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Nonparametric Identification of Two-Way Unobserved Heterogeneity
\maketitle
\begin{abstract}
We study identification of two-way unobserved heterogeneity in the
nonparametric panel regression $G_{it}=g(\alpha_i,\gamma_t)+\varepsilon_{it}$,
where identification of the latent types reduces to constructing identified,
\emph{injective} proxies for them. To this end we consider the singular value
decomposition (SVD) of the bivariate regression function
$g(\alpha,\gamma)$ on a product domain $\Omega_{\alpha}\times\Omega_{\gamma}$, whose left singular
functions $\{u_r\}$ serve as proxies for the unobserved heterogeneity parameter
$\alpha$.
The arguments are symmetric for $\{v_r\}$ vis-\`a-vis $\gamma$.
We work under an \emph{observational-equivalence simplification}: two
values of $\alpha$ that induce the same conditional response
$g(\alpha,\cdot)$ are identified, so that the response map
$\alpha\mapsto g(\alpha,\cdot)$ is injective by construction. We show two
things. First, this reduction is \emph{equivalent} to injectivity of the full
collection of left singular eigenfunctions, so no further condition is needed
over the infinite collection $\{u_r\}_{r\ge1}$. Second, under a single additional
\emph{local injectivity} condition, a finite collection of leading eigenfunctions
$U_R=(u_1^{\top},\dots,u_R^{\top})^{\top}$ is injective for all sufficiently large $R$. The proof
reduces a global univalence question to a local first-order condition plus a
topological compactness argument, bypassing the global
Jacobian conditions usually required.
\end{abstract}
\section{Introduction}
Suppose we observe $Y_{it}\in\mathbb{R}^{d_Y}$ from
\begin{equation}
Y_{it}=m\left(\alpha_i,\gamma_t,\eta_{it}\right),
\qquad i=1,\dots,N,\quad t=1,\dots,T,
\label{eq:DGP}
\end{equation}
where $\alpha_i\in\mathbb{R}^{d_{\alpha}}$ and $\gamma_t\in\mathbb{R}^{d_{\gamma}}$ are unobserved
finite-dimensional fixed effects and $\eta_{it}$ is an idiosyncratic shock.
We are interested in identifying (transformations of) the realised values of
the fixed effects, together with functionals of the distribution of
$Y_{it}\mid(\alpha_i,\gamma_t)$, under large $N$--large $T$ asymptotics. Our
identification relies on the fundamental requirement that there is a
known transformation $G:\mathbb{R}^{d_Y}\to\mathbb{R}^{d_G}$ such that variation in the
conditional mean of $G_{it}:=G(Y_{it})$ separates the types. Writing
$g(\alpha_i,\gamma_t):=\mathbb{E}[G_{it}\mid\alpha_i,\gamma_t]$, this is the
nonparametric two-way regression
\begin{equation}
\text{Model}:\qquad
G_{it}=g\left(\alpha_i,\gamma_t\right)+\varepsilon_{it},
\qquad \mathbb{E}\left[\varepsilon_{it}\mid\alpha_i,\gamma_t\right]=0,
\label{eq:modelI}
\end{equation}
with $g$ a nonparametric object subject only to smoothness
conditions, no parametric restrictions on the distribution of
$(\alpha_i,\gamma_t)$, and the leading case being continuously distributed
types.
This problem has received considerable attention:
\citet{bonhomme2021discretizing} developed first-step nonparametric
estimators of the fixed effects by discretisation based on
\eqref{eq:modelI}, used in a second step to estimate a parametric model for
$m$; \citet{beyhum2025inference} employed that first step for a
semiparametric model; \citet{freeman2023linear,freeman2026inference}
estimated the same model with alternative estimators of $g$ and
$(\alpha_i,\gamma_t)$.
This paper isolates the identification core of this program. As
Section~\ref{sect:proxy} makes precise, identification of $g$ and the types
reduces to constructing \emph{identified and injective} proxies for
$\alpha_i$ and $\gamma_t$. We construct such proxies from the singular value
decomposition (SVD) of the regression function $g$ and establish their
injectivity.
\section{Identification via Proxies}\label{sect:proxy}
The identification scheme rests on proxies that span the variation in
$g(\alpha_i,\gamma_t)$, as formalised in the following high-level assumption.
\begin{assumption}
\label{ass:proxyID}
(i) The distribution of $G_{it}\mid\{\alpha_i,\gamma_t\}$ is invariant with
respect to $(i,t)$; (ii) there exist $\lambda_i=U(\alpha_i)\in\mathbb{R}^{d_\lambda}$
and $f_t=V(\gamma_t)\in\mathbb{R}^{d_f}$, with $U,V$ measurable and
$d_\lambda,d_f<\infty$, that are
identified from the distribution of $\{Y_{it}:i\ge1,\,t\ge1\}$, and are injective; (iii) for any
given $(\alpha,\gamma)$ there exist $\delta=\delta(\alpha,\gamma)>0$ and
$C=C(\alpha,\gamma)<\infty$ such that,
whenever $\lVert (\alpha,\gamma)-(\alpha',\gamma')\rVert<\delta$,
\[
\lVert g(\alpha,\gamma)-g(\alpha',\gamma')\rVert
\le C\bigl\{\lVert \alpha-\alpha'\rVert+\lVert \gamma-\gamma'\rVert\bigr\}.
\]
\end{assumption}
Part (i) makes $g(\alpha_i,\gamma_t)=\mathbb{E}[G_{it}\mid\alpha_i,\gamma_t]$
invariant in $(i,t)$, while part (ii) lets the identified proxies
track the variation in $g$, with the injectivity component proved in this paper.
Part (iii) is a low-level Lipschitz condition over the primitive domain.
Note proxies $\lambda$ and $f$ are identified in \cite{freeman2023linear}. In the following we target identification of $g, \alpha,\gamma$ from these proxies.
\begin{theorem}
\label{th:proxyID}
Under Assumption~\ref{ass:proxyID} (i)-(ii),
$g_0(\lambda_i,f_t):=\mathbb{E}[G_{it}\mid\lambda_i,f_t]$ satisfies
\[
g_0(\lambda_i,f_t)
=g(\alpha_i,\gamma_t)
=g\bigl(U^{-1}(\lambda_i),V^{-1}(f_t)\bigr),
\]
so $g$ at the realised types is identified. Moreover $g_0$ inherits the
modulus of Assumption~\ref{ass:proxyID}(iii) in the primitives since whenever
$\lVert (\alpha,\gamma)-(\alpha',\gamma')\rVert<\delta(\alpha,\gamma)$,
\[
\lVert g_0(\lambda,f)-g_0(\lambda',f')\rVert
\le C(\alpha,\gamma)\bigl\{\lVert \alpha-\alpha'\rVert+\lVert \gamma-\gamma'\rVert\bigr\},
\]
with $(\lambda,f)=(U(\alpha),V(\gamma))$ and
$(\lambda',f')=(U(\alpha'),V(\gamma'))$. If in addition, for some $q\ge1$,
$g$ is $C^q$ and $U$ and $V$ are injective $C^q$ immersions, then $g_0$ is
$C^q$, differentiability being read on the image manifolds $U(\Omega_{\alpha})$ and
$V(\Omega_{\gamma})$.
\end{theorem}
\begin{proof}
Since $(\lambda_i,f_t)$ are identified they may be treated as observed, so
$g_0$ is identified, and \eqref{eq:modelI} with
Assumption~\ref{ass:proxyID}(i) gives
$g_0(\lambda_i,f_t)=\mathbb{E}[g(\alpha_i,\gamma_t)\mid\lambda_i,f_t]$. By
injectivity in part (ii), every nonempty level set of $U$ or $V$ is a
singleton, hence trivially connected, and the joint level set
$S=U^{-1}(\lambda_i)\times V^{-1}(f_t)$ is the singleton
$\{(U^{-1}(\lambda_i),V^{-1}(f_t))\}$, and since
$\lambda_i=U(\alpha_i)$ and $f_t=V(\gamma_t)$, its sole element is
$(\alpha_i,\gamma_t)$ itself. Thus $(\alpha_i,\gamma_t)$, and with it
$g(\alpha_i,\gamma_t)$, is a fixed function of $(\lambda_i,f_t)$; this
function is measurable, because a Borel injection between Borel subsets of
Euclidean spaces has Borel image and Borel inverse (the Lusin--Suslin
theorem), and it is continuous in the case relevant below, where $U$ and $V$
are continuous with compact domains. The conditional expectation of a
$\sigma(\lambda_i,f_t)$-measurable variable is the variable itself, whence
$g_0(\lambda_i,f_t)=g(\alpha_i,\gamma_t)=g(U^{-1}(\lambda_i),V^{-1}(f_t))$.
Part (iii) is not used for this identity, and no constancy argument on $S$
is required.
The modulus is now immediate as the identity at both points and part (iii) give,
whenever $\lVert (\alpha,\gamma)-(\alpha',\gamma')\rVert<\delta(\alpha,\gamma)$,
\[
\lVert g_0(\lambda,f)-g_0(\lambda',f')\rVert
=\lVert g(\alpha,\gamma)-g(\alpha',\gamma')\rVert
\le C(\alpha,\gamma)\bigl\{\lVert \alpha-\alpha'\rVert+\lVert \gamma-\gamma'\rVert\bigr\}.
\]
For differentiability, an injective $C^q$ immersion on a compact domain is a
homeomorphism onto its image, a $C^q$ submanifold, with $C^q$ inverse there
(the rank theorem holds in the $C^q$ category for every $q\ge1$), so the
composition $g_0=g\circ(U^{-1},V^{-1})$ is $C^q$.
\end{proof}
We call $(\lambda_i,f_t)$ satisfying Assumption~\ref{ass:proxyID}
\emph{valid proxies}, and \emph{regular proxies} when $U$ and $V$ are in
addition smooth immersions. Validity alone does not guarantee that the
normalised regression function $g_0$ inherits the smoothness of $g$;
regularity does (Theorem~\ref{th:proxyID}), and injectivity is its
nontrivial ingredient.
Whatever common order of
differentiability $(g,U,V)$ possess transfers to $g_0$. For the
eigenfunction proxies constructed below, that order is supplied by the
Sobolev order $p$ of Theorem~\ref{thm:primitive}. Functions $g\in H^{p}$ with
$p>q+(d_{\alpha}+d_{\gamma})/2$ delivers a $C^q$ kernel and $C^q$
singular functions, hence a $C^q$ normalised regression $g_0$.
The
remainder of the paper constructs regular proxies
from the SVD of $g$, i.e. the eigenfunction proxies $\lambda_i=U_R(\alpha_i)$ and
$f_t=V_R(\gamma_t)$ of Sections~\ref{sect:setup} and~\ref{sec:gamma}, and establishes their
injectivity
pointwise almost
everywhere (Proposition~\ref{prop:ae})
and under stronger conditions, uniformly on the support for all sufficiently large finite $R$
(Theorem~\ref{thm:main}); the two regimes, joined with their $\gamma$-side
analogues into a single statement for the proxy pair $(U_R,V_R)$, are
collected up front in Theorem~\ref{thm:headline}. When the kernel is additive across coordinates of
the types, the construction respects that structure and the normalised
regression decomposes componentwise, removing the curse of dimensionality in
the associated nonparametric regression
(Section~\ref{sec:additive}, Theorem~\ref{thm:additive}).
\section{Setup and Notation}\label{sect:setup}
Let $g$ be the two-way regression function of \eqref{eq:modelI}.
