Description

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