Stochastic Semantic Fields for Sentiment-Driven Models

This article studies a time-dependent latent sentiment representation governed by a semilinear stochastic evolution equation on a separable Hilbert space. The objective is to derive uncertainty, interpretability, and robustness from one stochastic law. The mathematical analysis establishes covariance propagation and spectral uncertainty attribution, Fréchet differentiability of the forcing-to-state and forcing-to-readout maps with quadratic remainders, trace-norm differentiation of the covariance, and an adjoint influence kernel whose norm equals the exact worst-case displacement over an energy-bounded forcing ball in the linear regime. The dependence of the certificates on dissipativity, forecast horizon, noise geometry, and spectral truncation is made explicit. A Galerkin state-space reduction yields exact linear transitions, a likelihood-based calibration scheme, and structural identifiability conditions. A reproducible three-mode study verifies the covariance and duality identities, evaluates the sensitivity constants, and compares Gaussian with split-conformal predictive intervals under controlled synthetic conditions. The results provide a rigorous operator calculus and a tractable finite approximation. They establish internal mathematical and numerical validity without asserting a generalised superiority.