Renormalization Curvature

Meaning

Renormalization curvature measures the failure of refinement and charged passage transport to commute on the selected branch.

For horizontal passage and vertical refinement , the elementary curvature is

It compares two routes around one refinement–passage square. The routes need not agree exactly. In a corridor containing order passages, even a small defect at each square can accumulate to order . The quantity needed by the confinement theorem is therefore not exact flatness and not a bounded total error. It is vanishing curvature density in the prescribed refinement and large-separation limit.

Example (illustrative)

One may first move a charged datum from one separator to the next and then refine it, or refine first and then move it. Renormalization curvature measures the difference between those represented outcomes.

Scope and limits

Renormalization curvature is not the local passage energy. The Dirichlet pencil already owns that positive spectral fact. Curvature controls whether the fact survives scale comparison. The word “renormalization” here does not by itself define a beta function, a running coupling, or perturbative renormalization-group flow.

Source

Phenomenological Implications I, §9, especially equations (36)–(38) and Remark 9.2.

Read the source manuscript.