L-space and S-space existence is asymmetric

In the ZFC branch, apply [F1]. It gives the required L-space without CH, $V=L$, PFA, or a large cardinal.

In the ZFC+CH branch, [F2] gives a strong S-space and therefore an S-space. This conclusion uses CH and is not transferred to the other branches.

In the ZFC+$V=L$ branch, [F3] gives $\mathrm{♢}$; since this branch includes AC, [F4] gives CH; and then [F2] gives a strong S-space.

In the ZFC+PFA branch, [F5] says that no S-space exists. This is incompatible with the conclusions of steps 1.2 and 1.3, so those hypotheses are not conjoined.

For the final metatheoretic qualification, assume $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C}+\text{a supercompact})$. Then [F6] gives $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C}+\text{no S-spaces})$. If ZFC proved that an S-space exists, appending that fixed proof to the latter theory would refute it, contrary to its consistency. Thus, under the displayed source-consistency assumption, S-space existence is not a ZFC theorem.

Steps 1.1--2.1 establish exactly the three branchwise assertions and the qualified asymmetry. Empty or singleton spaces do not create an exception: [F1] has underlying set ${\omega }_1$, [F2] likewise produces a nonempty strong S-space, and [F5] applies to the complete definition. The first power in [F2] supplies the ordinary S-space; the zeroth power is excluded by definition. All uses of AC are declared in [F7], and no converse consistency implication is claimed.