Two Information Losses in Weil Positivity

This research letter studies information loss in proof architectures for the Riemann Hypothesis based on Weil positivity. Rather than proposing a new RH-equivalent criterion, it separates two distinct reductions of arithmetic information: **phase uniformization**, which replaces the actual arithmetic phase orbit by uniform control over the full admissible phase torus, and **positive-part majorization**, which discards signed cancellation. The resulting framework distinguishes three proof architectures: phase-uniform positive-part certificates, exact-phase positive-part certificates, and true signed positivity. Within this framework, the note records two exact consequences. First, for a real measurable symbol \(d\) bounded from below, the positive-part replacement satisfies the sharp quadratic-form inequality \|ΠₐM(4a-d)₊Πₐ\|-4a≥-λ, where \(λ\) is the bottom of the corresponding signed quadratic form on \(PW_a\). This makes precise the intrinsic loss caused by discarding signed cancellation. Second, using finite Euler-factor modification of Beurling generalized prime systems of Diamond--Montgomery--Vorhauer type, the note shows that any prescribed finite set of classical prime Euler data can be matched while retaining failure of the corresponding RH analogue. Thus, within the Beurling multiplicative comparison class, finite multiplicative prime data alone cannot force RH-type positivity. The conclusions are deliberately limited. They do not imply any obstruction to the Weil criterion itself, do not address provability of RH in ZFC, and do not exclude methods using the actual arithmetic phase orbit, infinitely many but compressed phase data, direct signed-operator analysis, or arithmetic information tied to the additive structure of the integers.