Complex Algebraic Stratified Topology

Put $d=\dim S$ and let $G$ be the graph of the map $(x,y)\mapsto (T_{x}S,[x-y])$ from $S\times Y$ to $\mathrm{Gr}(d,n)\times {\mathbb{P}}^{n-1}$; it is algebraic by [F5], because on chart domains the tangent plane is the kernel of the Jacobian matrix of local equations, whose basis entries are regular ratios on each fixed nonzero-minor chart and the secant coordinates are regular since $x\mathrm{\ne }y$. Let $Z$ be the reduced closure of $G$ in $\bar{S}\times Y\times \mathrm{Gr}(d,n)\times {\mathbb{P}}^{n-1}$. The graph $G$ is isomorphic to $S\times Y$, hence irreducible by [F6], and $\bar{S}$, $Y$, $\mathrm{Gr}(d,n)$ and ${\mathbb{P}}^{n-1}$ are irreducible; the ambient product is therefore irreducible by [F6], and its closed subset $Z$, the closure of the irreducible $G$, is irreducible as well. Let $B=Z\setminus G$, a proper closed subset, and let $g:Z\to Y$ be the projection; it is dominant because it is onto on $G$.

Apply [F2] to $g$ to obtain a dense open subset of $Y$ over which the fibres $Z_y$ are nonempty and pure of dimension $r=\dim Z-\dim Y$. This number is positive, because each fibre contains $G_y=\{(x,y,T_{x}S,[x-y]):x\in S\}\cong S$ of dimension $\dim S\ge 1$. Each irreducible component $B_i$ of the proper closed subset $B$ has $\dim B_i<\dim Z$ by [F4]; if $B_i\to Y$ is dominant, [F2] gives a further dense open of $Y$ over which its fibres have pure dimension $\dim B_i-\dim Y\le r-1$, and if it is not dominant we remove its image closure. Intersecting the finitely many dense opens leaves a dense open $Y^0\subseteq Y$ on which $Z_y$ is pure $r$-dimensional while $B_y$ has dimension at most $r-1$; hence no irreducible component of $Z_y$ is contained in $B_y$, and for every component $V$ of $Z_y$ the set $V\cap G_y=V\setminus B_y$ is a nonempty dense open subset of the irreducible $V$.

Fix $y\in Y^0$ and an incidence point $z_0=(y,y,T,[u])\in Z_y$ with $u\mathrm{\ne }0$. Let $V$ be an irreducible component of $Z_y$ containing $z_0$ and let $\gamma$ be the holomorphic arc supplied by [F3] for the variety $V$, the nonempty Zariski open $V\setminus B_y$ and the point $z_0$. Since $\gamma$ maps into $Z_y$ and $Z_y\subseteq g^{-1}(y)$, its second coordinate is the constant $y$; write $\gamma (s)=(x(s),y,T(s),[u(s)])$ with $\gamma (0)=z_0$ and $\gamma (s)\in G$ for $s\mathrm{\ne }0$. Then $T(s)=T_{x(s)}S$ and $[u(s)]=[x(s)-y]$ for $s\mathrm{\ne }0$, so $x(s)\mathrm{\ne }y$ for $s\mathrm{\ne }0$. Choosing a projective chart containing $[u]$, the normalized representative of the arc is holomorphic, and since $x(s)-y$ is not identically zero some coordinate has finite positive vanishing order; with $m\ge 1$ the minimum of these orders, $x(s)-y=s^{m}v(s)$ with $v$ holomorphic and $v(0)\mathrm{\ne }0$, and $[v(0)]=[u]$.

Let $p_0$ be a complex covector annihilating the complex plane $T$. In a Grassmannian matrix chart around $T$ write $T(s)$ as the graph of a holomorphic matrix $A(s)$ with $A(0)=A_0$, possible for small $s$ because $T(s)\to T$; every annihilator of the graph of $A(s)$ has the form $q(s)=(-A(s{)}^{t}b,b)$ for $b\in ({\mathbb{C}}^{n-d}{)}^{\ast }$, and $b$ can be chosen with $(-A_0^{t}b,b)=p_0$ because $p_0$ annihilates $T$. The resulting $q(s)$ is holomorphic and annihilates each $T(s)$. Since $x(s)\in S$ for $s\mathrm{\ne }0$, its derivative lies in $T_{x(s)}S=T(s)$ by [F5], so $q(s)(x^{\mathrm{\prime }}(s))=0$ for $s\mathrm{\ne }0$. Differentiating $x(s)-y=s^{m}v(s)$ gives $x^{\mathrm{\prime }}(s)=s^{m-1}(mv(s)+s{v}^{\mathrm{\prime }}(s))$, and dividing by $s^{m-1}$ and letting $s\to 0$ yields $p_0(mv(0))=0$, hence $p_0(v(0))=0$. Every annihilator of $T$ therefore kills $v(0)$, so $v(0)\in T$ and the limiting complex secant line $[u]$ lies in $T$.

