canonicalRecognitionMesh_side

This module builds the Recognition gate of the 4D continuum-closure campaign: a canonical mesh carrier on the periodic Freudenthal 4-torus, together with a value-level action whose amplitude Hessian is the geometric Option-C midpoint Bloch symbol on the same torus family.

The carrier is packaged as RecognitionFreudenthalMesh4D. The canonical family is the one-parameter construction that sends a natural index $j$ to the mesh structure with that index as its sole field. The side length is the discrete linear size of that mesh (number of cells along an edge of the 4-torus).

Preferred limit shape in the module is amplitude Hessian at fixed mesh, then $N\to\infty$. Arbitrary test-variation pullbacks are excluded by the preflight; the module does not consume the Exact-J refinement family as a continuum premise.

One-line term proof by rfl. The canonical mesh at $j$ is defined by the structure constructor with field value $j$, and the side projection on that structure is definitionally $j+3$, so the equality is by reflexivity of definitional equality. No lemmas are invoked.

Pins the discrete size of the canonical Recognition mesh used throughout the exact-J / Option-C midpoint Bloch bridge. Downstream mesh objects in the same module (mesh wave covectors, true-weight Regge quadratic Hessian on edge classes, exact-J action on the mesh, and the equality of that action's amplitude Hessian with the true-Regge Hessian) all sit on this carrier and inherit the side formula.

In the broader QG campaign this is bookkeeping for the Recognition gate: it does not itself close the continuum limit or flip gap_action_recovery, but it fixes the $N$-scale that later theorems compose with the discrete torus bridge and the exact midpoint $m^2$ TT / gauge faces to reach the scale-explicit Option-C Einstein-Hilbert face. No parent theorem currently lists this declaration as a direct edge; it is infrastructure for the mesh section that follows.