Reflected Dynamics Induced by Deterministic Safety Projections — Public Companion Deposit

Public companion deposit for a peer-reviewed journal submission (planned: Nature Machine Intelligence) on the reflected stochastic dynamics induced by a deterministic safety projection operator ΠΩ that maps any proposed action into the safe set Ω at every invocation. The deposit contains the derived observables, the figure-generation script, and the publication-ready figures used in the manuscript and its supplementary information. Using only four abstract properties of the projection operator (feasibility, idempotence, minimal Euclidean correction, local regularity), the projected Euler–Maruyama scheme converges weakly to a reflected stochastic differential equation on Ω, whose stationary density obeys a Fokker–Planck equation with a no-flux boundary condition. This predicts three phenomenological signatures: S1 (hard truncation of the state distribution at ∂Ω), S2 (excess probability mass near ∂Ω), and S3 (intervention intensity scaling with distance to boundary). Phenomenological validation across twelve heterogeneous domains (thermodynamic, biomedical, energy, financial, linguistic, nuclear, aviation, agentic AI, and embedded hardware) confirms the predicted structure, with zero safety violations across a combined sample of 930,096 logged invocations (plus more than 2.42 × 10⁹ embedded-hardware iterations). Included are derived observables only—distance-to-boundary distributions (S1, S2), violation-margin/correction-magnitude (δ, λ) pairs (S3), aggregate efficiency metrics η (S4), a self-contained figure-generation script (NumPy, Pandas, Matplotlib, SciPy; no network, data loaders, or controller code), and publication-ready PNG/PDF figures. The deposit follows a three-tier policy: (1) openly available under CC BY 4.0 are the derived observables, script, and figures; (2) third-party source datasets (Kaggle PMU, continuous glucose monitoring databases, the Kaggle renewable-energy dataset, and Yahoo Finance prices) are not redistributed and remain subject to their respective licenses, with only derived arrays deposited and the authors’ prospective trading log available upon request; and (3) the internal implementation of the safety projection operator is not publicly released and is deliberately withheld pending publication of U.S. patent application USPTO 19/535,932, with international rights reserved under the Paris Convention and PCT.