= Solution
Let $T_{v',v''}=T_e+G'(v')-G''(v'')$. For a fixed row, moving one column to the right therefore gives
$$
T_{v',v''+1}-T_{v',v''}=-\big[G''(v''+1)-G''(v'')\big]
=-\widetilde\omega_e''+2\widetilde\omega_e''x_e''(v''+1).
$$
Thus \b[adjacent-column entries decrease by the lower-state vibrational spacing], independent of the upper-state row in the ideal model. If the positive difference is instead defined as left minus right, it is $\widetilde\omega_e''-2\widetilde\omega_e''x_e''(v''+1)$. Stating the subtraction direction prevents an artificial negative vibrational <frequency>.
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