  We resolve a mathematically precise SYZ conjecture for $A_n$ singularity by
building a quantum-corrected T-duality between two singular torus fibrations
related to the Kahler geometry of the $A_n$-smoothing and the Berkovich
geometry of the $A_n$-resolution, respectively. Our approach involves heavy
computations that embody a non-archimedean version of the partition of unity,
and it confirms the strategy that patching verified local singularity models
brings global SYZ conjecture solutions (like K3 surfaces) within reach. There
is also remarkably explicit extra evidence concerning the collision of singular
fibers and braid group actions. On one hand, we address the central challenge
of matching SYZ singular loci identified by Joyce. In reality, we construct not
merely an isolated SYZ mirror fibration partner, but a parameter-dependent one
that always keeps the matching singular loci plus integral affine structure,
even when the collision of singular fibers occurs. On the other hand, our SYZ
result surprisingly displays a visible tie, regardless of the parameter choice,
between the $(A_n)$-configuration of Lagrangian spheres occurred as vanishing
cycles in the $A_n$-smoothing and the exceptional locus of rational
$(-2)$-curves in the $A_n$-resolution. It closely aligns with the celebrated
works of Khovanov, Seidel, and Thomas from around 20 years ago in a somewhat
distant subject, providing geometric evidence for the family Floer functor
approach explored by Abouzaid and Fukaya. Intriguingly, these discoveries use
certain explicit realizations by order statistics.
