Mathematical Foundations
Every SmartHaus innovation starts with a proof. Explore the mathematics behind our AI systems.
Research Areas
Holographic Memory
H(x,t) = ∫∫∫ M(ξ,τ) K(x-ξ, t-τ) dξdτ
Mathematical framework for memory systems where every fragment contains relational information about the whole. Perfect recall from partial queries.
Read Paper →LQL Calculus
L: Cat → Cat, L(f∘g) = L(f) ⊗ L(g)
Category theory foundations for AI composition. Functorial transformations ensuring mathematical consistency across all operations.
Read Paper →Biological Calculus
dS/dt = ∇·(D∇S) + f(S)
Differential equations modeling cognitive processes. How mathematical biology informs AI architecture and emergent intelligence.
Read Paper →Published Research
Holographic Memory Systems for Conversational AI
SmartHaus Research • 2024 • Patent Pending
A novel approach to AI memory using holographic principles from quantum physics. Demonstrates perfect recall with partial queries and context preservation across sessions.
AIUCP: Multi-Model Orchestration Architecture
SmartHaus Research • 2024 • Patent Pending
Unified control plane for orchestrating multiple AI models with mathematical guarantees on consistency, fallback chains, and optimal model selection.
LEF: Quantum-Inspired Execution Framework
SmartHaus Research • 2024 • Patent Pending
Particle-based execution model for AI operations. Breaks complex tasks into quantum-like particles with superposition, entanglement, and collapse mechanics.
LQL DAG Calculus: Compositional AI
SmartHaus Research • 2024 • Patent Pending
Category theory approach to AI composition. Directed acyclic graphs with functorial properties ensuring mathematical consistency in complex workflows.
AIOS Biology Calculus: Cognitive Modeling
SmartHaus Research • 2024 • In Review
Mathematical modeling of biological intelligence for AI systems. Differential equations describing learning, adaptation, and emergent cognitive behaviors.
Open Research Problems
We're actively researching these mathematical challenges. Join us in solving the fundamental problems of AI.
Holographic Superposition
H(ψ₁ + ψ₂) ?= H(ψ₁) + H(ψ₂)
Can multiple memory states exist in quantum superposition? How do we collapse to the correct memory state?
Categorical AGI
AGI: Universe → Mind
Is there a functorial relationship between physical reality and artificial general intelligence?
Proof Complexity
P ?= NP for AI verification
Can we efficiently verify AI behavior in polynomial time? What are the complexity bounds on AI proofs?
Consciousness Metrics
Φ = ∫ I(S) dS
Mathematical definition of machine consciousness. Can integrated information theory apply to AI?
Research Tools & Resources
Proof Library
Coq and Lean proofs for all our mathematical claims. Formally verified, open source.
View Proofs →Calculus Notebooks
Interactive Jupyter notebooks demonstrating our mathematical concepts with live code.
Open Notebooks →Research Collaboration
Partner with us on fundamental AI research. Academic and industry collaborations welcome.
Collaborate →Mathematics is Our Foundation
Every SmartHaus system is built on mathematical proof. Join us in proving the future of AI.