Throughout, $\Omega_{\alpha}\subseteq\mathbb{R}^{d_{\alpha}}$ and $\Omega_{\gamma}\subseteq\mathbb{R}^{d_{\gamma}}$ are the supports
of the two heterogeneity parameters, and the probability measures
$\pi_\alpha$ on $\Omega_{\alpha}$ and $\pi_\gamma$ on $\Omega_{\gamma}$ have densities (with respect
to Lebesgue measure) bounded away from $0$ and $\infty$. The kernel $g$ is
$\mathbb{R}^{d_G}$ valued, where $d_G$ is the dimension of the chosen transformation $G$,
applied to the full observed vector $Y_{it}$, and it is the only free
dimension on the response side, stacked moments or covariates being
accommodated by enlarging $G$ itself.
Notation $L^{2}$ means scalar valued, $\mathcal{H}$ means $\mathbb{R}^{d_G}$ valued, and the subscript names
the domain. The subscript $\pi$ always denotes the measure of the indicated
domain, i.e. $\pi_\alpha$ on $\Omega_{\alpha}$, $\pi_\gamma$ on $\Omega_{\gamma}$, and
$\pi_\alpha\otimes\pi_\gamma$ on $\Omega_{\alpha}\times\Omega_{\gamma}$. The scalar spaces then
need no further symbol.
For the vector-valued spaces we abbreviate the $L^{2}_{\pi}$ mappings to $\mathbb{R}^{d_G}$ as
\[
\mathcal{H}_{\gamma}:=L^{2}_{\pi}(\Omega_{\gamma};\mathbb{R}^{d_G}),
\qquad
\mathcal{H}_{\alpha}:=L^{2}_{\pi}(\Omega_{\alpha};\mathbb{R}^{d_G}),
\]
with inner products
$\langle \phi,\,\psi\rangle_{L^{2}_{\pi}(\Omega_{\gamma})}=\int_{\Omega_{\gamma}}\phi\psi\,d\pi_\gamma$ and
$\langle \phi,\,\psi\rangle_{\mathcal{H}_{\gamma}}=\sum_{j=1}^{d_G}\langle \phi_j,\,\psi_j\rangle_{L^{2}_{\pi}(\Omega_{\gamma})}$, the
$\alpha$-side case being analogous. These are the only spaces used in the
main text, and each hosts one kind of object.
The scalar right singular
functions $v_r$ below live in $L^{2}_{\pi}(\Omega_{\gamma})$, the $\mathbb{R}^{d_G}$-valued left singular
functions $u_r$ and the slices $g(\cdot,\gamma)$ live in $\mathcal{H}_{\alpha}$, and the
responses $g(\alpha,\cdot)$ live in $\mathcal{H}_{\gamma}$. Mean-square convergence on the
product domain is convergence in $L^{2}_{\pi}(\Omega_{\alpha}\times\Omega_{\gamma};\mathbb{R}^{d_G})$. We
write $\lVert \cdot\rVert$ for the Euclidean norm on $\mathbb{R}^{k}$, for the Frobenius
norm of a matrix, and for the operator norm of an operator, the meaning fixed
by the argument.
\subsection{The integral operator and its SVD}
Let $g\in L^{2}_{\pi}(\Omega_{\alpha}\times\Omega_{\gamma};\mathbb{R}^{d_G})$, that is,
$\int_{\Omega_{\alpha}}\int_{\Omega_{\gamma}}\lVert g\rVert^2\,d\pi_\gamma\,d\pi_\alpha<\infty$. Define
the Hilbert-Schmidt integral operator $T_g$
\begin{align}
T_g:L^{2}_{\pi}(\Omega_{\gamma})\longrightarrow\mathcal{H}_{\alpha},
\qquad
(T_g\phi)(\alpha)=\int_{\Omega_{\gamma}} g(\alpha,\gamma)\,\phi(\gamma)\,d\pi_\gamma(\gamma)
\ \in\mathbb{R}^{d_G},
\label{eq:Tg}
\end{align}
and its associated dual $T_g^*$
\begin{align*}
T_g^*:\mathcal{H}_{\alpha}\longrightarrowL^{2}_{\pi}(\Omega_{\gamma}),
\qquad
(T_g^*\psi)(\gamma)=\int_{\Omega_{\alpha}} g(\alpha,\gamma)^{\top}\psi(\alpha)\,d\pi_\alpha(\alpha).
\end{align*}
By the singular value decomposition of a compact operator between Hilbert
spaces, there exist a nonincreasing sequence of singular values
$\sigma_1\ge\sigma_2\ge\cdots\geq0$,
an orthonormal system $\{u_r\}_{r\ge1}\subset\mathcal{H}_{\alpha}$ spanning
$\overline{\operatorname{range}(T_g)}$, and an orthonormal system
$\{v_r\}_{r\ge1}\subsetL^{2}_{\pi}(\Omega_{\gamma})$ spanning $\overline{\operatorname{range}(T_g^*)}$,
such that by \cite{reed1980methods},
\begin{equation}
g(\alpha,\gamma)=\sum_{r = 1}^\infty\sigma_r\,u_r(\alpha)\,v_r(\gamma),
\label{eq:svd}
\end{equation}
with convergence in $L^{2}_{\pi}(\Omega_{\alpha}\times\Omega_{\gamma};\mathbb{R}^{d_G})$.
Functions $u_r/v_r$ are the \emph{left/right singular functions}; the left
ones are $\mathbb{R}^{d_G}$ valued and the right ones scalar.
Section~\ref{sec:gamma} studies identification of $\gamma$ where the role of $u_r$ and $v_r$ are reversed.
The asymmetry in \eqref{eq:svd} reflects the choice of operator.
The domain of $T_g$ is scalar and its codomain is $\mathbb{R}^{d_G}$ valued,
and each family of singular functions lives in this space. Coordinatewise, \eqref{eq:svd} says
$g_j(\alpha,\gamma)=\sum_r\sigma_r u_{r,j}(\alpha)v_r(\gamma)$ for
$j=1,\dots,d_G$.
At each rank $r$ a \emph{single} scalar $v_r$ is shared by
all $d_G$ components of the kernel, and the component-specific information
sits in the vector $u_r(\alpha)$. The $v_r$ are eigenfunctions of
$T_g^{*}T_g$, whose kernel
$\int_{\Omega_{\alpha}}g(\alpha,\gamma)^{\top}g(\alpha,\gamma')\,d\pi_\alpha$ pools
variation across components, so in general $v_r$ is not a singular function
of any single component $g_j$. The pooling is what the proxy needs, and a pair
of time types separated by any one component is then separated by the common
system. Section~\ref{sec:gamma} derives $\gamma$-side proxies through the adjoint, a distinct decomposition, with
the component index attached to $\gamma$ instead.
For $p\in\mathbb{N}$ and a bounded domain $\Omega$, $H^p(\Omega)$ denotes the order-$p$
$L^2$ Sobolev space under Lebesgue measure and, for a measure $\Pi$ on
$\Omega$ with density $\pi$, its weighted counterpart is
\[
H^p(\Pi;\mathbb{R}^{k}):=\bigl\{u:\Omega\to\mathbb{R}^{k}\ :\ D^s u\in L^2(\Pi),\
|s|\le p\bigr\},
\qquad
\lVert u\rVert_{H^p(\Pi)}^2:=\sum_{|s|\le p}\int_\Omega\lVert D^s u\rVert^2\,d\Pi ,
\]
where $L^2(\Pi)$ is $L^2$ with respect to joint measure $\Pi$.
The scalar case $k=1$ being written $H^p(\Pi)$.
The two coincide under the standing bounds on the densities. In particular
\emph{no smoothness of the measures is required},
since $\pi_\alpha$ and $\pi_\gamma$ are only ever integrated against, not
differentiated.
\begin{lemma}
\label{lem:weighted}
If $0<c\le\pi\le C$ on $\Omega$, then $H^p(\Pi)=H^p(\Omega)$ as sets, with
\[
c\,\lVert u\rVert_{H^p(\Omega)}^2
\;\le\;\lVert u\rVert_{H^p(\Pi)}^2
\;\le\;C\,\lVert u\rVert_{H^p(\Omega)}^2 .
\]
This applies to $\pi_\alpha$, $\pi_\gamma$, and the product
$\pi_\alpha\otimes\pi_\gamma$, whose densities are bounded away from $0$ and
$\infty$.
\end{lemma}
\begin{proof}
Each term of the norm is $\int_\Omega|D^s u|^2\,\pi\,dx$ with $|s|\le p$, and
the bounds on $\pi$ sandwich it between $c\int|D^s u|^2dx$ and
$C\int|D^s u|^2dx$. No derivative falls on $\pi$, so no regularity of $\pi$
is used.
\end{proof}
Lemma~\ref{lem:weighted} states that the sets induced by the weighed and non-weighted Sobolev spaces of eigenfunctions are the same, i.e. $H^p(\Pi)=H^p(\Omega)$, and that the weighting also induces equivalent norms over these spaces.
Hence, eigenfunctions admitted by $H^p(\Pi)$, and related spectral decay, inherit the properties of the unweighted $H^p(\Omega)$, which have been studied thoroughly, see e.g. \cite{griebel2014approximation,griebel2019singular}.
We collect results relevant to our setting in Appendix~\ref{sect:sobolev}.
\subsection{Eigenfunction maps and truncations}
For $R\in\mathbb{N}\cup\{\infty\}$ define the (possibly infinite) \emph{left
eigenfunction map}
\begin{equation}
U_R:\Omega_{\alpha}\longrightarrow\mathbb{R}^{d_G R},
\qquad
U_R(\alpha):=\bigl(u_1(\alpha)^{\top},\dots,u_R(\alpha)^{\top}\bigr)^{\top},
\label{eq:UR}
\end{equation}
with the convention $U_\infty(\alpha)=\{u_r(\alpha)\}_{r=1}^\infty$. We call $U_R$ for finite $R$ the \emph{leading truncation} of order
$R$, since by \eqref{eq:svd} the components are ordered by decreasing singular
value. The \emph{response map} is
\begin{equation}
\Phi:\Omega_{\alpha}\longrightarrow\mathcal{H}_{\gamma},
\qquad
\Phi(\alpha):=g(\alpha,\cdot).
\label{eq:Phi}
\end{equation}
The truncated function is $g_R(\alpha,\gamma):=\sum_{r=1}^R\sigma_r u_r(\alpha)
v_r(\gamma)$ and the truncated response map is $\Phi_R(\alpha):=g_R(\alpha,\cdot)$.
\subsection{Gradient information matrices}
Define for
$R\in\mathbb{N}\cup\{\infty\}$ the \emph{gradient information matrix}
\begin{equation}
M_R(\alpha):=\sum_{r=1}^R\sigma_r^2\,Du_r(\alpha)^{\top}Du_r(\alpha)
\;\in\;\mathbb{R}^{d_{\alpha}\timesd_{\alpha}},
\label{eq:MR}
\end{equation}
where $Du_r(\alpha)\in\mathbb{R}^{d_G\timesd_{\alpha}}$ is the Jacobian of $u_r$ at $\alpha$,
a row vector transposed to a gradient when $d_G=1$. Each
summand is positive semidefinite, so $M_R(\alpha)\succeq0$ and
$R\mapsto M_R(\alpha)$ is nondecreasing in the positive semidefinite order.
The limit $M_\infty(\alpha):=\lim_{R\to\infty}M_R(\alpha)$ exists in the positive-semidefinite order.
