Given a generic solution $\g_0$ of the Einstein-null dusts system without restriction on the number of dusts, we construct families of solutions $(\g_\la)_{\la\in(0,1]}$ of the Einstein vacuum equations such that $\g_\la-\g_0$ and $\dr(\g_\la-\g_0)$ converges respectively strongly and weakly to 0 when $\la\to0$. Our construction, based on a multiphase geometric optics ansatz, thus extends the validity of the reverse Burnett conjecture without symmetry to a large class of massless kinetic spacetimes. In order to deal with the finite but arbitrary number of direction of oscillations we work in a generalised wave gauge and control precisely the self-interaction of each wave but also the interaction of waves propagating in different null directions, relying crucially on the non-linear structure of the Einstein vacuum equations. We also provide the construction of oscillating initial data solving the vacuum constraint equations and which are consistent with the spacetime ansatz.

§ INTRODUCTION

In this article we construct oscillatory solutions of the Einstein vacuum equations

\begin{align}
\end{align}

where $\g$ is a Lorentzian metric on a 4-dimensional manifold $\mathcal{M}$ and $R_{\mu\nu}(\g)$ is the Ricci tensor of $\g$. These solutions, denoted $\g_\la$, depend on a small wavelength $\la>0$ and oscillate at frequency $\la^{-1}$. We are particularly interested in their behaviour in the high-frequency limit $\la\to 0$ and our construction, based on a geometric optics expansion of the solutions, seems to be the first to allow a finite but arbitrary number of directions of oscillation with no symmetry assumption. It builds on previous works by the author [28, 29], where only one direction of oscillation was allowed. It is motivated by Burnett's conjecture in general relativity and by the study of high-frequency gravitational waves initiated by Choquet-Bruhat. We discuss further these two aspects in Sections <ref> and <ref> below where we also give two rough versions of our main result (see Theorem <ref> for the precise version) from these two perspectives.

§.§ The Burnett conjecture

The first motivation of our work is the description of the closure of the set of solutions to (<ref>) under appropriate weak limits. This question was first raised by Burnett in [1]. In this article, he considers sequences $(\g_\la)_{\la\in(0,1]}$ of solutions to (<ref>) on a given manifold $\mathcal{M}$ such that the following convergence holds when $\la\to 0$:

\begin{equation}\label{CV}
\begin{aligned}
\g_\la & \rightarrow \g_0 \quad\text{uniformly on compact sets},
\\ \dr\g_\la & \rightharpoonup \dr \g_0 \quad\text{weakly in $L^2_{loc}$},
\end{aligned}
\end{equation}

where $\dr$ denotes a partial derivative in any coordinates system on $\mathcal{M}$ and where $\g_0$ is a metric on $\mathcal{M}$. The question of interest here is the following: what matter model can the limit metric $\g_0$ describe? More precisely, what are the possible effective stress-energy tensors $R_{\mu\nu}(\g_0) - \half R(\g_0)(\g_0)_{\mu\nu}$? If the convergence of the derivatives $\dr\g_\la$ towards $\dr\g_0$ is strong then $\g_0$ also solves (<ref>), but if this convergence is only weak then products of derivatives of the metric in the Ricci tensor might produce backreaction, i.e a non-trivial effective stress-energy tensor. Burnett proposed a two-sided conjecture to answer the above questions:

* The direct Burnett conjecture. The effective stress-energy tensor is the one of a massless Vlasov field on $\mathcal{M}$, i.e there exists a density $f_0:\mathcal{M}\times T\mathcal{M}\longrightarrow \R$ such that $(\g_0,f_0)$ solves the massless Einstein-Vlasov system on $\mathcal{M}$

\begin{equation}\label{EV}
\left\{
\begin{aligned}
R_{\mu\nu}(\g_0) - \half R(\g_0)(\g_0)_{\mu\nu} & = \int_{\g_0(p,p)=0}f_0^2 p_\mu p_\nu,
\\ p^\alpha \dr_\alpha f_0 - p^\alpha p^\beta \Gamma(g_0)^\rho_{\alpha\beta} \dr_{p^\rho}f_0 & =0.
\end{aligned}
\right.
\end{equation}

* The reverse Burnett conjecture. For every solution $(\mathcal{M},\g_0,f_0)$ of (<ref>), there exists a sequence $(\g_\la)_{\la\in(0,1]}$ of solutions of (<ref>) on $\mathcal{M}$ with the convergence properties (<ref>).

Put together, these two parts of Burnett's conjecture affirm that the closure of the set of solutions to (<ref>) for the topology of the convergence (<ref>) identifies with the set of solutions to (<ref>).

The direct conjecture has been first considered in [10] and then proved in $\mathbb{U}(1)$ symmetry, i.e when $\mathcal{M}$ admits a spacelike Killing field, in [12] and [9], where the effective stress-energy tensor is identified by means of microlocal defect measures. The reverse conjecture, which is this article's topic, is opposite in spirit to the direct one since it requires solving (<ref>) with a given target $\g_0$. However, not only does it give a more complete and satisfying understanding of backreaction as a generic phenomenon, it is also highly connected to the direct conjecture. Indeed, the lack of strong convergence in (<ref>) does not allow boundedness of the sequence $(\g_\la)_{\la\in(0,1]}$ in $H^2_{loc}$ so that the celebrated bounded $L^2$ curvature conjecture proved in [18] does not apply and the mere existence of sequences $(\g_\la)_{\la\in(0,1]}$ of solutions to (<ref>) displaying the pathological behaviour (<ref>) is not guaranteed by the general theory. Solving the reverse conjecture precisely amounts to producing examples of such sequences and thus prevents the theorems on the direct conjecture to be empty.

The reverse conjecture has been proved first in $\mathbb{U}(1)$ symmetry in [11] where the authors consider targets $\g_0$ solving a discretized version of (<ref>), namely the Einstein-null dusts system

\begin{equation}\label{null dusts}
\left\{
\begin{aligned}
R_{\mu\nu}(\g_0) & = \sum_\A F_\A^2 \dr_\mu u_\A \dr_\nu u_\A,
\\ \g_0^{-1}(\d u_\A , \d u_\A) &  = 0,
\\ -2 L_\A F_\A + (\Box_{\g_0}u_\A)F_\A & = 0,
\end{aligned}
\right.
\end{equation}

where $\A$ runs through a finite set $\mathcal{A}$ of arbitrary cardinal denoted $|\mathcal{A}|$ in the sequel. In the setting of $\mathbb{U}(1)$ symmetry, the same authors prove in [13] that one can take the limit $|\mathcal{A}|\to+\infty$ and reach generic solutions to (<ref>) as targets for backreaction. In both these works, the $\mathbb{U}(1)$ symmetry plays an important role since it allows for the construction of an elliptic gauge, in which (<ref>) reduces to a wave-map system coupled to semi-linear elliptic equations for the metric.

Outside of symmetry, the two sides of Burnett's conjecture have been tackled in double null gauge in [23], where the authors also shed new lights on the well-posedness theory for null dust shell and the formation of trapped surfaces. Their proof relies on a low-regularity result for (<ref>) in double null gauge proved in [22], where derivatives of the metric in the two null directions are allowed to be only in $L^2_{loc}$. Due to the structure of (<ref>) in double null gauge, this has to be compensated by higher regularity in the angular directions. Therefore, [23] proves both sides of Burnett's conjecture with limits $\g_0$ solving (<ref>) with $|\mathcal{A}|=2$ and thus leaves open the question of whether there exists a setting for (<ref>) capable of handling more than two null dusts. The present article's main goal is to show that wave gauges provide such a setting. From this perspective, we obtain the following:

[Rough version of the result from the point of view of Burnett's conjecture]
The reverse Burnett conjecture holds true when $\g_0$ is a generic solution of (<ref>) in wave gauge with arbitrary $|\mathcal{A}|<+\infty$.

The intuitive reason behind the fact that wave gauges are good candidates to handle the superposition of more than two null dusts is that in these gauges the Einstein vacuum equations (<ref>) rewrite as a system of quasi-linear wave equations which doesn't single out any null direction, as opposed to double null gauges. We mention two interesting open problems:

* As we will see, in this article we construct sequences $(\g_\la)_{\la\in(0,1]}$ where the desired lack of strong convergence is purely due to oscillations. One of the strengths of [23] is to include concentration effects as the cause of weak convergence. It is an interesting open problem to prove Burnett's reverse conjecture outside of symmetry with $|\mathcal{A}|\geq 3$ and with concentration-based backreaction.

* More importantly, this article completely leaves open the question of sending the number of null dusts to infinity and thus proving the reverse Burnett conjecture with the highest level of generality. In particular, our local existence result in generalised wave gauge is not uniform in $|\mathcal{A}|$ and obtaining such uniformity seems to require ideas beyond the present work.

§.§ Multiphase geometric optics

The second motivation of our work is the multiphase geometric optics approximation for the Einstein vacuum equations (<ref>). Geometric optics seeks a description of how waves propagate as solutions to a given system of PDEs. In general relativity, one can wonder how gravitational waves propagate as solutions to (<ref>). As explained in depth in Chapter 35 of [25], one would also like to go beyond the linearized gravity setting which by definition cannot describe the energy of gravitational waves, a quadratic quantity by nature, and thus misses the global impact of gravitational waves on a background spacetime. The framework of geometric optics, as presented in [26] or [24], provides such a setting, as we will now describe.

In [2], Choquet-Bruhat is the first to construct WKB approximate solutions of (<ref>). More precisely, she constructs a family of metrics of the form

\begin{align}\label{rough CB}
\g_\la(x) & = \g_0(x) + \la \g^{(1)}(x,\th)_{|_{\th=\frac{\phi(x)}{\la}}} + \la^2 \g^{(2)}(x,\th)_{|_{\th=\frac{\phi(x)}{\la}}}
\end{align}

where the $\g^{(i)}$'s are periodic in the $\th$ variable and such that $R_{\mu\nu}(\g_\la) = \GO{\la}$. The typical geometric optics phenomena are recovered: $\phi$ must solve the background eikonal equation and $\g^{(1)}$ is transported along the rays. Less common is the fact that there should exist a scalar function $\tau>0$ such that $R_{\mu\nu}(\g_0) = \tau \dr_\mu \phi \dr_\nu \phi$, thus making [2] one of the earliest examples of backreaction. As pointed out in [24], it remained to prove or disprove the stability of the geometric optics approximation, i.e to answer the following: does the approximate solution (<ref>) stay close to an exact one on a uniform time scale in $\la$? A positive answer has been given in the articles [28, 29]. As the approximate construction of [2], these articles deal with the singlephase geometric optics approximation, when only one direction of oscillation is allowed. We extend these results to the following:

[Rough version of the result from the point of view of geometric optics]
Under a strong coherence assumption, the multiphase geometric optics approximation is stable for (<ref>).

Note that this result cannot follow from general results on multiphase geometric optics such as [15] mainly because the hyperbolicity of (<ref>) comes at the cost of a gauge choice, due to the invariance by diffeomorphism of the Ricci tensor. Here, the gauge is part of the geometric optics construction and in particular the metrics $\g_\la$ will be such that $\Box_{\g_\la}x^\a$, i.e the term defining the wave gauge, is oscillating and of order $\la$. Note that geometric optics for semi-linear gauge invariant equations have been studied in [14], here the situation is different since (<ref>) are quasi-linear.

§.§ A discussion of transparency

The two motivations described above are obviously very much connected. Both are concerned with describing the non-linear interactions through (<ref>) of small scale inhomogeneities in the metric, and geometric optics is used as a strategy to attack Burnett's reverse conjecture. The non-linear structure of the Ricci tensor plays a central role in our proof, and we would like to bring to the reader's attention a truly astounding aspect of (<ref>), illustrated by Theorems <ref> and <ref>.

As it was already the case in the approximate construction of [2], the transport equation along the rays of the optical function for the first profile in the WKB ansatz constructed in the present article is linear. At first glance, this is very surprising since the Einstein vacuum equations have a complicated non-linear structure. In the geometric optics literature, the fact that the transport equations for the first profile in a WKB ansatz turns out to be linear despite the waves interacting non-linearly is called transparency, see [16]. Note that transparency can also be seen from Burnett's conjecture's perspective, since the transport equation for the density in (<ref>) is linear.  However, Burnett's conjecture describes a truly non-linear effect since the effective stress-energy tensor, or alternatively the energy of the gravitational wave $\tau$ in Choquet-Bruhat's approximate construction, precisely originates in the quadratic self-interaction of derivatives of the metric. It is truly astounding that this interaction produces a global quadratic effect while being linearly propagated.

As pointed out in [19], transparency is directly linked to the null condition introduced in [6, 17] for the study of global existence for small data for non-linear wave equations. Elaborating on [2] in her article [3], Choquet-Bruhat uses geometric optics to show that (<ref>) cannot truly satisfy the null condition, otherwise we would have $\tau=0$ above or alternatively Burnett's conjecture would reduce to an uninteresting statement about vacuum spacetimes necessarily approaching vacuum spacetimes. Later, Lindblad and Rodnianski introduced in [20] a weakened version of the null condition, rightly called the weak null condition. They show that (<ref>) in wave gauge satisfy this condition and uses this to prove the stability of Minkowski in wave gauge in [21], a result already proved in the seminal [7]. However, as noticed in Section 3.1.6 in [28], there exist systems with the weak null condition but without the transparency property. The Einstein vacuum equations (<ref>) thus seem to be very much unique in the sense that both a linear and a non-linear behaviour can be simultaneously exhibited.

§.§ Notations and tools

In this section we introduce various notations and tools which will be used throughout the article.

Geometric notations.

* Our construction takes place on the fixed manifold $\mathcal{M}\vcentcolon=[0,1]\times\R^3$, endowed with the standard coordinates $(t,x^1,x^2,x^3)$. We define $\Si_t\vcentcolon=\{t\}\times \R^3$ for $t\in[0,1]$. On $\mathcal{M}$ we denote by $\mathbf{m}$ the Minkowski metric while on each $\Si_t$ the Euclidean metric is denoted by $e$. Greek indices will refer to the coordinates $(t,x^1,x^2,x^3)$ and will thus run from 0 to 3. Latin indices will refer to the coordinates $(x^1,x^2,x^3)$ and will thus run from 1 to 3. Repeated indices (with one up and one down) will be always summed over. If $r\geq 0$ then $B_r$ denotes the closed ball in $\R^3$ centered at 0 and of radius $r$ in the Euclidean metric, i.e $B_r=\{|x|\leq r \}$.

* If $T$ and $S$ are symmetric 2-tensors and if $\g$ is a Lorentzian metric on $\mathcal{M}$ then we define

\begin{align*}
|T\cdot S|_{\g} & = \g^{\a\b}\g^{\mu\nu}T_{\a\mu}S_{\b\nu},
\\ |T|_{\g} & = \sqrt{ \g^{\a\b}\g^{\mu\nu}T_{\a\mu}T_{\b\nu}},
\\ \tr_\g T & = \g^{\a\b}T_{\a\b}.
\end{align*}

These notations have their natural 1-tensor equivalent: $|X\cdot Y|_\g = \g^{\a\b}X_\a Y_\b$. We don't distinguish between the covariant or contravariant forms of tensors, and if $T$ is a general 2-tensor then its symmetric and anti-symmetric part are denoted
\begin{align*}
T_{(\a\b)} & = T_{\a\b} + T_{\b\a},
\\ T_{[\a \b]} & = T_{\a\b} - T_{\b\a}.
\end{align*}
All these notations extend naturally to tensors defined only on $_0$ endowed with a Riemannian metric.
\item For $f$ a scalar function on $ℳ$, $f$ will denote the Euclidean gradient of $f$ or, alternatively and with a slight abuse of notations, any spatial derivatives $_i f$ while $f$ will denote any spacetime derivatives $_f$. We associate to a Lorentzian metric $$ on $ℳ$ its wave operator $_$ which acts on scalar function as $_f  = ^μν _μ_νf - ()^ρ_μν_ρf$, where $()^ρ_μν$ denotes the usual Christoffel symbols of the metric $$. If $h$ is a Riemannian metric the same definition in spacelike coordinates defined its Laplace-Beltrami operator $_h$. If $v$ is a scalar function on $_0$ then we define its gradient with respect to the metric $h$ by the vector field $_h v = h^ij_i v _j$.
\end{itemize}

\paragraph{Analytic tools.}
\begin{itemize}
\item On each slice $\Si_t$ we define the usual Lebesgues and Sobolev spaces $L^p$ and $W^{k,p}$ with respect to the Euclidean element of integration. We also define the weighted Sobolev spaces $W^{k,p}_\de$ to be the completion of smooth and compactly supported functions on $\Si_t$ for the norm
\begin{align}
\l f \r_{W^{k,p}_\de} = \sum_{0\leq |m| \leq k} \l \pth{1+|x|^2}^{\frac{\de+|m|}{2}} \nab^m f  \r_{L^p}\label{norm WSS}
\end{align}
with the standard special case $H^k_\de\vcentcolon = W^{k,2}_\de$ and $L^2_\de=H^0_\de$. By replacing $L^p$ in \eqref{norm WSS} by $L^\infty$ we define the weighted Hölder spaces $C^k_\de$. These norms are extended to tensors by summing over their components in the coordinates $(t,x^1,x^2,x^3)$.
\item From [4], these spaces satisfy the continuous embeddings
\begin{align*}
W^{s_1,p}_{\de_1}\times W^{s_2,p}_{\de_2} \subset W^{s,p}_\de
\end{align*}
where $s\leq \min(s_1,s_2)$, $s<s_1+s_2-\frac{3}{p}$ and $\de<\de_1+\de_2+\frac{3}{p}$ and
\begin{align*}
W^{s,p}_\de \subset C^m_{\de+\frac{3}{p}}
\end{align*}
where $m<s-\frac{3}{p}$. From [4] we also get that $\De:H^2_\de\longrightarrow L^2_{\de+2}$ is an isomorphism if $-\frac{3}{2}<\de<-\half$ and where $\De$ is the flat Laplacian, sometimes denoted $\De_e$.
\item Estimates with no mention of the time will always refer to estimates holding uniformly in time over $[0,1]$, i.e $\l f\r_X \leq C$ will denote
\begin{align*}
\sup_{t\in[0,1]}\l f\r_{X(\Si_t)} \leq C,
\end{align*}
where $X$ is any of the function spaces defined here.
\end{itemize}

\paragraph{High-frequency and schematic notations.} Since our construction is based on a geometric optics expansion of the metric, we will encounter objects defined or expressed via an expansion in terms of $$ with oscillating coefficients, where by object we mean tensors of any types including scalar functions, vector fields, 1-forms and higher order tensors. We introduce some notations to describe and manipulate these expansions.
\begin{itemize}
\item If an object $S$ admits an expansion in terms of powers (non-negative or negative) of $\la$, we denote by $S^{(i)}$ the coefficient of $\la^i$ in this expansion. If $j\in\Z$, we define
\begin{align*}
S^{(\geq j)} = \sum_{k\geq j}\la^{k-j}S^{(k)},
\end{align*}
so that
\begin{align*}
S = \sum_{k\leq j-1} \la^k S^{(k)} + \la^j S^{(\geq j)}.
\end{align*}
\item A coefficient $S^{(i)}$ in the expansion of an object $S$ might oscillate at the frequency $\la^{-1}$, i.e be a linear combination of terms of the form
\begin{align*}
\end{align*}
for some $S^{(i,T,z)}$ independent of $\la$ (the notation $S^{(i,T,z)}$ will not be used systematically, it just serves our purpose here), $T\in\{\cos,\sin\}$ and $z$ a phase, i.e a scalar function on $\mathcal{M}$.
\item In order to manipulate complicated non-linear expressions where high-frequency expansions might be differentiated and multiplied, we introduce a schematic notation: $\cro{f}_\a$ will denote any quantity with no oscillating function, depending on the index $\a$ (this obviously extends to any number of indices) and of the form $f$. For instance the Christoffel symbol $\Ga(\g)^\rho_{\mu\nu}$ of a metric $\g$ might be rewritten $\cro{\g^{-1}\dr \g}^\rho_{\mu\nu}$.
\item We extend this notation to include undescribed oscillations:  $\cro{f}_\a^\osc$ will denote any quantity of the form
\begin{align*}
\sum_{\substack{i\geq 0 \\ z \text{ scalar function}\\T\in\{1,\cos,\sin\}}} \la^i T\pth{\frac{z}{\la}} \cro{f}_\a.
\end{align*}
\end{itemize}

\paragraph{Some important tensor operators.} We conclude this section by introducing several operators acting on symmetric 2-tensors and related to the action of the Ricci tensor on oscillating tensors.

\begin{mydef}\label{def pola}
Let $v$ be a scalar function on $\mathcal{M}$ and $S$ a symmetric 2-tensor.
\begin{itemize}
\item[(i)] We define the polarization tensor of $S$ with respect to $v$ by
\begin{align*}
\Pol{S}{v}_\a \vcentcolon = \g_0^{\mu\nu}S_{\a\mu}\dr_\nu v  - \half \tr_{\g_0}S \dr_\a v .
\end{align*}
\item[(ii)] We define the operator $\mathcal{P}_v$ by setting
\begin{align*}
\mathcal{P}_v(S)_{\a\b} \vcentcolon = - \g_0^{-1}(\d v,\d v) S_{\a\b} + \dr_{(\a}v \Pol{S}{v}_{\b)}.
\end{align*}
\end{itemize}
\end{mydef}

Note that if $_0^-1(ụ,ụ)=0$, then in any null frame $L,,e^(1),e^(2)$ associated to $u$ (see the next section for the definition of a null frame) we have
\begin{align}
\Pol{S}{u}_{L} & = - S_{L L} , \label{Pol L}
\\ \Pol{S}{u}_{e^{(\i)}} & = - S_{L e^{(\i)}} , \label{Pol e}
\\ \Pol{S}{u}_{\Lb} & = - S_{e^{(1)} e^{(1)}} - S_{e^{(2)} e^{(2)}} . \label{Pol Lb}
\end{align}

\begin{remark}
In our previous work [28], $\Pol{S}{v}$ was denoted by $\mathrm{Pol}(S)$ (there was no need to include the phase function in the notation since only one was considered). This latter notation can be misleading since a 2-tensor satisfying $\mathrm{Pol}(S)=0$ is physically polarized, while the notation suggests that it has 'zero' polarization. We thank Thibault Damour for this remark, according to which we changed our notation.
\end{remark}

The operator $𝒫_v$ will play a very important role in our construction in the case $_0^-1(ṿ, ṿ)≠0$ since it will describe the leading term in the Ricci tensor when acting on a tensor oscillating with the phase function $v$. The next lemma shows how one can invert this operator modulo its kernel (which we won't need). Its straightforward proof is left to the reader, it relies on the identity
\begin{align}\label{Pol dv S}
\Pol{\dr_{(\cdot}vQ_{\cdot)}}{v}=\g_0^{-1}(\d v,\d v) Q.
\end{align}

\begin{lemma}\label{lem Pv}
Let $v$ be a scalar function on $\mathcal{M}$ with $\g_0^{-1}(\d v, \d v)\neq0$. We have
\begin{align*}
\mathrm{ran}\mathcal{P}_v = \enstq{A\text{ symmetric 2-tensor}}{ \Pol{A}{v}=0}.
\end{align*}
Moreover, if $ \Pol{A}{v}=0$ then
\begin{align*}
\PP_v\pth{-\frac{1}{\g_0^{-1}(\d v,\d v)}A}=A.
\end{align*}
\end{lemma}

\subsection{The background null dusts spacetime}\label{section BG}

We assume that a regular solution of the Einstein-null dusts system is given on the manifold $ℳ$. More precisely, we consider given on $ℳ$ a Lorentzian metric $_0$ and two families of scalar functions $ u__∈𝒜$ and $ F__∈𝒜$, where the index $$ runs through a finite set $𝒜$, solving the Einstein-null dusts system
\begin{equation}\label{BG system}
\left\{
\begin{aligned}
R_{\mu\nu}(\g_0) & = \sum_\A F_\A^2 \dr_\mu u_\A \dr_\nu u_\A,
\\ \g_0^{-1}(\d u_\A , \d u_\A) &  = 0,
\\ -2 L_\A F_\A + (\Box_{\g_0}u_\A)F_\A & = 0,
\end{aligned}
\right.
\end{equation}
where $L_= - _0^αβ_αu__β$ is the spacetime gradient and is assumed to be future-directed. Thanks to the eikonal equation in \eqref{BG system}, $L_$ is null and geodesic i.e $_0(L_,L_)=0$ and
\begin{align}
\D_{L_\A}L_\A = 0 \label{geodesic equation},
\end{align}
where $$ denotes the Levi-Civita connection associated to $_0$.

