Associator-Gated Covariance Transport: A G₂-Octonionic Framework for Hysteretic Uncertainty Geometry in Viscous Time Theory

This manuscript formalizes a mathematical construction for examining whether a Viscous Time Theory (VTT) description of complex-system memory can be organized through an octonionic/\(G_2\) internal geometry while leaving the canonical uncertainty bound unchanged. VTT is treated as an analysis framework in which measurable or estimable trajectories exhibit coherence change, informational viscosity, attractor alignment, anisotropy, and hysteresis; these quantities are declared model coordinates, to be linked eventually to an operational measurement map. A fixed octonion convention, a three-coordinate control manifold \(u = (\Delta C, \Phi_\alpha, \eta)\), a positive 8-dimensional Hessian, an independently declared symplectic form, and genuine \(\mathfrak{g}_2\) derivations are combined to construct a deterministic reference model. The central diagnostic is the octonion associator norm: \(\Xi_{\mathrm{oct}} = \bigl\| [D_{\Delta C}q, D_{\Phi}q, D_{\eta}q] \bigr\|^2\). A matched associative null gives \(\Xi_{\mathrm{oct}} = 0\), while the declared active triple gives \(\Xi_{\mathrm{oct}} \approx 4\). A closed-loop calculation shows that both the null and active constructions generate a nonzero internal plane-holonomy residual, which establishes that holonomy alone cannot serve as evidence for a \(G_2\)/non-associative sector. A local Gaussian complete-positivity gate is then imposed on the reduced covariance illustration. The manuscript closes with falsification gates, gauge and baseline stress requirements, and a physical-bridge boundary for future empirical work.