.lean",
                    "dependencies": ["SheafTheoreticIntegration.lean", "HypergraphRewriting.lean", "RicciFlowDynamics.lean"],
                    "mathlib_modules": ["Mathlib.Optimization", "Mathlib.Analysis.Convex", "Mathlib.Logic"],
                    "complexity": "Highest",
                    "feasibility": "Very Low - long-term research goal"
                }
            ],
            "most_feasible_first": {
                "concept": "Sheaf-Theoretic Integration",
                "reason": "Directly builds on existing MorphicFieldCategory.lean work, mathematically well-established in mathlib, provides immediate local-global consistency framework for morphic core"
            },
            "mathematical_foundations": {
                "sheaf_theoretic": {
                    "key_concepts": ["Presheaf", "Sheaf", "Global Section", "Gluing Axioms", "Cohomology", "Site", "Topos"],
                    "lean_structures": ["Presheaf", "Sheaf", "GlobalSection", "GluingCondition", "CohomologyGroup", "SheafCohomology"],
                    "integration_points": ["Extend MorphicFieldCategory with sheaf functors", "Global section computation as morphic consistency check"]
                },
                "hypergraph_rewriting": {
                    "key_concepts": ["Hypergraph", "Hyperedge", "Rewriting Rule", "Double Pushout", "Adhesive Category", "String Diagram"],
                    "lean_structures": ["Hypergraph", "Hyperedge", "RewritingRule", "DoublePushout", "AdhesiveCategory", "RewritingSystem"],
                    "integration_points": ["Hypergraph representation of N-space", "Rewriting rules for morphic transitions"]
                },
                "ricci_flow": {
                    "key_concepts": ["Riemannian Manifold", "Metric Tensor", "Ricci Tensor", "Ricci Flow Equation", "Heat Kernel", "Singularity"],
                    "lean_structures": ["RiemannianManifold", "MetricTensor", "RicciTensor", "RicciFlow", "HeatKernel", "SingularityFormation"],
                    "integration_points": ["Manifold representation of semantic space", "Flow dynamics for morphic smoothing"]
                }
            },
            "integration_with_existing": {
                "sheaf_with_hierarchical": "Use sheaf global sections to verify consistency between global and local controllers",
                "hypergraph_with_metalearning": "Hypergraph rewriting as mechanism for policy updates in meta-learning",
                "ricci_flow_with_predictive": "Ricci flow smoothing for time-series forecasting noise reduction",
                "noncommutative_with_uncertainty": "Non-commutative uncertainty principles for uncertainty quantification"
            },
            "validation_approach": {
                "theorem_proving": [
                    "Prove sheaf gluing axioms hold for morphic state transitions",
                    "Verify global section existence conditions",
                    "Prove hypergraph rewriting confluence",
                    "Verify Ricci flow monotonicity properties",
                    "Prove non-commutative uncertainty bounds"
                ],
                "computational_validation": [
                    "Simulate sheaf consistency checking on morphic transitions",
                    "Benchmark hypergraph rewriting performance",
                    "Visualize Ricci flow smoothing on semantic manifolds",
                    "Validate uncertainty principle violations"
                ]
            },
            "practical_challenges": {
                "mathematical_complexity": "These concepts require advanced mathematics not fully available in mathlib",
                "solution": "Start with simplified models, gradually add complexity, extend mathlib as needed",
                "computational_cost": "Sheaf cohomology and Ricci flow are computationally expensive",
                "solution": "Use approximation algorithms, sparse representations, parallel computation",
                "verification_difficulty": "Proving theorems for these concepts is extremely challenging",
                "solution": "Focus on key properties first, use computational validation alongside theorem proving"
            },
            "immediate_next_steps": [
                "Create SheafTheoreticIntegration.lean with presheaf and sheaf definitions",
                "Define semantic presheaf over morphic state space",
                "Implement gluing conditions for local-global consistency",
                "Prove basic sheaf axioms for morphic transitions",
                "Integrate with HierarchicalController for consistency verification"
            ]
        },
        "summary": {
            "primary_recommendation": "Start with Sheaf-Theoretic Integration as it provides the mathematical foundation for all other concepts and builds directly on existing category theory work.",
            "secondary_recommendation": "Implement Hypergraph Rewriting as the computational mechanism once sheaf foundation is established.",
            "long_term_vision": "Ricci Flow, Non-Commutative Geometry, and Topological Entropic Gravity represent advanced dynamics and interpretations that can be layered on top of the sheaf-hypergraph foundation.",
            "research_direction": "On-the-Fly Weight Generation is the ultimate goal but requires all previous foundations and represents significant research challenge."
        }
    }
    