\paragraph{Gauge condition.} We assume that $_0$ satisfies the wave gauge condition
\begin{align}
\g_0^{\mu\nu}\Gamma(\g_0)^\rho_{\mu\nu} =0\label{wave condition BG},
\end{align}
in the coordinates $(t,x^1,x^2,x^3)$ of $ℳ$. This allows us to rewrite the first equation of \eqref{BG system} as
\begin{align}
-\tBox_{\g_0}(\g_0)_{\alpha\beta} + P_{\alpha\beta}(\g_0)(\dr \g_0,\dr \g_0) = 2 \sum_\A F_\A^2 \dr_\mu u_\A \dr_\nu u_\A. \label{eq BG coordinates}
\end{align}
\begin{align*}
P_{\alpha\beta}(\g_0)(\dr \g_0 ,\dr \g_0) & = \g_0^{\mu\rho}\g_0^{\nu\sigma} \bigg( \dr_{(\alpha}(\g_0)_{\rho\sigma}\dr_\mu (\g_0)_{\beta)\nu}  - \frac{1}{2} \dr_\alpha (\g_0)_{\rho\sigma} \dr_\beta (\g_0)_{\mu\nu}
\\&\hspace{3cm} - \dr_\rho (\g_0)_{\alpha\nu}\dr_\sigma (\g_0)_{\beta\mu} +  \dr_\rho (\g_0)_{\sigma\alpha} \dr_\mu (\g_0)_{\nu\beta} \bigg).
\end{align*}

\paragraph{Regularity and decay.} We will measure the regularity by an integer $N≥10$ and the decay by a real constant $$ satisfying $-3/2<<-$. We also introduce a smallness constant $>0$.
\begin{itemize}
\item We assume that the metric $\g_0$ is close to Minkowski in the following sense
\begin{align}\label{estim g0}
\l \g_0 - \mathbf{m} \r_{H^{N+1}_{\de}} + \l \dr_t \g_0 \r_{H^N_{\de+1}} + \l \dr_t^2\g_0 \r_{H^{N-1}_{\de+2}} \leq \e,
\end{align}
where $\mathbf{m}$ is the Minkowski metric on $\mathcal{M}$.
\item We assume that there exist constant non-zero vector fields $\mathfrak{z}_\A$ in $\R^3$ such that for all $\A\in\mathcal{A}$ we have
\begin{align}\label{estim uA}
\l \nab u_\A - \mathfrak{z}_\A \r_{H^N_{\de+1}} \leq \e.
\end{align}
We associate to each $u_\A$ a transport operator acting on tensors of all type
\begin{align*}
\Ll_\A \vcentcolon = -2\D_{L_\A} + \Box_{\g_0} u_\A.
\end{align*}
\item We assume that all densities $F_{\A|_{\Si_0}}$ are initially supported in a ball $B_R\subset \Si_0$ for some fixed $R>0$. The transport equation in \eqref{BG system} then implies that each $F_\A$ is supported in $J^+_0\pth{ B_R}$, i.e the causal future associated to $\g_0$. Thanks to \eqref{estim g0}-\eqref{estim uA} there exists a constant $C_{\mathrm{supp}}>0$ such that $J^+_0\pth{ B_R}\subset \enstq{(t,x)\in\mathcal{M}}{|x|\leq C_{\mathrm{supp}}R}$. We also assume that for all $\A\in\mathcal{A}$ we have
\begin{align}\label{estim FA}
\l F_\A \r_{H^N} \leq \e.
\end{align}
\end{itemize}

\paragraph{Strong coherence.} We introduce some sets of phases. First, the null harmonics that will appear in the metric $_$:
\begin{align*}
\mathcal{N}_k & \vcentcolon = \enstq{ k u_\A}{\A\in\mathcal{A}}.
\\ \mathcal{N} & \vcentcolon = \mathcal{N}_1  \cup \mathcal{N}_2 \cup\mathcal{N}_3.
\end{align*}
Second, the mixed harmonics that will appear in the metric $_$:
\begin{align*}
\mathcal{I}_2 & \vcentcolon = \enstq{u_\A\pm u_\B}{\A\neq \B \text{ and } \pm\in \{-1,+1\}} ,
\\ \mathcal{I}_3 & \vcentcolon = \enstq{u_\A\pm 2 u_\B}{\A\neq \B \text{ and } \pm\in \{-1,+1\}}
\\&\hspace{1cm} \cup \enstq{u_\A\pm_1 u_\B \pm_2 u_\C}{\A, \B, \C \text{ all distincts and } \pm_1,\pm_2\in \{-1,+1\}} ,
\\ \mathcal{I} & \vcentcolon = \mathcal{I}_2 \cup \mathcal{I}_3.
\end{align*}
We also define $𝒲 =𝒩∪ℐ$. Finally, the mixed harmonics that will appear using the contracted Bianchi identities in Section \ref{section propa 2}:
\begin{align*}
\mathcal{I}_4 & \vcentcolon =\enstq{u_\A\pm 3 u_\B}{\A\neq \B \text{ and } \pm\in \{-1,+1\}}
\\&\hspace{1cm} \cup \enstq{u_\A\pm_1 u_\B \pm_2 2 u_\C}{\A, \B, \C \text{ all distincts and } \pm_1,\pm_2\in \{-1,+1\}}
\\&\hspace{1cm} \cup \enstq{u_\A\pm_1 u_\B \pm_2 u_\C \pm_3 u_\D}{\A, \B, \C ,\D \text{ all distincts and } \pm_1,\pm_2,\pm_3\in \{-1,+1\}},
\\ \mathcal{I}_5 & \vcentcolon = \enstq{u_\A\pm 4u_\B}{\A\neq \B \text{ and } \pm\in \{-1,+1\}}
\\&\hspace{1cm}\cup \enstq{3u_\A\pm 2u_\B}{\A\neq \B \text{ and } \pm\in \{-1,+1\}}
\\&\hspace{1cm} \cup \enstq{u_\A\pm_1 2u_\B \pm_2 2u_\C}{\A, \B, \C \text{ all distincts and } \pm_1,\pm_2\in \{-1,+1\}}
\\&\hspace{1cm} \cup \enstq{u_\A\pm_1 u_\B \pm_2 3u_\C}{\A, \B, \C \text{ all distincts and } \pm_1,\pm_2\in \{-1,+1\}}
\\&\hspace{1cm} \cup \enstq{u_\A\pm_1 u_\B \pm_2 u_\C \pm_3 2u_\D}{\A, \B, \C ,\D \text{ all distincts and } \pm_1,\pm_2,\pm_3\in \{-1,+1\}},
\end{align*}
We also define $𝒵= 𝒲 ∪ℐ_4∪ℐ_5  $. Our geometric optics construction will rely on the following strong coherence assumption: there exists a constant $c_coherence>0$ such that
\begin{align}
\min_{v\in\mathcal{I}} |\g_0^{-1}(\d v,\d v)| \geq c_{\mathrm{coherence}}. \label{coherence}
\end{align}
We also assume that there exists a constant $c_spatial>0$ such that
\begin{align}\label{spatial}
\min_{z\in\mathcal{Z}}\pth{ \inf_{\R^3}|\nab z| } > c_{\mathrm{spatial}}.
\end{align}

\begin{remark}\label{remark adapted}
The assumptions \eqref{coherence}-\eqref{spatial} can be proved to hold under the assumption that the phases $u_\A$ are initially angularly separated, as was first noticed in [11]. More precisely, if there exists $\eta\in(0,1)$ such that for $\A\neq \B$ we have
\begin{align*}
\frac{\nabla u_\A \cdot \nabla u_\B}{|\nabla u_\A| |\nabla u_\B|} \leq 1 - \eta
\end{align*}
on $\Si_0$ and if \eqref{spatial} holds on $\Si_0$, then Huneau and Luk show that if $\e$ is small enough then there exists positive constants $c_\A$ such that if we replace the phases $u_\A$ by $u_\A'\vcentcolon = c_\A u_\A$ then \eqref{coherence}-\eqref{spatial} hold for some constants $c_{\mathrm{coherence}}$ and $c_{\mathrm{spatial}}$ (note that $(\g_0,c_\A^{-1} F_\A,  u_\A')$ would still solve \eqref{BG system}). Note that strictly speaking [11] consider less mixed phases in $\mathcal{Z}$ than the present article but their argument extends easily. Finally, note that ensuring angular separation and \eqref{spatial} on $\Si_0$ can be done by choosing some angularly separated $\mathfrak{z}_\A$ (which is a generic choice) and then choosing $\e$ small enough in \eqref{estim uA}. In particular, the assumptions \eqref{coherence}-\eqref{spatial} are generic.
\end{remark}

We mention a important consequence of \eqref{spatial}: together with a stationary phase argument we can show that for all $T:⟶$ smooth, $2π$-periodic with $∫_0^2πT=0$ and for all $z∈𝒵$ the sequence of functions $Tz/_∈(0,1]$ converges weakly to 0 in $L^2(K)$ when $→0$ and where $K$ is any compact subset of $^3$.

\paragraph{Initial data.} The background spacetime $(ℳ,_0,u_,F_)$ induces an initial data set \[\pth{\Si_0,g_0,k_0,u_{\A|_{\Si_0}},F_{\A|_{\Si_0}}}\] on $_0$ solving the null dusts constraint equations
\begin{align}
R(g_0) - |k_0|^2_{g_0} + (\tr_{g_0} k_0)^2 & = 2\sum_\A (\dr_t u_\A)^2 F_\A^2 , \label{BG hamiltonian}
\\ (\div_{g_0} k_0)_i - \dr_i \tr_{g_0} k_0 & = -\sum_\A \dr_t u_\A F_\A^2  \dr_i u_\A, \label{BG momentum}
\end{align}
and satisfying
\begin{align}\label{estim IDS}
\l g_0 - e \r_{H^{N+1}_{\de}} + \l k_0 \r_{H^N_{\de+1}} +  \lesssim \e.
\end{align}
To simplify the resolution of the constraint equations on $_0$ and the definition of initial data for the geometric optics hierarchy of equations, we make the generic assumption that $_t$ is the future-directed unit normal to $_0$ for $_0$. This implies that the second fundamental form $k_0$ of $_0$ in $(ℳ,_0)$ is given in coordinates by $(k_0)_ij=-_t(g_0)_ij$ and that the spatial components of \eqref{wave condition BG} rewrite
\begin{align}
\dr_t(\g_0)_{0\ell}   & = g_0^{ij}\pth{\dr_{i}(g_0)_{j\ell} - \half \dr_\ell (g_0)_{ij} },\label{dt g 0i}
\end{align}
on $_0$. This simplifying assumption also has consequences regarding optical functions $u_$. Since they solve the eikonal equations and that their spacetime gradient is assumed to be future-directed we have on $_0$ and for all $$
\begin{align}\label{eikonal init}
\dr_t u_\A = |\nab_{g_0}u_\A|_{g_0}.
\end{align}
On $_0$ we define the following vector fields
\begin{equation}\label{def vector fields init}
\begin{aligned}
N_\A & \vcentcolon = \frac{\nab_{g_0}u_\A}{ |\nab_{g_0}u_\A|_{g_0}},
\\ N^{(\pm)}_{\A\B} & \vcentcolon = \frac{\nab_{g_0}(u_\A\pm u_\B)}{ |\nab_{g_0}(u_\A\pm u_\B)|_{g_0}},
\end{aligned}
\end{equation}
where $≠$ in the second definition and where we used \eqref{estim g0} and \eqref{spatial} to divide by $|_g_0u_|_g_0$ or $|_g_0(u_±u_)|_g_0$. Note that \eqref{eikonal init} implies that
\begin{align*}
L_{\A|_{\Si_0}} & = |\nab_{g_0}u_\A|_{g_0} (\dr_t - N_\A).
\end{align*}
Moreover, if we project the geodesic equation \eqref{geodesic equation} onto $_0$, i.e compute $_0(_L_L_,_ℓ)$, we obtain
\begin{align}\label{geodesic spatial}
-   N_\A  (N_\A)_\ell    +\half N_\A^k N_\A^a \dr_\ell (g_0)_{ka} + \frac{1}{|\nab_{g_0} u_\A|_{g_0}} \dr_\ell |\nab_{g_0} u_\A|_{g_0}   =  \frac{1}{|\nab_{g_0} u_\A|_{g_0}} (N_\A)_\ell N_\A |\nab_{g_0} u_\A|_{g_0}  .
\end{align}
This identity will be used in the proof of Lemma \ref{lem ka11}.

\paragraph{Null frames.} To each optical function $u_$ we associate a null frame $L_,_,e^(1)_,e^(2)_$ on $ℳ$, which thus satisfies
\begin{align*}
\g_0(L_\A,L_\A) & = \g_0(\Lb_\A,\Lb_\A) = \g_0\pth{L_\A,e^{(\i)}_\A} = \g_0\pth{\Lb_\A,e^{(\i)}_\A}=0
\end{align*}
\begin{align*}
\g_0(L_\A,\Lb_\A) & = -2, \qquad \g_0\pth{e^{(\i)}_\A,e^{(\j)}_\A} = \de_{\i\j}.
\end{align*}
Moreover, we ask that $e^(1)_,e^(2)_$ is an orthonormal frame for $_0$ of $TP_,t,u$ where
\begin{align*}
P_{\A,t,u} = \Si_t \cap \enstq{(\tau,x)\in\mathcal{M}}{u_\A(\tau,x)=u},
\end{align*}
which, thanks to \eqref{estim uA}, has the topology of a plane in $^3$. More details on the definition of such null frames can be found in [27]. Finally, note that $N_,e^(1)_,e^(2)_$ is an orthonormal frame for $g_0$ of $T_0$ and that $_|__0=|_g_0u_|_g_0 (_t + N_)$.

\begin{remark}
In this article, we don't prove that such a background spacetime $(\mathcal{M},\g_0,u_\A,F_\A)$ can be constructed, this follows from [5] adapted to the null dusts case.
\end{remark}

\subsection{Statement of the result}

The following theorem is the main result of this article.

\begin{thm}\label{theo evol}
Let $(\mathcal{M},\g_0, u_\A,F_\A)$ be a solution of the Einstein-null dusts system as described in Section \ref{section BG}. There exists $\e_0=\e_0(N,\de,R)>0$ such that if $0<\e\leq \e_0$ then there exists a family $(\g_\la)_{\la\in(0,\e_0]}$ of solutions to the Einstein vacuum equations on $\mathcal{M}$ of the form
\begin{align}\label{ansatz theo}
\g_\la & = \g_0 + \la \sum_\A \cos\pth{\frac{u_\A}{\la}} F^{(1)}_\A + \la^2\pth{ \sum_{\substack{w\in\mathcal{N}_1\cup\mathcal{N}_2\cup\mathcal{I}_2\\T\in\{\cos,\sin\}}} T\pth{\frac{w}{\la}}F^{(2,w,T)}_\la +  \widetilde{\g_\la}}.
\end{align}
\begin{itemize}
\item[(i)] the metrics $\g_\la$ solve \eqref{EVE} in generalised wave gauge and in the coordinates $(t,x^1,x^2,x^3)$ we have
\begin{equation}\label{gauge theo}
\begin{aligned}
\g_\la^{\mu\nu}\Ga(\g_\la)^\rho_{\mu\nu} & \rightarrow 0 \quad\text{uniformly on compact sets},
\\ \dr_\a\pth{ \g_\la^{\mu\nu}\Ga(\g_\la)^\rho_{\mu\nu}} & \rightharpoonup 0 \quad\text{weakly in $L^2_{loc}$},
\end{aligned}
\end{equation}
when $\la\to 0$,
\item[(ii)] the tensors $F^{(1)}_\A$ and $F^{(2,w,T)}_\la$ are supported in $J^+_0\pth{ B_R}$ and $F^{(1)}_\A$ satisfies
\begin{align}
-2\D_{L_\A} F^{(1)}_\A + (\Box_{\g_0}u_\A)F^{(1)}_\A & = 0, \label{transport theo}
\\ \left| F^{(1)}_\A \right|^2_{\g_0} & = 8 F_\A^2,\label{energy theo}
\end{align}
\item[(iii)] there exists $C=C(N,\de,R)>0$ such that
\begin{align}
\l F^{(1)}_\A \r_{H^N} & \leq C\e,\label{estim theo 1}
\\ \l F^{(2,w,T)}_\la \r_{L^2} + \max_{r\in\llbracket 0,5\rrbracket}\la^r\l\dr\nab^r F^{(2,w,T)}_\la \r_{L^2} & \leq C\e,\label{estim theo 2}
\\ \l \widetilde{\g_\la} \r_{L^2_{\de}} + \max_{r\in\llbracket 0, 4 \rrbracket} \la^{r} \l\dr \nab^r \widetilde{\g_\la} \r_{L^2_{\de+1+r}} & \leq C \e. \label{estim theo 3}
\end{align}
\end{itemize}
\end{thm}

\noindent Some comments are in order.
\begin{itemize}
\item \textbf{The reverse Burnett conjecture.} The estimates stated in Theorem \ref{theo evol} are consistent with Burnett's weak convergence as in \eqref{CV}. Indeed if $K$ is a compact subset of $\R^3$ then \eqref{estim theo 2}-\eqref{estim theo 3} and the standard Sobolev embedding $H^2\subset L^\infty$ imply
\begin{align*}
\l F^{(2,w,T)}_\la \r_{L^\infty(K)} + \l \widetilde{\g_\la} \r_{L^\infty(K)}\lesssim \la^{-1},
\end{align*}
and thus the convergence of $\g_\la$ towards $\g_0$ uniformly on compact sets. Moreover, we have
\begin{align*}
\dr(\g_\la-\g_0) & = -\sum_\A \sin\pth{\frac{u_\A}{\la}}\dr u_\A F^{(1)}_\A
\\&\quad +  \la \pth{ \sum_\A \cos\pth{\frac{u_\A}{\la}} \dr F^{(1)}_\A + \sum_{\substack{w\in\mathcal{N}_1\cup\mathcal{N}_2\cup\mathcal{I}_2\\T\in\{\cos,\sin\}}} T'\pth{\frac{w}{\la}}\dr w F^{(2,w,T)}_\la }
\\&\quad +  \la^2 \pth{ \sum_{\substack{w\in\mathcal{N}_1\cup\mathcal{N}_2\cup\mathcal{I}_2\\T\in\{\cos,\sin\}}} T\pth{\frac{w}{\la}}\dr F^{(2,w,T)}_\la +  \dr \widetilde{\g_\la}}.
\end{align*}
Thanks to \eqref{spatial} and \eqref{estim theo 1}, $\sin\pth{\frac{u_\A}{\la}}\dr u_\A F^{(1)}_\A$ converges weakly to 0 in $L^2_{loc}$ and \eqref{estim theo 2}-\eqref{estim theo 3} again implies that the remaining terms in $\dr(\g_\la-\g_0)$ converge actually strongly to 0 in $L^2_{loc}$. Therefore, Theorem \ref{theo evol} indeed implies Theorem \ref{theo rough burnett}, where the genericity mentioned in the latter refers to the assumptions \eqref{coherence}-\eqref{spatial} (see Remark \ref{remark adapted}).
\item \textbf{Parametrization of the $(\g_\la)_{\la\in(0,1]}$.} For a given background spacetime $(\mathcal{M},\g_0, u_\A,F_\A)$ as described in Section \ref{section BG}, we actually construct several families $(\g_\la)_{\la\in(0,\e_0]}$ as in Theorem \ref{theo evol}. These families are parametrized by a set of seeds living on the initial hypersurface $\Si_0$ and defined in Definition \ref{def seed} below. Each seed is parametrized by two real numbers, which correspond to the coefficients of two possible polarizations for the leading oscillating term in \eqref{ansatz theo}, as in the linearized gravity setting in TT gauge (see Remark \ref{remark seed}).
\item \textbf{The generalised wave gauge.} The exact expression of the generalised wave gauge term $\g_\la^{\mu\nu}\Ga(\g_\la)^\rho_{\mu\nu}$ is somehow irrelevant since we don't really prescribe what it should contain but rather what it should not contain, namely first order derivatives of some of the metric components solving (or coupled with) a wave equation. A novelty compared to [28] is that we also use this gauge term to control the interaction of waves propagating in different directions, see the discussion on $\gg^{(3,e)}_{v,T}$ in Section \ref{section gauge term}. Overall, the gauge term helps us recover true hyperbolicity and ellipticity of the Ricci tensor.
\item \textbf{The constraint equations.} As in every resolution of \eqref{EVE} from a spacelike hypersurface, one should solve first the constraint equations on the initial hypersurface. In our previous work on the singlephase case, we separated the resolution of the constraint equations on $\Si_0$ from the resolution of \eqref{EVE} on $\mathcal{M}$, resulting in the two articles [28] and [29]. For the multiphase case, we decided to associate the elliptic and hyperbolic aspects of the construction in one self-consistent article. In particular, we want to highlight the surprising connections between the elliptic and hyperbolic procedures due to the oscillatory aspect of the metric. Some of these connections were already present in the singlephase case, but some are specific to the multiphase case, see the discussion after Proposition \ref{prop constraint main}.
\item \textbf{Gauge-independent transparency.} We conclude these comments by insisting on the astounding structures in the Einstein vacuum equations leading in particular to \eqref{transport theo}-\eqref{energy theo}. These relations satisfied by the first profiles in the ansatz \eqref{ansatz theo} for $\g_\la$ illustrate perfectly the discussion of Section \ref{section transparence}: the self-interaction of gravitational waves produces a quadratic macroscopic effect \eqref{energy theo} while each wave is linearly transported \eqref{transport theo}. Moreover the interaction of gravitational waves propagating in different null directions don't produce a macroscopic effect, in particular since the first mixed harmonics $v\in\mathcal{I}$ only appear at order $\la^2$ in \eqref{ansatz theo}. Finally, note that both the linear propagation and non-linear backreaction are phenomena that are independent of the choice of coordinates since \eqref{transport theo} and \eqref{BG system} are tensorial equations.
\end{itemize}
The rest of this article is devoted to the proof of Theorem \ref{theo evol}.