Under Assumption~\ref{ass:domain} it is finite and admits the closed form
\begin{equation}
M_\infty(\alpha)
=\int_{\Omega_{\gamma}}J(\alpha,\gamma)^{\top}J(\alpha,\gamma)\,d\pi_\gamma(\gamma),
\qquad J(\alpha,\gamma):=D_\alpha g(\alpha,\gamma)\in\mathbb{R}^{d_G\timesd_{\alpha}};
\label{eq:Minf}
\end{equation}
this, and the underlying gradient expansion
$J(\alpha,\cdot)=\sum_r\sigma_r Du_r(\alpha)v_r$, converging entrywise in
$L^{2}_{\pi}(\Omega_{\gamma})$, is established in Lemma~\ref{lem:gradexp}. We use $J$ and
$D_\alpha g$ interchangeably.
\section{Assumptions}
We collect sufficient conditions. Assumption~\ref{ass:domain} is
standard, and we show $u_r$ is $C^1$ on $\Omega_{\alpha}$ in Theorem~\ref{thm:primitive} under weak regularity.
Assumption~\ref{ass:obseq} is the observational-equivalence reduction.
Assumption~\ref{ass:ae-immersion} is used to show a finite set of
eigenfunctions inherits the injectivity property pointwise almost-everywhere,
and Assumption~\ref{ass:multi-index} is its uniform strengthening.
The price of the weaker Assumption~\ref{ass:ae-immersion} is that the result
is demoted to pointwise almost-everywhere.
\begin{assumption}[Domain and smoothness]
\label{ass:domain}
Support $\Omega_{\alpha}\subseteq\mathbb{R}^{d_{\alpha}}$ is compact, convex, and connected, and
$\Omega_{\gamma}\subseteq\mathbb{R}^{d_{\gamma}}$ is compact. The function $g$ is continuously
differentiable in $\alpha$, with $D_\alpha g$ jointly continuous on
$\Omega_{\alpha}\times\Omega_{\gamma}$; in particular each $u_r$ is $C^1$ on $\Omega_{\alpha}$.
\end{assumption}
\begin{lemma}
\label{lem:Phi-cont}
Under Assumption~\ref{ass:domain}, $\Phi$ is Lipschitz from $\Omega_{\alpha}$ into $\mathcal{H}_{\gamma}$,
$\lVert \Phi(\alpha_1)-\Phi(\alpha_2)\rVert_{\mathcal{H}_{\gamma}}\le L\,\lVert \alpha_1-\alpha_2\rVert$ with
$L=\sup_{\Omega_{\alpha}\times\Omega_{\gamma}}\lVert J\rVert<\infty$. Hence
$\Delta(\alpha_1,\alpha_2):=\lVert \Phi(\alpha_1)-\Phi(\alpha_2)\rVert_{\mathcal{H}_{\gamma}}^2$ is
continuous on $\Omega_{\alpha}\times\Omega_{\alpha}$.
\end{lemma}
\begin{proof}
The increment admits the integral form $g(\alpha_1,\gamma)-g(\alpha_2,\gamma)
=\int_0^1 J(\alpha_2+t(\alpha_1-\alpha_2),\gamma)
(\alpha_1-\alpha_2)\,dt$; since $J$ is bounded on the compact
$\Omega_{\alpha}\times\Omega_{\gamma}$, taking $\mathcal{H}_{\gamma}$ norms gives the bound.
\end{proof}
\begin{lemma}
\label{lem:gradexp}
Under Assumption~\ref{ass:domain}, for every $\alpha\in\Omega_{\alpha}$:
\begin{enumerate}
\item[\textup{(i)}] the differentiated series reproduces the Jacobian,
\begin{equation}
J(\alpha,\cdot)=\sum_{r = 1}^\infty\sigma_r\,Du_r(\alpha)\,v_r,
\qquad\text{with convergence entrywise in }L^{2}_{\pi}(\Omega_{\gamma});
\label{eq:gradseries}
\end{equation}
\item[\textup{(ii)}] consequently $\sum_{r\ge1}\sigma_r^2\lVert Du_r(\alpha)\rVert^2
\le\int_{\Omega_{\gamma}}\lVert J(\alpha,\gamma)\rVert^2d\pi_\gamma<\infty$, the monotone
limit $M_\infty(\alpha)=\sum_{r\ge1}\sigma_r^2 Du_r(\alpha)^{\top}
Du_r(\alpha)$ is finite, and it equals the closed form
\eqref{eq:Minf}.
\end{enumerate}
\end{lemma}
\begin{proof}
It suffices to establish \eqref{eq:fourier}, and with it (i)--(ii), for
$\alpha$ in the interior of $\Omega_{\alpha}$, which is dense in $\Omega_{\alpha}$ (a compact convex
set of positive Lebesgue measure, guaranteed by the density bounds, equals
the closure of its interior); the extension to boundary points follows by
continuity, since $Du_r$ and $\alpha\mapsto D_\alpha g(\alpha,\cdot)$ are
continuous and the subspace $\mathcal{V}$ below is closed.
Fix, then, $\alpha$ in the interior of $\Omega_{\alpha}$. Because $D_\alpha g$ is
jointly continuous on the
compact $\Omega_{\alpha}\times\Omega_{\gamma}$ it is bounded there, so differentiation under the integral
sign is justified in
$u_r(\alpha)=\sigma_r^{-1}\int_{\Omega_{\gamma}}g(\alpha,\gamma)v_r(\gamma)\,d\pi_\gamma$,
giving, entrywise,
\begin{equation}
\int_{\Omega_{\gamma}}J(\alpha,\gamma)\,v_r(\gamma)\,d\pi_\gamma
=\sigma_r\,Du_r(\alpha)\ \in\mathbb{R}^{d_G\timesd_{\alpha}}.
\label{eq:fourier}
\end{equation}
Fix an entry $(i,j)$. By \eqref{eq:fourier},
$\{\sigma_r\,\partial_{\alpha_j}u_{r,i}(\alpha)\}_{r\ge1}$ are the Fourier
coefficients of $\partial_{\alpha_j}g_i(\alpha,\cdot)\inL^{2}_{\pi}(\Omega_{\gamma})$ against the
orthonormal system $\{v_r\}$, so the scalar Bessel inequality gives
\[
\sum_{r\ge1}\sigma_r^2\,\bigl(\partial_{\alpha_j}u_{r,i}(\alpha)\bigr)^2
\le\lVert \partial_{\alpha_j}g_i(\alpha,\cdot)\rVert_{L^{2}_{\pi}(\Omega_{\gamma})}^2
=\int_{\Omega_{\gamma}}\bigl(\partial_{\alpha_j}g_i(\alpha,\gamma)\bigr)^2\,d\pi_\gamma .
\]
Now sum over the $d_Gd_{\alpha}$ entries. Using the Frobenius identities
$\lVert Du_r(\alpha)\rVert^2=\sum_{i,j}\bigl(\partial_{\alpha_j}u_{r,i}(\alpha)
\bigr)^2$ and $\lVert J(\alpha,\gamma)\rVert^2
=\sum_{i,j}\bigl(\partial_{\alpha_j}g_i(\alpha,\gamma)\bigr)^2$, this gives
\[
\sum_{r\ge1}\sigma_r^2\lVert Du_r(\alpha)\rVert^2
\le\int_{\Omega_{\gamma}}\lVert J(\alpha,\gamma)\rVert^2\,d\pi_\gamma
\le\sup_{\Omega_{\alpha}\times\Omega_{\gamma}}\lVert J\rVert^2<\infty,
\]
the final bound because $\pi_\gamma$ is a probability measure. This is the
summability in (ii); in particular the monotone psd limit $M_\infty(\alpha)$ is
finite.
It remains to identify the limit of $J(\alpha,\cdot)$. Let
$\mathcal{V}:=\overline{\operatorname{span}}\{v_r:r\ge1\}\subseteqL^{2}_{\pi}(\Omega_{\gamma})$, an orthonormal basis of which
is $\{v_r\}$, and let
$\mathcal{V}^{\,d_G}:=\{\phi\in\mathcal{H}_{\gamma}:\phi_i\in\mathcal{V}\text{ for each }i\}$, the
$d_G$-fold product of $\mathcal{V}$ and a closed subspace of $\mathcal{H}_{\gamma}$. Since the SVD
\eqref{eq:svd} converges in $L^{2}_{\pi}(\Omega_{\alpha}\times\Omega_{\gamma};\mathbb{R}^{d_G})$, we have
$g(\alpha,\cdot)\in\mathcal{V}^{\,d_G}$ for $\pi_\alpha$-almost every $\alpha$; as
$\alpha\mapsto g(\alpha,\cdot)$ is continuous into $\mathcal{H}_{\gamma}$
(Lemma~\ref{lem:Phi-cont}) and $\mathcal{V}^{\,d_G}$ is closed, in fact
$g(\alpha,\cdot)\in\mathcal{V}^{\,d_G}$
for \emph{every} $\alpha$. For a coordinate direction $e_j$ the difference
quotients $h^{-1}\bigl(g(\alpha+he_j,\cdot)-g(\alpha,\cdot)\bigr)$ lie in the
subspace $\mathcal{V}^{\,d_G}$ and, by the mean value theorem together with dominated
convergence (this is dominant since the constant
$\sup_{\Omega_{\alpha}\times\Omega_{\gamma}}\lVert J\rVert$ is integrable
since $\pi_\gamma$ is finite), converge in $\mathcal{H}_{\gamma}$ to $\partial_{\alpha_j}
g(\alpha,\cdot)$. As $\mathcal{V}^{\,d_G}$ is closed,
$\partial_{\alpha_j}g(\alpha,\cdot)\in\mathcal{V}^{\,d_G}$ and equals its orthogonal
projection onto $\mathcal{V}^{\,d_G}$, namely
$\sum_r\bigl(\int_{\Omega_{\gamma}}\partial_{\alpha_j}g(\alpha,\cdot)v_r\,d\pi_\gamma
\bigr)v_r$, which by \eqref{eq:fourier} is the $j$-th column of
$\sum_r\sigma_r Du_r(\alpha)v_r$. Collecting the columns proves (i).
Finally, expanding $M_\infty(\alpha)=\sum_r\sigma_r^2 Du_r(\alpha)^{\top}Du_r(\alpha)$ and using \eqref{eq:gradseries} with orthonormality of $\{v_r\}$ in
$L^{2}_{\pi}(\Omega_{\gamma})$ yields
$M_\infty(\alpha)=\int_{\Omega_{\gamma}}J^{\top}J\,d\pi_\gamma$,
the closed form \eqref{eq:Minf}.
\end{proof}
\begin{remark}
\label{rem:gradseries}
Lemma~\ref{lem:gradexp} is the only convergence fact used below.
It follows from
Assumption~\ref{ass:domain} alone and is weaker than uniform $C^1$ convergence of
\eqref{eq:svd}.
\end{remark}
\begin{assumption}[Observational Equivalence]
\label{ass:obseq}
Response map $\Phi$ in \eqref{eq:Phi} is injective by construction. That is, for all
$\alpha_1,\alpha_2\in\Omega_{\alpha}$,
\[
g(\alpha_1,\gamma)=g(\alpha_2,\gamma)\ \text{ for all }\gamma\in\Omega_{\gamma}
\quad\Longrightarrow\quad
\alpha_1=\alpha_2.
\]
\end{assumption}
\begin{remark}
\label{rem:obseq-justification}
Observationally equivalent values induce identical conditional responses and cannot be distinguished by
any estimator based on $g$. Passing to $\Omega_{\alpha}$ after removing observationally equivalent domain points is therefore
without loss and renders $\Phi$ injective by construction; we write $\Omega_{\alpha}$ for the
restricted domain.
