Finite Field Realisation of the Classical Constants π and e

We formulate the structural equivalents of the classical constants \( \pi \) and \( e \) as exact residues over finite fields: the half-period \( \piP \), \( \gen^{\piP}=-1 \), and the exponential unit \( \eP=\gen^{\lambda(\im)} \), where \( \gen \) is a primitive root. We determine the exact relation between these carriers and their corresponding classical values, as the horizon readouts of a chain of framed rationals (\( n!/\dr{n} \) for \( e \); the Wallis, arcsin and Machin chains for \( \pi \)); at IEEE-754 double precision the constants are the readouts of \( 18!/\dr{18} \) and the Machin partial \( M_{10} \). Inside the shell the same chains carry exact residue lines that the readout deletes: the derangement line of \( e \) is antiperiodic and its residue line terminates on Kurepa's left factorial, universal existence being equivalent to Kurepa's hypothesis; the line of \( \pi \) is legible exactly up to the angular address of \( -1 \) and terminates on the calibration face \( \piA^{-1}\equiv-2 \). Angularly the constants are dual: \( \chi(-1)=e^{\imR\pi} \) holds exactly in every shell, while radian calibration of \( \eP \) is impossible in every shell and abundant across shells. \( \pi \) is a structural constant and \( e \) a statistical one; transcendence belongs to the external completion and not to any element of the shell. All major claims are machine-verified in integer arithmetic and formalised in Lean 4.