\section{The high-frequency ansatz}\label{section expansion Ricci}

In this section we present the high-frequency ansatz for the metric $_$. Its expression is given in Section \ref{section expansion metric}, its Ricci tensor is computed in Section \ref{section ricci} and finally in Section \ref{section hierarchy} a hierarchy of equations and conditions is deduced from the requirement that $_$ solves the Einstein vacuum equations \eqref{EVE}.

\subsection{Expansion of the metric}\label{section expansion metric}

For $>0$, we consider the following metric
\begin{equation}\label{expansion metric}
\begin{aligned}
\g_\la & = \g_0 + \la \sum_\A  \cos\pth{\frac{u_\A}{\la}} F^{(1)}_\A  +   \lambda^2 \sum_\A \pth{ \sin\pth{\frac{u_\A}{\la}} \pth{ \F_\A + F^{(2,1)}_\A } + \cos\pth{\frac{2u_\A}{\la}} F^{(2,2)}_\A }
\\&\quad + \la^2 \sum_{\A\neq \B,\pm}   \cos\pth{\frac{u_\A\pm u_\B}{\la}} F^{(2,\pm)}_{\A\B} + \la^2 \h_\la
\\&\quad + \la^3 \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}} \gg^{(3,h)}_{u,T} + \la^3 \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}} T\pth{\frac{v}{\la}} \gg^{(3,e)}_{v,T}.
\end{aligned}
\end{equation}
In this expression,
\begin{itemize}
\item $F^{(1)}_\A$, $F^{(2,1)}_\A$, $F^{(2,2)}_\A$, $F^{(2,\pm)}_{\A\B}$, are symmetric 2-tensors which don't depend on $\la$ and we already assume $F^{(2,\pm)}_{\A\B}  = F^{(2,\pm)}_{\B\A}$,
\item the remainder $\h_\la$ and the amplitude $\F_\A$ are symmetric 2-tensors and do depend on $\la$. The higher order symmetric 2-tensors $\gg^{(3,h)}_{u,T}$ and $\gg^{(3,e)}_{v,T}$ will depend on $\F_\A$ and thus on $\la$.
\end{itemize}

The ansatz \eqref{expansion metric} differs from the singlephase ansatz in [28] only because of the terms $F^(2,±)_$ and $≫^(3,e)_v,T$. They will crucially be used to control the creation of mixed harmonics, i.e terms oscillating with phases $v∈ℐ$, at order $^0$ and $^1$ in the Ricci tensor of $_$ respectively.

\subsection{Expansion of the Ricci tensor}\label{section ricci}

In this section, we compute the Ricci tensor of the metric $_λ$. Since we want $_$ to solve \eqref{EVE} in generalised wave gauge we use the following decomposition of the Ricci tensor of Lorentzian metric $$:
\begin{align}\label{Ricci tensor GWC}
2R_{\a\b}(\g) = - \tBox_\g \g_{\a\b} + P_{\a\b}(\g)(\dr\g,\dr\g) + \g_{\rho(\a}\dr_{\b)}H^\rho + H^\rho\dr_\rho\g_{\a\b},
\end{align}
\begin{align}
\tBox_\g & \vcentcolon = \g^{\mu\nu}\dr_\mu \dr_\nu, \label{wave heur}
\\ H^\rho & \vcentcolon = \g^{\mu\nu}\Ga(\g)^\rho_{\mu\nu},\label{gauge heur}
\\ P_{\a\b}(\g)(\dr\g,\dr\g) & \vcentcolon = \g^{\mu\rho}\g^{\nu\si} \pth{ \dr_{(\a}\g_{\rho\si}\dr_\mu \g_{\b)\nu}  - \frac{1}{2} \dr_\a \g_{\rho\si} \dr_\b \g_{\mu\nu}  - \dr_\rho \g_{\a\nu}\dr_\si \g_{\b\mu} +  \dr_\rho \g_{\si\a} \dr_\mu\g_{\nu\b} }. \label{P dg dg}
\end{align}
We expand the wave part $__λ(_λ)_αβ$ in Section \ref{section wave part}, the quadratic non-linearity $P_αβ(_)(_λ,_λ) $ in Section \ref{section quadratic} and the gauge term $H^ρ$ in Section \ref{section gauge term}. We combine the results in Section \ref{section ricci final} to obtain the expansion of $R_(_λ)$. The computations will be performed under the following assumption on $F^(1)_$:
\begin{equation}\label{assumption}
\begin{aligned}
\Pol{F^{(1)}_\A}{u_\A} & =0,
\\ F^{(1)}_{L_\A \Lb_\A} & = 0.
\end{aligned}
\end{equation}
Note that thanks to \eqref{Pol L}-\eqref{Pol Lb} the assumption \eqref{assumption} is equivalent to $__0F^(1)_=0$ and $(F^(1)_)_L_ =0$. They are consistent with the singlephase case of [28] and the approximate construction of [2]. More precisely, the first assumption in \eqref{assumption} ensures that $R_(_λ)=1$. The second assumption is convenient to simplify the computations and is not necessary. They will be proved to be consistent with the hierarchy derived from \eqref{EVE}, as in [28].

\begin{remark}
We introduce two conventions on the use of the bracket schematic notations introduced in Section \ref{section notation}:
\begin{itemize}
\item In a bracket we will never write down the quantities depending on the background spacetime introduced in Section \ref{section BG}, since we are not solving for it. This includes metric or inverse metric coefficients, phases $z\in\mathcal{Z}$, vector fields from the null frames.
\item In a bracket we will never specify the indices referring to the set $\mathcal{A}$. For instance, $\cro{F^{(1)}}$ simply denotes any of the $\cro{F^{(1)}_\A}$ for $\A\in\mathcal{A}$. Similarly $\cro{\gg^{(3,h)}}$ denotes any of the $\cro{\gg^{(3,h)}_{u,T}}$ for $u\in\mathcal{N}$ and $T\in\{\cos,\sin\}$ (same for $\gg^{(3,e)}$).
\end{itemize}
These conventions also apply to the notation $\cro{\cdot}^\osc$ and will be used throughout the article.
\end{remark}

\subsubsection{The wave part}\label{section wave part}

In this section we expand the wave term in \eqref{Ricci tensor GWC}, i.e $__λ(_λ)_αβ$. The result is contained in the next proposition, whose proof is postponed to Appendix \ref{appendix proof lemma W}.

\begin{lemma}\label{lemma W}
Under the assumptions \eqref{assumption}, the wave part of the Ricci tensor of $\g_\la$ admits the expansion
\begin{equation}\label{exp W}
\begin{aligned}
\tBox_{\g_\la}(\g_\la)_{\a\b} & =  \tBox_{\g_0}(\g_0)_{\a\b} + \sum_\A \sin\pth{\frac{u_\A}{\la}} (W^{(0,1)}_\A)_{\a\b}  +  \sum_{\A\neq\B,\pm} \cos\pth{\frac{u_\A\pm u_\B}{\la}}  (W^{(0,\pm)}_{\A\B})_{\a\b}
\\&\quad + \la \sum_\A\pth{\cos\pth{\frac{u_\A}{\la}} (W^{(1,1)}_\A)_{\a\b} + \sin\pth{\frac{2u_\A}{\la}} (W^{(1,2)}_\A)_{\a\b}}
\\&\quad + \la\sum_\A\cos\pth{\frac{u_\A}{\lambda}} \cos\pth{ \frac{2u_\A}{\lambda} } (F^{(2,2)}_\A)_{L_\A L_\A} (F^{(1)}_\A)_{\alpha\beta}
\\&\quad +  \frac{\la}{2}\sum_\A \sin\pth{\frac{2u_\A}{\la}} (\F_\A)_{L_\A L_\A} (F^{(1)}_\A)_{\a\b} + \la \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}} T\pth{\frac{v}{\la}}(W^{(1)}_{v,T})_{\a\b}
\\&\quad + \la^2 W^{(\geq 2)}_{\a\b}.
\end{aligned}
\end{equation}
The terms of order 0 are given by
\begin{align}
(W^{(0,1)}_\A)_{\a\b} & = 2 L_\A (F^{(1)}_\A)_{\a\b} - (\tBox_{\g_0} u_\A)(F^{(1)}_\A)_{\a\b} ,\label{W 0 1}
\\ (W^{(0,\pm)}_{\A\B})_{\a\b} & = - \g_0^{-1}(\d (u_\A\pm u_\B), \d(u_\A\pm u_\B))  (F^{(2,\pm)}_{\A\B})_{\alpha\beta} \label{W 0 pm}
\\&\quad + \frac{1}{4}   \pth{ (F^{(1)}_\B)_{L_\A L_\A}   (F^{(1)}_\A)_{\alpha\beta} + (F^{(1)}_\A)_{L_\B L_\B}   (F^{(1)}_\B)_{\alpha\beta} }.\non
\end{align}
The terms of order 1 are given by
\begin{align}
(W^{(1,1)}_\A)_{\a\b} & =   (-2 L_\A + \tBox_{\g_0}u_\A ) \left( (\F_\A)_{\alpha\beta} + (F^{(2,1)}_\A)_{\alpha\beta} \right)   + (\h_\lambda)_{L_\A L_\A} (F^{(1)}_\A)_{\alpha\beta}     \label{W 1 1}
\\&\quad + \cro{\pth{1 + F^{(2,\pm)} + (F^{(1)})^2 }F^{(1)} + \dr^2 F^{(1)}}_{\a\b} , \non
\\ (W^{(1,2)}_\A)_{\a\b} & = -2 (-2 L_\A + \tBox_{\g_0}u_\A ) (F^{(2,2)}_\A)_{\alpha\beta}  +  \cro{ \pth{ F^{(2,1)} + \dr F^{(1)} + F^{(1)} }F^{(1)} }_{\a\b} , \label{W 1 2}
\\ (W^{(1)}_{v,T})_{\a\b} & = - \g_0^{-1}(\d v, \d v) (\gg^{(3,e)}_{v,T})_{\a\b}  \label{W 1 v T}
\\&\quad +  \Big\{\dr^{\leq 1}F^{(2,\pm)} + \pth{ \F + F^{(2,1)} + F^{(2,2)} + F^{(2,\pm)}  + (F^{(1)})^2 + F^{(1)} + \dr F^{(1)} }F^{(1)} \Big\}_{\a\b} .\non
\end{align}
The higher order terms are given by
\begin{align}
W^{(\geq 2)}_{\a\b} & = \tBox_{\g_\la}(\h_\la)_{\a\b}  + \sum_\A \sin\pth{\frac{u_\A}{\la}}\tBox_{\g_\la}(\F_\A)_{\a\b} \label{W 2}
\\&\quad +  \la \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}}\tBox_{\g_\la} (\gg^{(3,h)}_{u,T})_{\a\b}  + \la  \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}} T\left( \frac{v}{\lambda} \right)\tBox_{\g_\la} (\gg^{(3,e)}_{v,T})_{\a\b}\non
\\&\quad + \Big\{ \g_\la^{-1}\pth{\dr^{\leq 1}\pth{\gg^{(3,h)} + \gg^{(3,e)} + \F} + \dr^{\leq 2}\pth{ F^{(1)}  + F^{(2,1)} + F^{(2,2)} +  F^{(2,\pm)}} } \Big\}_{\a\b}^\osc \non.
\end{align}
\end{lemma}

We draw the reader's attention on several structural facts from Lemma \ref{lemma W}:
\begin{enumerate}
\item The main term produced by the wave part of the Ricci is a transport term along the rays: at order $\la^0$ we see the transport of $F^{(1)}_\A$ by the vector field $L_\A$ and at order $\la^1$ we see the transport of $\F_\A$, $F^{(2,1)}_\A$ and $F^{(2,2)}_\A$ by $L_\A$ again. By imposing transport equations for these tensors we will be able to absorb most terms at these orders in the Ricci tensor. Therefore, the admissible terms at order $\la^0$ oscillate like $\sin\pth{\frac{u_\A}{\la}}$ and the admissible terms at order $\la^1$ oscillate like $\cos\pth{\frac{u_\A}{\la}}$ and $\sin\pth{\frac{2u_\A}{\la}}$. We already see in \eqref{exp W} the presence at order $\la^1$ of forbidden term such as $(F^{(2,2)}_\A)_{L_\A L_\A} (F^{(1)}_\A)_{\alpha\beta}$ oscillating like $\cos\pth{\frac{3u_\A}{\la}}$, we will deal with it differently.
\item At order $\la^2$, we obviously recover the wave operator acting on the remainder $\h_\la$. Note how we made a difference in \eqref{W 2} between second order derivatives of the various tensors in \eqref{expansion metric}. Second order derivatives of $F^{(1)}_\A$, $F^{(2,1)}_\A$, $F^{(2,2)}_\A$ and $F^{(2,\pm)}_{\A\B}$ are considered error terms and put in the brackets, while second order derivatives of $\F_\A$, $\gg^{(3,h)}_{u,T}$ and $\gg^{(3,e)}_{v,T}$ are carefully conserved and in particular the wave operator structure will crucially be used to prove well-posedness.
\item Opposite to the terms oscillating in null directions, the wave operator acts 'elliptically' on terms oscillating with phase $v\in\mathcal{I}$ since $\g_0^{-1}(\d v,\d v)\neq 0$ whenever $v\in\mathcal{I}$ (see \eqref{coherence}). This explains the presence of $F^{(2,\pm)}_{\A\B}$ in \eqref{W 0 pm} and of $\gg^{(3,e)}_{v,T}$ in \eqref{W 1 v T}.
\end{enumerate}

\subsubsection{The quadratic non-linearity}\label{section quadratic}

In this section we expand the quadratic non-linearity in \eqref{Ricci tensor GWC}, i.e $P_αβ(_)(_λ,_λ) $. The result is contained in the next proposition, whose proof is postponed to Appendix \ref{appendix proof lemma P}.

\begin{lemma}\label{lemma P}
Under the assumptions \eqref{assumption}, the quadratic part of the Ricci tensor of $\g_\la$ admits the expansion
\begin{equation}\label{exp P}
\begin{aligned}
P_{\alpha\beta}(\g_\la)(\dr \g_\lambda, \dr \g_\lambda) & = P_{\alpha\beta}(\g_0)(\dr \g_0, \dr \g_0)  - \frac{1}{4}  \sum_{\A}  \left| F^{(1)}_\A \right|^2_{\g_0} \dr_\alpha u_\A \dr_\beta u_\A   + \sum_\A \sin\pth{\frac{u_\A}{\la}} (P^{(0,1)}_\A)_{\a\b}
\\&\quad + \sum_\A \cos\pth{\frac{2u_\A}{\la}} (P^{(0,2)}_\A)_{\a\b} + \sum_{\A\neq \B,\pm}\cos\pth{\frac{u_\A\pm u_\B}{\la}} (P^{(0,\pm)}_{\A\B})_{\a\b}
\\&\quad + \la \sum_\A\pth{ \cos\pth{\frac{u_\A}{\la}} (P^{(1,1)}_\A)_{\a\b} + \sin\pth{ \frac{2u_\A}{\la}} (P^{(1,2)}_\A)_{\a\b} }
\\&\quad + \la  \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}} \dr_{(\a}u (\hat{P}^{(1)}_{u,T})_{\b)} + \la \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}}T\pth{ \frac{v}{\la}} (P^{(1)}_{v,T})_{\a\b}
\\&\quad + \la^2  P^{(\geq 2)}_{\alpha\beta} .
\end{aligned}
\end{equation}
The terms of order 0 are given by
\begin{align}
(P^{(0,1)}_\A)_{\a\b} & = - 2\Ga(\g_0)^\mu_{(\a \rho} \dr^\rho u_\A (F^{(1)}_\A)_{\beta)\mu}  - (F^{(1)}_\A)^{\mu\nu} \dr_{(\a}u_\A \pth{ \dr_\mu (\g_0)_{\b)\nu}  - \frac{1}{2} \dr_{\b)} (\g_0)_{\mu\nu} },\label{P 0 1}
\\ (P^{(0,2)}_\A)_{\a\b} & = \frac{1}{4}\left| F^{(1)}_\A \right|^2_{\g_0}\dr_\alpha u_\A \dr_\beta u_\A  , \label{P 0 2}
\\ (P^{(0,\pm)}_{\A\B})_{\a\b} & = \frac{1}{4}\bigg( \pm\dr_{(\alpha}u_\A (F^{(1)}_\A)_{L_\B\si} (F^{(1)}_\B)_{\beta)}^\si  \pm\dr_{(\alpha}u_\B (F^{(1)}_\B)_{L_\A\si} (F^{(1)}_\A)_{\beta)}^\si  \label{P 0 pm}
\\&\hspace{2cm} \pm\frac{1}{2} \dr_{(\alpha} u_\A\dr_{\beta)} u_\B   \left| F^{(1)}_\A\cdot F^{(1)}_\B \right|_{\g_0}  \pm (F^{(1)}_\A)_{(\alpha L_\B} (F^{(1)}_\B)_{\beta) L_\A}\non
\\&\hspace{6cm}  \mp \g_0^{-1}(\d u_\A,\d u_\B)  (F^{(1)}_\A)_{(\alpha}^\nu (F^{(1)}_\B)_{\nu\beta)}   \bigg).\non
\end{align}
The terms of order 1 are given by
\begin{align}
(P^{(1,1)}_\A)_{\a\b} & = 2\Ga(\g_0)^{\mu}_{(\a\rho}  \dr^\rho u_\A\pth{(\F_\A)_{\beta)\mu} +  (F^{(2,1)}_\A)_{\beta)\mu}}  \label{P 1 1}
\\&\quad +  \cro{ \dr F^{(1)} + F^{(1)} F^{(2,\pm)} + F^{(1)}+(F^{(1)})^3 }_{\a\b} , \non
\\ (P^{(1,2)}_\A)_{\a\b} & = -4\Ga(\g_0)^{\mu}_{(\a\rho}  \dr^\rho u_\A (F^{(2,2)}_\A)_{\beta)\mu} + \cro{F^{(1)} \dr F^{(1)}+ (F^{(1)})^2}_{\a\b}, \label{P 1 2}
\\ (\hat{P}^{(1)}_{u,T})_{\b} & = \cro{\pth{1+F^{(1)}}\pth{(F^{(1)})^2 +\F + F^{(2,1)} +  F^{(2,2)}}}_{\b}, \label{Phat 1 u T}
\\ (P^{(1)}_{v,T})_{\a\b} & = \Big\{ F^{(2,\pm)} + (F^{(1)})^2+(F^{(1)})^3  + F^{(1)} \pth{\dr F^{(1)} + \F + F^{(2,1)} + F^{(2,2)} + F^{(2,\pm)} }  \Big\}_{\a\b}. \label{P 1 v T}
\end{align}
The terms of order 2 are simply of the form
\begin{align}
P^{(\geq 2)}_{\alpha\beta} & =   \cro{ \pth{ \g_\la^{-1}\g_\la^{-1}\dr\g_\la\dr\g_\la }^{(\geq 2)} }_{\a\b}^\osc .\label{P 2}
\end{align}
\end{lemma}

We draw the reader's attention on several structural facts from Lemma \ref{lemma P}:
\begin{enumerate}
\item It is clear from the resonant term in \eqref{exp P} that the quadratic part of the Ricci tensor is responsible for backreaction, i.e the fact that the Ricci tensor of the background spacetime $\g_0$ cannot vanish and needs to absorb the sum of $ \left| F^{(1)}_\A \right|^2_{\g_0}  \dr_\alpha u_\A \dr_\beta u_\A $, which in particular makes clear the null dusts structure of backreaction.
\item Following the first comment after Lemma \ref{lemma W}, we see the presence at order $\la^0$ of a forbidden term due to $P^{(0,2)}_\A$. This term is the high-high counterpart of backreaction and its particular null dust tensorial structure $\dr_\a u_\A \dr_\b u_\A$ given in \eqref{P 0 2} will be crucially used. Similarly, the forbidden harmonics $3u_\A$ is present at order $\la^1$ with a particular tensorial structure $\dr_{(\a}u (\hat{P}^{(1)}_{u,T})_{\b)}$ (recall that $\mathcal{N}$ includes the third harmonics $3u_\A$).
\end{enumerate}

\subsubsection{The gauge part}\label{section gauge term}

In this section we expand the gauge term $H^ρ$ which appears differentiated in \eqref{Ricci tensor GWC}. In the usual wave gauge, it is simply set to 0. In our case, we will set to 0 only some problematic terms in $H^ρ$ which we regroup and call $Υ^ρ$. The terms that we will put in $Υ^ρ$ will serve two main purposes: first recover hyperbolicity and obtain a well-posed final system and second simplify the control of the mixed harmonics at order $^1$. As a secondary purpose it will allow $≫^(3,h)_u,T$ not to depend on the remainder $_$. For now, we simply give the definition of $Υ^ρ$:
\begin{equation}\label{Upsilon}
\begin{aligned}
\Upsilon^\rho & \vcentcolon =  \g_\la^{\rho\sigma} \g_\la^{\mu\nu} \Bigg( \dr_\mu (\h_\lambda)_{\sigma\nu} - \half \dr_\sigma (\h_\lambda)_{\mu\nu} +  \sum_\A  \sin\left(\frac{u_\A}{\lambda} \right) \pth{\dr_\mu (\F_\A)_{\sigma\nu} - \half \dr_\sigma (\F_\A)_{\mu\nu} }  \Bigg)
\\&\quad + \la \g_\la^{\rho\sigma} \g_\la^{\mu\nu} \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}} \pth{\dr_\mu (\gg^{(3,h)}_{u,T})_{\sigma\nu} - \half \dr_\sigma (\gg^{(3,h)}_{u,T})_{\mu\nu}}
\\&\quad + \la \g_\la^{\rho\sigma} \g_\la^{\mu\nu} \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}}T\pth{\frac{v}{\la}} \pth{\dr_\mu (\gg^{(3,e)}_{v,T})_{\sigma\nu} - \half \dr_\sigma (\gg^{(3,e)}_{v,T})_{\mu\nu}}
\\&\quad +  \g_\la^{\rho\sigma} \g_\la^{\mu\nu}  \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}}T'\pth{\frac{v}{\la}}\pth{ \dr_\mu v(\gg^{(3,e)}_{v,T})_{\sigma\nu} - \half \dr_\sigma v(\gg^{(3,e)}_{v,T})_{\mu\nu} }
\\&\quad  -  \g_0^{\rho\sigma} \h_\la^{\mu\nu} \Bigg(\dr_\mu (\g_0)_{\sigma\nu} - \half \dr_\sigma (\g_0)_{\mu\nu} -  \sum_\A  \sin\left( \frac{u_\A}{\lambda} \right) \pth{ \dr_\mu u_\A (F^{(1)}_\A)_{\sigma\nu} - \half \dr_\sigma u_\A (F^{(1)}_\A)_{\mu\nu}  }\Bigg).
\end{aligned}
\end{equation}
The gauge term is expanded in the next proposition, whose proof is postponed to Appendix \ref{appendix proof lemma H}.

