$\|\psi^{\text{Re}}+u\|_{rH^{1}}\leq 2\|h\|_{\mathcal{H}}$ as in the proof of
the previous lemma, it suffices to prove the corollary with $\psi$ tacitly
standing in for $\psi+u$.

(A) Let $\mathcal{Q}=Q_{\Phi}+Q_{A}+Q_{a.\varphi}+Q_{\mu}$ be the latter four
nonlinear terms in Proposition (8.8). Definition (10.8) means
$\varepsilon\|\eta\|_{3/2+\underline{\gamma}}\leq
C\varepsilon^{1/2}\|\eta\|_{\mathfrak{W}}$, and Definition 10.5 and Theorem
7.5 imply
$\|(\varphi,a)\|_{H^{1}_{\varepsilon}}\leq\|(\varphi,a)\|_{\mathcal{H}}$. That
the same holds for $(\psi,b)$ is immediate from Definition 10.5. Item (II) and
(III) of Lemma 9.4 therefore imply

$\|\mathcal{Q}(h)\|_{L^{2}}\leq
C\varepsilon^{1/2-\gamma}\|h\|_{\mathcal{H}}^{2}.$

and because the weights in the $\mathfrak{L}$-norm are larger than the
$L^{2}$-norm by at most $\varepsilon^{-1/6}$, it follows that

$\|\mathcal{Q}(h)\|_{\mathfrak{L}}\leq
C\varepsilon^{3/12-\gamma}\|h\|_{\mathcal{H}}^{2}.$

Proceeding now to $Q_{\text{SW}}$, there are four sub-terms coming from
$(\chi^{+})^{2}Q_{\text{SW}}(\varphi,a)$,
$(\chi^{-})^{2}Q_{\text{SW}}(\psi,b)$ and the cross-terms. We bound two pieces
of each coming from the partition of unity $\zeta^{\pm}$ in Proposition 9.4.
By Item (I) of Proposition 9.4 and the above,

$\|Q_{\text{SW}}(\varphi,a)\zeta^{+}\|_{\mathfrak{L}}\leq\varepsilon^{-1/6}\|\zeta^{+}Q_{\text{SW}}(\varphi,a)\zeta^{+}\|_{L^{2}}\leq
C\varepsilon^{-1/6}\varepsilon^{4/12-\gamma}\|(\varphi,a)\|_{H^{1,+}_{\varepsilon}}^{2}\leq
C\varepsilon^{2/12-\gamma}\|h\|_{\mathcal{H}}^{2}$

and likewise for the other three sub-terms where $\zeta^{+}>0$.

Where $\zeta^{-}>0$, the $S^{\text{Re}}\oplus\Omega$-components of
$(\varphi,a)$ are exponentially small by Lemma 7.13 applied with compact sets
$K_{\varepsilon}$ whose boundary lies halfway between
$\text{supp}(\mathbb{1}^{+})$ and $\text{supp}(d\zeta^{+})$. In this same
region, the $S^{\text{Im}}\oplus\Omega$-components of $(\psi,b)$ are smaller
by a factor of $\varepsilon^{1/6}$ by (10.10). Thus in this region, the right
colum of Item (I) of Lemma 9.4 provdies the necessary bounds (in fact with
$\varepsilon^{3/12-\gamma})$.

(B) The derivative $\text{d}_{h_{N}}\mathbb{Q}(h)$ is given by the five terms
of Proposition 8.8 viewed as multilinear functions of the arguments, with
precisely one argument being chosen from the components of $h$. The proof of
the bound in (A) applies equally well in the bilinear setting to show that

$\|\text{d}\mathbb{Q}_{h_{N}}(h)\|_{\mathfrak{L}}\leq
C\varepsilon^{2/12-\gamma}\|h_{N}\|_{\mathcal{H}}\|h\|_{\mathcal{H}}$

and the assertion follows. More specifically, the requirement that
$\|h_{N}\|\leq C\varepsilon^{-1/20}$, means
$\varepsilon\|\eta_{N}\|_{3/2+\underline{\gamma}}\leq C$, hence the
assumptions of Items (II) and (III) in Lemma 9.4 are satisfied, and the terms
of $\text{d}\mathbb{Q}_{h_{N}}$ multi-linear in $h_{N}$ can be bounded just as
in that lemma.

(C) Follows from Item (B) applied to the family configurations
$h_{N}=th_{1}+(1-t)h_{2}$ and integration. ∎

##  11\. The Alternating Iteration

This section proves Proposition 10.1 by carrying out the cyclic iteration
outlined in Section 2.4. This is done by constructing three parametrices
$P_{\xi},P^{+},P^{-}$, one corresponding to each of the three stages of the
iteration (2.17).

###  11.1. The Deformation Step

This subsection constructs the deformation parametrix $P_{\xi}$, and
establishes the first of the three induction steps in the cyclic iteration
(2.17). This step (which corresponds to the horizontal arrow in the diagram
below (2.17)) shows the singular set $\mathcal{Z}_{\xi,\tau}$ can be adjusted
to effectively cancel the obstruction components of an error term, without the
error term growing much larger (in particular, this requires controlling the
key term discussed in Section 2.5.1).

The following proposition is applied to the error terms inductively, beginning
with the error $\mathfrak{e}_{1}$ of the initial approximate solutions
$(\Phi_{1},A_{1})$ in Theorem 7.5. For the remainder of Section 11, we fix,
once and for all, a choice of
$(\varepsilon,\tau)\in(0,\varepsilon_{0})\times(-\tau_{0},\tau_{0})$ and omit
this dependence from the subscripts in the notation where no confusion will
arise.

Let $\Pi^{\perp}:L^{2}\to\text{Ob}^{\perp}(\mathcal{Z}_{\tau})$ be the
$L^{2}$-orthogonal projection, so that $\Pi_{\tau}=(\Pi^{\perp},\pi_{\tau})$
in the orthogonal decomposition in Definition 5.1 where $\pi_{\tau}$ is the
$L^{2}$-orthogonal projection to the span of $\Phi_{\tau}$. Define the
deformation parametrix

$P_{\xi}:\mathfrak{L}\longrightarrow\mathfrak{W}\hskip
56.9055ptP_{\xi}:=\left(\underline{T}_{\tau}^{\circ},\not{D}_{A_{\tau}}\right)^{-1}\circ\Pi^{\perp}\circ\mathbb{1}^{-}$
(11.1)

where $(\underline{T}_{\tau}^{\circ},\not{D}_{A_{\tau}})$ is the map from
Definition 10.4, with the map ob now kept implicit in the notation. In the
following proposition, $\overline{\mathbb{SW}}_{\Lambda}$ is as in (11.30).

######  Proposition 11.1.

$P_{\xi}$ is a linear operator uniformly bounded in $\varepsilon,\tau$ and
satisfies the following property. If for $N\in\mathbb{N}$,
$h_{N}\in\mathcal{H}$ is a configuration with

  1. (I)

$\|h_{N}\|_{\mathcal{H}}\leq
C\varepsilon^{-\gamma}\|\mathfrak{e}_{1}\|_{\mathfrak{L}}$

  2. (II)

$\overline{\mathbb{SW}}_{\Lambda}(h_{N})=\mathfrak{e}_{N}$ (resp.
$\text{d}\overline{\mathbb{SW}}_{h_{1}}(h_{N})=\mathfrak{e}_{N}$) where
$\|\mathfrak{e}_{N}\|_{\mathfrak{L}}\leq
C\delta^{N-1}\|\mathfrak{e}_{1}\|_{\mathfrak{L}}$,

then the updated configuration

$h_{N}^{\prime}=(\text{Id}-P_{\xi}\circ\overline{\mathbb{SW}}_{\Lambda})h_{N}$
(11.2)

satisfies

$\overline{\mathbb{SW}}_{\Lambda}(h^{\prime}_{N})=(1-\Pi_{\tau})\mathfrak{e}_{N}^{\prime}\
+\ \mathfrak{g}^{\prime}_{N}\ +\ \lambda\varepsilon^{-1}\Phi_{\tau}\ +\
\mathfrak{e}_{N+1}$

(resp. the same for $\text{d}\overline{\mathbb{SW}}_{h_{1}}$), where
$\mathfrak{e}_{N}^{\prime},\mathfrak{e}_{N+1}\in\mathfrak{L}$, and
$\lambda\in\mathbb{R}$ obey

  1. (1)

$\|\mathfrak{e}_{N}^{\prime}\|_{\mathfrak{L}}\leq
C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$, and
$\Pi(\mathfrak{e}^{\prime}_{N})=0$.

  2. (2)

$\|\mathfrak{g}_{N}^{\prime}\|_{\mathfrak{L}}\leq
C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and
$\mathfrak{g}_{N}^{\prime}=\mathfrak{g}_{N}^{\prime}\mathbb{1}^{+}$.

