$\mathcal{D}=\mathcal{B}\cup\mathcal{G}$ can be performed with any
$\epsilon>0$ sufficiently small to meet these requirements, which eventually
just depend on $M$.

Before attempting a "global" decomposition of
$\mathcal{D}=\mathcal{D}\cup\mathcal{B}$, we perform a localised version. Fix
$\bar{Q}\in\mathcal{D}$ arbitrary, and write
$\mathcal{D}(\bar{Q}):=\\{Q\in\mathcal{D}:Q\subset\bar{Q}\\}$. We claim that
$\mathcal{D}(\bar{Q})$ can be partitioned to collections
$\mathcal{B}(\bar{Q})$ and $\mathcal{G}(\bar{Q})$ with the properties
(D1)-(D4).

We simply let $\mathcal{B}(\bar{Q})$ consist of all the rectangles
$Q\in\mathcal{D}(\bar{Q})$ with

$\beta_{f}(HQ)\geq H^{-1}A^{-2}(1+\Sigma)^{-2}\epsilon.$ (6.2)

Let
$\mathcal{G}(\bar{Q}):=\mathcal{D}(\bar{Q})\,\setminus\,\mathcal{B}(\bar{Q})$.
For $Q\in\mathcal{G}(\bar{Q})$, let $\mathbb{V}_{Q}$ be the best approximating
plane for $f$ in $HQ$, so
$\beta_{f}(\mathbb{V}_{Q};HQ)<H^{-1}A^{-1}(1+\Sigma)^{-2}\epsilon$. In
particular, $\beta_{f}(\mathbb{V};Q)<\epsilon$, so if
$\theta\in[-\tfrac{\pi}{2},\tfrac{\pi}{2})$ is the unique parameter with
$\angle(\mathbb{V}_{Q},\mathbb{W}_{\theta})=0$, and $\epsilon,\delta>0$ are
small enough, depending only on $M$, Corollary 5.13 implies
$\mathcal{H}^{3}(\Pi_{\theta}[f(Q))]\geq\delta\mathcal{H}^{3}(Q)$. We also
define $\mathbb{W}_{Q}:=\mathbb{W}_{\theta}$ for $Q\in\mathcal{G}(\bar{Q})$,
so

$\angle(\mathbb{V}_{Q},\mathbb{W}_{Q})=0,\qquad Q\in\mathcal{G}(\bar{Q}).$
(6.3)

We now perform the tree decomposition of $\mathcal{G}(\bar{Q})$ in the only
possible way. Pick some maximal rectangle $Q\in\mathcal{G}(\bar{Q})$ which has
not yet been assigned to any tree. If either the $\mathcal{D}$-parent
"$\hat{Q}$" of $Q$ lies in $\mathcal{D}\,\setminus\,\mathcal{G}(\bar{Q})$, or
$\hat{Q}\in\mathbf{Leaves}(\hat{\mathcal{T}})$ for some existing tree
$\hat{\mathcal{T}}$, then $Q$ becomes the top rectangle of a new tree
$\mathcal{T}$, and we set
$\mathbb{W}_{\mathcal{T}}:=\mathbb{W}_{Q}=\mathbb{W}_{Q(\mathcal{T})}$.
Otherwise,
$\hat{Q}\in\hat{\mathcal{T}}\,\setminus\,\mathbf{Leaves}(\hat{\mathcal{T}})$
for some existing tree $\hat{\mathcal{T}}$, and we add $Q$ to
$\hat{\mathcal{T}}$. Now that $Q$ has been assigned to some tree
$\mathcal{T}$, we declare when, exactly, $Q$ is a minimal rectangle in
$\mathcal{T}$:

$Q\in\mathbf{Leaves}(\mathcal{T})\quad\Longleftrightarrow\quad\mathbf{ch}(Q)\cap\mathcal{B}(\bar{Q})\neq\emptyset\quad\text{or}\quad\angle(\mathbb{V}_{Q},\mathbb{W}_{\mathcal{T}})\geq\Sigma/2.$

This completes the construction of the trees. Note that, for every tree
$\mathcal{T}$, the family $\mathbf{Leaves}(\mathcal{T})$ is a union of the
collections

$\mathbf{Leaves}_{1}(\mathcal{T}):=\\{Q\in\mathbf{Leaves}(\mathcal{T}):\mathbf{ch}(Q)\cap\mathcal{B}(\bar{Q})\neq\emptyset\\}$

and

$\mathbf{Leaves}_{2}(\mathcal{T}):=\\{Q\in\mathbf{Leaves}(\mathcal{T}):\angle(\mathbb{V}_{Q},\mathbb{W}_{\mathcal{T}})\geq\Sigma/2\\}.$

Let us mention that, while it is possible that
$Q(\mathcal{T})\in\mathbf{Leaves}_{1}(\mathcal{T})$, we always have
$Q(\mathcal{T})\in\mathcal{T}\,\setminus\,\mathbf{Leaves}_{2}(\mathcal{T})$,
because $\angle(\mathbb{V}_{Q(\mathcal{T})},\mathbb{W}_{\mathcal{T}})=0$ by
(6.3) and the choice $\mathbb{W}_{\mathcal{T}}:=\mathbb{W}_{Q(\mathcal{T})}$.

Which conditions among (D1)-(D4) are satisfied? The Carleson packing condition
for $\mathcal{B}(\bar{Q})$ required in (D1) is satisfied by Corollary 4.20
(also because $\Sigma\sim_{M}1$, as we will soon see). The property (D3) of
$\mathbf{Leaves}(\mathcal{T})$ is evidently satisfied by construction. The
projection condition (6.1) and the strong $\beta$-number bound in (D2) are
also satisfied by construction (and because
$\mathcal{T}\subset\mathcal{G}(\bar{Q})$), so the only part of (D2) in doubt
is the slope bound $\angle(\mathbb{V}_{Q},\mathbb{W}_{\mathcal{T}})\leq\Sigma$
for $Q\in\mathcal{T}$. We handle this in the next lemma before turning to the
most difficult property (D4).

######  Lemma 6.4.

Let $\mathcal{T}$ be one of the trees constructed above. If the constant
$\epsilon>0$ is chosen small enough, depending on $M,\Sigma$, then
$\angle(\mathbb{V}_{Q},\mathbb{W}_{\mathcal{T}})\leq\Sigma$ for all
$Q\in\mathcal{T}$.

###### Proof.

The inequality is only in doubt for $Q\in\mathbf{Leaves}_{2}(\mathcal{T})$, so
fix $Q\in\mathbf{Leaves}_{2}(\mathcal{T})$. Note that the $\mathcal{D}$-parent
$\hat{Q}$ of $Q$ lies in
$\mathcal{T}\,\setminus\,\mathbf{Leaves}_{2}(\mathcal{T})$, hence satisfies
$\angle(\mathbb{V}_{\hat{Q}},\mathbb{W}_{\mathcal{T}})<\Sigma/2$, because
$Q(\mathcal{T})\notin\mathbf{Leaves}_{2}(\mathcal{T})$.

To show that $\angle(\mathbb{V}_{Q},\mathbb{W}_{\mathcal{T}})\leq\Sigma$, pick
two points $w_{1},w_{2}\in Q\subset\hat{Q}$ which lie on a common horizontal
line and satisfy $d_{\mathrm{par}}(w_{1},w_{2})=\ell(Q)$. Since
$\beta_{f}(Q)<\epsilon$ and $\beta_{f}(\hat{Q})<\epsilon$, there exist
horizontal lines $L_{Q}\subset\mathbb{V}_{Q}$ and
$L_{\hat{Q}}\subset\mathbb{V}_{\hat{Q}}$ with the property

$\\{f(w_{1}),f(w_{2})\\}\subset N(L_{Q},\epsilon\ell(Q))\cap
N(L_{\hat{Q}},2\epsilon\ell(Q)),$

and moreover
$\angle(L_{\hat{Q}},\mathbb{W}_{\mathcal{T}})=:\angle(\mathbb{V}_{\hat{Q}},\mathbb{W}_{\mathcal{T}})<\Sigma/2$.
So, it now suffices to prove the following geometric statement for
$r:=2\ell(Q)>0$. Assume that $L_{1},L_{2}\subset\mathbb{H}$ are horizontal
lines, $\mathbb{W}^{\prime}$ is a vertical subgroup, and
$p_{1},p_{2}\in\mathbb{H}$ are points satisfying

  * (a)

$\angle(L_{1},\mathbb{W}^{\prime})<\Sigma/2$,

  * (b)

$r/M\leq d(p_{1},p_{2})\leq Mr$,

  * (c)

$\\{p_{1},p_{2}\\}\subset N(L_{1},\epsilon r)\cap N(L_{2},\epsilon r)$.

Then $\angle(L_{2},\mathbb{W}^{\prime})\leq\Sigma$ if $\epsilon>0$ is chosen
small enough, depending on $M,\Sigma$. After rotating $\mathbb{W}^{\prime}$ to
$\mathbb{W}$, if necessary, this statement follows directly from Lemma 3.6
(more precisely the first part about "$|a_{2}|\leq 2\Sigma$"). ∎

We finally onsider condition (D4). Let $\kappa\sim_{M}\delta\sim_{M}1$ be a
suitable small constant to be determined soon, and classify the trees
according to the following three types:

  1. (i)

Those with
$|Q(\mathcal{T})\,\setminus\,\cup\mathbf{Leaves}(\mathcal{T})|\geq\kappa\cdot|Q(\mathcal{T})|$.

  2. (ii)

Those with
$|\cup\mathbf{Leaves}_{1}(\mathcal{T})|\geq\kappa\cdot|Q(\mathcal{T})|$.

  3. (iii)

Those with
$|\cup\mathbf{Leaves}_{2}(\mathcal{T})|\geq(1-2\kappa)\cdot|Q(\mathcal{T})|$.

Every tree has one of the types (i)-(iii), because

$Q(\mathcal{T})\subset[Q(\mathcal{T})\,\setminus\,\cup\mathbf{Leaves}(\mathcal{T})]\cup[\cup\mathbf{Leaves}_{1}(\mathcal{T})]\cup[\cup\mathbf{Leaves}_{2}(\mathcal{T})].$

We next claim that _trees of type (iii) do not exist_ if the "approximation
parameter" $\epsilon>0$ is chosen small enough, and and the "slope parameter"
$\Sigma\geq 1$ is chosen large enough, depending only on $\delta\sim_{M}1$. To
see this, let $\mathcal{T}$ be a hypothetical tree of type (iii), and recall
from (6.1) that

$\mathcal{H}^{3}(\Pi_{\theta}[f(Q(\mathcal{T}))])\geq\delta\mathcal{H}^{3}(Q(\mathcal{T}))$
(6.5)

for the unique parameter $\theta\in[-\tfrac{\pi}{2},\tfrac{\pi}{2})$ such that
$\mathbb{W}_{\mathcal{T}}=\mathbb{W}_{\theta}$. We claim that

$\mathcal{H}^{3}(\Pi_{\theta}[f(Q)])=o(\epsilon,\Sigma)\cdot\mathcal{H}^{3}(Q),\qquad
Q\in\mathbf{Leaves}_{2}(\mathcal{T}),$ (6.6)

where $o(\epsilon,\Sigma)$ denotes a constant which tends to $0$ as
$\epsilon\to 0$ and $\Sigma\to\infty$. Let us complete the non-existence proof
of trees of type (iii), assuming that (6.6) has already been established. In
addition to (6.6), we will need the following estimate:

$\displaystyle\mathcal{H}^{3}(\Pi_{\theta}[f(Q(\mathcal{T})\,\setminus\,\cup\mathbf{Leaves}_{2}(\mathcal{T}))])$
$\displaystyle\lesssim\mathcal{H}^{3}(f(Q(\mathcal{T})\,\setminus\,\cup\mathbf{Leaves}_{2}(\mathcal{T})))$
$\displaystyle\lesssim_{M}\kappa\cdot|Q(\mathcal{T})|.$ (6.7)

The first inequality follows from the well-known fact that vertical
projections do not increase $3$-dimensional Hausdorff measure by more than a
constant, see [7, Lemma 3.6]. The second inequality follows from the the
$M$-bilipschitz property of $f$, and the hypothesis that $\mathcal{T}$ has
type (iii). By (6.7), we may choose $\kappa\sim_{M}\delta$ so small that
$\mathcal{H}^{3}(\Pi_{\theta}[f(Q(\mathcal{T})\,\setminus\,\cup\mathbf{Leaves}_{2}(\mathcal{T}))])<(\delta/2)\cdot\mathcal{H}^{3}(Q(\mathcal{T}))$.
Using this estimate, (6.5), and the disjointness of the rectangles in
$\mathbf{Leaves}(\mathcal{T})$, we have

$\displaystyle\delta\mathcal{H}^{3}(Q(\mathcal{T}))$
$\displaystyle\leq\mathcal{H}^{3}(\Pi_{\theta}[f(\cup\mathbf{Leaves}_{2}(\mathcal{T}))])+\mathcal{H}^{3}(\Pi_{\theta}[f(Q(\mathcal{T})\,\setminus\,\cup\mathbf{Leaves}_{2}(\mathcal{T}))])$
$\displaystyle\leq\sum_{Q\in\mathbf{Leaves}_{2}(\mathcal{T})}\mathcal{H}^{3}(\Pi_{\theta}[f(Q)])+\frac{\delta}{2}\cdot|Q(\mathcal{T})|$
$\displaystyle\leq
o(\epsilon,\Sigma)\sum_{Q\in\mathbf{Leaves}(\mathcal{T})}\mathcal{H}^{3}(Q)+\frac{\delta}{2}\cdot|Q(\mathcal{T})|<\delta|Q(\mathcal{T})|,$

and choosing $\epsilon>0$ so small and $\Sigma\geq 1$ so large, depending on
$\delta\sim_{M}1$, that $o(\epsilon,\Sigma)<\delta/2$. This contradiction will
show that trees of type (iii) exist, once (6.6) has been established.

To prove (6.6), fix $Q\in\mathbf{Leaves}_{2}(\mathcal{T})$. Applying the
rotation $R_{\theta}^{-1}$, we may assume $\theta=0$, so $\Pi_{\theta}=\Pi$,
$\mathbb{W}_{\mathcal{T}}=\mathbb{W}$, and
$\angle(\mathbb{V}_{Q})=\angle(\mathbb{V}_{Q},\mathbb{W}_{\mathcal{T}})\geq\Sigma/2$,
where $\mathbb{V}_{Q}$ is a vertical plane with $f(Q)\subset
N(\mathbb{V},\epsilon\ell(Q))$. Of course we also have $f(Q)\subset
B(f(c_{Q}),M\ell(Q))$. Putting this information together, (6.6) will follow
from a geometric estimate of the form

$\mathcal{H}^{3}(\Pi[B(p,r)\cap N(\mathbb{V},\epsilon
r)])=o(\epsilon,\Sigma)\cdot
r^{3},\qquad\angle(\mathbb{V})\geq\Sigma/2,\,p\in\mathbb{H},\,r>0.$

The claim is invariant under scaling and left translation, so we may assume
that $\mathbb{V}$ is a vertical subgroup, and $r=1$. Also, since we only need
to consider the situation where $B(p,r)\cap N(\mathbb{V},\epsilon
r)\neq\emptyset$, we may assume that $p\in\mathbb{V}$, replacing "$r$" by
"$2r$" if necessary. After this reduction, we may apply one more left
translation which keeps the subgroup $\mathbb{V}$ fixed, but sends $p$ to $0$;
in other words, it suffices to show

$\mathcal{H}^{3}(\Pi[B(0,1)\cap
N(\mathbb{V},\epsilon)])=o(\epsilon,\Sigma),\qquad\angle(\mathbb{V})\geq\Sigma/2.$

This is so obvious that we leave further calculations to the reader (if
$\epsilon=0$ and $\angle(\mathbb{V})=\infty$, the projection
$\Pi(B(0,1)\cap\mathbb{V})$ is just a segment of length $\sim 1$ on the
$t$-axis).

We have now proven that trees of type (iii) do not exist. So, to verify the
Carleson packing condition for the tops $Q(\mathcal{T})$ in property (D4), it
suffices to consider trees of type (i)-(ii). First, the sets
$Q(\mathcal{T})\,\setminus\,\cup\mathbf{Leaves}(\mathcal{T})\subset
Q(\mathcal{T})$ are disjoint for distinct trees, so

$\mathop{\sum_{\mathcal{T}\text{ type (i)}}}_{Q(\mathcal{T})\subset
Q_{0}}|Q(\mathcal{T})|\leq\frac{1}{\kappa}\mathop{\sum_{\mathcal{T}\text{ type
(i)}}}_{Q(\mathcal{T})\subset
Q_{0}}|Q(\mathcal{T})\,\setminus\,\cup\mathbf{Leaves}(\mathcal{T})|\leq\frac{1}{\kappa}\cdot|Q_{0}|\sim_{M}|Q_{0}|,\quad
Q_{0}\in\mathcal{D}.$

To handle trees of type (ii), we use the Carleson packing condition for the
strong vertical $\beta$-numbers in Corollary 4.20, recalling that every leaf
$Q\in\mathbf{Leaves}_{1}(\mathcal{T})$ has a child
$Q^{\prime}\in\mathbf{ch}(Q)$ with
$\beta_{f}(HQ^{\prime})\gtrsim_{M}\epsilon$:

$\mathop{\sum_{\mathcal{T}\text{ type (ii)}}}_{Q(\mathcal{T})\subset
Q_{0}}|Q(\mathcal{T})|\leq\frac{1}{\kappa}\mathop{\sum_{\mathcal{T}\text{ type
(ii)}}}_{Q(\mathcal{T})\subset
Q_{0}}\sum_{Q\in\mathbf{Leaves}_{1}(\mathcal{T})}|Q|\lesssim\frac{1}{\kappa}\mathop{\sum_{Q\subset
Q_{0}}}_{\beta(HQ)\gtrsim_{M}\epsilon}|Q|\lesssim_{\epsilon,H,M}|Q_{0}|,\quad
Q_{0}\in\mathcal{D}.$

This completes the proofs of (D1)-(D4) for the decomposition
$\mathcal{D}(\bar{Q})=\mathcal{B}(\bar{Q})\cup\mathcal{G}(\bar{Q})$, where
$\bar{Q}\in\mathcal{D}$ was an arbitrary "starting rectangle".

To infer, from this special case, a decomposition
$\mathcal{D}=\mathcal{B}\cup\mathcal{G}$ satisfying properties (D1)-(D4), one
follows [15, p. 38] verbatim. Choose a sequence of disjoint "starting
rectangles" $\bar{Q}_{1},\bar{Q}_{2},\ldots$ with the following property:
every rectangle in $\mathcal{D}_{n}$ is contained in $\bigcup\bar{Q}_{j}$ for
$n\geq 0$, and for $n<0$, the same is still true for every rectangle in
$\mathcal{D}_{n}\,\setminus\,\mathcal{E}_{n}$, where
$\operatorname{card}\mathcal{E}_{n}\lesssim 1$. Such a sequence
$\bar{Q}_{1},\bar{Q}_{2}\ldots$ is easy to pick by hand, or see [15, p. 38]
for a more abstract construction; the idea is roughly to cover $B(0,1)$ by
disjoint rectangles of side-length $1$, then $B(0,2)\,\setminus\,B(0,1)$ by
disjoint rectangles of side-length $2$ (without overlapping the unit
rectangles), and so on. Once this is done properly, all the rectangles in
$Q\in\mathcal{D}_{n}$ with $\operatorname{dist}(Q,0)\gg 2^{-n}$ will be
contained in the union $\bigcup\bar{Q}_{j}$, and of course
$\operatorname{card}\\{Q\in\mathcal{D}_{n}:\operatorname{dist}(Q,0)\lesssim
2^{-n}\\}\lesssim 1$.

Now, we define $\mathcal{G}:=\cup\mathcal{G}(\bar{Q}_{j})$ and
$\mathcal{B}:=\mathcal{D}\,\setminus\,\mathcal{G}$. The tree partition of
$\mathcal{G}$ is given as the union of the tree partitions of the families
$\mathcal{G}(\bar{Q}_{j})\subset\mathcal{D}(\bar{Q}_{j})$. The conditions
(D2)-(D3) concerning individual trees are valid without problems, and the
Carleson packing condition (D4) follows from the disjointness of the
rectangles $\bar{Q}_{1},\bar{Q}_{2},\ldots$ The Carleson packing condition for
$\mathcal{B}$ postulated in (D1) is also valid, because

$\mathcal{B}=\bigcup_{j\geq
1}\mathcal{B}(\bar{Q}_{j})\cup\bigcup_{n\in\mathbb{Z}}\mathcal{E}_{n}.$

This completes the construction of the trees satisfying conditions (D1)-(D4).