Now let $x_j\in S$ and $y_j\in Y$ be sequences converging to the same point $y\in Y^0$, with tangent planes $T_{x_j}S$ converging to a plane $T$ and real secant lines through $x_j,y_j$ converging to a real line $\mathrm{\ell }$. After choosing their signs, the secant directions normalized to unit vectors converge to a unit vector $w$ spanning $\mathrm{\ell }$, and the complex projective points $[x_j-y_j]=[w_j]$ converge to $[w]$, the complex line spanned by $\mathrm{\ell }$; moreover $(x_j,y_j,T_{x_j}S,[x_j-y_j])\in G\subseteq Z$ converges to $(y,y,T,[w])\in Z_y$ because $Z$ is closed. Applying steps 3.1 and 4.1 to this incidence point shows that its limiting complex secant line lies in $T$, so $w\in T$ and $\mathrm{\ell }=\mathbb{R}w\subseteq T$. This is Whitney (b) at $y$ in the full moving-point form.

For Whitney (a) at $y$, let $x_j\in S$ converge to $y$ with tangent planes converging to $T$, and let $w\in T_{y}Y$ be a nonzero real tangent vector (the zero vector already belongs to $T$). In a holomorphic chart of the smooth manifold $Y$ near $y$ choose points $y_j\in Y$ of the form $y+{ϵ}_{j}w+o({ϵ}_j)$ with ${ϵ}_j>0$, ${ϵ}_j\to 0$ and $\mathrm{\parallel }{x}_j-y\mathrm{\parallel }\mathrm{/}{ϵ}_j\to 0$, for instance ${ϵ}_j=\sqrt{\mathrm{\parallel }{x}_j-y\mathrm{\parallel }}$ and a straight segment in the chart. Then $\mathrm{\parallel }{x}_j-y_j\mathrm{\parallel }\ge {ϵ}_j\mathrm{\parallel }w\mathrm{\parallel }\mathrm{/}2$ for large $j$, and the secant directions from $y_j$ to $x_j$ converge to the real line spanned by $w$. Since the tangent planes $T_{x_j}S\to T$, Whitney (b) from step 5.1 places that real line in $T$, so $w\in T$. Every real tangent vector of $T_{y}Y$ arises in this way, so $T_{y}Y\subseteq T$, proving Whitney (a).

Set $E=Y\setminus Y^0$; it is a proper closed algebraic subset by step 2.1, and steps 5.1 and 6.1 show that $(S,Y)$ satisfies (a) and (b) at every point of $Y\setminus E$. For finitely many prescribed incident upper strata $S_1,\dots ,S_m$ over the same irreducible $Y$, apply the construction to each and take the union of the finitely many proper closed exceptional sets; since $Y$ is irreducible, that finite union is still proper, by [F4]. For a quasi-projective ambient manifold, use the finitely many affine ambient charts $U_{\alpha }$ that meet $Y$; their intersections with the irreducible $Y$ are nonempty opens. Inside each chart apply the affine result to every pair of irreducible components $(S_i,Y_j)$ of $(S\cap U_{\alpha },Y\cap U_{\alpha })$ with $Y_j\subseteq \overline{S_i}$ and $\dim Y_j<\dim S_i$, and for each point of an irreducible component of $Y\cap U_{\alpha }$ one of these pairs accounts for every sequence in $S$ converging to it after passing to a subsequence in a component. Each chart exceptional set is closed in $Y\cap U_{\alpha }$ and has smaller dimension than $Y$, so its closure in $Y$ is proper by [F4]. The union of these finitely many closures and the proper closed complements $Y\setminus U_{\alpha }$ is still proper because $Y$ is irreducible. Outside this union every chart argument applies, giving the required closed $E⊊Y$.

