$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho^{3};\mathsf{Q})_{\bar{a}}+\mathscr{O}_{M-3}(\rho,\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{2};\mathsf{D}\mathsf{B})_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{W}^{2})_{\bar{a}}\text{.}$

Now, for the remaining term $\mathscr{A}_{2}$, we begin by expanding

(7.42) $\displaystyle\mathscr{A}_{2}$
$\displaystyle=-\mathsf{g}^{de}\sum_{j=1}^{d}\check{\mathsf{w}}_{\bar{a}_{j}[d]}\,\bar{\Box}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}-\mathsf{g}^{de}\sum_{j=1}^{d}(\bar{\Box}+\sigma_{\mathsf{w}})\check{\mathsf{w}}_{\bar{a}_{j}[d]}\,(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}$
$\displaystyle\qquad-\rho^{2}\mathsf{g}^{de}\sum_{j=1}^{d}\bar{\mathsf{D}}_{\rho}\check{\mathsf{w}}_{\bar{a}_{j}[d]}\,\bar{\mathsf{D}}_{\rho}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}-\rho^{2}\mathsf{g}^{bc}\mathsf{g}^{de}\sum_{j=1}^{d}\mathsf{D}_{b}\check{\mathsf{w}}_{\bar{a}_{j}[d]}\,\mathsf{D}_{c}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}$
$\displaystyle=\mathscr{A}_{2,1}+\mathscr{A}_{2,2}+\mathscr{A}_{2,3}+\mathscr{A}_{2,4}\text{.}$

We now expand each of the terms on the right-hand side of (7.42). First, by
Proposition 3.8,

(7.43)
$\mathscr{A}_{2,4}=\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}\mathsf{Q})_{\bar{a}}\text{.}$

Next, for $\mathscr{A}_{2,3}$, we recall (2.20), along with (3.11) and the
first part of (3.16):

(7.44) $\displaystyle\mathscr{A}_{2,3}$
$\displaystyle=\mathscr{O}_{M-3}(\rho^{2};\mathscr{L}_{\rho}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathscr{L}_{\rho}\mathsf{Q})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{3};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{3};\mathsf{Q})_{\bar{a}}$
$\displaystyle=\mathscr{O}_{M-3}(\rho^{3};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{3};\mathsf{Q})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}\text{.}$

For $\mathscr{A}_{2,2}$, we begin by applying the wave equation (3.6)
satisfied by $\mathsf{w}$. By inspecting the right-hand of the appropriate
equation and recalling Proposition 3.8, we obtain

$\displaystyle(\bar{\Box}+\sigma_{\mathsf{w}})\mathsf{w}$
$\displaystyle=\sum_{k=0}^{2}\mathscr{O}_{M-2}(\rho^{3};\mathsf{D}\mathsf{w}^{k})+\sum_{k=0}^{2}\mathscr{O}_{M-3}(\rho^{2};\mathsf{w}^{k})$
$\displaystyle=\mathscr{O}_{M-3}(\rho^{2})\text{.}$

From the above, we then obtain the following asymptotics:

(7.45)
$\mathscr{A}_{2,2}=\mathscr{O}_{M-3}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{Q})_{\bar{a}}\text{.}$

For $\mathscr{A}_{2,1}$, we decompose $\bar{\Box}$ using (2.35) and apply
Proposition 3.8:

(7.46) $\displaystyle\mathscr{A}_{2,1}$
$\displaystyle=-\rho^{2}\mathsf{g}^{bc}\mathsf{g}^{de}\sum_{j=1}^{d}\check{\mathsf{w}}_{\bar{a}_{j}[d]}\,\mathsf{D}_{bc}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}+\mathscr{O}_{M-2}(\rho^{2};\mathscr{L}_{\rho}^{2}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q}))_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho;\mathscr{L}_{\rho}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q}))_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{\bar{a}}\text{.}$

