Existence of a symmetry-axis compatible background blow-up profile

There exist a time $T>0$ and smooth functions

$$V,U,G,P\in C^\infty\bigl([0,T)\times \mathbb{R}\times[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\bigr)$$

with smooth initial data

$$a(x,\theta)=f_1(x^2,\phi(\theta)),\quad b(x,\theta)=f_2(x^2,\phi(\theta)),\quad c(x,\theta)=f_3(x^2,\phi(\theta)),$$

for smooth $f_1,f_2,f_3$ and a smooth even angular profile $\phi(\theta)$ adapted to $\theta=0,\pm\frac{\pi}{2}$, satisfying the compatibility, regularity, parity, and ridge-flatness conditions of the rev3 formulation. Existence of a symmetry-axis compatible background blow-up profile. The quadruple $(V,U,G,P)$ solves the background system on $[0,T)\times\mathbb{R}\times[-\frac{\pi}{2},\frac{\pi}{2}]$ with the stated initial data, preserves evenness in $(r,z)$ and the associated symmetry-axis flatness, reduces along $x=0$ to the closed apex ODE dynamics, and obeys

$$|V(t,0,\theta_*)|\sim\frac{c_*}{T-t}$$

as $t\uparrow T$ for some $c_*>0$ and each $\theta_*\in\\{0,\pm\tfrac{\pi}{2}\\}$. Moreover, the global size bounds

$$\\|V(t)\\|_{L^\infty_{x,\theta}}\leq\frac{C_1}{T-t},\quad \\|U(t)\\|_{L^\infty_{x,\theta}}\leq\frac{C_2}{T-t},\quad \\|G(t)\\|_{L^\infty_{x,\theta}}\leq\frac{C_3}{T-t}$$

hold, the required adapted derivatives satisfy bounds at the same critical scale, and sufficiently small perturbations close the bootstrap up to time $T$ while preserving the same apex singularity scenario. This would provide a full background and perturbative construction extending the rigorously established apex blow-up mechanism away from the apex and to the full wedge; the source does not establish these assertions.