Quantum Probability as the Sum of Holonomies of the Canonical 1-Form

We examine the representation of quantum transition amplitudes and Born-rule probabilities in terms of prequantum U(1) gauge holonomies within Kostant-Souriau geometric quantization. On extended phase space, the Poincare-Cartan canonical one-form (p dq - H dt) serves directly as the connection one-form of the prequantum line bundle. We establish two complementary realizations of phase-space histories. First, for a single smooth phase-space trajectory tracked by rank-one Kahler/coherent-state projection operators, the Balachandran-Roy Anti-Zeno product converges to unitary prequantum parallel transport governed by the connection one-form. Second, for the configuration-space transition amplitude, we employ Fujiwara's time-sliced Blattner-Kostant-Sternberg (BKS) pairing along piecewise-Hamiltonian segments. Although time-sliced histories exhibit intermediate momentum jumps and distinct endpoint momenta, these discontinuities lie along the vertical momentum fibers of constant position and time, where the canonical one-form vanishes identically. Bridging the momentum jumps and endpoint fibers by vertical segments lifts every pair of broken histories to a genuinely closed loop in extended phase space without altering its action. Consequently, the time-sliced transition amplitude and Born-rule probability are oscillatory integrals of prequantum U(1) holonomies around closed phase-space loops, converging in the continuum limit and reducing in the semiclassical stationary-phase limit to a sum over inter-branch classical holonomies---given by the integral of p dq along the time-evolved Lagrangian fiber---weighted by half-form Van Vleck-Pauli-Morette determinants.