For the asympototic terms on the right-hand side, we recall (3.11) and (3.16):

$\displaystyle\mathscr{L}_{\rho}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})$
$\displaystyle=\mathsf{m}-\check{\mathsf{m}}+\mathscr{O}_{M-2}(\rho;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-2}(\rho;\mathsf{Q})\text{,}$
$\displaystyle\mathscr{L}_{\rho}^{2}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})$
$\displaystyle=-2\mathsf{W}^{2}+\mathscr{O}_{M-3}(1;\mathsf{g}-\check{\mathsf{g}})+\mathscr{O}_{M-3}(1;\mathsf{Q})+\mathscr{O}_{M-2}(\rho^{-1};\mathsf{m}-\check{\mathsf{m}})\text{.}$

Thus, combining (7.46) and the above yields

$\displaystyle\mathscr{A}_{2,1}$
$\displaystyle=-\rho^{2}\mathsf{g}^{bc}\mathsf{g}^{de}\sum_{j=1}^{d}\check{\mathsf{w}}_{\bar{a}_{j}[d]}\,\mathsf{D}_{bc}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{Q})_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho;\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{W}^{2})_{\bar{a}}\text{,}$

Combining (7.42)–(7.45) and the above, we have that

(7.47) $\displaystyle\mathscr{A}_{2}$
$\displaystyle=-\rho^{2}\mathsf{g}^{bc}\mathsf{g}^{de}\sum_{j=1}^{d}\check{\mathsf{w}}_{\bar{a}_{j}[d]}\,\mathsf{D}_{bc}(\mathsf{g}-\check{\mathsf{g}}+\mathsf{Q})_{a_{j}e}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{g}-\check{\mathsf{g}})_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho^{2};\mathsf{Q})_{\bar{a}}+\mathscr{O}_{M-2}(\rho;\mathsf{m}-\check{\mathsf{m}})_{\bar{a}}+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{\bar{a}}$
$\displaystyle\qquad+\mathscr{O}_{M-3}(\rho^{2};\mathsf{D}\mathsf{Q})_{\bar{a}}+\mathscr{O}_{M-2}(\rho^{2};\mathsf{W}^{2})_{\bar{a}}\text{.}$

Finally, both (3.18) and (3.19) follow from (7.34), (7.38), (7.41), and
(7.47).

###  7.10. Proof of Proposition 3.15

The first part of (3.20) is simply the first equation in (3.16), while the
second identity in (3.20) follows from the final part of (3.3), along with
Proposition 3.8 and (3.14). (Note this is just the third part of (3.16),
except we do not renormalize $\mathsf{w}^{2}-\check{\mathsf{w}}^{2}$.)

For the remaining identities, we begin by rewriting the first, third, and
fourth parts of (3.5) as

(7.48) $\displaystyle\mathscr{L}_{\rho}(\rho^{2-n}\mathsf{w}^{2})_{ab}$
$\displaystyle=\rho^{2-n}\mathsf{g}^{cd}\mathsf{D}_{c}\mathsf{w}^{1}_{bad}+\rho^{2-n}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{w}^{0})_{ab}+\rho^{2-n}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{2})_{ab}\text{,}$
$\displaystyle\mathscr{L}_{\rho}(\rho^{-1}\mathsf{w}^{1})_{abc}$
$\displaystyle=\rho^{-1}\mathsf{D}_{b}\mathsf{w}^{2}_{ac}-\rho^{-1}\mathsf{D}_{c}\mathsf{w}^{2}_{ab}+\rho^{-1}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{1})_{abc}\text{,}$
$\displaystyle\mathscr{L}_{\rho}\mathsf{w}^{0}_{abcd}$
$\displaystyle=\mathsf{D}_{a}\mathsf{w}^{1}_{bcd}-\mathsf{D}_{b}\mathsf{w}^{1}_{acd}+\rho^{-1}\mathsf{g}_{ad}\mathsf{w}^{2}_{bc}+\rho^{-1}\mathsf{g}_{bc}\mathsf{w}^{2}_{ad}-\rho^{-1}\mathsf{g}_{ac}\mathsf{w}^{2}_{bd}-\rho^{-1}\mathsf{g}_{bd}\mathsf{w}^{2}_{ac}$
$\displaystyle\qquad+\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{0})_{abcd}+\mathscr{S}(\mathsf{m},\mathsf{w}^{2})_{abcd}\text{.}$

