An Arc Hilbert-Poincaré Criterion for Rational Double Points

For a normal complex algebraic surface germ, we prove that its focused scheme-theoretic arc Hilbert-Poincare series equals (1-q)^(-3) times the product of (1-q^j)^(-2) for j >= 2 if and only if it is an ADE rational double point. The known forward formula is complemented by a converse: low-weight coefficients recover a hypersurface of multiplicity two, and a graded annihilator argument forces every finite prefix of its jet equations to be a regular sequence. The resulting fiber dimension bound gives irreducibility of all jet schemes, so Mustata's criterion applies. The same argument gives a converse for isolated normal hypersurface singularities of maximal canonical multiplicity. Normality is essential for the surface statement: the known normal-crossing example xy=0 already has the same series. The forward formulas and that example are attributed to Bruschek, Mourtada and Schepers. This four-page English preprint addresses the explicitly normal category of UnsolvedMath OWR-12697689-012; it does not claim a first historical or unrestricted source-question resolution. Exact rational computations through weight ten supplement the written proof. AI-assisted and self-audited; not independently peer reviewed or formally verified, and no exhaustive priority certification is asserted.