Establish the computational interpretation of OHTT gaps

Develop a computational interpretation for Open Horn Type Theory by giving operational meaning to gap witnesses (e.g., as certificates, effects, or constraints), defining evaluation rules that respect the Exclusion Law, and relating gap constructs to programming-language mechanisms so that the proof-relevant gap structure integrates with the coherent fragment’s computational content.

Background

While MLTT/HoTT have well-studied computational interpretations, OHTT introduces proof-relevant gap witnesses as primitive objects. The authors suggest several possible computational roles for gaps—certificates, exceptions/effects, constraints—but do not develop them.

They explicitly state that the computational meaning of gaps and their integration into a programming-language setting remain to be worked out.