Admissible Yang–Mills Fibration

Source result: GAT III, Proposition 4.1 and Theorem 4.2.
Status: Source-bound theorem in the public preprint.

Catalog summary

When certified physical continuation supplies coherent reindexing functors on a selected admissible branch, the realizations and their comparisons form a Grothendieck fibration over its refinement category. The lifting property packages continuation already established by the source theory.

What the theorem says

At each refinement stage there is a category of recognized physical realizations. To relate stages, one needs actual comparison operations that carry the declared physical data. On a branch where these operations are functors and compose coherently, the stagewise categories assemble into one category over the refinement system.

The resulting projection is a cloven Grothendieck fibration: an ordinary categorical fibration with specified cartesian lifts. “Cartesian” describes a universal factorization property, not a choice of spatial coordinates. This is standard category theory applied to the certified Yang–Mills continuation data.

Hypotheses and scope

Fix the admitted refinement category for a selected Cat-valued branch. For each stage , let be its category of recognized physical realizations. The branch must supply:

• For every admitted arrow , a certified contravariant reindexing functor .
• Invertible comparisons between iterated reindexing and reindexing along a composite, together with unit comparisons satisfying the pentagon and unit laws.
• The physical certification required by GAT III §3: each in-scope coordinate is carried with its declared preservation, controlled-defect, externalized, or out-of-scope status. The physical carrier, complete ground, sectors, forms, descent data, Ward structure, and provenance are separately accounted for.

These data define a contravariant pseudofunctor. A pseudofunctor composes up to specified coherent isomorphisms; it need not compose by literal equality.

Precise conclusion

The Grothendieck construction has objects . An arrow from to over consists of a fiber morphism . Its projection

is a cloven Grothendieck fibration. The chosen cartesian lift of at is

Why it works

The certified reindexing tells us what becomes at the source stage. Any comparison over factors through that reindexed object by a comparison within the source fiber. The coherence cells ensure that doing this through intermediate stages respects composition.

Example

For admitted refinements , a realization at can be reindexed as or as . The source-certified compositor relates these descriptions. The fibration records that relation together with the stagewise realizations; it does not silently identify the two descriptions by notation.

What the conclusion does not supply

The theorem does not create missing physical comparisons, assert reindexing on every conceivable refinement, or trivialize the represented coefficient data. Its Cat-valued hypothesis matters. The broader distributor-valued continuation of GAT VIII need not be representable by such ordinary reindexing functors.

Source

GAT III — Admissible Yang–Mills Fibrations and Thin Haag–Kastler Reconstruction, §§3–4, especially Definition 3.3, Proposition 4.1, and Theorem 4.2. Public version v1.0-rc1-webpub1.