  3. (3)

$\|\mathfrak{e}_{N+1}\|_{\mathfrak{L}}\leq
C\varepsilon^{1/48}\delta\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$

  4. (4)

$|\lambda|\leq C\varepsilon^{1-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$.

Moreover, $h_{N}^{\prime}$ continues to satisfy (I).

###### Proof.

The error term may be written $\mathfrak{e}_{N}=e_{N}+f_{N}+g_{N}+\Psi_{N}$
where

$\displaystyle g_{N}$ $\displaystyle:=$
$\displaystyle\mathfrak{e}_{N}\mathbb{1}^{+}\hskip 105.2751pt\
f_{N}:=\mathfrak{e}_{N}^{\text{Im}}\mathbb{1}^{-}+\pi^{\text{med}}\Pi(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})$
$\displaystyle\Psi_{N}$ $\displaystyle:=$
$\displaystyle\pi^{\text{high}}\Pi(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})\hskip
71.13188pte_{N}:=(1-\Pi)\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\pi^{\text{low}}\Pi(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-})+\pi_{\tau}(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}).$

The definition of the $\mathfrak{L}$-norm in (10.5) implies that
$\|g_{N}\|_{L^{2}}\leq
C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ ,
$\|f_{N}\|_{L^{2}}\leq C\varepsilon^{1/6}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$
and $\|\Pi(e_{N}+f_{N}+\Psi_{N})\|_{L^{2}_{-\nu}}\leq
C\varepsilon^{-\nu/2}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$. Moreover, (10.3)
implies $\|\Psi_{N}\|_{L^{2}}\leq
C\varepsilon^{10}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$.

Define

$(\eta,u):=P_{\xi}(\Pi^{\perp}(e_{N}+f_{N}+\Psi_{N}))$

so that

$\displaystyle\underline{T}^{\circ}(\eta^{\text{low}})\ $ $\displaystyle=$
$\displaystyle\Pi^{\perp}(e_{N})\hskip 2.84544pt\ =\
\pi^{\text{low}}\Pi^{\perp}(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})$
(11.3) $\displaystyle\underline{T}^{\circ}(\eta^{\text{med}})$
$\displaystyle=$ $\displaystyle\Pi^{\perp}(f_{N})\hskip 2.84544pt\ =\
\pi^{\text{med}}\Pi^{\perp}(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})$
(11.4) $\displaystyle\not{D}_{A_{\tau}}u$ $\displaystyle=$
$\displaystyle\Pi^{\perp}(\Psi_{N})\ =\
\pi^{\text{high}}\Pi^{\perp}(\mathfrak{e}^{\text{Re}}_{N}\mathbb{1}^{-})$
(11.5)

where $\eta=\eta^{\text{low}}+\eta^{\text{med}}$. The above estimates and the
uniform bounds on $(\underline{T}^{\circ},\not{D})^{-1}$ coming from Corollary
6.8 (the version in Corollary 6.10) and Corollary 5.6 show that $\eta,u$
satisfy $\|\eta\|_{\mathfrak{W}}\leq C\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and
$\|u\|_{\mathfrak{W}}\leq
C\varepsilon^{8}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$.

We now proceed to calculate
$\overline{\mathbb{SW}}_{\Lambda}(h^{\prime}_{N})$, where
$h_{N}^{\prime}=h_{N}-(\eta,u)$ as in (11.2). First, with
$\xi=\varepsilon\eta$ as previously, we compute:

$\displaystyle\text{d}\overline{\mathbb{SW}}_{h_{1}}(\xi,0,0,0,0)$
$\displaystyle=$
$\displaystyle\text{d}\not{\mathbb{D}}(\eta,0)+\Xi^{+}(\eta)+\Xi^{-}(\eta)$
(11.6) $\displaystyle=$
$\displaystyle\underline{T}^{\circ}(\eta)+\mathfrak{t}^{\circ}(\eta)+(1-\Pi)\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)+\pi_{\tau}\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)+\Xi^{+}(\eta)+\Xi^{-}(\eta)$
$\displaystyle\text{d}\overline{\mathbb{SW}}_{h_{1}}(0,0,0,0,u)$
$\displaystyle=$ $\displaystyle\not{D}_{A_{\tau}}(\chi^{-}u)+K_{1}$ (11.7)
$\displaystyle=$
$\displaystyle\not{D}_{A_{\tau}}u-(1-\chi^{-})\not{D}_{A_{\tau}}u+d\chi^{-}.u+K_{1}(\chi^{-}u)$

where $K_{1}$ is as in Step 2 of the proof of Lemma 10.14.

With these, we now compute:

$\displaystyle\overline{\mathbb{SW}}_{\Lambda}(h_{N}^{\prime})$
$\displaystyle=$
$\displaystyle\overline{\mathbb{SW}}(h_{1})+\text{d}_{h_{1}}\mathbb{SW}(h_{N})\
-\ \text{d}_{h_{1}}\mathbb{SW}(\xi,u)\ +\ \mathbb{Q}(h_{N}^{\prime})\ -\
(\mu+\Lambda)\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}$ (11.9) $\displaystyle=$
$\displaystyle\ \overline{\mathbb{SW}}_{\Lambda}(h_{N})\ -\
\text{d}_{h_{1}}\mathbb{SW}(\xi,u)\ +\
\mathbb{Q}(h^{\prime}_{N})-\mathbb{Q}(h_{N})$ $\displaystyle=$
$\displaystyle\mathfrak{e}_{N}\ -\ \underline{T}^{\circ}(\eta^{\text{low}})\
-\ \underline{T}^{\circ}(\eta^{\text{med}})\ -\ \not{D}_{A_{\tau}}u\ -\
(1-\Pi)\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)-\pi_{\tau}\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)$
$\displaystyle\ +\ \Xi^{+}(\eta)\ +\ \text{{Zh}}(\eta,u)$ $\displaystyle=$
$\displaystyle g_{N}\ +\ \Xi^{+}(\eta)\ +\
\mathfrak{e}^{\text{Im}}_{N}\mathbb{1}^{-}\ +\
(1-\Pi)(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))+\pi_{\tau}(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))$
$\displaystyle\ +\ \text{{Zh}}(\eta,u)$

where (11.9) is obtained by substituting (11.6) and (11.7) with

$\text{{Zh}}(\eta,u):=\left(\mathbb{Q}(h^{\prime}_{N})-\mathbb{Q}(h_{N})\right)\
-\ \Xi^{-}(\eta)\ -\ \mathfrak{t}^{\circ}(\eta)\ +\
(1-\chi^{-})\not{D}_{A_{\tau}}u\ -\ d\chi^{-}.u\ -\ K_{1}(\chi^{-}u)$ (11.10)

and (11.9) from substituting (11.4)–(11.5).

Splitting up the terms of (11.9), define

$\displaystyle\mathfrak{e}_{N}^{\prime}$ $\displaystyle:=$
$\displaystyle\mathfrak{e}^{\text{Im}}_{N}\mathbb{1}^{-}\ +\
(1-\Pi)(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))$
$\displaystyle\mathfrak{g}_{N}^{\prime}$ $\displaystyle=$ $\displaystyle
g_{N}\ +\ \Xi^{+}(\eta)$ $\displaystyle\lambda$ $\displaystyle:=$
$\displaystyle\varepsilon\langle\pi_{\tau}(\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}+\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))\
,\ \Phi_{\tau}\rangle_{L^{2}}$ $\displaystyle\mathfrak{e}_{N+1}$
$\displaystyle:=$ $\displaystyle\text{{Zh}}(\eta,u).$

It now suffices to show that these satisfy the conclusions (1)–(4) of the
proposition.

Beginning with (3), we have that

$\|\text{{Zh}}(\eta,u)\|_{\mathfrak{L}}\leq
C\varepsilon^{1/48}\delta\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.$ (11.11)

By the bound $\|u\|_{\mathfrak{W}}\leq
C\varepsilon^{8}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ from above and Corollary
10.3, all the terms of (11.10) except those involving
$\mathbb{Q},\Xi^{+},\Xi^{-}$ are bounded by, say,
$\varepsilon^{5}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and may be safely
ignored. Corollary 9.4, and repeating the argument that obtained (10.19) for
$\Xi^{-}$ shows that $\|\text{{Zh}}(\eta,u)\|_{\mathfrak{L}}\leq
C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$. Since
$\gamma<<1$ and $\varepsilon^{1/48}\delta=\varepsilon^{1/24}$, (11.11)
follows, which is conclusion (2).

