Normal Deformation as Constrained Refoliation

Description Normal Deformation as Constrained Refoliation is the third paper in the Wheeler–DeWitt ARC. It establishes the normal-deformation and refoliation structure required for the representation-stable canonical gravity realization obtained through WDW‑1 and WDW‑2. Its central question is: When an authenticated Einstein–source realization is presented on different admissible hypersurfaces, under what conditions does normal deformation preserve one constrained physical realization rather than generate a slicing-dependent replacement? The manuscript treats normal deformation as a constrained refoliation problem. Building on the connection-typed canonical projection of WDW‑1 and the proper-redescription groupoid descent of WDW‑2, it develops the lapse-directed normal and shift-directed tangential deformation roles together with the constraint relations required for their lawful composition. Its central passage is: [ \text{representation-stable canonical gravity realization} \longrightarrow \text{constrained normal deformation} \longrightarrow \text{refoliation-faithful canonical continuation}. ] The result distinguishes a genuine alteration of the realized Einstein–source structure from a lawful change of hypersurface presentation. Foliation, lapse, shift, canonical chart, and local representative may organize a presentation, but they do not acquire primitive authority to alter the inherited standing-bearing content, source attachment, boundary-standing data, or canonical constraint roles. WDW‑3 is therefore not a fixed-slice construction and does not make any preferred foliation physically fundamental. Its load-bearing contribution is to establish a controlled refoliation structure on the globally descended canonical realization, so that subsequent gravity–matter and quantum-constraint stages inherit lawful continuation rather than an uncontrolled collection of slice-labelled representatives. The paper does not yet integrate classical matter occupancy, construct the configuration-space Wheeler–DeWitt candidate family, establish quantum constraint survival, or provide final physical realization. Those are obligations of the later WDW papers.