\begin{lemma}\label{lemma H}
Under the assumptions \eqref{assumption}, the gauge term $H^\rho$ satisfies
\begin{align*}
H^\rho & = \mathring{H}^\rho + \la^2 \Upsilon^\rho
\end{align*}
where $\mathring{H}^\rho$ admits the expansion
\begin{align*}
\mathring{H}^\rho & = \la  \sum_\A\pth{\cos\pth{\frac{u_\A}{\la}} (H^{(1,1)}_\A)^\rho + \sin\pth{\frac{2u_\A}{\la}} (H^{(1,2)}_\A)^\rho}  +\la \sum_{\A\neq\B,\pm}\sin\pth{\frac{u_\A\pm u_\B}{\la}}(H^{(1,\pm)}_{\A\B})^\rho
\\&\quad + \la^2 \pth{ (H^{(2)})^\rho + \Upsilon^\rho } + \la^3 (H^{(\geq 3)})^\rho.
\end{align*}
The terms of order 1 are given by
\begin{align}
(H^{(1,1)}_\A)^\rho & = \g_0^{\rho\sigma} \Pol{\F_\A + F^{(2,1)}_\A}{u_\A}_\si + \g_0^{\rho\si} \g_0^{\mu\nu} \pth{ \dr_\mu (F^{(1)}_\A)_{\sigma\nu} - \half \dr_\sigma (F^{(1)}_\A)_{\mu\nu} } \label{H 1 1}
\\&\quad - (F^{(1)}_\A)^{\mu\nu} \g_0^{\rho\sigma} \pth{\dr_\mu (\g_0)_{\sigma\nu} - \half \dr_\sigma (\g_0)_{\mu\nu} },\non
\\ (H^{(1,2)}_\A)^\rho  & = -2 \g_0^{\rho\sigma} \Pol{F^{(2,2)}_\A}{u_\A}_\si - \frac{1}{4}  \dr^\rho u_\A  \left| F^{(1)}_\A\right|^2_{\g_0} , \label{H 1 2}
\\ (H^{(1,\pm)}_{\A\B})^\rho  & = -\g_0^{\rho\sigma} \Pol{F^{(2,\pm)}_{\A\B}}{u_\A\pm u_\B}_\si - \frac{1}{8} \dr^\rho (u_\A\pm u_\B) \left| F^{(1)}_\A\cdot F^{(1)}_\B\right|_{\g_0}  \label{H 1 pm}
\\&\quad  - \frac{1}{4}  (F^{(1)}_\B)^{\nu}_{L_\A} (F^{(1)}_\A)_{\nu}^\rho \mp \frac{1}{4}  (F^{(1)}_\A)^{\nu}_{L_\B} (F^{(1)}_\B)_{\nu}^\rho  .\non
\end{align}
The terms of order 2 are given by
\begin{equation}\label{H 2 gauge}
\begin{aligned}
\\& =  \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T'\pth{\frac{u}{\la}}\g_0^{\rho\sigma}  \Pol{\gg^{(3,h)}_{u,T}}{u}_\si
\\&\quad +  \sum_{\substack{u\in\mathcal{N}\\T\in\{1,\cos,\sin\}}}T\pth{\frac{u}{\la}} \Big\{\dr F^{(2,1)} + \dr F^{(2,2)}
\\&\hspace{4cm} +  \pth{ \F + F^{(2,1)} + F^{(2,2)} + F^{(2,\pm)} + (F^{(1)})^2 + \dr F^{(1)} } \pth{1+F^{(1)}} \Big\}^\rho
\\&\quad +  \sum_{\substack{v\in\mathcal{I}\\T\in\{1,\cos,\sin\}}}T\pth{ \frac{v}{\la}} \Big\{ \dr F^{(2,\pm)}
\\&\hspace{4cm} +  \pth{ \F + F^{(2,1)} + F^{(2,2)} + F^{(2,\pm)} + (F^{(1)})^2 + \dr F^{(1)} } \pth{1+F^{(1)}} \Big\}^\rho.
\end{aligned}
\end{equation}
The terms of order 3 are of the form
\begin{align}
(H^{(\geq 3)})^\rho & = \Big\{ \pth{ (\g_\la^{-1} (\g_\la)^{(\geq 2)} + (\g_\la^{-1})^{(\geq 1)}(\g_\la^{-1})^{(\geq 1)}}
\\&\hspace{1cm}\times \pth{ \dr F^{(1)}  + \F + \dr^{\leq 1} F^{(2,1)} +  \dr^{\leq 1}F^{(2,2)} +  \dr^{\leq 1}F^{(2,\pm)} + \gg^{(3,h)} } \Big\}^{\rho,\osc} \non.
\end{align}
\end{lemma}

We draw the reader's attention on several structural facts from Lemma \ref{lemma H}:
\begin{enumerate}
\item The gauge term $H^\rho$ contains first order derivatives of the metric. Some appear in $\mathring{H}^\rho$ and some appear in $\Upsilon^\rho$, i.e are considered as problematic terms from the well-posedness point of view. In particular, first order derivatives of $\h_\la$, $\F_\A$, $\gg^{(3,h)}_{u,T}$ and $\gg^{(3,e)}_{v,T}$ have been included in $\Upsilon^\rho$, see the first three lines of \eqref{Upsilon}. By doing so, the second order derivatives of these tensors due to derivatives of $H^\rho$ in the Ricci tensor won't appear in the final system, where only the wave terms from \eqref{W 2} will be allowed (recall the second comment after Lemma \ref{lemma W}).
\item At order $\la^1$ and $\la^2$, we see the polarization tensors of $\F_\A$, $F^{(2,1)}_\A$, $F^{(2,2)}_\A$ and $\gg^{(3,h)}_{u,T}$ appearing as leading terms, where we recall that polarization tensors have been defined in Definition \ref{def pola}. Note that the polarization tensor of $\gg^{(3,e)}_{v,T}$ should also appear in \eqref{H 2 gauge}. However, we included the polarization tensor of $\gg^{(3,e)}_{v,T}$ in $\Upsilon^\rho$ in order to remove it from the final hierarchy, see the fourth line in \eqref{Upsilon}. The reason to do so will be explained after Proposition \ref{prop expression of the ricci}.
\end{enumerate}

\subsubsection{The Ricci tensor}\label{section ricci final}

In this section, we put together the results of Lemmas \ref{lemma W}, \ref{lemma P} and \ref{lemma H} and obtain the final expression of $R_(_)$ in the next proposition, whose proof is postponed to Appendix \ref{appendix proof prop ricci}.

\begin{prop}\label{prop expression of the ricci}
Under the assumptions \eqref{assumption}, the Ricci tensor of $\g_\la$ admits the expansion
\begin{align*}
R_{\a\b} (\g_\la) & = R^{(0)}_{\a\b} + \la R^{(1)}_{\a\b} + \la^2 R^{(\geq 2)}_{\a\b}.
\end{align*}
The term $R^{(0)}_{\a\b}$ is given by
\begin{align}
R^{(0)}_{\a\b} & = \sum_\A \pth{ F_\A^2 -  \frac{1}{8} \left| F^{(1)}_\A \right|^2_{\g_0} } \dr_\mu u_\A \dr_\nu u_\A+ (R^{(0)}_{\mathrm{null}})_{\a\b} +  (R^{(0)}_{\mathrm{mixed}})_{\a\b}  , \label{R 0}
\end{align}
\begin{align}
2 (R^{(0)}_{\mathrm{null}})_{\a\b} & =  \sum_\A \sin\pth{\frac{u_\A}{\la}}\bigg( (\Ll_\A F^{(1)}_\A)_{\alpha\beta} -  \dr_{(\a}u_\A  \Pol{\F_\A }{u_\A}_{\b)} \non
\\&\hspace{4cm} -  \dr_{(\a}u_\A \pth{  \Pol{ F^{(2,1)}_\A}{u_\A}_{\b)} +(Q^{(0)}_\A)_{\b)} }\bigg)\label{R 0 null}
\\&\quad - 4 \sum_\A \cos\pth{\frac{2u_\A}{\la}} \dr_{(\alpha} u_\A\pth{\Pol{F^{(2,2)}_\A}{u_\A}_{\b)}  +   \frac{3}{32} \dr_{\beta)} u_\A  \left| F^{(1)}_\A \right|^2_{\g_0} }\non,
\\ 2(R^{(0)}_{\mathrm{mixed}})_{\a\b} & = \sum_{\A\neq\B,\pm} \cos\pth{\frac{u_\A\pm u_\B}{\la}}\pth{  -\mathcal{P}_{u_\A\pm u_\B}\pth{ F^{(2,\pm)}_{\A\B}}_{\a\b} + (\mathcal{I}^{(0,\pm)}_{\A\B})_{\a\b} }, \label{R 0 mixed}
\end{align}
\begin{align}
(Q^{(0)}_\A)_{\b} & \vcentcolon=  \g_0^{\mu\nu} \pth{ \dr_\mu (F^{(1)}_\A)_{\b\nu} - \half \dr_\b (F^{(1)}_\A)_{\mu\nu} }  ,\label{Q 0}
\\ (\mathcal{I}^{(0,\pm)}_{\A\B})_{\a\b} & \vcentcolon= - \frac{1}{4}   \pth{ (F^{(1)}_\B)_{L_\A L_\A}   (F^{(1)}_\A)_{\alpha\beta} + (F^{(1)}_\A)_{L_\B L_\B}   (F^{(1)}_\B)_{\alpha\beta} }   + (P^{(0,\pm)}_{\A\B})_{\a\b} \label{I 0 pm}
\\&\quad +\dr_{(\a}(u_\A\pm u_\B)\bigg(  - \frac{1}{8} \dr_{\b)} (u_\A\pm u_\B) \left| F^{(1)}_\A\cdot F^{(1)}_\B\right|_{\g_0}  \non
\\&\hspace{4cm}  - \frac{1}{4} (F^{(1)}_\B)^{\nu}_{L_\A} (F^{(1)}_\A)_{\nu\b)} \mp \frac{1}{4} (F^{(1)}_\A)^{\nu}_{L_\B} (F^{(1)}_\B)_{\nu\b)} \bigg). \non
\end{align}
The term $R^{(1)}_{\a\b}$ is given by
\begin{align}
R^{(1)}_{\a\b} & = (R^{(1)}_{\mathrm{null}})_{\a\b} +  (R^{(1)}_{\mathrm{mixed}})_{\a\b} + \half  (\g_0)_{\rho(\a}(\dr_{\b)}\Upsilon^\rho)^{(-1)}  ,\label{R 1}
\end{align}
\begin{align}
2(R^{(1)}_{\mathrm{null}})_{\a\b} & = -\sum_\A \cos\pth{\frac{u_\A}{\lambda}} \bigg(   (\Ll_\A\F_\A)_{\alpha\beta} + (\Ll_\A F^{(2,1)}_\A)_{\alpha\beta} + (\h_\lambda)_{L_\A L_\A} (F^{(1)}_\A)_{\alpha\beta} \label{R 1 null}
\\&\hspace{5cm}    - \D_{(\a} \Pol{\F_\A+F^{(2,1)}_\A}{u_\A}_{\b)} +  (\tilde{R}^{(1,1)}_\A)_{\a\b}    \bigg) \non
\\&\quad + 2\sum_\A\sin\pth{\frac{2u_\A}{\lambda}} \bigg(    (\Ll_\A F^{(2,2)}_\A)_{\alpha\beta}  -  \D_{(\a} \Pol{F^{(2,2)}_\A}{u_\A}_{\b)} +  (\tilde{R}^{(1,2)}_\A)_{\a\b}   \bigg) \non
\\&\quad + \sum_\A\bigg( 2  \sin\pth{\frac{u_\A}{\la}}  \sin\pth{\frac{2u_\A}{\la}}  - \cos\pth{\frac{u_\A}{\lambda}} \cos\pth{ \frac{2u_\A}{\lambda} } \bigg) (F^{(2,2)}_\A)_{L_\A L_\A} (F^{(1)}_\A)_{\alpha\beta} \non
\\&\quad  - \sum_\A \sin\pth{\frac{2u_\A}{\la}} (\F_\A)_{L_\A L_\A} (F^{(1)}_\A)_{\a\b}  \non
\\&\quad +  \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}} \dr_{(\a}u \pth{  -  \Pol{\gg^{(3,h)}_{u,T} }{u}_{\b)} + (\tilde{R}^{(1,h,T,u)}_\A)_{\b)} }  , \non
\\ 2(R^{(1)}_{\mathrm{mixed}})_{\a\b} & = \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\} }} T\left( \frac{v}{\lambda} \right) \pth{ \g_0^{-1}(\d v, \d v) (\gg^{(3,e)}_{v,T})_{\a\b} + (\tilde{R}^{(1,e,T,v)}_\A)_{\a\b} }. \label{R 1 mixed}
\end{align}
\begin{align}
\tilde{R}^{(1,1)}_\A & =  \cro{\pth{F^{(2,\pm)} + (F^{(1)})^2 }F^{(1)} + \dr^{\leq 2} F^{(1)} + (F^{(1)})^2},\label{Rtilde 1 1}
\\ \tilde{R}^{(1,2)}_\A & = \cro{ \pth{ F^{(2,1)} + \dr^{\leq 1} F^{(1)} }F^{(1)} }_{\a\b},\label{Rtilde 1 2}
\\ \tilde{R}^{(1,h,T,u)}_\A & = \Big\{\dr F^{(2,1)} + \dr F^{(2,2)} \label{R 1 h}
\\&\hspace{1cm} +  \pth{ \F + F^{(2,1)} + F^{(2,2)} + F^{(2,\pm)} + (F^{(1)})^2 + \dr F^{(1)} } \pth{1+F^{(1)}} \Big\}_{\b)}, \non
\\ \tilde{R}^{(1,e,T,v)}_\A & =  \Big\{\dr^{\leq 1}F^{(2,\pm)} + (F^{(1)})^2 \label{R 1 e}
\\&\hspace{1cm}+ \pth{ \F + F^{(2,1)} + F^{(2,2)} + F^{(2,\pm)}  + (F^{(1)})^2 + \dr^{\leq 1} F^{(1)} }\pth{1+F^{(1)}} \Big\}.\non
\end{align}
The term $R^{(\geq 2)}_{\a\b}$ is given by
\begin{equation}\label{R 2}
\begin{aligned}
2R^{(\geq 2)}_{\a\b} & =  -\tBox_{\g_\la}(\h_\la)_{\a\b}  - \sum_\A \sin\pth{\frac{u_\A}{\la}}\tBox_{\g_\la}(\F_\A)_{\a\b}
\\&\quad -  \la \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}}\tBox_{\g_\la} (\gg^{(3,h)}_{u,T})_{\a\b}  - \la  \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}} T\left( \frac{v}{\lambda} \right)\tBox_{\g_\la} (\gg^{(3,e)}_{v,T})_{\a\b}
\\&\quad  + \tilde{R}^{(2)}_{\a\b} + \pth{ H^\rho \dr_\rho (\g_\la)_{\a\b} + (\g_\la)_{\rho(\a} \dr_{\b)}H^\rho }^{(\geq 2)}.
\end{aligned}
\end{equation}
\begin{equation}\label{R 2 w T}
\begin{aligned}
\tilde{R}^{(2)}_{\a\b} & = \Big\{\pth{ \g_\la^{-1}\g_\la^{-1}\dr\g_\la\dr\g_\la }^{(\geq 2)}  + \g_\la^{-1}\Big(\dr^{\leq 1}\gg^{(3,h)} + \dr^{\leq 1}\gg^{(3,e)} + \dr^{\leq 2} F^{(1)} + \dr^{\leq 1} \F
\\&\hspace{6cm}+ \dr^{\leq 2}F^{(2,1)} + \dr^{\leq 2}F^{(2,2)} + \dr^{\leq 2} F^{(2,\pm)} \Big)  \Big\}_{\a\b}^\osc .
\end{aligned}
\end{equation}
\end{prop}

In Section \ref{section hierarchy} and \ref{section reformulation} below we will precisely state the hierarchy of equations and relations that will ensure $R_(_)=0$. Before that, we briefly discuss the mechanisms at stake, following the comments made after Lemmas \ref{lemma W}, \ref{lemma P} and \ref{lemma H}:
\begin{enumerate}
\item \textbf{Admissible terms.} The admissible terms at order $\la^0$ and $\la^1$, i.e the $\sin\pth{\frac{u_\A}{\la}}$ terms in \eqref{R 0 null} and the $\cos\pth{\frac{u_\A}{\la}}$ and $\sin\pth{\frac{2u_\A}{\la}}$ terms in \eqref{R 1 null} can be set to zero by imposing transport equations for $F^{(1)}_\A$, $\F_\A$, $F^{(2,1)}_\A$ and $F^{(2,2)}_\A$. The seemingly redundant presence of $\F_\A$ is justified by the term $(\h_\lambda)_{L_\A L_\A} (F^{(1)}_\A)_{\alpha\beta}$ in \eqref{R 1 null}, which will be absorbed by the RHS of the transport equation for $\F_\A$.
\item \textbf{Forbidden terms at order $\la^0$.} In \eqref{R 0 null}, the forbidden terms in $\cos\pth{\frac{2u_\A}{\la}}$ can be set to zero by imposing a polarization condition for $F^{(2,2)}_\A$. This is only possible if the forbidden term that need to be absorbed has the tensorial structure $\dr_{(\a}u_\A Q_{\b)}$ for some 1-form $Q$. Here, this is even better since $Q=\d u_\A$ (recall \eqref{P 0 2})! If the polarization condition for $F^{(2,2)}_\A$ holds, this extra bit of structure would imply $(F^{(2,2)}_\A)_{L_\A L_\A}=0$ (where we used \eqref{Pol L}).
\item \textbf{Forbidden terms at order $\la^1$.} In \eqref{R 1 null}, two types of forbidden terms occur. First, the term $(F^{(2,2)}_\A)_{L_\A L_\A} (F^{(1)}_\A)_{\alpha\beta}$ coming from the quasi-linear wave operator oscillates like $\cos\pth{\frac{3u_\A}{\la}}$. However, the previous comment precisely says that it vanishes thanks to the structure of the semi-linear forbidden terms that the polarization condition for $F^{(2,2)}_\A$ is absorbing. Second, the term $\dr_{(\a}u(\tilde{R}^{(1,h,T,u)}_\A)_{\b)}$ precisely has the required structure to be absorbed by a polarization condition for $\gg^{(3,h)}_{u,T}$.
\item \textbf{Mixed harmonics at order $\la^0$.} The term $R^{(0)}_{\mathrm{mixed}}$ given by \eqref{R 0 mixed} will be simply canceled by asking $F^{(2,\pm)}_{\A\B}$ to solve
\begin{align*}
\mathcal{P}_{u_\A\pm u_\B}\pth{ F^{(2,\pm)}_{\A\B}} = \mathcal{I}^{(0,\pm)}_{\A\B}.
\end{align*}
Since $\g_0^{-1}(\d(u_\A\pm u_\B),\d(u_\A\pm u_\B))\neq 0$ (recall \eqref{coherence}), Lemma \ref{lem Pv} shows that this equation is solvable if and only if $\Pol{\mathcal{I}^{(0,\pm)}_{\A\B}}{u_\A\pm u_\B}=0$. We will check this in Section \ref{section ensuring R 0} but we draw the reader's attention on the importance of this simple algebraic fact which is a reminder of Einstein vacuum equations' transparency.
\item \textbf{Mixed harmonics at order $\la^1$.} The structure of $R^{(1)}_{\mathrm{mixed}}$ differs from the one of $R^{(0)}_{\mathrm{mixed}}$ since instead of having the operator $\PP_v$ acting on $\gg^{(3,e)}_{v,T}$, we only have the part of $\PP_v$ coming from the wave part of the Ricci tensor, i.e $-\g_0^{-1}(\d v,\d v)\mathrm{Id}$. Opposite to $\PP_v$, this operator has full range and no condition on the RHS needs to hold, which simplifies greatly our construction since we don't need the exact expression of $\tilde{R}^{(1,e,T,v)}_\A$. The absence of the other half of $\PP_v$ is due to our choice of generalised wave gauge, i.e our choice of $\Upsilon^\rho$ (recall the second comment after Lemma \ref{lemma H}). We see here how the gauge helps us recover true ellipticity when considering mixed harmonics, as it usually does help us recover true hyperbolicity when solving \eqref{EVE}.
\end{enumerate}

\subsection{The hierarchy}\label{section hierarchy}

In this section, following the general mechanisms presented after Proposition \ref{prop expression of the ricci}, we derive a hierarchy of equations and relations for the various terms in $_$ which will ensure $R_ (_) =0$ by ensuring $R^(0)_=0$, $R^(1)_=0$ and $R^(≥2)_=0$.