##  7\. Constructing intrinsic bilipschitz graphs

Recalling the the decomposition $\mathcal{D}=\mathcal{B}\cup\mathcal{G}$
achieved in the previous section, the proof of Theorem 1.3 is only one
component short: to every tree $\mathcal{T}:=\mathcal{T}_{j}$, satisfying the
properties stated in (D2), we need to associate an intrinsic bilipschitz graph
$\Gamma_{\mathcal{T}}$ which $\eta$-approximates $f$ in the sense specified in
(1.8). Performing a rotation around the $t$-axis if necessary, we may assume
that the tree $\mathcal{T}$ is $(\epsilon,H,\Sigma)$-vertical, where
$\epsilon,H$ may be chosen freely (as long as $\epsilon>0$ was taken small
enough, depending on $M$), and $\Sigma$ is a parameter determined by $M$. The
existence of $\Gamma_{\mathcal{T}}$ is the content of the following
proposition:

######  Proposition 7.1.

For every $M\geq 1$ and $\eta,\Sigma>0$, there exist constants $\epsilon>0$
and $H\geq 10$ such that the following holds. Let
$f\colon\mathbb{W}\to\mathbb{H}$ be an $M$-bilipschitz map, and let
$\mathcal{T}\subset\mathcal{D}$ be an $(\epsilon,H,\Sigma)$-vertical tree
associated with $f$. Then, there exists an intrinsic Lipschitz graph
$\Gamma_{\mathcal{T}}\subset\mathbb{H}$ over the plane $\mathbb{W}$ such that

$\operatorname{dist}(f(w),\Gamma_{\mathcal{T}})\leq\eta\cdot\ell(Q),\qquad
w\in 2Q,\,Q\in\mathcal{T}.$ (7.2)

The intrinsic Lipschitz constant of $\Gamma_{\mathcal{T}}$ only depends on $M$
and $\Sigma$. The graph $\Gamma_{\mathcal{T}}$ can be chosen so that it admits
a $C$-bilipschitz parametrisation by $\mathbb{W}$, where $C$ also only depends
on $M$ and $\Sigma$.

The intrinsic graph $\Gamma_{\mathcal{T}}$ will be constructed by specifying
its _characteristic curves_ , but any general theory of these curves is not
required to understand the following proposition:

######  Proposition 7.3.

Let $M\geq 1$, $\Sigma,\eta>0$, $f\colon\mathbb{W}\to\mathbb{H}$, and
$\mathcal{T}$ have the same meaning as in Proposition 7.1. Then, the following
holds if $H,K\geq 1$ are chosen large enough depending on $M,\Sigma,\eta$, and
$\epsilon\in(0,1]$ is chosen small enough depending on all the previous
parameters. There exists a map
$\tau=\tau_{\mathcal{T}}\colon\mathbb{W}\to\mathbb{R}^{2}$ with the following
properties.

  * ($\tau_{1}$)

For every $t\in\mathbb{R}$ fixed, $y\mapsto\tau(y,t)\in C^{1,1}(\mathbb{R})$,
and _a fortiori_ $\partial_{y}\tau(y,t)$ is $2\Sigma$-Lipschitz:

$|\partial_{y}\tau(y_{2},t)-\partial_{y}\tau(y_{2},t)|\leq
2\Sigma|y_{2}-y_{1}|,\qquad y_{1},y_{2}\in\mathbb{R}.$

  * ($\tau_{2}$)

There exists a constant $\theta>0$, depending only on $M,\Sigma$, such that

$\theta\cdot|t_{2}-t_{1}|\leq|\tau(y,t_{2})-\tau(y,t_{1})|\leq\theta^{-1}\cdot|t_{2}-t_{1}|,\qquad
t_{1},t_{2},y\in\mathbb{R}.$

In particular, the graphs $\\{(y,\tau(y,t)):y\in\mathbb{R}\\}$,
$t\in\mathbb{R}$, foliate $\mathbb{R}^{2}$.

  * ($\tau_{3}$)

For every $y\in\mathbb{R}$ fixed, the map $t\mapsto\partial_{y}\tau(y,t)$ is
$\tfrac{1}{2}$-Hölder continuous with constant $\eta$:

$|\partial_{y}\tau(y,t_{2})-\partial_{y}\tau(y,t_{1})|\leq\eta\sqrt{|t_{2}-t_{1}|},\qquad
t_{1},t_{2}\in\mathbb{R}.$

  * ($\tau_{4}$)

Let $Q\in\mathcal{T}$ be a rectangle with corner $w:=(x,t)\in\mathcal{C}(Q)$,
and let $\ell\subset\mathbb{W}$ be the unique horizontal line containing $w$.
Let $q:=q_{Q,\ell}$ be the $(Q,\ell)$-approximate quadric introduced in
Section 4.3. Then,

$|\tau(y,t)-q(y)|\leq\eta\cdot\ell(Q)^{2}\quad\text{and}\quad|\dot{\tau}(y,t)-\dot{q}(y)|\leq\eta\cdot\ell(Q)$

for all $y\in B(f_{2}(w),K\ell(Q))$, where $f_{2}$ is the second component of
$f$.

######  Remark 7.4.

Property ($\tau_{4}$) captures, roughly speaking, that the values of $\tau$
are dictated by approximate quadrics associated with rectangles
$Q\in\mathcal{T}$. This statement encodes the relation between $\tau$ and $f$,
because, by Corollary 4.40, the values of the approximate quadrics evaluated
at $f_{2}(w)$ are nearly determined by $f(w)$. The property ($\tau_{4}$) is
eventually used to verify the $\eta$-approximation in (7.2), and for this
purpose, we will need to choose $K\geq A(1+\Sigma)M$ for a sufficiently large
absolute constant $A\geq 1$.

During the remainder of this section, we will reduce the proof of Proposition
7.1 (hence Theorem 1.3) to Proposition 7.3, which we now take for granted. The
proof of Proposition 7.3 is contained in the next section. To prove
Proposition 7.1, we define a function $\phi_{\mathcal{T}}$ implicitly by

$\phi_{\mathcal{T}}(0,y,\tau(y,t)):=\dot{\tau}(y,t),\qquad(y,t)\in\mathbb{W},$

where $\dot{\tau}(y,t):=\partial_{y}\tau(y,t)$, here and always. The
definition of $\phi_{\mathcal{T}}$ is well-posed by properties
($\tau_{1}$)-($\tau_{2}$). We also define the _intrinsic graph map_

$\displaystyle\Phi_{\mathcal{T}}(0,y,\tau(y,t))$
$\displaystyle:=(0,y,\tau(y,t))\cdot(\dot{\tau}(y,t),0,0),\qquad(y,t)\in\mathbb{W},$

and the intrinsic graph
$\Gamma_{\mathcal{T}}:=\\{\Phi_{\mathcal{T}}(0,y,t):(0,y,t)\in\mathbb{W}\\}$.

######  Remark 7.5.

I explain, briefly and informally, how the properties
($\tau_{1}$)-($\tau_{4}$) of the map "$\tau$" translate into geometric
properties of $\Gamma_{\mathcal{T}}$. The property ($\tau_{1}$) says nothing
beyond "$\phi_{\mathcal{T}}$ is an intrinsic Lipschitz function". Property
($\tau_{2}$) is the deepest property, and allows us to find a bilipschitz
parametrisation for $\Gamma_{\mathcal{T}}$; more on this very soon. Property
$(\tau_{3})$ means that the graph $\Gamma_{\mathcal{T}}$ looks everywhere, and
at every scale, like a "vertical Lipschitz flag", up to an error quantified by
the parameter $\eta$. Namely, if $\eta=0$, then $\dot{\tau}(y,t)$ is
independent of $t$, and therefore also
$(0,y,t)\mapsto\phi_{\mathcal{T}}(0,y,t)$ is independent of $t$. And the
intrinsic graph of an intrinsic Lipschitz function, which does not depend on
the $t$-variable, has the form
$\Gamma_{\mathbb{R}^{2}}\times\mathbb{R}\subset\mathbb{H}$, where
$\Gamma_{\mathbb{R}_{2}}\subset\mathbb{R}^{2}$ is a Lipschitz graph.

Finally, condition ($\tau_{4}$) means that the "Lipschitz flag" mentioned
above looks like a vertical plane near all points of the form
$\Phi_{\mathcal{T}}(0,f_{2}(w),t)$, where $w=(x,t)\in\mathcal{C}(Q)$ and
$Q\in\mathcal{T}$; to see this, try replacing "$\tau$" and "$\dot{\tau}$" by
"$q$" and "$\dot{q}$" in the definition of $\Phi_{\mathcal{T}}$, and notice
(or check form (7.15)) that the image of $\Phi_{\mathcal{T}}$ is a vertical
plane in this case.

The map $\Phi_{\mathcal{T}}$ is the "standard" intrinsic graph map of
$\phi_{\mathcal{T}}$, and is almost never a bilipschitz map
$\mathbb{W}\to\Gamma_{\mathcal{T}}$, as explained in Section 3.1. To obtain a
bilipschitz parametrisation of $\Gamma_{\mathcal{T}}$, one instead needs to
consider $\Psi\colon\mathbb{W}\to\mathbb{H}$ defined by

$\Psi(y,t):=\Phi_{\mathcal{T}}(0,y,\tau(y,t)),\qquad(y,t)\in\mathbb{W}.$ (7.6)

It is not a new idea that the map "$\Psi$" above is a good candidate for
bilipschitz parametrising $\Gamma_{\mathcal{T}}$ by $\mathbb{W}$; the same map
was used earlier in, at least, [13] and [4] in a scenario where
$\phi_{\mathcal{T}}$ is assumed _a priori_ Euclidean Lipschitz regular. This
need not be true in our setting, but condition ($\tau_{2}$) is an adequate
substitute.

######  Proposition 7.7.

The function $\phi_{\mathcal{T}}$ is an intrinsic Lipschitz function. Further,
the map $\Psi$ is a bilipschitz homeomorphism
$(\mathbb{W},d_{\mathrm{par}})\to\Gamma_{\mathcal{T}}$. The intrinsic
Lipschitz, and bilipschitz constants, here, depend only on
$\eta,\Sigma,\theta$ in the properties $(\tau_{1})$-$(\tau_{3})$.

Recalling that $\theta\sim_{M,\Sigma}1$, and choosing
$0<\eta<\min\\{\Sigma,\theta\\}$, Proposition 7.7 implies the claims in
Proposition 7.1 about the intrinsic Lipschitz constant of
$\Gamma_{\mathcal{T}}$, and the bilipschitz constant of its
$\mathbb{W}$-parametrisation $\Psi$.

###### Proof of Proposition 7.7.

Fix $y_{1},y_{2},t_{1},t_{2}\in\mathbb{R}$, and write
$p_{j}:=\Psi(y_{j},t_{j})$ for $j\in\\{1,2\\}$. These are two arbitrary points
on $\Gamma_{\mathcal{T}}$. We record immediately that $p^{-1}\cdot
p_{2}=\Psi(y_{1},t_{1})^{-1}\cdot\Psi(y_{2},t_{2})=(a,b,c)$, where

$\begin{cases}a=\dot{\tau}(y_{2},t_{2})-\dot{\tau}(y_{1},t_{1}),\\\
b=y_{2}-y_{1},\\\
c=\tau(y_{2},t_{2})-\tau(y_{1},t_{1})-\tfrac{1}{2}(y_{2}-y_{1})\cdot[\dot{\tau}(y_{1},t_{1})+\dot{\tau}(y_{2},t_{2})].\end{cases}$
(7.8)

To prove that $\phi_{\mathcal{T}}$ is intrinsic Lipschitz, we need to show
(recall (3.2)) that

$|\dot{\tau}(y_{2},t_{2})-\dot{\tau}(y_{1},t_{1})|\lesssim_{\eta,\Sigma,\theta}\|\Pi(p_{1}^{-1}\cdot
p_{2})\|,$ (7.9)

and to prove that $\Psi$ is bilipschitz, we need to show that

$d(p_{1},p_{2})=\|p_{1}^{-1}\cdot
p_{2}\|\sim_{\eta,\Sigma,\theta}\max\\{|y_{1}-y_{2}|,\sqrt{|t_{1}-t_{2}|}\\}.$
(7.10)

We start with the latter task. The third component "$c$" in (7.8) simplifies
if either $y_{1}=y_{2}$ or $t_{1}=t_{2}$. First, if $y_{1}=y=y_{2}$, then

$|c|=|\tau(y,t_{2})-\tau(y,t_{1})|\in[\theta|t_{2}-t_{1}|,\theta^{-1}|t_{2}-t_{1}|]$
(7.11)

by condition ($\tau_{2}$). Second, if $t_{1}=t=t_{2}$, then

$|c|=\left|\int_{y_{1}}^{y_{2}}\frac{2\dot{\tau}(y,t)-[\dot{\tau}(y_{1},t)+\dot{\tau}(y_{2},t)]}{2}\,dy\right|\lesssim\Sigma|y_{2}-y_{1}|^{2}$
(7.12)

by condition ($\tau_{1}$). Since moreover, by conditions ($\tau_{1})$ and
($\tau_{3}$),

$|a|\leq|\dot{\tau}(y_{2},t_{2})-\dot{\tau}(y_{1},t_{2})|+|\dot{\tau}(y_{1},t_{2})-\dot{\tau}(y_{1},t_{1})|\leq
2\Sigma|y_{2}-y_{1}|+\eta\sqrt{|t_{2}-t_{1}|},$

we can now use the triangle inequality to conclude from (7.11)-(7.12) that

$d(p_{1},p_{2})\lesssim\max\\{\Sigma,\sqrt{\Sigma},\eta,\theta^{-1/2}\\}\cdot
d_{\mathrm{par}}((y_{1},t_{1}),(y_{2},t_{2})).$

Hence $\Psi\colon\mathbb{W}\to\Gamma_{\mathcal{T}}$ is Lipschitz. To prove a
lower bound, observe first that

$d(\Psi(y_{1},t_{1}),\Psi(y_{2},t_{2}))=d(p_{1},p_{2})\geq|b|=|y_{2}-y_{1}|.$

So, if $\sqrt{|t_{2}-t_{1}|}\lesssim_{\Sigma,\theta}|y_{2}-y_{1}|$, the lower
bound is clear. If, on the other hand $|y_{2}-y_{1}|\ll\sqrt{|t_{1}-t_{2}|}$,
or more precisely
$(1+10\max\\{\Sigma,\sqrt{\Sigma}\\})|y_{2}-y_{1}|\leq\tfrac{1}{2}\sqrt{\theta}\sqrt{|t_{2}-t_{1}|}$,
then we use (7.11)-(7.12), and the upper bound for $|a|$ (with $t_{1}=t_{2}$)
to deduce

$\displaystyle d(p_{1},p_{2})$ $\displaystyle\geq
d(\Psi(y_{1},t_{1}),\Psi(y_{1},t_{2}))-d(\Psi(y_{1},t_{2}),(y_{2},t_{2}))$
$\displaystyle\geq\sqrt{\theta}\sqrt{|t_{2}-t_{1}|}-(1+10\max\\{\Sigma,\sqrt{\Sigma}\\})|y_{2}-y_{1}|$
$\displaystyle\geq\tfrac{1}{2}\sqrt{\theta}\sqrt{|t_{2}-t_{1}|}\gtrsim\sqrt{\theta}\cdot
d_{\mathrm{par}}((y_{1},t_{1}),(y_{2},t_{2})).$

This completes the proof of the bilipschitz property (7.10).

We then turn to the proof of the intrinsic Lipschitz condition (7.9). First,
we spell out

$\Pi(p_{1}^{-1}\cdot
p_{2})=(0,y_{2}-y_{1},\tau(y_{2},t_{2})-\tau(y_{1},t_{1})-(y_{2}-y_{1})\dot{\tau}(y_{1},t_{1})).$

Denote the third component "$h$", and estimate it from below as follows:

$\displaystyle|h|$
$\displaystyle\geq|\tau(y_{2},t_{2})-\tau(y_{2},t_{1})|-|\tau(y_{2},t_{1})-\tau(y_{1},t_{1})-(y_{2}-y_{2})\dot{\tau}(y_{1},t_{1})|$
$\displaystyle\geq\theta\cdot|t_{2}-t_{1}|-\left|\int_{y_{1}}^{y_{2}}\dot{\tau}(y,t_{1})-\dot{\tau}(y_{1},t_{1})\,dy\right|\geq\theta\cdot|t_{2}-y_{1}|-2\Sigma\cdot|y_{2}-y_{1}|^{2}.$

We used here properties ($\tau_{1}$)-($\tau_{2}$). Consequently, if
$2\Sigma\cdot|y_{1}-y_{2}|^{2}\leq\tfrac{\theta}{2}\cdot|t_{2}-t_{1}|$, then
$\|\Pi(p_{1}^{-1}\cdot
p_{2})\|\geq\sqrt{|h|}\gtrsim_{\theta}\sqrt{|t_{2}-t_{1}|}$. In the opposite
case,

$\|\Pi(p^{-1}\cdot
p_{2})\|\geq|y_{2}-y_{1}|\gtrsim_{\Sigma,\theta}\sqrt{|t_{2}-t_{1}|}.$

In both cases, $\|\Pi(p^{-1}\cdot
p_{2})\|\gtrsim_{\Sigma,\theta}|y_{2}-y_{1}|+\sqrt{|t_{2}-t_{1}|}$.

Next, we use properties ($\tau_{2}$)-($\tau_{3}$):

$\displaystyle|\dot{\tau}(y_{2},t_{2})-\dot{\tau}(y_{1},t_{1})|$
$\displaystyle\leq|\dot{\tau}(y_{2},t_{2})-\dot{\tau}(y_{1},t_{2})|+|\dot{\tau}(y_{1},t_{2})-\dot{\tau}(y_{1},t_{1})|$
$\displaystyle\leq 2\Sigma\cdot|y_{2}-y_{1}|+\eta\cdot\sqrt{|t_{2}-t_{1}|}.$

Comparing this upper bound with the lower bound $\|\Pi(p^{-1}\cdot
p_{2})\|\gtrsim_{\Sigma,\theta}|y_{2}-y_{1}|+\sqrt{|t_{2}-t_{1}|}$ completes
the proof of (7.9), and the proof of the proposition. ∎

The next proposition shows that the intrinsic Lipschitz graph
$\Gamma_{\mathcal{T}}$ satisfies the main requirement of Proposition 7.1,
namely (7.2). This will be based on properties ($\tau_{3}$)-($\tau_{4}$).

######  Proposition 7.13.

Let $Q\in\mathcal{T}$, and assume that the constant $K$ in ($\tau_{4}$)
satisfies $K\geq A(1+\Sigma)M$. Then,
$\operatorname{dist}(f(w),\Gamma_{\mathcal{T}})=o(\eta)\cdot\ell(Q)$ for all
$w\in 2Q$.

The notation $o(\eta)$ here refers to a constant which can be chosen
arbitrarily small by taking $\eta>0$ in Proposition 7.3 sufficiently small,
depending only on $M$ and $\Sigma$. Hence, Proposition 7.13 implies (7.2),
modulo the effort of re-naming small constants.

###### Proof of Proposition 7.13.

The idea is simple: since $Q\in\mathcal{T}$ is an
$(\epsilon,H,\Sigma)$-vertical rectangle, we know that $f(2Q)$ is approximated
by the vertical plane $\mathbb{V}:=\mathbb{V}_{Q}$ up to an error
$\epsilon\ell(Q)$. So, choosing $0<\epsilon\leq\eta$, it suffices to show that
$\mathbb{V}$ is further approximated by $\Gamma_{\mathcal{T}}$ in a
sufficiently large neighbourhood of $f(Q)$, up to an error
$o(\eta)\cdot\ell(Q)$.

For this purpose, let $w_{0}\in\mathcal{C}(Q)$ be an arbitrary corner. Then,
$2Q\subset B(w_{0},5\ell(Q))$, so

$f(2Q)\subset B(f(w_{0}),5M\ell(Q))\cap N(\mathbb{V},\epsilon\ell(Q))\subset
N(\mathbb{V}\cap B(f(w_{0}),6M\ell(Q)),\epsilon\ell(Q)),$

using $M\geq 1\geq\epsilon$ in the last inclusion. Abbreviate
$B(f(w_{0}),6M\ell(Q))=:B_{0}$. It remains to prove

$\mathbb{V}\cap B_{0}\subset N(\Gamma_{\mathcal{T}},o(\eta)\cdot\ell(Q)),$
(7.14)

because then $f(2Q)\subset N(\Gamma_{\mathcal{T}},o(\eta)\cdot\ell(Q))$.

To prove (7.14), let $\ell=\\{(y,t_{0}):y\in\mathbb{R}\\}\subset\mathbb{W}$ be
the unique horizontal line containing $w_{0}$, and let $q:=q_{Q,\ell}$ be the
$(Q,\ell)$-approximate quadric. Thus $q(y)=\tfrac{1}{2}ay^{2}+by+c$ for some
$|a|\leq\Sigma$ and $b,c\in\mathbb{R}$. With this notation, we parametrise
$\mathbb{V}$ with the map

$V(y,t)=(0,y,q(y)+t)\cdot(\dot{q}(y),0,0)=(ay+b,y,\tfrac{1}{2}by+c+t),$ (7.15)

which has the special property $V(y,t)=V(y,0)\cdot(0,0,t)$. The map $V$ is an
analogue of the bilipschitz parametrisation "$\Psi$" for $\mathbb{V}$; replace
$\tau(y,t)$ by $q(y)+t$ in (7.6) to see the connection). It is easy to see
that $V$ is a bilipschitz embedding with constant $\lesssim 1+\Sigma$; one can
either use the formula (7.15) directly, or repeat the proof of Proposition
7.7, noting that $t\mapsto q(y)+t$ is $1$-bilipschitz, and
$t\mapsto\partial_{y}[q(y)+t]=\dot{q}(y)$ is independent of $t$.

