Octonionic Isothermic Surfaces and the E₈ Bonnet Problem: A Geometric Mechanism for Modular Spacetime Reconstruction

We extend the recent resolution of the Global Bonnet Problem by Bobenko, Hoffmann, and Sageman-Furnas to the setting of octonionic geometry. We define the class of octonionic isothermic surfaces in ⁷ () and prove that the Kamberov--Pedit--Pinkall spectral deformation formula yields a valid Bonnet pair if and only if the extrinsic position vector associates with the intrinsic tangent frame, a constraint linking surfaces to G₂-calibrated geometries. We show that the periodicity conditions for compact tori are governed by theta functions on rhombic lattices, and we introduce the principle of dynamical rationality: the geometry closes not because of arithmetic properties of the period ratio (no classical CM points lie in the allowed range), but because of the quantization of the monodromy angle via the spectral curve, a geometric analog of Bohr--Sommerfeld quantization. The Bonnet ambiguity connects the discrete spectrum of compact Bonnet pairs to the E₈ theta function and suggests a geometric origin for gauge redundancy in spacetime reconstruction.