Atiyah-Hirzebruch Integral Hodge Counterexamples and Torsion Obstructions: Formal Separation from the Rational Clay Hodge Conjecture

Standalone formal documentation and mathematical ledger of the 1962 Atiyah-Hirzebruch torsion counterexamples to the Integral Hodge Conjecture on smooth projective varieties over C, constructed via Steenrod operations in topological K-theory on Godeaux-Serre quotients of products of elliptic curves by elementary abelian p-groups (p >= 3) in complex dimension >= 7. Explicitly documents why torsion classes vanish under tensoring with Q (H²ᵖ(X, Z)_tors (x) Q = 0) and therefore do not constitute counterexamples to the Clay Mathematics Institute Rational Hodge Conjecture. This repository records known 1962 integral torsion obstructions (Atiyah-Hirzebruch) and their vanishing after rationalization. It does not prove or disprove the rational Clay Hodge Conjecture. Copyright (c) 2026 Benjamin Stanley Frohman. Licensed under the Apache License, Version 2.0.