Cohomologically Calibrated Affine Connections and Forced Irreducibility

Abstract: We establish a principle of forced geometric irreducibility on product manifolds. We prove that for any product manifold $M=M_1\times M_2$, a cohomologically calibrated affine connection, $\nabla^{\mathcal{C}}$, is necessarily holonomically irreducible, provided its calibration class $[\omega] \in H^3(M;\mathbb{R})$ is mixed. The core of the proof relies on Hodge theory; we show that the algebraic structure of the harmonic part of the torsion generates non-zero off-diagonal components in the full Riemann curvature tensor, which cannot be globally cancelled. This non-cancellation is formally proven via an integral argument. We illustrate the main theorem with explicit constructions on $S^2\times \Sigma_g$, showing that this result holds even in special cases where the Ricci tensor is diagonal, such as the Einstein-calibrated connection. Finally, we briefly discuss speculative analogies between forced irreducibility and quantum entanglement.