\end{remark}
\begin{assumption}
\label{ass:ae-immersion}
Degeneracy set
$Z:=\{\alpha\in\Omega_{\alpha}:\det M_\infty(\alpha)=0\}$ has $\pi_\alpha(Z)=0$.
\end{assumption}
\noindent
Since $M_\infty(\alpha)\succeq0$ always, and a symmetric positive-semidefinite
matrix is positive definite if and only if its determinant is nonzero,
$\alpha\notin Z$ if and only if $M_\infty(\alpha)\succ0$:
Assumption~\ref{ass:ae-immersion} is equivalent to requiring
$M_\infty(\alpha)\succ0$ for $\pi_\alpha$-almost every $\alpha$.
It permits a rank drop on the $\pi_\alpha$-null set $Z$ (closed, by
continuity of $M_\infty$; Lemma~\ref{lem:Minf-uniform} below) and is the
condition of the almost-everywhere result, Proposition~\ref{prop:ae}.
Assumption~\ref{ass:multi-index} next is its uniform strengthening.
\begin{assumption}
\label{ass:multi-index}
There exists
$b_0=(\gamma_{0,1},\dots,\gamma_{0,p_\alpha})\in\Omega_{\gamma}^{p_\alpha}$ with
$d_{\alpha} \le d_G\,p_\alpha$ and $p_\alpha<\infty$ such that for every
$\alpha\in\Omega_{\alpha}$ the stacked Jacobian
\[
D_\alpha G_1(\alpha,b_0)
:=\Bigl[\,D_\alpha g(\alpha,\gamma_{0,1})\,;\ \cdots\ ;\,
D_\alpha g(\alpha,\gamma_{0,p_\alpha})\,\Bigr]
\in\mathbb{R}^{d_G p_\alpha\timesd_{\alpha}}
\]
has full column rank $d_{\alpha}$.
\end{assumption}
\begin{remark}
\label{rem:support}
Because $\Omega_{\gamma}$ and $\Omega_{\alpha}$ are, by the convention of
Section~\ref{sect:setup}, the supports of
$\pi_\gamma$ and $\pi_\alpha$, the evaluation points automatically lie in the
supports; this is what transfers the finite-design full rank to $M_\infty$ in
\eqref{eq:Minf} (Lemma~\ref{lem:Minf-uniform}). The condition is local (an
immersion condition), and is stronger than $M_\infty(\alpha)\succ0$
pointwise, since one design $b_0$ must deliver rank $d_{\alpha}$ uniformly in
$\alpha$. Only this $\alpha$-side condition enters
Theorem~\ref{thm:main}; the symmetric $\gamma$-side condition, on a stacked
Jacobian $D_\gamma G_2(a_0,\gamma)$, is stated as the $\gamma$-analogue in
Section~\ref{sec:gamma}, where it is first used.
\end{remark}
\begin{remark}
\label{rem:independence}
Assumptions~\ref{ass:obseq} and~\ref{ass:multi-index} are logically independent. With $d_{\alpha}=1$, $\Omega_{\alpha}=[-1,1]$, and
$g(\alpha,\gamma)=\alpha^3 h(\gamma)$ (with $h:\Omega_{\gamma}\to\mathbb{R}^{d_G}$ nonconstant), $\Phi$ is injective
(Assumption~\ref{ass:obseq} holds) yet $D_\alpha g(0,\gamma)\equiv0$, so
Assumption~\ref{ass:multi-index} fails at $\alpha=0$. Conversely, an everywhere
full-rank Jacobian may self-intersect globally, so
Assumption~\ref{ass:multi-index} does not imply Assumption~\ref{ass:obseq}.
\end{remark}
\begin{lemma}
\label{lem:Minf-uniform}
Under Assumption~\ref{ass:domain}, $\alpha\mapsto M_\infty(\alpha)$ is continuous on
$\Omega_{\alpha}$. If in addition Assumption~\ref{ass:multi-index} holds, then
$M_\infty(\alpha)\succ0$ for every $\alpha\in\Omega_{\alpha}$ and, where $\lambda_{\min}$ denotes minimum eigenvalue,
\begin{equation}
c_0:=\min_{\alpha\in\Omega_{\alpha}}\lambda_{\min}\bigl(M_\infty(\alpha)\bigr)>0,
\qquad
M_\infty(\alpha)\succeq c_0 I_{d_{\alpha}}\ \text{ for all }\alpha\in\Omega_{\alpha}.
\label{eq:c0}
\end{equation}
\end{lemma}
\begin{proof}
Continuity follows from \eqref{eq:Minf} (Lemma~\ref{lem:gradexp}) and joint
continuity of $J$. For a unit vector $v$,
$v^{\top}M_\infty(\alpha)v=\int_{\Omega_{\gamma}}\lVert J(\alpha,\gamma)v\rVert^2\,
d\pi_\gamma$; were this zero, $J(\alpha,\cdot)v$ would vanish
$\pi_\gamma$-a.e., hence, by continuity, at the evaluation points
$\gamma_{0,1},\dots,\gamma_{0,p_\alpha}\in\Omega_{\gamma}=\operatorname{supp}\pi_\gamma$,
contradicting $\operatorname{rank} D_\alpha G_1(\alpha,b_0)=d_{\alpha}$. Thus $M_\infty(\alpha)\succ0$,
and $\lambda_{\min}(M_\infty(\cdot))$ is continuous and strictly positive on the
compact $\Omega_{\alpha}$, so attains a positive minimum $c_0$.
\end{proof}
\noindent
Assumption~\ref{ass:multi-index} forces $Z=\varnothing$, and
almost-everywhere immersion in Assumption~\ref{ass:ae-immersion} holds
everywhere.
\section{Main Result}\label{sect:mainResult}
This section splits the leading eigenfunction injectivity into a pointwise
result and a uniform result.
The pointwise result uses the weaker Assumption~\ref{ass:ae-immersion} and
holds $\pi_\alpha$-almost everywhere.
The uniform result uses the stronger Assumption~\ref{ass:multi-index} to show
a uniform set of eigenfunctions is globally injective.
A corollary then quantifies the trade-off between the two.
We state the headline result first. It collects the full
set of assumptions under which the truncated pair $(U_R,V_R)$ is injective,
and hence supplies valid proxies in the sense of
Assumption~\ref{ass:proxyID}\textup{(ii)}. Its
proof is given at the end of Section~\ref{sec:gamma}, once preliminary results are developed.
\begin{theorem}[Injective eigenfunction proxies]
\label{thm:headline}
Let Assumptions~\ref{ass:domain} and~\ref{ass:obseq} hold together with
their $\gamma$-analogues (Section~\ref{sec:gamma}).
\begin{enumerate}
\item[\textup{(i)}] \textup{(Almost-everywhere regime.)} If in addition
Assumption~\ref{ass:ae-immersion} and its $\gamma$-analogue hold, then for
every $\delta>0$ there exist a finite $R(\delta)$ and compact sets
$\Omega_\delta^\alpha\subseteq\Omega_{\alpha}\setminus Z$ and
$\Omega_\delta^\gamma\subseteq\Omega_{\gamma}\setminus Z_\gamma$ with
$\pi_\alpha(\Omega_\delta^\alpha)\ge1-\delta$ and
$\pi_\gamma(\Omega_\delta^\gamma)\ge1-\delta$, such that for every
$R\ge R(\delta)$ the truncations $U_R$ and $V_R$ are injective on
$\Omega_\delta^\alpha$ and $\Omega_\delta^\gamma$, respectively.
\item[\textup{(ii)}] \textup{(Uniform regime.)} If in addition
Assumption~\ref{ass:multi-index} and its $\gamma$-analogue hold, then there
is a finite $R^{*}$ such that for every $R\ge R^{*}$, $U_R$ is injective on
$\Omega_{\alpha}$ and $V_R$ is injective on $\Omega_{\gamma}$.
\end{enumerate}
In either regime $(\lambda_i,f_t)=(U_R(\alpha_i),V_R(\gamma_t))$ are valid
proxies in the sense of Assumption~\ref{ass:proxyID}\textup{(ii)}: on all of
$\Omega_{\alpha}\times\Omega_{\gamma}$ at any fixed $R\ge R^{*}$ under \textup{(ii)}, and on
$\Omega_\delta^\alpha\times\Omega_\delta^\gamma$ with $\delta$ arbitrarily
small, at the price of $R(\delta)\to\infty$ as $\delta\downarrow0$, under
\textup{(i)}.
\end{theorem}
\subsection{Choosing Between the Regimes}
\label{sec:regimes}
The two regimes of Theorem~\ref{thm:headline} trade the strength of the
maintained assumptions against the behaviour of the truncation order.
For identification of $g$ at the realised types (Theorem~\ref{th:proxyID}),
either serves.
Under \textup{(i)} the identity $g_0(\lambda_i,f_t)=g(\alpha_i,\gamma_t)$ holds
for types in $\Omega_\delta^\alpha\times\Omega_\delta^\gamma$ for every
$\delta$, and so for almost every type once $R$ is allowed to grow along
$\delta\downarrow0$. For estimation the comparison is governed by the mass and diameter of the undetermined set. Fix $R$ and collect the conflated types
\[
B_R:=\bigl\{\alpha\in\Omega_{\alpha}:\ U_R(\alpha)=U_R(\alpha')\text{ for some }
\alpha'\ne\alpha\bigr\},
\qquad
\delta_\alpha(R):=\pi_\alpha(B_R),
\]
\[
\epsilon_\alpha(R):=\sup\bigl\{\lVert \alpha_1-\alpha_2\rVert:\ \alpha_1\neq\alpha_2,\
U_R(\alpha_1)=U_R(\alpha_2)\bigr\},
\]
with $B_R^\gamma$, $\delta_\gamma(R)$, $\epsilon_\gamma(R)$ defined
symmetrically.
Each $B_R$ is Borel (a countable union of projections of
compact sets) and the sets are nested downward in $R$.
Under Assumptions~\ref{ass:domain}
and~\ref{ass:obseq} alone, $\epsilon_\alpha(R)\downarrow0$, i.e. the
away-from-the-diagonal step in the proof of Theorem~\ref{thm:main} uses
neither immersion condition, so for every $\epsilon>0$ every conflation of
separation at least $\epsilon$ is removed at some finite order. Hence,
for large $R$ any conflated pair is
$\epsilon_\alpha(R)$-close. Further, under the assumptions of regime
\textup{(i)}, $\delta_\alpha(R)\downarrow0$, hence if $\alpha\notin Z$, its
partners $\alpha'_R$ must accumulate at $\alpha$ itself (any other limit
point would conflate with $\alpha$ under $U_\infty$, contradicting
Proposition~\ref{prop:part1} and Assumption~\ref{ass:obseq}), so they
eventually enter a compact neighbourhood of $\alpha$ on which
Proposition~\ref{prop:ae} makes some finite truncation injective. Hence
$\bigcap_R B_R\subseteq Z$ and continuity from above gives
$\pi_\alpha(B_R)\downarrow\pi_\alpha(\bigcap_R B_R)=0$.
A unit $i$
with $\alpha_i\in B_R$ potentially has $g_0(\lambda_i,f_t)\neq g(\alpha_i,\gamma_t)$
at \emph{every} $t$, since $g_0$ averages $g$ over the conflated
fibre at each $\gamma_t$, and symmetrically for a time $t$ with
$\gamma_t\in B_R^\gamma$. Hence, misclassification contaminates whole rows and
columns of an $N\times T$ array, so the fraction of affected cells is of
order $\delta(R):=\delta_\alpha(R)+\delta_\gamma(R)$. The size of the error on an affected
cell is bounded by response variation across the conflated set. With
$L$ the joint Lipschitz constant of $g$,
\[
\bigl|g_0(\lambda_i,f_t)-g(\alpha_i,\gamma_t)\bigr|
\;\le\; L\,\epsilon(R),
\qquad
\epsilon(R):=\epsilon_\alpha(R)+\epsilon_\gamma(R),
\]
so the error shrinks with the spatial scale of the misclassifications
(Remark~\ref{rem:blend-sharp}'s grid of size $2^{-R}$).