We next wish to parametrise $\mathbb{V}\cap B_{0}$ by $V$, and we claim the
following: there exists an absolute constant $A\geq 1$ such that

$\mathbb{V}\cap B_{0}\subset V(B[(f_{2}(w_{0}),0),AM(1+\Sigma)\ell(Q)]),$
(7.16)

The proof of (7.16) is based on recalling that $q=q_{Q,\ell}$ is the
approximate quadric associated with the horizontal line $\ell\ni w_{0}$.
Recalling the definition of the map $\iota_{Q}$ from (4.30), we in fact have
$V(f_{2}(w_{0}),0)=\iota_{Q}(w_{0})$ (use the formula (7.15) to verify this),
and hence

$d(f(w_{0}),V(f_{2}(w_{0}),0))=d(f(w_{0}),\iota_{Q}(w_{0}))\lesssim(1+\Sigma)\epsilon\ell(Q)$
(7.17)

by Lemma 4.29 (or see the discussion below (4.30)). Since $V$ is an
$A(1+\Sigma)$-bilipschitz parametrisation of $\mathbb{V}$, and $f(w_{0})$ is
the centre of $B_{0}$, (7.16) follows from (7.17).

With (7.16) in hand, (7.14) will evidently follow from

$V(B[(f_{2}(w_{0}),0),AM(1+\Sigma)\ell(Q)])\subset
N(\Gamma_{\mathcal{T}},o(\eta)\cdot\ell(Q)).$ (7.18)

To prove (7.18), pick $(y,h)\in B[(f_{2}(w_{0}),0),AM(1+\Sigma)\ell(Q)]$,
hence

$|y-f_{2}(w_{0})|\leq A(1+\Sigma)M\ell(Q)\quad\text{and}\quad\sqrt{|h|}\leq
A(1+\Sigma)M\ell(Q).$ (7.19)

We need to prove that
$\operatorname{dist}(V(y,t),\Gamma_{\mathcal{T}})=o(\eta)\cdot\ell(Q)$. We
start by verifying

$d(V(y,0),\Psi(y,t_{0}))=o(\eta)\cdot\ell(Q).$ (7.20)

Recall that $t_{0}$ is second coordinate of $w_{0}$. To prove (7.20), compute
explicitly

$\Psi(y,t_{0})^{-1}\cdot
V(y,0)=(\dot{q}(y)-\dot{\tau}(y,t_{0}),0,q(y)-\tau(y,t_{0})).$

Property ($\tau_{4}$), combined with the first part of (7.19) and $K\geq
A(1+\Sigma)M$, now implies

$|\tau(y,t_{0})-q(y)|\leq\eta\cdot\ell(Q)^{2}\quad\text{and}\quad|\dot{\tau}(y,t_{0})-\dot{q}(y)|\leq\eta\cdot\ell(Q),$

and (7.20) follows. As a consequence,

$d(V(y,h),\Psi(y,t_{0})\cdot(0,0,h))=d(V(y,0)\cdot(0,0,h),\Psi(y,t_{0})\cdot(0,0,h))\leq
o(\eta)\cdot\ell(Q).$

Now $\operatorname{dist}(V(y,h),\Gamma_{\mathcal{T}})=o(\eta)\cdot\ell(Q)$,
and hence (7.18), will follow as soon as we prove

$\operatorname{dist}(\Psi(y,t_{0})\cdot(0,0,h),\Gamma_{\mathcal{T}})=o(\eta)\cdot\ell(Q).$
(7.21)

To see this, let us spell out the definition:

$\Psi(y,t_{0})\cdot(0,0,t)=(\dot{\tau}(y,t_{0}),y,\tau(y,t_{0})+t+\tfrac{1}{2}y\cdot\dot{\tau}(y,t_{0})).$

By property ($\tau_{2}$), the map $t\mapsto\tau(y,t)$ is a
$\theta^{-1}$-bilipschitz homeomorphism $\mathbb{R}\to\mathbb{R}$, for any
fixed $y\in\mathbb{R}$, where $\theta\sim_{M,\Sigma}1$. Recalling from (7.19)
that $|h|\lesssim_{M,\Sigma}\ell(Q)^{2}$, there consequently exists
$t_{h}\in\mathbb{R}$ with $|t_{h}-t_{0}|\lesssim_{M,\Sigma}\ell(Q)^{2}$ such
that $\tau(y,t_{h})=\tau(y,t_{0})+h$. To complete the proof of (7.21), and the
proposition, we claim that

$d(\Psi(y,t_{0})\cdot(0,0,h),\Psi(y,t_{h}))\leq_{M,\Sigma}\eta\cdot\ell(Q).$

This follows immediately from computing explicitly

$[\Psi(y,t_{0})\cdot(0,0,h)]^{-1}\cdot\Psi(y,t_{h})=(\dot{\tau}(y,t_{h})-\dot{\tau}(y,t_{0}),0,0),$

recalling that $|t_{h}-t_{0}|\lesssim_{M,\Sigma}\ell(Q)^{2}$, and using the
$\tfrac{1}{2}$-Hölder continuity of $t\mapsto\dot{\tau}(y,t)$ with constant
$\eta$, recall property ($\tau_{3}$). The proof of Proposition 7.13 is
complete. ∎

##  8\. Reduction to an inductive construction

The purpose of this section is to reduce Proposition 7.3 to yet another
result, Proposition 8.3 below. We first describe this proposition, and the
details of the reduction are contained in Section 8.1. With no loss of
generality, we assume that the top rectangle of the given
$(\epsilon,H,\Sigma)$-vertical tree $\mathcal{T}$ is
$Q(\mathcal{T})=[0,1)^{2}\subset\mathbb{W}$. Also, with no loss of generality,
we assume that $\mathcal{T}$ is a positive vertical tree, that is, the
signature of $\mathcal{T}$ is $+$. The case of negative signature could be
reduced to the positive case, but in fact it would be easier to make minor
changes in the argument whenever the positive signature is used.

Instead of constructing $\tau\colon\mathbb{W}\to\mathbb{R}^{2}$ directly, we
will prove, inductively, the existence of a sequence of maps $\tau_{n}$,
$n\geq 0$, defined on the dyadic horizontal lines of generation $n$:

$\mathcal{L}_{n}=\\{\mathbb{R}\times\\{k\cdot 2^{-n}\\}:k\in\mathbb{Z}\\}.$

The lines in $\mathcal{L}_{n}$ will be in one-to-one correspondence with
_characteristics of generation $n$_, denoted $\mathcal{X}_{n}$, each of which
is a graph of a $C^{2}$-function, with $2\Sigma$-Lipschitz first derivative.
For $n\geq 0$, we will construct a map $\tau_{n}\colon\mathcal{L}_{n}\to
C^{2}(\mathbb{R})$. Thus, the $\tau_{n}$-image of every line in
$\mathcal{L}_{n}$ will be a real-valued $C^{2}$-function defined on
$\mathbb{R}$. The relationship between $\tau_{n}$ and $\tau$ will, eventually,
be that if $\ell=\mathbb{R}\times\\{k\cdot 2^{-n}\\}\in\mathcal{L}_{n}$, then

$\tau(y,k\cdot 2^{-n})\approx\tau_{n}(\ell)(y),\qquad y\in\mathbb{R},$

where the error tends geometrically to $0$ as $n\to\infty$.

The construction of the maps $\tau_{n}$ involves a number of constant
parameters; we gather here some reminders and information. We have already
encountered the bilipschitz constant $M$, the "slope" constant $\Sigma$, and
the constants $B\geq 5$ (such that $F_{Q}$ is vertically
$\epsilon^{\prime}\ell(Q)$-increasing on $BQ$ by Lemma 4.45), $H\geq
A(1+\Sigma)M^{2}B$, and $N=A(1+\Sigma)M$. Also, the constant $K\geq 1$ already
appeared in condition ($\tau_{4}$) of Proposition 7.3, and we mentioned in
Remark 7.4 that we need to choose $K\geq A(1+\Sigma)M$ for the purpose of
verifying (7.2). Given such a constant "$K$", we will below need to assume

$H\geq A\max\\{N^{2},KN\\}.$ (8.1)

The letter "$A$" will continue to denote a generic large absolute constant,
whose value may change without separate mention. In addition to all the
previous constants, there will be a new small constant $\delta>0$, which may
be specified by the "user", but must be sufficiently small in a manner
depending on $M,\Sigma$, and $\eta$ (as in property ($\tau_{3}$) of
Proposition 7.3). The constant "$\delta$" here has nothing to do with the
"projection constant" which appeared in (6.1); this previous "$\delta$" will
no longer be used.

Finally, the constant $\epsilon>0$ in the definition of
$(\epsilon,H,\Sigma)$-vertical rectangles needs to be chosen small in a manner
depending on all the previous constants. All in all, the relations between the
constants are given by the following informal sequence:

$1\lesssim_{\delta,K,M,\Sigma}\epsilon\ll\\{\delta,M,\Sigma\\}\ll K\ll
H\lesssim_{\delta,K,M,\Sigma}1.$ (8.2)

After this introduction, here is the statement containing the properties of
the maps $\tau_{n}$:

######  Proposition 8.3.

Let $M\geq 1$, $\Sigma>0$, let $\delta>0$ be sufficiently small relative to
$M,\Sigma$, let $K\geq 1$ be sufficiently large depending on
$\delta,M,\Sigma$, let $H\geq 1$ be sufficiently large depending $K,M,\Sigma$,
and let $\epsilon>0$ be sufficiently small depending on $\delta,H,K,M,\Sigma$.
Let $\mathcal{T}$ be an $(\epsilon,H,\Sigma)$-vertical tree with top
$Q(\mathcal{T})=[0,1)^{2}$. Then, for every $n\geq 0$, there exists a map
$\tau_{n}\colon\mathcal{L}_{n}\to C^{2}(\mathbb{R})$ satisfying the following
axioms (X1)-(X5).

The axioms (X1)-(X5) closely mirror the properties ($\tau_{1}$)-($\tau_{4}$)
listed in Proposition 7.3; in fact, (X1)-(X4) correspond directly to
($\tau_{1}$)-($\tau_{4}$), and (X5) is an additional axiom which ensures that
the maps $\tau_{n}$ "form a Cauchy sequence in $C^{1}$" – a deliberately vague
statement to be clarified in Section 8.1. We move to the axioms (X1)-(X5).

The first axiom requires that $\dot{\tau}_{n}(\ell)$ is $2\Sigma$-Lipschitz
for all $\ell\in\mathcal{L}_{n}$:

$|\dot{\tau}_{n}(\ell)(y_{1})-\dot{\tau}_{n}(\ell)(y_{2})|\leq
2\Sigma|y_{1}-y_{2}|,\qquad y_{1},y_{2}\in\mathbb{R},\,n\geq
0,\,\ell\in\mathcal{L}_{n}.$ (X1)

The characteristics will have the same "vertical ordering" as the lines in
$\mathcal{L}_{n}$. Since $\pi_{2}(\ell)\in\mathbb{R}$ is a singleton for all
$\ell\in\mathcal{L}_{n}$, we may define
$\ell_{2}-\ell_{1}:=\pi(\ell_{2})-\pi(\ell_{1})\in\mathbb{R}$. We write
$\ell_{2}>\ell_{1}$ if $\ell_{2}-\ell_{1}>0$. If
$\ell_{1},\ell_{2}\in\mathcal{L}_{n}$ with $\ell_{2}>\ell_{1}$, the second
axiom states that there exists a constant $\theta=\theta(M,\Sigma)>0$ such
that

$\theta(\ell_{2}-\ell_{1})\leq\tau_{n}(\ell_{2})(y)-\tau_{n}(\ell_{1})(y)\leq\theta^{-1}(\ell_{2}-\ell_{1}),\qquad
y\in\mathbb{R}.$ (X2)

This is the only axiom where the positive signature of $\mathcal{T}$ is
visible. If the signature were negative, one would simply replace
$\tau_{n}(\ell_{2})(y)-\tau_{n}(\ell_{1})(y)$ by
$\tau_{n}(\ell_{1})(y)-\tau_{n}(\ell_{2})(y)$.

The slopes of neighbouring characteristics do not differ much: the third axiom
states that if $\ell\in\mathcal{L}_{n}$, and
$\bar{\ell}\in\\{\ell,\ell^{\uparrow},\ell_{\downarrow}\\}$, then

$|\dot{\tau}_{n}(\ell)(y)-\dot{\tau}_{n}(\bar{\ell})(y)|\leq\delta\cdot
2^{-n/2},\qquad y\in\mathbb{R}.$ (X3)

Here $\ell^{\uparrow}$ and $\ell_{\downarrow}$ refer to the _$\mathcal{L}_{n}$
-neighbours of $\ell$_: by definition, $\ell^{\uparrow}:=\ell+2^{-n}$ and
$\ell_{\downarrow}:=\ell-2^{-n}$.

The fourth axiom nails the connection between the characteristics in
$\mathcal{X}_{n}$ and the map
$f=(f_{1},f_{2},f_{3})\colon\mathbb{W}\to\mathbb{H}$ via the
$(Q,\ell)$-approximate quadrics, recall Section 4.3. For $Q\in\mathcal{D}$,
recall the notation $\mathcal{C}(Q)$ for the four corners of $Q$.

######  Definition 8.4 ($\mathcal{C}$ and $\mathcal{C}_{m}$).

For $m\in\mathbb{Z}$, we write

$\mathcal{C}_{m}:=\bigcup_{Q\in\mathcal{T}_{m}}\mathcal{C}(Q)\quad\text{and}\quad\mathcal{C}:=\bigcup_{m\in\mathbb{Z}}\mathcal{C}_{m}.$

For a horizontal line $\ell\subset\mathbb{W}$, we also write
$\mathcal{C}_{m}(\ell):=\mathcal{C}_{m}\cap\ell$.

So, the set $\mathcal{C}$ consists of the corners of rectangles only in
$\mathcal{T}$. Since we assume that the top of $\mathcal{T}$ is $[0,1)^{2}$,
we have $\mathcal{T}_{m}=\emptyset$, hence $\mathcal{C}_{m}=\emptyset$, for
all $m<0$, and $\mathcal{C}_{0}=\\{(0,0),(1,0),(0,1),(1,1)\\}$. The axiom
(X4), below, together with the "stability" axiom (X5), implies that the
characteristics in $\mathcal{X}_{n}$ resemble $(Q,\ell)$-approximate quadrics
in large neighbourhoods of points of the form $f_{2}(w)$, where
$w\in\mathcal{C}_{\lceil n/2\rceil}(\ell)$ is a corner of
$Q\in\mathcal{T}_{\lceil n/2\rceil}$.

Fix $\ell\in\mathcal{L}_{n}$ and $y\in\mathbb{R}$. Let

$w\in\mathcal{C}_{\lceil n/2\rceil}(\ell)\cup\mathcal{C}_{\lceil
n/2\rceil}(\ell^{\uparrow})\cup\mathcal{C}_{\lceil
n/2\rceil}(\ell_{\downarrow})$

be a corner point such that $y\in B(f_{2}(w),K2^{-n/2})$. Let
$Q\in\mathcal{T}_{\lceil n/2\rceil}$ be any rectangle with corner $w$, and let
$q_{Q,\ell}$ be the $(Q,\ell)$-approximate quadric. We require that

$\begin{cases}|\tau_{n}(\ell)(y)-q_{Q,\ell}(y)|=o(\epsilon)\cdot 2^{-n},\\\
|\dot{\tau}_{n}(\ell)(y)-\dot{q}_{Q,\ell}(y)|=o(\epsilon)\cdot 2^{-n/2},\\\
|\ddot{\tau}_{n}(\ell)(y)-\ddot{q}_{Q,\ell}(y)|=o(\epsilon).\end{cases}$ (X4)

The $o(\epsilon)$-notation here means the following: we write
$|g(y)-h(y)|=o(\epsilon)\cdot\mathrm{const}$ if for all $\bar{\epsilon}>0$,
there exists $\epsilon>0$, depending only on $K,M,\Sigma$, such that
$|g(y)-h(y)|\leq\bar{\epsilon}\cdot\mathrm{const}$. The "triple control" in
(X4) is such a common notion in the proof below that we give it a separate
abbreviation: for $g,h\in C^{2}(\mathbb{R})$, and $y\in\mathbb{R}$, we write
$g(y)\approx_{n}h(y)$ if

$\begin{cases}|g(y)-h(y)|=o(\epsilon)\cdot 2^{-n},\\\
|\dot{g}(y)-\dot{h}(y)|=o(\epsilon)\cdot 2^{-n/2},\\\
|\ddot{g}(y)-\ddot{h}(y)|=o(\epsilon).\end{cases}$ (8.5)

Thus, (X4) is abbreviated to $\tau_{n}(y)\approx_{n}q_{Q,\ell}(y)$, for
appropriate $Q,\ell,y$.

Characteristics of generation $n-1$ are generally not characteristics of
generation $n$, for $n\geq 1$, but this is almost true: the final axiom states
that

$|\tau_{n}(\ell)(y)-\tau_{n-1}(\ell)(y)|\leq\delta\cdot
2^{-n}\quad\text{and}\quad|\dot{\tau}_{n}(\ell)(y)-\dot{\tau}_{n-1}(\ell)(y)|\leq\delta\cdot
2^{-n/2}$ (X5)

for all $\ell\in\mathcal{L}_{n-1}\subset\mathcal{L}_{n}$ and all
$y\in\mathbb{R}$.

######  Remark 8.6.

The appearance of $\lceil n/2\rceil$ in axiom (X4) may appear arbitrary and
awkward, but there is a reason for the numerology: for $n\geq 0$, we,
informally speaking, want $\tau_{n}\colon\mathcal{L}_{n}\to C^{2}(\mathbb{R})$
to keep records of all the data of the tree $\mathcal{T}$ down to a certain
depth $d(n)$. We choose $d(n)$ to be the _smallest integer such that that all
the lines in $\mathcal{L}_{n}$ contain corner points of rectangles in
$\mathcal{D}_{d(n)}$_. This means that we have to take $d(n):=\lceil
n/2\rceil$; see Figure 2 for an illustration of the case $n=2$. Note, for
example, that the horizontal line $\ell$ in the middle lies in
$\mathcal{L}_{1}$, and the smallest generation of rectangles with corners on
$\ell$ is $\lceil 1/2\rceil=1$. Most likely any fixed choice $d(n)\sim n/2$
would work equally well!

\begin{overpic}[scale={0.8}]{Image5.pdf} \put(8.0,1.0){$(0,0)$}
\put(72.0,1.0){$(1,0)$} \put(8.0,80.0){$(0,1)$} \put(72.0,80.0){$(1,1)$}
\put(84.0,9.0){$n=0$} \put(84.0,72.0){$n=0$} \put(84.0,41.0){$n=1$}
\put(84.0,25.0){$n=2$} \put(84.0,57.0){$n=2$} \put(5.0,41.0){$\ell$}
\end{overpic} Figure 2. The points in $\mathcal{C}_{0}$ depicted are in red,
and the points in $\mathcal{C}_{1}\,\setminus\,\mathcal{C}_{0}$ are depicted
in orange. The generations of the horizontal lines are indicated with "$n=j$"
for $j\in\\{0,1,2\\}$.

###  8.1. From Proposition 8.3 to Proposition 7.3

Let us take the maps $\tau_{n}\colon\mathcal{L}_{n}\to C^{2}(\mathbb{R})$ for
granted, and use them to complete the proof of Proposition 7.3. We start by
quantifying how the maps $\tau_{n}$ "form a Cauchy sequence in
$C^{1}(\mathbb{R})$". We claim that if $N>n\geq 0$,
$\ell_{n}\in\mathcal{L}_{n}$, and $\ell_{N}\in\mathcal{L}_{N}$ satisfy
$|\ell_{n}-\ell_{N}|\leq 100\cdot 2^{-n}$, then

$\|\tau_{n}(\ell_{n})-\tau_{N}(\ell_{N})\|_{L^{\infty}(\mathbb{R})}\lesssim\max\\{\delta,\theta\\}\cdot
2^{-n}\quad\text{and}\quad\|\dot{\tau}_{n}(\ell_{n})-\dot{\tau}_{N}(\ell_{N})\|_{L^{\infty}(\mathbb{R})}\lesssim\delta\cdot
2^{-n/2}.$ (8.7)

To see this, choose a sequence $\ell_{n+1},\ldots,\ell_{N}$ of horizontal
lines such that $\ell_{j}\in\mathcal{L}_{j}$, and
$\ell_{j-1}\in\\{\ell_{j},\ell_{j}^{\uparrow},\ell_{j\downarrow}\\}$ for all
$j\in\\{n+2,\ldots,N\\}$. Then $|\ell_{n+1}-\ell_{N}|\leq 2^{-n}$, so
$|\ell_{n}-\ell_{n+1}|\leq 101\cdot 2^{-n}$. It follows that $\ell_{n}$ is
$\leq 101$ steps away in $\mathcal{L}_{n}$ from $\ell_{n+1}$ (if
$\ell_{n+1}\in\mathcal{L}_{n}$) or one of the $\mathcal{L}_{n+1}$-neighbours
of $\ell_{n+1}$ (if
$\ell_{n+1}\in\mathcal{L}_{n+1}\,\setminus\,\mathcal{L}_{n}$). For example, if
$\ell_{n+1}\in\mathcal{L}_{n}$, then

$\|\dot{\tau}_{n}(\ell_{n})-\dot{\tau}_{N}(\ell_{N})\|_{L^{\infty}(\mathbb{R})}\leq\|\dot{\tau}_{n}(\ell_{n})-\dot{\tau}_{n}(\ell_{n+1})\|_{L^{\infty}(\mathbb{R})}+\sum_{j=n+2}^{N}\|\dot{\tau}_{j}(\ell_{j})-\dot{\tau}_{j-1}(\ell_{j-1})\|_{L^{\infty}(\mathbb{R})}.$

The first term can be estimated by $\leq 101$ applications of axiom (X3), and
the second term can be estimated using both axioms (X3) and (X5). After
summing a geometric series, the result is
$\|\dot{\tau}_{n}(\ell_{n})-\dot{\tau}_{N}(\ell_{N})\|_{L^{\infty}(\mathbb{R})}\lesssim\delta\cdot
2^{-n/2}$. The other cases, and the first inequality in (8.7), are proven such
a similar manner (also using axiom (X2)) that we omit the details.