(4) follows directly from the definition of $\lambda$ and Cauchy-Schwartz,
since
$\|\mathfrak{e}_{N}^{\text{Re}}\mathbb{1}^{-}\|_{L^{2}}\leq\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$
and

$\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)\|_{L^{2}}\leq C\|\eta\|_{1/2}\leq
C\|\eta\|_{\mathfrak{W}}$

by Proposition 9.3.

For (1)–(2), the fact the $\Pi(\mathfrak{e}_{N}^{\prime})=0$ and
$\mathfrak{g}^{\prime}_{N}=\mathfrak{g}_{N}^{\prime}\mathbb{1}^{+}$ are
immediate from the definitions. It remains to show that the asserted bounds
hold. To re-iterate the cancellation that led to (11.9), adding and
subtracting (11.4),(11.5), $\Psi_{N}$ and
$\lambda\varepsilon^{-1}\Phi_{\tau}$, then using
$\Pi(\text{d}\not{\mathbb{D}})=T^{\circ}+\mathfrak{t}^{\circ}$ yields

$\mathfrak{e}_{N}^{\prime}=\mathfrak{e}_{N}^{\text{Im}}\mathbb{1}^{-}\ +\
e_{N}^{\text{Re}}\mathbb{1}^{-}\ +\ \text{d}\not{\mathbb{D}}_{h_{0}}(\eta)\
-\mathfrak{t}^{\circ}(\eta)\ -\ \Psi_{N}\ -\
\lambda\varepsilon^{-1}\Phi_{\tau}$

Because $\mathfrak{t}^{\circ}(\eta),\Psi_{N}$ are $O(\varepsilon^{8})$ these
may be safely ignored as in the proof of (3). Since
$\mathfrak{e}^{\text{Im}}_{N}$ is unchanged, and
$\pi^{\text{med}}\Pi(\mathfrak{e}_{N}^{\prime})=0$, it suffices to bound the
three terms on the top line in the norm (10.13).

$\displaystyle\|\text{d}\not{\mathbb{D}}_{h_{0}}(\eta)\mathbb{1}^{+}\|_{L^{2}}\
$ $\displaystyle\leq$ $\displaystyle\ \ \
C\varepsilon^{1/12-\gamma}\|\eta\|_{\mathfrak{W}}\ \hskip 73.97733pt\leq\
C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$
$\displaystyle\|(\mathfrak{e}_{N}^{\text{Re}}+\
\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))\mathbb{1}^{-}\|_{L^{2}}\ $
$\displaystyle\leq$ $\displaystyle\ \ \ C\|\mathfrak{e}_{N}\|_{\mathfrak{L}}\
\ \ \ +\ \ \ \ C\|\eta\|_{\mathfrak{W}}\ \ \ \ \ \ \ \ \ \ \ \leq\ \
C\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$
$\displaystyle\|(\mathfrak{e}_{N}^{\text{Re}}+\
\text{d}\not{\mathbb{D}}_{h_{0}}(\eta))\mathbb{1}^{-}\|_{L^{2}_{-\nu}}$
$\displaystyle\leq$ $\displaystyle\
C\varepsilon^{-\nu/2}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}\
+C\varepsilon^{-\nu/2-\gamma}\|\eta\|_{\mathfrak{W}}\ \ \ \leq\ \
C\varepsilon^{-\gamma}\varepsilon^{-\nu/2}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$
$\displaystyle\|g_{N}+\Xi^{+}(\eta)\|_{L^{2}}\ $ $\displaystyle\leq$
$\displaystyle C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}\
+\ C\varepsilon^{1/12-\gamma}\|\eta\|_{\mathfrak{W}}\ \leq\
C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$

where each line, the definition of the $\mathfrak{L}$-norm is used in
conjunction with, (10.22), (10.17), and (10.18). For the
$\lambda\varepsilon^{-1}\Phi_{\tau}$, the same bounds hold by (2) and the fact
that $\Phi_{\tau}\in L^{2}\cap L^{2}_{-\nu}$. Conclusion (1) follows from the
first three lines, and (2) from the fourth.

That (I) holds for $h_{N}^{\prime}$ is immediate from the bounds
$\|\eta\|_{\mathfrak{W}}\ +\ \|u\|_{\mathcal{X}}\leq
C\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$, and the fact that it holds for $h_{N}$.
Finally, the proof of the (resp. $\text{d}\overline{\mathbb{SW}}$) statements
is identical, omitting any mention of $\mathbb{Q}$ and $\Lambda$. ∎

###  11.2. The Outside Step

This subsection covers the second of the three stages of the cyclic induction
(2.17). Now that the error terms are essentially orthogonal to the
obstruction, solving in the outside can proceed using Lemma 4.4 and
Proposition 7.11.

Define the outside parametrix $P^{-}$ by

$\displaystyle
P^{-}:\mathfrak{L}\longrightarrow\mathcal{H}^{-}\oplus\mathbb{R}\hskip
56.9055ptP^{-}$ $\displaystyle:=$
$\displaystyle\left(\mathcal{L}^{-1}_{(\Phi_{\tau},A_{\tau})}(1-\Pi^{\perp})\
,\ -\varepsilon\pi_{\tau}\right)\circ\mathbb{1}^{-}$ (11.12)

where $\pi_{\tau}:\mathfrak{L}\to\mathbb{R}$ is understood to mean the
coefficient of $\Phi_{\tau}$ in $\mathbb{R}\Phi_{\tau}$. Notice that the first
component indeed lands in $\mathcal{H}^{-}$ by Definition (10.5).

######  Proposition 11.2.

$P\\!^{-}$ is a linear operator uniformly bounded in $\varepsilon,\tau$ and
satisfies the following property. If for $N\in\mathbb{N}$,
$h^{\prime}_{N}\in\mathcal{H}$ is a configuration satisfying the conclusions
of Proposition 11.1, then the updated configuration

$h_{N}^{\prime\prime}=(\text{Id}-P^{-}\circ\overline{\mathbb{SW}}_{\Lambda})h_{N}^{\prime}$
(11.13)

satisfies

$\overline{\mathbb{SW}}_{\Lambda}(h^{\prime\prime}_{N})=\mathfrak{e}_{N}^{\prime\prime}\mathbb{1}^{+}\
+\ \mathfrak{e}_{N+1}$

(resp. the same for $\text{d}_{h_{1}}\overline{\mathbb{SW}}$), where
$\mathfrak{e}_{N}^{\prime\prime},\mathfrak{e}_{N+1}\in\mathfrak{L}$ obey

  1. (1’)

$\|\mathfrak{e}_{N}^{\prime\prime}\|_{\mathfrak{L}}\leq
C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$

  2. (2’)

Item (2) from Proposition 11.1 continues to hold.

Moreover, $h_{N}^{\prime\prime}$ continues to satisfy (I) from Proposition
11.1.

###### Proof.

As in the proof of Proposition 11.1, write
$\mathfrak{e}_{N}^{\prime}+\mathfrak{g}_{N}^{\prime}=e^{\prime}_{N}+f^{\prime}_{N}+g^{\prime}_{N}$,
where

$g_{N}^{\prime}:=\mathfrak{g}_{N}^{\prime}\mathbb{1}^{+}\hskip
56.9055ptf_{N}^{\prime}:=(\mathfrak{e}^{\prime}_{N})^{\text{Im}}\mathbb{1}^{-}\hskip
56.9055pte_{N}^{\prime}:=(\mathfrak{e}_{N}^{\prime})^{\text{Re}}.$ (11.14)

Conclusions (1)–(4) of Proposition 11.1 mean that $\Pi(e_{N}^{\prime})=0$,
that $\|e_{N}\|_{L^{2}_{-\nu}}\leq
C\varepsilon^{-\gamma}\varepsilon^{-\nu/2}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$,
for both $\nu=0,\nu^{-}$, and that $\|f_{N}\|_{L^{2}}\leq
C\varepsilon^{1/6}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ and
$\|g_{N}\|_{L^{2}}\leq
C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$, where
$\mathfrak{e}_{N}$ is the original error from Proposition 11.1.

Set

$(\psi,b,\mu):=P^{-}(e^{\prime}_{N}+f^{\prime}_{N}+\lambda\varepsilon^{-1}\Phi_{\tau})$

so that

$\displaystyle\not{D}_{A_{\tau}}\psi^{\text{Re}}$ $\displaystyle=$
$\displaystyle e_{N}^{\prime}$
$\displaystyle\mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{\text{Im}}(\psi^{\text{Im}},b)$
$\displaystyle=$ $\displaystyle f_{N}^{\prime}$ $\displaystyle\mu$
$\displaystyle=$
$\displaystyle-\varepsilon\pi(\lambda\varepsilon^{-1}\Phi_{\tau})=-\lambda$

where $\psi=(\psi^{\text{Re}},\psi^{\text{Im}})\in S^{\text{Re}}\oplus
S^{\text{Im}}$.

