Trajectory Trichotomy for Linear Vector Fields

Definitions and mathematical statement

Let $M$ be a finite set of real linear vector fields. In the usual homogeneous convention a field on ${\mathbb{R}}^r$ has the form $x\mapsto Ax$ for a real matrix $A$, and its full trajectory through $v$ is the image of $t\mapsto e^{tA}v$, $t\in{\mathbb{R}}$. Let $G$ be a family of trajectories of members of $M$. Write ${\mathbb{R}}_G$ for the real field expanded by these trajectory sets, subject to the source's convention about how families are named. Two such expansions are interdefinable over the real field if each set or function added in either expansion is definable in the other, with the same specified parameter convention.

The supplied corrected statement of Theorem 1 gives the quantifiers \[ \forall M\text{ finite}\ \exists W\subseteq ({\mathbb{R}}\setminus{\mathbb{Q}})\cup\mathrm i({\mathbb{R}}\setminus\{0\})\text{ finite}\ \forall G\ \exists W'\subseteq W: \] ${\mathbb{R}}_G$ is interdefinable with at least one of \[ ({\mathbb{R}};+,{\cdot},<,\exp),\qquad ({\mathbb{R}};+,{\cdot},<,{\mathbb{Z}}),\qquad ({\mathbb{R}};+,{\cdot},<,(P_w)_{w\in W'}). \] Here the uniform finite set $W$ depends on $M$, while $W'$ may depend on $G$, and \[ P_{a+\mathrm i b}(t)=t^a\left(\cos(b\log t),\sin(b\log t)\right), \qquad t>0. \] The displayed formula defines the real two-coordinate representation of the complex power. The permitted exponents in $W$ are specifically irrational real numbers or nonzero purely imaginary numbers, not arbitrary mixed complex numbers. The empty subset $W'$ is allowed by the supplied statement.

The question is whether these three interdefinability alternatives form a trichotomy: beyond the stated "at least one", are the three types mutually exclusive in the intended sense? Precisely fixing this requires the parameter convention, the representation of $G$ as named sets or as a uniform family, and any intended identifications among power-family expansions. No disjointness result or counterexample is asserted here.

There is a source-field tension: the corrected_verified statement and verification say the primary report was checked and the missing antecedent, exponent set and power definition restored. The separately supplied literature triage, checked 21 August 2026, still says that an omitted "the above" prevents identifying a uniquely triageable conjecture. Both claims are retained; the dated triage must not be mistaken for a literature determination of the restored formulation.

Short English statement

A supplied classification says that trajectory expansions of a finite collection of linear vector fields have at least one of three interdefinability types: exponential, integer, or finitely many specified power functions. The question asks whether those alternatives are mutually exclusive. Parameter and family conventions, plus conflicting restoration and triage notes, require source review.

Sources

• Local UnsolvedMath registry, permanent ID 30000097, exact source code catalogue entry; Trajectory Trichotomy for Linear Vector Fields. Proposed year 2004, status open. The supplied corrected_verified statement expands the original reference to Theorem 1 with the three types and exponent formula.
• Supplied report citation: Model Theory and Complex Analytic Geometry (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/35, pp. 1845--1882, https://doi.org/10.4171/owr/2004/35. Title year differs from DOI/proposed year. The verification's claimed direct report check is retained as metadata, not an inspection performed here.
• Supplied literature triage checked 2026-08-21, citing only "original Oberwolfach report", https://doi.org/10.4171/owr/2004/35. It reports an omitted antecedent and no uniquely triageable conjecture, in tension with the supplied restored statement. This dated OWR-derived assessment requests independent verification of time-sensitive status and cited claims; no separate theorem-author or later-paper citation is supplied.