Formal meta-conjecture for Picard–Fuchs equations and integrality

Determine that, for a Picard–Fuchs equation (E,∇) over a finitely generated Z-algebra R⊂C, if (E,∇) admits an (ω(p))-integral isomonodromic deformation over the formal completion \widehat{S} at a point s, then the formal isomonodromic deformation of (E,∇) to \widehat{S}_K (the completion over the fraction field K of R) satisfies any property P conjecturally equivalent to geometric origin.

Background

This formal version ties the authors’ integrality criterion to motivic predictions in the formal neighborhood, asserting that arithmetic integrality along \widehat{S} forces motivic-type properties on the formal isomonodromic deformation.

It is intended to bridge the arithmetic criteria developed in the paper with broader motivic stability expectations in families.