Euler Collapse in Viscous Time Theory: An Informational Spiral Framework for Coherence, Phase Inversion, and Endogenous Collapse Dynamics

Euler’s identity eiπ +1 = 0 is traditionally regarded as a static algebraic relation linking fundamental mathematical constants. In this work, we propose a different interpretation within the framework of Viscous Time Theory (VTT) and the Informational Resonance Spiral Viscous Time (IRSVT): Euler’s identity is treated not as a timeless equality, but as the limiting case of a structured dynamical collapse process in an informational manifold. We introduce the Euler Collapse Spiral (ECS) as a geometric object describing coherence-driven spiral trajectories toward collapse, and define Euler Collapse States (ECS nodes) as discrete, detectable events characterized by phase inversion and coherence nullification. A collapse tension functional is formulated, and an Euler collapse exponent is introduced to quantify the approach to the singular limit. We further formalize the convergence mechanism between the coherence gradient ∆C and the attractor field Φα, providing a geometric basis for irreversible logical bifurcations in the IRSVT manifold.To make this framework testable, we introduce a set of numerical diagnostics, including scaling-law estimation, inverse-vorticity collapse-time reconstruction, enstrophy growth analysis, spectral compatibility tests, similarity-coordinate rescaling, and spiral manifold reconstruction, together with robustness and falsification controls. Using time-resolved numerical data, we obtain a consistent validation signature: vorticity exhibits exact power-law blow-up with measured exponent γ = 1.000000 ± O(10-6), inverse vorticity evolves linearly enabling deterministic collapse-time recovery, enstrophy diverges quadratically, inertial-range -5/3 spectral scaling is preserved, and the dynamics collapse onto a smooth logarithmic spiral manifold in similarity coordinates with machine-precision residuals. Multi-precision arithmetic, noise injection, and parameter perturbations confirm that these features are structurally stable and not numerical artifacts. These results support the interpretation of Euler’s identity as the asymptotic signature of an endogenous informational collapse process, in which apparent singularity corresponds to a geometrically ordered contraction in similarity time rather than uncontrolled instability. The ECS framework thus provides a unified geometric and computational perspective on collapse phenomena, with implications for informational geometry, non-stochastic computation, and the emergence of structured attractors in recursive dynamical systems.