A Mathematical Framework for High-Energy Astrophysical Neutrino Astronomy: Flux Reconstruction, Detector Response, and Statistical Inference

High-energy astrophysical neutrino astronomy provides a means of investigating energetic cosmic environments through particles that interact weakly with matter and can travel over astronomical distances. The mathematical interpretation of neutrino observations requires a coherent description of source fluxes, flavor propagation, neutrino interactions, detector response, and statistical inference. In this paper, we develop a unified mathematical framework for these tasks. We introduce differential flux models, describe standard three-flavor neutrino propagation, formulate detector response as a positive integral operator, and construct likelihood functions for event-level data. We establish a consistency result relating the response operator to the expected event count and examine the inverse problem of recovering a flux from reconstructed observables. A symbolic example illustrates the calculation of expected event counts for a power-law spectrum with an exponential cutoff. We also discuss maximum-likelihood estimation, Bayesian inference, regularization, identifiability, point-source searches, and multi-messenger analysis. The purpose is to provide a self-contained mathematical treatment of established methods in high-energy neutrino astronomy. The formulations are intended as a foundation for reproducible analyses and further work on inverse problems, uncertainty quantification, and source reconstruction.