We now show that $P^{-}$ is uniformly bounded. Lemma 4.4 and Proposition 7.11
and the above bounds on $e^{\prime}_{N},f^{\prime}_{N}$ show that these unique
solutions (where $\pi_{\tau}(\psi^{\text{Re}})=0$), satisfy

$\displaystyle\|\psi^{\text{Re}}\|_{rH^{1}_{e}}$ $\displaystyle\leq$
$\displaystyle C\|e_{N}^{\prime}\|_{L^{2}}\ \ \ \ \ \ \ \ \ \ \leq\
C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ (11.15)
$\displaystyle\varepsilon^{\nu/2}\|\psi^{\text{Re}}\|_{r^{1+\nu}H^{1}_{e}}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{\nu/2}\|e_{N}^{\prime}\|_{L^{2}_{-\nu}}\ \ \hskip 5.12128pt\leq\
C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ (11.16)
$\displaystyle\varepsilon^{-1/6}\|(\psi^{\text{Im}},b)\|_{\mathcal{H}^{1,-}_{\varepsilon}}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{-1/6}\|f_{N}^{\prime}\|_{L^{2}}\ \ \ \leq\
C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$ (11.17)
$\displaystyle\varepsilon^{-1}|\mu|$ $\displaystyle\leq$ $\displaystyle
C\|\lambda\varepsilon^{-1}\Phi_{\tau}\|_{L^{2}}\ \hskip 7.96674pt\leq\
C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.$ (11.18)

To explain further how to obtain the bounds in the middle column, the
condition that $\pi_{\tau}(\psi^{\text{Re}})=0$ means that the $\nu=0$ version
of Lemma 4.4 applies without the projection term, which yields (11.15). In
turn, (11.15) and Cauchy-Schwartz show that $\pi_{\nu}(\psi^{\text{Re}})\leq
C\|\psi\|_{L^{2}}\leq C\|e^{\prime}_{N}\|_{L^{2}}$ since $\Phi_{\tau}\in
L^{2}_{-2\nu}$, after which (11.16) follows from Lemma 4.4 taking
$\nu=\nu^{-}$. (11.17) and (11.18) are immediate from Proposition 7.11 and the
definition of $\mu$. The final column follows immediately from the bounds on
(11.14) and conclusion (4) of Proposition 11.1.

To show that $P^{-}$ is uniformly bounded, it must be shown that
$\|\mathfrak{e}_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau}\|_{\mathfrak{L}}$
dominates each term in the middle column (since $P^{-}$ ignores
$\mathfrak{g}_{N}^{\prime}$). Indeed, $\mathfrak{e}_{N}^{\prime}$ is, by its
construction in the proof of Proposition 11.1, $L^{2}$-orthogonal to
$\Phi_{\tau}$. Therefore,

$\displaystyle\|e_{N}^{\prime}\|_{L^{2}}+\|\lambda\varepsilon^{-1}\Phi_{\tau}\|_{L^{2}}$
$\displaystyle\leq$ $\displaystyle\ \ \ \ \ \
\|\mathfrak{e}_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau}\|_{L^{2}}\hskip
93.89418pt\ \leq
C\|\mathfrak{e}_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau}\|_{\mathfrak{L}}\
\ \ \ \ \ \ $ (11.19) $\displaystyle\|e_{N}^{\prime}\|_{L^{2}_{-\nu}}$
$\displaystyle\leq$
$\displaystyle\|(e_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau})\mathbb{1}^{-}\|_{L^{2}_{-\nu}}+\|(\lambda\varepsilon^{-1}\Phi_{\tau})\mathbb{1}^{-}\|_{L^{2}_{-\nu}}\leq
C\|\mathfrak{e}_{N}^{\prime}+\lambda\varepsilon^{-1}\Phi_{\tau}\|_{\mathfrak{L}}.$
(11.20)

The first of these is simply orthogonality along with fact that the
$\mathfrak{L}$-norm dominates the $L^{2}$-norm. The second is the triangle
inequality, then invoking the first one along with the fact that the
$L^{2}_{-\nu}$ and $L^{2}$-norms are equivalent on the 1-dimensional span of
$\Phi_{\tau}$. That $\varepsilon^{-1/6}\|f_{N}^{\prime}\|_{L^{2}}$ in (11.17)
is likewise bounded by the right side of (11.19) is immediate from the
Definition of the $\mathfrak{L}$-norm. This completes the claim that $P^{-}$
is uniformly bounded.

We now proceed to calculate
$\overline{\mathbb{SW}}_{\Lambda}(h^{\prime\prime}_{N})$ where
$h_{N}^{\prime\prime}=h_{N}-(\psi,b,\mu)$ as in (11.13). First,

$\displaystyle\text{d}\overline{\mathbb{SW}}_{h_{1}}(\psi,b,\mu)$
$\displaystyle=$
$\displaystyle\mathcal{L}_{(\Phi_{\tau},A_{\tau})}\chi^{-}(\psi,b)\ +\
\mathcal{K}_{1}(\psi,b)\ -\ \mu\varepsilon^{-1}\chi^{-}\Phi_{\tau}$ (11.21)
$\displaystyle=$ $\displaystyle\not{D}_{A_{\tau}}\psi^{\text{Re}}\ +\
\mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{\text{Im}}(\psi^{\text{Im}},b)\ -\
\mu\varepsilon^{-1}\Phi_{\tau}$ $\displaystyle\ +\ d\chi^{-}(\psi,b)\ +\
K_{1}(\psi,b)\ +\ (1-\chi^{-})\mu\varepsilon^{-1}\Phi_{\tau},$

where we have used the fact that $\chi^{-}=1$ on the support of
$\mathbb{1}^{-}$. Using the conclusion of Proposition 11.1 and the definitions
(11.14), then substituting (11.21) yields

$\displaystyle\overline{\mathbb{SW}}_{\Lambda}(h_{N}^{\prime\prime})$
$\displaystyle=$ $\displaystyle\text{d}_{h_{1}}\mathbb{SW}(h_{N}^{\prime})\ -\
\text{d}_{h_{1}}\mathbb{SW}(\psi,b,\mu)\ +\ \mathbb{Q}(h_{N}^{\prime\prime})\
-\ (\mu_{N}+\mu+\Lambda)\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}$
$\displaystyle=$ $\displaystyle\
\overline{\mathbb{SW}}_{\Lambda}(h_{N}^{\prime})\ -\
\text{d}_{h_{1}}\mathbb{SW}(\psi,b,\mu)\ +\
\mathbb{Q}(h_{N}^{\prime\prime})-\mathbb{Q}(h_{N}^{\prime})$ $\displaystyle=$
$\displaystyle e_{N}^{\prime}\ +\ f_{N}^{\prime}\ +\ g_{N}^{\prime}\ +\
\lambda\varepsilon^{-1}\Phi_{\tau}\ -\ \not{D}_{A_{\tau}}\psi^{\text{Re}}\ -\
\mathcal{L}_{(\Phi_{\tau},A_{\tau})}^{\text{Im}}(\psi^{\text{Im}},b)\ +\
\mu\varepsilon^{-1}\Phi_{\tau}\ \ \ $ $\displaystyle\ +\
\left(\mathbb{Q}(h_{N}^{\prime\prime})-\mathbb{Q}(h_{N}^{\prime})\right)\ +\
d\chi^{-}(\psi,b)\ +\ K_{1}(\psi,b)\ +\
(1-\chi^{-})\mu\varepsilon^{-1}\Phi_{\tau}\ +\ \mathfrak{e}_{N+1}$
$\displaystyle=$ $\displaystyle g_{N}^{\prime}+\
\left(\mathbb{Q}(h_{N}^{\prime\prime})-\mathbb{Q}(h_{N}^{\prime})\right)\ +\
d\chi^{-}(\psi,b)\ +\ K_{1}(\psi,b)\ +\
(1-\chi^{-})\mu\varepsilon^{-1}\Phi_{\tau}\ +\ \mathfrak{e}_{N+1}.$

Then (re)-define

$\displaystyle\mathfrak{e}_{N}^{\prime\prime}$ $\displaystyle:=$
$\displaystyle g_{N}^{\prime}\ +\ d\chi^{-}(\psi,b)\ +\
(1-\chi^{-})\mu\varepsilon^{-1}\Phi_{\tau}$ (11.22)
$\displaystyle\mathfrak{e}_{N+1}$ $\displaystyle\mapsto$
$\displaystyle\mathfrak{e}_{N+1}\ +\
\left(\mathbb{Q}(h_{N}^{\prime})-\mathbb{Q}(h_{N}^{\prime\prime})\right)\ +\
K_{1}(\psi,b).$ (11.23)

Notice that $\mathfrak{e}_{N}^{\prime\prime}$ now includes the crucial
alternating error term $d\chi^{-}(\psi,b)$.

