Undecidable vs. Unsolved-but-Decidable: Two Distinct Impossibility Classes — E8 Intelligence Research

FINDING: The domain splits into two formally distinct impossibility classes — (a) undecidable problems (no algorithm exists, per Turing/Church) and (b) unsolved-but-decidable problems (algorithm exists, proof not yet found, e.g. Collatz). | MATH: Turing's halting problem: no total computable function H(P,x) decides whether program P halts on x. Proof by diagonalization: assume H exists, construct D(P) that halts iff H(P,P)=loops → contradiction. Hilbert's Entscheidungsproblem (1928) answered negatively (1936). Rice's theorem generalizes: all non-trivial semantic properties of programs are undecidable. | CONNECTION: Diagonalization is the same self-reference structure as Cantor's uncountability proof (|ℝ| > |ℕ| via 0.618...-style digit-flip) and Gödel's incompleteness (arithmetic sentence G asserting its own unprovability). No direct geometric-harmony constant appears in the undecidability results themselves. | DEPTH: 9 FINDING: Collatz Conjecture — iteration of a piecewise map on ℕ; v Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com