We now construct the map $\tau\colon\mathbb{W}\to\mathbb{R}^{2}$. Fix
$t\in\mathbb{R}$, and choose a sequence of integers
$\\{k_{n}\\}_{n\in\mathbb{N}}$ such that $|t-k_{n}\cdot 2^{-n}|\leq 2^{-n}$
for all $n\in\mathbb{N}$. Then, writing
$\ell_{n}:=\mathbb{R}\times\\{k_{n}\cdot 2^{-n}\\}\in\mathcal{L}_{n}$, the
inequalities (8.7) imply that $\\{\tau_{n}(\ell_{n})\\}_{n\in\mathbb{N}}$ is a
Cauchy sequence in $C^{1}(\mathbb{R})$. Hence the limit

$\tau(\cdot,t):=\lim_{n\to\infty}\tau_{n}(\ell_{n})\in C^{1}(\mathbb{R})$

exists in the $C^{1}$-norm, and

$\dot{\tau}(y,t)=\lim_{n\to\infty}\dot{\tau}_{n}(\ell_{n})(y),\qquad
y\in\mathbb{R}.$

In particular, we see that $y\mapsto\dot{\tau}(y,t)=\partial_{y}\tau(y,t)$ is
$2\Sigma$-Lipschitz by axiom (X1), which gives property ($\tau_{1}$). We
remark that, as a corollary of (8.7), we have

$\|\tau_{n}(\ell)-\tau(\cdot,t)\|_{L^{\infty}(\mathbb{R})}\lesssim\max\\{\delta,\theta\\}\cdot
2^{-n}\quad\text{and}\quad\|\dot{\tau}_{n}(\ell)-\dot{\tau}(\cdot,t)\|_{L^{\infty}(\mathbb{R})}\lesssim\delta\cdot
2^{-n/2}$ (8.8)

whenever $n\geq 0$, $\ell\in\mathcal{L}_{n}$ and $|\pi_{2}(\ell)-t|\leq
100\cdot 2^{-n}$. This implies that the definition of $\tau(\cdot,t)$ does not
depend on the choice of the sequence $\\{k_{n}\\}_{n\in\mathbb{N}}$, as long
as $|t-k_{n}\cdot 2^{-n}|\leq 2^{-n}$ for all $n\in\mathbb{N}$. This leads to
an observation, which will strengthen first inequality in (8.8). Namely, if
$t=\pi_{2}(\ell)$ for some $\ell\in\mathcal{L}_{n}$, i.e. $t$ is a dyadic
rational, then $\tau_{m}(\ell)$ is defined for all $m\geq n$, and
$\tau(\cdot,t)$ can be expressed as
$\tau(\cdot,t)=\lim_{m\to\infty}\tau_{m}(\ell)$. It follows from (X5) that

$\|\tau_{n}(\ell)-\tau(\cdot,t)\|_{L^{\infty}(\mathbb{R})}\leq\sum_{m\geq
n}\|\tau_{m+1}(\ell)-\tau_{m}(\ell)\|_{L^{\infty}(\mathbb{R})}\lesssim\delta\cdot
2^{-n}.$ (8.9)

We then turn to the task of verifying properties ($\tau_{2}$)-($\tau_{4}$) for
$\tau$. The separation property ($\tau_{2}$) follows immediately from axiom
(X2):

$\tau(y,t_{2})-\tau(y,t_{1})=\lim_{n\to\infty}\tau_{n}(\ell^{1}_{n})(y)-\tau_{n}(\ell^{2}_{n})(y)\in[\theta(t_{2}-t_{1}),\theta^{-1}(t_{2}-t_{1})],$

where
$\\{\ell_{n}^{1}\\}_{n\in\mathbb{N}},\\{\ell_{n}^{2}\\}_{n\in\mathbb{N}}$ are
sequences of horizontal lines with $|\pi_{2}(\ell_{n}^{1})-t_{1}|\leq 2^{-n}$
and $|\pi_{2}(\ell_{n}^{2})-t_{2}|\leq 2^{-n}$. The Hölder continuity property
($\tau_{3}$) follows easily from (8.8) if we choose $\delta\leq c\eta$ for a
sufficiently small absolute constant $c>0$. To see this, let
$t_{1},t_{2}\in\mathbb{R}$. The case $|t_{1}-t_{2}|\geq 1$ is a little
special: then one needs to check from the case $n=0$ of the construction of
$\tau_{n}$ that the characteristics $\tau_{0}(\ell)$,
$\ell\in\mathcal{L}_{0}$, only differ by a constant, hence
$\dot{\tau}_{0}(\ell_{1})\equiv\dot{\tau}_{0}(\ell_{2})$ for
$\ell_{1},\ell_{2}\in\mathcal{L}_{0}$. This will indeed be so, see
(9.1)-(9.2), and the discussion below (9.3). Hence, if
$\ell_{1},\ell_{2}\in\mathcal{L}_{0}$ satisfy $|\pi_{2}(\ell_{j})-t_{j}|\leq
1$, we have

$\|\dot{\tau}(\cdot,t_{2})-\dot{\tau}(\cdot,t_{1})\|_{L^{\infty}(\mathbb{R})}\leq\|\dot{\tau}_{0}(\ell_{2})-\dot{\tau}(\cdot,t_{2})\|_{L^{\infty}(\mathbb{R})}+\|\dot{\tau}_{0}(\ell_{1})-\dot{\tau}(\cdot,t_{1})\|_{L^{\infty}(\mathbb{R})}\leq\eta\sqrt{|t_{1}-t_{2}|}.$

by (8.8). Consider then the case $|t_{1}-t_{2}|<1$, and pick $n\geq 0$ such
that $2^{-(n+1)}<|t_{1}-t_{2}|<2^{-n}$. Let
$\ell_{1},\ell_{2}\in\mathcal{L}_{n}$ be lines with
$|\pi_{2}(\ell_{j})-t_{j}|\leq 2^{-n}$. Then $\ell_{1},\ell_{2}$ are
neighbours, or perhaps two steps away, in $\mathcal{L}_{n}$, and hence axiom
(X3) implies

$\|\dot{\tau}_{n}(\ell_{2})-\dot{\tau}_{n}(\ell_{1})\|_{L^{\infty}(\mathbb{R})}\lesssim\delta\cdot
2^{-n/2}\lesssim\delta\cdot\sqrt{|t_{2}-t_{1}|}.$

It then follows from (8.8), and the triangle inequality, that
$\|\dot{\tau}(\cdot,t_{2})-\dot{\tau}(\cdot,t_{1})\|_{L^{\infty}(\mathbb{R})}\leq\eta\sqrt{|t_{2}-t_{1}|}$.
The proof of property ($\tau_{3}$) is complete.

Finally, to verify property ($\tau_{4}$), let $Q\in\mathcal{T}$. More
precisely, let $m\in\mathbb{N}$ be such that $Q\in\mathcal{T}_{m}$, so
$\ell(Q)=2^{-m}$. Let $w=(x,t)\in\mathcal{C}(Q)$ be a corner, and let
$\ell=\pi_{2}^{-1}\\{t\\}\subset\mathbb{W}$ be the unique horizontal line
containing $w$. Since the horizontal edges of all rectangles in
$\mathcal{D}_{m}$ are contained on lines in $\mathcal{L}_{2m}$, we have
$\ell\in\mathcal{L}_{n}$ and $w\in\mathcal{C}_{\lceil n/2\rceil}(\ell)$ with
$n:=2m$. Next, let $q=q_{Q,\ell}$ be the $(Q,\ell)$-approximate quadric, and
fix

$y\in B(f_{2}(w),K\ell(Q))=B(f_{2}(w),K2^{-n/2}).$

According to axiom (X4), we have

$|\tau_{n}(\ell)(y)-q(y)|=o(\epsilon)\cdot
2^{-n}\quad\text{and}\quad|\dot{\tau}_{n}(\ell)(y)-\dot{q}(y)|=o(\epsilon)\cdot
2^{-n/2}.$ (8.10)

Choosing $\delta\leq c\eta$ as before, $\epsilon>0$ small depending on
$\eta,K,M,\Sigma$, and combining (8.8)-(8.9) with (8.10), we find

$|\tau(y,t)-q(y)|\leq\eta\cdot\ell(Q)^{2}\quad\text{and}\quad|\dot{\tau}(y,t)-\dot{q}(y)|\leq\eta\cdot\ell(Q),$

as desired in property ($\tau_{4}$). This completes the proof of Proposition
7.3, modulo the proof of Proposition 8.3.

##  9\. Constructing the characteristics

In this section, we prove Proposition 8.3.

###  9.1. Case $n=0$

The construction of the characteristics in $\mathcal{X}_{n}$ is inductive, and
incorporates information about the characteristics already constructed in
$\mathcal{X}_{n-1}$. The only exception is the case $n=0$, where we now begin;
then $\lceil n/2\rceil=0$, and

$\mathcal{L}_{0}=\\{\mathbb{R}\times\\{k\\}:k\in\mathbb{Z}\\}=:\\{\ell_{k}:k\in\mathbb{Z}\\}.$

Let $Q_{0}:=[0,1)^{2}$ be the only rectangle in $\mathcal{T}_{0}$. For
$k\in\\{-1,0,1,2\\}$, let $q_{k}:=q_{Q_{0},\ell_{k}}$ be the
$(Q_{0},\ell_{k})$-approximate quadric. For the same indices
$k\in\\{-1,0,1,2\\}$, we also define

$\tau_{0}(\ell_{k})(y):=q_{k}(y).$ (9.1)

For the indices $k\notin\\{-1,0,1,2\\}$, the definition of
$\tau_{0}(\ell_{k})$ is a bit arbitrary: let us set

$\tau_{0}(\ell_{k}):=\tau_{0}(\ell_{0})(y)+C_{M,\Sigma}k,\qquad
k\notin\\{-1,0,1,2\\},$ (9.2)

where $C_{M,\Sigma}\sim N^{1/2}$ is a constant to be determined shortly.

This completes the definition of $\tau_{0}\colon\mathcal{L}_{0}\to
C^{1,1}(\mathbb{R})$, and now we need to check the axioms (X1)-(X5). Axiom
(X1) is easy: since $\mathcal{T}$ is an $(\epsilon,H,\Sigma)$-vertical tree,
every $(Q,\ell)$-approximate quadric $q_{Q,\ell}$ with $Q\in\mathcal{T}$
satisfies $\|\ddot{q}_{Q,\ell}\|_{L^{\infty}(\mathbb{R})}\leq\Sigma$. In
particular,

$\|\ddot{\tau}_{0}(\ell_{k})\|_{L^{\infty}(\mathbb{R})}\leq\Sigma,\qquad
k\in\mathbb{Z}.$ (9.3)

which is even better than (X1). Axiom (X2) requires some work, so we postpone
it for a moment. Axiom (X3) is a triviality: the quadrics $q_{k}$ are
associated to the same rectangle $Q_{0}$, hence only differ by a constant (the
graphs of every $q_{k}$ arise as $\Pi$-projections of certain horizontal in
the vertical plane $\mathbb{V}_{Q_{0}}$). Hence
$\dot{\tau}_{0}(\ell_{k})\equiv\dot{\tau}_{0}(\ell_{k+1})$ for all
$k\in\mathbb{Z}$.

There is very little to verify in axiom (X4), but let us do this over-formally
to get some familiarity with the axiom. Let $\ell\in\mathcal{L}_{0}$ and fix
$y\in\mathbb{R}$. Assume that there exists a rectangle $Q\in\mathcal{T}_{0}$
with corner
$w\in\mathcal{C}_{0}(\ell_{\downarrow})\cup\mathcal{C}_{0}(\ell)\cup\mathcal{C}_{0}(\ell^{\uparrow})$
such that $y\in B(f_{2}(w),K)$. The claim is that
$\tau_{0}(\ell)(y)\approx_{0}q_{Q,\ell}(y)$, where $q_{Q,\ell}$ is the
$(Q,\ell)$-approximate quadric.

Since $Q_{0}$ is the only rectangle in $\mathcal{T}_{0}$, we have $Q=Q_{0}$,
and $\ell\in\\{\ell_{-1},\ell_{0},\ell_{1},\ell_{2}\\}$. For such lines, in
(9.1), we defined $\tau_{0}(\ell)$ to be the $(Q_{0},\ell)$-approximate
quadric. Hence $\tau_{0}(\ell):=q_{Q_{0},\ell}\approx_{0}q_{Q_{0},\ell}$, and
the verification of axiom (X4) is complete.

Axiom (X5) is meaningless for the time being, since it compares
$\tau_{n-1}(\ell)$ with $\tau_{n}(\ell)$ for $n\geq 1$. So, we finally turn to
axiom (X2). We first claim that for $k\in\\{0,1,2\\}$,

$\tau_{0}(\ell_{k})(y)-\tau_{0}(\ell_{k-1})(y)=q_{k}(y)-q_{k-1}(y)\in[cN^{-1/2},CN^{1/2}],\qquad
y\in\mathbb{R},$ (9.4)

where $0<c\leq C<\infty$ are absolute constants. To see this, we recall that
the signature of $\mathcal{T}$, and hence $Q_{0}\in\mathcal{T}$ is $+$, which
meant that the $(N,\epsilon)$-QIE

$F:=F_{Q_{0}}:=\pi_{\mathbb{V}_{Q_{0}}}\circ\iota_{Q_{0}}=\psi_{\mathbb{V}_{Q_{0}}}\circ\Pi\circ\iota_{Q_{0}}\colon
HQ_{0}\to\mathbb{W}$

is vertically $\epsilon^{\prime}$-increasing on the rectangle $BQ_{0}\subset
HQ_{0}$, where $\epsilon^{\prime}=2N\epsilon$. This fact was established in
Lemma 4.45, assuming that $H\geq A(1+\Sigma)^{2}MB$. The quadric straightening
map $\psi_{0}:=\psi_{\mathbb{V}_{Q_{0}}}\colon\mathbb{W}\to\mathbb{W}$
appearing in the definition of $F$ was introduced just above (4.31). It had
the form

$\psi_{0}(y,t)=(y,t-\tfrac{1}{2}ay^{2}-by),$

where $a=a_{Q_{0}}$ and $b=b_{Q_{0}}$ are the common coefficients of
approximate $(Q_{0},\ell)$-quadrics, all of which have the form
$q_{\ell}(y)=\tfrac{1}{2}ay^{2}+by+c_{\ell}$. In particular, the quadrics
$q_{-1},q_{0},q_{1},q_{2}$ can be written in this general form for certain
coefficients $c_{j}:=c_{\ell_{j}}\in\mathbb{R}$, and $q_{k}-q_{k-1}\equiv
c_{k}-c_{k-1}$. So, proving (9.4) is equivalent to showing that
$c_{k}-c_{k-1}\sim_{N}1$. For this purpose, we define
$w_{k}:=(0,k)\in\mathbb{W}$. Since $F$ is vertically
$\epsilon^{\prime}$-increasing on $BQ_{0}$, and $B\geq 5$, in particular the
map $t\mapsto\pi_{2}(F(0,t))$ is $\epsilon^{\prime}$-increasing on $[k-1,k]$
for $k\in\\{0,1,2\\}$. Since
$\|\pi_{2}(w_{k})-\pi_{2}(w_{k-1})\|>\epsilon^{\prime}$ (recall from (4.43)
that $\epsilon\ll N^{-10}$), and $(\pi_{2}\circ F)|_{\ell}\equiv c_{\ell}$ by
the expression (4.35), this implies

$c_{k}=\pi_{2}(F(w_{k}))>\pi_{2}(F(w_{k-1}))>c_{k-1},\qquad k\in\\{0,1,2\\}.$

To establish the more quantitative estimate needed for (9.4), we use Lemma
3.30, which implies that the map $t\mapsto\pi_{2}(F(0,t))$ is an
$(N,2\epsilon)$-QIE on $([k-1,k],\|\cdot\|)$ (for applying Lemma 3.30, we also
use that $F$ is a QIE on $HQ_{0}$, a large neighbourhood of $[-1,2]^{2}$).
Hence, for $k\in\\{0,1,2\\}$,

$\displaystyle(2N)^{-1}$ $\displaystyle\leq
N^{-1}-2\epsilon\leq\|\pi_{2}(F(w_{k}))-\pi_{2}(F(w_{k-1}))\|=\|c_{k}-c_{k-1}\|\leq
N+2\epsilon\leq 2N.$

This completes the proof of (9.4). To complete the proof of axiom (X2), we
recall the definition

$\tau_{0}(\ell_{k})(y):=\tau_{0}(\ell_{0})(y)+C_{M,\Sigma}k=q_{0}(y)+C_{M,\Sigma}k,\qquad
k\notin\\{-1,0,1,2\\}.$

We see from (9.4), and recalling that all the quadrics $q_{k}$ only differ by
a constant, that axiom (X2) is satisfied if $C_{M,\Sigma}\sim N^{1/2}$ is
chosen sufficiently large. This completes the proof of Proposition 8.3 in the
case $n=0$.

###  9.2. Case $n\geq 1$

We next assume that the map $\tau_{n-1}\colon\mathcal{L}_{n-1}\to
C^{1,1}(\mathbb{R})$ has already been constructed for some $n\geq 1$, and
satisfies axioms (X1)-(X5). We now describe the construction of
$\tau_{n}\colon\mathcal{L}_{n}\to C^{1,1}(\mathbb{R})$. Fix
$\ell\in\mathcal{L}_{n}$. The construction is a little different in the cases
where also $\ell\in\mathcal{L}_{n-1}$ (so $\tau_{n-1}(\ell)$ is defined), and
where $\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$. The
constructions can, however, be carried out simultaneously.

Write $m:=\lceil n/2\rceil$, and let
$\mathcal{C}_{m}^{+}(\ell):=\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})\cup\mathcal{C}_{m}(\ell_{\downarrow})$.
We would like to define

$\mathcal{I}(\ell):=\\{I_{n}(w):w\in\mathcal{C}^{+}_{m}(\ell)\\},\quad\text{where}\quad
I_{n}(w):=\bar{B}(f_{2}(w),K2^{-n/2}),$

but the intervals $I_{n}(w)$, $w\in\mathcal{C}_{m}^{+}(\ell)$, may have plenty
of overlap, which leads to technical issues. To avoid these, we prune
$\mathcal{I}(\ell)$ as follows. Pick $I\in\mathcal{I}(\ell)$ arbitrarily and
see if it is covered by the intervals in
$\mathcal{I}(\ell)\,\setminus\,\\{I\\}$. If it is, discard it; otherwise keep
it. The process terminates in
$\leq\operatorname{card}\mathcal{I}(\ell)<\infty$ steps. The union of the
intervals never changed during the process, and the remaining intervals have
overlap bounded by $2$: if three intervals of the same length meet at a common
point, then one is covered by the two others.

We keep the notation $\mathcal{I}(\ell)$ for the sub-collection obtained
above. The key facts are these:

$\bigcup_{w\in\mathcal{C}_{m}^{+}(\ell)}I_{n}(w)=\bigcup_{I\in\mathcal{I}(\ell)}I\quad\text{and}\quad\sum_{I\in\mathcal{I}(\ell)}\mathbf{1}_{I}\leq
2.$ (9.5)

From the second property in (9.5), we infer that also

$\sum_{I\in\mathcal{I}(\ell)}\mathbf{1}_{2I}\leq 4,$ (9.6)

because if $y\in 2I_{1}\cap\cdots\cap 2I_{5}$, then
$I_{j}\subset[x-|I|,x+|I|]$ for all $1\leq j\leq 5$, hence
$5|I|\leq\int_{x-|I|}^{x+|I|}\sum\mathbf{1}_{I}\leq 4|I|$ by (9.5), a
contradiction.