To conclude, we show $\mathfrak{e}_{N}^{\prime\prime}$ and
$\mathfrak{e}_{N+1}$ satisfy conclusions (1)–(2). That (1) holds for the first
term of $\mathfrak{e}_{N}^{\prime\prime}$ in (11.22) is immediate from the
bound on $g^{\prime}_{N}$ from 11.14. The bounds on $e_{N}^{\prime}$ in 11.14
also imply that $e_{N}^{\prime}$ has $\varepsilon^{1/2}$-effective support,
hence Lemma 4.7 applies to show that

$\|d\chi^{-}.\psi^{\text{Re}}\|_{L^{2}}\leq
C\varepsilon^{1/12-\gamma}\|e_{N}^{\prime}\|_{L^{2}}\leq
C\varepsilon^{1/12-2\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.$ (11.24)

Note that the $(1-\chi^{-})\mathfrak{e}_{N}^{\prime}$ term in Lemma 4.7
vanishes because $\chi^{-}=1$ on $\text{supp}(\mathbb{1}^{-})$. (11.24) also
holds for $(\psi^{\text{Im}},b)$ simply because of the $\varepsilon^{1/6}$
weight on these components in (11.17). Finally, for the third term of
$\mathfrak{e}_{N}^{\prime\prime}$ in (11.22) , notice that direct integration
using the fact that $|\Phi_{\tau}|\leq Cr^{1/2}$ shows that
$\|(1-\chi^{-})\Phi_{\tau}\|_{L^{2}}\leq C\varepsilon^{1-\gamma}$, and (11.18)
therefore shows the third term of $\mathfrak{e}_{N}^{\prime\prime}$ is bounded
by the right hand side of (11.24) with an extra factor of
$\varepsilon^{11/12-\gamma}$. Combining these three shows that

$\|\mathfrak{e}_{N}^{\prime\prime}\|_{\mathfrak{L}}\leq
C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$

and
$\mathfrak{e}_{N}^{\prime\prime}=\mathfrak{e}_{N}^{\prime\prime}\mathbb{1}^{-}$
because $\chi^{-}=1$ outside $\text{supp}(\mathbb{1}^{+})$.

Using the bounds (11.15)–(11.17), the terms involving $\mathbb{Q}$ can be
bounded identically to in the proof of Proposition 11.1. As in the proof of
Lemma 10.14, $K_{1}$ is exponentially small, hence negligible. Conclusion (2)
follows after increasing $\gamma$ slightly.

Finally, that (I) continues to hold for $h_{N}^{\prime\prime}$ and of the
(resp. $\text{d}\overline{\mathbb{SW}}_{h_{1}}$) statements follows
identically to in Proposition 11.1 using the uniform boundedness of $P^{-}$. ∎

###  11.3. The Inside Step

The section completes the third stage of the cyclic induction by constructing
$P^{+}$. Define

$\displaystyle P^{+}:\mathfrak{L}\longrightarrow\mathcal{H}^{+}\hskip
56.9055ptP^{+}$ $\displaystyle:=$
$\displaystyle\mathcal{L}^{-1}_{(\Phi_{1},A_{1})}\mathbb{1}^{+}.$ (11.25)

Definition 10.5 ensures that $P^{+}$ lands in $\mathcal{H}^{+}$.

######  Proposition 11.3.

$P^{+}$ is a linear operator uniformly bounded in $\varepsilon,\tau$ and
satisfies the following property. If for $N\in\mathbb{N}$,
$h^{\prime\prime}_{N}\in\mathcal{H}$ is a configuration satisfying the
conclusions of Proposition 11.2, then the updated configuration

$h_{N+1}=(\text{Id}-P^{-}\circ\overline{\mathbb{SW}}_{\Lambda})h_{N}^{\prime\prime}$
(11.26)

(resp. the same for $\text{d}\overline{\mathbb{SW}}_{h_{1}})$ satisfies the
hypotheses of Proposition 11.1 with $N^{\prime}=N+1$.

###### Proof.

Uniform boundedness is immediate from the definition of the
$\mathcal{H}^{+}$-norm in Definition 10.5. Let
$\mathfrak{e}_{N}^{\prime\prime}$ be as in the conclusion of Proposition 11.2,
so that we may write

$g_{N}^{\prime\prime}:=\mathfrak{e}_{N}^{\prime\prime}\mathbb{1}^{+}=\mathfrak{e}_{N}^{\prime\prime}$

where $\|g_{N}^{\prime\prime}\|_{L^{2}}\leq
C\varepsilon^{1/12-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}$.

Set

$(\varphi,a):=P^{+}(g_{N}^{\prime\prime})$

so that

$\mathcal{L}_{(\Phi_{1},A_{1})}(\varphi,a)=g_{N}^{\prime\prime},$

and

$\|(\varphi,a)\|_{H^{1,+}_{\varepsilon}}\ \leq\
\|(\varphi,a)\|_{\mathcal{H}^{+}}=C\varepsilon^{-1/12-\gamma}\|g_{N}^{\prime\prime}\|\
\leq\ C\varepsilon^{-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.$ (11.27)

We now proceed to calculate $\overline{\mathbb{SW}}_{\Lambda}(h_{N+1})$ where
$h_{N+1}=h_{N}-(\varphi,a)$ as in (11.13).

$\displaystyle\overline{\mathbb{SW}}_{\Lambda}(h_{N+1})$ $\displaystyle=$
$\displaystyle\text{d}_{h_{1}}\mathbb{SW}(h_{N}^{\prime\prime})\ -\
\text{d}_{h_{1}}\mathbb{SW}(\varphi,a)\ +\ \mathbb{Q}(h_{N+1})\ -\
(\mu_{N}+\mu+\Lambda)\chi^{-}\frac{\Phi_{\tau}}{\varepsilon}$ (11.28)
$\displaystyle=$ $\displaystyle\
\overline{\mathbb{SW}}_{\Lambda}(h_{N}^{\prime})\ -\
\text{d}_{h_{1}}\mathbb{SW}(\psi,b,\mu)\ +\
\mathbb{Q}(h_{N+1})-\mathbb{Q}(h_{N}^{\prime\prime})$ $\displaystyle=$
$\displaystyle g_{N}^{\prime\prime}\ -\
\mathcal{L}_{(\Phi_{1},A_{1})}(\varphi,a)\ +\
\left(\mathbb{Q}(h_{N+1})-\mathbb{Q}(h_{N}^{\prime\prime})\right)\ +\
d\chi^{+}(\varphi,a)\ +\ \mathfrak{e}_{N+1}$ $\displaystyle=$
$\displaystyle\left(\mathbb{Q}(h_{N+1})-\mathbb{Q}(h_{N}^{\prime\prime})\right)\
+\ d\chi^{+}(\varphi,a)\ +\ \mathfrak{e}_{N+1}$

The alternating error from $d\chi^{+}$ has now been shifted back to the
outside region.

Re-defining $\mathfrak{e}_{N+1}$ to include all three terms of (11.28), we
claim that it satisfies (II) of Proposition 11.1 with $N^{\prime}=N+1$. In
fact, one has the slightly stronger bound

$\|\mathfrak{e}_{N+1}\|_{\mathfrak{L}}\leq
C\varepsilon^{1/48-\gamma}\delta\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.$ (11.29)

Indeed, the terms involving $\mathbb{Q}$ in (11.28) may be bounded as in the
proof of Proposition 11.1. Then, since $d\chi^{+}$ is supported where
$r=O(\varepsilon^{1/2})$, Lemma 7.12 implies

$\displaystyle\|d\chi^{+}(\varphi,a)\|_{\mathfrak{L}}$ $\displaystyle\leq$
$\displaystyle\|d\chi^{+}(\varphi,a)\|_{L^{2}}+\varepsilon^{\nu}\|d\chi^{+}(\varphi,a)\|_{L^{2}_{-\nu}}$
$\displaystyle\leq$ $\displaystyle C\|d\chi^{+}(\varphi,a)\|_{L^{2}}\leq
C\varepsilon^{-1/24-\gamma}\|g_{N}^{\prime\prime}\|_{L^{2}}\leq
C\varepsilon^{1/24-\gamma}\|\mathfrak{e}_{N}\|_{\mathfrak{L}}.$

Finally, the original $\mathfrak{e}_{N+1}$ in (11.28) already satisfies
(11.29) by virtue of (2’) in Proposition 11.2. (11.29) follows.

