An Arithmetic Resolvent and Heat Kernel for the Riemann Xi Function in the Variable \(s(1-s)\)

We descend the completed Riemann Xi function through the quotient map s ↦ s(1-s) and obtain an entire function 𝓓 of order 1/2 and genus zero. Its zeros are the functional-equation orbit values ρ(1-ρ), and the Riemann hypothesis is equivalent to their support on [1/4, ∞). The logarithmic derivative on the negative axis yields an arithmetic resolvent that admits both an explicit prime-Gamma expression and a zero-orbit Stieltjes expansion. We then construct an arithmetic heat kernel whose prime term is a Gaussian in log n and prove an exact Laplace transform identity linking the heat kernel to the resolvent. Finally, we identify the same heat kernel with the normally convergent zero-orbit heat sum. All interchanges of series and integrals are justified without assuming the Riemann hypothesis. The paper packages classical ingredients into a single quotient-variable resolvent/heat formalism.