\subsubsection{Ensuring $R^{(0)}_{\a\b}=0$}\label{section ensuring R 0}

Looking at \eqref{R 0}, we first want the non-oscillating terms to vanish. For that, we impose the following backreaction condition for $F^(1)_$:
\begin{align*}
\left| F^{(1)}_\A \right|^2_{\g_0} & = 8 F_\A^2.
\end{align*}
We wish to have $R^(0)_null=0$. Looking at \eqref{R 0 null}, we cancel the admissible harmonics by imposing the following transport equation for $F^(1)_$
\begin{align*}
\Ll_\A F^{(1)}_\A & = 0,
\end{align*}
as well as a polarization condition for $_$ and $F^(2,1)_$
\begin{align*}
\Pol{\F_\A }{u_\A} & = 0,
\\ \Pol{F^{(2,1)}_\A}{u_\A} & = - Q^{(0)}_\A,
\end{align*}
where $Q^(0)_$ is defined in \eqref{Q 0} (this is not clear that \eqref{Q 0} actually defines a tensor, see Remark \ref{remark Q0} for a justification of this fact). We cancel the forbidden harmonics in $R^(0)_null$ by imposing a polarization condition for $F^(2,2)_$
\begin{align*}
\Pol{F^{(2,2)}_\A}{u_\A} & = -  \frac{3}{32} \left| F^{(1)}_\A \right|^2_{\g_0} \d u_\A.
\end{align*}
For convenience, we define the following tensors
\begin{align}
V^{(2,1)}_\A & \vcentcolon=  \Pol{F^{(2,1)}_\A}{u_\A} + Q^{(0)}_\A,\label{def 1 V 2 1}
\\ V^{(2,2)}_\A & \vcentcolon= \Pol{F^{(2,2)}_\A}{u_\A} +  \frac{3}{32} \left| F^{(1)}_\A \right|^2_{\g_0} \d u_\A,\label{def 1 V 2 2}
\end{align}
so that the polarization conditions for $F^(2,1)_$ and $F^(2,2)_$ rewrite
\begin{align}
V^{(2,1)}_\A & = 0,\label{premiere V 2 1 = 0}
\\ V^{(2,2)}_\A & = 0.\label{premiere V 2 2 = 0}
\end{align}

\begin{remark}\label{remark Q0}
Note that $Q^{(0)}_\A$ defined by \eqref{Q 0} is actually a tensor. More generally, one can check that
\begin{align*}
S^{\mu\nu} \pth{ \dr_\mu T_{\si\nu} - \half \dr_\si T_{\mu\nu} } & = S^{\mu\nu} \pth{ \D_\mu T_{\si\nu} - \half \D_\si T_{\mu\nu} + \Ga(\g)^\rho_{\mu\nu} T_{\si\rho}}
\end{align*}
so that
\begin{align*}
S^{\mu\nu} \pth{ \dr_\mu T_{\si\nu} - \half \dr_\si T_{\mu\nu} }
\end{align*}
is the $\dr_\si$ component of a 1-form. In particular, $V^{(2,1)}_\A$ and $V^{(2,2)}_\A$ defined in \eqref{def 1 V 2 1} and \eqref{def 1 V 2 2} are tensors so that the polarization conditions \eqref{premiere V 2 1 = 0} and \eqref{premiere V 2 2 = 0} are not gauge conditions.
\end{remark}

We cancel the mixed term $R^(0)_mixed$ given by \eqref{R 0 mixed} by imposing the following equation for $F^(2,±)_$
\begin{align}
\PP_{u_\A\pm u_\B}\pth{F^{(2,\pm)}_{\A\B}} & = \mathcal{I}^{(0,\pm)}_{\A\B}, \label{eq F2pm first}
\end{align}
where $ℐ^(0,±)_$ is defined by \eqref{I 0 pm}. Thanks to Lemma \ref{lem Pv}, this equation is solvable if and only its RHS satisfies some polarization condition.

\begin{lemma}\label{lem F2pm}
We assume that \eqref{assumption} holds. If we define
\begin{align*}
F^{(2,\pm)}_{\A\B} & \vcentcolon = -\frac{\mathcal{I}^{(0,\pm)}_{\A\B} }{\g_0^{-1}(\d(u_\A\pm u_\B),\d(u_\A\pm u_\B))},
\end{align*}
then \eqref{eq F2pm first} holds.
\end{lemma}

\begin{proof}
According to Lemma \ref{lem Pv} it suffices to show that
\begin{align}
\Pol{ \mathcal{I}_{\A\B}^{(0,\pm)} }{ u_\A\pm u_\B } = 0.\label{Pol I 0 pm}
\end{align}
For clarity we define
\begin{align*}
A_{\a\b} & \vcentcolon =  - \frac{1}{4}   \pth{ (F^{(1)}_\B)_{L_\A L_\A}   (F^{(1)}_\A)_{\alpha\beta} + (F^{(1)}_\A)_{L_\B L_\B}   (F^{(1)}_\B)_{\alpha\beta} },
\\ B_{\a\b} & \vcentcolon = \dr_{(\a}(u_\A\pm u_\B)\bigg(  - \frac{1}{8} \dr_{\b)} (u_\A\pm u_\B) \left| F^{(1)}_\A\cdot F^{(1)}_\B\right|_{\g_0}
- \frac{1}{4} (F^{(1)}_\B)^{\nu}_{L_\A} (F^{(1)}_\A)_{\nu\b)} \mp \frac{1}{4} (F^{(1)}_\A)^{\nu}_{L_\B} (F^{(1)}_\B)_{\nu\b)} \bigg).
\end{align*}
so that $ (\mathcal{I}^{(0,\pm)}_{\A\B})_{\a\b} = A_{\a\b}  + B_{\a\b} + (P^{(0,\pm)}_{\A\B})_{\a\b}$. Thanks to \eqref{assumption} we first have
\begin{align*}
\Pol{A}{u_\A\pm u_\B}_\a & =   \frac{1}{4}   \pth{ \pm(F^{(1)}_\B)_{L_\A L_\A}   (F^{(1)}_\A)_{\alpha L_\B} + (F^{(1)}_\A)_{L_\B L_\B}   (F^{(1)}_\B)_{\alpha L_\A} }.
\end{align*}
For $B_{\a\b}$, \eqref{Pol dv S} gives
\begin{align*}
\Pol{B}{u_\A\pm u_\B}_\a & = \mp \half \g_0^{-1}(\d u_\A, \d u_\B)   \bigg( (F^{(1)}_\B)^{\nu}_{L_\A}  (F^{(1)}_\A)_{\a\nu} \pm (F^{(1)}_\A)^{\nu}_{L_\B} (F^{(1)}_\B)_{\a\nu}
\\&\hspace{5cm} + \half  \left| F^{(1)}_\A \cdot F^{(1)}_\B \right|_{g_0}  \dr_\a (u_\A\pm u_\B)  \bigg).
\end{align*}
For $(P^{(0,\pm)}_{\A\B})_{\a\b}$, \eqref{P 0 pm} and \eqref{assumption} give
\begin{align*}
\Pol{P^{(0,\pm)}_{\A\B}}{u_\A \pm u_\B}_\a & =  - (P^{(0,\pm)}_{\A\B})_{\a (L_\A \pm L_\B)} + \half \tr_{\g_0}P^{(0,\pm)}_{\A\B} ((L_\A)_\a \pm (L_\B)_\a)
\\& = \pm \frac{1}{4} \dr_{\a} (u_\A\pm u_\B) \g_0^{-1}(\d u_\A,\d u_\B) \left| F^{(1)}_\A\cdot F^{(1)}_\B \right|_{\g_0}
\\&\quad + \frac{1}{2}\g_0^{-1}(\d u_\A,\d u_\B) \pth{ (F^{(1)}_\A)_{L_\B\si} (F^{(1)}_\B)_{\a}^\si  \pm (F^{(1)}_\B)_{L_\A\si} (F^{(1)}_\A)_{\a}^\si }
\\&\quad - \frac{1}{4} (F^{(1)}_\A)_{ L_\B L_\B} (F^{(1)}_\B)_{\a L_\A}  \mp\frac{1}{4} (F^{(1)}_\A)_{\alpha L_\B} (F^{(1)}_\B)_{L_\A L_\A}
\\& = - \Pol{A}{u_\A\pm u_\B}_\a - \Pol{B}{u_\A\pm u_\B}_\a,
\end{align*}
which concludes the proof of \eqref{Pol I 0 pm} and thus of the lemma.
\end{proof}

\begin{remark}
If one is interested in the strict equivalent to Choquet-Bruhat's singlephase approximate construction from [2] in the multiphase case, the above conditions are sufficient since they indeed lead to $R_{\a\b}(\g_\la)=\GO{\la}$.
\end{remark}

\subsubsection{Ensuring $R^{(1)}_{\a\b}=0$}\label{section ensuring R 1}

Looking at \eqref{R 1}, we wish to have $R^(1)_null=0$ and $R^(1)_mixed=0$. We first consider $R^(1)_null$ and rewrite it with the help of the tensors $V^(2,1)_$ and $V^(2,2)_$ defined by \eqref{def 1 V 2 1} and \eqref{def 1 V 2 2}. Thanks to \eqref{R 1 null} we obtain
\begin{equation}\label{R 1 null bis}
\begin{aligned}
2(R^{(1)}_{\mathrm{null}})_{\a\b} & = -\sum_\A \cos\pth{\frac{u_\A}{\lambda}} \bigg(   (\Ll_\A\F_\A)_{\alpha\beta} + (\Ll_\A F^{(2,1)}_\A)_{\alpha\beta} + (\h_\lambda)_{L_\A L_\A} (F^{(1)}_\A)_{\alpha\beta} +  (\tilde{R}^{(1,1)}_\A)_{\a\b}
\\&\hspace{5cm}   - \D_{(\a} \Pol{\F_\A}{u_\A}_{\b)}  - \D_{(\a} (V^{(2,1)}_\A)_{\b)}  + \D_{(\a} (Q^{(0)}_\A)_{\b)}     \bigg)
\\&\quad + 2\sum_\A\sin\pth{\frac{2u_\A}{\lambda}} \bigg(    (\Ll_\A F^{(2,2)}_\A)_{\alpha\beta}  -  \D_{(\a} (V^{(2,2)}_\A)_{\b)}
\\&\hspace{4.5cm} + \frac{3}{32}\D_{(\a} \pth{ \left| F^{(1)}_\A \right|^2_{\g_0}\d u_\A}_{\b)} +  (\tilde{R}^{(1,2)}_\A)_{\a\b}   \bigg)
\\&\quad - \sum_\A\bigg( 2  \sin\pth{\frac{u_\A}{\la}}  \sin\pth{\frac{2u_\A}{\la}}  - \cos\pth{\frac{u_\A}{\lambda}} \cos\pth{ \frac{2u_\A}{\lambda} } \bigg) (V^{(2,2)}_\A)_{L_\A} (F^{(1)}_\A)_{\alpha\beta}
\\&\quad  - \sum_\A \sin\pth{\frac{2u_\A}{\la}} (\F_\A)_{L_\A L_\A} (F^{(1)}_\A)_{\a\b}
\\&\quad +  \sum_{u\in\mathcal{N},T\in\{\cos,\sin\}}T\pth{\frac{u}{\la}} \dr_{(\a}u \pth{  -  \Pol{\gg^{(3,h)}_{u,T} }{u}_{\b)} + (\tilde{R}^{(1,h,T,u)}_\A)_{\b)} }  .
\end{aligned}
\end{equation}
Since the above polarization conditions would imply $V^(2,1)_=V^(2,2)_=_u_=0$, we cancel the admissible harmonics in $(R^(1)_null)_$ by imposing transport equations for $F^(2,1)_$ and $F^(2,2)_$:
\begin{align}
(\Ll_\A F^{(2,1)}_\A)_{\a\b} & = -\D_{(\a} (Q^{(0)}_\A)_{\b)} -  (\tilde{R}^{(1,1)}_\A)_{\a\b} , \label{premiere eq F 2 1}
\\ (\Ll_\A F^{(2,2)}_\A)_{\alpha\beta}  & = - \frac{3}{32}\D_{(\a} \pth{ \left| F^{(1)}_\A \right|^2_{\g_0}\d u_\A}_{\b)} -  (\tilde{R}^{(1,2)}_\A)_{\a\b}, \label{premiere eq F 2 2}
\end{align}
The equation for $_$ needs to be adjusted so that the coupling with the equation for $_$ leads to a well-posed system in the high-frequency limit $→0$. As in [28] we impose the following transport equation
\begin{align}
\Ll_\A\F_\A & = - \Pi_\leq \pth{ (\h_\lambda)_{L_\A L_\A} }F^{(1)}_\A ,\label{premiere eq F}
\end{align}
where the operator $Π_≤$ is defined by
\begin{align*}
\Pi_\leq (f) = \mathcal{F}^{-1}\pth{\chi_\la \mathcal{F}(f)},
\end{align*}
where $χ_(ξ)=χ(ξ)$ for $χ:^3⟶[0,1]$ a smooth function supported in ${|ξ|≤2}$ and such that $χ_|_{|ξ|≤1}=1$ and where $ℱ$ is the standard Fourier transform on $^3$. In order to cancel the non-tangential terms containing forbidden harmonics we impose a polarization condition for $≫^(3,h)_u,T$
\begin{align}
\Pol{\gg^{(3,h)}_{u,T} }{u}& = \tilde{R}^{(1,h,T,u)}_\A. \label{premiere pola g 3 h}
\end{align}
In order to cancel $R^(1)_mixed$ given by \eqref{R 1 mixed}, we simply define $≫^(3,e)_v,T$ by
\begin{align}
\gg^{(3,e)}_{v,T} & = - \frac{1}{\g_0^{-1}(\d v, \d v)} \tilde{R}^{(1,e,T,v)}_\A,\label{premiere def g 3 e}
\end{align}
where we recall that if $v∈ℐ$, then $_0^-1(ṿ, ṿ)≠0$ (see Section \ref{section BG}).

\subsubsection{Ensuring $R^{(\geq 2)}_{\a\b}=0$}\label{section ensuring R 2}

Thanks to its expression given by \eqref{R 2}, in order to cancel $R^(≥2)_$ it suffices to impose a wave equation for the remainder $_$. This equation needs to be adjusted to handle the coupling with the equation for $_$, and we also need to set aside the problematic derivatives all contained in $Υ^ρ$. Therefore we impose the following wave equation for $_$
\begin{align*}
\tBox_{\g_\la}(\h_\la)_{\a\b}  & = - \sum_\A \sin\pth{\frac{u_\A}{\la}}\tBox_{\g_\la}(\F_\A)_{\a\b} -  \la \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}}\tBox_{\g_\la} (\gg^{(3,h)}_{u,T})_{\a\b}
\\&\quad  - \la  \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}} T\left( \frac{v}{\lambda} \right)\tBox_{\g_\la} (\gg^{(3,e)}_{v,T})_{\a\b} + \tilde{R}^{(2)}_{\a\b}  +\pth{ \mathring{H}^\rho \dr_\rho (\g_\la)_{\a\b} + (\g_\la)_{\rho(\a} \dr_{\b)}\mathring{H}^\rho }^{(\geq 2)}
\\&\quad - \frac{1}{\la}\sum_\A \cos\pth{\frac{u_\A}{\la}}\Pi_\geq\pth{(\h_\lambda)_{L_\A L_\A}}(F^{(1)}_\A)_{\a\b},
\end{align*}
where the operator $Π_≥$ is defined by $Π_≥=Id-Π_≤$. We also impose the following generalised wave gauge condition
\begin{align*}
\Upsilon^\rho & = 0 .
\end{align*}

\subsection{Reformulation of Theorem \ref{theo evol}}\label{section reformulation}

The conclusion of Sections \ref{section ensuring R 0}, \ref{section ensuring R 1} and \ref{section ensuring R 2} is the following: if the tensors $F^(1)_$, $_$, $F^(2,1)_$, $F^(2,2)_$ and $_$ solve the system
\begin{align}
\Ll_\A F^{(1)}_\A & = 0,\label{eq F1}
\\ (\Ll_\A F^{(2,1)}_\A)_{\a\b} & = -\D_{(\a} (Q^{(0)}_\A)_{\b)} -  (\tilde{R}^{(1,1)}_\A)_{\a\b} , \label{eq F21}
\\ (\Ll_\A F^{(2,2)}_\A)_{\alpha\beta}  & = - \frac{3}{32}\D_{(\a} \pth{ \left| F^{(1)}_\A \right|^2_{\g_0}\d u_\A}_{\b)} -  (\tilde{R}^{(1,2)}_\A)_{\a\b},\label{eq F22}
\\ \Ll_\A\F_\A & = - \Pi_\leq \pth{ (\h_\lambda)_{L_\A L_\A} }F^{(1)}_\A,\label{eq F}
\\ \tBox_{\g_\la}(\h_\la)_{\a\b} & = - \sum_\A \sin\pth{\frac{u_\A}{\la}}\tBox_{\g_\la}(\F_\A)_{\a\b} -  \la \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}}\tBox_{\g_\la} (\gg^{(3,h)}_{u,T})_{\a\b} \label{eq h}
\\&\quad  - \la  \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}} T\left( \frac{v}{\la} \right)\tBox_{\g_\la} (\gg^{(3,e)}_{v,T})_{\a\b} + \tilde{R}^{(2)}_{\a\b}  +\pth{ \mathring{H}^\rho \dr_\rho (\g_\la)_{\a\b} + (\g_\la)_{\rho(\a} \dr_{\b)}\mathring{H}^\rho }^{(\geq 2)}\non
\\&\quad - \frac{1}{\la}\sum_\A \cos\pth{\frac{u_\A}{\la}}\Pi_\geq\pth{(\h_\la)_{L_\A L_\A}}(F^{(1)}_\A)_{\a\b},\non
\end{align}
together with the polarization conditions
\begin{align}
\Pol{F^{(1)}_\A}{u_\A} & = 0,\label{pola F1}
\\ F^{(1)}_{L_\A \Lb_\A} & = 0,\label{pola F1 bis}
\\ \Pol{\F_\A}{u_\A} & = 0,\label{pola F}
\\ V^{(2,1)}_\A & = 0,\label{pola F21}
\\ V^{(2,2)}_\A & = 0,\label{pola F22}
\\ \Pol{\gg^{(3,h)}_{u,T} }{u} & =  \tilde{R}^{(1,h,T,u)}_\A,\label{pola g3h}
\end{align}
the backreaction condition
\begin{align}
\left| F^{(1)}_\A \right|^2_{\g_0} = 8 F_\A^2 \label{energy F1},
\end{align}
the generalised wave gauge condition
\begin{align}
\Upsilon^\rho & =0,\label{GWC}
\end{align}
with $F^(2,±)_$ and $≫^(3,e)_v,T$ defined by
\begin{align}
(F^{(2,\pm)}_{\A\B})_{\a\b} & \vcentcolon = - \frac{(\mathcal{I}^{(0,\pm)}_{\A\B})_{\a\b} }{\g_0^{-1}(\d(u_\A\pm u_\B),\d(u_\A\pm u_\B))}, \label{def F2pm}
\\ \gg^{(3,e)}_{v,T} & \vcentcolon = - \frac{1}{\g_0^{-1}(\d v, \d v)} \tilde{R}^{(1,e,T,v)}_\A, \label{def g3e}
\end{align}
then the metric $_$ defined by \eqref{expansion metric} solves the Einstein vacuum equations \eqref{EVE}. The initial data for the transport-wave system \eqref{eq F1}-\eqref{eq h} consist in initial values on $_0$ for
\begin{align*}
(F^{(1)}_\A)_{\a\b},\quad (F^{(2,1)}_\A)_{\a\b},\quad (F^{(2,2)}_\A)_{\a\b},\quad (\F_\A)_{\a\b},\quad (\h_\la)_{\a\b} \quad \text{and} \quad \T_\la (\h_\la)_{\a\b},
\end{align*}
where $_$ is the future-directed unit normal to $_0$ for $_$ (which can be computed from $_|__0$). Therefore, the task of proving Theorem \ref{theo evol} is divided into three steps:
\begin{enumerate}
\item In Section \ref{section initial data}, we construct initial data such that the algebraic conditions \eqref{pola F1}-\eqref{GWC} and the constraint equations are satisfied on $\Si_0$.
\item In Section \ref{section solving the system}, we solve the system \eqref{eq F1}-\eqref{eq h} on $[0,1]\times \R^3$.
\item In Section \ref{section propagation}, we show that the algebraic conditions \eqref{pola F1}-\eqref{GWC} actually hold on the full spacetime $[0,1]\times \R^3$.
\end{enumerate}

\begin{remark}\label{remark F2pm}
Thanks to \eqref{def F2pm}, \eqref{I 0 pm} and \eqref{P 0 pm} we have $F^{(2,\pm)}_{\A\B} = \cro{ (F^{(1)})^2 }$. This would be used without mention in the rest of this article.
\end{remark}

\begin{remark}\label{remark g3h}
The tensors $\gg^{(3,h)}_{u,T}$ only need to satisfy the polarization condition \ref{pola g3h} and in particular are not obtained by solving partial differential equations. Therefore, we can simply define $\gg^{(3,h)}_{u,T}$ for $u=ku_\A$ by setting
\begin{align*}
(\gg^{(3,h)}_{u,T})_{L_\A L_\A} & \vcentcolon = - \frac{1}{k} (\tilde{R}^{(1,h,T,u)}_\A)_{L_\A},
\\ (\gg^{(3,h)}_{u,T})_{L_\A e^{(\i)}_\A} & \vcentcolon = - \frac{1}{k} (\tilde{R}^{(1,h,T,u)}_\A)_{e^{(\i)}_\A},
\\ (\gg^{(3,h)}_{u,T})_{e^{(1)}_\A e^{(1)}_\A} & \vcentcolon = - \frac{1}{k} (\tilde{R}^{(1,h,T,u)}_\A)_{\Lb_\A},
\end{align*}
and by setting all the other components of $\gg^{(3,h)}_{u,T}$ in the frame $\pth{L_\A,\Lb_\A,e^{(1)}_\A,e^{(2)}_\A}$ to 0. In particular, thanks to \eqref{R 1 h} this implies
\begin{align}
\gg^{(3,h)}_{u,T} & = \Big\{\dr F^{(2,1)} + \dr F^{(2,2)} +  \pth{ \F + F^{(2,1)} + F^{(2,2)} + (F^{(1)})^2 + \dr F^{(1)} } \pth{1+F^{(1)}} \Big\} , \label{g3h}
\end{align}
where we already applied Remark \ref{remark F2pm} and replaced $F^{(2,\pm)}$ by $(F^{(1)})^2$ in the schematic notation.
\end{remark}

\section{Construction of initial data}\label{section initial data}

In this section, we construct initial data for the system \eqref{eq F1}-\eqref{eq h} on $_0$.  They need to be such that the algebraic conditions \eqref{pola F1}-\eqref{GWC} hold on $_0$ and also such that the constraint equations are satisfied by the induced metric on $_0$ and the second fundamental form of $_0$. These equations are
\begin{equation}\label{constraint equations}
\begin{aligned}
R(g) - |k|^2_g + (\tr_g k)^2 & = 0,
\\ \div_g k - \d \tr_g k & = 0,
\end{aligned}
\end{equation}
where $R(g)$ is the scalar curvature of $g$.

