Finite Field Realisation of the Classical Constants π and e

Over a finite prime shell \(\mathbb F_p\), \(p=4\kappa+1\), the roles of \(\pi\) and \(e\) are exact residues: the half-period \(\pi=2\kappa\), and the exponential marker \(e=g^{\lambda(i)}\). We determine the exact relation between these carriers and the classical values. Each classical value is the horizon readout of a chain of framed rationals \(n!/!n\) for \(e\); the Wallis, arcsin and Machin chains for \(\pi\); at IEEE-754 double precision the constants are the readouts of \(18!/!18\) and the Machin partial \(M_{10}\). Inside the shell the same chains carry exact residue lines that the readout deletes: the line of \(e\) is antiperiodic and 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 \(\pi^{-1}\equiv-2\). Angularly the constants are dual: \(\chi(-1)=e^{i\pi}\) holds exactly in every shell, while radian calibration of \(e\) is impossible in every shell and abundant across shells. \(\pi\) is structural, \(e\) statistical; transcendence belongs to the external completion, never to the shell element. All exact claims are machine-verified in integer and rational arithmetic.