= Solution
The <symplectic neighborhood theorem> says that a <symplectomorphism> between compact <symplectic submanifolds>, together with a <symplectic vector bundle> isomorphism of their <symplectic normal bundles>, extends to a <symplectomorphism> of neighborhoods. To outline the proof, use <tubular neighborhoods> to extend the bundle identification smoothly. The two pulled-back <symplectic forms> agree as bilinear forms along the zero section. The <relative Poincaré lemma> writes their difference as $d\alpha$, with $\alpha$ vanishing there. Their linear interpolation $\omega_t$ remains <nondegenerate> on a sufficiently small neighborhood. Solving
$$
\iota_{V_t}\omega_t=-\alpha
$$
and integrating $V_t$ gives the <relative Moser theorem>; the flow fixes the submanifold and carries one form to the other.
For the <symplectic fiber sum along a square-zero surface>, the <self-intersection> is the <Euler number> of the oriented rank-two <normal bundle>. Thus each normal bundle is trivial. Rescale one ambient <symplectic form> if necessary to equalize the areas of the two copies of $C$. The <Moser theorem> for the surface then supplies a base identification preserving their area forms. The <symplectic neighborhood theorem> identifies each neighborhood with a product carrying
$$
\omega_C+\frac12d(r^2)\wedge d\theta.
$$
Remove small disk neighborhoods and glue collars with the base identification and
$$
\theta_Y=-\theta_X,\qquad
r_Y^2=a-r_X^2
$$
for a suitable positive constant $a$. Both signs reverse, so the normal two-forms match; the base forms already match. They define a closed <nondegenerate> two-form across the neck, agreeing with the ambient forms elsewhere. This constructs the <symplectic sum>. Its construction uses the area normalization and the chosen normal-bundle gluing.
For a smooth counterexample, take $X=Y=\mathbb{CP}^2$, and in a small four-ball in each choose an unknotted <sphere> bounding a three-ball. These spheres are <null-homologous> and have <self-intersection> zero. Their standard smooth fiber sum is
$$
Z\cong\mathbb{CP}^2\mathbin{\#}\mathbb{CP}^2
\mathbin{\#}(S^1\times S^3).
$$
Indeed, the complement of the standard $S^2\times D^2$ in $S^4$ is $S^1\times D^3$; doubling these complements gives $S^1\times S^3$, and the two ambient projective-plane summands remain as <connected sums>. This is the <smooth fiber sum along unknotted null-homologous spheres>.
In the induced <orientation>, both summands in $Z=\mathbb{CP}^2\#(\mathbb{CP}^2\#(S^1\times S^3))$ have positive <positive index of the intersection form>. The <symplectic connected-sum obstruction> therefore rules out a <symplectic form>: connected-sum vanishing of the <Seiberg–Witten invariant of a four-manifold> contradicts <Taubes nonvanishing theorem>. In the opposite orientation the <intersection form> is negative definite, also ruling out a <symplectic form>, whose class has positive square. Thus \b[the smooth fiber sum need not be symplectic].
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