Over the course of the proofs of (a single cycle of) Propositions 11.1, 11.2,
and 11.3, the constants $C,\gamma$ have been increased a finite number of
times. Let $C_{0}$ being the original constant in Item (II) of Proposition
11.1 and $C_{1},\gamma_{1}$ be the final versions of the constants appearing
in (11.29). Since $\gamma_{1}<<1$ still, we may assume that

$C_{1}\varepsilon^{1/48-\gamma_{1}}\leq C_{0}$

once $\varepsilon$ is sufficiently small, which reduces (11.29) to hypothesis
(II) of Proposition 11.1 with $N^{\prime}=N+1$ as desired. The proof of (I)
and of the (resp. $\text{d}\overline{\mathbb{SW}}_{h_{1}}$) statements again
follows as in Propositions 11.1 and 11.2 using (11.27). ∎

###  11.4. Proof of Proposition 10.1

This section completes the proof of Proposition 10.1.

Analogously to (2.23), define $\mathbb{A}$ and
$\mathbb{P}_{1}=\text{d}\mathbb{A}$ by

$\displaystyle\mathbb{A}$ $\displaystyle=$ $\displaystyle P_{\xi}\ +\
P^{-}(\text{Id}-\ \overline{\mathbb{SW}}_{\Lambda}P_{\xi})\ \ +\ \
P^{+}\left(\text{Id}-\ \overline{\mathbb{SW}}_{\Lambda}P_{\xi}\ -\ \
\overline{\mathbb{SW}}_{\Lambda}P^{-}(\text{Id}\ -\
\overline{\mathbb{SW}}_{\Lambda}P_{\xi})\right),$ (11.30)
$\displaystyle\mathbb{P}_{1}$ $\displaystyle=$ $\displaystyle P_{\xi}\ +\
P^{-}(\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}P_{\xi})\ +\
P^{+}\left(\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}P_{\xi}-\
\text{d}\overline{\mathbb{SW}}_{h_{1}}P^{-}(\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}P_{\xi})\right),$
(11.31)

So that (analogously to 2.24),

$\displaystyle T=\text{Id}-\mathbb{A}\circ\ \overline{\mathbb{SW}}_{\Lambda}\
\ $ $\displaystyle=$ $\displaystyle(\text{Id}-P^{+}\
\overline{\mathbb{SW}}_{\Lambda}\
)(\text{Id}-P^{-}\overline{\mathbb{SW}}_{\Lambda}\ )(\text{Id}-P_{\xi}\
\overline{\mathbb{SW}}_{\Lambda}\ ),$ (11.32)
$\displaystyle\text{dT}=\text{Id}-\mathbb{P}_{1}\circ\text{d}\overline{\mathbb{SW}}_{h_{1}}$
$\displaystyle=$
$\displaystyle(\text{Id}-P^{+}\text{d}\overline{\mathbb{SW}}_{h_{1}})(\text{Id}-P^{-}\text{d}\overline{\mathbb{SW}}_{h_{1}})(\text{Id}-P_{\xi}\text{d}\overline{\mathbb{SW}}_{h_{1}}).$
(11.33)

Thus applying $T$ carries one complete cycle of the iteration (2.17), and
$\text{d}T$ one complete cycle of the linearized iteration (the resp.
statements in Propositions 11.1–11.3).

The next lemma is the analogue of Proposition 2.4.

######  Lemma 11.4.

The linearization
$\text{d}\overline{\mathbb{SW}}_{h_{1}}:\mathcal{H}\to\mathfrak{L}$ is
invertible, and there is a constant $C$ independent of $\varepsilon,\gamma$
such that

$\|h\|_{\mathcal{H}}\leq
C\varepsilon^{-\gamma}\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(h)\|_{\mathfrak{L}}$
(11.34)

holds.

###### Proof.

Let $\mathbb{P}_{0}:\mathfrak{L}\to\mathcal{H}$ be defined by

$\mathbb{P}_{0}:=P^{+}\oplus\begin{pmatrix}P_{\xi}&0\\\ *&P^{-}\end{pmatrix}$

where $*=-P^{-}(1-\Pi^{\perp})\text{d}\not{\mathbb{D}}_{h_{0}}P_{\xi}$. Since
the map $\beta:\mathfrak{L}(Y)\to\mathfrak{L}(Y^{+})\oplus\mathfrak{L}(Y^{-})$
defined by
$\beta(\mathfrak{e}):=(\mathfrak{e}\mathbb{1}^{+},\mathfrak{e}\mathbb{1}^{-})$,
which is built into the definitions of $P_{\xi},P^{\pm}$ is an isomorphism,
and so is each of $P_{\xi},P^{-},P^{+}$, it follows that $\mathbb{P}_{0}$ is
an isomorphism hence Fredholm with index 0.

Then, the calculations (11.6), (11.7), and (11.21) and the subsequent bounds
(e.g. 11.11) in the proofs of Propositions 11.1 and 11.2 show that

$\displaystyle\mathbb{P}_{1}-\mathbb{P}_{0}$ $\displaystyle=$
$\displaystyle-(P^{+}+P^{-})(\Xi+\mathfrak{t}^{\circ})P_{\xi}\ -\
P^{+}(d\chi^{-}P^{-}(\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}P_{\xi}))\
+\ \star$ (11.35) $\displaystyle-(P^{+}+P^{-})O(\varepsilon^{8}).$

where $\star$ consists of terms involving the 1-dimensional span of
$\Phi_{\tau}$, and the $O(\varepsilon^{8})$ accounts for terms involving $u$
and $K_{1}$. The terms in the top line of (11.35) involving are compact,
because they factor through either the compact inclusion
$\mathcal{H}^{1,-}\hookrightarrow L^{2}$ or through the finite-dimensional
spaces spanned by $\eta\in\mathfrak{W}$ and $\Phi_{\tau}$ respectively. Since
$\|\mathbb{P}_{0}^{-1}\|\leq C\varepsilon^{-\gamma}$, the $O(\varepsilon^{8})$
terms may be safely ignored. It follows that $\mathbb{P}_{1}$ is Fredholm of
index 0 once $\varepsilon$ is sufficiently small.

The same argument shows that $\mathbb{P}_{N}$ defined by

$\text{Id}-\mathbb{P}_{N}\circ\text{d}\overline{\mathbb{SW}}_{h_{1}}=\text{d}T^{N}$

is likewise Fredholm of Index 0. By construction (cf 2.21–2.24),
$\mathbb{P}_{N}$ applies $N$ stages of the (resp.
$\text{d}\overline{\mathbb{SW}}_{h_{1}}$) alternating iteration from
Propositions 11.1–11.3 starting from
$\mathfrak{e}_{1}=\text{Id}-\text{d}\overline{\mathbb{SW}}_{h_{1}}\mathbb{P}_{0}$.
We conclude that

$\text{d}\overline{\mathbb{SW}}_{h_{1}}\mathbb{P}_{N}=\text{Id}+O(\delta^{N-1})$

It follows as in the proof of Proposition 2.4 that
$\text{d}\overline{\mathbb{SW}}_{h_{1}}$ is an isomorphism. (11.34) follows
from the bound $\|\mathbb{P}_{0}^{-1}\|\leq C\varepsilon^{-\gamma}$ above, and
statement (I) in Propositions 11.1–11.3 with $h_{N}$ being the correction from
$\mathbb{P}_{N}-\mathbb{P}_{0}$, since
$\|\mathfrak{e}_{1}\|_{\mathfrak{L}}\leq C\varepsilon^{-\gamma}$. ∎

###### Proof of Proposition 10.1.

Let $\mathcal{C}=B_{r}(0)\subseteq\mathcal{H}$ of radius
$r=\varepsilon^{-1/20}$, and let $T$ be given by (10.1) with $\mathbb{A}$
defined by (11.30).

(A) is deduced assuming (C) as follows. Let $h\in\mathcal{C}$, then since
$\|\overline{\mathbb{SW}}_{\Lambda}(0)\|_{\mathfrak{L}}\leq
C\varepsilon^{-1/24-\gamma}$ by Theorem 7.5,

$\|T(h)\|_{\mathcal{H}}\leq\|T(h)-T(0)\|_{\mathcal{H}}+\|T(0)\|_{\mathcal{H}}\leq
C\sqrt{\delta}\|h\|_{\mathcal{H}}+C\varepsilon^{-1/24-2\gamma}\leq r,$

where the bound on $T(0)$ is a consequence of (I) in Propositions 11.1–11.3
with $\mathfrak{e}_{1}=\overline{\mathbb{SW}}_{\Lambda}(0)$. Thus
$T:\mathcal{C}\to\mathcal{C}$ preserves $\mathcal{C}$. Continuity of $T$ is
immediate from (C). Smooth dependence on $(\varepsilon,\tau)$ is immediate
from the smooth dependence of the Seiberg–Witten equations on $p_{\tau}$, and
the smooth dependence of $(\Phi_{1},A_{1})$ and $(\Phi_{\tau},A_{\tau})$ and
the linearized equations at these and their inverses used to construct $T$.

