Additive Sieve in the Logarithmic Lattice: A Geometric Reading of the Multiplicative Structure of ℕ

We introduce the Logarithmic Lattice $L$, a geometric reformulation of the multiplicative monoid of positive integers through the prime factorization map

ϕ:N→L,n=∏ipiai↦(a1,a2,… ) ,\phi : \mathbb{N} \to L, \quad n = \prod_i p_i^{a_i} \mapsto (a_1, a_2, \dots) \,,ϕ:N→L,n=i∏piai↦(a1,a2,…),

which converts multiplication into vector addition. Within this framework, we provide:

1. A restatement of primality as a lattice vacancy condition.
2. A simple derivation of the Vacancy Density Observation, ρvac(R)∼1/R\rho_{\text{vac}}(R) \sim 1/Rρvac(R)∼1/R, from the Prime Number Theorem.
3. Lattice-based characterizations of the von Mangoldt and Möbius functions, and a geometric reformulation of the Prime Number Theorem.
4. A geometric interpretation of the Ulam spiral pattern via fixed-divisor obstructions and the conjectural Bateman–Horn constant, including numerical comparisons.
5. A preliminary lattice potential Φ(N)=eΩ(N)ln⁡N\Phi(N) = e^{\Omega(N)} \ln NΦ(N)=eΩ(N)lnN in which primes occupy the trivial minimum.

This work does not claim new theorems in analytic number theory. Instead, it provides a self-contained lattice viewpoint, complementing analytic-sieve approaches and serving as a reference for two companion preprints exploring the lattice potential and high-dimensional concentration phenomena.