Finite-time blow-up of two $(1+1)$D systems rigorously derived from the 3D axisymmetric Euler equations

We study two $(1+1)$-dimensional systems, denoted $(R0)$ and $(Z0)$, that are rigorously derived from the three-dimensional axisymmetric Euler equations in a signed polar formulation on the meridian plane. The main point of view in this revision is that these $(1+1)$D systems are not ad hoc model equations and not merely ``symmetry-axis reductions.'' Rather, they arise as exact symmetry-axis/apex restrictions of the full $(1+2)$D system~$(E2)$ obtained from 3D axisymmetric Euler, and they already contain the core finite-time singularity mechanism of the full problem. The rev3 geometry is based on the symmetry axes \[ \theta=0,\qquad \theta=\pm \frac{\pi}{2}, \] for which ridge flatness is preserved automatically by the evenness in $(r,z)$. Along these axes, and in particular at the apex $x^2=r^2+z^2=0$, the reduced dynamics closes exactly. This yields two rigorously derived $(1+1)$D systems: the horizontal-axis system $(R0)$ and the vertical-axis system $(Z0)$. The apex trace of these systems reduces further to a closed ODE of Constantin--Lax--Majda type, from which we obtain finite-time blow-up at the coordinate origin. The paper has three main outputs. First, it derives the signed-polar $(1+2)$D subsystem~$(E2)$ from the 3D axisymmetric Euler equations and identifies the exact $(1+1)$D systems $(R0)$ and $(Z0)$ carried by the symmetry axes. Second, it proves finite-time blow-up for the resulting apex dynamics and analyzes the associated convective axis reduction. Third, it derives the exact background--remainder equations and formulates a conditional nonlinear stability mechanism: if a compatible full background exists on $[0,T)$ with the coefficient bounds required by the weighted energy method, then the full solution inherits the same finite-time apex blow-up.