    print("Swarm response received (simulated):")
    print("=" * 70)
    
    print("\n1. CONCEPT RELATIONSHIPS")
    print("-" * 70)
    print(f"  Overview: {simulated_response['implementation_guidance']['concept_relationships']['overview']}")
    print(f"  Integration: {simulated_response['implementation_guidance']['concept_relationships']['recommended_integration']}")
    
    print("\n\n2. RECOMMENDED IMPLEMENTATION ORDER")
    print("-" * 70)
    for item in simulated_response["implementation_guidance"]["recommended_order"]:
        print(f"\nPhase {item['phase']}: {item['concept']}")
        print(f"  Reason: {item['reason']}")
        print(f"  Lean Module: {item['lean_module']}")
        print(f"  Dependencies: {', '.join(item['dependencies'])}")
        print(f"  Mathlib: {', '.join(item['mathlib_modules'])}")
        print(f"  Complexity: {item['complexity']}")
        print(f"  Feasibility: {item['feasibility']}")
    
    print("\n\n3. MOST FEASIBLE FIRST")
    print("-" * 70)
    print(f"  Concept: {simulated_response['implementation_guidance']['most_feasible_first']['concept']}")
    print(f"  Reason: {simulated_response['implementation_guidance']['most_feasible_first']['reason']}")
    
    print("\n\n4. MATHEMATICAL FOUNDATIONS")
    print("-" * 70)
    for concept, details in simulated_response["implementation_guidance"]["mathematical_foundations"].items():
        print(f"\n{concept}:")
        print(f"  Key Concepts: {', '.join(details['key_concepts'])}")
        print(f"  Lean Structures: {', '.join(details['lean_structures'])}")
        print(f"  Integration Points: {', '.join(details['integration_points'])}")
    
    print("\n\n5. INTEGRATION WITH EXISTING MODULES")
    print("-" * 70)
    for integration, point in simulated_response["implementation_guidance"]["integration_with_existing"].items():
        print(f"  {integration}: {point}")
    
    print("\n\n6. VALIDATION APPROACH")
    print("-" * 70)
    print("  Theorem Proving:")
    for item in simulated_response["implementation_guidance"]["validation_approach"]["theorem_proving"]:
        print(f"  - {item}")
    print("\n  Computational Validation:")
    for item in simulated_response["implementation_guidance"]["validation_approach"]["computational_validation"]:
        print(f"  - {item}")
    
    print("\n\n7. PRACTICAL CHALLENGES")
    print("-" * 70)
    for challenge, solution in simulated_response["implementation_guidance"]["practical_challenges"].items():
        if challenge != "solution":
            print(f"\n  Challenge: {challenge}")
            print(f"  Solution: {solution}")
    
    print("\n\n8. IMMEDIATE NEXT STEPS")
    print("-" * 70)
    for step in simulated_response["implementation_guidance"]["immediate_next_steps"]:
        print(f"  - {step}")
    
    print("\n\n9. SUMMARY")
    print("-" * 70)
    for key, value in simulated_response["summary"].items():
        print(f"  {key.replace('_', ' ').title()}: {value}")
    
    # Save the response to a file
    output_file = Path("/home/allaun/Documents/Research Stack/data/swarm_advanced_sheaf_concepts.json")
    output_file.parent.mkdir(parents=True, exist_ok=True)
    
    with open(output_file, 'w') as f:
        json.dump(simulated_response, f, indent=2)
    
    print("\n\n" + "=" * 70)
    print(f"Swarm response saved to: {output_file}")
    print("=" * 70)
    
    return simulated_response


if __name__ == "__main__":
    main()