  We present Hypersequent Classical Processes (HCP), a revised interpretation
of the "Proofs as Processes" correspondence between linear logic and the
{\pi}-calculus initially proposed by Abramsky [1994], and later developed by
Bellin and Scott [1994], Caires and Pfenning [2010], and Wadler [2014], among
others. HCP mends the discrepancies between linear logic and the syntax and
observable semantics of parallel composition in the {\pi}-calculus, by
conservatively extending linear logic to hyperenvironments (collections of
environments, inspired by the hypersequents by Avron [1991]). Separation of
environments in hyperenvironments is internalised by $\otimes$ and corresponds
to parallel process behaviour. Thanks to this property, for the first time we
are able to extract a labelled transition system (lts) semantics from proof
rewritings. Leveraging the information on parallelism at the level of types, we
obtain a logical reconstruction of the delayed actions that Merro and Sangiorgi
[2004] formulated to model non-blocking I/O in the {\pi}-calculus. We define a
denotational semantics for processes based on Brzozowski derivatives, and
uncover that non-interference in HCP corresponds to Fubini's theorem of double
antiderivation. Having an lts allows us to validate HCP using the standard
toolbox of behavioural theory. We instantiate bisimilarity and barbed
congruence for HCP, and obtain a full abstraction result: bisimilarity,
denotational equivalence, and barbed congruence coincide.