Replace every prescribed locally closed subset $A$ by the two prescribed closed subsets $\bar{A}$ and $\bar{A}\setminus A$; their difference recovers $A$, and it suffices to respect the resulting finite list of closed subsets. Refine the smooth-partition construction of [F2] to construct, for a closed residual variety $R$ together with a finite list of prescribed closed subsets and finitely many previously fixed smooth strata outside $R$ whose closures must be respected, a Whitney partition of $R$ compatible with all of them: take the irreducible components $R_i$ of $R$, remove their nonsmooth loci and their intersections with the other components, and remove every prescribed closed subset not containing $R_i$, obtaining dense smooth opens $L_i$; then for every previously fixed upper stratum $S$, remove $R_i\cap \bar{S}$ when $R_i\subseteq ̸\bar{S}$, and when $R_i\subseteq \bar{S}$ use [F3] to remove the closure in $R_i$ of the exceptional subset of [F3], which is proper because it has dimension smaller than $\dim R_i$. Finitely many upper strata leave nonempty dense smooth opens $U_i\subseteq R_i$, and [F3] makes each $U_i$ Whitney-regular with every incident previously fixed upper stratum.

Emit the $U_i$ as strata, add their closures $R_i$ to the list of closed subsets to be respected, and recurse on the closed complement $R\setminus {\bigcup }_i{U}_i$, whose dimension drops because every component lost a dense open subset. The recursion terminates by induction on dimension, producing a finite partition of $R$ into smooth irreducible locally closed strata. Strata of the same round have no frontier relation, since each $U_i$ misses all other components $R_j$. Every pair of incident strata has its upper stratum emitted earlier, and when the lower stratum was emitted step 1.1 imposed Whitney (a) and (b) with that upper stratum; boundaries have strictly smaller dimension by [F2]. The respected closures make every stratum closure a union of strata. In a quasi-projective ambient space, cover it by finitely many affine coordinate charts and apply the affine-chart version of [F3]; a sequence in $R$ converging to a point meets only finitely many components of the chart intersections, so passing to a subsequence in one component reduces the conditions to the chart version. Thus Whitney partitions compatible with a finite list of closed sets and with fixed external upper strata exist.

Construct the map-compatible partitions by decreasing target residual dimension. Start with target residual $R=Y$ and source residual $f^{-1}(R)=X$, and at each later round maintain the original closed-set lists together with the closures of all previously emitted source and target strata. Partition the source residual $f^{-1}(R)$ by step 2.1, so that it is Whitney-compatible with all previously emitted source strata. For each irreducible component $R_i$ of $R$, choose a dense smooth open $L_i\subseteq R_i$ as in step 1.1, respecting the target prescriptions and Whitney-regular with all previously emitted target strata. For every new source stratum $Z_{\alpha }$ compute the closure of its image in $R$; if that closure is not $R_i$, remove its intersection with $R_i$, a proper closed subset; if the closure is $R_i$, then $Z_{\alpha }\to R_i$ is dominant and [F4] supplies a dense open subset of $R_i$ on which the restriction is smooth and onto. Intersecting the finitely many resulting dense opens with $L_i$ gives a nonempty dense open $V_i\subseteq R_i$.

Emit the $V_i$ as target strata and the nonempty pieces $Z_{\alpha }\cap f^{-1}(V_i)$ as source strata. Each such piece is open in the smooth irreducible stratum $Z_{\alpha }$, hence smooth irreducible and locally closed, and it is Whitney-regular with every incident previously fixed upper source stratum, because Whitney conditions are local and were imposed on all of $Z_{\alpha }$ in step 2.1; with the pieces of the same round it has no frontier relation except through closures, which the later rounds make unions of strata because every closure is added to the respected list. Each piece maps smoothly and onto $V_i$ by its selection in step 3.1, and target strata are Whitney-regular with earlier target strata since $V_i\subseteq L_i$. Source frontier relations inside $f^{-1}({\bigcup }_i{V}_i)$ are inherited from the source partition, and a stratum closure over $R_i$ misses every $V_j$ with $j\mathrm{\ne }i$.

Remove ${\bigcup }_i{V}_i$ from $R$, restrict both residuals and both lists, add all newly emitted stratum closures to the respective lists, and repeat. The target dimension strictly decreases each round, so after finitely many rounds every target point and every source point has been placed: a source point lies over one emitted target stratum and is contained in one emitted piece. Future source partitions are chosen Whitney-compatible with every earlier source stratum by step 2.1, and future target opens are chosen Whitney-compatible with earlier target strata in step 3.1, so every incident pair is covered; the closure-bookkeeping makes all stratum closures unions of strata. Each emitted piece is irreducible and nonempty, so [F7] makes it analytically connected and identifies its analytic closure with the algebraic closure tracked by the construction. This yields finite smooth Whitney frontier stratifications of $X$ and $Y$, respecting all prescribed subsets (including the strata of any prescribed finite algebraic stratification, which were treated as prescribed sets in step 1.1), with each source stratum mapping smoothly and surjectively onto one target stratum.