\subsection{From a seed to solutions to the constraint equations}

The construction of initial data is based on a seed, with which we first construct a family of solutions to the constraint equations and then pick a particular element of this family to define appropriate initial data for the system \eqref{eq F1}-\eqref{eq h}.

\begin{mydef}\label{def seed}
A seed is a family $\pth{ \bar{F}^{(1)}_\A}_{\A\in\mathcal{A}}$ of symmetric 2-tensors on $\Si_0$ such that for all $\A\in\mathcal{A}$ we have
\begin{align}\label{seed form}
\bar{F}^{(1)}_\A = F_\A \pth{ \th^+_\A \pth{ e^{(1),\flat}_\A\otimes e^{(1),\flat}_\A - e^{(2),\flat}_\A\otimes e^{(2),\flat}_\A } + \th^\times_\A \pth{ e^{(1),\flat}_\A\otimes e^{(2),\flat}_\A + e^{(2),\flat}_\A\otimes e^{(1),\flat}_\A }  },
\end{align}
where $e^{(\i),\flat}_\A$ denotes the 1-form canonically associated to the vector field $e^{(\i)}_\A$ by $g_0$ and where $\pth{ \th^+_\A, \th^\times_\A}$ are numbers satisfying $\pth{\th^+_\A}^2  + \pth{\th^\times_\A}^2  = 4$.
\end{mydef}

The expression \eqref{seed form} directly implies that $F̅^(1)_$ satisfies
\begin{equation}\label{assum F1}
\begin{aligned}
\tr_{g_0}\bar{F}^{(1)}_\A & = 0,
\\ (\bar{F}^{(1)}_\A)_{N_\A i} & = 0,
\\ \left|\bar{F}^{(1)}_\A\right|^2_{g_0} & = 8F_\A^2.
\end{aligned}
\end{equation}
Moreover, the assumptions made on $F_$ in Section \ref{section BG} imply that $ F̅^(1)_$ is supported in $B_R$ and satisfies
\begin{align}\label{estim seed}
\l  \bar{F}^{(1)}_\A \r_{H^N(\Si_0)} \lesssim \e.
\end{align}

\begin{remark}\label{remark seed}
Note the analogy between \eqref{seed form} and the TT gauge of linearized gravity, where $\th^+_\A$ and $\th^\times_\A$ would play the role as the coefficients of the two possible polarizations for the gravitational wave.
\end{remark}

\noindent Before stating our main result on the constraint equations, we introduce two operators acting on the space of symmetric 2-tensors on $_0$:
\begin{align}
\bar{\PP}^{[1]}_v(S) & = S - \frac{1}{2} \pth{ \tr_{g_0}S - S_{N_vN_v} }g_0,\label{def P1(T)}
\\ \bar{\PP}^{[2]}_v(S)_{ij} & = S_{ij}  +  (N_v)_{(i}  \pth{(N_v)_{j)} \tr_{g_0}S - S_{N_v j)}} - \half \pth{ \tr_{g_0}S - S_{N_v N_v}} (g_0)_{ij}, \label{def P2(T)}
\end{align}
where $v$ is a scalar function on $_0$ such that $|_g_0 v|_g_0≠0$ and
\begin{align*}
N_v \vcentcolon =\frac{\nab_{g_0} v}{|\nab_{g_0} v|_{g_0}}.
\end{align*}
Note that $N_=N_u_$ and $N^(±)_=N_u_±u_$ where $N_$ and $N^(±)_$ are defined in \eqref{def vector fields init}. Next lemma gives important properties of these operators which will be used in Section \ref{section initial data final}, its proof is left to the reader.

\begin{lemma}\label{lem P 1et2}
Let $v$ a scalar function on $\Si_0$ such that $|\nab_{g_0} v|_{g_0}\neq 0$. We have $\bar{\PP}^{[1]}_v\circ\bar{\PP}^{[1]}_v=\bar{\PP}^{[1]}_v$ and $\bar{\PP}^{[2]}_v\circ\bar{\PP}^{[2]}_v=\bar{\PP}^{[2]}_v$. Moreover we have
\begin{align}
\mathrm{ran}\bar{\PP}^{[1]}_v & =  \enstq{S \text{ symmetric 2-tensors on $\Si_0$}}{\tr_{g_0}S = S_{N_vN_v}} ,\label{image P1}
\\ \mathrm{ran}\bar{\PP}^{[2]}_v & =  \enstq{S \text{ symmetric 2-tensors on $\Si_0$}}{(\tr_{g_0}S) (N_v)_i  = S_{iN_v}} .\label{image P2}
\end{align}
\end{lemma}

\noindent Finally, if $X$ and $N$ are two vector fields on $_0$ we define
\begin{align}
(N\tilde{\otimes}X)_{ij} \vcentcolon= N_{(i} X_{j)} - \half X_N (g_0)_{ij},\label{def produit}
\end{align}
where $X_N= g_0(X,N)$. We are now ready to state our main result on the constraint equations.

\begin{prop}\label{prop constraint main}
Let $\pth{ \bar{F}^{(1)}_\A}_{\A\in\mathcal{A}}$ be a seed. Let $\pth{\ka^{(1,1)}_\A, \ka^{(1,2)}_\A}_{\A\in\mathcal{A}}$ and $\pth{\ga^{(2,\pm)}_{\A\B},\ka^{(1,\pm)}_{\A\B}}_{\A,\B\in\mathcal{A}, \A\neq \B}$ be families of symmetric 2-tensors on $\Si_0$ supported in $B_R$ with the symmetry property $\ga^{(2,\pm)}_{\A\B}=\ga^{(2,\pm)}_{\B\A}$ and $\ka^{(1,\pm)}_{\A\B}=\ka^{(1,\pm)}_{\B\A}$ and such that
\begin{align}
\l \ga^{(2,\pm)}_{\A\B} \r_{H^N(\Si_0)} + \l  \ka^{(1,1)}_\A\r_{H^{N-1}(\Si_0)} + \l  \ka^{(1,2)}_\A \r_{H^{N-1}(\Si_0)} + \l  \ka^{(1,\pm)}_{\A\B} \r_{H^{N-1}(\Si_0)} \leq \e,\label{estim ga kappa param}
\end{align}
If $\e$ is small enough, there exists $(g_\la,k_\la)$ a solution of the constraint equations on $\Si_0$ of the form
\begin{equation}\label{g la}
\begin{aligned}
g_\la & = g_0 + \la \sum_\A \cos\pth{ \frac{u_\A}{\la}} \bar{F}^{(1)}_\A +  \lambda^2 \sum_\A \sin\pth{\frac{u_\A}{\la}}  4\wc{\ffi}^{(2,1)}_{\A} g_0  +  \lambda^2 \sum_\A \cos\pth{\frac{2u_\A}{\la}} \frac{3}{4} F_\A^2  g_0
\\&\quad + \la^2\sum_{\A\neq\B,\pm}\cos\pth{\frac{u_\A\pm u_\B}{\la}}\pth{ \bar{\PP}^{[1]}_{u_\A\pm u_\B}\pth{\ga^{(2,\pm)}_{\A\B}}  + 4\wc{\ffi}^{(2,\pm)}_{\A\B}g_0  } + \la^2 h_\la
\end{aligned}
\end{equation}
\begin{equation}\label{k la}
\begin{aligned}
k_\la & = k_0 + \half \sum_\A \sin\pth{ \frac{u_\A}{\la}} |\nab_{g_0} u_\A|_{g_0}\bar{F}^{(1)}_\A
\\&\quad + \la\sum_\A \cos\pth{\frac{u_\A}{\la}} \pth{ \bar{\PP}^{[2]}_{u_\A}\pth{\ka^{(1,1)}_\A} +  \frac{N_\A\tilde{\otimes} \wc{X}^{(2,1)}_\A}{|\nab_{g_0} u_\A|_{g_0}} }
\\&\quad + \la\sum_\A \sin\pth{\frac{2u_\A}{\la}} \pth{ \bar{\PP}^{[2]}_{u_\A}\pth{\ka^{(1,2)}_\A}  -\frac{3}{2} |\nab_{g_0} u_\A|_{g_0} F_\A^2 N_\A\tilde{\otimes} N_\A  }
\\&\quad + \la\sum_{\A\neq\B,\pm}\sin\pth{\frac{u_\A\pm u_\B}{\la}} \pth{  \bar{\PP}^{[2]}_{u_\A\pm u_\B}\pth{\ka^{(1,\pm)}_{\A\B}} - \frac{N^{(\pm)}_{\A\B}\tilde{\otimes} \wc{X}^{(2,\pm)}_{\A\B}}{|\nab_{g_0}\pth{ u_\A\pm u_\B}|_{g_0}} } + \la^2 \tilde{k}^{cons}_\la,
\end{aligned}
\end{equation}
where $\wc{\ffi}^{(2,1)}_{\A}$, $\wc{\ffi}^{(2,\pm)}_{\A\B}$, $\wc{X}^{(2,1)}_\A$ and $\wc{X}^{(2,\pm)}_{\A\B}$ are defined by
\begin{align}
\wc{\ffi}^{(2,1)}_{\A} & \vcentcolon = \frac{1}{8|\nab_{g_0} u_\A|_{g_0}^2}(\bar{F}^{(1)}_\A)_{ij} \pth{- \dr^i \dr^j u_\A + |\nab_{g_0} u_\A|_{g_0}k_0^{ij} + \half  \dr_{\ell} u_\A  \dr^\ell g_0^{ij} },  \label{wcffi 21}
\\ \wc{\ffi}^{(2,\pm)}_{\A\B} & \vcentcolon =  \frac{ \left| \bar{F}^{(1)}_\A \cdot \bar{F}^{(1)}_\B \right|_{g_0}}{64 |\nab_{g_0}(u_\A\pm u_\B)|_{g_0}^2} \Big( 2 |\nab_{g_0} u_\A|_{g_0}^2 + 2 |\nab_{g_0} u_\B|_{g_0}^2 \label{wcffi 2pm}
\\&\hspace{5cm} \pm 3 |\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0} \mp |\nab_{g_0} u_\A|_{g_0} |\nab_{g_0} u_\B|_{g_0} \Big)  \non
\\&\quad \mp \frac{ (\bar{F}^{(1)}_\A)_{i \nab_{g_0} u_\B} (\bar{F}^{(1)}_\B)_{\nab_{g_0} u_\A}^i}{32 |\nab_{g_0}(u_\A\pm u_\B)|_{g_0}^2} ,\nonumber
\end{align}
\begin{align}
(\wc{X}^{(2,1)}_\A)_i & \vcentcolon = \half  \dr^b \pth{|\nab_{g_0} u_\A|_{g_0} \bar{F}^{(1)}_\A}_{bi} +\frac{1}{4} |\nab_{g_0} u_\A|_{g_0} (\bar{F}^{(1)}_\A)_{bc} \dr_i g_0^{bc} \label{wcX 21}
\\&\quad +  \half |\nab_{g_0} u_\A|_{g_0} \pth{ \dr_a g_0^{ca} + \half g_0^{ab}g_0^{cd}\dr_d (g_0)_{ab} } (\bar{F}^{(1)}_\A)_{ic}   - \half \dr_i u_\A (\bar{F}^{(1)}_\A)_{bc} (k_0)^{bc}\nonumber,
\\ (\wc{X}^{(2,\pm)}_{\A\B})_i & \vcentcolon = - \frac{1}{8} |\nab_{g_0} u_\B|_{g_0} (\bar{F}^{(1)}_\A)^{b}_{\nab_{g_0} u_\B} (\bar{F}^{(1)}_\B)_{bi} - \frac{1}{8}  |\nab_{g_0} u_\A|_{g_0} (\bar{F}^{(1)}_\B)^{b}_{\nab_{g_0} u_\A} (\bar{F}^{(1)}_\A)_{bi} \non
\\&\quad + \bigg( \frac{1}{8} \pth{|\nab_{g_0} u_\A|_{g_0} \dr_i u_\A+ |\nab_{g_0} u_\B|_{g_0} \dr_i u_\B} \label{wcX 2pm}
\\&\hspace{2cm} +\frac{1}{16}  \pth{ \pm  |\nab_{g_0} u_\B|_{g_0}   \dr_i u_\A    \pm  |\nab_{g_0} u_\A|_{g_0}   \dr_i u_\B }\bigg) \left| \bar{F}^{(1)}_\A \cdot \bar{F}^{(1)}_\B \right|_{g_0}.\nonumber
\end{align}
Moreover, the remainders $h_\la$ and $ \tilde{k}^{cons}_\la$ belong to the spaces $H^5_\de(\Si_0)$ and $H^4_{\de+1}(\Si_0)$ respectively and satisfy
\begin{align}
\max_{r\in\llbracket 0,4 \rrbracket} \la^r \l \nab^{r+1}h_\la \r_{L^2_{\de+r+1}(\Si_0)} & \leq C_{\mathrm{cons}} \e, \label{estim reste cons 1}
\\ \max_{r\in\llbracket 0,4 \rrbracket} \la^r \l \nab^{r}\tilde{k}^{cons}_\la \r_{L^2_{\de+r+1}(\Si_0)} & \leq C_{\mathrm{cons}} \e, \label{estim reste cons 2}
\end{align}
for some $C_{\mathrm{cons}}=C_{\mathrm{cons}}(N,\de,R)>0$.
\end{prop}

The result of Proposition \ref{prop constraint main} will be used as a black box in the final construction of initial data in Proposition \ref{prop ID main}, and its proof, based on the conformal method, is postponed to Appendix \ref{appendix proof prop constraint main}. The freedom in the choice of the tensors $^(2,±)_$, $^(1,1)_$, $^(1,2)_$ and $^(1,±)_$ will be crucially used to match the solution of the constraint equations and the data induced by the spacetime metric. This is related to an inherent redundancy in the geometric optics approach for \ref{EVE}. Indeed, solving the constraint roughly provides the zeroth and first derivatives of the metric on $_0$. However, our geometric optics construction implies that the first terms in the ansatz for $_$ actually solve first order transport equation, so that their first order derivatives are not free. This difficulty was already present in the singlephase case of [28, 29], but the multiphase case brings a new one: the tensor $F^(2,±)_$ solves an algebraic equation and is defined by \eqref{def F2pm} so that even its zeroth derivatives is not free. Therefore, we need to make sure that the solution $(g_,k_)$ of the constraint equations matches the expression of the spacetime ansatz by choosing appropriate tensors $^(2,±)_$, $^(1,1)_$, $^(1,2)_$ and $^(1,±)_$.

\subsection{Initial data for the spacetime metric $\g_\la$}\label{section initial data final}

In this section, we define initial data for the system \eqref{eq F1}-\eqref{eq h}. The properties of these initial data are summarized in Proposition \ref{prop ID main} at the end of the section.

\subsubsection{Initial data for the metric and the induced metric}\label{section induced metric}

We start by defining initial data for the induced metric $_|__0$. In order to ensure that the induced metric on $_0$ is given by $g_$ from Proposition \ref{prop constraint main}, we will make a particular choice of tensor $^(2,±)_$.

\paragraph{Initial data for $F^{(1)}_\A$.} For $F^(1)_$ we define the initial data
\begin{align}
(F^{(1)}_\A)_{ij|_{\Si_0}} & \vcentcolon = (\bar{F}^{(1)}_\A)_{ij},\label{ID F1 ij}
\\ (F^{(1)}_\A)_{0\a|_{\Si_0}} & \vcentcolon = 0. \label{ID F1 0}
\end{align}
Thanks to \eqref{assum F1}, these initial data are such that the conditions \eqref{pola F1}-\eqref{pola F1 bis} and \eqref{energy F1} hold on $_0$. Moreover, they allow us to obtain the initial value of derivatives of $F^(1)_$. While spacelike derivatives can be obtained by directly differentiating \eqref{ID F1 ij} or \eqref{ID F1 0}, time derivatives can be deduced from the transport equation we want $F^(1)_$ to solve in the spacetime, i.e \eqref{eq F1}. Indeed, an equation of the form $_T = S$ rewrites on $_0$:
\begin{align}\label{dt via transport}
\dr_t T_{\mu\nu}  & = N_\A T_{\mu\nu} + (\dr_t - N_\A)^\a \Ga(\g_0)^\rho_{\a(\mu}T_{\nu)\rho} + \frac{1}{2|\nab_{g_0} u_\A|_{g_0}} (\Box_{\g_0}u_\A)T_{\mu\nu}  - \frac{1}{2|\nab_{g_0} u_\A|_{g_0}} S_{\mu\nu} ,
\end{align}
where we used $L_|__0 = |_g_0 u_|_g_0 (_t - N_)$. In the case of $F^(1)_$, this implies in particular
\begin{align}
\dr_t (F^{(1)}_\A)_{00|_{\Si_0}} & = 0, \label{dt F1 00}
\\ \dr_t (F^{(1)}_\A)_{0i|_{\Si_0}} & =  (\bar{F}^{(1)}_\A)_{i k} g_0^{k\ell} \pth{  \dr_t (\g_0)_{0\ell} +  (k_0)_{N_\A\ell} } \label{dt F1 0i},
\\ \dr_t (F^{(1)}_\A)_{ij|_{\Si_0}} & =  N_\A (\bar{F}^{(1)}_\A)_{ij}  +   (\dr_t - N_\A)^\rho \Gamma(\g_0)^k_{\rho (i} (\bar{F}^{(1)}_\A)_{k j)}+ \frac{1}{2|\nabla_{g_0} u_\A|_{g_0}}(\Box_{\g_0}u_\A)  (\bar{F}^{(1)}_\A)_{ij},  \label{dt F1 ij}
\end{align}
where we used \eqref{ID F1 ij}-\eqref{ID F1 0}. Using again \eqref{assum F1}, \eqref{dt F1 ij} implies
\begin{align}
g_0^{ij}\dr_t (F^{(1)}_\A)_{ij|_{\Si_0}} & = -2 (\bar{F}^{(1)}_\A)_{ij}  k_0^{ij}, \label{trace dt F1}
\\ N_\A^i N_\A^j  \dr_t (F^{(1)}_\A)_{ij|_{\Si_0}} & = 0,\label{NN dt F1}
\\ N_\A^i (e^{(\i)}_\A)^j \dr_t(F^{(1)}_\A)_{ij|_{\Si_0}} & =\pth{  -  N_\A (N_\A)_\ell  -   (k_0)_{N_\A\ell}    + \half  N_\A^i  N_\A^j  \dr_\ell (g_0)_{j i} } g_0^{k\ell}(\bar{F}^{(1)}_\A)_{k e^{(\i)}_\A}.  \label{Ne dt F1}
\end{align}
Together with the definition \eqref{Q 0}, the identities \eqref{dt F1 00}, \eqref{dt F1 0i} and \eqref{trace dt F1} allow us to compute $Q^(0)_|__0$:
\begin{align}
(Q^{(0)}_{\A})_{\ell |_{\Si_0}} & = -  (\bar{F}^{(1)}_\A)_{\ell k} g_0^{ki} \pth{  \dr_t (\g_0)_{0i} +  (k_0)_{N_\A i} }+ g_0^{ij}\dr_i (\bar{F}^{(1)}_\A)_{\ell j} + \half (\bar{F}^{(1)}_\A)_{ij}   \dr_\ell g_0^{ij} ,\label{Q 0 ell}
\\ (Q^{(0)}_{\A})_{0|_{\Si_0}} & = (\bar{F}^{(1)}_\A)_{ij}  k_0^{ij} .\label{Q 0 0}
\end{align}

\paragraph{Initial data for $\F_\A$, $F^{(2,1)}_\A$ and $F^{(2,2)}_\A$.} For $_$, we simply define
\begin{align}
\F_{\A |_{\Si_0}} & \vcentcolon= 0,\label{ID F}
\end{align}
which clearly implies that \eqref{pola F} holds on $_0$. For $F^(2,1)_$ we define
\begin{align}
(F^{(2,1)}_\A)_{ij|_{\Si_0}} & \vcentcolon =    4\wc{\ffi}^{(2,1)}_{\A} (g_0)_{ij},\label{ID F21 ij}
\\ (F^{(2,1)}_\A)_{00|_{\Si_0}} & \vcentcolon= 0,\label{ID F21 00}
\\ (F^{(2,1)}_\A)_{0N_\A|_{\Si_0}} & \vcentcolon=  2\wc{\ffi}^{(2,1)}_{\A} - \frac{1}{2|\nab_{g_0} u_\A|_{g_0}^2}(Q^{(0)}_\A)_{{L_\A}|_{\Si_0}},\label{ID F21 0N}
\\ (F^{(2,1)}_\A)_{0 e^{(\i)}_\A|_{\Si_0}} & \vcentcolon=  \frac{1}{|\nab_{g_0} u_\A|_{g_0}} (Q^{(0)}_\A)_{{e^{(\i)}_\A}|_{\Si_0}},\label{ID F21 0e}
\end{align}
where we used \eqref{Q 0 ell}-\eqref{Q 0 0}, and for $F^(2,2)_$ we define
\begin{align}
(F^{(2,2)}_\A)_{ij|_{\Si_0}} & \vcentcolon = \frac{3}{4} F_\A^2  (g_0)_{ij},\label{ID F22 ij}
\\ (F^{(2,2)}_\A)_{00|_{\Si_0}} & \vcentcolon= 0,\label{ID F22 00}
\\ (F^{(2,2)}_\A)_{0N_\A|_{\Si_0}} & \vcentcolon=  \frac{3}{8} F_\A^2, \label{ID F22 0N}
\\ (F^{(2,2)}_\A)_{0 e^{(\i)}_\A|_{\Si_0}} & \vcentcolon= 0. \label{ID F22 0e}
\end{align}

\begin{lemma}\label{lem pola F2 init}
The initial data \eqref{ID F21 ij}-\eqref{ID F22 0e} are such that \eqref{pola F21}-\eqref{pola F22} hold on $\Si_0$.
\end{lemma}

