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Source. Section 3.8, printed p. 11, physical p. 17 of the selected thesis.

Target and construction

Work on the $2\pmod4$ target in the first prime-$5$ input of the $4$-hole. All prime-$5$ packages below are restricted to this target. Begin with the four packages

$$1,\quad2,\quad4,\quad8^\uparrow. \tag{1}$$

Using their compatible prime-$5$ children and one $25^\uparrow$ creates five more packages. The source reports that the sixth regular input of the earlier prime-$11$ package is already covered on this target, so that these nine packages fill its other nine inputs and create one complete $11^\uparrow$. Under the concrete Owens prime-$5$ permutation, however, that sixth-input $x$ mask is not by itself a complete cover of the whole $2\pmod4$ target used here. The claimed completion therefore depends on the common precoverage/allocation certificate. Numerically, it raises the source's package count to ten.

Partition those ten packages into five ordered pairs and place each pair in the two required children of a $3^\uparrow$. This creates five complete prime-$3$ packages, for a total of fifteen. Twelve of the available packages fill a $13^\uparrow$, and sixteen fill a $17^\uparrow$, raising the total to seventeen.

On this branch, the third input of each $7^\uparrow$ is already covered except for one prime-$3$ child. Three groups of the available packages therefore create three complete prime-$7$ packages. One of them must have the source's displayed form

$$7^\uparrow(1,2,3(x,1,x),4,8^\uparrow,3^\uparrow(2,4)). \tag{2}$$

The pool now has twenty packages. Removing the atomic packages $1$ and $2$, whose prospective outer moduli $19$ and $38$ are below $42$, leaves exactly eighteen regular inputs for a $19^\uparrow$.

The count is therefore

$$4+5+1+5+1+1+3-2=18=19-1. \tag{3}$$

Every completion in this argument is relative to the displayed target and the earlier black or gray children. Five of the eighteen packages cover the whole $4$-hole rather than only its first prime-$5$ input. Consequently a later prime-$19$ arrow on that larger target needs only thirteen new inputs.

Exact scope

Formula (3), the branch capacities, and the necessary special package (2) give a complete package-count argument at the level printed by Owens. The relative-coverage claim is conditional at the reported prime-$11$ mask above, and the thesis does not list which earlier package occupies each input of the five prime-$3$ arrows, the prime-$13$ and prime-$17$ arrows, or the two prime-$7$ arrows other than (2). Counts alone do not prove that all resulting unbounded prime-exponent signatures are distinct. Thus this page does not promote the prime-$19$ pool to an independently certified ordered modulus list. The missing precoverage and ordering are included in the common allocation interface on the construction ledger.

In the acknowledgments (physical p. 4), Owens thanks the thesis adviser, Pace Nielsen, for suggestions, especially on the prime-$19$ step, and credits that step with reducing the number of primes the construction needs. That historical statement is reported as Owens's account, not as a priority claim independently established here.