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Variational meta-conjecture for properties characterizing geometric origin

Establish that, for a smooth proper morphism f: X→S of smooth \overline{k}-schemes and a semisimple local system V on X, if V|_{X_s} satisfies a property P that is conjecturally equivalent to being of geometric origin, then V|_{X_t} satisfies P for all t∈S(\overline{k}).

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Background

This meta-conjecture abstracts the variational motivic principle to any property P expected to characterize geometric origin (e.g., arithmeticity, Hodge-theoretic conditions).

It frames a general persistence statement across fibers, supplying a template to derive concrete predictions from the motivic conjecture.

References

Conjecture (Variational meta-conjecture). Let f: X\to S be a smooth proper morphism of smooth \overline{k}-schemes, and let \mathbb{V} be a semisimple local system on X. Suppose there exists s\in S(\overline{k}) such that \mathbb{V}|{X_s} has property P. Then for all t\in S(\overline{k}), \mathbb{V}|{X_t} has property P.

— Algebraicity and integrality of solutions to differential equations (2501.13175 - Lam et al., 22 Jan 2025) in Conjecture, Section “A motivic variational principle”