arXiv:2610.35940v1 Announce Type: new Abstract: Let g be a Lie algebra of dimension 2n+1, let beta be a contact form, and let alpha be a nonzero integrable 1-form. We study when the affine path alpha_t = (1-t)alpha + t beta consists of contact forms with the orientation determined by beta \wedge (d beta)^n. We first establish an algebraic criterion: affine deformability is equivalent to the non-negativity of a scalar obtained from the interaction form Theta(alpha, beta) = alpha \wedge (d beta)^n + n beta \wedge d alpha \wedge (d beta)^{n-1}. For unimodular Lie algebras this scalar is (n+1)alpha(Z_beta), where Z_beta is the Reeb vector. We also derive a formula involving the modular character tau(X) = tr(ad_X) in the non-unimodular case. On nilpotent contact Lie algebras, every integrable 1-form is closed; using the standard structural properties of nilpotent contact Lie algebras, this implies that every left-invariant foliation is an affine origin for every fixed contact form. An explicit three-dimensional example illustrates that the modular term can yield affine deformability even when alpha(Z_beta) = 0. We discuss the implications for left-invariant forms on Lie groups and compact quotients. We do not claim a general obstruction example in the non-unimodular case; establishing such an example requires a separate verified construction.
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