Mathematical Content
Each integer n ∈ ℕ is represented by two canonical normalized states in ℓ²(ℕ):
- Multiplicative state: |ψₙ⟩ = (1/√d(n)) Σ_{d|n} |d⟩, where d(n) denotes the divisor count function
- Additive state: |φₙ⟩ = (1/√(n-1)) Σ_{k=1}^{n-1} |k⟩
Both representations induce inner products and Fubini-Study distances on ℕ.
Proven Results
The framework establishes five main theorems:
- The divisor kernel K(n,m) = d(gcd(n,m)) is positive-definite on ℕ
- The multiplicative distance d_mult(n,m) = √(2 - 2⟨ψₙ|ψₘ⟩) satisfies metric axioms
- The additive distance d_add(n,m) = √(2 - 2⟨φₙ|φₘ⟩) satisfies metric axioms
- The combined distance d_codex(n,m) = √(α·d_mult² + β·d_add²) is a metric for α,β ≥ 0
- The multiplicative entropy S_mult(n) = log₂(d(n)) equals the bipartite von Neumann entropy of |ψₙ⟩
All theorems are proven using standard techniques from functional analysis and number theory.
Computational Implementation
The work includes complete algorithmic implementations with computational complexity O(√n) for:
- Divisor enumeration
- Inner product calculation
- Distance matrix computation
- Entropy calculation
- Network construction
- Clustering algorithms
Mathematical Mappings
The framework provides exact correspondences between arithmetic structure and:
- Quantum state spaces (via Hilbert space embedding)
- Discrete energy spectra (via divisor-indexed levels)
- Crystallographic lattices (via prime factorization)
- Information encoding capacity (via entropy measures)
These are mathematical correspondences, not physical models.
Applications
The codex enables:
- Geometric analysis of number-theoretic properties
- Network-based study of divisibility relationships
- Clustering of integers by arithmetic structure
- Distance-based classification methods
- Graph-theoretic approaches to multiplicative number theory
