  As a continuation of Parts I \cite{Part-1:2020}, II \cite{Part-2:2021}, III
\cite{Part-3:2022}, where ultra-compact wave functions were constructed for a
few low-lying states of He-like and Li-like sequences, the family of
spin-singlet $(1s\,ns)$ type excited states $n\,{}^1 S$ of the He-like sequence
is studied with an emphasis on the $n=3,4,5$: $3\,{}^1 S, 4\,{}^1 S, 5\,{}^1 S$
states, for nuclear charges $Z \leq 20$. Particular attention is given to
finding of critical charges $Z=Z_B$ at which the ultra-compact wave functions
lose their square-integrability.
  For each ${}^1 S$ state an ultra-compact, seven-parametric trial function is
constructed, which describes the domain of applicability of the
non-relativistic Quantum Mechanics of Coulomb Charges (QMCC) for the total
energies (4-5 significant digits (s.d.)) and reproduces 3 decimal digits (d.d.)
of the spin-singlet states $n\,{}^1 S$ of He-like ions (in the static
approximation with point-like, infinitely heavy nuclei) for $n=1,2,3,\ldots$
and any $Z \leq 20$\,. All energies are well described by second degree
polynomials in $Z$ (the Majorana formula).
  Critical charges $Z=Z_B^{(n)}$, where the ultra-compact trial function for
the $n^1 S, n=1,2,3,\ldots$ states loses its square-integrability, are
estimated: for all studied states $Z_B^{(n)}$ increases slowly with $n$; it
seems they lie in the interval $Z_B(n^1 S) \sim 0.90 - 0.95$, in particular,
with $Z_B^{(1)}=Z_B^{(2)}\,=\,0.904$, $Z_B^{(3)}=Z_B^{(4)}\,=\,0.928$,
$Z_B^{(5)}\ =\ 0.939$.
