with some sort of integer enumerative counts of curves are rarely clear. However, it is well-known that genus $0$ GW-invariants of Fano manifolds are precisely counts of rational curves; this observation is key to enumerating rational curves in projective space in [@KoM Section 5] and [@RT Section 10]. String theory predicts an amazing integral structure for GW-invariants of Calabi-Yau threefolds. These predictions originate in [@AsMo], [@GV1], and [@GV2] and are extended to all threefolds in [@P1].\
GW-invariants of a symplectic manifold $X$ are obtained by evaluating natural coholomogy classes on the virtual fundamental class () of the space of stable $J$-holomorphic maps to $X$. The main statement of this paper, Theorem \[main\_thm\], compares GW-invariants counting stable maps meeting specified constraints in the ambient manifold with analogous counts of such maps to a submanifold containing the images of all such constrained maps to first order. It leads immediately to Corollary \[LeP\_crl\], which in a way is a succinct re-formulation of the main conclusion of [@LeP], and with a bit more work to Theorem \[FanoGV\_thm\], which confirms the “Fano case" of the Gopakumar-Vafa prediction of [@P1 Section 0.2]. Theorem \[main\_thm\] is obtained by deforming the Cauchy-Riemann equation in two stages so that 