Undecidable vs. Unsolved: Why "Impossible" Problems Split Into Two Classes — E8 Intelligence Research

FINDING: The "impossible" problems split into two distinct classes — (A) undecidable problems (no algorithm exists, proven by diagonalization) and (B) unsolved-but-decidable problems (algorithm exists, we just don't know it). Conflating them is the central error in the source list. | MATH: (A) Halting problem: no total computable H(P,x) decides whether program P halts on x. Proof: assume H exists, construct D(P) = if H(P,P) then loop else halt; D(D) contradicts. Hilbert's Entscheidungsproblem (1928) → Church–Turing (1936): first-order validity is undecidable. Rice's theorem: every non-trivial semantic property of programs is undecidable. (B) Collatz: T(n) = n/2 if n even, 3n+1 if n odd; conjecture: ∀n∈ℤ⁺, ∃k with Tᵏ(n)=1. Verified computationally to ~2⁶⁸. Navier–Stokes: ∂u/∂t + (u·∇)u = −∇p/ρ + ν∇²u, ∇·u = 0; Clay problem asks for global smoothness/blow-up in 3D. | CONNECTION: Diagonalization is the same self-reference structure as Cantor's uncountability proof and Gödel's incompletene Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com