With (9.5)-(9.6) in hand, we construct a partition of unity for
$\ell\in\mathcal{L}_{n}$ fixed. Define the (possibly empty) closed sets

$U(\ell):=\bigcup_{I\in\mathcal{I}(\ell)}I=\bigcup_{w\in\mathcal{C}_{m}(\ell)}I_{n}(w)\quad\text{and}\quad
U^{\sharp}(\ell):=\bigcup_{I\in\mathcal{I}(\ell)}\kappa_{0}I=\bigcup_{w\in\mathcal{C}_{m}(\ell)}\kappa_{0}I_{n}(w),$
(9.7)

where $\kappa_{0}\in(1,\tfrac{4}{5}\sqrt{2})$ is an arbitrary fixed constant;
let us for concreteness set

$\kappa_{0}:=\tfrac{1}{2}(1+\tfrac{4}{5}\sqrt{2}).$

The last set equation in (9.7) requires a little argument, since the selection
of the intervals in $\mathcal{I}(\ell)$ had, in principle, nothing to do with
the enlargements $\kappa_{0}I$. Fortunately, the same selection process
simultaneously yields the last equation in (9.7). Indeed, if some interval
$\kappa_{0}I_{n}(w)$, $w\in\mathcal{C}_{m}(\ell)$, were not covered by the
intervals $\kappa_{0}I$ with $I\in\mathcal{I}(\ell)$, then in particular
$I_{n}(w)\notin\mathcal{I}(\ell)$. Hence $I_{n}(w)\subset I_{1}\cup\ldots\cup
I_{k}$ for certain intervals $I_{1},\ldots,I_{k}\in\mathcal{I}(\ell)$, and it
is easy to check that consequently
$\kappa_{0}I_{n}(w)\subset\kappa_{0}I_{1}\cup\ldots\cup\kappa_{0}I_{k}$.

Let $\\{\varphi_{I,\ell}\\}_{I\in\mathcal{I}(\ell)}\subset
C^{\infty}(\mathbb{R})$ be a family of functions satisfying the following
properties:

$\mathbf{1}_{U(\ell)}\leq\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}\leq\mathbf{1}_{U^{\sharp}(\ell)}\quad\text{and}\quad\operatorname{spt}\varphi_{I,\ell}\subset\kappa_{0}I\text{
for all }I\in\mathcal{I}(\ell).$

Define also

$\varphi_{\ell}=1-\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell},\quad\text{hence}\quad\mathbf{1}_{\mathbb{R}\,\setminus\,U^{\sharp}(\ell)}\leq\varphi_{\ell}\leq\mathbf{1}_{\mathbb{R}\,\setminus\,U(\ell)}.$

Since $\kappa_{0}\leq 2$, the overlap condition (9.6) holds, and
$|I|=2K2^{-n/2}$ for all $I\in\mathcal{I}(\ell)$, the functions
$\varphi_{I,\ell}$ and $\varphi_{\ell}$ can be easily chosen to satisfy the
following bounds for all $y\in\mathbb{R}$:

$|\dot{\varphi}_{\ell}(y)|+\sum_{I\in\mathcal{I}(\ell)}|\dot{\varphi}_{I,\ell}(y)|\lesssim\tfrac{1}{K}\cdot
2^{n/2}\quad\text{and}\quad|\ddot{\varphi}_{\ell}(y)|+\sum_{I\in\mathcal{I}(\ell)}|\ddot{\varphi}_{I,\ell}(y)|\lesssim\tfrac{1}{K^{2}}\cdot
2^{n}.$ (9.8)

Before defining $\tau_{n}(\ell)$, we associate a quadric to each interval
$I\in\mathcal{I}(\ell)$. Recall that every interval $I\in\mathcal{I}(\ell)$
has the form $I=I_{n}(w)$ for some
$w_{I}\in\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})\cup\mathcal{C}_{m}(\ell_{\downarrow})$.
Let $Q_{I}\in\mathcal{T}_{m}$ be any rectangle with corner $w_{I}$, and let

$q_{I,\ell}:=q_{Q_{I},\ell}$

be the $(Q_{I},\ell)$-approximate quadric. There may be several (up to $4$)
admissible choices of $Q_{I}$, and they might produce different quadrics
$q_{I,\ell}$. However, we will see in Proposition 9.16 below that the
different quadrics nearly agree, so the particular choice does not matter. We
are now prepared to define $\tau_{n}(\ell)$. If $\ell\in\mathcal{L}_{n-1}$,
set

$\tau_{n}(\ell):=\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}\cdot
q_{I,\ell}+\varphi_{\ell}\cdot\tau_{n-1}(\ell).$ (9.9)

If $\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$, set instead

$\tau_{n}(\ell):=\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}\cdot
q_{I,\ell}+\tfrac{1}{2}\cdot\varphi_{\ell}\cdot[\tau_{n-1}(\ell^{\uparrow})+\tau_{n-1}(\ell_{\downarrow})],$
(9.10)

observing that $\ell^{\uparrow},\ell_{\downarrow}\in\mathcal{L}_{n-1}$.

####  9.2.1. Preparatory lemmas

Before beginning to verify axioms (X1)-(X5), we prove a few lemmas, which will
be needed in the verification of multiple axioms. In these lemmas, the
definitions of, and relationships between, the constants
$\epsilon,H,K,M,N,\Sigma$ are as stated before. We recall from (8.2) that
$\epsilon\ll\\{M,\Sigma\\}\ll K\ll H$.

The first lemma uses nothing but the $M$-Lipschitz property of $f$, hence
$f_{2}$.

######  Lemma 9.11.

Let $w_{1},w_{2}\in\mathbb{W}$ with $d_{\mathrm{par}}(w_{1},w_{2})\leq
100\cdot 2^{-n/2}$. Then, assuming that $K\geq 100AM$ for
$A>(\tfrac{4}{5}\sqrt{2}-\kappa_{0})^{-1}$, we have

$B(f_{2}(w_{2}),\kappa_{0}K2^{-n/2})\subset
B(f_{2}(w_{1}),\tfrac{4}{5}K2^{-(n-1)/2})\subset
B(f_{2}(w_{1}),\kappa_{0}K2^{-(n-1)/2}).$

###### Proof.

Since $f_{2}$ is $M$-Lipschitz, and $d_{\mathrm{par}}(w_{1},w_{2})\leq
100\cdot 2^{-n/2}$, we have

$B(f_{2}(w_{2}),\kappa_{0}K2^{-n/2})\subset
B(f_{2}(w_{1}),\kappa_{0}K2^{-n/2}+100\cdot M2^{-n/2}).$

Since $K\geq 100AM$, and $\kappa_{0}=\tfrac{1}{2}(1+\tfrac{4}{5}\sqrt{2})$,
here $\kappa_{0}K+100M\leq(\kappa_{0}+A^{-1})K<\tfrac{4}{5}\sqrt{2}\cdot K$,
hence

$B(f_{2}(w_{1}),\kappa_{0}K2^{-n/2}+M2^{-(n-1)/2})\subset
B(f_{2}(w_{1}),\tfrac{4}{5}K2^{-(n-1)/2}).$

The second inclusion simply follows from $\tfrac{4}{5}\leq\kappa_{0}$. ∎

The next lemma implies that the "$\approx_{n}$"-notation introduced in (8.5)
interacts well with partitions of unity. For technical purposes, we record a
slightly more quantitative result:

######  Lemma 9.12.

Let $\\{\varphi_{I}\\}_{I\in\mathcal{I}}\subset C^{2}(\mathbb{R})$ be a family
of functions satisfying

$\sum_{I\in\mathcal{I}}\varphi_{I}(y_{0})=1,\quad\sum_{I\in\mathcal{I}}|\dot{\varphi}_{I}(y_{0})|\leq
c_{1}\cdot
2^{n/2},\quad\text{and}\quad\sum_{I\in\mathcal{I}}|\ddot{\varphi}_{I}(y_{0})|\leq
c_{2}\cdot 2^{n}$ (9.13)

for some fixed $y_{0}\in\mathbb{R}$. Let
$\\{p_{I}\\}_{I\in\mathcal{I}}\cup\\{q_{I}\\}_{I\in\mathcal{I}}\subset
C^{2}(\mathbb{R})$ be further families of functions satisfying

$|p_{I}(y_{0})-q_{I}(y_{0})|\leq\epsilon_{1}\cdot
2^{-n},\quad|\dot{p}_{I}(y_{0})-\dot{q}_{I}(y_{0})|\leq\epsilon_{2}\cdot
2^{-n/2},\quad\text{and}\quad|\ddot{p}_{I}(y_{0})-\ddot{q}_{I}(y_{0})|\leq\epsilon_{3}$

for all $I\in\mathcal{I}$ with $y_{0}\in\operatorname{spt}\varphi_{I}$. Here
$c_{1},c_{2},\epsilon_{1},\epsilon_{2},\epsilon_{3}>0$ are constants. Then,
writing $Q_{p}:=\sum\varphi_{I}p_{I}$ and $Q_{q}:=\sum\varphi_{I}q_{I}$, we
have

$\begin{cases}|Q_{p}(y_{0})-Q_{q}(y_{0})|\leq\epsilon_{1}\cdot 2^{-n},\\\
|\dot{Q}_{p}(y_{0})-\dot{Q}_{q}(y_{0})|\leq(c_{1}\epsilon_{1}+\epsilon_{2})\cdot
2^{-n/2},\\\ |\ddot{Q}_{p}(y_{0})-\ddot{Q}_{q}(y_{0})|\leq
c_{2}\epsilon_{1}+2c_{1}\epsilon_{2}+\epsilon_{3}.\end{cases}$ (9.14)

###### Proof.

Write
$Q_{p}(y_{0})-Q_{q}(y_{0})=\sum_{I\in\mathcal{I}}\varphi_{I}(y_{0})\cdot[p_{I}(y_{0})-q_{I}(y_{0})]$,
and note that the sum can be restricted to those $I\in\mathcal{I}$ with
$y_{0}\in\operatorname{spt}\varphi_{I}$. Then (9.14) follows by direct
computation. ∎

Quite often the following corollary is enough:

######  Corollary 9.15.

Let $\\{\varphi_{I}\\}_{I\in\mathcal{I}}$ be a partition of unity satisfying
(9.13) with $c_{1},c_{2}\lesssim 1$. Let
$\\{q_{I}\\}_{I\in\mathcal{I}(\ell)}\subset C^{2}(\mathbb{R})$. Assume that,
for some $y_{0}\in\mathbb{R}$ fixed, there exists a function $q=q_{y_{0}}\in
C^{2}(\mathbb{R})$ such that

$q_{I}(y_{0})\approx_{n}q(y_{0})\quad\text{for all}\quad
I\in\mathcal{I}(\ell)\quad\text{with}\quad
y_{0}\in\operatorname{spt}\varphi_{I,\ell}.$

Then, $q(y_{0})\approx_{n}Q(y_{0})$, where
$Q=\sum_{I\in\mathcal{I}}\varphi_{I}q_{I}$.

The next proposition gives a condition for checking that the hypothesis of
Corollary 9.15 is valid. The notation $q_{Q,\ell}$ will refer to the
$(Q,\ell)$-approximate quadric associated with an
$(\epsilon,H,\Sigma)$-vertical rectangle $Q\in\mathcal{D}$.

######  Proposition 9.16.

Let $w_{1},w_{2}\in\mathcal{C}_{\lceil
n/2\rceil}\cup\mathcal{C}_{\lceil(n-1)/2\rceil}$ be corners of
$Q_{1},Q_{2}\in\mathcal{T}_{\lceil
n/2\rceil}\cup\mathcal{T}_{\lceil(n-1)/2\rceil}$. Assume that
$w_{1}\in\ell_{1}$ and $w_{2}\in\ell_{2}$, where
$\ell_{1},\ell_{2}\subset\mathbb{W}$ are horizontal lines with
$|\ell_{1}-\ell_{2}|\leq 100\cdot 2^{-n}$. Then, if $\ell\subset\mathbb{W}$ is
an additional horizontal line with $|\ell-\ell_{1}|\leq 100\cdot 2^{-n}$, we
have

$q_{Q_{1},\ell}(y)\approx_{n}q_{Q_{2},\ell}(y),\qquad y\in
B(f_{2}(w_{1}),100K2^{-n/2})\cap B(f_{2}(w_{2}),100K2^{-n/2}).$ (9.17)

###### Proof.

Write $m:=\lceil n/2\rceil$. Then $2^{-m}\leq\ell(Q_{j})\leq 2^{-m+1}$ for
both $j\in\\{1,2\\}$. If the intersection in (9.17) is empty, there is nothing
to prove. In the opposite case, we have

$|f_{2}(w_{1})-f_{2}(w_{2})|\leq 200K\cdot 2^{-m}.$ (9.18)

Based on this, we claim that

$d_{\mathrm{par}}(w_{1},w_{2})\lesssim\max\\{KN,N^{2}\\}\cdot 2^{-m}.$ (9.19)

Writing $w_{j}:=(y_{j},t_{j})$, $j\in\\{1,2\\}$, the hypothesis of the lemma
implies that $\|t_{1}-t_{2}\|\lesssim 2^{-m}$. Therefore, if

$|y_{1}-y_{2}|\leq\max\\{\tfrac{1}{100}\cdot 2^{-m},4N^{2}\|t_{1}-t_{2}\|\\},$
(9.20)

the estimate (9.19) follows immediately. In the opposite case, we rely on
Proposition 4.52, which we recall here for the reader’s convenience (with
terminology slightly adapted to the present situation):

######  Proposition 9.21.

Let $Q_{1},Q_{2}\in\mathcal{T}$, and let
$w_{j}=(y_{j},t_{j})\in\tfrac{1}{2}HQ_{j}$, $j\in\\{1,2\\}$, with

$|y_{1}-y_{2}|>\max\\{\tfrac{1}{100}\cdot\min\\{\ell(Q_{1}),\ell(Q_{2})\\},4N^{2}\|t_{1}-t_{2}\|\\}.$

Then,

$|f_{2}(w_{2})-f_{2}(w_{1})|\geq\tfrac{1}{2N}d_{\mathrm{par}}(w_{1},w_{2}).$

With the proposition in hand, recalling (9.18), and assuming that (9.20)
fails, we estimate

$d_{\mathrm{par}}(w_{1},w_{2})\leq 2N|f_{2}(w_{1})-f_{2}(w_{2})|\lesssim
KN2^{-m},$

and (9.19) follows. The main consequence of (9.19) is that
$\operatorname{dist}_{\mathrm{par}}(Q_{1},Q_{2})\leq
d_{\mathrm{par}}(w_{1},w_{2})\lesssim\max\\{KN,N^{2}\\}\ell(Q_{j})$ for
$j\in\\{1,2\\}$, recalling $Q_{1},Q_{2}\in\mathcal{T}_{m}$. Therefore, if
$H\gg\max\\{KN,N^{2}\\}$ is sufficiently large, then
$\tfrac{1}{2}HQ_{1}\cap\tfrac{1}{2}HQ_{2}\cap\ell\neq\emptyset$, and we may
even pick two "horizontally aligned" points

$v_{1},v_{2}\in\ell\cap\tfrac{1}{2}HQ_{1}\cap\tfrac{1}{2}HQ_{2}\quad\text{with}\quad
d_{\mathrm{par}}(v_{1},v_{2})=2^{-m}.$

These points will help us to show that the quadrics $q_{1}:=q_{Q_{1},\ell}$
and $q_{2}:=q_{Q_{2},\ell}$ are almost the same, which is what we claimed. To
make this precise, we will appeal to the following _quadric comparison lemma_
, whose proof will be postponed to Appendix A:

######  Lemma 9.22 (Quadric comparison lemma).

Let $q\colon\mathbb{R}\to\mathbb{R}$ be a quadric, and let
$I=[a_{1},a_{2}]\subset\mathbb{R}$ be an interval. Write

$m:=\max\\{|q(a_{1})|,|q(a_{2})|\\}\quad\text{and}\quad\dot{m}:=\max\\{|\dot{q}(a_{1})|,|\dot{q}(a_{2})|\\}.$

Then, for all $r\geq 1$ and $t\in rI$,

$|q(t)|\lesssim(m+|I|\dot{m})r^{2},\quad|\dot{q}(t)|\lesssim(\dot{m}+\tfrac{m}{|I|})r,\quad\text{and}\quad|\ddot{q}(t)|\lesssim\tfrac{m}{|I|^{2}}+\tfrac{\dot{m}}{|I|}.$
(9.23)

We aim to apply the lemma to the quadric $q:=q_{1}-q_{2}$ on the interval
$I:=[a_{1},a_{2}]:=[f_{2}(v_{1}),f_{2}(v_{2})]$ (or
$I=[f_{2}(v_{2}),f_{2}(v_{1})]$), where
$v_{1},v_{2}\in\ell\cap\tfrac{1}{2}HQ_{1}\cap\tfrac{1}{2}HQ_{2}$ were chosen
above. To this end, we need to have upper and lower bounds for $|I|$, plus
estimates for $|q(a_{j})|,|\dot{q}(a_{j})|$, $j\in\\{1,2\\}$. Let us start by
estimating $|I|$. Since
$v_{1},v_{2}\in\ell\cap\tfrac{1}{2}HQ_{1}\cap\tfrac{1}{2}HQ_{2}$, and
$d_{\mathrm{par}}(v_{1},v_{2})=\ell(Q_{1})=\ell(Q_{2})$, Proposition 9.21 can
be applied, once more:

$|I|=|f_{2}(v_{2})-f_{2}(v_{1})|\geq\tfrac{1}{2N}d_{\mathrm{par}}(v_{1},v_{2})=\tfrac{1}{2N}\cdot
2^{-m}.$

Evidently $|I|\leq M2^{-m}$, since $f_{2}$ is $M$-Lipschitz.

To estimate $|q(a_{j})|,|\dot{q}(a_{j})|$, $j\in\\{1,2\\}$, we observe that
$v_{1},v_{2}\in\ell\cap HQ_{1}\cap HQ_{2}$, and then apply Corollary 4.40 to
both quadrics $q_{i}$ at both points $a_{j}=f_{2}(v_{j})$:

$|f_{t}(v_{j})-q_{i}(a_{j})|\lesssim_{\Sigma}\epsilon\cdot
2^{-n}\quad\text{and}\quad|f_{x}(v_{j})-\dot{q}_{i}(a_{j})|\lesssim_{\Sigma}\epsilon\cdot
2^{-m},\qquad i,j\in\\{1,2\\}.$

Using the triangle inequality, and recalling that $q=q_{1}-q_{2}$, we infer

$m:=\max_{j\in\\{1,2\\}}|q(a_{j})|\lesssim_{\Sigma}\epsilon\cdot
2^{-n}\quad\text{and}\quad\dot{m}:=\max_{j\in\\{1,2\\}}|q(a_{j})|\lesssim_{\Sigma}\epsilon\cdot
2^{-m}.$

Now, combining these bounds with the length estimate $|I|\sim_{N}2^{-m}\sim
2^{-n/2}$, we may apply the quadric comparison lemma: for any $r\geq 1$ and
$t\in rI$,

$\begin{cases}|q_{1}(t)-q_{2}(t)|\lesssim_{r,N,\Sigma}\epsilon\cdot 2^{-n},\\\
|\dot{q}_{1}(t)-\dot{q}_{2}(t)|\lesssim_{r,N,\Sigma}\epsilon\cdot 2^{-m}\\\
|\ddot{q}_{1}(t)-\ddot{q}_{2}(t)|\lesssim_{N,\Sigma}\epsilon.\end{cases}$
(9.24)

The bound for the second derivatives is naturally independent or $r$, since
the second derivative of a quadric is constant. As for the first two
estimates, the point "$t$" that we are really interested in is the point $y\in
B(f_{2}(w_{1}),100K2^{-n/2})\cap B(f_{2}(w_{2}),100K2^{-n/2})$ given by
(9.17). Since $d_{\mathrm{par}}(v_{j},w_{j})\lesssim H2^{-m}$, it follows from
the $M$-Lipschitz property of $f_{2}$ that

$\operatorname{dist}(y,I)\leq|y-f_{2}(w_{1})|+|f_{2}(v_{1})-f_{2}(w_{1})|\lesssim(K+HM)2^{-m}.$

Since $|I|\sim_{N}2^{-m}$, and $M\leq N$, we have $y\in rI$ for some
$r\sim_{K,N}1$, and it finally follows from (9.24) that

$|q_{1}(y)-q_{2}(y)|\lesssim_{K,N,\Sigma}\epsilon\cdot
2^{-n}\quad\text{and}\quad|\dot{q}_{1}(y)-\dot{q}_{2}(y)|\lesssim_{K,N,\Sigma}\epsilon\cdot
2^{-m}.$

The correct bound for $|\ddot{q}_{1}(y)-\ddot{q}_{2}(y)|$ already follows from
(9.24). Since $N\sim_{M,\Sigma}1$, this completes the proof of the
proposition. ∎

The final result in this section is virtually a restatement of Corollary 3.5;

######  Corollary 9.25.

Let $\ell_{1}<\ell_{2}<\ell_{3}<\ell_{4}<\ell_{5}$ be consecutive lines in
$\mathcal{L}_{n}$, $n\geq 1$, such that
$\ell_{2},\ell_{4}\in\mathcal{L}_{n-1}$. Assume that one of the lines
$\ell_{j}$ contains the corner "$w$" of a rectangle $Q\in\mathcal{T}_{\lceil
n/2\rceil}$. Then, the dyadic parent of $Q$ has a corner
$\widehat{w}\in\ell_{2}\cup\ell_{4}$ which lies in $\mathcal{C}_{\lceil
n/2\rceil}\cap\mathcal{C}_{\lceil(n-1)/2\rceil}$.

