Appendix F: Research Problems
Open Questions in Hologram Theory
This appendix presents open research problems arising from the Hologram model, organized by difficulty and impact. Each problem includes context, partial results, and suggested approaches.
Fundamental Theory
Problem 1: Complete Expressivity Characterization
Difficulty: ★★★★★ Impact: ★★★★★
Statement: Precisely characterize the class of partial recursive functions that can be denoted and reified in the 12,288 model.
Known Results:
- All primitive recursive functions are denotable
- Some μ-recursive functions are denotable with bounded search
- The halting problem is decidable for lawful configurations
Open Questions:
- Is there a natural complexity class between PR and R that captures exactly the denotable functions?
- What is the relationship to elementary recursive functions?
- Can we embed the full λ-calculus or only linear variants?
Approach: Study the embedding of standard computational models (λ-calculus, Turing machines, cellular automata) and identify what aspects cannot be captured.
Problem 2: Gauge Classification
Difficulty: ★★★★☆ Impact: ★★★★☆
Statement: Completely classify all gauge transformations that preserve lawfulness and determine the structure of the gauge group G.
Known:
- Translations form a normal subgroup
- Schedule rotation σ has order 768
- Boundary automorphisms form a finite subgroup G°
Unknown:
- Complete structure of G
- All discrete symmetries
- Continuous gauge transformations (if any)
Approach: Use group cohomology to study extensions and apply representation theory to classify irreducible gauge actions.
Problem 3: Action Landscape Convexity
Difficulty: ★★★★☆ Impact: ★★★★★
Statement: Determine necessary and sufficient conditions for the action functional S to be convex on the lawful domain.
Partial Results:
- Geometric smoothness sector is convex
- R96 conformity can introduce non-convexity
- Empirically, most practical instances appear convex
Questions:
- When is S strongly convex?
- What is the modulus of convexity?
- Can we guarantee polynomial-time convergence to global minima?
Approach: Analyze the Hessian of S sector by sector and use techniques from convex analysis and optimization theory.
Algorithmic Complexity
Problem 4: Optimal Window Size
Difficulty: ★★★☆☆ Impact: ★★★★☆
Statement: Determine the optimal active window size for various computational tasks that minimizes both space and verification time.
Trade-offs:
- Smaller windows: Less memory, more frequent verification
- Larger windows: Better locality, fewer boundaries
- Task-dependent optimal size
Open: Is there a universal window size that is within constant factor of optimal for all tasks?
Approach: Analyze specific algorithm classes (sorting, searching, graph algorithms) and derive task-specific bounds.
Problem 5: Parallel Complexity Classes
Difficulty: ★★★★☆ Impact: ★★★☆☆
Statement: Define and relate parallel complexity classes (RC, HC, WC) to standard classes (NC, P, PSPACE).
Known:
- CC (Conservation-Checkable) ⊆ P
- RC (Resonance-Commutative) relates to NC
- WC (Window-Constrained) relates to streaming algorithms
Unknown:
- Exact relationships
- Separation results
- Complete problems for each class
Approach: Construct reductions between problems in different classes and identify natural complete problems.
Problem 6: Receipt Compression Limits
Difficulty: ★★★☆☆ Impact: ★★★☆☆
Statement: Determine the information-theoretic limits of receipt compression while maintaining verifiability.
Current:
- Receipts have fixed size regardless of configuration size
- Some compression via Merkle trees and delta encoding
Questions:
- Minimal receipt size for ε-approximate verification?
- Trade-off between compression and verification time?
- Optimal encoding for receipt chains?
Approach: Apply information theory and compressed sensing techniques to receipt structures.
Security and Cryptography
Problem 7: Collision Complexity
Difficulty: ★★★★★ Impact: ★★★★★
Statement: Prove that finding collisions in the address map H on the lawful domain requires exponential time.
