Renormalization as Admissible Transport: Ground-Lineage Compatibility, Dirichlet Covariance, and Refinement Descent

Renormalization as Admissible Transport: Ground-Lineage Compatibility, Dirichlet Covariance, and Refinement Descent

J. Scott Little
Collection: Mathematical Physics · Interface Papers
Research status: Author manuscript · not peer reviewed
Source date: 7 September 2026
Website release candidate: v1.0-webpub1
Copyright: © 2026 J. Scott Little. All rights reserved.

The question

When does a coarse-graining map become a genuine, presentation-independent renormalization morphism on the represented carrier selected by the theory?

A coarse-graining map can be completely positive, state preserving, nuclear, compositional, and still be the wrong carrier map. Even after the carrier is identified, one still has to know whether the invariant analytic structure survives scale transport and whether two finite refinement presentations give the same arrow.

Main result

The paper separates those issues rather than hiding them inside one use of the word renormalization.

1. Ground-lineage compatibility (GLC) identifies observable coarse-graining with the selected carrier transport:

2. Dirichlet covariance certifies transport of the specified fixed sector:

3. Raw composition gives a functor on finite refinement presentations.

4. Common-refinement invariance removes presentation dependence and gives

The finite conditional-expectation model shows that all of these hypotheses can hold at once. Three counterexamples then show that the three structural requirements are genuinely separate.

Why this paper exists

The continuum limit is not being asked to decide which finite-scale carrier map was correct. The interface is settled before the endpoint is constructed. That is the precise sense in which the paper says that renormalization compatibility is a morphism problem before it is a limit problem.

What it does not claim

• It does not introduce a new renormalization-group algorithm.
• It does not claim novelty for conditional expectations, GNS carriers, Dirichlet forms, nuclearity, or functorial renormalization separately.
• It does not identify SNC with Dirichlet certification.
• It does not construct a continuum AQFT or prove the Haag–Kastler axioms.
• The word admissible is relative to the specified Dirichlet interface in this paper; it is not a claim of every possible physical admissibility condition.

Finite model

For nested finite von Neumann algebras with faithful trace, the trace-preserving conditional expectation implements the observable and carrier scale map simultaneously on the corresponding -spaces. Compatible depolarizing Dirichlet semigroups provide the certification law, while the tower property supplies raw composition and common-refinement independence.

Reading route

Start with Sections 2–3 for the exact interface. Sections 4–5 contain the functor and descent theorem. Section 6 is the quickest stress test because it gives one successful noncommutative model and three compact failures. Section 7 is the local-net extension; Section 8 states the scope boundaries.

Citation

J. Scott Little, Renormalization as Admissible Transport: Ground-Lineage Compatibility, Dirichlet Covariance, and Refinement Descent, author manuscript, v1.0-webpub1, 7 September 2026, J. Scott Little Research Archive.