Signed Möbius Off-Diagonal Cancellation, One-Sided Prime Smoothing, and Spectral Positivity: A Consolidated Candidate Route to the Riemann Hypothesis (RH-58)

We consolidate the developments following PIHR–RH LVII (RH-57), The Signed Off-Diagonal in a Weighted Möbius Criterion for the Riemann Hypothesis. The predecessor manuscript established a finite convolution representation of a weighted summatory Möbius function, evaluated its diagonal contribution through an explicit Euler product, and identified a signed off-diagonal remainder whose critical error estimate would imply the Riemann hypothesis (RH). The present work compares three complementary routes: weighted Möbius inversion, the one-sided Chebyshev criterion, and Weil positivity. We establish an exact hyperbola decomposition for general convolution-invertible arithmetic coefficients, an exact triangular smoothing identity for the prime-counting error, and a quantitative endpoint-transfer lemma. We also formulate the correctly projected spectral problem for pole-free Weil test functions and derive an operator-theoretic certification framework. These results clarify which difficulties are genuinely arithmetic and which are consequences of an incomplete analytic reduction. In particular, finite-dimensional positivity, compactness of a negative spectral component, and factorially convergent approximation errors do not by themselves establish RH. Our preferred candidate route combines the signed Möbius off-diagonal structure of RH-57 with a one-sided, endpoint-uniform prime-error estimate. The missing step is a quantitative cancellation theorem that preserves the signed coefficients and attains the critical square-root exponent.