Preserving Meaning, Discarding Detail: Möbius Inversion, Adjunctions, Fourier Duality, and Invariants as Related but Distinct Perspective Shifts in Mathematics

This repository contains the LaTeX source and compiled PDF for the paper Preserving Meaning, Discarding Detail: Möbius Inversion, Adjunctions, Fourier Duality, and Invariants as Related but Distinct Perspective Shifts in Mathematics. The paper is an expository and pedagogical work examining several classical mathematical constructions that are often treated independently, but which are frequently felt to play similar explanatory roles. It considers Möbius inversion, Galois connections and adjunctions, Fourier duality, and functorial invariants as related but fundamentally distinct responses to a common explanatory challenge: managing complexity through deliberate changes of representation. Rather than claiming a single unifying mechanism, the paper adopts a comparative approach. It analyses how different areas of mathematics trade descriptive detail for structural clarity in different ways, sometimes reversibly, sometimes irreversibly, and sometimes by intentional abstraction. Throughout, the emphasis is on clarifying both affinities and limits, and on preventing category errors that arise from over-generalising superficial similarities. The contribution is explicitly explanatory rather than foundational. The paper proposes no new formal theory and makes no unification claims. Instead, it offers a heuristic framework for pedagogical comparison, together with counterexamples that delimit its scope. It is intended for mathematicians, students, and researchers interested in conceptual clarification, mathematical practice, and the role of perspective shifts across combinatorics, algebra, analysis, and category-theoretic settings. The repository includes a self-contained LaTeX source file, the compiled PDF, and minimal documentation for reuse and citation.