###### Proof.

Only the last statement was not part of Corollary 3.5. If $n$ is odd, then
$\lceil(n-1)/2\rceil=\lceil n/2\rceil-1$, and the corners of the dyadic parent
$\hat{Q}\in\mathcal{T}_{\lceil n/2\rceil-1}$ of $Q$ lie in
$\mathcal{C}_{\lceil(n-1)/2\rceil}$ by definition (here we used $n\geq 1$, so
the dyadic parent of $Q$ still lies in $\mathcal{T}$). Moreover, by the
coherence axiom (T3), all the children of $\hat{Q}$ lie in
$\mathcal{T}_{\lceil n/2\rceil}$, and consequently the corners of $\hat{Q}$
are corners of some rectangles in $\mathcal{T}_{\lceil n/2\rceil}$.

If $n$ is even, then $\lceil(n-1)/2\rceil=\lceil n/2\rceil$, hence the
siblings of $Q\in\mathcal{T}_{\lceil n/2\rceil}$ lie simultaneously in
$\mathcal{T}_{\lceil n/2\rceil}$ and $\mathcal{T}_{\lceil(n-1)/2\rceil}$. In
particular, the corners of $\hat{Q}$ lie in $\mathcal{C}_{\lceil
n/2\rceil}\cap\mathcal{C}_{\lceil(n-1)/2\rceil}$. ∎

######  Lemma 9.26.

Let $n\geq 0$ and $Q\in\mathcal{D}_{\lceil n/2\rceil}$. Then at least one of
the horizontal edges of $Q$ is contained on a line in $\mathcal{L}_{n}$.

###### Proof.

The height of the rectangle $Q$ is $4^{-\lceil
n/2\rceil}\in\\{2^{-n},2^{-n-1}\\}$, depending on the parity of $n$. If the
height is $2^{-n}$, then both horizontal edges of $Q$ are contained on lines
in $\mathcal{L}_{n}$. If the height is $2^{-n-1}$, then both horizontal edges
of $Q$ are contained on lines in $\mathcal{L}_{n+1}$, and one observes that
every second line in $\mathcal{L}_{n+1}$ is contained in $\mathcal{L}_{n}$. ∎

We then begin to verify the axioms (X1)-(X5) in order. While verifying axiom
(Xi) for $i\in\\{1,\ldots,5\\}$ and $\tau_{n}$, we may inductively use all of
the axioms (Xj) for $\tau_{n-1}$. Moreover, to verify axioms (X1) for
$\tau_{n}$, we will use axioms (X3) and (X5) for $\tau_{n}$. However, the
proofs of axioms (X3),(X5) for $\tau_{n}$ will not use axiom (X1) for
$\tau_{n}$, so there is no circular reasoning.

####  9.2.2. Verifying axiom (X1)

Fix $\ell\in\mathcal{L}_{n}$. The claim is, then, that

$|\dot{\tau}_{n}(\ell)(y_{1})-\dot{\tau}_{n}(\ell)(y_{2})|\leq
2\Sigma|y_{1}-y_{2}|,\qquad y_{1},y_{2}\in\mathbb{R},$ (9.27)

and we assume inductively that this holds for $\tau_{n-1}(\ell)$,
$\tau_{n-1}(\ell^{\uparrow})$, and $\tau_{n-1}(\ell_{\downarrow})$. An easy
case occurs if
$\\{y_{1},y_{2}\\}\subset\mathbb{R}\,\setminus\,U^{\sharp}(\ell)$, so
$\varphi_{\ell}\equiv 1$ on a neighbourhood of $y_{1},y_{2}$ (this is why we
defined $U,U^{\sharp}$ as closed sets). Then, depending on whether
$\ell\in\mathcal{L}_{n-1}$ or
$\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$, we have either

$\dot{\tau}_{n}(\ell)(y_{1})-\dot{\tau}_{n}(\ell)(y_{2})=\dot{\tau}_{n-1}(\ell)(y_{1})-\dot{\tau}_{n-1}(\ell)(y_{2})$

or

$\dot{\tau}_{n}(\ell)(y_{1})-\dot{\tau}_{n}(\ell)(y_{2})=\tfrac{1}{2}[\dot{\tau}_{n-1}(\ell^{\uparrow})(y_{1})-\dot{\tau}_{n-1}(\ell^{\uparrow})(y_{2})]+\tfrac{1}{2}[\dot{\tau}_{n-1}(\ell^{\downarrow})(y_{1})-\dot{\tau}_{n-1}(\ell^{\downarrow})(y_{2})].$

In both cases, (9.27) follows from induction and the triangle inequality.

Let us then assume that $y_{1}\in U^{\sharp}(\ell)$, for example. Then, it
does not matter if $y_{2}\in U^{\sharp}(\ell)$ or
$y_{2}\in\mathbb{R}\,\setminus\,U^{\sharp}(\ell)$, and the verification of
(9.27) will not use the inductive assumption (X1). Since $y_{1}\in
U^{\sharp}(\ell)$, we have

$y_{1}\in\kappa_{0}I_{n}(w)=B(f_{2}(w),\kappa_{0}K2^{-n/2})$ (9.28)

for some
$w\in\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})\cup\mathcal{C}_{m}(\ell_{\downarrow})$.
This means that $w\in\ell\cup\ell^{\uparrow}\cup\ell_{\downarrow}$ and
$w\in\mathcal{C}(Q)$ for some $Q\in\mathcal{T}_{m}=\mathcal{T}_{\lceil
n/2\rceil}$. There are three different cases to consider:

$|y_{1}-y_{2}|\geq\tfrac{1}{10}K\quad\text{or}\quad\tfrac{1}{5}K\cdot
2^{-n/2}<|y_{1}-y_{2}|<\tfrac{1}{10}K\quad\text{or}\quad|y_{1}-y_{2}|\leq\tfrac{1}{5}K\cdot
2^{-n/2}.$ (9.29)

We handle these cases in order. First, assume
$|y_{1}-y_{2}|\geq\tfrac{1}{10}K$. Let
$\ell=:\ell_{n}\leq\ell_{n-1}\leq\ldots\leq\ell_{0}$ be a sequence of
horizontal lines such that $\ell_{k}\in\mathcal{L}_{k}$ and
$\ell_{k+1}\in\\{\ell_{k},\ell_{k}^{\uparrow}\\}$. Then, by axioms (X3) and
(X5) for $\tau_{0},\ldots,\tau_{n}$, we have

$\displaystyle|\dot{\tau}_{n}(\ell_{n})(y_{j})-\dot{\tau}_{0}(\ell_{0})(y_{j})|$
$\displaystyle\leq\sum_{k=0}^{n-1}|\dot{\tau}_{k+1}(\ell_{k+1})(y_{j})-\dot{\tau}_{k}(\ell_{k})(y_{j})|$
$\displaystyle\lesssim\delta\sum_{k=0}^{n-1}2^{-k/2}\sim\delta\leq\frac{10\delta}{K}|y_{1}-y_{2}|,\qquad
j\in\\{1,2\\}.$

Since $K>1$ in Proposition 8.3 is chosen small in a manner depending on both
$\delta$ and $\Sigma$, it follows that

$|\dot{\tau}_{n}(\ell)(y_{2})-\dot{\tau}_{n}(\ell)(y_{1})|\leq|\dot{\tau}_{0}(\ell_{0})(y_{2})-\dot{\tau}_{0}(\ell_{0})(y_{1})|+\tfrac{A\delta}{K}|y_{1}-y_{2}|\leq
2\Sigma|y_{2}-y_{1}|.$

Here we also used the special definition of $\tau_{0}$, which implies, as seen
in (9.3), that $\dot{\tau}_{0}(\ell)$ is $\Sigma$-Lipschitz, not just
$2\Sigma$-Lipschitz, for all $\ell\in\mathcal{L}_{0}$.

Recalling the case distinction (9.29), we next assume $\tfrac{1}{5}K\cdot
2^{-n/2}<|y_{1}-y_{2}|<\tfrac{1}{10}K$. Note that this forces $n\geq 2$. We
then claim that there exists $0\leq\hat{n}\leq n-2$, and a rectangle
$\hat{Q}\in\mathcal{T}_{\lceil\hat{n}/2\rceil}$ with $\hat{Q}\supset Q$ such
that

$|y_{1}-y_{2}|\geq 2^{-\hat{n}/2}\quad\text{and}\quad y_{1},y_{2}\in
B(f_{2}(\widehat{w}),K2^{-\hat{n}/2})$ (9.30)

for some
$\widehat{w}\in\mathcal{C}(\hat{Q})\subset\mathcal{C}_{\lceil\hat{n}/2\rceil}$.
Let $n_{1}\in\\{0,\ldots,n\\}$ be the greatest index with the property
$|y_{1}-y_{2}|\leq\tfrac{1}{5}K2^{-n_{1}/2}$. Then $1\leq n_{1}<n$ by our case
assumption, so $\hat{n}:=n_{1}-1\in\\{0,\ldots,n-2\\}$, and

$|y_{1}-y_{2}|>\tfrac{1}{5}K2^{-(n_{1}+1)/2}\geq 2^{-\hat{n}/2}.$

So, the first assertion of (9.30) holds.

To prove the second assertion, let $Q_{1}\in\mathcal{T}_{\lceil
n_{1}/2\rceil}$ and $\hat{Q}\in\mathcal{T}_{\lceil\hat{n}/2\rceil}$ be
rectangles with $Q\subset Q_{1}\subset\hat{Q}$. Let
$w_{1}\in\mathcal{C}(Q_{1})$ be an arbitrary corner. We also choose a corner
$\widehat{w}\in\mathcal{C}(\hat{Q})$, but with some care: recall from Lemma
9.26 that one of the horizontal edges of
$\hat{Q}\in\mathcal{D}_{\lceil\hat{n}/2\rceil}$ lies on a line
$\ell_{\hat{n}}\in\mathcal{L}_{\hat{n}}$. We choose $\widehat{w}$ to be one of
the two corners of $\hat{Q}$ on $\ell_{\hat{n}}$.

Certainly $d_{\mathrm{par}}(w_{1},\widehat{w})\leq 100\cdot 2^{-n_{1}/2}$.
Then, (9.28) and Lemma 9.11 (applied with $w_{1}$ and $w_{2}:=\widehat{w}$)
first imply that

$y_{1}\in B(f_{2}(w),\kappa_{0}K2^{-n/2})\subset
B(f_{2}(w_{1}),\kappa_{0}K2^{-n_{1}/2})\subset
B(f_{2}(\widehat{w}),\tfrac{4}{5}K2^{-\hat{n}/2}),$

and next

$|y_{2}-f_{2}(w_{1})|\leq|y_{1}-y_{2}|+\tfrac{4}{5}K2^{-\hat{n}/2}\leq\tfrac{1}{5}K2^{-\hat{n}/2}+\tfrac{4}{5}K2^{-\hat{n}/2}=K2^{-\hat{n}/2}.$

This verifies (9.30) with $\widehat{w},\hat{Q}$, and $\hat{n}$, as above.

This information enables us to use the inductive assumption (X4) on the
characteristics of generation $0\leq\hat{n}\leq n-2$. Let us construct a
sequence of horizontal lines $\ell_{n},\ldots,\ell_{\hat{n}-1}$:

  * (a)

$\ell_{n}\in\\{\ell,\ell^{\uparrow},\ell_{\downarrow}\\}$ is the line
containing $w$, so $\ell_{n}$ intersects the closures of $Q$ and
$\hat{Q}\supset Q$,

  * (b)

$\ell_{k}\in\mathcal{L}_{k}$ for all $n\leq k\leq\hat{n}-1$, and
$\ell_{k-1}\in\\{\ell_{k},\ell_{k}^{\uparrow},\ell_{k\downarrow}\\}$ for
$\hat{n}+2\leq k\leq n$ (this notation refers to the
$\mathcal{L}_{k}$-neighbours of $\ell_{k}$),

  * (c)

Every line $\ell_{k}$, $n\leq k\leq\hat{n}-1$, intersects the closure of
$\hat{Q}$,

The line $\ell_{n}$ is dictated by (a). Assume that
$\ell_{k}\in\mathcal{L}_{k}$ has already been chosen for some $\hat{n}+2\leq
k\leq n$. Then either $\ell_{k}\in\mathcal{L}_{k-1}$, and we simply set
$\ell_{k-1}:=\ell_{k}$, or otherwise both $\mathcal{L}_{k}$-neighbours of
$\ell_{k}$ are in $\mathcal{L}_{k-1}$. Since the height of the rectangle
$\hat{Q}\in\mathcal{T}_{\lceil\hat{n}/2\rceil}\subset\mathcal{D}_{\lceil\hat{n}/2\rceil}$
is $4^{-\lceil\hat{n}/2\rceil}\in\\{2^{-\hat{n}},2^{-(\hat{n}+1)}\\}$,
depending on the parity of $\hat{n}$, and the vertical separation of
$\mathcal{L}_{k}$-neighbours is $2^{-k}\leq 2^{-\hat{n}+2}$, one of these
neighbours meets the closure of $\hat{Q}$. So,
$\ell_{k-1}\in\mathcal{L}_{k-1}$ may be chosen as in (b) and (c) for
$\hat{n}+2\leq k\leq n$.

We earlier defined $\ell_{\hat{n}}\in\mathcal{L}_{\hat{n}}$ to be a line
containing the corner $\widehat{w}\in\mathcal{C}(\hat{Q})$. According to
(b)-(c), both $\ell_{\hat{n}+1}\in\mathcal{L}_{\hat{n}+1}$ and
$\ell_{\hat{n}}\in\mathcal{L}_{\hat{n}}\subset\mathcal{L}_{\hat{n}+1}$
intersect the closure of $\hat{Q}$. If the height of $\hat{Q}$ is
$2^{-\hat{n}+1}$, it follows that
$\ell_{\hat{n}}\in\\{\ell_{\hat{n}+1},\ell_{\hat{n}+1}^{\uparrow},\ell_{\hat{n}+1\downarrow}\\}$.
Otherwise, the height of $\hat{Q}$ is $2^{-\hat{n}}$, and $\ell_{\hat{n}}$ may
be up to two steps away from $\ell_{\hat{n}+1}$ in $\mathcal{L}_{\hat{n}+1}$.
In both cases, axioms (X3)-(X5) yield

$\|\dot{\tau}_{k}(\ell_{k})-\dot{\tau}_{k+1}(\ell_{k+1})\|_{L^{\infty}(\mathbb{R})}\lesssim\delta\cdot
2^{-k/2},\qquad\hat{n}\leq k\leq n-1.$ (9.31)

From (9.31), we deduce that for both $j\in\\{1,2\\}$,

$\displaystyle|\dot{\tau}_{n}(\ell)(y_{j})-\dot{\tau}_{\hat{n}}(\ell_{\hat{n}})(y_{j})|$
$\displaystyle\leq\sum_{k=\hat{n}}^{n-1}|\dot{\tau}_{k+1}(\ell_{k+1})(y_{j})-\dot{\tau}_{k}(\ell_{k})(y_{j})|$
$\displaystyle\leq\delta\sum_{k=\hat{n}}^{n-1}2^{-k/2}\sim\delta\cdot
2^{-\hat{n}/2}\leq\delta|y_{1}-y_{2}|,$ (9.32)

using the lower bound in (9.30) in the final inequality. Next, we estimate the
difference
$|\dot{\tau}_{\hat{n}}(\ell_{\hat{n}})(y_{2})-\dot{\tau}_{\hat{n}}(\ell_{\hat{n}})(y_{1})|$
using the inclusion $y_{1},y_{2}\in B(f_{2}(\widehat{w}),K2^{-\hat{n}/2})$
from (9.30), and also
$\widehat{w}\in\mathcal{C}_{\lceil\hat{n}/2\rceil}(\ell_{\hat{n}})$. With
these facts in hand, axiom (X4) lets us compare
$\dot{\tau}_{\hat{n}}(\ell_{\hat{n}})$ to the
$(\hat{Q},\ell_{\hat{n}})$-approximate quadric
$q:=q_{\hat{Q},\ell_{\hat{n}}}$:

$|\dot{\tau}_{\hat{n}}(\ell_{\hat{n}})(y_{2})-\dot{\tau}_{\hat{n}}(\ell_{\hat{n}})(y_{1})|\leq|\dot{q}(y_{2})-\dot{q}(y_{1})|+o(\epsilon)\cdot
2^{-\hat{n}/2}\leq|\dot{q}(y_{2})-\dot{q}(y_{1})|+o(\epsilon)\cdot|y_{1}-y_{2}|.$

Since $\|\ddot{q}\|_{L^{\infty}}\leq\Sigma$ by the assumption that
$\hat{Q}\in\mathcal{T}$ is an $(\epsilon,H,\Sigma)$-vertical rectangle, we
combine the estimate above with (9.32) to deduce

$|\dot{\tau}_{n}(\ell_{n})(y_{2})-\dot{\tau}_{n}(\ell_{n})(y_{1})|\leq\Sigma|y_{1}-y_{2}|+A(\delta+o(\epsilon))|y_{1}-y_{2}|,$

where $A\geq 1$ is an absolute constant. Finally, the same estimate holds for
$|\dot{\tau}_{n}(\ell)(y_{2})-\dot{\tau}_{n}(\ell)(y_{1})|$, since
$\ell_{n}\in\\{\ell,\ell^{\uparrow},\ell_{\downarrow}\\}$, and axiom (X3)
implies
$\|\dot{\tau}_{n}(\ell)-\dot{\tau}_{n}(\ell_{n})\|_{L^{\infty}(\mathbb{R})}\leq\delta\cdot
2^{-n/2}$. Choosing $\delta,\epsilon>0$ small enough, depending on $\Sigma$,
we see that $|\dot{\tau}_{n}(\ell)(y_{2})-\dot{\tau}_{n}(\ell)(y_{1})|\leq
2\Sigma|y_{1}-y_{2}|$, as desired. This completes the proof of axiom (X1) in
the case where $\tfrac{1}{5}K\cdot 2^{-n/2}<|y_{1}-y_{2}|<\tfrac{1}{10}K$.

Recalling the case distinction in (9.29), it remains to treat the case
$|y_{1}-y_{2}|\leq\tfrac{1}{5}K\cdot 2^{-n/2}$. Axioms (X3)-(X5) will not be
used here. Recall from (9.28) that
$w\in\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})\cup\mathcal{C}_{m}(\ell_{\downarrow})$
is the corner of some rectangle $Q\in\mathcal{T}_{m}$ which satisfies
$y_{1}\in B(f_{2}(w),\kappa_{0}K2^{-n/2})$. Here $m=\lceil n/2\rceil$, as
always. We will next apply Corollary 9.25 to find a "parent" corner point
$\widehat{w}$ close to $w$. The details are slightly differently depending on
whether $\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$ or
$\ell\in\mathcal{L}_{n-1}$.

If $\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$, let
$\ell_{1}<\ldots<\ell_{5}$ be the five consecutive lines of $\mathcal{L}_{n}$
with $\ell_{2}=\ell_{\downarrow}\in\mathcal{L}_{n-1}$ and
$\ell_{4}=\ell^{\uparrow}\in\mathcal{L}_{n-1}$. Then, recalling that
$w\in\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})\cup\mathcal{C}_{m}(\ell_{\downarrow})$
is the corner of $Q\in\mathcal{T}_{m}$, Corollary 9.25 produces a corner
$\widehat{w}\in\mathcal{C}_{m}\cap\mathcal{C}_{\lceil(n-1)/2\rceil}$ of the
dyadic parent of $Q$ which lies on $\ell^{\uparrow}\cup\ell_{\downarrow}$.

If, on the other hand, $\ell\in\mathcal{L}_{n-1}$, then choose the five lines
in $\mathcal{L}_{n}$ so that $\ell\in\\{\ell_{2},\ell_{4}\\}$. Then
$\ell=\ell_{2}$ and $\ell_{4}$ are consecutive in $\mathcal{L}_{n-1}$, and
Corollary 9.25 produces a corner
$\widehat{w}\in\mathcal{C}_{m}\cap\mathcal{C}_{\lceil(n-1)/2\rceil}$ of the
dyadic parent of $Q$ which lies on $\ell\cup\ell_{4}$.

