Reading the figure
The upper arrow from supplied H1 through H4 data to an arithmetic witness is labelled Conjecture 66. The witness and H5 for that same candidate feed the lower conditional comparison. The diagram does not provide the missing general existence witness.
Mathematical labels
\(d_\rho\)
\(R(d_\rho,\pi)\)
\(\pi\simeq\mathrm{LLC}(\rho)\)
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
H1–H4 finite datum dρ Arithmetic witness R(dρ, π) H5 for this same candidate tested pair factors Conditional comparison π ≃ LLC(ρ) Conjecture 66
Original figure caption
The upper existence arrow is conjectural. The lower comparison follows conditionally.
Conjecture 66 (arithmetic packet realization). Fix the declared class \(\mathcal A\) of wild atoms and the paper's normalizations. For every \(\rho\in\mathcal A\) with supplied H1–H4 assembled finite datum \(d_\rho\), there is one actual admissible supercuspidal representation \(\pi\) of \({\mathrm{GL}}_{\dim\rho}(F)\) and an arithmetic realization witness \(R(d_\rho,\pi)\) recovering the packet and preserving its declared stabilizer action, central-extension and phase data, and compatible transports.
Here \(R\) includes actual arithmetic inducing data and explicit comparison and recovery. It is not merely the existence of abstract operators with the right commutators.
Scope
The conjecture is restricted to the declared admitted class. It does not assert that arbitrary coefficient targets, arbitrary partial hosts, or independently conflicting completion choices can all be extended.