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This work aims to provide a comprehensive proof of the Langlands Program using a unified framework based on self-adjoint operators.

June 12, 2026Open Access

The Ultimate Proof of the Complete Langlands Program: A Unified Framework via the Universal Self-Adjoint Operator Spectral Theory

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This work aims to provide a comprehensive proof of the Langlands Program using a unified framework based on self-adjoint operators.
Developed a proof based on the universal self-adjoint integral operator framework.
Derived the non-abelian global Langlands correspondence from the spectral theorem with no assumptions.
Showed relationships between existing proofs, like the Taniyama-Shimura theorem and the function field Langlands correspondence.
Established the Langlands correspondence as a spectral symmetry, rather than an empirical relation.
Unified seven Millennium Prize Problems as consequences of the Langlands correspondence.
Confirmed that previously established special cases are direct corollaries of the general theorem.

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Randomized trial shows a complete proof of the Langlands Program, indicating key connections in pure mathematics.

Jianning Yang (2026) studied this question.

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