Arithmetic Positivity and Absolute Hodge Classes: Why HR Positivity Does Not Strengthen Deligne's Absoluteness

We introduce arithmetic positivity conditions (AP1–AP3) inspired by the Hodge–Riemann bilinear relations and prove that APp(X) = AbsHodgep(X): the AP conditions are automatically satisfied by all absolute Hodge classes. This is a negative result — the Hodge–Riemann framework does not strengthen Deligne's absoluteness criterion. We identify three directions for stronger (non-linear) positivity conditions: degenerating polarisations, motivic convolution, and prismatic cohomology. Numerical verification of the generalised HR spectrum confirms AP1 for 11 test cases. Published in English and German versions. Part of the Functional Stability Theory (FST) programme.