In procedures that orthogonalise with respect to the regression function,
first-order errors cancel and the conflation enters root-$NT$ normalised
sums only through second-order products of nuisance errors, both factors
carrying the bias on the same affected cells; the contribution is of order
$\sqrt{NT}\,\delta(R)\,\bigl(L\epsilon(R)\bigr)^2$, while integrated
squared-error (consistency) conditions pick up
$\delta(R)(L\epsilon(R))^2=o(1)$ automatically. For parametric rate convergence, regime
\textup{(i)} therefore needs
\[
\sqrt{NT}\;\delta(R)\,\epsilon(R)^2\;\longrightarrow\;0
\]
along the chosen truncation sequence, which shrinks with mass and squared diameter jointly. In the
construction of Remark~\ref{rem:blend-sharp} with a bounded type density,
$\delta(R)$ and $\epsilon(R)$ are both of order $2^{-R}$, so the display
holds once $R$ grows like $\log NT$.
No universal rate exists, however.
The Dini argument is rate-free, and the same construction can be slowed so
that $\delta(R)$ and $\epsilon(R)$ vanish arbitrarily slowly. Under regime
\textup{(i)} the display therefore remains a genuine restriction tying the
truncation sequence to the unknown collision geometry near $Z$; and near
$Z$ the inherited derivatives of $g_0$ degenerate, so smoothness conditions
on $g_0$ are likewise $\Omega_\delta$-relative. Regime \textup{(ii)}
removes all of this at once, since $\delta(R)=\epsilon(R)=0$ for
$R\ge R^{*}$.
\subsection{Infinite Eigenfunction Collection}
The first result shows that injectivity of the full
eigenfunction collection requires only
Assumption~\ref{ass:obseq}.
\begin{proposition}
\label{prop:part1}
Let $g$ be square-integrable on $\Omega_{\alpha}\times\Omega_{\gamma}$ with the SVD
\eqref{eq:svd}, where $r$ ranges over
$\sigma_r>0$ throughout; let $\alpha\mapsto g(\alpha,\cdot)$ be continuous
into $\mathcal{H}_{\gamma}$ and take the continuous versions $u_r=\sigma_r^{-1}T_gv_r$ (both
supplied by Assumption~\ref{ass:domain}). Then for all
$\alpha_1,\alpha_2\in\Omega_{\alpha}$,
\begin{equation}
\lVert \Phi(\alpha_1)-\Phi(\alpha_2)\rVert_{\mathcal{H}_{\gamma}}^2
=\sum_{r\ge1}\sigma_r^2\,\lVert u_r(\alpha_1)-u_r(\alpha_2)\rVert^2.
\label{eq:parseval}
\end{equation}
Consequently the following are equivalent:
\begin{enumerate}
\item[\textup{(i)}] $\Phi$ is injective (Assumption~\ref{ass:obseq});
\item[\textup{(ii)}] the full eigenfunction map $U_\infty$ is injective
\textup{(}equal to $U_M$ when $\operatorname{rank} T_g=M<\infty$\textup{)}.
\end{enumerate}
\end{proposition}
\begin{proof}
Fix $\alpha\in\Omega_{\alpha}$. The element
$\Phi(\alpha)=g(\alpha,\cdot)\in\mathcal{H}_{\gamma}$ has the expansion
$\Phi(\alpha)=\sum_{r\ge1}\bigl(\sigma_r u_r(\alpha)\bigr)v_r$ with vector
coefficients: as in the proof of
Lemma~\ref{lem:gradexp}, $\Phi(\alpha)\in\mathcal{V}^{\,d_G}$ for
\emph{every} $\alpha$ (the $L^2$-convergence of \eqref{eq:svd} gives this
$\pi_\alpha$-a.e., and $\Phi$ is continuous with $\mathcal{V}^{\,d_G}$ closed), so
$\Phi(\alpha)=\sum_r c_r(\alpha)v_r$ with
$c_r(\alpha)=\int_{\Omega_{\gamma}}\Phi(\alpha)v_r\,d\pi_\gamma=(T_gv_r)(\alpha)
=\sigma_r u_r(\alpha)\in\mathbb{R}^{d_G}$ by
the continuous versions. Since $\{v_re_i\}_{r\ge1,\,i\le d_G}$ is orthonormal in
$\mathcal{H}_{\gamma}$, $e_1,\dots,e_{d_G}$ being the coordinate vectors of $\mathbb{R}^{d_G}$,
Parseval's identity applied to $\Phi(\alpha_1)-\Phi(\alpha_2)
=\sum_r\sigma_r\bigl(u_r(\alpha_1)-u_r(\alpha_2)\bigr)v_r$ yields
\eqref{eq:parseval}.
For the equivalence: the right-hand side of \eqref{eq:parseval} is zero if and
only if $\sigma_r\lVert u_r(\alpha_1)-u_r(\alpha_2)\rVert=0$ for every $r$,
which, because $\sigma_r>0$, holds if and only if
$u_r(\alpha_1)=u_r(\alpha_2)$ for every $r$, i.e.\ $U_\infty(\alpha_1)=
U_\infty(\alpha_2)$. The left-hand side is zero if and only if
$\Phi(\alpha_1)=\Phi(\alpha_2)$. Therefore $\Phi(\alpha_1)=\Phi(\alpha_2)$ iff
$U_\infty(\alpha_1)=U_\infty(\alpha_2)$, and (i)$\Leftrightarrow$(ii) follows.
\end{proof}
\begin{remark}
\label{rem:positive-sigma}
The collection $\{u_r\}$ ranges over the positive singular values, equivalently
over an orthonormal basis of $\overline{\operatorname{range}(T_g)}$, which is what
``$\sigma_r>0$ throughout'' records. This is a convention, not a restriction: a
finite-rank kernel ($\sigma_r=0$ for $r>M$) is covered with
$U_\infty=U_M=(u_1,\dots,u_M)$, the proof unchanged since the dropped terms vanish
in \eqref{eq:parseval}. Positivity cannot simply be waived, however: were
zero-$\sigma$ directions (an arbitrary orthonormal completion of $\mathcal{H}_{\alpha}$) appended
to the collection, the step ``$\sigma_r(u_r(\alpha_1)-u_r(\alpha_2))=0$ for all $r$
$\Rightarrow u_r(\alpha_1)=u_r(\alpha_2)$ for all $r$'' would fail and
\textup{(ii)}$\Rightarrow$\textup{(i)} would break, as such directions carry no
information about $g$. Finally, when $M<\infty$ one has $\Delta_R=\Delta$ for all
$R\ge M$, so Theorem~\ref{thm:main} holds with $R_0=M$ directly from
Assumption~\ref{ass:obseq}, with no recourse to Assumption~\ref{ass:multi-index}.
\end{remark}
\begin{corollary}
\label{cor:part1}
Under Assumption~\ref{ass:obseq}, the map $U_\infty$ is injective on $\Omega_{\alpha}$.
\end{corollary}
\begin{proof}
Immediate from Proposition~\ref{prop:part1}, (i)$\Rightarrow$(ii).
\end{proof}
\subsection{Finite Truncations}
We now show that a finite leading truncation $U_R$ inherits injectivity once
$R$ is large enough. This is the substantive step: injectivity of the infinite
collection (Corollary~\ref{cor:part1}) does not by itself guarantee that any
finite truncation separates all pairs of points, and an additional local
condition is required.
It is convenient to introduce, for $R\in\mathbb{N}\cup\{\infty\}$, the
\emph{separation function}
\begin{equation}
\Delta_R(\alpha_1,\alpha_2)
:=\sum_{r\le R}\sigma_r^2\,\lVert u_r(\alpha_1)-u_r(\alpha_2)\rVert^2
=\lVert \Phi_R(\alpha_1)-\Phi_R(\alpha_2)\rVert_{\mathcal{H}_{\gamma}}^2,
\label{eq:sep}
\end{equation}
where the second equality follows as in Proposition~\ref{prop:part1} applied to
$g_R$. Three properties are immediate:
\begin{itemize}
\item[(a)] \emph{Continuity}: each $\Delta_R$ ($R\in\mathbb{N}$) is continuous on
$\Omega_{\alpha}\times\Omega_{\alpha}$, being a finite sum of continuous functions; and
$\Delta_\infty=\Delta$ is continuous, equal to
$\lVert \Phi(\alpha_1)-\Phi(\alpha_2)\rVert^2_{\mathcal{H}_{\gamma}}$, by
Lemma~\ref{lem:Phi-cont}.
\item[(b)] \emph{Monotonicity}: each summand $\ge0$, so
$\Delta_{R}\le\Delta_{R+1}$ and $\Delta_R\nearrow\Delta$ pointwise as
$R\to\infty$.
\item[(c)] \emph{Injectivity criterion}: since $\sigma_r>0$,
$U_R$ is injective if and only if $\Delta_R(\alpha_1,\alpha_2)>0$ for
all $\alpha_1\ne\alpha_2$.
\end{itemize}
Under Assumption~\ref{ass:obseq} and Proposition~\ref{prop:part1}, we have
$\Delta(\alpha_1,\alpha_2)>0$ for all $\alpha_1\ne\alpha_2$.
\begin{lemma}
\label{lem:near}
Under Assumptions~\ref{ass:domain} and~\ref{ass:multi-index}, with $c_0$ as in
\eqref{eq:c0}, there exist a finite $R_1$ and a radius $\epsilon>0$ such that
\[
\Delta_{R_1}(\alpha_1,\alpha_2)\ \ge\ \frac{c_0}{8}\,\lVert \alpha_1-\alpha_2\rVert^2
\qquad\text{whenever }\ \lVert \alpha_1-\alpha_2\rVert<\epsilon .
\]
\end{lemma}
\begin{proof}
\emph{Choice of $R_1$.} Each $M_R$ is continuous (finite sum of continuous
matrix-valued functions) and $M_R\nearrow M_\infty$ pointwise in the positive
semidefinite order (Lemma~\ref{lem:gradexp}), with $M_\infty$ continuous by
Lemma~\ref{lem:Minf-uniform}. The scalar functions
$\alpha\mapsto v^{\top}M_R(\alpha)v$ increase monotonically to
$\alpha\mapsto v^{\top}M_\infty(\alpha)v$ for each fixed unit vector $v$; by
Dini's theorem in \cite{rudin1976} this convergence is uniform on the compact $\Omega_{\alpha}$, and a finite
cover of the unit sphere in $\mathbb{R}^{d_{\alpha}}$ makes the convergence
$M_R\to M_\infty$ uniform in operator norm on $\Omega_{\alpha}$. Hence we may fix $R_1$
with
\begin{equation}
M_{R_1}(\alpha)\succeq\tfrac{c_0}{2}\,I_{d_{\alpha}}
\qquad\text{for all }\alpha\in\Omega_{\alpha}.
