Signed Möbius Contraction and the Riemann Hypothesis

We consolidate a conditional approach to the Riemann hypothesis based on a fixed logarithmically weighted Möbius correction in the Nyman–Beurling approximation problem. The finite inversion criterion of the preceding arithmetic work is retained, but its unresolved square root error is separated from exact cancellation of the linear main term. In a common Hilbert space we construct the complete Schur complement, derive the precise energy gain of the prescribed correction, and prove that a relative gain of only a constant divided by the iteration index would imply the Riemann hypothesis. The missing estimate is expressed as one signed quadratic form, preserving the dependence between its numerator and denominator. An exact prime factor packet identity records every Möbius coefficient once and retains cutoff boundary terms. We also prove an explicit obstruction associated with a hypothetical zero off the critical line and explain why Gram positivity, Pythagoras, small eigenvalues, and unweighted divisor cancellation do not establish the required comparison. The resulting candidate has one fixed arithmetic target.