Constructive Rice's Theorem via Hilbert's Tenth Problem Without LEM — E8 Intelligence Research

FINDING: Rice's theorem generalizes the halting problem — every non-trivial semantic property of partial recursive functions is undecidable; the constructive proof via Hilbert's Tenth Problem eliminates the two uses of the Law of Excluded Middle (LEM) in the classical proof. MATH: - Halting set: H = {⟨M,w⟩ : M halts on w}. H is recursively enumerable but not recursive. H̄ is not r.e. - Rice: For any P ⊆ RE (partial recursive functions) with P ≠ ∅ and P ≠ RE, the index set S_P = {e : φ_e ∈ P} is undecidable. - Reduction schema: H ≤_m S_P for every non-trivial P (via padding lemma: for each e there is a total recursive f with φ_{f(e)} = φ_e). - Constructive form (arXiv 2604.16477): replaces the two LEM invocations — (i) diagonalization over the halting predicate, (ii) case split on whether the always-diverging program b ∈ P — with explicit witnesses. Yields a program that, given a decision procedure for S_P, constructs a contradiction. - Hilbert's Tenth Problem (Matiyasevich–Robinson–Da Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com