\begin{proof}
We recall that $L_{\A|_{\Si_0}} = |\nab_{g_0} u_\A|_{g_0} (\dr_t - N_\A)$ and $\Lb_{\A|_{\Si_0}} = |\nab_{g_0} u_\A|_{g_0} (\dr_t + N_\A)$. We start with \eqref{pola F21}. From \eqref{Pol L} and \eqref{def 1 V 2 2} we have
\begin{align*}
(V^{(2,1)}_\A)_{L_\A|_{\Si_0}} & = \Pol{F^{(2,1)}_\A}{u_\A}_{L_\A|_{\Si_0}} + (Q^{(0)}_{\A})_{L_\A |_{\Si_0}}
\\& = - (F^{(2,1)}_\A)_{L_\A L_\A|_{\Si_0}} + (Q^{(0)}_{\A})_{L_\A |_{\Si_0}}
\\& =  |\nab_{g_0} u_\A|_{g_0}^2\pth{ -  (F^{(2,1)}_\A)_{0 0|_{\Si_0}} +  2 (F^{(2,1)}_\A)_{0 N_\A|_{\Si_0}}  -  (F^{(2,1)}_\A)_{N_\A N_\A|_{\Si_0}} }  + (Q^{(0)}_{\A})_{L_\A |_{\Si_0}}
\\& = |\nab_{g_0} u_\A|_{g_0}^2\pth{  4\wc{\ffi}^{(2,1)}_{\A} - \frac{1}{|\nab u_\A|^2}(Q^{(0)}_\A)_{{L_\A}|_{\Si_0}} -  (F^{(2,1)}_\A)_{N_\A N_\A|_{\Si_0}} } + (Q^{(0)}_{\A})_{L_\A |_{\Si_0}}
\\& = 0,
\end{align*}
and from \eqref{Pol e} we also have
\begin{align*}
(V^{(2,1)}_\A)_{e^{(\i)}_\A|_{\Si_0}} & = \Pol{F^{(2,1)}_\A}{u_\A}_{e^{(\i)}_\A|_{\Si_0}} + (Q^{(0)}_{\A})_{e^{(\i)}_\A |_{\Si_0}}
\\& = - (F^{(2,1)}_\A)_{e^{(\i)}_\A L_\A|_{\Si_0}} + (Q^{(0)}_{\A})_{e^{(\i)}_\A |_{\Si_0}}
\\& = -|\nab_{g_0} u_\A|_{g_0} \pth{  (F^{(2,1)}_\A)_{e^{(\i)}_\A 0|_{\Si_0}} - (F^{(2,1)}_\A)_{e^{(\i)}_\A N_\A|_{\Si_0}} } + (Q^{(0)}_{\A})_{e^{(\i)}_\A |_{\Si_0}}
\\& = -|\nab_{g_0} u_\A|_{g_0} \pth{ \frac{1}{|\nab_{g_0} u_\A|_{g_0}} (Q^{(0)}_\A)_{{e^{(\i)}_\A}|_{\Si_0}} - (F^{(2,1)}_\A)_{e^{(\i)}_\A N_\A|_{\Si_0}} } + (Q^{(0)}_{\A})_{e^{(\i)}_\A |_{\Si_0}}
\\& = 0,
\end{align*}
where we used \eqref{ID F21 ij} and \eqref{ID F21 00}-\eqref{ID F21 0e}. From \eqref{Pol Lb} we also have
\begin{align*}
(V^{(2,1)}_\A)_{\Lb_\A|_{\Si_0}} & = \Pol{F^{(2,1)}_\A}{u_\A}_{\Lb_\A|_{\Si_0}} + (Q^{(0)}_{\A})_{\Lb_\A |_{\Si_0}}
\\& =   |\nab_{g_0} u_\A|_{g_0}^2\pth{ (F^{(2,1)}_\A)_{ N_\A N_\A|_{\Si_0}} - \tr_{g_0} F^{(2,1)}_{\A\;\;|_{\Si_0}}}
+ |\nab_{g_0} u_\A|_{g_0} \pth{  (Q^{(0)}_{\A})_{0 |_{\Si_0}} + N_\A^\ell  (Q^{(0)}_{\A})_{\ell |_{\Si_0}} }.
\end{align*}
Using \eqref{ID F21 ij} together with \eqref{image P1} on one hand and \eqref{Q 0 ell}-\eqref{Q 0 0} on the other hand we obtain
\begin{align*}
(V^{(2,1)}_\A)_{\Lb_\A|_{\Si_0}} & =  -8 |\nab_{g_0} u_\A|_{g_0}^2 \wc{\ffi}^{(2,1)}_{\A}
+ |\nab_{g_0} u_\A|_{g_0} \pth{ (\bar{F}^{(1)}_\A)_{ij}  k_0^{ij} - \frac{(\bar{F}^{(1)}_\A)_{\ell j}}{|\nab_{g_0} u_\A|_{g_0}} g_0^{ij}\dr_i \dr^\ell u_\A + \half (\bar{F}^{(1)}_\A)_{ij} N_\A g_0^{ij} }
\\& = 0,
\end{align*}
where we used \eqref{wcffi 21}. We now turn to \eqref{pola F22}. From \eqref{def 1 V 2 2} we have
\begin{align*}
(V^{(2,2)}_\A)_{L_\A|_{\Si_0}} & =  \Pol{F^{(2,2)}_\A}{u_\A}_{L_\A|_{\Si_0}}
\\& = - (F^{(2,2)}_\A)_{L_\A L_\A|_{\Si_0}}
\\& = |\nab_{g_0} u_\A|_{g_0}^2\pth{ -  (F^{(2,2)}_\A)_{0 0|_{\Si_0}} +  2 (F^{(2,2)}_\A)_{0 N_\A|_{\Si_0}}  -  (F^{(2,2)}_\A)_{N_\A N_\A|_{\Si_0}} }
\\& =  |\nab_{g_0} u_\A|_{g_0}^2\pth{ \frac{3}{4} F_\A^2  -  \frac{3}{4} F_\A^2  }
\\& = 0,
\end{align*}
\begin{align*}
(V^{(2,2)}_\A)_{e^{(\i)}_\A|_{\Si_0}} & =  \Pol{F^{(2,2)}_\A}{u_\A}_{e^{(\i)}_\A|_{\Si_0}}
\\& = - (F^{(2,2)}_\A)_{e^{(\i)}_\A L_\A|_{\Si_0}}
\\& = - |\nab_{g_0} u_\A|_{g_0} \pth{ (F^{(2,2)}_\A)_{e^{(\i)}_\A 0|_{\Si_0}} - (F^{(2,2)}_\A)_{e^{(\i)}_\A N_\A|_{\Si_0}} }
\\& = 0,
\end{align*}
where we used \eqref{ID F22 ij} and \eqref{ID F22 00}-\eqref{ID F22 0e}. Using now only \eqref{ID F22 ij} we finally obtain
\begin{align*}
(V^{(2,2)}_\A)_{\Lb_\A|_{\Si_0}} & = \Pol{F^{(2,2)}_\A}{u_\A}_{\Lb_\A|_{\Si_0}} + \frac{3}{2}F_\A^2 |\nab u_\A|^2
\\& =   |\nab_{g_0} u_\A|_{g_0}^2\pth{ (F^{(2,2)}_\A)_{ N_\A N_\A|_{\Si_0}} - \tr_{g_0} F^{(2,2)}_{\A\;\;|_{\Si_0}}} + \frac{3}{2}F_\A^2 |\nab_{g_0} u_\A|_{g_0}^2
\\& = - \frac{3}{2}F_\A^2 |\nab_{g_0} u_\A|_{g_0}^2 + \frac{3}{2}F_\A^2 |\nab_{g_0} u_\A|_{g_0}^2
\\&=0,
\end{align*}
which concludes the proof of the lemma.
\end{proof}

\paragraph{Initial data for $\h_\la$.} In order to define initial data for $_$, we first need to obtain the values of $(≫^(3,h)_u,T)_|̱__0$ and $(≫^(3,e)_v,T)_|̱__0$. For $(≫^(3,h)_u,T)_|̱__0$, we use \eqref{g3h}, and note that the initial values of $F^(1)_$, $_$, $F^(2,1)_$ and $F^(2,2)_$ have been defined in \eqref{ID F1 ij}-\eqref{ID F1 0} and \eqref{ID F21 ij}-\eqref{ID F22 0e}, the initial values of $F^(1)_$ and $_t F^(1)_$ can be deduced from \eqref{ID F1 ij}-\eqref{ID F1 0} and \eqref{dt F1 00}-\eqref{dt F1 ij} respectively, the initial values of $F^(2,1)_$ and $F^(2,2)_$ can be deduced from \eqref{ID F21 ij}-\eqref{ID F22 0e} and for the time derivatives $_t F^(2,1)_$ and $_t F^(2,2)_$ we use the transport equations \eqref{eq F21}-\eqref{eq F22} we want $F^(2,1)_$ and $F^(2,2)_$ to solve in the spacetime. Thanks to \eqref{dt via transport} we obtain
\begin{align*}
\dr_t (F^{(2,1)}_\A)_{\a\b} & = \cro{ \nab^{\leq 1} F^{(2,1)} + \dr^{\leq 2} F^{(1)} + (F^{(1)})^2 + (F^{(1)})^3}_{\a\b},
\\ \dr_t (F^{(2,2)}_\A)_{\a\b} & = \cro{ \nab^{\leq 1} F^{(2,2)} +  \pth{ F^{(2,1)} + \dr^{\leq 1} F^{(1)} }F^{(1)} }_{\a\b},
\end{align*}
where we also used \eqref{Rtilde 1 1}-\eqref{Rtilde 1 2}. This allows us to completely define the initial value of $(≫^(3,h)_u,T)_|̱__0$. For $(≫^(3,e)_v,T)_|̱__0$, we use \eqref{def g3e} and \eqref{R 1 e} to obtain
\begin{align*}
\gg^{(3,e)}_{v,T} & =  \Big\{ \pth{ \F + F^{(2,1)} + F^{(2,2)}  + (F^{(1)})^2 + \dr^{\leq 1} F^{(1)} }\pth{1+F^{(1)}} \Big\}.
\end{align*}
Therefore, using the various definitions of initial data above, this allows us to completely define the initial value of $(≫^(3,e)_v,T)_|̱__0$. We can now define the initial data for $_$:
\begin{align}
(\h_\la)_{ij|_{\Si_0}} & \vcentcolon= (h_\la)_{ij} - \la \sum_{u\in\mathcal{N},T\in\{\cos,\sin\}}T\pth{\frac{u}{\la}} (\gg^{(3,h)}_{u,T})_{ij|_{\Si_0}}   -\la \sum_{v\in\mathcal{I},T\in\{\cos,\sin\}} T\pth{\frac{v}{\la}} (\gg^{(3,e)}_{v,T})_{ij|_{\Si_0}},  \label{ID h ij}
\\ (\h_\la)_{0\a|_{\Si_0}} & \vcentcolon= - \la \sum_{u\in\mathcal{N},T\in\{\cos,\sin\}}T\pth{\frac{u}{\la}} (\gg^{(3,h)}_{u,T})_{0\a|_{\Si_0}} -\la \sum_{v\in\mathcal{I},T\in\{\cos,\sin\}} T\pth{\frac{v}{\la}} (\gg^{(3,e)}_{v,T})_{0\a|_{\Si_0}}.   \label{ID h 0}
\end{align}

\paragraph{The induced metric.} Thanks to \eqref{ID F1 ij}, \eqref{ID F}, \eqref{ID F21 ij}, \eqref{ID F22 ij} and \eqref{ID h ij} we can compute the induced metric on $_0$ by $_$:
\begin{align*}
(\g_\lambda)_{ij|_{\Si_0}} & = (g_0)_{ij} + \la \sum_\A  \cos\left( \frac{u_\A}{\lambda} \right) (\bar{F}^{(1)}_\A)_{ij} + \lambda^2\sum_\A  \sin\left(\frac{u_\A}{\lambda} \right) 4\wc{\ffi}^{(2,1)}_{\A} (g_0)_{ij}
\\&\quad + \lambda^2\sum_\A  \cos\left( \frac{2u_\A}{\lambda}\right)  \frac{3}{4} F_\A^2  (g_0)_{ij} + \la^2 \sum_{\A\neq \B,\pm} \cos\left(\frac{u_\A\pm u_\B}{\lambda}\right) (F^{(2,\pm)}_{\A\B})_{ij|_{\Si_0}}  + \lambda^2 (h_\lambda)_{ij} .
\end{align*}
Therefore, $g_$ given by \eqref{g la} is the induced metric if and only if the following holds
\begin{align}\label{eq ga2pm 1}
(F^{(2,\pm)}_{\A\B})_{ij|_{\Si_0}} & = \bar{\PP}^{[1]}_{u_\A\pm u_\B}\pth{\ga^{(2,\pm)}_{\A\B}}_{ij}  + 4\wc{\ffi}^{(2,\pm)}_{\A\B}(g_0)_{ij}.
\end{align}
Since $(F^(2,±)_)_ij|__0$ can be computed from \eqref{def F2pm} and \eqref{ID F1 ij}-\eqref{ID F1 0} and $^(2,±)_$ from \eqref{wcffi 2pm}, \eqref{eq ga2pm 1} is an equation for $^(2,±)_$. The following lemma shows how one can solve it.

\begin{lemma}\label{lem ga2pm}
If we define
\begin{align}
\ga^{(2,\pm)}_{\A\B} & \vcentcolon= (F^{(2,\pm)}_{\A\B})_{ij|_{\Si_0}}  - 4\wc{\ffi}^{(2,\pm)}_{\A\B}(g_0)_{ij},\label{def ga2pm}
\end{align}
then \eqref{eq ga2pm 1} holds.
\end{lemma}

\begin{proof}
If $\ga^{(2,\pm)}_{\A\B}$ is defined by \eqref{def ga2pm}, then \eqref{eq ga2pm 1} rewrites $\bar{\PP}^{[1]}_{u_\A\pm u_\B}\pth{\ga^{(2,\pm)}_{\A\B}}=\ga^{(2,\pm)}_{\A\B}$. Therefore, thanks to Lemma \ref{lem P 1et2}, proving the lemma is equivalent to
\begin{align}\label{condition ga2pm}
(\ga^{(2,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}N^{(\pm)}_{\A\B}} - \tr_{g_0}\ga^{(2,\pm)}_{\A\B} & = 0,
\end{align}
with $\ga^{(2,\pm)}_{\A\B}$ defined by \eqref{def ga2pm}. From \eqref{def ga2pm} we have
\begin{align*}
(\ga^{(2,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}N^{(\pm)}_{\A\B}} - \tr_{g_0}\ga^{(2,\pm)}_{\A\B} & = \mp\frac{(\mathcal{I}^{(0,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}N^{(\pm)}_{\A\B}} - \tr_{g_0}\mathcal{I}^{(0,\pm)}_{\A\B} }{2\pth{|\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0} - |\nab_{g_0} u_\A|_{g_0}|\nab_{g_0} u_\B|_{g_0}}} + 8\wc{\ffi}^{(2,\pm)}_{\A\B}.
\end{align*}
From \eqref{I 0 pm}, \eqref{P 0 pm} and \eqref{ID F1 ij}-\eqref{ID F1 0} we find on $\Si_0$
\begin{align*}
&(\mathcal{I}^{(0,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}N^{(\pm)}_{\A\B}} - \tr_{g_0}\mathcal{I}^{(0,\pm)}_{\A\B}
\\& =   - \frac{|\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0} - |\nab_{g_0} u_\A|_{g_0}|\nab_{g_0} u_\B|_{g_0}}{2|\nab_{g_0} (u_\A\pm u_\B)|_{g_0}^2}  (\bar{F}^{(1)}_\B)_{\nab_{g_0} u_\A}^k (\bar{F}^{(1)}_\A)_{\nab_{g_0} u_\B k}
\\&\quad \pm  \frac{\left|\bar{F}^{(1)}_\A\cdot \bar{F}^{(1)}_\B\right|_{g_0}}{4|\nab_{g_0}(u_\A\pm u_\B)|_{g_0}^2} \Big( 2|\nab_{g_0} u_\A|_{g_0}^2 |\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0} \pm |\nab_{g_0} u_\A|_{g_0}^2  | \nab_{g_0} u_\B|_{g_0}^2
\\&\hspace{5cm} \pm3 |\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0}^2  + 2 |\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0} | \nab_{g_0} u_\B|_{g_0}^2
\\&\hspace{5cm}     - 2  |\nab_{g_0} u_\A|_{g_0}^3|\nab_{g_0} u_\B|_{g_0}    - 2   |\nab_{g_0} u_\A|_{g_0}|\nab_{g_0} u_\B|_{g_0}^3
\\&\hspace{7cm}  \mp4 |\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0}  |\nab_{g_0} u_\A|_{g_0}|\nab_{g_0} u_\B|_{g_0}     \Big).
\end{align*}
Using now \eqref{wcffi 2pm} we finally obtain \eqref{condition ga2pm} and thus the lemma.
\end{proof}

\subsubsection{Initial data for derivatives of the metric and the second fundamental form}\label{section second fundamental form}

In terms of initial data for the system \eqref{eq F1}-\eqref{eq h} it only remains to define the initial data for $_(_)_$, where $_$ is the future-directed unit normal to $_0$ for $_$. As usual when solving the Einstein vacuum equations in wave coordinates, $_(_)_ij$ will set the second fundamental form and $_(_)_0$ will ensure that \eqref{GWC} holds on $_0$. In order to ensure that the second fundamental form is given by $k_$ from Proposition \ref{prop constraint main}, we will also make a particular choice of $^(1,1)_$, $^(1,2)_$ and $^(1,±)_$. We first need to compute $_$.

\paragraph{The future-directed unit normal and the second fundamental form.}
Using the fact that $_t$ is the future-directed unit normal to $_0$ for $_0$ and using also \eqref{ID F1 0}, \eqref{ID F} and \eqref{ID h 0} we obtain
\begin{align}
\T_\la & = \dr_t + \la^2 \sum_\A\pth{ \sin\pth{\frac{u_\A}{\la}}\T^{(2,1)}_\A + \cos\pth{\frac{2u_\A}{\la}}\T^{(2,2)}_\A } \label{unit normal}
\\&\quad + \la^2\sum_{\A\neq\B,\pm}\cos\pth{\frac{u_\A\pm u_\B}{\la}}\T^{(2,\pm)}_{\A\B} + \la^3 \mathfrak{T}_\la, \non
\end{align}
where the vector fields $^(2,1)_$, $^(2,2)_$ and $^(2,±)_$ are tangent to $_0$ and satisfy
\begin{align}
g_0\pth{\T^{(2,1)}_\A,\dr_i} & = - (F^{(2,1)}_\A)_{0i}, \label{T21}
\\ g_0\pth{\T^{(2,2)}_\A,\dr_i} & = - (F^{(2,2)}_\A)_{0i}, \label{T22}
\\ g_0\pth{\T^{(2,\pm)}_{\A\B},\dr_i} & = - (F^{(2,\pm)}_{\A\B})_{0i},\label{T2pm}
\end{align}
and where the vector field $𝔗_=𝔗_^_$ satisfies
\begin{align}
\mathfrak{T}_\la^\a = \cro{\pth{ F^{(2,1)} + F^{(2,2)} + (F^{(1)})^2 }\pth{ (\g_\la^{-1})^{(\geq 1)}  + \g_\la^{-1}\pth{ F^{(2,1)} + F^{(2,2)} + (F^{(1)})^2 }} }^\osc.\label{mathfrak T}
\end{align}
We can now compute the second fundamental form of $_0$ with respect to the spacetime metric $_$, using in particular \eqref{eikonal init}, \eqref{ID F1 ij}, \eqref{ID F21 ij}, \eqref{ID F22 ij} and \eqref{T21}-\eqref{mathfrak T}:
\begin{equation}\label{second fundamental form}
\begin{aligned}
-\half (\LL_{\T_\la}\g_\la)_{ij} & =  (k_0)_{ij} + \half \sum_\A \sin\pth{\frac{u_\A}{\la}}|\nab_{g_0} u_\A|_{g_0} (\bar{F}^{(1)}_\A)_{ij}
\\&\quad - \frac{\la}{2} \sum_\A \cos\pth{\frac{u_\A}{\la}}\Big( \dr_t (F^{(1)}_\A)_{ij} + 4 |\nab_{g_0} u_\A|_{g_0} \wc{\ffi}^{(2,1)}_{\A} (g_0)_{ij} - \dr_{(i}u_\A(F^{(2,1)}_\A)_{0j)}\Big)
\\&\quad + \la \sum_\A \sin\pth{\frac{2u_\A}{\la}} \pth{ \frac{3}{4}|\nab_{g_0} u_\A|_{g_0} F_\A^2  (g_0)_{ij} - \dr_{(i}u_\A(F^{(2,2)}_\A)_{0j)}}
\\&\quad + \frac{\la}{2} \sum_{\A\neq \B,\pm} \sin\pth{\frac{u_\A\pm u_\B}{\la}}\bigg( \pth{ |\nab_{g_0} u_\A|_{g_0} \pm |\nab_{g_0} u_\B|_{g_0}} (F^{(2,\pm)}_{\A\B})_{ij}
\\&\hspace{7cm} - \dr_{(i}(u_\A\pm u_\B)(F^{(2,\pm)}_{\A\B})_{0j)} \bigg)
\\&\quad - \frac{\la^2}{2} \T_\la (\h_\la)_{ij} + \la^2 \tilde{k}^{evol}_\la,
\end{aligned}
\end{equation}
where the remainder is of the form
\begin{align}
\tilde{k}^{evol}_\la & = \cro{\pth{ \g_\la \dr \T_\la + \T_\la \dr \pth{\g_\la - \la^2 \h_\la} }^{(\geq 2)}}^\osc,\label{ktilde evol}
\end{align}
where by $_$ we denote a component of the vector field $_$ in coordinates. Note that the only terms involved in \eqref{ktilde evol} that have not yet been defined are time derivatives of $$. They can be deduced from the transport equation \eqref{eq F} we want $_$ to satisfy in the spacetime. Thanks to \eqref{dt via transport} and \eqref{ID F}, \eqref{eq F} becomes on $_0$
\begin{align*}
\dr_t (\F_\A)_{\mu\nu|_{\Si_0}}  & =  \frac{1}{2|\nab_{g_0} u_\A|_{g_0}} \Pi_\leq \pth{ (\h_\lambda)_{L_\A L_\A|_{\Si_0}} } (F^{(1)}_\A)_{\mu\nu|_{\Si_0}} .
\end{align*}
Therefore, $k_$ given by \eqref{k la} is the second fundamental form if and only if the expressions \eqref{k la} and \eqref{second fundamental form} coincide, that is if and only if the following four identities hold:
\begin{align}
\bar{\PP}^{[2]}_{u_\A}\pth{\ka^{(1,1)}_\A}_{ij} &+  \frac{(N_\A\tilde{\otimes} \wc{X}^{(2,1)}_\A)_{ij}}{|\nab_{g_0} u_\A|_{g_0}} \label{eq ka11}
\\& = -\half \pth{ 4|\nab_{g_0} u_\A|_{g_0} \wc{\ffi}^{(2,1)}_{\A} (g_0)_{ij}  - \dr_{(i} u_\A (F^{(2,1)}_\A)_{0j)} + \dr_t(F^{(1)}_\A)_{ij|_{\Si_0}} }, \non
\end{align}
\begin{align}
\bar{\PP}^{[2]}_{u_\A}\pth{\ka^{(1,2)}_\A}_{ij}   -\frac{3}{2} |\nab_{g_0} u_\A|_{g_0} F_\A^2 (N_\A\tilde{\otimes} N_\A)_{ij} = \frac{3}{4} |\nab_{g_0} u_\A|_{g_0} F_\A^2  (g_0)_{ij} - \dr_{(i} u_\A (F^{(2,2)}_\A)_{0j)} ,  \label{eq ka12}
\end{align}
\begin{align}
\bar{\PP}^{[2]}_{u_\A\pm u_\B}&\pth{\ka^{(1,\pm)}_{\A\B}}_{ij} - \frac{(N^{(\pm)}_{\A\B}\tilde{\otimes} \wc{X}^{(2,\pm)}_{\A\B})_{ij}}{|\nab_{g_0}\pth{ u_\A\pm u_\B}|_{g_0}} \label{eq ka1pm}
\\&  = \half \pth{ (|\nab_{g_0} u_\A|_{g_0}\pm|\nab_{g_0} u_\B|_{g_0}) (F^{(2,\pm)}_{\A\B})_{ij|_{\Si_0}}  - \dr_{(i} (u_\A\pm u_\B)  (F^{(2,\pm)}_{\A\B})_{0j)} }  \non,
\end{align}
\begin{align} \label{ID dth ij first}
-  \frac{1}{2} (\T_\la(\h_\lambda)_{ij})_{|_{\Si_0}} + (\tilde{k}^{evol}_\la)_{ij} & = (\tilde{k}^{cons}_\la)_{ij}.
\end{align}

