Octonionic Isothermic Surfaces and the $E_8$ 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 $\R^7 \cong \im(\OO)$ and prove that the Kamberov--Pedit--Pinkall spectral deformation formula yields a valid Bonnet pair if and only if the \emph{extrinsic position vector} associates with the \emph{intrinsic tangent frame}, a constraint linking surfaces to $G_2$-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 \textbf{dynamical rationality}: the geometry closes not because of arithmetic properties of the period ratio $\tau$ (no classical CM points lie in the allowed range), but because of the quantization of the monodromy angle $\theta$ via the spectral curve, a geometric analog of Bohr--Sommerfeld quantization. The Bonnet ambiguity connects the discrete spectrum of compact Bonnet pairs to the $E_8$ theta function and suggests a geometric origin for gauge redundancy in spacetime reconstruction.