In both cases, $d_{\mathrm{par}}(w,\widehat{w})\leq 100\cdot 2^{-n/2}$, so
Lemma 9.11 implies

$y_{1}\in B(f_{2}(w),\kappa_{0}K2^{-n/2})\subset
B(f_{2}(\widehat{w}),\tfrac{4}{5}K\cdot 2^{-(n-1)/2}).$

Consequently $|y_{2}-f_{2}(\widehat{w})|\leq\tfrac{1}{5}K\cdot
2^{-n/2}+\tfrac{4}{5}K\cdot 2^{-(n-1)/2}<K2^{-(n-1)/2}$, so in fact

$[y_{1},y_{2}]\subset B(f_{2}(\widehat{w}),K2^{-(n-1)/2}).$ (9.33)

We are now in a position to apply the inductive hypothesis (X4) to the quadric
$\tau_{n-1}(\ell)$ (in case $\ell\in\mathcal{L}_{n-1}$) or both quadrics
$\tau_{n-1}(\ell^{\uparrow})$ and $\tau_{n-1}(\ell_{\downarrow})$ (in case
$\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$). We treat the cases
separately for clarity. Let us first assume $\ell\in\mathcal{L}_{n-1}$, so
$\widehat{w}\in\ell\cup\ell_{4}$, where $\ell$ and $\ell_{4}$ are consecutive
in $\mathcal{L}_{n-1}$. Let $\hat{Q}\in\mathcal{T}_{\lceil(n-1)/2\rceil}$ be a
rectangle with corner $\widehat{w}$. Let $q:=q_{\hat{Q},\ell}$ be the
$(\hat{Q},\ell)$-approximate quadric. Then, axiom (X4) applied inductively to
$\tau_{n-1}(\ell)$ implies

$\tau_{n-1}(y)\approx_{n-1}q(y),\qquad y\in[y_{1},y_{2}].$ (9.34)

Similar reasoning works if
$\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$, so
$\widehat{w}\in\ell^{\uparrow}\cup\ell_{\downarrow}$. Then, (X4) is applied
inductively to both $\tau_{n-1}(\ell^{\uparrow})$ and
$\tau_{n-1}(\ell_{\downarrow})$, and to the $(\hat{Q},\ell^{\uparrow})$\- and
$(\hat{Q},\ell_{\downarrow})$-approximate quadrics $q^{\uparrow}$ and
$q_{\downarrow}$; the meaning of $\hat{Q}\in\mathcal{T}_{\lceil(n-1)/2\rceil}$
is the same as before. The results are

$\tau_{n-1}(\ell^{\uparrow})(y)\approx_{n-1}q^{\uparrow}(y)\quad\text{and}\quad\tau_{n-1}(\ell_{\downarrow})(y)\approx_{n-1}q_{\downarrow}(y),\qquad
y\in[y_{1},y_{2}].$ (9.35)

To conclude the proof of axiom (X1), we will use (9.34)-(9.35), and
additionally the following claim; in the two possible definitions

$\begin{cases}\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}(y)q_{I,\ell}(y)+\varphi_{\ell}(y)\cdot\tau_{n-1}(\ell)(y),\\\
\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}(y)q_{I,\ell}(y)+\tfrac{1}{2}\varphi_{\ell}(y)\cdot[\tau_{n-1}(\ell^{\uparrow})(y)+\tau_{n-1}(\ell_{\downarrow})(y)]\end{cases}$

of $\tau_{n}(\ell)(y)$, we have $q_{I,\ell}(y)\approx_{n}q(y)$ for all
$y\in[y_{1},y_{2}]$ and for all $I\in\mathcal{I}(\ell)$ such that
$y\in\operatorname{spt}\varphi_{I,\ell}$. To accomplish this, we appeal to
Proposition 9.16. Note that if $y\in\operatorname{spt}\varphi_{I,\ell}$ for
some $y\in[y_{1},y_{2}]$, then

$y\in\kappa_{0}I_{n}(w_{I})=\bar{B}(f_{2}(w_{I}),\kappa_{0}K2^{-n/2})\subset
B(f_{2}(w_{I}),100K2^{-n/2})$

for some
$w_{I}\in\mathcal{C}_{m}(\ell_{\downarrow})\cup\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})$,
and $q_{I,\ell}=q_{Q_{I},\ell}$ is the $(Q_{I},\ell)$-approximate quadric for
some rectangle $Q_{I}\in\mathcal{T}_{m}=\mathcal{T}_{\lceil n/2\rceil}$ with
corner $w_{I}$. On the other hand, from (9.33) we know that
$y\in[y_{1},y_{2}]\subset B(f_{2}(\widehat{w}),100K2^{-n/2})$. Since the
vertical separation between the lines containing $w_{I}$ and $\widehat{w}$ is
certainly $\leq 100\cdot 2^{-n}$, Proposition 9.16 is applicable with
$(Q_{1},w_{1}):=(Q_{I},w_{I})\in\mathcal{T}_{\lceil
n/2\rceil}\times\mathcal{C}(Q_{I})$ and
$(Q_{2},w_{2}):=(\hat{Q},\widehat{w})\in\mathcal{T}_{\lceil(n-1)/2\rceil}\times\mathcal{C}(\hat{Q})$,
and implies

$q_{I,\ell}(y)=q_{Q_{I},\ell}(y)\approx_{n}q_{\hat{Q},\ell}(y)=q(y),\qquad
y\in[y_{1},y_{2}].$ (9.36)

Now, all the pieces are in place to prove (X1) in the final case
$|y_{1}-y_{2}|\leq\tfrac{1}{5}K\cdot 2^{-n/2}$. Assume, first, that
$\ell\in\mathcal{L}_{n-1}$, so

$\tau_{n}(\ell)(y)=\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}(y)q_{I,\ell}(y)+\varphi_{\ell}(y)\cdot\tau_{n-1}(\ell)(y).$

Combining (9.34) and (9.36), we see that
$q_{I,\ell}(y)\approx_{n}q(y)\approx_{n-1}\tau_{n-1}(\ell)(y)$ for all
$I\in\mathcal{I}(\ell)$ such that $y\in\operatorname{spt}\varphi_{I,\ell}$.
Now Corollary 9.15 implies

$\tau_{n}(\ell)(y)\approx_{n-1}q(y),\qquad y\in[y_{1},y_{2}].$ (9.37)

In particular, since $\|\ddot{q}\|_{L^{\infty}}\leq\Sigma$, we have
$|\ddot{\tau}_{n}(\ell)(y)|\leq\Sigma+o(\epsilon)$ for $y\in[y_{1},y_{2}]$,
and (X1) then follows by choosing $\epsilon>0$ small enough, depending on
$K,M,\Sigma$.

Assume finally that $\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$, so

$\tau_{n}(\ell)(y)=\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}(y)q_{I,\ell}(y)+\tfrac{1}{2}\varphi_{\ell}(y)\cdot[\tau_{n-1}(\ell^{\uparrow})(y)+\tau_{n-1}(\ell_{\downarrow})(y)].$

We claim that

$\begin{cases}|\dot{\tau}_{n}(\ell)(y)-\dot{q}(y)|\lesssim_{M}\left(\tfrac{1}{K}+o(\epsilon)\right)2^{-n/2},\\\
|\ddot{\tau}_{n}(\ell)(y)-\ddot{q}(y)|\lesssim_{M}\tfrac{1}{K}+o(\epsilon),\end{cases}\quad
y\in[y_{1},y_{2}].$ (9.38)

Only the latter estimate will be needed here, but the first one is useful
later. Together with $\|\ddot{q}\|_{L^{\infty}(\mathbb{R})}\leq\Sigma$, it
implies
$\|\ddot{\tau}_{n}(\ell)\|_{L^{\infty}([y_{1},y_{2}])}\leq\Sigma+C_{M}(o(\epsilon)+\tfrac{1}{K})$.
Consequently $|\dot{\tau}_{n}(\ell)(y_{2})-\dot{\tau}_{n}(\ell)(y_{1})|\leq
2\Sigma$ if we choose $K$ large enough depending on $M,\Sigma$, and
$\epsilon>0$ small enough depending on $K,M,\Sigma$, as we assumed in
Proposition 8.3.

To prove (9.38), we will use Lemma 9.12 with
$\\{\varphi_{I}\\}_{I\in\mathcal{I}}:=\\{\varphi_{I,\ell}\\}_{I\in\mathcal{I(\ell)}}\cup\\{\tfrac{1}{2}\varphi_{\ell}\\}\cup\\{\tfrac{1}{2}\varphi_{\ell}\\}$,

$p_{I}\in\\{\varphi_{I,\ell},\tau_{n-1}(\ell^{\uparrow}),\tau_{n-1}(\ell_{\downarrow})\\}\quad\text{and}\quad
q_{I}\equiv q.$

We start by recapping from (9.35) that
$\tau_{n-1}(\ell^{\uparrow})(y)\approx_{n-1}q^{\uparrow}(y)$ and
$\tau_{n-1}(\ell_{\downarrow})(y)\approx_{n-1}q{\downarrow}(y)$ for all
$y\in[y_{1},y_{2}]$, where $q^{\uparrow}$ and $q_{\downarrow}$ are the
$(\hat{Q},\ell^{\uparrow})$ and $(\hat{Q},\ell_{\downarrow})$-approximate
quadrics, respectively. The quadrics $q,q^{\uparrow},q^{\downarrow}$ are all
associated with the common $(\epsilon,H,\Sigma)$-vertical rectangle $\hat{Q}$,
so they only differ by a constant. In particular, the first and second
derivatives of $q,q^{\uparrow},q^{\downarrow}$ agree. Also, it is not hard to
see that

$\|q-q^{\uparrow}\|_{L^{\infty}(\mathbb{R})},\|q-q_{\downarrow}\|_{L^{\infty}(\mathbb{R})}\lesssim_{M}2^{-n},$

because the graphs of $q,q^{\uparrow},q_{\downarrow}$ are $\Pi$-projections of
horizontal lines on $\mathbb{V}_{\hat{Q}}$ which best approximate $f(\ell\cap
H\hat{Q})$, $f(\ell^{\uparrow}\cap H\hat{Q})$, and $f(\ell_{\downarrow}\cap
H\hat{Q})$ respectively. We will derive much more precise information at
(9.55), so we refer the reader there for more details. Consequently, for
$\hat{\ell}\in\\{\ell^{\uparrow},\ell_{\downarrow}\\}$, and the appropriate
choice of $\hat{q}\in\\{q^{\uparrow},q_{\downarrow}\\}$, we have

$\begin{cases}|\tau_{n-1}(\hat{\ell})(y)-q(y)|\leq|\tau_{n-1}(\hat{\ell})(y)-\hat{q}(y)|+\|\hat{q}-q\|_{L^{\infty}(\mathbb{R})}\lesssim_{M}2^{-n},\\\
|\dot{\tau}_{n-1}(\hat{\ell})(y)-\dot{q}(y)|=|\dot{\tau}_{n-1}(\hat{\ell})(y)-\dot{\hat{q}}(y)|=o(\epsilon)\cdot
2^{-n/2},\\\
|\ddot{\tau}_{n-1}(\hat{\ell})(y)-\ddot{q}(y)|=|\ddot{\tau}_{n-1}(\hat{\ell})(y)-\ddot{\hat{q}}(y)|=o(\epsilon),\qquad\qquad
y\in[y_{1},y_{2}].\end{cases}$

With this information (and of course (9.36)), and recalling the estimates
(9.8) for the first and second derivatives of the functions $\varphi_{I,\ell}$
and $\varphi_{\ell}$, we are in a position to apply Lemma 9.12 with constants
$c_{1},c_{2}\sim K^{-1}$, $\epsilon_{1}\lesssim_{M}1$, and
$\epsilon_{2},\epsilon_{3}=o(\epsilon)$. The estimates (9.38) follow
immediately, and the proof of axiom (X1) is complete.

####  9.2.3. Verying axioms (X2)-(X3)

Recall that axiom (X2) asks to prove that if
$\ell_{1},\ell_{2}\in\mathcal{L}_{n}$ satisfy $\ell_{2}>\ell_{1}$, then

$\theta(\ell_{2}-\ell_{1})\leq\tau_{n}(\ell_{2})(y)-\tau_{n}(\ell_{1})(y)\leq\theta^{-1}(\ell_{2}-\ell_{1}),\qquad
y\in\mathbb{R}.$ (9.39)

By trivial induction, it suffices to prove these inequalities for any pair of
consecutive lines $\ell_{1},\ell_{2}\in\mathcal{L}_{n}$ with
$\ell_{2}>\ell_{1}$. Therefore $\ell_{2}-\ell_{1}=2^{-n}$ (this notation meant
that $\pi_{2}(\ell_{2})-\pi_{2}(\ell_{1})=2^{-n}$). So, fix consecutive lines
$\ell_{1},\ell_{2}\in\mathcal{L}_{n}$ with $\ell_{2}>\ell_{1}$. Then, either

$\ell_{1}\in\mathcal{L}_{n-1}\quad\text{and}\quad\ell_{2}\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1},$
(9.40)

or the same holds with the roles of $\ell_{1},\ell_{2}$ reversed. We assume
with no loss of generality that (9.40) holds. Under these assumptions, we
will, in addition to (9.39), prove that

$|\dot{\tau}_{n}(\ell_{2})(y)-\dot{\tau}_{n}(\ell_{1})(y)|\leq\delta\cdot
2^{-n/2},\qquad y\in\mathbb{R}.$ (9.41)

This will establish axiom (X3).

To get started, we introduce separate notation for the upper
$\mathcal{L}_{n}$-neighbour of $\ell_{2}$:

$\ell_{3}:=\ell_{2}^{\uparrow}\in\mathcal{L}_{n-1}.$

Thus, $\ell_{1}<\ell_{2}<\ell_{3}$, where $\ell_{j}$ is consecutive to
$\ell_{j+1}$ in $\mathcal{L}_{n}$, and $\ell_{1},\ell_{3}$ are neighbours in
$\mathcal{L}_{n-1}$. For $j\in\\{1,2,3\\}$, we abbreviate

$\varphi_{I,j}:=\varphi_{I,\ell_{j}},\quad\varphi_{j}:=\varphi_{\ell_{j}}=1-\sum_{I\in\mathcal{I}(\ell_{j})}\varphi_{I,j},\quad\text{and}\quad
q_{I,j}:=q_{I,\ell_{j}}.$

Now, to prove (9.39) and (9.41), fix $y\in\mathbb{R}$. An easy case occurs if
$y\in\mathbb{R}\,\setminus\,[U^{\sharp}(\ell_{1})\cup U^{\sharp}(\ell_{2})]$.
Then $\varphi_{I,j}\equiv 0$ and $\varphi_{j}\equiv 1$ for $j\in\\{1,2\\}$ and
$I\in\mathcal{I}(\ell_{j})$, so

$\tau_{n}(\ell_{2})(y)=\tfrac{1}{2}\cdot[\tau_{n-1}(\ell_{1})+\tau_{n-1}(\ell_{3})]\quad\text{and}\quad\tau_{n}(\ell_{1})(y)=\tau_{n-1}(\ell_{1})$

by the definitions (9.9)-(9.10), and recalling (9.40). Using the inductive
hypothesis (X2) for $\tau_{n-1}(\ell_{3})-\tau_{n-1}(\ell_{1})$, we find

$\tau_{n}(\ell_{2})(y)-\tau_{n}(\ell_{1})(y)=\tfrac{1}{2}\cdot[\tau_{n-1}(\ell_{3})(y)-\tau_{n-1}(\ell_{1})(y)]\geq\tfrac{\theta}{2}(\ell_{3}-\ell_{1})=\theta(\ell_{2}-\ell_{1}).$

The upper bound
$\tau_{n}(\ell_{2})(y)-\tau_{n}(\ell_{1})(y)\leq\theta^{-1}(\ell_{2}-\ell_{1})$
is proved in the same manner. Similarly, we use the inductive hypothesis (X3)
for $\tau_{n-1}$ to prove (9.41):

$|\dot{\tau}_{n}(\ell_{2})(y)-\dot{\tau}_{n}(\ell_{1})(y)|=\tfrac{1}{2}\cdot|\dot{\tau}_{n-1}(\ell_{3})(y)-\dot{\tau}_{n-1}(\ell_{1})(y)|\leq\tfrac{1}{2}\cdot\delta\cdot
2^{-(n-1)/2}\leq\delta\cdot 2^{-n/2}.$

We are now done in the case
$y\in\mathbb{R}\,\setminus\,[U^{\sharp}(\ell_{1})\cup U^{\sharp}(\ell_{2})]$.

Assume next that $y\in U^{\sharp}(\ell_{1})\cup U^{\sharp}(\ell_{2})$. In this
case, the inductive assumptions (X2)-(X3) on $\tau_{n-1}(\ell_{1})$ and
$\tau_{n-1}(\ell_{3})$ will not be used to verify (9.39) and (9.41). Instead,
we will rely solely on (X4). By definition of the sets $U^{\sharp}(\ell_{j})$,
the following is true for at least one index $j\in\\{1,2\\}$: there exists
$w\in\mathcal{C}_{m}(\ell_{j})\cup\mathcal{C}_{m}(\ell_{j}^{\uparrow})\cup\mathcal{C}_{m}(\ell_{j\downarrow})$,
where $m=\lceil n/2\rceil$, such that

$|y-f_{2}(w)|\leq\kappa_{0}K2^{-n/2}.$ (9.42)

The possible lines $\ell_{j},\ell_{j}^{\uparrow},\ell_{j\downarrow}$ occurring
here, for $j\in\\{1,2\\}$, are
$\ell_{1\downarrow},\ell_{1},\ell_{2},\ell_{3}$, but to make the following
discussion symmetric, we artificially add the possibility
$w\in\mathcal{C}_{m}(\ell_{3}^{\uparrow})$. In other words, we only use the
information

$w\in\mathcal{C}_{m}(\ell_{1\downarrow})\cup\mathcal{C}_{m}(\ell_{1})\cup\mathcal{C}_{m}(\ell_{2})\cup\mathcal{C}_{m}(\ell_{3})\cup\mathcal{C}_{m}(\ell_{3}^{\uparrow}),$
(9.43)

see Figure 3.

\begin{overpic}[scale={0.61}]{Image4.pdf} \put(56.0,29.5){$\hat{Q}$}
\put(74.0,30.0){$\widehat{w}$} \put(56.0,6.0){$w$} \end{overpic} Figure 3. A
possible configuration of the points $w\in\mathcal{C}_{m}$,
$\widehat{w}\in\mathcal{C}_{m-1}\cap\mathcal{C}_{m}$ and the rectangle
$\hat{Q}\in\mathcal{T}_{\lceil(n-1)/2\rceil}$.

Let $Q\in\mathcal{T}_{m}$ be a rectangle with corner $w$. Assumption (9.43)
enables us to use Corollary 9.25 much like we did while verifying axiom (X1).
The conclusion is that there exists a corner $\widehat{w}$ of the dyadic
parent of $Q$ which lies on $\ell_{1}\cup\ell_{3}$, and is also the corner of
some rectangle $\hat{Q}\in\mathcal{T}_{\lceil(n-1)/2\rceil}$ (which is either
the dyadic parent or sibling of $Q$, depending on the parity of $n$, as we saw
in the proof of Corollary 9.25). We write
$\hat{\ell}\in\\{\ell_{1},\ell_{3}\\}$ for the line such that
$\widehat{w}\in\hat{\ell}$.

Recall from (9.42) that $|y-f_{2}(w)|\leq\kappa_{0}K2^{-n/2}$. Since
$d_{\mathrm{par}}(w,\widehat{w})\leq 100\cdot 2^{-m}=100\cdot 2^{-n/2}$, we
deduce from Lemma 9.11 that

$y\in B(f_{2}(w),\kappa_{0}K2^{-n/2})\subset
B(f_{2}(\widehat{w}),K2^{-(n-1)/2}).$ (9.44)

This inclusion enables us to apply the inductive hypothesis (X4) on
$\tau_{n-1}(\ell_{1})(y)$ and $\tau_{n-1}(\ell_{3})(y)$. Namely, for both
$\ell:=\ell_{1}$ and $\ell:=\ell_{3}$, the line
$\hat{\ell}\in\\{\ell_{1},\ell_{3}\\}$ satisfies
$\hat{\ell}\in\\{\ell,\ell^{\uparrow},\ell_{\downarrow}\\}$, where
$\ell^{\uparrow},\ell_{\downarrow}$ refer to the generation $n-1$ neighbours
of $\ell$. It follows that

$\widehat{w}\in\mathcal{C}_{\lceil(n-1)/2\rceil}(\ell)\cup\mathcal{C}_{\lceil(n-1)/2\rceil}(\ell^{\uparrow})\cup\mathcal{C}_{\lceil(n-1)/2\rceil}(\ell_{\downarrow}),\qquad\ell\in\\{\ell_{1},\ell_{3}\\}.$

Since also (9.44) holds, we may infer inductively from axiom (X4) that

$\tau_{n-1}(\ell)(y)\approx_{n-1}q_{\hat{Q},\ell}(y),\qquad\ell\in\\{\ell_{1},\ell_{3}\\},$
(9.45)

where $q_{\hat{Q},\ell}$ is the $(\hat{Q},\ell)$-approximate quadric. Let us,
at this point, recall the definitions

$\tau_{n}(\ell_{1})(y)=\sum_{I\in\mathcal{I}(\ell_{1})}\varphi_{I,1}(y)\cdot
q_{I,1}(y)+\varphi_{1}(y)\cdot\tau_{n-1}(\ell_{1})(y)$ (9.46)

and

$\tau_{n}(\ell_{2})(y)=\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}(y)\cdot
q_{I,2}(y)+\tfrac{1}{2}\varphi_{2}(y)\cdot[\tau_{n-1}(\ell_{1})(y)+\tau_{n-1}(\ell_{3})(y)].$
(9.47)

We just deduced in (9.45) that
$\tau_{n-1}(\ell_{j})(y)\approx_{n-1}q_{\hat{Q},\ell_{j}}(y)$ for
$j\in\\{1,3\\}$. We would like to know, in addition, that whenever
$y\in\operatorname{spt}\varphi_{I,j}$ and $j\in\\{1,2\\}$, then
$q_{I,j}(y)\approx_{n}q_{\hat{Q},\ell_{j}}(y)$. This can be deduced from
Proposition 9.16, essentially by the same argument we used at (9.36). We
repeat the brief details.