\label{eq:MR1}
\end{equation}
\emph{Local quadratic lower bound.} Fix $\alpha_1,\alpha_2\in\Omega_{\alpha}$ and write
$\delta=\alpha_1-\alpha_2$. By convexity of $\Omega_{\alpha}$
(Assumption~\ref{ass:domain}), the segment $\zeta(t)=\alpha_2+t\delta$ lies in
$\Omega_{\alpha}$ for $t\in[0,1]$. Let
$\Sigma_{R_1}=\operatorname{diag}(\sigma_1 I_{d_G},\dots,\sigma_{R_1}I_{d_G})\in\mathbb{R}^{d_G R_1\times d_G R_1}$
and $P(t)=\Sigma_{R_1}\,DU_{R_1}(\zeta(t))\in\mathbb{R}^{d_G R_1\timesd_{\alpha}}$, where
$DU_{R_1}=\bigl(Du_1^{\top},\dots,Du_{R_1}^{\top}\bigr)^{\top}
\in\mathbb{R}^{d_G R_1\timesd_{\alpha}}$ is the Jacobian of $U_{R_1}$.
By the fundamental theorem of calculus,
\[
\Sigma_{R_1}\bigl(U_{R_1}(\alpha_1)-U_{R_1}(\alpha_2)\bigr)
=\int_0^1 P(t)\,\delta\,dt,
\]
so, by the triangle inequality,
\begin{equation}
\Delta_{R_1}(\alpha_1,\alpha_2)^{1/2}
=\Bigl\|\int_0^1 P(t)\delta\,dt\Bigr\|
\ge\lVert P(0)\delta\rVert-\int_0^1\lVert P(t)-P(0)\rVert\,dt\;\lVert \delta\rVert.
\label{eq:segment}
\end{equation}
For the first term, $\lVert P(0)\delta\rVert^2
=\delta^{\top}DU_{R_1}(\alpha_2)^{\top}\Sigma_{R_1}^2 DU_{R_1}(\alpha_2)\delta
=\delta^{\top}M_{R_1}(\alpha_2)\delta\ge\tfrac{c_0}{2}\lVert \delta\rVert^2$ by
\eqref{eq:MR1}, hence $\lVert P(0)\delta\rVert\ge\sqrt{c_0/2}\,\lVert \delta\rVert$.
For the second term, $P$ is continuous on the compact $\Omega_{\alpha}$ (as
$DU_{R_1}$ is continuous), hence uniformly continuous; choose $\epsilon>0$ so
that $\lVert \alpha-\alpha'\rVert<\epsilon$ implies
$\lVert \Sigma_{R_1}DU_{R_1}(\alpha)-\Sigma_{R_1}DU_{R_1}(\alpha')\rVert
\le\tfrac12\sqrt{c_0/2}$. Then whenever $\lVert \delta\rVert<\epsilon$ we have
$\lVert \zeta(t)-\alpha_2\rVert=t\lVert \delta\rVert<\epsilon$ and so
$\lVert P(t)-P(0)\rVert\le\tfrac12\sqrt{c_0/2}$ for all $t$. Substituting into
\eqref{eq:segment},
\[
\Delta_{R_1}(\alpha_1,\alpha_2)^{1/2}
\ge\sqrt{c_0/2}\,\lVert \delta\rVert-\tfrac12\sqrt{c_0/2}\,\lVert \delta\rVert
=\tfrac12\sqrt{c_0/2}\,\lVert \delta\rVert,
\]
whence $\Delta_{R_1}(\alpha_1,\alpha_2)\ge\tfrac{c_0}{8}\lVert \delta\rVert^2$ for
$\lVert \delta\rVert<\epsilon$, as claimed.
\end{proof}
\subsection{Pointwise Almost-everywhere Injectivity}
\label{app:ae}
Assumption~\ref{ass:multi-index} requires full rank at \emph{every} $\alpha$.
The example $g(\alpha,\gamma)=\alpha^3 h(\gamma)$ of Remark~\ref{rem:independence}
shows this can fail on a $\pi_\alpha$-null set without harming identification:
there $U_\infty$ stays injective and the drop occurs only at $\alpha=0$, with
$\pi_\alpha(\{0\})=0$. Assumption~\ref{ass:ae-immersion} is the weaker notion that
tolerates such a drop, which is the eigenfunction-map analogue of injectivity almost
everywhere (global invertibility) in the sense of \cite{traver2025global}. Its
degeneracy set $Z$ is closed, since $M_\infty$ is continuous
(Lemma~\ref{lem:Minf-uniform}).
\begin{proposition}
\label{prop:ae}
Under Assumptions~\ref{ass:domain}, \ref{ass:obseq},
and~\ref{ass:ae-immersion}, for every compact $K\subseteq\Omega_{\alpha}\setminus Z$ there is
a finite $R_K$ with $U_R$ injective on $K$ for all $R\ge R_K$. Hence $\pi_\alpha$-almost every
$\alpha$ has a neighborhood on which a finite leading truncation is injective.
\end{proposition}
\begin{proof}
On compact $K$, matrix $M_\infty$ is continuous and pd, so
$c_K:=\min_{\alpha\in K}\lambda_{\min}(M_\infty(\alpha))>0$.
Lemma~\ref{lem:near} applies with the global floor \eqref{eq:c0} replaced by
$M_\infty\succeq c_K I_{d_{\alpha}}$ on $K$: its proof uses the floor only at the
base point $\alpha_2$ and through the uniform convergence $M_R\to M_\infty$
on the compact $\Omega_{\alpha}$, which is unaffected. This yields finite $R_1$ and
$\epsilon>0$ with $\Delta_{R_1}(\alpha_1,\alpha_2)>0$ whenever
$0<\lVert \alpha_1-\alpha_2\rVert<\epsilon$ and $\alpha_2\in K$. Away from the
diagonal, on the compact set
$\{(\alpha_1,\alpha_2)\in K\times K:\lVert \alpha_1-\alpha_2\rVert\ge\epsilon\}$
the continuous function $\Delta$ is strictly positive
(Proposition~\ref{prop:part1} and Assumption~\ref{ass:obseq}), hence attains
a positive minimum, and by Dini's theorem $\Delta_R\nearrow\Delta$ uniformly
there; choose $R_K\ge R_1$ with $\Delta_R>0$ on this set for all $R\ge R_K$.
By monotonicity (property (b)) and the injectivity criterion (property (c)),
$U_R$ is injective on $K$ for every $R\ge R_K$. As $Z$ is closed,
$\Omega_{\alpha}\setminus Z$ is open and, since $\pi_\alpha(Z)=0$, almost every $\alpha$
lies in such a $K$.
\end{proof}
The order $R_K$ grows as $K$ approaches $Z$.
A measure-zero rank drop costs
the single uniform truncation order, restored under the stronger
Assumption~\ref{ass:multi-index} in Theorem~\ref{thm:main} below, but not
identification. As in
\cite{traver2025global}, the global condition is supplied by Assumption~\ref{ass:obseq}, which is the
analogue of prescribed homeomorphic boundary data, and
Assumption~\ref{ass:ae-immersion} replaces nondegeneracy everywhere.
\subsection{Uniform Finite Leading Eigenfunctions}
Under the stronger Assumption~\ref{ass:multi-index} the degeneracy set is
empty (Lemma~\ref{lem:Minf-uniform}) and the compact exhaustion of
Proposition~\ref{prop:ae} is unnecessary: a single finite truncation order
delivers injectivity on all of $\Omega_{\alpha}$.
\begin{theorem}
\label{thm:main}
Suppose Assumptions~\ref{ass:domain}, \ref{ass:obseq},
and~\ref{ass:multi-index} hold. Then there exists a finite $R_0$ such that for
every $R\ge R_0$ the leading truncation $U_R$ of \eqref{eq:UR} is injective on $\Omega_{\alpha}$.
\end{theorem}
\begin{proof}
Let $R_1$ and $\epsilon>0$ be as in Lemma~\ref{lem:near}, so that
\begin{equation}
\Delta_{R_1}(\alpha_1,\alpha_2)>0
\qquad\text{whenever }0<\lVert \alpha_1-\alpha_2\rVert<\epsilon.
\label{eq:nearpos}
\end{equation}
\emph{Away from the diagonal.} The set
$K_\epsilon:=\{(\alpha_1,\alpha_2)\in\Omega_{\alpha}\times\Omega_{\alpha}:\lVert \alpha_1-\alpha_2\rVert
\ge\epsilon\}$ is closed in the compact $\Omega_{\alpha}\times\Omega_{\alpha}$, hence compact. The
function $\Delta$ is continuous on $K_\epsilon$ and, by
Proposition~\ref{prop:part1} together with Assumption~\ref{ass:obseq}, strictly
positive there; therefore it attains a minimum
\[
m_\epsilon:=\min_{(\alpha_1,\alpha_2)\in K_\epsilon}\Delta(\alpha_1,\alpha_2)>0.
\]
The functions $\Delta_R$ increase monotonically to the continuous limit
$\Delta$ on the compact $K_\epsilon$, with each $\Delta_R$ continuous; by
Dini's theorem the convergence is uniform on $K_\epsilon$. Choose
$R_2\ge R_1$ such that $\sup_{K_\epsilon}(\Delta-\Delta_R)\le m_\epsilon/2$ for
all $R\ge R_2$, so that
\begin{equation}
\Delta_R(\alpha_1,\alpha_2)\ge\Delta(\alpha_1,\alpha_2)-\tfrac{m_\epsilon}{2}
\ge\tfrac{m_\epsilon}{2}>0
\qquad\text{on }K_\epsilon,\ \text{for all }R\ge R_2.
\label{eq:farpos}
\end{equation}
\emph{Conclusion.} Set $R_0:=R_2$ and take any $R\ge R_0$ and any
$\alpha_1\ne\alpha_2$. If $\lVert \alpha_1-\alpha_2\rVert<\epsilon$, then by
monotonicity (property (b)) and \eqref{eq:nearpos},
\[
\Delta_R(\alpha_1,\alpha_2)\ge\Delta_{R_1}(\alpha_1,\alpha_2)>0.
\]
If $\lVert \alpha_1-\alpha_2\rVert\ge\epsilon$, then $(\alpha_1,\alpha_2)\in
K_\epsilon$ and \eqref{eq:farpos} gives $\Delta_R(\alpha_1,\alpha_2)>0$. In
both cases $\Delta_R(\alpha_1,\alpha_2)>0$. By the injectivity criterion
(property (c)), $U_R$ is injective.
\end{proof}
\subsection{Quantifying the Trade-off}
The trade-off between Theorem~\ref{thm:main} and Proposition~\ref{prop:ae} can
be quantified: a single finite truncation order suffices on all but an
arbitrarily small mass of types.
\begin{corollary}
\label{cor:blend}
Under Assumptions~\ref{ass:domain}, \ref{ass:obseq},
and~\ref{ass:ae-immersion}, for every $\delta>0$ there exist a finite
$R(\delta)$ and a compact $\Omega_\delta\subseteq\Omega_{\alpha}\setminus Z$ with
$\pi_\alpha(\Omega_\delta)\ge1-\delta$ such that $U_R$ is injective on
$\Omega_\delta$ for every $R\ge R(\delta)$.
\end{corollary}
\begin{proof}
For $\eta>0$ let
$T_\eta:=\{\alpha\in\Omega_{\alpha}:\operatorname{dist}(\alpha,Z)<\eta\}$. Since $Z$ is
closed, $\bigcap_{\eta>0}T_\eta=Z$, so continuity from above of the finite
measure $\pi_\alpha$ gives $\pi_\alpha(T_\eta)\downarrow\pi_\alpha(Z)=0$ as
$\eta\downarrow0$. Fix $\eta$ with $\pi_\alpha(T_\eta)\le\delta$ and set
$\Omega_\delta:=\Omega_{\alpha}\setminus T_\eta$, a compact subset of $\Omega_{\alpha}\setminus Z$ with
$\pi_\alpha(\Omega_\delta)\ge1-\delta$. Proposition~\ref{prop:ae} applied to
$K=\Omega_\delta$ yields the finite $R(\delta):=R_K$.