(B) By (11.32), applying $T$ constitutes a full cycle of the three-stage
iteration carried out by Propositions 11.1–11.3. The conclusion follows from
applying these inductively.

(C) Let $\mathfrak{q}$ be such that

$T(h)=\text{d}T(h)+\mathfrak{q}(h).$

where $\text{d}T$ is as in (11.33). The same argument as (B), now using the
(resp. $\text{d}\overline{\mathbb{SW}}_{h_{1}}$) statements in Propositions
11.1–11.3 shows that

$\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(\text{d}T(h))\|_{\mathfrak{L}}\leq
C\delta\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(h)\|_{\mathfrak{L}}.$

Therefore since $\text{d}T$ is linear, Lemmas 10.14 and 11.4 shows

$\displaystyle\|T(h_{1})-T(h_{2})\|_{\mathcal{H}}$ $\displaystyle\leq$
$\displaystyle
C\varepsilon^{-\gamma}\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(T(h_{1})-T(h_{2}))\|_{\mathfrak{L}}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{-\gamma}\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(\text{d}T(h_{1})-\text{d}T(h_{2}))\|_{\mathfrak{L}}\
+\
C\varepsilon^{-\gamma}\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(\mathfrak{q}(h_{1})-\mathfrak{q}(h_{2}))\|_{\mathfrak{L}}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{-\gamma}\delta\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(h_{1}-h_{2})\|_{\mathfrak{L}}\
+\
C\varepsilon^{-\gamma}\|\text{d}\overline{\mathbb{SW}}_{h_{1}}(\mathfrak{q}(h_{1})-\mathfrak{q}(h_{2}))\|_{\mathfrak{L}}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon^{-2\gamma}\delta\|h_{1}-h_{2}\|_{\mathcal{H}}\ +\
C\varepsilon^{-2\gamma}\|\mathfrak{q}(h_{1})-\mathfrak{q}(h_{2})\|_{\mathcal{H}}.$

Since $C\varepsilon^{-2\gamma}\delta<\sqrt{\delta}$ once $\varepsilon$ is
sufficiently, small, the following bound completes (C):

$\|\mathfrak{q}(h_{1})-\mathfrak{q}(h_{2})\|_{\mathcal{H}}\leq
C\varepsilon^{1/12-6\gamma}\|h_{1}-h_{2}\|_{\mathcal{H}}.$ (11.36)

To prove (11.36), we calculate $\mathfrak{q}=T-\text{d}T$ by expanding (11.32)
and (11.33), with
$\overline{\mathbb{SW}}_{\Lambda}=\text{d}\overline{\mathbb{SW}}_{h_{1}}+\mathbb{Q}+\Lambda$
where the latter is shorthand for
$\Lambda=\Lambda(\tau)\varepsilon^{-1}\chi^{-}\Phi_{\tau}$. Writing
$\mathbb{L}=\text{d}\overline{\mathbb{SW}}_{h_{1}},$ this shows

$\displaystyle\mathfrak{q}$ $\displaystyle=$ $\displaystyle P^{-}\mathbb{Q}\
+\ P_{\xi}\mathbb{Q}\ +\ P^{+}\mathbb{Q}\ +\ P^{-}\mathbb{Q}P_{\xi}\mathbb{L}\
+\ P^{-}\mathbb{Q}P_{\xi}\mathbb{Q}\ +\ P^{-}\mathbb{L}P_{\xi}\mathbb{Q}$
$\displaystyle+$ $\displaystyle P^{+}\mathbb{Q}P^{-}\mathbb{L}\ +\
P^{+}\mathbb{Q}P_{\xi}\mathbb{L}\ +\ P^{+}\mathbb{Q}P^{-}\mathbb{Q}\ +\
P^{+}\mathbb{Q}P_{\xi}\mathbb{Q}\ +\ P^{+}\mathbb{L}P^{-}\mathbb{Q}\ +\
P^{+}\mathbb{L}P_{\xi}\mathbb{Q}$ $\displaystyle+$ $\displaystyle
P^{+}\mathbb{L}P^{-}\mathbb{Q}P_{\xi}\mathbb{L}\ +\
P^{+}\mathbb{L}P^{-}\mathbb{L}P_{\xi}\mathbb{Q}+P^{+}\mathbb{Q}P^{-}\mathbb{L}P_{\xi}\mathbb{L}\
+\ P^{+}\mathbb{L}P^{-}\mathbb{Q}P_{\xi}\mathbb{Q}$ $\displaystyle+$
$\displaystyle P^{+}\mathbb{Q}P^{-}\mathbb{Q}P_{\xi}\mathbb{L}\ +\
P^{+}\mathbb{Q}P^{-}\mathbb{L}P_{\xi}\mathbb{Q}\ +\
P^{+}\mathbb{Q}P^{-}\mathbb{Q}P_{\xi}\mathbb{Q}$ $\displaystyle+$
$\displaystyle\mathfrak{q}_{\Lambda}$

where $\mathfrak{q}_{\Lambda}$ is the same collection of terms replacing each
instance of $\mathbb{Q}$ with $\Lambda$. Since $\Lambda$ is constant,
$q_{\Lambda}(h_{1})-q_{\Lambda}(h_{2})=0$. Because each remaining term has at
least one factor of $\mathbb{Q}$, (11.36) now follows from the boundedness of
$P_{\xi},P^{\pm}$ in Propositions 11.1–11.3, the boundedness of $\mathbb{L}$
from Lemma 10.14, along with Items (A) and (C) of Corollary 10.8. ∎

##  12\. Gluing

This final section concludes the proof Theorem 1.6 using the cyclic iteration
from Section 9. The iteration leads to glued solutions of a “Seiberg–Witten
eigenvector” equation for every pair of parameters $(\varepsilon,\tau)$. These
fail to solve the true Seiberg–Witten equations by a 1-dimensional obstruction
coming from a multiple of the eigenvector. The condition that this obstruction
vanish defines $\tau(\varepsilon)$ implicitly as a function of $\varepsilon$,
completing the proof of the theorem.

###  12.1. Glued Configurations

Proposition 10.1 and the Banach fixed-point theorem (with smooth dependence on
parameters) immediately imply the following.

######  Corollary 12.1.

For every pair
$(\varepsilon,\tau)\in(-\varepsilon_{0},\varepsilon_{0})\times(-\tau_{0},\tau_{0})$
with $\varepsilon_{0},\tau_{0}$ sufficiently small, there exist tuples
$(\mathcal{Z},\Phi,A,\mu)$ depending smoothly on $\varepsilon,\tau$ where

  1. (1)

$\mathcal{Z}(\varepsilon,\tau)=\mathcal{Z}_{\tau,\xi(\varepsilon,\tau)}$ is
the singular set arising from a linearized deformation
$\xi(\varepsilon,\tau)\in C^{\infty}(\mathcal{Z}_{\tau},N\mathcal{Z}_{\tau})$,

  2. (2)

$\mu(\varepsilon,\tau)\in\mathbb{R}$, and

  3. (3)

$(\Phi(\varepsilon,\tau),A(\varepsilon,\tau))\in
C^{\infty}(Y;S_{E}\oplus\Omega)$

that satisfy

$\text{SW}\left(\frac{\Phi(\varepsilon,\tau)}{\varepsilon},A(\varepsilon,\tau)\right)=(\Lambda(\tau)+\mu(\varepsilon,\tau)\chi^{-}_{\varepsilon}\cdot\frac{\Phi_{\tau,\xi(\varepsilon,\tau)}}{\varepsilon}$
(12.1)

where $\Phi_{\tau,\xi(\varepsilon,\tau)}$ is as in Definition 8.3, and
$\chi^{-}_{\varepsilon}=\chi^{-}_{\varepsilon,\tau,\xi(\varepsilon,\tau)}$
cut-off function (8.10).

###### Proof.

Part (C) of Proposition 10.1 and the Banach fixed-point theorem give a
smoothly varying family of fixed points
$h=(\xi,\varphi,a,\psi,b,\mu,u)\in\mathbb{H}^{1}_{e}\times\mathcal{X}$. By (B)
of Proposition 10.1, these satisfy, $\overline{\mathbb{SW}}_{\Lambda}(h)=0$.
Setting

$(\Phi(\varepsilon,\tau),A(\varepsilon,\tau))=(\Phi_{1}({\varepsilon,\tau,\xi}),A_{1}({\varepsilon,\tau,\xi}))\
+\ \chi_{\varepsilon}^{+}(\varphi,a)\ +\ \chi_{\varepsilon}^{-}(\psi+u,b),$

where $\xi=\xi(\varepsilon,\tau)$, the conclusion follows the Definition
(8.12) of $\overline{\mathbb{SW}}$ and Corollary 8.5 with right-hand side
$g=\varepsilon^{-1}\chi_{\varepsilon}^{-}\cdot\Lambda(\tau)\Phi_{\tau,\xi(\varepsilon,\tau)}$,
which satisfies $g\in C^{\infty}(Y)$ as in the proof of that corollary. ∎

###  12.2. The One-Dimensional Obstruction

The configurations (12.1) solve the Seiberg–Witten equations if and only if

$\Lambda(\tau)+\mu(\varepsilon,\tau)=0$ (12.2)

is satisfied. The next lemma shows that the assumption of transverse spectral
crossing (Definition 1.5) means the condition (12.2) defines $\tau$ implicitly
as a function of $\varepsilon$.

