The Arithmetic Geometry of Integer Divisor Lattices: A Classification Framework for Prime Powers and Composite Classes

This repository contains the preprint and computational proofs for a new classification framework in multiplicative number theory. By mapping integers into the phase space of divisor-sum coordinates (n, M1, M2), we demonstrate that prime factorization properties constrain integers to live on specific, rigid algebraic surfaces.

Key Discoveries:

• The Universal Cubic Surface: We derive a single cubic equation that perfectly unifies Prime Numbers (tau=2) and Semiprimes (tau=4). This proves that while these numbers are arithmetically distinct, they inhabit the exact same topological manifold in the observable space.

• The Linear Law for Sphenics: We prove that for sphenic integers (pqr) on the modular grid n = 0 (mod 6), the typically chaotic sum-of-divisors function collapses to a strict linear invariant: M1 = 2n + 12.

• Hypercube Detection: We identify a degree-3 algebraic variety that characterizes 4-dimensional divisor lattices (tau=16), extending the geometric hierarchy.

Contents:

• Preprint PDF: Full derivation of the theorems and algebraic proofs.

• master_geometry_engine.py: The discovery algorithm that uses Nullspace Analysis to mine polynomial laws from raw integer data.

• verify_arithmetic_geometry.py: A rigorous verification suite that tests the theorems against N=100,000 integers to confirm the "Unification" hypothesis.

Methodology: The coordinates used in this work are derived from the Fourier coefficients of Eisenstein series (Quasimodular Forms), providing a bridge between Modular Form theory and the Arithmetic Geometry of integer lattices.