The analytic clauses follow from [F5] and the same locally standard smooth Jacobian charts: the strata are smooth complex manifolds, and each smooth restriction is, in those charts, a holomorphic map whose differential is onto and whose rank is locally constant, hence a submersion. If $f$ is proper in the analytic topology and $T$ is a target stratum, then for every compact subset $C\subseteq T$ the inclusion $T\hookrightarrow Y$ is continuous, so $C$ is compact in $Y$, and $f^{-1}(C)\subseteq f^{-1}(T)$ is compact by properness of $f$; hence the restriction $f^{-1}(T)\to T$ is proper, although individual source strata inside $f^{-1}(T)$ may fail to be closed. Empty spaces and empty prescribed families add no strata, and no tube control, isotopy or triangulation existence is claimed.

For the projective case factor $f$ as a closed algebraic embedding $i:X\hookrightarrow Y\times {\mathbb{P}}^N(\mathbb{C})$ followed by the projection $\pi :Y\times {\mathbb{P}}^N(\mathbb{C})\to Y$. Complex projective space is the continuous image of the unit sphere in ${\mathbb{C}}^{N+1}$ under $z\mapsto [z]$, hence compact by [F6]. A closed algebraic embedding is analytically closed, because in each chart its defining polynomial equations have continuous coefficients and their common zero set is closed. For a compact $C\subseteq Y$ the set ${\pi }^{-1}(C)=C\times {\mathbb{P}}^N(\mathbb{C})$ is compact by [F6], and $f^{-1}(C)=i^{-1}(C\times {\mathbb{P}}^N(\mathbb{C}))$ is a closed subset of it, hence compact. Therefore $f$ is analytically proper, and step 6.1 applies; this is the analytic properness used by the consumers.

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.

Fix $s\in S$ and put $I(s)=\{i:s\in C_i\}$, a finite set. Since the $C_i$ with $i\mathrm{\notin }I(s)$ are closed in $S$ and avoid $s$, and the $U_i$ with $i\in I(s)$ are open, choose a source neighbourhood $V_s\subseteq S$ that avoids every $C_i$ with $i\mathrm{\notin }I(s)$ and lies in every $U_i$ with $i\in I(s)$; shrink it so that the active maps factor smoothly through a single maximal map $F_j$. Since $F_j{\mathrm{\mid }}_S$ is a submersion, there are ambient coordinates $(b,u,z)$ on a neighbourhood of $s$ with $S=\{z=0\}$ and $F_j=b$: straighten $S$, use [F1] on $F_j{\mathrm{\mid }}_S$ for coordinates $(b,u)$, and then apply [F1] in the full-rank case to the map $(b,u,z)\mapsto (F_j(b,u,z),u,z)$, whose differential at $s$ is invertible because $\mathrm{\partial }{F}_j\mathrm{/}\mathrm{\partial }b$ is the identity at $z=0$. If $I(s)$ is empty, ordinary straightened coordinates suffice.

Put $Q=T{\mathbb{R}}^N{\mathrm{\mid }}_S\mathrm{/}TS$ and identify it with the Euclidean orthogonal normal bundle. The $z$-directions of the adapted charts give smooth local splittings $j_s:Q\to T{\mathbb{R}}^N{\mathrm{\mid }}_S$ of the quotient map whose images are vertical for every active map. Choose a locally finite smooth partition of unity ${\theta }_{\alpha }$ on $S$ subordinate to the neighbourhoods of step 1.1. Near each chart extend its vertical coordinate fields to the ambient chart. For a parameter $(s,v)\in Q$ let $W_{\alpha }(s,v;x)$ be the linear combination of these fields with the $Q$-coordinates of $v$ at $s$, multiplied by a smooth local cutoff in $(s,x)$ equal to one near the diagonal over the support of ${\theta }_{\alpha }$, and set $W(s,v;x)={\sum }_{\alpha }{\theta }_{\alpha }(s)W_{\alpha }(s,v;x)$. The sum is locally finite in $s$, smooth in all variables, and linear in $v$; at $x=s$ it equals $j_s(v)={\sum }_{\alpha }{\theta }_{\alpha }(s)j_{\alpha }(s)(v)$, whose quotient is $v$ because every summand splits the quotient map. Whenever $s\in C_i$, every summand of nonzero weight lies in a neighbourhood carrying that constraint, so $W(s,v;\cdot )$ is vertical for $F_i$ near $x=s$.

