Cosine-kernel representations and a conditional criterion for the Riemann Hypothesis

All steps through the cosine-transform representation are proved in full detail by classical means (Poisson summation, the theta modular identity, and a Mellin-transform argument). The zero-distribution conclusion is formulated as a kernel-class criterion: it holds whenever the kernel belongs to any class whose cosine transform is known to lie in the Laguerre-P'olya class. We establish evenness, reality, rapid decay, and large-|x| positivity for 𝚽.