A Para-Hermitian Geometric System for Quantum Mechanics and General Relativity: Time as a Mode of Each Fundamental Dimension

In this paper, we propose a six-dimensional para-Hermitian manifold in which each spatial dimension carries an intrinsic temporal mode via split-complex coordinates with signature (3,3). The geometry is restricted to Minkowski space on the four-dimensional submanifold defined by synchronized temporal modes, with no rescaling of the speed of light required. A single action is analyzed in two complementary dynamical limits. In the phase-locked sector, Kaluza–Klein reduction recovers four-dimensional Einstein gravity, with Newton’s constant determined by the ratio of the six-dimensional gravitational coupling to the transverse fiber volume. In the phase-fluctuating sector, the non-relativistic limit of the bulk Klein–Gordon equation for the misalignment doublet leads to the Schrödinger equation, with Planck’s constant identified as the canonical normalization of the diagonal Goldstone field. The Born rule probability measure coincides with the geometric [Formula: see text] norm on the misalignment plane, as a consistency condition associated with that Goldstone mode, which serves as a Page–Wootters relational clock specified by the geometry. The Wheeler–DeWitt frozen-time problem is thereby reformulated in relational terms. A single new mass scale separates the quantum and classical regimes; the conservative model-independent lower bound from LHC non-observation of new scalar states places this scale at a few TeV.