Let $E(s,v)$ be the time-one endpoint of the flow of $x\mapsto W(s,v;x)$ starting at $s$. By [F3] applied to the jointly smooth field $W$ this is defined and smooth on an open neighbourhood of the zero section: for small $v$ the field is small, and rescaling the parameter or restricting a common short-time interval gives the time-one solution. It satisfies $E(s,0)=s$ and $d_{v}E(s,0)=j_s$, because the field vanishes at $v=0$ and its first parameter derivative is the constant initial vector $j_s(v)$; moreover $d_{s}E(s,0)$ is the identity, so the differential of $E$ at the zero section is the isomorphism $(a,v)\mapsto a+j_s(v)$. Hence $E$ is a local diffeomorphism at every point of the zero section. If $s\in C_i$, a sufficiently small parameter keeps the whole trajectory in the region where $W$ is vertical for $F_i$, so $F_i(E(s,v))=F_i(s)$ there by the chain rule. All radius requirements so far are positive local requirements on $s$.

For $s\in S$ let $r(s)$ be the supremum of $a\in (0,1]$ such that $E$ is defined and injective on $D_a(s)=\{(t,v):t\in S,\mathrm{\parallel }t-s\mathrm{\parallel }<a,\mathrm{\parallel }v\mathrm{\parallel }<a\}$. It is positive: $E$ is a local diffeomorphism at each point of the zero section, and an injectivity neighbourhood of $(s,0)$ contains $D_a(s)$ for sufficiently small $a>0$ in a local bundle trivialization. If $a<r(s)$ then $D_a(s)$ is eligible by restriction, and $D_{a-\mathrm{\parallel }s-t\mathrm{\parallel }}(t)\subseteq D_a(s)$ shows $r(t)\ge r(s)-\mathrm{\parallel }s-t\mathrm{\parallel }$; exchanging $s$ and $t$ makes $r$ 1-Lipschitz. Choose a positive smooth radius $\delta (s)$ with $\delta (s)<r(s)\mathrm{/}16$, with $\mathrm{\parallel }E(s,v)-s\mathrm{\parallel }<r(s)\mathrm{/}16$ whenever $\mathrm{\parallel }v\mathrm{\parallel }<\delta (s)$, and with any prescribed positive size bounds enforced; such a smooth minorant exists by [F2], summing suitably small positive constants with a subordinate partition of unity on a locally finite cover on which the required local flow, identity and size bounds have positive lower bounds; the prescribed size bounds have this property by hypothesis.

Suppose $E(s,v)=E(t,w)$ with $d=\mathrm{\parallel }s-t\mathrm{\parallel }$ inside this radius domain. Then $d<(r(s)+r(t))\mathrm{/}16\le (2r(s)+d)\mathrm{/}16$, so $d<2r(s)\mathrm{/}15$; also $\mathrm{\parallel }v\mathrm{\parallel }<r(s)\mathrm{/}16$ and $\mathrm{\parallel }w\mathrm{\parallel }<r(t)\mathrm{/}16<(17\mathrm{/}240)r(s)$. Both parameter points therefore lie in the eligible domain $D_{r(s)\mathrm{/}2}(s)$, on which $E$ is injective, so $(s,v)=(t,w)$. After shrinking the radius further to stay inside the local-diffeomorphism locus, $E$ is an injective local diffeomorphism on its full variable-radius domain, hence a diffeomorphism onto an open neighbourhood $T$ of $S$; its inverse composed with the projection of $Q$ onto $S$ defines $\pi :T\to S$, the tube map is exactly $E$, and step 3.1 gives $F_i(x)=F_i(\pi (x))$ whenever $\pi (x)\in C_i$ and $x$ lies close enough to its zero-section point. The prescribed displacement and fiber-radius bounds were built into $\delta$ in step 4.1.

