Triple Products of Eigenfunctions and Spheres

Author

Joe Schaefer

Dedication

To Autumn.

Abstract

Let $P_\alpha=-\Delta_g+\langle\nabla\alpha,\nabla\cdot\rangle$ be a Witten operator on a closed connected Riemannian manifold, acting in $L^2(e^{-\alpha}\ d\mathrm{vol}_g)$. The spectrum of $P_\alpha$ does not determine $(g,\alpha)$. The full multiplication table of eigenfunctions does. This paper isolates a finite piece of that table.

Fix cutoffs $\Lambda\le\Lambda’<\infty$ and let $\mathcal{V}$ be the sum of eigenspaces of eigenvalue at most $\Lambda$. The visible packet is the restriction of the triple products $\Phi$ to pairs from $\mathcal{V}$ against modes up to $\Lambda’$, together with the weighted tail

$$R(u,v)=-P(I-\Pi_{\Lambda’})(uv)=-\sum_{\lambda>\Lambda’}\lambda\ \Phi(u,v,\bar e_\lambda)\ e_\lambda$$

and its uniform size $\Delta=\sup\lVert R(u,v)\rVert_2$ over unit $u,v\in\mathcal{V}$. Write $t=(I-\Pi_{\Lambda’})(uv)$. No Weyl law, no potential, and no curvature bound is used.

If $R=0$, the packet determines $(X,g,\alpha)$ up to a $C^\infty$ isometry of $g$ and an additive constant on $\alpha$. If $\lVert t\rVert_{C^1}$ is smaller than a threshold depending on the embedding of $X$ in $\mathcal{V}^*$, evaluation realises $X$ as a $C^1$ submanifold $\Sigma\subset\mathcal{V}^*$ on which

$$\lVert\iota^*g_\Sigma-g\rVert_{C^k}+\lVert\alpha_\Sigma\circ\iota-\alpha-c\rVert_{C^k}\le C(d,k)\ \lVert R\rVert_{C^k(\Sigma)},$$

with $d=\dim\mathcal{V}$. That bound is not a statement about $\Delta$. When the tail of $\Phi$ occupies a finite slot, Sobolev embedding on $\Sigma$ converts $\lVert R\rVert_{C^k}$ into a multiple of $\Delta$, with constants depending only on window dimensions and $k$.

If in addition $\alpha=0$, an exact packet realises $X$ as a union of orbits of a compact Lie group of isometries of the reconstructed metric (Theorem 21). When $\dim\mathcal{V}=n+1$ and the quadratic $Q=\sum u_a^2$ is constant, that action is transitive and $X$ is isometric to a sphere of radius $\sqrt Q$.

There is a correspondence between slot-sphere exact packets and compact connected Riemannian homogeneous spaces. Whether every exact unweighted window is of slot-sphere type is left open.

The unweighted first window on the sphere is the minimal finite-tail case: $\Lambda=n$, $\dim\mathcal{V}=n+1$, and $R=0$ already at $\Lambda’=2n+2$. That frame recognizes the unit round sphere exactly, and recognizes it with linear $C^0$ stability for an isolated cluster of first $n+1$ harmonics with an almost spherical table. Those statements are frame-side. They are not Obata theorems.

1. Introduction

A Witten operator on a closed Riemannian manifold $(X,g)$ is the weighted Laplacian

$$P_\alpha=-\Delta_g+\langle\nabla\alpha,\nabla\cdot\rangle$$

acting in $L^2(e^{-\alpha}\ d\mathrm{vol}_g)$ \cite{Wit82}. Its spectrum is discrete, nonnegative, and unbounded, with kernel the constant functions. The geometric data of the pair are the metric $g$ and the weight $\alpha$; equivalently, Riemannian volume and the measure $\mu=e^{-\alpha}\ d\mathrm{vol}_g$. Ordinary Laplace–Beltrami operators are the case $\alpha=0$.

The spectrum alone does not determine $(g,\alpha)$. What does determine them, in the large, is the multiplication table of eigenfunctions

$$\Phi(u,v,w)=\int_X uvw\ d\mu$$

together with the list of eigenvalues \cite{Sch24,Sch26}. That statement consumes every eigenspace. The purpose of this article is a finite substitute.
Fix cutoffs $\Lambda\le\Lambda’<\infty$ and let

$$\mathcal{V}=\bigoplus_{\lambda\le\Lambda}\ker(P-\lambda)$$

be the corresponding spectral window. Products of window functions need not lie in $\mathcal{V}$. Write $m(u,v)=\Pi_{\Lambda’}(uv)$ for the projection onto modes up to $\Lambda’$, and

$$R(u,v)=-P\bigl(I-\Pi_{\Lambda’}\bigr)(uv)=-\sum_{\lambda>\Lambda’}\lambda\ \Phi(u,v,\bar e_\lambda)\ e_\lambda$$

for the weighted tail of those products. Throughout, $u,v\in\mathcal{V}$. The uniform size of the tail is

$$\Delta=\sup_{\lVert u\rVert_2=\lVert v\rVert_2=1}\lVert R(u,v)\rVert_2.$$

The exact subclass is $R=0$ at a finite $\Lambda’$, which is an algebraic constraint on the window, not an asymptotic one.

A Wyman bound controls the unweighted $\ell^2$ mass of $\Phi(u,v,e_\lambda)$ off the frequency triangle. It is not used in Theorems 1–4. The coefficients of $R$ exist by completeness of the eigenbasis. The bound is recorded to mark the class: compact hyperbolic surfaces satisfy it, and W1–W2, for all large windows, and still have $R\neq 0$. Exactness is a constraint on the visible table, not a consequence of triple-product decay.

