Paper 134DY: Direct Theta Bounds for Compatible Screening From the First Quantified Matrix Inequality to the Full Operator

This paper proves the strengthened first screening inequality directly for the normalized Riemanntheta response on the entire nonnegative real axis. The inequality was previously a conditionalconsequence of bounded screening in revised Paper 134CQ. An exact cancellation converts it toa shifted two-by-two inverse-response moment inequality and to positivity of an explicit entirenumerator. A directed-rounding Taylor certificate proves the numerator and the decreasing-responsecondition on a compact interval. A separate analytic estimate, derived from Binet’s integral andthe Euler product, supplies an explicit overlapping tail. The result implies strict positivity of everytwo-point Loewner matrix for both the negative response and the negative logarithm of the positiveresponse, without assuming a positive screening measure or any zero locations. The argument extendsthe finite-interval and qualitative-tail results of Papers 134CR–134CU; it does not present theirearlier order-two reductions as new. The certificate uses real arithmetic and the Python standardlibrary, includes analytic integration and coefficient tails, and is independently replayed with exactrational cover comparisons. The full bounded screening operator of revised Paper 134BC still requiresarbitrary-order positivity. An exact analytic control demonstrates why the new first-order matrixinequality cannot be promoted to that conclusion by itself.