Taylor expansion for synthetic G-structures

Establish a Taylor-polynomial local presentation for a synthetic $G$-structure $Phi_G$ on a set $M$, with coefficients determined by the synthetic torsion and the covariant derivatives of the synthetic curvature at a point, assuming suitable definitions of synthetic torsion and curvature.

Background

The paper proposes extending the synthetic Taylor formula for Riemannian metrics to general GG-structures. The conjectured expansion uses the canonical flat GG-structure as its constant term, torsion in the linear term, and covariant derivatives of curvature in higher-order terms. The conjecture presupposes suitable definitions of synthetic Lie groups, synthetic torsion, and synthetic curvature.

References

We conjecture that $\Phi_G$ has a local presentation in terms of a Taylor polynomial approximation with Taylor coefficients that codify obstructions to integrability.

— Retro-synthetic Riemannian geometry in Rocq I : Taylor series of a metric  (2609.20127 - Clemente et al., 17 Sep 2026) in Section 4, item 2; Conjecture (Taylor's theorem for synthetic $G$-structures), Conjecture \ref{Gconjecture}

The goal would be to use this new synthetic theorem to obtain a classical differential geometry version, which is currently an open problem.

— Retro-synthetic Riemannian geometry in Rocq I : Taylor series of a metric  (2609.20127 - Clemente et al., 17 Sep 2026) in Section 4, item 2 (paragraph following Conjecture \ref{Gconjecture})