Exact Gram Inversion for Fractional-Part Approximation

Exact finite Gram inversion, ordinary sawtooth least-squares errors and rigorous reciprocal-dilate certificates. The Riemann hypothesis and the limiting reciprocal approximation criterion remain unproved. We solve the finite least-squares problem for ordinary sawtooth dilates by explicit Gram inversion and obtain its exact error and asymptotic constant. For the distinct reciprocal-dilate problem, we derive rational Gram certificates, rigorous residual intervals and Hardy-space zero bounds. The Riemann hypothesis remains open.