= Solution
Averaging the left-minus-right differences across the three rows gives lower-state spacings of about $1613.0$ and $1591.7\ \mathrm{cm^{-1}}$. Their difference is $2\widetilde\omega_e''x_e''$, so
$$
\boxed{\widetilde\omega_e''x_e''\simeq10.7\ \mathrm{cm^{-1}},\qquad
\widetilde\omega_e''\simeq1634\ \mathrm{cm^{-1}}.}
$$
Similarly, averaging successive down-column differences across the three columns gives upper-state spacings of about $1750.3$ and $1710.7\ \mathrm{cm^{-1}}$. Hence
$$
\boxed{\widetilde\omega_e'x_e'\simeq19.8\ \mathrm{cm^{-1}},\qquad
\widetilde\omega_e'\simeq1790\ \mathrm{cm^{-1}}.}
$$
Using the first row and first column alone would give approximately $(1637,11)$ and $(1790,20.5)\ \mathrm{cm^{-1}}$, respectively. The small inconsistency between nominally identical spacings is already present in the entries; it should be treated as limited data/model precision rather than fitted by assigning different lower-state constants to different rows. Values near \b[$(1.63\times10^3,11)$ for the lower state and $(1.79\times10^3,20)$ for the upper state] express the supported precision.
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