The problem of hidden variables in quantum mechanics, which concerns whether quantum theory can be embedded into a classical theory, remains a controversial topic forty years after the advent of quantum mechanics. While most physicists consider this possibility remote, some have continued to explore it. There are conflicting results: some claim proofs of the non-existence of hidden variables, such as von Neumann's, while others attempt to introduce hidden variables, like de Broglie and Bohm. However, there is no exact mathematical criterion to evaluate these proposals. This paper aims to prove the non-existence of hidden variables. It provides a precise necessary condition for their existence, which is discussed in Sections I and II. The proposals in the literature usually introduce a phase space of hidden states, but they fail to account for the algebraic structure of quantum observables. This structure, formalized in Section II as a partial algebra, is essential for a classical reinterpretation. The paper shows that there exists a finite partial algebra of quantum observables that cannot be embedded in a commutative algebra, such as the algebra of real-valued functions on a phase space. The result can be understood intuitively through an example involving an atom of orthohelium in a rhombic electric field. The classical interpretation must predict the atom's energy state changes in various directions, but some predictions are contradicted by experimental results. The paper also discusses the logic of quantum mechanics, showing that the embedding problem is equivalent to whether quantum logic is essentially classical. A classical tautology can be false under meaningful substitutions of quantum propositions, indicating that quantum logic differs from classical logic. Section V compares the present proof with von Neumann's proof, which is based on the non-existence of a real-valued function.The problem of hidden variables in quantum mechanics, which concerns whether quantum theory can be embedded into a classical theory, remains a controversial topic forty years after the advent of quantum mechanics. While most physicists consider this possibility remote, some have continued to explore it. There are conflicting results: some claim proofs of the non-existence of hidden variables, such as von Neumann's, while others attempt to introduce hidden variables, like de Broglie and Bohm. However, there is no exact mathematical criterion to evaluate these proposals. This paper aims to prove the non-existence of hidden variables. It provides a precise necessary condition for their existence, which is discussed in Sections I and II. The proposals in the literature usually introduce a phase space of hidden states, but they fail to account for the algebraic structure of quantum observables. This structure, formalized in Section II as a partial algebra, is essential for a classical reinterpretation. The paper shows that there exists a finite partial algebra of quantum observables that cannot be embedded in a commutative algebra, such as the algebra of real-valued functions on a phase space. The result can be understood intuitively through an example involving an atom of orthohelium in a rhombic electric field. The classical interpretation must predict the atom's energy state changes in various directions, but some predictions are contradicted by experimental results. The paper also discusses the logic of quantum mechanics, showing that the embedding problem is equivalent to whether quantum logic is essentially classical. A classical tautology can be false under meaningful substitutions of quantum propositions, indicating that quantum logic differs from classical logic. Section V compares the present proof with von Neumann's proof, which is based on the non-existence of a real-valued function.