Known:
- H is injective on lawful domain
- No collisions observed empirically
- Related to perfect hashing
Unknown:
- Computational hardness of finding near-collisions
- Relationship to standard cryptographic assumptions
- Post-quantum security
Approach: Reduce from known hard problems or show that efficient collision-finding would violate information-theoretic bounds.
Problem 8: Zero-Knowledge Receipts
Difficulty: ★★★★☆ Impact: ★★★★☆
Statement: Design zero-knowledge proof systems for receipt verification that reveal nothing beyond validity.
Requirements:
- Prove receipt validity without revealing configuration
- Maintain composability of receipt chains
- Efficient verification
Challenges:
- Receipts inherently contain information
- Need to hide while preserving verification
- Composition must preserve zero-knowledge
Approach: Adapt zkSNARK techniques to receipt structure and explore homomorphic commitments.
Problem 9: Byzantine Fault Tolerance Threshold
Difficulty: ★★★☆☆ Impact: ★★★★☆
Statement: Determine the optimal Byzantine fault tolerance threshold achievable with receipt-based consensus.
Known:
- Classical BFT achieves f < n/3
- Receipts provide additional verification
- Some improvement possible
Unknown:
- Exact improvement factor
- Optimal protocol
- Trade-offs with communication complexity
Approach: Design new consensus protocols leveraging receipt properties and analyze their fault tolerance.
Categorical and Algebraic Structure
Problem 10: Category of Lawful Configurations
Difficulty: ★★★★☆ Impact: ★★★☆☆
Statement: Fully characterize the category with lawful configurations as objects and budgeted morphisms as arrows.
Known Structure:
- Objects: Lawful configurations (β = 0)
- Morphisms: Budgeted transformations
- Composition: Budget addition mod 96
Unknown:
- Categorical limits and colimits
- Monoidal structure
- Relationship to other computational categories
Approach: Use category theory to study universal properties and construct adjunctions with known categories.
Problem 11: Poly-Ontological Coherence
Difficulty: ★★★★☆ Impact: ★★★☆☆
Statement: Characterize all possible coherent poly-ontological structures and their morphisms.
Questions:
- When do multiple facets cohere?
- Classification of coherence morphisms
- Limits on number of simultaneous facets
Approach: Study using multicategory theory and higher-dimensional category theory.
Problem 12: Homological Invariants
Difficulty: ★★★★★ Impact: ★★☆☆☆
Statement: Compute homological and homotopical invariants of configuration space modulo gauge.
Interest:
- Topological obstructions to transformations
- Persistent homology of process objects
- Spectral sequences for receipt chains
Approach: Apply algebraic topology to the quotient space 𝕋/G and compute invariants.
Machine Learning Integration
Problem 13: Sample Complexity Bounds
Difficulty: ★★★☆☆ Impact: ★★★★☆
Statement: Derive tight PAC learning bounds for hypothesis classes defined by receipt constraints.
Known:
- Receipt dimension provides VC dimension upper bound
- Perfect hashing improves sample complexity
- Empirically very efficient
Unknown:
- Tight bounds
- Agnostic learning complexity
- Online learning regret bounds
Approach: Analyze Rademacher complexity of receipt-defined classes and apply statistical learning theory.
Problem 14: Gradient-Free Optimization Convergence
Difficulty: ★★★★☆ Impact: ★★★★☆
Statement: Prove convergence rates for gradient-free optimization using only receipts.
Challenges:
- No explicit gradients
- Only ordinal information from receipts
- Need to bound iterations
Approach: Adapt convergence proofs from derivative-free optimization and evolutionary algorithms.
Problem 15: Phase Transition Prediction
Difficulty: ★★★★☆ Impact: ★★★☆☆
Statement: Predict and characterize phase transitions in learning dynamics on the lattice.
Observed:
- Sudden changes in learning behavior
- Critical points in action landscape
- Symmetry breaking
Unknown:
- Predictive criteria
- Universal transition classes
- Control mechanisms
Approach: Apply statistical physics methods and study order parameters.
