December 6, 2025Open Access

The Rough Langlands Correspondence: Unifying Galois Representations and Automorphic Forms via Roughness Symmetry

Key Points

Automorphic forms are derived from prime numbers as they behave in rough geometric spaces, suggesting a new interpretation of their relationship.
The exploration highlights a manifestation of the Roughness Symmetry Principle in number theory and harmonic analysis by connecting energy spectra and topological structures.
Assessment using rough operator algebra provides a framework to understand the relationship between objects in number theory and analysis with respect to their geometric properties.
This approach may enable further insights into the interplay between discrete and continuous mathematical constructs in advanced theories.

Abstract

The Langlands Program, often called the "Grand Unified Theory of Mathematics," conjectures a deep correspondence between Number Theory (Galois representations) and Harmonic Analysis (Automorphic forms). However, the underlying mechanism of this duality remains elusive. In this paper, we propose a physical solution using Rough Operator Algebra (ROA). We interpret Prime Numbers not as static points, but as Topological Knots in a rough geometric space (α < 1). We prove that Automorphic Forms are the Energy Spectra released when these knots unravel during the Roughness Renormalization process (α → 1). This establishes the Roughness Symmetry Principle: the discrete complexity of number theory is the mirror image of the continuous energy of analysis.

The Rough Langlands Correspondence: Unifying Galois Representations and Automorphic Forms via Roughness Symmetry

Key Points

Abstract

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