Paper 125F: The Abel–Hardy Abscissa of a Möbius Moving-Window Field A Positive-Energy Criterion for the Rightmost Zeta Zero, Zero Multiplicity, and the Riemann Hypothesis

A positive logarithmically weighted energy of a smoothed Möbius moving-window field has aconvergence boundary equal to the right edge of the nontrivial zeta zeros, shifted by one half. The fieldretains the primitive observation geometry of Paper 125E while smoothing both the shell boundaryand the observation window. Its Hilbert-valued Laplace transform is the Mellin observation vectordivided by the Riemann zeta function. A convolution argument shows that the vector cannot vanishidentically at any spectral parameter, so every zeta zero remains observable. Explicit vertical-growthand contour arguments establish the weighted-energy abscissa, and Plancherel gives the associatedpositive vertical-line identity on its valid half-plane. The result is an exact Riemann-hypothesiscriterion without a simplicity assumption. A multiplicative correlation formula holds absolutely inits initial range; finite-time Gram forms provide an unambiguous monotone extension wherever thefull energy is finite or infinite. A separate local calculation identifies the spectral growth caused bya zero of given multiplicity. The retained fifty-zero figures illustrate finite Abel averaging, and acomplete standard-library implementation reproduces their original discretization while distinguishingnumerical refinement from an infinite-spectrum estimate. The required energy finiteness for everypositive weight remains unproved. No uniform bound at zero weight, sharp-window error estimate,new zero-free half-plane, or proof of the Riemann hypothesis is claimed.