Arbitrary-Cover Extension of Selected Cofinal Transport

Existing page: Arbitrary-Cover Extension of Selected Cofinal Transport
Status: Open research problem; the larger cover system still requires precise specification.
Suggested display title: Extending Admissible Transport Beyond Selected Covers

Catalog summary

GAT VIII constructs physical comparison data on a selected system of admissible covers and refinements. Determine when that construction extends to a larger, precisely specified cover system while preserving its physical data and compatibility relations.

What the problem concerns

A cover describes a region through overlapping smaller regions. A refinement replaces those pieces with finer ones. Changing the cover changes the local description, so the physical data on the pieces and their overlaps must still be compared consistently.

GAT VIII treats these passages through a comparison distributor: for a permitted change of description and a pair of physical realizations, it records the certified comparisons between them and the equivalences between those comparisons. A single preferred transported object is not required.

The construction uses a selected source-free system at positive thickness. Here, “selected” identifies the demands and comparison operations supplied by the source construction. “Cofinal” concerns that selected demand system. It does not mean that every open cover or every geometric refinement is covered by the theorem.

What is already established

Theorem 8.1 assembles the selected comparisons into a normal lax homomorphism valued in groupoid-enriched profunctors. Its hypotheses control certificate refinement, removal of auxiliary presentation data, and the compatibility of the declared comparison cells.

Appendix A explains the common-refinement mechanism on chosen finite admissible protected covers. Certificates for a whole finite comparison diagram can be brought to a common refinement, where the prescribed composition and coherence relations can be checked. This removes dependence on the choice of certificate without removing substantive physical data.

What remains open

Specify a larger system of covers, intersections, refinements, and comparison cells, and determine geometric and analytic conditions under which the selected comparison distributor extends to that system. On the original selected system, the extension must recover the existing construction up to the appropriate coherent equivalence.

The extension must establish:

1. Which new comparisons exist. A geometric refinement does not itself prove that a given physical realization has a certified continuation. The domain on which physical lifting is possible must be identified.
2. Which finite diagrams admit common certification. Certifying each arrow separately is insufficient. Their composites, overlap relations, and declared higher compatibility cells must be handled together.
3. What the comparisons retain. The extension must preserve the declared carrier and algebra types, projections, Hamiltonian forms and domains, cohomological and selector data, Ward and net data, provenance, and comparison defects. An absent comparison remains absent.
4. Why auxiliary choices remain auxiliary. Enlarging the cover system must not make certificate or presentation choices into new physical distinctions without accounting for that change.

Why the existing result does not settle it

The selected construction is closed under its admitted comparison operations. A newly allowed cover can introduce intersections or mixed comparison diagrams for which that closure and common-certification argument has not been established. Cofinality within the selected system does not supply the missing comparisons or their coherence on a larger system.

Example

Suppose two overlapping descriptions of a region can each be refined into pieces handled by the existing construction. Their individual refinements may be valid, yet the comparison routes on their overlap still require a joint certificate. To extend the theory, one must show that those routes fit into a common certified diagram and satisfy the prescribed relations. Any coefficient reassociation or comparison defect must remain visible in that calculation.

This example illustrates the proof obligation. It is not a counterexample to the existence of an extension.

What would count as a resolution

A positive result would define the enlarged cover system and give verifiable conditions under which the full comparison construction extends to it. Simply assuming all the desired extension data already exist would not identify the missing geometric or analytic mechanism.

A negative result for a specified cover class would exhibit a precise obstruction: for example, failure of physical liftability, common certification, or compatibility of the required comparison cells. It would establish a limitation on that class; additional restrictions might still permit an extension.

Neither outcome automatically settles representability, preservation of a uniform positive coercive floor, or a zero-thickness limit.

Source

GAT VIII — Indexed Admissible Distributors and Selected Cofinal Čech Transport in Semisimple Yang–Mills Theory, public version v0.4-webpub1: §§2–8, especially Theorem 8.1; Appendix A.1–A.3; §19. The extension question is motivated by the manuscript’s explicit scope boundary. The checklist above makes that question more explicit; it is not a new theorem in the source.