\end{proof}
\begin{remark}
\label{rem:blend-sharp}
Corollary~\ref{cor:blend} cannot be strengthened to a null exceptional set:
there exist kernels satisfying Assumptions~\ref{ass:domain}, \ref{ass:obseq},
and~\ref{ass:ae-immersion} for which, for \emph{every} finite $R$, the
truncation $U_R$ fails to be injective on every set of full
$\pi_\alpha$-measure. The mechanism is that collisions may accumulate at $Z$
at ever finer scales, each resolved only by ever later eigenfunctions.
With
$d_{\alpha}=1$, $d_G=1$, and $Z=\{0\}$, take $u_1\propto\alpha^2$ near the origin and let each
$u_{k+1}$ carry an odd component supported on
$\{|\alpha|\asymp2^{-k}\}$.
Then $U_\infty$ is injective and
$M_\infty(\alpha)>0$ for $\alpha\neq0$, yet for each fixed $R$ every mirror
pair $(\alpha,-\alpha)$ at scales finer than $2^{-R}$ collides under
$U_R$, a positive-measure family of collisions meeting every full-measure
set.
Hence $R(\delta)\to\infty$ as $\delta\downarrow0$ is unavoidable in
general, and Corollary~\ref{cor:blend} interpolates between
Theorem~\ref{thm:main} ($\delta=0$, under the everywhere floor of
Assumption~\ref{ass:multi-index}) and Proposition~\ref{prop:ae}.
\end{remark}
\subsection{Symmetric Statements for the Right Eigenfunctions}
\label{sec:gamma}
Both results transfer to the right singular functions $\{v_r\}$ and the
parameter $\gamma$, using the \emph{same} singular system. The $\gamma$-side
response map is $\Psi(\gamma):=g(\cdot,\gamma)\in\mathcal{H}_{\alpha}$, and pairing it with the
orthonormal system $\{u_r\}\subset\mathcal{H}_{\alpha}$ gives, by the adjoint,
\begin{equation}
\langle \Psi(\gamma),\,u_r\rangle_{\mathcal{H}_{\alpha}}
=\int_{\Omega_{\alpha}}g(\alpha,\gamma)^{\top}u_r(\alpha)\,d\pi_\alpha
=(T_g^{*}u_r)(\gamma)=\sigma_r\,v_r(\gamma),
\label{eq:gammacoef}
\end{equation}
a \emph{scalar} coefficient. Hence $\Psi(\gamma)=\sum_r\sigma_r v_r(\gamma)u_r$,
lying in $\overline{\operatorname{span}}\{u_r\}\subseteq\mathcal{H}_{\alpha}$ for every $\gamma$ by the argument
of Lemma~\ref{lem:gradexp}, and Parseval gives the $\gamma$-analogue of
\eqref{eq:parseval},
$\lVert \Psi(\gamma_1)-\Psi(\gamma_2)\rVert_{\mathcal{H}_{\alpha}}^2
=\sum_{r\ge1}\sigma_r^2\bigl(v_r(\gamma_1)-v_r(\gamma_2)\bigr)^2$.
Differentiating \eqref{eq:gammacoef} in $\gamma$ and applying the entrywise
Bessel and projection argument of Lemma~\ref{lem:gradexp} to $D_\gamma g$ gives
the $\gamma$-side gradient information matrix
\[
N_\infty(\gamma)=\sum_{r\ge1}\sigma_r^2\,Dv_r(\gamma)^{\top}Dv_r(\gamma)
=\int_{\Omega_{\alpha}}D_\gamma g(\alpha,\gamma)^{\top}D_\gamma g(\alpha,\gamma)\,
d\pi_\alpha .
\]
Write $V_R(\gamma)=(v_1(\gamma),\dots,v_R(\gamma))^{\top}\in\mathbb{R}^{R}$,
$N_R(\gamma):=\sum_{r\le R}\sigma_r^2\,Dv_r(\gamma)^{\top}Dv_r(\gamma)$, and
$Z_\gamma:=\{\gamma\in\Omega_{\gamma}:\det N_\infty(\gamma)=0\}$. The $\gamma$-analogue of an
Assumption is that Assumption with the roles of $\alpha$ and $\gamma$
exchanged. Two are used below. The $\gamma$-analogue of
Assumption~\ref{ass:obseq} asks that $g(\cdot,\gamma_1)=g(\cdot,\gamma_2)$ on
$\Omega_{\alpha}$ imply $\gamma_1=\gamma_2$, i.e.\ that $\Psi$ be injective. The
$\gamma$-analogue of Assumption~\ref{ass:multi-index} asks that there exist
$a_0=(\alpha_{0,1},\dots,\alpha_{0,p_\gamma})\in\Omega_{\alpha}^{p_\gamma}$ with
$d_{\gamma}\le d_G\,p_\gamma$ and $p_\gamma<\infty$ such that for every
$\gamma\in\Omega_{\gamma}$ the stacked Jacobian
\[
D_\gamma G_2(a_0,\gamma)
:=\Bigl[\,D_\gamma g(\alpha_{0,1},\gamma)\,;\ \cdots\ ;\,
D_\gamma g(\alpha_{0,p_\gamma},\gamma)\,\Bigr]
\in\mathbb{R}^{d_G p_\gamma\timesd_{\gamma}}
\]
has full column rank $d_{\gamma}$. Since both sides read off the
same singular values, $\max(R_0,R_0')$ is a truncation order in a single
ordering.
\begin{corollary}[Finite truncation]
\label{cor:gamma}
Under the $\gamma$-analogues of Assumptions~\ref{ass:domain}, \ref{ass:obseq},
and~\ref{ass:multi-index}, there is a finite $R_0'$ such that $V_R$ is injective
on $\Omega_{\gamma}$ for all $R\ge R_0'$. If the assumptions of Theorem~\ref{thm:main}
hold as well, taking $R\ge\max(R_0,R_0')$ makes $U_R$ and $V_R$
simultaneously injective.
\end{corollary}
\begin{proof}
The proof of Theorem~\ref{thm:main} applies verbatim under the dictionary
\[
(\Phi,\{u_r\},M_R,M_\infty,Z,\Omega_{\alpha},\pi_\alpha)\;\longrightarrow\;
(\Psi,\{v_r\},N_R,N_\infty,Z_\gamma,\Omega_{\gamma},\pi_\gamma):
\]
the Parseval identity
displayed above replaces \eqref{eq:parseval}, so
Proposition~\ref{prop:part1} transfers with the $\gamma$-analogue of
Assumption~\ref{ass:obseq} supplying injectivity of $\Psi$ and hence of
$V_\infty$; the gradient expansion for $D_\gamma g$ noted above replaces
Lemma~\ref{lem:gradexp}; the $\gamma$-analogue of
Assumption~\ref{ass:multi-index} yields the floor
$N_\infty(\gamma)\succeq c_0' I_{d_{\gamma}}$ on $\Omega_{\gamma}$ by the argument of
Lemma~\ref{lem:Minf-uniform}; and Lemma~\ref{lem:near} together with the
Dini argument of Theorem~\ref{thm:main} use only these objects and the
compactness of $\Omega_{\gamma}$.
\end{proof}
\begin{corollary}[Almost-everywhere injectivity]
\label{cor:gamma-ae}
Under the $\gamma$-analogues of Assumptions~\ref{ass:domain}, \ref{ass:obseq},
and~\ref{ass:ae-immersion}, for every compact $K\subseteq\Omega_{\gamma}\setminus Z_\gamma$
there is a finite $R_K'$ with $V_R$ injective on $K$ for all $R\ge R_K'$. Hence
$\pi_\gamma$-almost every $\gamma$ has a neighborhood on which a finite truncation
of eigenfunctions is injective.
\end{corollary}
\begin{proof}
The proof of Proposition~\ref{prop:ae} applies under the same dictionary as
in the proof of Corollary~\ref{cor:gamma}, with the $\gamma$-analogue of
Assumption~\ref{ass:ae-immersion} reading $\pi_\gamma(Z_\gamma)=0$.
\end{proof}
\begin{remark}
\label{rem:transpose}
Neither corollary is obtained by applying Theorem~\ref{thm:main} or
Proposition~\ref{prop:ae} to the transposed kernel
$\tilde g(\gamma,\alpha):=g(\alpha,\gamma)$. When $d_G=1$ that route is
available: $T_{\tilde g}=T_g^{*}$, and the singular system of $\tilde g$ is
that of $g$ with the roles of $u_r$ and $v_r$ exchanged. When $d_G>1$ it is
not: the operator
$T_{\tilde g}:L^{2}_{\pi}(\Omega_{\alpha})\to\mathcal{H}_{\gamma}$,
$(T_{\tilde g}\phi)(\gamma)=\int_{\Omega_{\alpha}}g(\alpha,\gamma)\phi(\alpha)\,d\pi_\alpha$,
carries its own singular system, and its scalar functions on $\Omega_{\alpha}$
diagonalise the pooled kernel
$\int_{\Omega_{\gamma}}g(\alpha_1,\gamma)^{\top}g(\alpha_2,\gamma)\,d\pi_\gamma$; in
general this is a different decomposition from \eqref{eq:svd}, with
different singular values and functions. The mirrored argument above stays
within the single system of \eqref{eq:svd}, whose $\gamma$-side coefficients
are read through the adjoint \eqref{eq:gammacoef}; this is also what makes
$\max(R_0,R_0')$ meaningful in one ordering.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:headline}]
(i) Corollary~\ref{cor:blend} provides $\Omega_\delta^\alpha$ and a finite
order $R_\alpha(\delta)$. The same construction on the $\gamma$-side, with
the sets $T_\eta$ built around the closed $Z_\gamma$ and
Corollary~\ref{cor:gamma-ae} in place of Proposition~\ref{prop:ae},
provides $\Omega_\delta^\gamma$ and $R_\gamma(\delta)$. Take
$R(\delta)=\max\{R_\alpha(\delta),R_\gamma(\delta)\}$, well defined in a
single ordering since both sides read the same singular values.
(ii) Theorem~\ref{thm:main} gives $R_0$ and Corollary~\ref{cor:gamma} gives
$R_0'$; take $R^{*}=\max(R_0,R_0')$. The proxy-validity conclusion is
Theorem~\ref{th:proxyID} applied with $U=U_R$ and $V=V_R$ on the indicated
domains.
\end{proof}
\subsection{Additive Kernels}
\label{sec:additive}
Estimating the normalised regression $g_0$ on the proxy coordinates is a
nonparametric regression in $(d_G+1)R$ variables, which for large $R$ suffers the
usual curse of dimensionality. This subsection shows that when the kernel is
\emph{additive} across types, the entire construction
respects that structure.
The eigenfunctions themselves are additive, their
coordinate components are injective coordinate-by-coordinate, and the
normalised regression decomposes into components each depending on one
scalar unit type and one scalar time type. This is the identification
content of Theorems~4 and~5 of \citet{freeman2026inference}, where it
licenses backfitting estimation of $g_0$.
\begin{assumption}[Additive kernel on a product support]
\label{ass:additive}
$d_{\alpha}=d_{\gamma}=d$; the supports are products of compact intervals,
$\Omega_{\alpha}=\prod_{k=1}^{d}\Omega_{\alpha,k}$ and
$\Omega_{\gamma}=\prod_{k=1}^{d}\Omega_{\gamma,k}$; and
\[
g(\alpha,\gamma)=\sum_{k=1}^{d}h_k(\alpha_k,\gamma_k)
\qquad\text{for functions }
h_k:\Omega_{\alpha,k}\times\Omega_{\gamma,k}\to\mathbb{R}^{d_G} .