######  Lemma 12.2.

The solutions
$(\Phi(\varepsilon,\tau),A(\varepsilon,\tau),\mu(\varepsilon,\tau))$ from
Proposition 12.1 depend smoothly on
$(\varepsilon,\tau)\in(0,\varepsilon_{0})\times(-\tau_{0},\tau_{0})$.
Moreover,

$|\mu(\varepsilon,\tau)|\ +\ |\partial_{\tau}\mu(\varepsilon,\tau)|\leq
C\varepsilon^{2/3}.$ (12.3)

holds uniformly.

###### Proof.

The standard proof of the Banach fixed point theorem with smooth dependence on
parameters shows that a smooth family of fixed points $h_{\tau}$ satisfy

$\|\partial_{\tau}h(\tau)\|_{\mathcal{H}}\leq
C\|(\partial_{\tau}T_{\tau})h(\tau)\|_{\mathcal{H}}.$

Because of the weight on $\mu$ in (10.6), it therefore suffices to show that

$\|(\partial_{\tau}T_{\tau})h(\tau)\|_{\mathcal{H}}\leq C\varepsilon^{-1/4}.$
(12.4)

By Theorem 7.5 Item (I), Lemma 7.11, the expression Proposition 8.7 (using
Item (II) of Theorem 7.5), and the proofs of Propositions 9.1, 9.2, and 9.3
(which may be repeated equally well with
$(\partial_{\tau}\Phi_{1},\partial_{\tau}A_{1})$) show that

$\|\partial_{\tau}\mathcal{L}_{(\Phi_{1},A_{1})}^{+}\|\ +\
\|\partial_{\tau}\mathcal{L}_{(\Phi_{1},A_{1})}^{-}\|+\|\partial_{\tau}\text{d}\overline{\mathbb{SW}}_{h_{1}}(\xi)\|\leq
C\varepsilon^{-\gamma}$

are uniformly bounded as maps the $H^{1,\pm}_{\varepsilon}\to L^{2}$ and
$\mathfrak{W}\to L^{2}$. A similar argument applies to
$\partial_{\tau}\mathbb{Q}$. Together with, Item (I) of Theorem 7.5, these
show that $\partial_{\tau}\overline{\mathbb{SW}}_{\Lambda}$ is bounded by
$C\varepsilon^{-\gamma}$ at $h\in\mathcal{C}$. Differentiating
$\text{Id}=P_{\tau}^{-1}P_{\tau}$ for all three parametrices and using the
bounds from Propositions 11.1–11.3 yields bounds on
$\partial_{\tau}P_{\xi},\partial_{\tau}P^{\pm}$ by at most
$C\varepsilon^{-1/12-\gamma}$. Using the product rule on (11.32) and combining
these yields (12.4). ∎

The proof of the following lemma is essentially identical the treatment of
[60, Eq. (10.6)].

######  Lemma 12.3.

If the family of parameters $p_{\tau}=(g_{\tau},B_{\tau})$ has a transverse
spectral crossing, then (12.2) implicitly defines a function
$\tau(\varepsilon)$ so that

$\Lambda(\tau(\varepsilon))+\mu(\varepsilon,\tau(\varepsilon))=0$

for $\varepsilon\in(0,\varepsilon_{0})$, and either $\tau(\varepsilon)>0$ or
$\tau(\varepsilon)<0$.

###### Proof.

The assumption that $\tau=0$ is a transverse spectral crossing means that
$\Lambda(0)=0$ and $\dot{\Lambda}(0)\neq 0$. By the inverse function theorem,
there is an inverse $\Lambda^{-1}$ defined on an open neighborhood of
$\tau=0$. Set $\Gamma(\varepsilon)=\Lambda\circ\tau(\varepsilon)$, so that
(12.2) becomes the condition that

$\Gamma(\varepsilon)+\mu(\varepsilon,\Lambda^{-1}\circ\Gamma(\varepsilon))=0.$

This is an equation for a real number $\Gamma$ depending on a parameter
$\varepsilon$. (12.3) implies that this equation can be solved for
$\Gamma\in\mathbb{R}$ using the Inverse Function Theorem with smooth
dependence on the parameter $\varepsilon$; thus
$\tau(\varepsilon)=\Lambda^{-1}\circ\Gamma(\varepsilon)$ solves (12.2).

To see the sign of $\tau(\varepsilon)$, we expand (12.2) in series. From the
proof of Proposition 10.1 implies that $\mu$ is a sum
$\mu(\varepsilon,\tau)=\mu_{1}(\varepsilon,\tau)+\delta\mu_{2}(\varepsilon,\tau)+\delta^{2}\mu_{3}(\varepsilon,\tau)+\ldots$.
Meanwhile $\Lambda(\tau)$ is smooth and may be expanded in Taylor series.

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\Lambda(\tau)+\mu(\varepsilon,\tau)$ $\displaystyle=$
$\displaystyle\dot{\Lambda}(0)\tau+\varepsilon^{2/3}\overline{\mu}_{1}(\varepsilon,0)+O(\tau^{2})+O(\varepsilon^{2/3}\delta)+O(\varepsilon^{2/3}\tau),$

where $\mu_{i}=\varepsilon^{2/3}\overline{\mu}_{i}$. It follows that for
$\varepsilon$ sufficiently small, $\tau$ has the opposite sign of
$\frac{\overline{\mu}_{1}(\varepsilon,0)}{\dot{\Lambda}(0)}$. ∎

###### Proof of Theorem 1.6.

Part (1) follows directly from Lemma 3.8. Along the family of parameters
$\tau(\varepsilon)$ satisfying (12.2) constructed in Lemma 12.3, the glued
configurations of Proposition 12.1 satisfy the (extended) Seiberg–Witten
equations. Integrating by parts shows the $0$-form component $a_{0}$ vanishes,
and setting
$(\Psi_{\varepsilon},A_{\varepsilon})=(\varepsilon^{-1}{\Phi(\varepsilon,\tau(\varepsilon))}\
,\ A(\varepsilon,\tau(\varepsilon)))$ yields the solutions in Part (2).

The glued configurations have
$\|\Psi_{\varepsilon}\|_{L^{2}}=\varepsilon^{-1}+O(\varepsilon^{-1/24-\gamma})$,
and this norm depends smoothly on $\varepsilon$. By re-parameterizing, it may
be assumed that $\|\varepsilon\Psi_{\varepsilon}\|_{L^{2}}=1$ as in (1.13). As
$\varepsilon\to 0$, Theorem 3.2 shows that after renormalization,
$(\Psi_{\varepsilon},A_{\varepsilon})$ converges in $C^{\infty}_{loc}$ to a
$\mathbb{Z}_{2}$-harmonic spinor for the parameter $p_{0}=(g_{0},B_{0})$ and
that $\|\varepsilon\Psi_{\varepsilon}\|\to|\Phi_{0}|$ in $C^{0,\alpha}$. Since
$(\mathcal{Z}_{0},A_{0},\Phi_{0})$ is regular, it is the unique
$\mathbb{Z}_{2}$-harmonic spinor for this parameter and must therefore be the
limit. This establishes Part (3). ∎

######  Remark 12.4.

Recalling the construction of $\mu$ from Step 5 of the proof of Proposition
11.2 with $N=1$, the leading order obstruction in the proof of Lemma 12.3
appears to be

$\varepsilon^{2/3}\overline{\mu}_{1}(\varepsilon,0)=\varepsilon\langle\text{d}_{h_{0}}\not{\mathbb{D}}(\eta_{1},0)\
+\ d\chi^{+}.\varphi_{1},\Phi_{0}\rangle,$ (12.5)

where $\varphi_{1}$ is as in Item (II) of Theorem 7.5 for $\tau=0$. In fact,
since $\Phi_{0}=O(r^{1/2})$, simple integration shows that the second term is
smaller by a factor of $\varepsilon^{1/2}$ than the bound on
$\text{d}_{h_{0}}\not{\mathbb{D}}(\eta_{1},0)$ obtained in the proof of
Proposition 11.1. The latter bound, however, might not be optimal.

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