2D inviscid Boussinesq equations and 3D axisymmetric Euler equations: (1) A unification ($Em$), (2) Finite-time blow-up of two unified $(1+1)$D systems rigorously derived from ($Em$)

We derive $(1+2)$D subsystems~$(E1,E2)$ from the (2D inviscid Boussinesq, 3D axisymmetric Euler) equations in the (meridian) plane. The integer $m=1,2$ only appears in two numerical coefficients of subsystem~$(Em)$. Thus we discover a unification. We then study two unified $(1+1)$-dimensional systems, denoted $(R0)$ and $(Z0)$, that are rigorously derived from the $(Em)$. 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~$(Em)$ obtained from 2D inviscid Boussinesq and 3D axisymmetric Euler, and they already contain the core finite-time singularity mechanism of the full problem. The paper has three main outputs. First, it derives the polar $(1+2)$D subsystem~$(Em)$ from the 2D inviscid Boussinesq equations and from the 3D axisymmetric Euler equations and identifies the exact unified $(1+1)$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 adapted coefficient bounds required by the weighted energy method, if the weighted elliptic estimate holds, and if a gap exponent $\sigma \in (C_{\mathrm{l}\mathrm{i}\mathrm{n}},1)$ is available so that the remainder remains below the background scale, then the same finite-time apex blow-up transfers to the full solution.