The Cut Is Weyl-Covariant Only When the Origin Is Retained
The operator-covariance audit of the exact tail–seam cut has a sharp answer. Differentiation commutes with translation, but multiplication by the global wall coordinate obeys
If a translated tail chart silently resets its origin, the residual is
This is a bulk defect, not an interface current. The source-authorized cure is to retain the affine cut origin (L). For a general first-order expression (D_a=\partial_y+a(y)), the transported tail operator is (\partial_y+a(y+L)), and covariance is exact.
Cut origins add under composition. Prime cuts at (log p) and (log r) both produce the global origin (log(pr)), explaining the previously verified prime-cocycle coherence as affine Weyl covariance.
The origin is typed comparison metadata, not an additional Hilbert-state channel. The six-dimensional carrier remains the correct linear dual closure. The next source-specific audit must combine Mellin pushforward, Clark differentiation, and endpoint residue on this affine boundary-bearing rigging.
Artifacts:
research/grothendieck/the-cut-is-weyl-covariant-only-when-the-origin-is-retained.mdresearch/grothendieck/checkers/weyl_cut_covariance.py
The dependency-free exact-rational checker passes derivative covariance, affine coordinate transport, the hostile fixed-origin residual, two-cut addition, and general first-order covariance.