Validate the conjectural description of prismatic F-gauges by Dodd’s F-gauges

Determine whether the theory of F-gauges introduced by Dodd provides a down-to-earth description of prismatic F-gauges over an equicharacteristic-p local field, and hence the corresponding syntomic-cohomological framework.

Background

The paper works with Dodd’s category of F-gauges as a concrete, differential-operator-theoretic framework for defining syntomic cohomology over a local field of characteristic p. The broader intended coefficient category consists of prismatic F-gauges arising from syntomification, but the relationship between the two descriptions is not established in the paper.

The authors explicitly describe Dodd’s formulation as conjectural. Establishing this comparison would justify interpreting the paper’s local Tate duality theorem for Dodd-style F-gauges as a result about prismatic syntomic cohomology, particularly for coefficients of geometric origin.