William Troiani - Gentzen-Mints-Zucker Duality: Rethinking Curry-Howard

See this 2020 paper Gentzen-Mints-Zucker duality by Daniel Murfet and William Troiani:

The Curry-Howard correspondence is often described as relating proofs (in intutionistic natural deduction) to programs (terms in simply-typed lambda calculus). However this narrative is hardly a perfect fit, due to the computational content of cut-elimination and the logical origins of lambda calculus. We revisit Howard's work and interpret it as an isomorphism between a category of proofs in intuitionistic sequent calculus and a category of terms in simply-typed lambda calculus. In our telling of the story the fundamental duality is not between proofs and programs but between local (sequent calculus) and global (lambda calculus or natural deduction) points of view on a common logico-computational mathematical structure. 


See also Program Synthesis and On Getting Machines to do Stuff and then Jean-Yves Girard - Lectures at IHP thematic trimester : Semantics of proofs and certified mathematics. 

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But I am not sure about the significance of the "computational content" of a sequent calculus proof. One gets a similar blow-up in translating from Lambda terms to combinatory logic terms in a Hilbert-style proof system, but maybe the same global/local view can explain some of the differences in behaviour of terms in combinatory logic and lambda terms.


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