\]
\end{assumption}
\noindent
A product of compact intervals is compact, convex, and connected, so
Assumption~\ref{ass:additive} is compatible with
Assumption~\ref{ass:domain}. No smoothness of the $h_k$ is assumed: each
$h_k$ equals a bivariate slice of $g$ net of a constant (fix a reference
point $(\alpha^{\circ},\gamma^{\circ})\in\Omega_{\alpha}\times\Omega_{\gamma}$ and vary only the
$k$-th coordinates), so the $h_k$ inherit the continuity and
differentiability of $g$ and are identified up to additive constants.
Write $\pi_{\gamma,k}$ for the $k$-th marginal of $\pi_\gamma$ and
$\bar v_{r,k}(c):=\mathbb{E}[v_r(\gamma_t)\mid\gamma_{t,k}=c]$; no
independence across the coordinates of $\alpha_i$ or of $\gamma_t$ is
imposed.
\begin{proposition}[Additive eigenfunctions]
\label{prop:additive-eigen}
Suppose Assumptions~\ref{ass:domain} and~\ref{ass:additive} hold. For
every $r$ with $\sigma_r>0$ there are continuous $\mathbb{R}^{d_G}$-valued functions
$u_{r,k}$ on $\Omega_{\alpha,k}$, $C^1$ under
Assumption~\ref{ass:domain}, such that
\[
u_r(\alpha)=\sum_{k=1}^{d}u_{r,k}(\alpha_k)
\quad\text{for every }\alpha\in\Omega_{\alpha},
\qquad
u_{r,k}(s)=\sigma_r^{-1}\int_{\Omega_{\gamma,k}}
h_k(s,c)\,\bar v_{r,k}(c)\,d\pi_{\gamma,k}(c),
\]
and symmetrically $v_r(\gamma)=\sum_{k=1}^{d}v_{r,k}(\gamma_k)$. The
decomposition is unique up to constants summing to zero: if
$\sum_{k}w_k(\alpha_k)=0$ on $\Omega_{\alpha}$ with each $w_k$ continuous, then every
$w_k$ is constant. In particular the components are identified from the
$u_r$ once normalised, say $u_{r,k}(\alpha^{\circ}_k)=0$ for $k\ge2$.
\end{proposition}
\begin{proof}
By Assumption~\ref{ass:additive}, for every
$\alpha\in\Omega_{\alpha}$,
\[
(T_g v_r)(\alpha)
=\int_{\Omega_{\gamma}}g(\alpha,\gamma)\,v_r(\gamma)\,d\pi_\gamma(\gamma)
=\sum_{k=1}^{d}\int_{\Omega_{\gamma}}h_k(\alpha_k,\gamma_k)\,v_r(\gamma)\,
d\pi_\gamma(\gamma),
\]
and since the $k$-th integrand depends on $\gamma$ only through its $k$-th
coordinate, conditioning on $\gamma_k$ (iterated expectations) reduces each
term to an integral against the marginal: with
$\bar v_{r,k}(c)=\mathbb{E}[v_r(\gamma_t)\mid\gamma_{t,k}=c]$,
\[
\int_{\Omega_{\gamma}}h_k(\alpha_k,\gamma_k)\,v_r(\gamma)\,d\pi_\gamma(\gamma)
=\int_{\Omega_{\gamma,k}}h_k(\alpha_k,c)\,
\bar v_{r,k}(c)\,d\pi_{\gamma,k}(c),
\]
each summand a function of $\alpha_k$ alone. Since
$u_r=\sigma_r^{-1}T_g v_r$ in $\mathcal{H}_{\alpha}$ and both sides are continuous while
$\pi_\alpha$ has full support (density bounded below), the identity holds at
every $\alpha$. Continuity of $u_{r,k}$ follows from dominated convergence
($h_k$ is bounded and continuous, $\bar v_{r,k}\in L^1(\pi_{\gamma,k})$),
and $C^1$ from differentiation under the integral, $\partial_s h_k$ being
bounded and continuous under Assumption~\ref{ass:domain}. For uniqueness,
fix the reference point $\alpha^{\circ}$ and, given $k$, evaluate
$\sum_j w_j(\alpha_j)=0$ at the point with $j$-th coordinate
$\alpha^{\circ}_j$ for $j\neq k$ and $k$-th coordinate $s$ free, which the
product support permits: $w_k(s)=-\sum_{j\neq k}w_j(\alpha^{\circ}_j)$ is
constant. The statement for $v_r$ follows from the adjoint by the same computation:
$(T_g^{*}u_r)(\gamma)=\int_{\Omega_{\alpha}}g(\alpha,\gamma)^{\top}u_r(\alpha)\,d\pi_\alpha
=\sum_k\int_{\Omega_{\alpha,k}}\bar u_{r,k}(s)^{\top}h_k(s,\gamma_k)\,
d\pi_{\alpha,k}(s)$, with
$\bar u_{r,k}(s):=\mathbb{E}[u_r(\alpha_i)\mid\alpha_{i,k}=s]$, each summand a
function of $\gamma_k$ alone; divide by $\sigma_r$.
\end{proof}
\begin{theorem}[Additive extension]
\label{thm:additive}
Suppose Assumptions~\ref{ass:domain}, \ref{ass:obseq},
\ref{ass:multi-index}, their $\gamma$-analogues, and
Assumption~\ref{ass:additive} hold, and let $R\ge\max(R_0,R_0')$ with $R_0$,
$R_0'$ as in Theorem~\ref{thm:main} and Corollary~\ref{cor:gamma}. Define
the coordinate truncations
\begin{align*}
U^{(k)}_R(s)&:=\bigl(u_{1,k}(s)^{\top},\dots,u_{R,k}(s)^{\top}\bigr)^{\top},
\qquad s\in\Omega_{\alpha,k},\\
V^{(k)}_R(c)&:=\bigl(v_{1,k}(c),\dots,v_{R,k}(c)\bigr)^{\top},
\qquad c\in\Omega_{\gamma,k},
\end{align*}
taking values in $\mathbb{R}^{d_G R}$ and $\mathbb{R}^{R}$, respectively. Then:
\begin{enumerate}
\item[\textup{(i)}] each $U^{(k)}_R$ is injective on $\Omega_{\alpha,k}$
and each $V^{(k)}_R$ is injective on $\Omega_{\gamma,k}$;
\item[\textup{(ii)}] the stacked proxies
$\lambda_i:=\bigl(U^{(1)}_R(\alpha_{i,1}),\dots,
U^{(d)}_R(\alpha_{i,d})\bigr)$ and
$f_t:=\bigl(V^{(1)}_R(\gamma_{t,1}),\dots,
V^{(d)}_R(\gamma_{t,d})\bigr)$ are identified, continuous, and
injective in $\alpha_i$ and $\gamma_t$, so they satisfy
Assumption~\ref{ass:proxyID}(ii);
\item[\textup{(iii)}] the normalised regression is additive: with
$\lambda_{i,k}:=U^{(k)}_R(\alpha_{i,k})$ and
$f_{t,k}:=V^{(k)}_R(\gamma_{t,k})$,
\[
g_0(\lambda_i,f_t):=\mathbb{E}[G_{it}\mid\lambda_i,f_t]
=\sum_{k=1}^{d}\tilde h_k(\lambda_{i,k},f_{t,k}),
\qquad
\tilde h_k:=h_k\circ\bigl((U^{(k)}_R)^{-1},(V^{(k)}_R)^{-1}\bigr),
\]
with each $\tilde h_k$ continuous on the compact product
$U^{(k)}_R(\Omega_{\alpha,k})\times V^{(k)}_R(\Omega_{\gamma,k})$
of two curves. Each component depends on one scalar unit type and
one scalar time type, whatever $d$ and $R$.
\end{enumerate}
\end{theorem}
\begin{proof}
\emph{(i).} Fix $k$ and $s,s'\in\Omega_{\alpha,k}$ with
$U^{(k)}_R(s)=U^{(k)}_R(s')$, and let $\alpha[s]\in\Omega_{\alpha}$ denote the point
with $k$-th coordinate $s$ and $j$-th coordinate $\alpha^{\circ}_j$ for
$j\neq k$, which lies in $\Omega_{\alpha}$ by the product structure. By
Proposition~\ref{prop:additive-eigen}, for every $r\le R$ the components in
coordinates $j\neq k$ cancel in the difference, so
\[
u_r(\alpha[s])-u_r(\alpha[s'])
=u_{r,k}(s)-u_{r,k}(s')=0 ,
\]
and note this holds for \emph{any} additive decomposition, the constants
cancelling as well. Hence $U_R(\alpha[s])=U_R(\alpha[s'])$, and
Theorem~\ref{thm:main} ($R\ge R_0$) gives $\alpha[s]=\alpha[s']$, i.e.\
$s=s'$. The $\gamma$-side is symmetric, via Corollary~\ref{cor:gamma}.
\emph{(ii).} The $u_r$ are identified, hence so are
$u_{r,k}$ by the uniqueness statement of
Proposition~\ref{prop:additive-eigen} under the stated normalisation.
Injectivity of each block in its own coordinate gives injectivity of the
stacked map: if $\lambda_i=\lambda_i'$ then
$U^{(k)}_R(\alpha_{i,k})=U^{(k)}_R(\alpha_{i,k}')$ for every $k$, so
$\alpha_{i,k}=\alpha_{i,k}'$ by (i).
\emph{(iii).} Assumption~\ref{ass:proxyID} holds: part (i) is the
maintained invariance of Model \eqref{eq:modelI}; part (ii) is established
in (ii) above; part (iii) holds with
$C=\sup_{\Omega_{\alpha}\times\Omega_{\gamma}}\lVert (D_\alpha g,D_\gamma g)\rVert<\infty$ under
Assumption~\ref{ass:domain} and its $\gamma$-analogue.
Theorem~\ref{th:proxyID} therefore gives
$g_0(\lambda_i,f_t)=g(\alpha_i,\gamma_t)
=\sum_k h_k(\alpha_{i,k},\gamma_{t,k})$. By (i) each $U^{(k)}_R$ is a
continuous injection on a compact set, hence a homeomorphism onto its image
with continuous inverse, and likewise $V^{(k)}_R$; substituting
$\alpha_{i,k}=(U^{(k)}_R)^{-1}(\lambda_{i,k})$ and
$\gamma_{t,k}=(V^{(k)}_R)^{-1}(f_{t,k})$ gives the display, with
$\tilde h_k$ continuous as a composition of continuous maps.
\end{proof}
\section{Conclusion}
Theorem~\ref{thm:main} replaces a global univalence problem by a local one.
Under Assumption~\ref{ass:obseq} the global injectivity of the limit map is
\emph{free} (Proposition~\ref{prop:part1}), so the only substantive assumption is
\emph{local} immersion (Assumption~\ref{ass:multi-index}).
The refinement to a finite truncation is then purely topological, via monotonicity of $\Delta_R$
and Dini's theorem on the compact $\Omega_{\alpha}\times\Omega_{\alpha}$.
The truncation order $R_0$ is governed by the rate at which $M_R\to M_\infty$
relative to the floor $c_0$ and by the decay of the singular values $\sigma_r$;
faster decay and a larger $c_0$ both reduce it.