We study multigrade deep learning (MGDL) as a principled framework for structured error refinement in deep neural networks. While the approximation power of neural networks is now relatively well understood, training very deep architectures remains challenging due to highly non-convex and often ill-conditioned optimization landscapes. In contrast, for relatively shallow networks, most notably one-hidden-layer $\texttt{ReLU}$ models, training admits convex reformulations with global guarantees, motivating learning paradigms that improve stability while scaling to depth. MGDL builds upon this insight by training deep networks grade by grade: previously learned grades are frozen, and each new residual block is trained solely to reduce the remaining approximation error, yielding an interpretable and stable hierarchical refinement process. We develop an operator-theoretic foundation for MGDL and prove that, for any continuous target function, there exists a fixed-width multigrade $\texttt{ReLU}$ scheme whose residuals decrease strictly across grades and converge uniformly to zero. To the best of our knowledge, this work provides the first rigorous theoretical guarantee that grade-wise training yields provable vanishing approximation error in deep networks. Numerical experiments further illustrate the theoretical results.

In this study, we focus on the aforementioned paper, "Examination Kubo-Matsubara Green's Function Of The Edwards-Anderson Model: Extreme Value Information Flow Of Nth-Order Interpolated Extrapolation Of Zero Phenomena Using The Replica Method (2024)". This paper also applies theoretical physics methods to better understand the filter bubble phenomenon, focusing in particular on loop expansions and truncation functions. Using the loop expansion method, the complexity of social interactions during the occurrence of filter bubbles will be discussed in order to introduce series, express mathematically, and evaluate the impact of these interactions. We analyze the interactions between agents and their time evolution using a variety of Green's functions, including delayed Green's functions, advanced Green's functions, and causal Green's functions, to capture the dynamic response of the system through local potential approximations. In addition, we apply truncation functions and truncation techniques to ensure incremental safety and evaluate the long-term stability of the system. This approach will enable a better understanding of the mechanisms of filter bubble generation and dissolution, and discuss insights into their prevention and management. This research explores the possibilities of applying theoretical physics frameworks to social science problems and examines methods for analyzing the complex dynamics of information flow and opinion formation in digital society.This paper is partially an attempt to utilize "Generative AI" and was written with educational intent. There are currently no plans for it to become a peer-reviewed paper.