A supercompact gives the relative consistency of no S-spaces

Let $T_{\mathrm{P}}$ be ZFC+PFA and let $T_{\mathrm{N}}$ be ZFC plus the sentence that there are no S-spaces. Expand the fixed finite mathematical derivation underlying F2 in the chosen calculus: expand its displayed definitions and abbreviations, and replace every invoked proved premise by its fixed derivation. F3 shows that the resulting finite concatenation is a $T_{\mathrm{P}}$-derivation of the extra axiom of $T_{\mathrm{N}}$. Fix its standard code $e$. The checker from F4 accepts this particular numeral, and F5 supplies a finite PA proof of that positive closed checker instance. Thus both $e$ and PA's verification of $e$ are constructed here; neither is attributed to F2's interface.

Given a purported $T_{\mathrm{N}}$-refutation $p$, use F4 to check it and to replace each use of the no-S-space axiom by a renamed copy of $e$. Retain the ZFC axiom lines and append the same logical inferences. This yields a $T_{\mathrm{P}}$-refutation $r(p)$. The construction is a bounded syntactic substitution into the finite code $p$, with a fixed default on malformed inputs, so the arithmetic base combines the fixed positive checker proof from step 1.1 with induction on the decoded line list to verify that $r$ is total and that ${\mathrm{Prf}}_{T_{\mathrm{N}}}(p,<mi\; mathvariant="normal">⌜</mi>\mathrm{\perp }<mi\; mathvariant="normal">⌝</mi>)⟶{\mathrm{Prf}}_{T_{\mathrm{P}}}(r(p),<mi\; mathvariant="normal">⌜</mi>\mathrm{\perp }<mi\; mathvariant="normal">⌝</mi>)$.

Apply [F6] to the reduction in step 2.1. The arithmetic base proves $\mathrm{Con}(T_{\mathrm{P}})\to \mathrm{Con}(T_{\mathrm{N}})$. Compose this implication with [F1] to obtain the displayed result.

The source theory's large-cardinal clause is exactly the one fixed by [F7]. The argument only transforms finite proof codes: it does not choose a generic filter, construct a model of the whole source theory, or infer a transitive model from its consistency. Empty spaces and singleton spaces need no special consistency argument—[F2]'s no-S-space theorem already treats the complete definition—and no converse implication is asserted.