Whitney normal tubes have submersive radius and bounded stratumwise lifts

Write $y=\pi (x)$ and $n=x-y$ for a point of the tube, and $r=\mathrm{\parallel }n\mathrm{\parallel }$. When $r>0$, differentiating the norm gives $d{r}_x(v)=⟨n\mathrm{/}r,v-d{\pi }_x(v)⟩=⟨n\mathrm{/}r,v⟩$, because $d{\pi }_x(v)\in T_{y}R$ is orthogonal to $n$; and at $y\in R$ the inverse differential of normal addition identifies $d{\pi }_y$ with the orthogonal projection onto $T_{y}R$ by [F2]. Fix a lower-stratum coordinate patch with a smooth orthonormal frame for $TR$, and use it to identify the target of $d\pi$ with ${\mathbb{R}}^m$, $m=\dim R$.

For $x$ in an incident upper stratum $S$ let $A_x:T_{x}S\to {\mathbb{R}}^m\oplus \mathbb{R}$ be the matrix of $(d{\pi }_x,d{r}_x)$ in that frame. Suppose no uniform lower bound $\mathrm{\parallel }{A}_x^{\ast }\lambda \mathrm{\parallel }\ge c\mathrm{\parallel }\lambda \mathrm{\parallel }$ held near a point $y_0\in R$. Then there are $x_j\to y_0$ in incident strata and unit vectors ${\lambda }_j=(a_j,b_j)$ with $\mathrm{\parallel }{A}_{x_j}^{\ast }{\lambda }_j\mathrm{\parallel }\to 0$; passing to one of the finitely many strata and to subsequences, the tangent planes $T_{x_j}S$ converge to a plane $T$, the unit normal directions $n_j\mathrm{/}{r}_j$ converge to a unit vector $u$, and the target vectors converge to a unit $(a,b)$. By smoothness of the tube projection, $d{\pi }_{x_j}$ converges to the orthogonal projection onto $T_{y_0}R$, and by step 1.1 the adjoints satisfy $A_{x_j}^{\ast }{\lambda }_j\to {\mathrm{\Pi }}_T(a)+bu$, where ${\mathrm{\Pi }}_T$ is the orthogonal projection onto $T$.

By Whitney (a), $T_{y_0}R\subseteq T$, so ${\mathrm{\Pi }}_T(a)=a$; by Whitney (b) applied to the secants $x_j-\pi (x_j)$, the unit limit $u$ lies in $T$. Hence the limit $a+bu$ lies in $T$ and has squared norm $\mathrm{\parallel }a{\mathrm{\parallel }}^2+b^2=1$, because $a\in T_{y_0}R$ is orthogonal to $u$. This contradicts $\mathrm{\parallel }{A}_{x_j}^{\ast }{\lambda }_j\mathrm{\parallel }\to 0$. Therefore there is a neighbourhood of $y_0$ in the tube and a constant $c>0$ with $\mathrm{\parallel }{A}_x^{\ast }\lambda \mathrm{\parallel }\ge c\mathrm{\parallel }\lambda \mathrm{\parallel }$ for all $x$ there in incident strata. In particular $A_x^{\ast }$ is injective, $A_x$ is surjective, and $(\pi ,r){\mathrm{\mid }}_S$ is a submersion throughout that neighbourhood.

Set $H_x=A_x^{\ast }(A_x{A}_x^{\ast }{)}^{-1}$. Its image lies in $T_{x}S$ and $A_x{H}_x=\mathrm{i}\mathrm{d}$, and it is smooth along $S$ because the frame, the tube differentials and their coefficients are smooth; the singular-value bound gives $\mathrm{\parallel }{H}_x\mathrm{\parallel }\le 1\mathrm{/}c$. For a smooth bounded field $w$ on $R$ put $V(x)=H_x(w(\pi (x)),0)$, and put $V_{\mathrm{r}\mathrm{a}\mathrm{d}}(x)=H_x(0,r(x))$. Then $d{\pi }_x(V)=w(\pi (x))$, $d{r}_x(V)=0$, $d{\pi }_x(V_{\mathrm{r}\mathrm{a}\mathrm{d}})=0$, $d{r}_x(V_{\mathrm{r}\mathrm{a}\mathrm{d}})=r(x)$, and $\mathrm{\parallel }V\mathrm{\parallel }\le \mathrm{\parallel }w\mathrm{\parallel }\mathrm{/}c$, $\mathrm{\parallel }{V}_{\mathrm{r}\mathrm{a}\mathrm{d}}\mathrm{\parallel }\le r(x)\mathrm{/}c$; all these fields are smooth along $S$ and locally bounded near the given point of $R$. Since there are finitely many incident strata, the neighbourhood and the constant can be chosen to work for all of them at once.

The construction applies near every point of $R$ and an empty stratum contributes no instances; when $m=0$ the same argument uses only the normal row of $A_x$ and proves radial submersivity, so no positive lower-stratum dimension is needed. What is proved is ordinary tube transversality together with one-stratum bounded lifts: on approaching a different intermediate stratum, independently chosen lifts need not agree with that stratum's choices, so simultaneous compatible control data and the gluing of controlled lifts remain separate obligations.