\paragraph{Adjusting the second fundamental form.}
First, note that \eqref{ID dth ij first} defines the initial data for $_(_λ)_ij$:
\begin{align}\label{ID dth ij}
(\T_\la(\h_\lambda)_{ij})_{|_{\Si_0}} & \vcentcolon = - 2(\tilde{k}^{cons}_\la)_{ij} + 2(\tilde{k}^{evol}_\la)_{ij} .
\end{align}
Second, all the terms in \eqref{eq ka11} can be computed from \eqref{wcffi 21}, \eqref{wcX 21}, \eqref{dt F1 ij}, \eqref{ID F21 0N} and \eqref{ID F21 0e}. All the terms in \eqref{eq ka12} can be computed from \eqref{ID F21 0N} and \eqref{ID F21 0e}. All the terms in \eqref{eq ka1pm} can be computed from \eqref{def F2pm}, \eqref{ID F1 ij}, \eqref{ID F1 0} and \eqref{wcX 2pm}. Therefore, \eqref{eq ka11}-\eqref{eq ka1pm} are equations for $^(1,1)_$, $^(1,2)_$ and $^(1,±)_$.

\begin{lemma}\label{lem ka11}
If we define
\begin{align}
(\ka^{(1,1)}_\A)_{ij} & \vcentcolon= - \frac{(N_\A\tilde{\otimes} \wc{X}^{(2,1)}_\A)_{ij}}{|\nab_{g_0} u_\A|_{g_0}} - \half \dr_t(F^{(1)}_\A)_{ij|_{\Si_0}} \label{def ka11}
- \frac{|\nab_{g_0} u_\A|_{g_0}}{2}\pth{ 4\wc{\ffi}^{(2,1)}_{\A} (g_0)_{ij}  - (N_\A)_{(i} (F^{(2,1)}_\A)_{0j)}  },
\end{align}
then \eqref{eq ka11} holds.
\end{lemma}

\begin{proof}
If $\ka^{(1,1)}_\A$ is defined by \eqref{def ka11}, then \eqref{eq ka11} rewrites $\bar{\PP}^{[2]}_{u_\A}\pth{\ka^{(1,1)}_\A}=\ka^{(1,1)}_\A$. Therefore, thanks to Lemma \ref{lem P 1et2}, proving the lemma is equivalent to
\begin{align*}
(\ka^{(1,1)}_\A)_{N_\A N_\A}-\tr_{g_0}\ka^{(1,1)}_\A & = 0,
\\ (\ka^{(1,1)}_\A)_{N_\A e^{(\i)}_\A} & = 0,
\end{align*}
with $\ka^{(1,1)}_\A$ defined by \eqref{def ka11}. Thanks to \eqref{wcffi 21}, \eqref{wcX 21}, \eqref{trace dt F1} and \eqref{NN dt F1} we first have
\begin{align*}
(\ka^{(1,1)}_\A)_{N_\A N_\A}-\tr_{g_0}\ka^{(1,1)}_\A & =  - \frac{1}{|\nab_{g_0} u_\A|_{g_0}}(\wc{X}^{(2,1)}_\A)_{N_\A} + \half g_0^{ij} \dr_t(F^{(1)}_\A)_{ij|_{\Si_0}}  + 4|\nab_{g_0} u_\A|_{g_0}\wc{\ffi}^{(2,1)}_{\A}
\\& =  \frac{1}{2|\nab_{g_0} u_\A|_{g_0}}(\bar{F}^{(1)}_\A)_{ij}\dr^i\dr^j u_\A   - \frac{1}{4}  (\bar{F}^{(1)}_\A)_{ij} N_\A g_0^{ij}   - \half  (\bar{F}^{(1)}_\A)_{ij} (k_0)^{ij}
\\&\quad + \frac{1}{2|\nab_{g_0} u_\A|_{g_0}}(\bar{F}^{(1)}_\A)_{ij} \pth{- \dr^i \dr^j u_\A + |\nab_{g_0} u_\A|_{g_0}k_0^{ij} + \half  \dr_{\ell} u_\A  \dr^\ell g_0^{ij} }
\\& = 0.
\end{align*}
Using now \eqref{wcffi 21}, \eqref{Ne dt F1}, \eqref{ID F21 0e} and \eqref{Q 0 ell} we have
\begin{align*}
(\ka^{(1,1)}_\A)_{N_\A e^{(\i)}_\A} & =  - \frac{1}{|\nab_{g_0} u_\A|_{g_0}}(\wc{X}^{(2,1)}_\A)_{e}   - \half N_\A^i (e^{(\i)}_\A)^j \dr_t(F^{(1)}_\A)_{ij}  + \frac{|\nab_{g_0} u_\A|_{g_0}}{2}  (F^{(2,1)}_\A)_{0e^{(\i)}_\A}
\\& =   - \half g_0^{k\ell}(\bar{F}^{(1)}_\A)_{k e^{(\i)}_\A}  \bigg( - \dr^b (g_0)_{\ell b} + \half g_0^{ab}\dr_\ell (g_0)_{ab}+\dr_t (\g_0)_{0\ell}
\\&\hspace{4cm} -  N_\A (N_\A)_\ell    + \half  N_\A^i  N_\A^j  \dr_\ell (g_0)_{j i} + \frac{1}{|\nab_{g_0} u_\A|_{g_0}} \dr_\ell |\nab_{g_0} u_\A|_{g_0}   \bigg) .
\end{align*}
We then use \eqref{dt g 0i} and \eqref{geodesic spatial} to obtain
\begin{align*}
(\ka^{(1,1)}_\A)_{N_\A e^{(\i)}_\A} & =  -  \frac{1}{2|\nab_{g_0} u_\A|_{g_0}} (\bar{F}^{(1)}_\A)_{N_\A e^{(\i)}_\A}  N_\A |\nab_{g_0} u_\A|_{g_0}
\\& = 0.
\end{align*}
This concludes the proof of the lemma.
\end{proof}

\begin{lemma}\label{lem ka12}
If we define
\begin{align}
(\ka^{(1,2)}_\A)_{ij} & \vcentcolon= \frac{3}{2}|\nab_{g_0} u_\A|_{g_0}F_\A^2(N_\A)_{i}(N_\A)_{j} \label{def ka12}
\end{align}
then \eqref{eq ka12} holds.
\end{lemma}

\begin{proof}
Thanks to \eqref{ID F22 0N} and \eqref{ID F22 0e}, the equation \eqref{eq ka12} rewrites
\begin{align*}
\bar{\PP}^{[2]}_{u_\A}\pth{\ka^{(1,2)}_\A}_{ij} & =\frac{3}{2}|\nab_{g_0} u_\A|_{g_0}F_\A^2(N_\A)_{i}(N_\A)_{j}.
\end{align*}
Therefore, Lemma \ref{lem P 1et2} implies that \eqref{def ka12} is a solution if and only if it satisfies
\begin{align*}
(\ka^{(1,2)}_\A)_{N_\A N_\A}-\tr_{g_0}\ka^{(1,2)}_\A & = 0,
\\ (\ka^{(1,2)}_\A)_{N_\A e^{(\i)}_\A} & = 0,
\end{align*}
which is obviously the case.
\end{proof}

\begin{lemma}\label{lem ka1pm}
If we define
\begin{align}
(\ka^{(1,\pm)}_{\A\B})_{ij} & \vcentcolon=  \frac{-(|\nab_{g_0} u_\A|_{g_0}\pm|\nab_{g_0} u_\B|_{g_0})(\mathcal{I}^{(0,\pm)}_{\A\B})_{ij|_{\Si_0}} + \dr_{(i} (u_\A\pm u_\B)(\mathcal{I}^{(0,\pm)}_{\A\B})_{0j)}    }{2\g_0^{-1}(\d(u_\A\pm u_\B),\d(u_\A\pm u_\B))}   \label{def ka1pm}
\\&\quad +  \frac{(N^{(\pm)}_{\A\B}\tilde{\otimes} \wc{X}^{(2,\pm)}_{\A\B})_{ij}}{|\nab_{g_0}\pth{ u_\A\pm u_\B}|_{g_0}}\non,
\end{align}
then \eqref{eq ka1pm} holds.
\end{lemma}

\begin{proof}
Since $F^{(2,\pm)}_{\A\B}$ is defined by \eqref{def F2pm}, the equation \eqref{eq ka1pm} with $\ka^{(1,\pm)}_{\A\B}$ defined by \eqref{def ka1pm} rewrites $\bar{\PP}^{[2]}_{u_\A\pm u_\B}\pth{\ka^{(1,\pm)}_{\A\B}}=\ka^{(1,\pm)}_{\A\B}$ and, thanks to Lemma \ref{lem P 1et2}, holds if and only if
\begin{align}
(\ka^{(1,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}N^{(\pm)}_{\A\B}} - \tr_{g_0}\ka^{(1,\pm)}_{\A\B} & = 0,\label{ka1pm 1}
\\ (\ka^{(1,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}Z} & = 0,\label{ka1pm 2}
\end{align}
for any vector field $Z$ such that $g_0\pth{N^{(\pm)}_{\A\B},Z}=0$. We start with the first identity. We have
\begin{align*}
(\ka^{(1,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}N^{(\pm)}_{\A\B}} &- \tr_{g_0}\ka^{(1,\pm)}_{\A\B}
\\& = \mp \frac{|\nab_{g_0} u_\A|_{g_0}\pm|\nab_{g_0} u_\B|_{g_0}  }{4\g_0^{-1}(\d u_\A,\d u_\B)}  \pth{ (\mathcal{I}^{(0,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}N^{(\pm)}_{\A\B}} - \tr_{g_0}\mathcal{I}^{(0,\pm)}_{\A\B} } + \frac{(\wc{X}^{(2,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}}}{|\nab_{g_0}\pth{ u_\A\pm u_\B}|_{g_0}}
\\& = -4 (|\nab_{g_0} u_\A|_{g_0}\pm|\nab_{g_0} u_\B|_{g_0} ) \wc{\ffi}^{(2,\pm)}_{\A\B}  + \frac{(\wc{X}^{(2,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}}}{|\nab_{g_0}\pth{ u_\A\pm u_\B}|_{g_0}},
\end{align*}
where we used \eqref{condition ga2pm} and \eqref{def ga2pm}. Using now \eqref{wcffi 2pm} and \eqref{wcX 2pm} we obtain \eqref{ka1pm 1}. We now turn to \eqref{ka1pm 2}. Let $Z$ such that $g_0\pth{N^{(\pm)}_{\A\B},Z}=0$, we have
\begin{align*}
(\ka^{(1,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}Z} & = \frac{-(|\nab_{g_0} u_\A|_{g_0}\pm|\nab_{g_0} u_\B|_{g_0})(\mathcal{I}^{(0,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}Z} + |\nab_{g_0}(u_\A\pm u_\B)|_{g_0} (\mathcal{I}^{(0,\pm)}_{\A\B})_{0Z}  }{\pm4\g_0^{-1}(\d u_\A,\d u_\B)}
\\&\quad  +  \frac{1}{|\nab_{g_0}\pth{ u_\A\pm u_\B}|_{g_0}} (\wc{X}^{(2,\pm)}_{\A\B})_{Z}.
\end{align*}
From \eqref{I 0 pm}, \eqref{P 0 pm} and \eqref{ID F1 ij}-\eqref{ID F1 0} we find on $\Si_0$
\begin{align*}
-(|\nab_{g_0} u_\A|_{g_0}&\pm|\nab_{g_0} u_\B|_{g_0})(\mathcal{I}^{(0,\pm)}_{\A\B})_{N^{(\pm)}_{\A\B}Z} + |\nab_{g_0}(u_\A\pm u_\B)|_{g_0} (\mathcal{I}^{(0,\pm)}_{\A\B})_{0Z}
\\& =  \pm \frac{(F^{(1)}_\A)^{\ell}_{\nab_{g_0} u_\B}  (F^{(1)}_\B)_{\ell Z}}{2|\nab_{g_0} (u_\A\pm u_\B)|_{g_0} }  |\nab_{g_0} u_\B|_{g_0} \Big( |\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0} - |\nab_{g_0} u_\A|_{g_0} |\nab_{g_0} u_\B|_{g_0} \Big)
\\&\quad \pm \frac{(F^{(1)}_\B)^{\ell}_{\nab_{g_0} u_\A}   (F^{(1)}_\A)_{\ell Z}}{2|\nab_{g_0}(u_\A\pm u_\B)|_{g_0}}  |\nab_{g_0} u_\A|_{g_0} \Big( |\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0} - |\nab_{g_0} u_\A|_{g_0} |\nab_{g_0} u_\B|_{g_0} \Big)
\\&\quad  +   \frac{1}{4}\left|\bar{F}^{(1)}_\A\cdot \bar{F}^{(1)}_\B\right|_{g_0}\frac{|\nab_{g_0} u_\A\cdot Z|_{g_0}}{|\nab_{g_0}(u_\A\pm u_\B)|_{g_0}} \Big( \mp |\nab_{g_0} u_\A|_{g_0} + |\nab_{g_0} u_\B|_{g_0} \Big)
\\&\hspace{6cm}\times \Big(|\nab_{g_0} u_\A\cdot \nab_{g_0} u_\B|_{g_0} - |\nab_{g_0} u_\A|_{g_0} |\nab_{g_0} u_\B|_{g_0} \Big) ,
\end{align*}
where we also used $|\nab_{g_0} u_\B\cdot Z|_{g_0}=\mp |\nab_{g_0} u_\A\cdot Z|_{g_0} $.  Using \eqref{wcX 2pm} we finally obtain \eqref{ka1pm 2} and thus the lemma.
\end{proof}

\paragraph{Setting the gauge.} We conclude this section by defining initial data for $_(_)_0 $. They will be chosen so that \eqref{GWC} holds on $_0$. Thanks to the definition of $Υ^ρ$ in \eqref{Upsilon} this condition rewrites
\begin{align}
\g_\la^{\mu\nu} \pth{ \dr_\mu (\h_\la)_{\si\nu} - \half \dr_\si (\h_\la)_{\mu\nu}} & = \tilde{\Upsilon}_\si ,\label{condition dth0}
\end{align}
where we defined
\begin{equation*}
\begin{aligned}
\tilde{\Upsilon}_\si & \vcentcolon = - \g_\la^{\mu\nu}   \sum_\A  \sin\left(\frac{u_\A}{\lambda} \right) \pth{\dr_\mu (\F_\A)_{\si\nu} - \half \dr_\si (\F_\A)_{\mu\nu} }
\\&\quad - \la  \g_\la^{\mu\nu} \sum_{\substack{u\in\mathcal{N}\\T\in\{\cos,\sin\}}}T\pth{\frac{u}{\la}} \pth{\dr_\mu (\gg^{(3,h)}_{u,T})_{\si\nu} - \half \dr_\si (\gg^{(3,h)}_{u,T})_{\mu\nu}}
\\&\quad - \la  \g_\la^{\mu\nu} \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}}T\pth{\frac{v}{\la}} \pth{\dr_\mu (\gg^{(3,e)}_{v,T})_{\si\nu} - \half \dr_\si (\gg^{(3,e)}_{v,T})_{\mu\nu}}
\\&\quad  -  \g_\la^{\mu\nu}  \sum_{\substack{v\in\mathcal{I}\\T\in\{\cos,\sin\}}}T'\pth{\frac{v}{\la}}\pth{ \dr_\mu v(\gg^{(3,e)}_{v,T})_{\si\nu} - \half \dr_\si v(\gg^{(3,e)}_{v,T})_{\mu\nu} }
\\&\quad +  (\g_\la)_{\rho\si} \g_0^{\rho\ga} \h_\la^{\mu\nu} \Bigg(\dr_\mu (\g_0)_{\ga\nu} - \half \dr_\ga (\g_0)_{\mu\nu} -  \sum_\A  \sin\left( \frac{u_\A}{\lambda} \right) \pth{ \dr_\mu u_\A (F^{(1)}_\A)_{\ga\nu} - \half \dr_\ga u_\A (F^{(1)}_\A)_{\mu\nu}  }\Bigg).
\end{aligned}
\end{equation*}
Note that $Υ̃^ρ_|__0$ can be completely computed. Moreover, \eqref{condition dth0} can be ensured by the following choice of initial data for $_t(_)_0$:
\begin{align}
\dr_t (\h_\la)_{00|_{\Si_0}}  & \vcentcolon = \frac{2}{\g_\la^{00}}\pth{\tilde{\Upsilon}_0 -  \g_\la^{0i}  \dr_i (\h_\la)_{00} - \g_\la^{ij} \pth{ \dr_i (\h_\la)_{0 j} - \half \dr_t (\h_\la)_{ij}}}, \label{ID dth 00}
\\ \dr_t (\h_\la)_{k0|_{\Si_0}}  & \vcentcolon = \frac{1}{\g_\la^{00}}\bigg( \tilde{\Upsilon}_k + \half \g_\la^{00}\dr_k (\h_\la)_{00} - \g_\la^{0i} \pth{ \dr_t (\h_\la)_{k i} - \dr_k (\h_\la)_{0i} +  \dr_i (\h_\la)_{k0}}  \label{ID dth 0k}
\\&\hspace{6cm}- \g_\la^{ij} \pth{ \dr_i (\h_\la)_{k j} - \half \dr_k (\h_\la)_{ij}}\bigg).\non
\end{align}
Note that the RHS of \eqref{ID dth 00} and \eqref{ID dth 0k} are fully known on $_0$ and that we used the fact that $_^00≠0$. Note also that the initial data for $_t(_)_0|__0$ allows us easily to obtain $_(_)_0|__0$.

\subsubsection{Conclusion}

Combining the results of the previous sections, we can prove the following.

\begin{prop}\label{prop ID main}
Let $\pth{ \bar{F}^{(1)}_\A}_{\A\in\mathcal{A}}$ be a seed. There exists a choice of initial data for the system \eqref{eq F1}-\eqref{eq h} such that
\begin{itemize}
\item[(i)] the algebraic conditions \eqref{pola F1}-\eqref{GWC} hold on $\Si_0$,
\item[(ii)] the induced metric and the second fundamental form computed from $\g_{\la|_{\Si_0}}$ solve the constraint equations on $\Si_0$.
\end{itemize}
Moreover, the initial data for $F^{(1)}_\A$, $F^{(2,1)}_\A$, $F^{(2,2)}_\A$ are supported in $B_R$ and there exists $C_{\mathrm{init}}=C_{\mathrm{init}}(N,\de,R)>0$ such that
\begin{align}
\l F^{(1)}_\A\r_{H^N(\Si_0)} + \l F^{(2,1)}_\A\r_{H^{N-1}(\Si_0)} + \l F^{(2,2)}_\A\r_{H^{N-1}(\Si_0)} & \leq C_{\mathrm{init}} \e,\label{estim ID 1}
\\ \max_{r\in\llbracket 0 ,4 \rrbracket}\la^r\l \dr \nab^r \h_\la \r_{L^2_{\de+r+1}(\Si_0)} & \leq C_{\mathrm{init}} \e.\label{estim ID 2}
\end{align}
\end{prop}

\begin{proof}
We start by defining initial data for $F^{(1)}_\A$ and $\F_\A$ following \eqref{ID F1 ij}-\eqref{ID F1 0} and \eqref{ID F}. Thanks to \eqref{Q 0 ell}-\eqref{Q 0 0}, we can then define initial data for $F^{(2,1)}_\A$ and $F^{(2,2)}_\A$ following \eqref{ID F21 ij}-\eqref{ID F22 0e}. This then allows us to define every terms on the RHS of \eqref{def ga2pm}, \eqref{def ka11}, \eqref{def ka12} and \eqref{def ka1pm} and thus allows to define $\ga^{(2,\pm)}_{\A\B}$, $\ka^{(1,1)}_\A$, $\ka^{(1,2)}_\A$ and $\ka^{(1,\pm)}_{\A\B}$ by \eqref{def ga2pm}, \eqref{def ka11}, \eqref{def ka12} and \eqref{def ka1pm} respectively. Note that these tensors only depend on the seed $\pth{ \bar{F}^{(1)}_\A}_{\A\in\mathcal{A}}$ and are all of the form $\cro{ (\bar{F}^{(1)})^2 }$, which thanks to \eqref{estim seed} implies
\begin{align}
\l \ga^{(2,\pm)}_{\A\B} \r_{H^N} + \l  \ka^{(1,1)}_\A\r_{H^N} + \l  \ka^{(1,2)}_\A \r_{H^N} + \l  \ka^{(1,\pm)}_{\A\B} \r_{H^N} \lesssim \e^2.
\end{align}
In particular, if $\e$ is small enough then \eqref{estim ga kappa param} holds and we can apply Proposition \ref{prop constraint main} and obtain a solution of the constraint equations $(g_\la,k_\la)$ given by \eqref{g la}-\eqref{k la}. This allows us to define every terms on the RHS of \eqref{ID h ij}-\eqref{ID h 0}, \eqref{ID dth ij} and \eqref{ID dth 00}-\eqref{ID dth 0k}, which in turn define initial data for $\h_\la$ and $\T_\la\h_\la$. This concludes the definition of initial data for the system \eqref{eq F1}-\eqref{eq h}. Moreover, thanks to Lemmas \ref{lem ga2pm}, \ref{lem ka11}, \ref{lem ka12} and \ref{lem ka1pm}, the induced metric and the second fundamental form computed from $\g_\la$ on $\Si_0$ are given by $(g_\la,k_\la)$ and thus are solving the constraint equations on $\Si_0$. We now verify that the algebraic conditions \eqref{pola F1}-\eqref{GWC} hold on $\Si_0$: \eqref{pola F1}-\eqref{pola F1 bis} and \eqref{energy F1} hold thanks to \eqref{ID F1 ij}-\eqref{ID F1 0} and \eqref{assum F1}, \eqref{pola F} holds thanks to \eqref{ID F}, \eqref{pola F21}-\eqref{pola F22} hold thanks to Lemma \ref{lem pola F2 init}, \eqref{pola g3h} already holds thanks to Remark \ref{remark g3h}, \eqref{GWC} holds thanks to the choice of initial data for $\dr_t(\h_\la)_{0\a|_{\Si_0}}$. Finally, the estimate \eqref{estim ID 1} follows from \eqref{estim seed}, \eqref{ID F1 ij}-\eqref{ID F1 0}, \eqref{ID F21 ij}-\eqref{ID F22 0e} and the estimate \eqref{estim ID 2} follows in addition from \eqref{estim reste cons 1}-\eqref{estim reste cons 2}, \eqref{ID h ij}-\eqref{ID h 0}, \eqref{ID dth ij} and \eqref{ID dth 00}-\eqref{ID dth 0k}.
\end{proof}

\section{Solving the system}\label{section solving the system}

In this section, we present our result on well-posedness for the system \eqref{eq F1}-\eqref{eq h} with the initial data constructed in Proposition \ref{prop ID main}. It is contained in the following proposition.

\begin{prop}\label{prop system}