SyCo-AE conjecture on approximating full-space constraints

Let $G(x)$ be a complex, not-yet-known constraint on the full space, let $\phi$ map reduced-order states to the full space, and let $g: \mathbb{R}^r \to \mathbb{R}^s$ be a synthetic constraint with $s \leq r$. Reduced-order states $u$ satisfy the synthetic constraint when

$$0 = g(u).$$

SyCo-AE conjecture. The full-space constraint can be approximated by

$$0 = G(x) \approx G(\phi(u))$$

for all reduced-order states $u$ satisfying $0 = g(u)$. This conjecture expresses the hope that synthetically constraining the reduced dynamics provides useful information about an otherwise unknown full-space constraint and thereby improves data-efficient modeling of chaotic dynamics.