Induct on stratum dimension, retaining the already constructed tube maps on outer collars while shrinking their final tube domains. Give each tube map $E_R(r,v)$ the displacement bound $\mathrm{\parallel }{E}_R(r,v)-r\mathrm{\parallel }<\mathrm{dist}(r,\bar{R}\setminus R)\mathrm{/}3$ when the frontier is nonempty, and cap displacement and radius by one; these are positive local bounds on the locally closed $R$ and can be enforced by [F2] and [F3]. Consequently a convergent sequence of tube points whose base points tend to the frontier has the same frontier limit, since the displacement tends to zero. In particular, the smaller closed-radius band of a tube, intersected with a stratum different from it, has relative closure inside the retained outer tube: if the base limit lies in $R$ this follows from the strict radius inequality and continuity, and if it lies on the frontier then the point limit would lie in a strictly lower-dimensional stratum, which is impossible.

Make the tubes of incomparable strata disjoint. Their ambient closures miss the other stratum by the frontier axiom $\bar{R}={\coprod }_{Q\le R}Q$. Impose on a tube based at $r\in R$ the displacement bound $\mathrm{dist}(r,\bar{S})\mathrm{/}3$ and the symmetric bound for a tube based at $s\in S$. If a point $x$ lay in both tubes, then $\mathrm{\parallel }r-s\mathrm{\parallel }\le \mathrm{\parallel }x-r\mathrm{\parallel }+\mathrm{\parallel }x-s\mathrm{\parallel }<(\mathrm{dist}(r,\bar{S})+\mathrm{dist}(s,\bar{R}))\mathrm{/}3\le 2\mathrm{\parallel }r-s\mathrm{\parallel }\mathrm{/}3$, a contradiction. There are finitely many pairs, and shrinking retains every previously established identity and submersion property. Hence the active lower-stratum controllers at any point of a stratum form a chain.

Suppose all tubes of lower-dimensional strata are constructed, and consider a new stratum $S$. Inside the currently retained tube of each $R<S$ choose three variable squared-radius bands with bounds ${ϵ}_R>{ϵ}_R\mathrm{/}2>{ϵ}_R\mathrm{/}4$, the outer bound smaller than the actual parameter radius squared. Let $C_R$ be the closure in $S$ of the middle band intersected with $S$; by step 1.1 it is relatively closed in $S$, contained in the outer tube, and the ambient closure $A_R$ of the inner band satisfies $A_R\cap S\subseteq {\mathrm{int}}_S{C}_R$. The map $F_R=({\pi }_R,{\rho }_R)$ is submersive on $S$ by the induction hypothesis. On an overlap with $R<T<S$ the old identities give $F_R=F_R{\mathrm{\mid }}_T\circ F_T$ wherever the compositions are defined, so every active lower controller factors through the maximal active one, and by step 2.1 the simultaneously active indices are totally ordered. Thus the chain hypothesis of [F2] holds. If the root map $f$ is present, treat it as an everywhere active controller: on lower tubes $f=f{\mathrm{\mid }}_R\circ {\pi }_R$, and it is submersive on $S$ where no lower controller is active.

Apply [F2] to the manifold $S$, the finitely many controllers $F_R$ with constraint sets $C_R$ (and $f$ if present), obtaining $E_S$ and ${\pi }_S$ that preserve $F_R$ whenever the base point lies in $C_R$, and preserve $f$ in the root case. Arrange that every point of the new tube lying in an inner $R$-band has its $S$-base in $C_R$: choose by [F3] a smooth nonnegative ${\psi }_R$ supported in ${\mathrm{int}}_S{C}_R$ and positive on the relatively closed $A_R\cap S$, put $b_R(s)=\mathrm{dist}(s,A_R)+{\psi }_R(s)>0$, and impose $\mathrm{\parallel }{E}_S(s,v)-s\mathrm{\parallel }<b_R(s)\mathrm{/}2$ for every $R$ using the size-bound clause of [F2]. If $s\mathrm{\notin }{C}_R$ and $E_S(s,v)\in A_R$, then ${\psi }_R(s)=0$ and $\mathrm{dist}(s,A_R)\le \mathrm{\parallel }{E}_S(s,v)-s\mathrm{\parallel }<\mathrm{dist}(s,A_R)\mathrm{/}2$, a contradiction; so $s\in C_R$ and $F_R(E_S(s,v))=F_R(s)$. Both the radius and its base-dependent bound depend only on $F_R$, so membership in the inner $R$-band is also preserved under ${\pi }_S$. Restricting every old final tube to its inner band keeps the compositions defined on the actual intersections and keeps ${\pi }_R{\pi }_S={\pi }_R$ and ${\rho }_R{\pi }_S={\rho }_R$ there.