Suppose \(\pi\) is an actual admissible supercuspidal representation of the correct rank and \(R(d_\rho,\pi)\) is supplied. If H5 gives equality of its required pair local factors with those of \({\operatorname{LLC}}(\rho)\) on the stipulated characterizing twist family, with the fixed normalizations, then
The proof applies the supplied local-factor characterization to these two actual representations. It does not need existence for other diagrams or H5 for every alternative completion.
There is a logical difference between
The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of \(d\) unproved.
Reading the diagram
H5 remains a conditional premise about the same independently constructed candidate. It is not renamed as an unconditional factor-comparison theorem.