Reading the figure
Four boxes distinguish Weil atoms, monodromy chains, finite packet comparison, and actual arithmetic realization. The horizontal arrow from finite comparison to arithmetic realization is marked conjectural. Finite comparison does not create the arithmetic witness.
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
Weil atoms Monodromy chains Finite packet comparison Actual arithmetic realization Conjectural realization The finite comparison does not create the arithmetic witness.
Original figure caption
An atom, its finite packet data, and its arithmetic realization have different roles.
A Weil atom is an irreducible Weil representation. A finite packet records selected quotient spaces, projectors, central extensions, phases, and transports. An arithmetic realization is an actual admissible representation of the required general linear group together with a comparison and recovery of that packet.
Monodromy organizes an atom into a finite Weil–Deligne chain. It does not supply an atom that is missing. Likewise a finite Heisenberg comparison identifies already supplied finite representations, rather than automatically constructing a representation of \({\mathrm{GL}}_n(K)\).
Reading the diagram
The final existence arrow is the content of Conjecture 66. Finite comparison, arithmetic construction, and local-factor characterization are separate steps.