A TOY MODEL for Generative Velocity and the Geometry of Stability-Loss

This technical report presents a minimal, operationally deployable toy model for detecting stability loss in streaming telemetry systems. The model provides two complementary detection mechanisms: Time-domain detection (Case 1): A Skyline-based Taylor Sensor identifies the emergence of a non-absorbable component in the signal history. A positive residual Φ(t)=Sky(t)−Upoly indicates that the telemetry signal has exited the convex-absorbable regime defined by healthy data. Context-domain detection (Case 2): The deformation of the operational geometry is quantified using the generative velocity Ugen(Qt)≈∥Qt−P(Qt)∥F, where Qt is the contextual overlap matrix and P(Qt) its polytope projection. Persistent curvature in this geometry signals long-term drift of the system’s technical spacetime. Both detection rules rely only on streaming histograms, overlap matrices, and Frobenius-norm deviations, making the method suitable for real-time onboard or ground-based telemetry pipelines. The approach requires no specialized mathematical infrastructure and is compatible with standard flight software architectures. Appendix B provides a rigorous measure-theoretic formulation of the critical regime, including absolute continuity breakdown, singular components, and the equivalence between overlap degeneration and operator-theoretic time-to-threshold. The polytope–sphere transition observed in telemetry is shown to be structurally analogous to the classical-to-Tsirelson transition in correlation bounds. This report serves as an operational companion to the theoretical framework introduced in Generative Velocity and the Geometry of Stability Loss, isolating the core mechanisms required for practical anomaly detection in streaming systems. Related work Generative Velocity and the Geometry of Stability- Loss Toward a Mathematical Describability of Systemic Breakdownhttps://doi.org/10.5281/zenodo.18195700 Generative Velocity as a Measure--Theoretic Invariant: Skyline Structure and the Radon--Nikodym Decompositionhttps://doi.org/10.5281/zenodo.18825940 Generative Foundations of Measure and Distance: Contextual Transformation and the Emergence of Structural Boundarieshttps://doi.org/10.5281/zenodo.18491118 Trans–Integral: Observer–Dependent Measure and the Emergence of Time — Lebesgue Theory as a Dirac–Limit of Assembler Densityhttps://doi.org/10.5281/zenodo.17671551 TI07 — Generative Time and the Reconstruction of the Path Integralhttps://doi.org/10.5281/zenodo.17864219 TI07 Companion, Vol. 1 — Split Generative Yoga under Generated Timehttps://doi.org/10.5281/zenodo.18195541 TI07 Companion, Vol. 2 — Quantitative Constraints on Generated Timehttps://doi.org/10.5281/zenodo.18195700 A Measure-Theoretic Unification of Statistical, Quantum, and Cosmological Doppler Shiftshttps://doi.org/10.5281/zenodo.17731306 TI06 — Generative Time and Emergent Lorentz Geometry: Observer–Dependent Kinematics, Energy, and Path–Integral Structurehttps://doi.org/10.5281/zenodo.17835394