This volume develops the computational compatibility layer of the Re-Phase framework.It introduces explicit computational queries for feasibility, witnesses, recoverability, obstruction, path feasibility, projection membership, projection gaps, and approximation certificates. The guiding principle is that computability is not a license to forget structure: computation and approximation must be performed only after a representation-sound object has been fixed.The volume separates projection error from approximation error. Projection error arises when required structural data are forgotten before compatibility is tested. Approximation error arises when sets are enlarged or reduced under a stated approximation direction. These mechanisms are illustrated by finite enumeration witnesses, including context search, forward–backward path computation, latent-mode false gluing, shared-parameter layered false gluing, and positive-part box approximation.No large-scale solver, neural-network training method, artificial-intelligence method, stochastic filtering algorithm, optimal-control method, or implementation-specific algorithm is developed here. The role of this volume is to define the computational grammar and certificate discipline for later Re-Phase layers.
Takashi Ito (Thu,) studied this question.