X. Advanced Mathematical Framework
ℤ₄-Voltage Graph
The Gross-Tucker voltage graph construction applied to OTH: the base graph is the chord-graph of B, with ℤ₄ voltage labels on edges encoding the fiber (inversion) shift at each step. The derived graph is the chord-graph of E. This is the exact discrete analogue of the continuous fiber bundle.
Metrized Holonomy Groupoid
The proposed unifying framework for all OTH motion types: a groupoid over the voltage graph whose morphisms encode all possible paths through E, equipped with the metric inherited from the L1 voice-leading norm. The holonomy of a path (the net fiber shift accumulated around a closed loop) captures the “geometric phase” phenomenon — progressions that return to the same base chord but in a different inversion.
Discrete Gauge Theory
The interpretation of the voltage graph as a discrete gauge field, where the ℤ₄ fiber plays the role of the gauge group and the voltage assignment is the discrete connection. Parallel transport around loops produces holonomy (fiber phase shift). This situates OTH within the framework of discrete differential geometry.
Ollivier-Ricci Curvature
A discrete analogue of Ricci curvature applicable to graphs. An open computational question for OTH: does the Ollivier-Ricci curvature of B correlate with the degree landscape and harmonic stability? Positive curvature at the Summit would indicate that geodesics converge there (a geometric formalization of tonic stability).