We present a unified theoretical framework connecting our recent non-local gauge phase corrections to the high-dimensional vectorial probability spaces recently uncovered in Bell tests. By analyzing the breakdown of the Schwarz integrability condition for pathological gauge phases inside active magnetic field regions, we demonstrate that second-order Taylor expansion terms leave a geometric residue that does not vanish upon boundary traversal. We show that what stochastic calculus achieves via a pathological, nowhere-differentiable trajectory, our framework achieves via a non-Schwarz, non-commuting gauge field geometry. Both mechanisms force the second-order expansion residue to remain alive in the continuum limit, structurally transforming a standard $1\text{D}$ scalar line integral into a higher-dimensional tensor mapping. This structural isomorphism offers a concrete physical mechanism for the hidden ensemble geometries capable of reproducing quantum correlations within a local realistic theory, expanding our foundational understanding of Bell test loopholes.
Publication Date: 2026-06-14