Let $j\in\\{1,2\\}$, and let $I:=I_{n}(w_{I})\in\mathcal{I}(\ell_{j})$ be an
interval such that

$w_{I}\in\mathcal{C}_{m}(\ell_{j})\cup\mathcal{C}_{m}(\ell_{j}^{\uparrow})\cup\mathcal{C}_{m}(\ell_{j\downarrow})\subset\ell_{1\downarrow}\cup\ell_{1}\cup\ell_{2}\cup\ell_{3},$

and

$y\in\operatorname{spt}\varphi_{I,j}\subset\kappa_{0}I=B(f_{2}(w_{I}),\kappa_{0}K2^{-n/2})\subset
B(f_{2}(w_{I}),100K2^{-n/2}).$ (9.48)

Let $Q_{I}\in\mathcal{T}_{m}$ be the rectangle with corner $w_{I}$ such that
$q_{I,j}$ is the $(Q_{I},\ell_{j})$-approximate quadric, and recall that
$\hat{Q}\in\mathcal{T}_{\lceil(n-1)/2\rceil}$ was a rectangle with corner
$\widehat{w}$. Combining (9.44) and (9.48), and noting that the lines
containing $w_{I}$, $\widehat{w}$, and $\ell_{j}$, all lie in
$\\{\ell_{1\downarrow},\ldots,\ell_{3}\\}$, we deduce from Proposition 9.16
that

$q_{I,j}(y):=q_{Q_{I},\ell_{j}}(y)\approx_{n}q_{\hat{Q},\ell_{j}}(y)\text{ for
}j\in\\{1,2\\}\text{ and }I\in\mathcal{I}(\ell_{j})\text{ with
}y\in\operatorname{spt}\varphi_{I,j}.$ (9.49)

Let us abbreviate $q_{j}:=q_{\hat{Q},\ell_{j}}$ for $j\in\\{1,2,3\\}$. We are
very close to completing the proof of (9.41), hence (X3). Combining (9.49)
with (9.45)-(9.46), it first follows that

$\tau_{n}(\ell_{1})(y)\approx_{n-1}q_{1}(y).$ (9.50)

(This uses Corollary 9.15 in the same way as (9.37)). Similarly, combining
(9.49) with (9.45), (9.47), one can deduce (using Lemma 9.12) that

$\tau_{n}(\ell_{2})(y)\approx_{n-1}\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}(y)q_{2}(y)+\tfrac{1}{2}\varphi_{2}(y)\cdot[q_{1}(y)+q_{3}(y)],$
(9.51)

and further

$|\dot{\tau}_{n}(\ell_{2})(y)-\dot{q}_{2}(y)|\lesssim_{M}\left(\tfrac{1}{K}+o(\epsilon)\right)\cdot
2^{-n/2}.$ (9.52)

This is proven in exactly the same way as the first assertion in (9.38), so we
omit the details; the main point is, again, that the quadrics
$q_{1},q_{2},q_{3}$ only differ by a constant $\lesssim_{M}2^{-n}$. Combining
(9.50) and (9.52), we estimate

$|\dot{\tau}_{n}(\ell_{1})(y)-\dot{\tau}_{n}(\ell_{2})(y)|\lesssim_{M}|\dot{q}_{1}(y)-\dot{q}_{2}(y)|+\left(o(\epsilon)+\tfrac{1}{K}\right)\cdot
2^{-n/2}=\left(o(\epsilon)+\tfrac{1}{K}\right)\cdot 2^{-n/2}.$

Now (9.41) follows by choosing $K\geq 1$ small enough depending on $\delta,M$,
and $\epsilon>0$ small enough depending on $K,M,\Sigma$. This completes the
proof of (X3).

We turn to proving (X2). Using (9.50), and that
$\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}+\varphi_{2}\equiv 1$, we expand

$\tau_{n}(\ell_{1})(y)\approx_{n-1}q_{1}(y)\left[\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}(y)+\varphi_{2}(y)\right]=\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}(y)q_{1}(y)+\varphi_{2}(y)q_{1}(y).$

To estimate the difference $\tau_{n}(\ell_{2})(y)-\tau_{n}(\ell_{1})(y)$ we
subtract the approximation (9.51) from the one above:

$\displaystyle\tau_{n}(\ell_{2})(y)-\tau_{n}(\ell_{1})(y)$
$\displaystyle\approx_{n-1}\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}(y)q_{2}(y)+\tfrac{1}{2}\varphi_{2}(y)\cdot[q_{1}(y)+q_{3}(y)]$
$\displaystyle\qquad\qquad-\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}(y)q_{1}(y)-\varphi_{2}(y)q_{1}(y)$
$\displaystyle=\tfrac{1}{2}\varphi_{2}(y)[q_{3}(y)-q_{1}(y)]+[q_{2}(y)-q_{1}(y)]\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}(y).$
(9.53)

To conclude the proof of (X2) from here, it suffices to show that

$q_{3}(y)-q_{1}(y)\sim_{N}2^{-n}\sim_{N}q_{2}(y)-q_{1}(y),$ (9.54)

recalling that $N\sim_{M,\Sigma}1$. Indeed, plugging these estimates to
(9.53), and using
$\varphi_{2}+\sum_{I\in\mathcal{I}(\ell_{2})}\varphi_{I,2}\equiv 1$, we find
that $\eqref{form56}\sim_{N}2^{-n}$. Consequently, the computation preceding
(9.53), and the definition of the "$\approx_{n}$" notation imply that
$\tau_{n}(\ell_{2})-\tau_{n}(\ell_{1})(y)$ is within additive error of
$o(\epsilon)\cdot 2^{-n}$, from the expression at (9.53), which is
$\sim_{N}2^{-n}$. Choosing $\epsilon>0$ small enough, depending only on
$K,M,\Sigma$, it follows that
$\tau_{n}(\ell_{2})(y)-\tau_{n}(\ell_{1})(y)\sim_{N}2^{-n}$, as claimed in
(X2). This completes the proof of (X2) in the case $y\in
U^{\sharp}(\ell_{1})\cup U^{\sharp}(\ell_{2})$.

It remains to establish (9.54). This argument is very similar to the one we
already saw at the end of case $n=0$. We first recall that the signature of
$\mathcal{T}$, and hence $\hat{Q}\in\mathcal{T}$ is $+$, which meant that the
$(N,\epsilon\ell(\hat{Q}))$-QIE

$F:=F_{\hat{Q}}:=\pi_{\mathbb{V}_{\hat{Q}}}\circ\iota_{\hat{Q}}=\psi_{\mathbb{V}_{\hat{Q}}}\circ\Pi\circ\iota_{\hat{Q}}\colon
H\hat{Q}\to\mathbb{W}$

is vertically $\epsilon^{\prime}\ell(\hat{Q})$-increasing on the rectangle
$B\hat{Q}\subset H\hat{Q}$, where $\epsilon^{\prime}=2N\epsilon$. This fact
was established in Lemma 4.45, assuming $H\geq A(1+\Sigma)^{2}MB$. The quadric
straightening map
$\psi:=\psi_{\mathbb{V}_{\hat{Q}}}\colon\mathbb{W}\to\mathbb{W}$ appearing in
the definition of $F$ was introduced just above (4.31). It had the form

$\psi(y,t)=(y,t-\tfrac{1}{2}ay^{2}-by),$

where $a$ and $b$ are the common coefficients of approximate
$(\hat{Q},\ell)$-quadrics, all of which have the form
$q_{\ell}(y)=\tfrac{1}{2}ay^{2}+by+c_{\ell}$. In particular, the quadrics
$q_{1},q_{2},q_{3}$ appearing in (9.54) can be written in this general form
for certain coefficients $c_{1},c_{2},c_{3}\in\mathbb{R}$, and
$q_{i}-q_{j}\equiv c_{i}-c_{j}$ for $i,j\in\\{1,2,3\\}$. So, proving (9.54) is
equivalent to showing that

$c_{2}-c_{1}\sim_{N}\sim 2^{-n}\sim_{N}c_{3}-c_{1}.$ (9.55)

To prove (9.55), we pick (any) $3$ vertically aligned points

$(y_{0},t_{1}),(y_{0},t_{2}),(y_{0},t_{3})\in
B\hat{Q}\quad\text{with}\quad(y_{0},t_{j})\in\ell_{j},\,j\in\\{1,2,3\\}.$

This can be done for $B\geq 5$, since $\hat{Q}$ had a corner
$\widehat{w}\in\ell_{1}\cup\ell_{3}$, and
$\operatorname{dist}(\ell_{i},\ell_{j})\leq 2\ell(\hat{Q})$ for $1\leq i,j\leq
3$. Evidently $t_{j+1}=t_{j}+2^{-n}$, so in particular
$\|t_{j+1}-t_{j}\|=2^{-n/2}$, for $j\in\\{1,2\\}$. Since $F$ is vertically
$\epsilon^{\prime}\ell(\hat{Q})$-increasing, in particular the map
$t\mapsto\pi_{2}(F(y_{0},t))$ is $\epsilon^{\prime}\ell(\hat{Q})$-increasing
on $([t_{1},t_{2}],\|\cdot\|)$. Since
$\|t_{j+1}-t_{j}\|>\epsilon^{\prime}\ell(\hat{Q})$ (recall from (4.43) that
$\epsilon\ll N^{-10}$), this implies

$c_{3}=\pi_{2}(F(y_{0},t_{3}))>c_{2}=\pi_{2}(F(y_{0},t_{2}))>c_{1}=\pi_{2}(F(y_{0},t_{1})).$

Here we also used formula (4.35), which says that $(\pi_{2}\circ
F)|_{\ell}\equiv c_{\ell}$. To establish the more quantitative estimates
(9.55), we use Lemma 3.30, which implies that the map
$t\mapsto\pi_{2}(F(y_{0},t))$ is an $(N,2\epsilon\ell(\hat{Q}))$-QIE on
$([t_{1},t_{3}],\|\cdot\|)$. Hence,

$\displaystyle(2N)^{-1}2^{-n/2}$ $\displaystyle\leq
N^{-1}2^{-n/2}-2\epsilon\ell(\hat{Q})$
$\displaystyle\leq\|\pi_{2}(F(y_{0},t_{j+1}))-\pi_{2}(F(y_{0},t_{j}))\|=\|c_{j+1}-c_{j}\|$
$\displaystyle\leq N2^{-n/2}+2\epsilon\ell(\hat{Q})\leq 2N\cdot 2^{-n/2},\quad
j\in\\{1,2\\}.$

This completes the proof of (9.55), and consequently the proofs of (9.54) and
(X2).

####  9.2.4. Verifying axiom (X4)

The definition of $\tau_{n}(\ell)$ has been set up so that axiom (X4) is easy
to verify. Write $m:=\lceil n/2\rceil$, $\ell\in\mathcal{L}_{n}$, and let
$w\in\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})\cup\mathcal{C}_{m}(\ell_{\downarrow})$.
Let $Q\in\mathcal{T}_{m}$ be a rectangle with corner $w$, and let
$q:=q_{Q,\ell}$ be the $(Q,\ell)$-approximate quadric. The claim is, then,
that

$\tau_{n}(y)\approx_{n}q(y),\qquad y\in B(f_{2}(w),K2^{-n/2})=:I_{0}.$ (9.56)

Fix $y\in I_{0}$. Recall from the definition of the functions
$\\{\varphi_{I,\ell}\\}_{I\in\mathcal{I}(\ell)}$ that

$\mathbf{1}_{U(\ell)}\leq\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}\leq 1,$

so consequently
$\varphi_{\ell}=1-\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}$ vanishes on
$U(\ell)$. Since $I_{0}\subset U(\ell)$, we have

$\tau_{n}(\ell)(y)=\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}(y)\cdot
q_{I,\ell}(y),\qquad y\in I_{0},$

regardless of whether $\ell\in\mathcal{L}_{n-1}$ or
$\ell\in\mathcal{L}_{n}\,\setminus\,\mathcal{L}_{n-1}$. Now, all that remains
to be verified is:

$q_{I,\ell}(y)\approx_{n}q(y)\qquad\text{whenever}\quad
I\in\mathcal{I}(\ell)\text{ and }y\in\operatorname{spt}\varphi_{I,\ell}.$
(9.57)

Indeed, once (9.57) is settled, the approximation (9.56) follows from
Corollary 9.15.

The verification of (9.57) is similar to arguments we have already recorded
multiple times, but let us record them once more, since (X4) is perhaps the
most fundamental axiom. Let $I\in\mathcal{I}(\ell)$ be such that
$y\in\operatorname{spt}\varphi_{I,\ell}$. Thus, there exists
$w_{I}\in\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})\cup\mathcal{C}_{m}(\ell_{\downarrow})$
such that

$y\in B(f_{2}(w_{I}),\kappa_{0}K2^{-n/2})\cap B(f_{2}(w),K2^{-n/2}).$

Let $Q_{I}\in\mathcal{T}_{m}$ be a rectangle with corner $w_{I}$ such that
$q_{I,\ell}=q_{Q_{I},\ell}$ is the $(Q_{I},\ell)$-approximate quadric. Since
$w,w_{I}\in\ell_{\downarrow}\cup\ell\cup\ell^{\uparrow}$, and the separation
of these lines is bounded by $\leq 100\cdot 2^{-n}$, Proposition 9.16 is
applicable and yields (9.57) directly. This completes the verification of
axiom (X4).

####  9.2.5. Verifying axiom (X5)

Let $\ell\in\mathcal{L}_{n-1}\subset\mathcal{L}_{n}$. The claim is that

$|\tau_{n}(\ell)(y)-\tau_{n-1}(\ell)(y)|\leq\delta\cdot
2^{-n}\quad\text{and}\quad|\dot{\tau}_{n}(\ell)(y)-\dot{\tau}_{n-1}(\ell)(y)|\leq\delta\cdot
2^{-n/2}.$ (9.58)

Since $\ell\in\mathcal{L}_{n-1}$, the definition of $\tau_{n}(\ell)$ is

$\tau_{n}(\ell)(y)=\sum_{I\in\mathcal{I}(\ell)}\varphi_{I,\ell}(y)\cdot
q_{I,\ell}(y)+\varphi_{\ell}(y)\cdot\tau_{n-1}(\ell)(y),\qquad
y\in\mathbb{R}.$ (9.59)

If $y\notin U^{\sharp}(\ell)$, then simply
$\tau_{n}(\ell)(y)=\tau_{n-1}(\ell)(y)$ and
$\dot{\tau}_{n}(\ell)(y)=\dot{\tau}_{n-1}(\ell)(y)$, and there is nothing to
prove. So, let us assume that $y\in U^{\sharp}(\ell)$. In other words, with
$m:=\lceil n/2\rceil$, there exists
$w\in\mathcal{C}_{m}(\ell)\cup\mathcal{C}_{m}(\ell^{\uparrow})\cup\mathcal{C}_{m}(\ell_{\downarrow})$
such that $y\in B(f_{2}(w),\kappa_{0}K2^{-n/2})$. Let $Q\in\mathcal{T}_{m}$ be
a rectangle with corner $w$. In this case, the verification of (9.58) will
only be based on the inductive information from axiom (X4), which implies that
$\tau_{n-1}(\ell)$ is close to a certain quadric around $y$. The details are
extremely similar to those already seen during the proof of axiom (X1) (more
precisely the final case where $|y_{1}-y_{2}|\leq\tfrac{1}{5}K\cdot
2^{-n/2}$), so we just recall the skeleton of the argument: one begins by
finding a corner point $\widehat{w}\in\mathcal{C}_{\lceil(n-1)/2\rceil}$ on
either $\ell$, or its $\mathcal{L}_{n-1}$-neighbour, with the property

$y\in B(f_{2}(\widehat{w}),K2^{-(n-1)/2}).$ (9.60)

Then, one selects a rectangle $\hat{Q}\in\mathcal{T}_{\lceil(n-1)/2\rceil}$
with corner $\widehat{w}$, and uses (9.60) and (X4) inductively to show that
$\tau_{n-1}(\ell)(y)\approx_{n-1}q(y)$, where $q:=q_{\hat{Q},\ell}$ is the
$(\hat{Q},\ell)$-approximate quadric; see (9.34) for comparison. Next, one
proves that all the quadrics $q_{I,\ell}$ appearing in (9.59) with
$y\in\operatorname{spt}\varphi_{I,\ell}$ must also satisfy
$q_{I,\ell}(y)\approx_{n}q(y)$. For the details, see the proof of (9.36). From
this, it follows from Corollary 9.15 that $\tau_{n}(\ell)(y)\approx_{n-1}q(y)$
(compare with (9.37)), and finally
$\tau_{n}(\ell)(y)\approx_{n-1}\tau_{n-1}(\ell)(y)$. This implies (9.58) if
$\epsilon>0$ is chosen sufficiently small, depending only on
$\delta,K,M,\Sigma$.

The axioms (X1)-(X5) have been verified, and the proof of Proposition 8.3 is
complete. Since the proof of Theorem 1.3 had previously been reduced to
establishing Proposition 8.3, also the proof of Theorem 1.3 is complete.

##  Appendix A Proof of the quadric comparison lemma

Here we prove Lemma 9.22, whose statement we now recall:

######  Lemma A.1.

Let $q\colon\mathbb{R}\to\mathbb{R}$ be a quadratic function, and let
$I=[a,b]\subset\mathbb{R}$ be an interval. Write

$m:=\max\\{|q(a)|,|q(b)|\\}\quad\text{and}\quad\dot{m}:=\max\\{|\dot{q}(a)|,|\dot{q}(b)|\\}.$

Then, for all $r\geq 1$ and $t\in rI$,

$\
|q(t)|\lesssim(m+|I|\dot{m})r^{2},\quad|\dot{q}(t)|\lesssim(\dot{m}+\tfrac{m}{|I|})r\quad\text{and}\quad|\ddot{q}(t)|\lesssim\tfrac{m}{|I|^{2}}+\tfrac{\dot{m}}{|I|}.$

###### Proof.

We first prove the case $[a,b]=[0,1]$. The quadric $q$ can be written as

$q(t)=q(0)+\dot{q}(0)t+\tfrac{1}{2}\ddot{q}(0)t^{2},$

where $|q(0)|\leq m$, $|\dot{q}(0)|\leq\dot{m}$, and

$|\ddot{q}(t)|\equiv|\ddot{q}(0)|\leq 2|q(1)|+2|q(0)|+2|\dot{q}(0)|\leq
4m+2\dot{m}.$ (A.2)

Consequently,

$|q(t)|\leq
m+\dot{m}|t|+(4m+2\dot{m})t^{2}\quad\text{and}\quad|\dot{q}(t)|\leq\dot{m}+(8m+4\dot{m})|t|.$
(A.3)

Next, consider the general case, and define $\bar{q}(t):=q(t(b-a)+a)$. Then,

$\bar{m}:=\max\\{|\bar{q}(0)|,|\bar{q}(1)|\\}=\max\\{|q(a)|,|q(b)|\\}=m,$

and

$\dot{\bar{m}}:=\max\\{|\dot{\bar{q}}(0)|,|\dot{\bar{q}}(1)|\\}=|I|\cdot\max\\{|\dot{q}(a)|,|\dot{q}(b)|\\}=|I|\dot{m},$

and $|\ddot{\bar{q}}(t)|\leq 4\bar{m}+2\dot{\bar{m}}=4m+2|I|\dot{m}$ by (A.2).
Using (A.3), we infer that for all $t\in rI$,

$|q(t)|=\left|\bar{q}\left(\tfrac{t-a}{b-a}\right)\right|\leq
m+|I|\dot{m}\tfrac{|t-a|}{|b-a|}+(4m+2|I|\dot{m})\left(\tfrac{t-a}{b-a}\right)^{2}\lesssim(m+|I|\dot{m})r^{2}$

and

$|\dot{q}(t)|=\tfrac{1}{|I|}\left|\dot{\bar{q}}\left(\tfrac{t-a}{b-a}\right)\right|\leq\dot{m}+\tfrac{1}{|I|}(8m+4|I|\dot{m})\tfrac{|t-a|}{|b-a|}\lesssim(\dot{m}+\tfrac{m}{|I|})r$

and

$|\ddot{q}(t)|=\tfrac{1}{|I|^{2}}\left|\ddot{\bar{q}}\left(\tfrac{t-a}{b-a}\right)\right|\lesssim\tfrac{m}{|I|^{2}}+\tfrac{\dot{m}}{|I|},$

as claimed. ∎

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