= Solution
Under <proportional hazards>, each coefficient is constant over event time. Examine the <Schoenfeld residuals>, preferably <scaled Schoenfeld residuals>, against time or a prespecified transformation such as log time, ranks or a fitted survival-time transformation. Plot a smooth trend and its uncertainty for each <covariate>. Under the working model there should be no systematic residual-time association; a persistent trend suggests a time-varying coefficient. In a coefficient-scale plot that adds $\widehat\beta$ to the scaled residuals, the null trend is horizontal at the fitted coefficient.
A formal <proportional hazards assumption test> can add a time interaction, for example $\beta_r(t)=\beta_r+\theta_r g(t)$, and test $\theta_r=0$ using a score test with information adjusted for the fitted constant coefficients. The score contributions involve the Schoenfeld functions weighted by $g(t_j)$. One can test each <covariate> and use a joint global test for a vector of time-interaction coefficients. Such an association would be possible despite the unweighted residual sum being zero, so that fitted identity does not establish <proportional hazards>.
\b[A systematic nonhorizontal coefficient trend or significant residual-time association is evidence against a constant <hazard ratio>.] Failure to reject is compatible with the assumption, not proof of it. These diagnostics target the proportionality restriction; they do not by themselves verify <covariate> functional form or independence of the censoring mechanism.
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