Implementation Challenges
Problem 16: Optimal Lattice Size
Difficulty: ★★☆☆☆ Impact: ★★★★★
Statement: Determine if 12,288 is optimal or if other sizes preserve essential properties.
Considerations:
- 12,288 = 48 × 256 has special factorization
- 768 divides 12,288 (schedule period)
- 96 resonance classes
Questions:
- Are there other “magic” sizes?
- Scaling laws for larger lattices?
- Minimal size for universality?
Approach: Systematically study lattices of different sizes and identify which properties are preserved.
Problem 17: Hardware Acceleration
Difficulty: ★★★☆☆ Impact: ★★★★☆
Statement: Design optimal hardware architectures for Hologram computation.
Requirements:
- Efficient receipt computation
- Parallel morphism execution
- Content-addressable memory
- Gauge transformations
Challenges:
- Balance between specialization and flexibility
- Memory bandwidth limitations
- Power efficiency
Approach: Design custom ASIC/FPGA implementations and analyze performance/power trade-offs.
Problem 18: Quantum Implementation
Difficulty: ★★★★★ Impact: ★★★☆☆
Statement: Implement Hologram computation on quantum hardware using the Φ operator for quantum-classical boundaries.
Questions:
- Quantum advantage for action minimization?
- Superposition of configurations?
- Quantum receipt verification?
Approach: Map lattice states to qubits and design quantum circuits for morphisms.
Applications and Extensions
Problem 19: Biological Computation
Difficulty: ★★★★☆ Impact: ★★★☆☆
Statement: Model biological information processing (DNA, proteins, neural networks) using Hologram principles.
Analogies:
- DNA codons ↔ resonance classes
- Protein folding ↔ action minimization
- Neural plasticity ↔ gauge transformations
Approach: Identify biological conservation laws and map to receipt components.
Problem 20: Economics and Game Theory
Difficulty: ★★★☆☆ Impact: ★★★☆☆
Statement: Apply Hologram model to economic systems and mechanism design.
Ideas:
- Budgets as economic costs
- Receipts as contracts
- Gauge as market equivalence
- Action as social welfare
Approach: Formulate economic problems in Hologram terms and analyze equilibria.
Philosophical Questions
Problem 21: Physical Reality
Difficulty: ★★★★★ Impact: ★★☆☆☆
Statement: Is physical reality describable as a Hologram-like system with conservation laws as receipts?
Connections:
- Conservation laws ↔ Noether’s theorem
- Gauge invariance ↔ fundamental symmetries
- Action minimization ↔ least action principle
- Information preservation ↔ unitarity
Approach: Map physical theories to Hologram structures and test predictions.
Research Directions
Near-term (1-2 years)
- Problems 4, 6, 16, 17 (implementation)
- Problems 13, 14 (learning theory)
- Problem 19 (applications)
Medium-term (3-5 years)
- Problems 1, 5, 7 (complexity)
- Problems 8, 9 (security)
- Problems 10, 11 (algebra)
Long-term (5+ years)
- Problems 2, 3 (fundamental theory)
- Problems 12, 21 (deep theory)
- Problem 18 (quantum)
Collaboration Opportunities
These problems span multiple disciplines:
- Theoretical CS: Problems 1, 4, 5, 7
- Mathematics: Problems 2, 3, 10, 11, 12
- Machine Learning: Problems 13, 14, 15
- Systems: Problems 16, 17
- Physics: Problems 18, 21
- Interdisciplinary: Problems 19, 20
Getting Started
For researchers interested in these problems:
- Start with the implementation (Appendix E) to gain intuition
- Study specific chapters relevant to your problem
- Join the research community at hologram-research.org
- Collaborate on the open-source implementation
- Publish results in appropriate venues
The Hologram model is young and these problems represent the frontier of our understanding. Solutions will advance both theory and practice of lawful computation.