A reduced original normal pair for a least-weight essential disk has no spherical summand

Suppose $G=S$ is a sphere and put $Y=H$. Their remaining intersection is nonempty, since otherwise their full resolution would be a disconnected union. It consists only of circles, because $S$ is closed. Each circle separates $S$, and therefore is nonseparating in $Y$ by [F1]. Each is also two-sided in $Y$, using only the coherent switching neighborhoods. Indeed, choose one of its two disk sides on $S$ by [F3]. In a transverse cross-section there are two $S$ rays and two $Y$ rays. The retained regular pairing pairs this chosen $S$ ray with one $Y$ ray. Transport it around the circle. The $S$ ray returns to itself, as the chosen disk side is global on $S$, and the regular pairing returns to itself by [F1]. Consequently the selected $Y$ ray returns to itself. The local rectangular collars on $Y$ thus glue without reversal and give an annular collar of the circle in $Y$. No ambient sphere collar was used.

The full resolution has boundary exactly $\mathrm{\partial }Y$ and Euler characteristic $2+\chi (Y)=1$, since interior-circle cutting and regluing preserves Euler characteristic. Hence $Y$ has one essential boundary circle $C$ and $\chi (Y)=-1$. Cut it along one intersection circle $c_0$, using its annular collar from step 1.1. It stays connected and has Euler characteristic $-1$ and three boundary circles, so is a pair of pants by [F2]. Any other intersection circle is inessential or boundary-parallel there by [F3]. It cannot be inessential or parallel to $C$, since then it would separate $Y$ after regluing the copies of $c_0$. Thus every other circle is parallel to one of those two copies. Cutting along the whole family consequently yields a single pair of pants $K$ containing $C$ and otherwise annuli. This follows by ordering the finitely many nested parallel circles at the two ends of the cut pair of pants.

If the intersection family consists of $c_0$ alone, cutting $S$ gives two disks $E_1,E_2$, and cutting $Y$ gives $K$. Either local switch along $c_0$ attaches one of these disks to each of the two cut boundary circles of $K$. Both choices therefore produce one connected disk with boundary $C$. With all double curves resolved its image is embedded. Choose the irregular switch. Its disk is essential, so [F4] produces a normal essential disk of weight less than $w(F)$, contradicting minimality.

If there are at least two intersection circles, the other two boundary circles $a,b$ of $K$ are distinct members of the family. On $S$ take the two disjoint disks $E_a,E_b$ bounded by them whose interiors avoid the other boundary circle; [F3] gives those disks and a remaining nonempty annulus. Resolve just $a,b$, choosing at each the pairing that joins the $K$ side to the corresponding cap side, and leave all other curves unswitched. Then $K\cup E_a\cup E_b$, rounded only in those switching neighborhoods, is one parametrizing disk component $P$ with boundary $C$. Its image is embedded: the cap interiors are disjoint subsets of $S$, the interior of $K$ misses $S$, and each local switch joins adjacent rays of the two transverse sheets by a disjoint strip. Other intersection circles inside a cap meet only portions of $Y$ outside $K$; their unswitched crossings do not join parametrizing components and do not give self-intersections of $P$. Thus no global side of $S$ or displacement of entire caps is needed.

The remaining spherical annulus supplies points outside $P$, so the normalization in step 3.2 is disconnected. If both selected pairings were regular, this would contradict the reduced property [F1]. At least one is irregular. Apply [F4] to its embedded essential disk component $P$. Previous regular switches and these two switches miss the original edges, and the total original weight is $w(S)+w(Y)=w(F)$. The resulting essential normal disk has weight less than $w(F)$, a contradiction. Together with step 3.1 this excludes a spherical $G$, and exchanging names excludes a spherical $H$. Every construction used the original diagram and its existing neighborhoods.