Reading the figure
A finite compact packet and a noncompact central-character and root choice feed one actual candidate with an arithmetic recovery witness. Completion is additional construction data.
Mathematical labels
\(d_\rho\)
\(R(d_\rho,\pi)\)
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
Finite compact packet dρ Noncompact completion central character + root One actual candidate R(dρ, π)
Original figure caption
This diagram starts with an already constructed compatible arithmetic host.
The full character is trivial on \(U_F^1\). Residue data and the noncompact scalar can then match a prescribed smooth central quasicharacter \(\omega\) trivial on \(U_F^1\), when completion choices are free or already compatible.
For \(F={\mathbb F}_2((t))\), the residue unit group is trivial, so these covered central characters are unramified. An \(N\)th root in \({\mathbb C}^\times\) fixes the action required by \(\omega(t)\). All choices are recorded before invoking H5.
Scope
An incompatible immutably supplied completion is not silently replaced. Nor is an arbitrary one-factor wild determinant assumed trivial. The source central character must belong to the covered class.
Reading the diagram
A root choice completes an existing candidate. It does not prove the general realization conjecture or the same candidate's pair-factor equalities.