= Solution
Under the <Riemann hypothesis>, every $\beta=1/2$, so for $\sigma>1/2$ the <absolutely convergent> formula gives
$$
\frac{\partial}{\partial\sigma}\log|\xi(\sigma+it)|
=\sum_\rho\frac{\sigma-1/2}{(\sigma-1/2)^2+(t-\gamma)^2}>0.
$$
The zero set is nonempty: otherwise <Hadamard factorization> would make $\xi$ an exponential of a linear <polynomial>, and its functional symmetry would force it to be constant, contrary to gamma growth on the positive real axis. Thus the inequality is strict in the open right half-plane. <Continuity> at $\sigma=1/2$, including at boundary zeros, proves the claimed increasing modulus on the closed half-line.
Conversely, suppose the modulus is nondecreasing for every fixed $t$. If a zero $\beta+i\gamma$ had $\beta>1/2$, then nonnegativity and monotonicity would force $|\xi(\sigma+i\gamma)|=0$ throughout $1/2\le\sigma\le\beta$. The <identity theorem> would make $\xi$ identically zero, a contradiction. A zero left of the line reflects to one right of the line by the <functional equation> and <complex conjugation> symmetry. Hence every zero lies on the <critical line>. This proves the <xi modulus criterion for the Riemann hypothesis>. The two following roman headers refer to supplied asymptotic assumptions, not further questions, and require no Solution sections.
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