In particular, here we converted $\bar{\mathsf{D}}_{\rho}$ to
$\mathscr{L}_{\rho}$ using (2.20). We then take the difference of each of the
equations in (7.48), applied once with respect to $\mathsf{g}$ and once with
respect to $\check{\mathsf{g}}$, and we treat the resulting difference terms
as in the proofs of Propositions 3.13 and 3.14. In particular, various
geometric quantities can be controlled using Proposition 3.8 and (3.14);
moreover, the difference $\mathsf{D}-\check{\mathsf{D}}$ can be controlled by
$\mathsf{D}(\mathsf{g}-\check{\mathsf{g}})$ using the last part of (3.15).

For example, from the first part of (7.48), we have

$\displaystyle\mathscr{L}_{\rho}[\rho^{2-n}(\mathsf{w}^{2}-\check{\mathsf{w}}^{2})]_{ab}$
$\displaystyle=\rho^{2-n}(\mathsf{g}^{cd}-\check{\mathsf{g}}^{cd})\mathsf{D}_{c}\mathsf{w}^{1}_{bad}+\rho^{2-n}\check{\mathsf{g}}^{cd}\mathsf{D}_{c}(\mathsf{w}^{1}-\check{\mathsf{w}}^{1})_{bad}+\rho^{2-n}\check{\mathsf{g}}^{cd}(\mathsf{D}-\check{\mathsf{D}})_{c}\check{\mathsf{w}}^{1}_{bad}$
$\displaystyle\qquad+[\rho^{2-n}\,\mathscr{S}(\mathsf{g}^{-2},\mathsf{m},\mathsf{w}^{0})-\rho^{2-n}\,\mathscr{S}(\check{\mathsf{g}}^{-2},\check{\mathsf{m}},\check{\mathsf{w}}^{0})]_{ab}$
$\displaystyle\qquad+[\rho^{2-n}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{2})-\rho^{2-n}\,\mathscr{S}(\mathsf{g}^{-1},\mathsf{m},\mathsf{w}^{2})]_{ab}\text{,}$
$\displaystyle=\mathscr{O}_{M}(\rho^{2-n};\mathsf{D}(\mathsf{w}^{1}-\check{\mathsf{w}}^{1}))_{ab}+\mathscr{O}_{M-4}(\rho^{3-n};\mathsf{g}-\check{\mathsf{g}})_{ab}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{2-n};\mathsf{m}-\check{\mathsf{m}})_{ab}+\mathscr{O}_{M-3}(\rho^{3-n};\mathsf{D}(\mathsf{g}-\check{\mathsf{g}}))_{ab}$
$\displaystyle\qquad+\mathscr{O}_{M-2}(\rho^{3-n};\mathsf{w}^{0}-\check{\mathsf{w}}^{0})+\mathscr{O}_{M-2}(\rho^{3-n};\mathsf{w}^{2}-\check{\mathsf{w}}^{2})\text{,}$

which yields the third equation in (3.20). Note that in the last step, we used
the improved vanishing rate for $\mathsf{w}^{1}$ in (3.9) when dealing with
terms involving $\mathsf{g}-\check{\mathsf{g}}$ and
$\mathsf{D}(\mathsf{g}-\check{\mathsf{g}})$. The remaining parts of (3.20) are
obtained analogously, once again using the improved rates for $\mathsf{w}^{1}$
and $\mathsf{w}^{2}$ in (3.9).

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