The visible packet is

$$\bigl(\Lambda,\ \Lambda’,\ \sigma(P)\cap[0,\Lambda’],\ \Phi\big|_{\mathcal{V}\otimes\mathcal{V}\otimes\mathcal{V}’},\ R\bigr).$$

The cometric $\Gamma$ of $g$ is not an extra input. The identity

$$P(uv)=u\ Pv+v\ Pu-2\Gamma(u,v)$$

computes $\Gamma$ on $\mathcal{V}$ from $(P,\Phi)$ with remainder $R$. Two nondegeneracy hypotheses close the packet: evaluation $X\to\mathcal{V}^\ast$ is a smooth embedding, and ${du:u\in\mathcal{V}}$ spans $T^*X$. No Weyl law is assumed, no potential is present, and no curvature bound is used.

If $R=0$, the packet determines $(X,g,\alpha)$ up to a $C^\infty$ isometry of $g$ and an additive constant on $\alpha$. If $\alpha=0$, that pair carries a compact Lie group of isometries whose orbits decompose $X$ (§9); if also $\dim\mathcal{V}=n+1$ and $Q$ is constant, $X$ is a sphere. If $\lVert t\rVert_{C^1}$ is smaller than a threshold depending on the embedding of $X$ in $\mathcal{V}^*$, evaluation realises $X$ as a $C^1$ submanifold $\Sigma\subset\mathcal{V}^*$ on which

$$\lVert\iota^*g_\Sigma-g\rVert_{C^k}+\lVert\alpha_\Sigma\circ\iota-\alpha-c\rVert_{C^k}\le C(d,k)\ \lVert R\rVert_{C^k(\Sigma)},$$

with $d=\dim\mathcal{V}$. When the tail of $\Phi$ occupies a finite slot, Sobolev embedding on $\Sigma$ converts this to a linear bound in $\Delta$ whose constants depend only on window dimensions and $k$. In both regimes the manifold is recovered as the space of multiplicative characters of the window, the metric by dualizing $\Gamma$, and the weight from the ratio of $\mu$ to Riemannian volume.

The theorem is recognition, not construction: $P$ is given as Witten, and the packet is asked to recover the geometry that produced it. It is not a spectral uniqueness theorem. Two Witten operators with the same spectrum, or even the same $\sigma(P)\cap[0,\Lambda’]$, need not be isometric; the missing data are $\Phi$ and $R$. Two operators with the same exact packet ($R=0$) are isometric, with $\alpha$ agreeing up to a constant. Two operators with the same approximate packet need not be isometric; they are $C^k$-close to isometric on the scale of $\lVert R\rVert_{C^k(\Sigma)}$.

A distinguished slice is the unweighted first window on the sphere. There one may take $\Lambda=n$, $\dim\mathcal{V}=n+1$, and $R=0$ already at $\Lambda’=2n+2$: products of linear coordinates on $S^n$ occupy only the constant slot and the quadratic harmonics. Theorem 3 recognizes the unit round sphere from that exact frame. Theorem 4 is linear $C^0$ stability for an isolated cluster of the first $n+1$ harmonics whose visible products lie in $\eta$-gaps about $0$ and $2n+2$; on a sufficiently small $C^2$ ball about the round metric those gaps persist and the table deficits are $O\bigl(\lVert g-g_{\mathrm{rd}}\rVert_{C^2}\bigr)$. Those two statements are compared with Obata’s theorem \cite{Oba62} and with quantitative Obata theorems on the Ricci side \cite{CMS}. They are not Ricci theorems. They use a linear frame and its products, not a lower bound on Ricci curvature and a single first eigenfunction.

If $\alpha=0$ and the exact window has an isometric embedding with constant slot quadratics, $(X,g)$ is a compact homogeneous space. Every compact homogeneous Riemannian manifold has some isometric embedding with an exact slot-sphere packet; §9.1 says that if the packet given is of that form, then $X$ is a single $K$-orbit and so $X$ is in that precise class.

The article is organized as follows. Section 2 fixes the Witten category and the window packet. Section 3 extracts $\Gamma$ from the multiplication rule with remainder $R$. Section 4 identifies points of $X$ with characters of the window. Section 5 dualizes $\Gamma$ to a metric and reads $\alpha$ from the two measures. Section 6 assembles Theorems 1 and 2. Section 7 specializes to the unweighted first window on the sphere. Section 8 records what the packet does not determine. Section 9 shows that an exact unweighted window is a union of orbits of a compact linear group of isometries, and a sphere when $d=n+1$ and $Q$ is constant. 9.1 gives a second openness criterion – the embedding is isometric into slot spheres.

The correspondence is between slot-sphere exact packets and compact connected Riemannian homogeneous spaces (Proposition~30). Whether every exact unweighted window is of slot-sphere type (equivalently: whether Theorem 21 always supplies an open orbit) is left open.

Geometry sits in the multiplication table, not in the list of eigenvalues. The contribution here is that a finite, explicitly tailed piece of that table already reconstructs a Witten pair, and that the unit sphere is the minimal unweighted instance of the same mechanism; an exact unweighted window is homogeneous once it has an open orbit.

A discrete analogue is immediate for low-energy eigenpairs of a finite-element Laplace or Witten operator. Let $\mathcal{V} _ h$ be the span of the first $d$ computed modes and let $\Phi_h$ be their triple products against a slightly larger computed window, with tail $R_h$ the residual of Leibniz for the discrete product. Theorem 2 applies verbatim to this packet: if $\lVert t _ h\rVert_{C^1}$ is below the embedding threshold, evaluation realises a $C^1$ mesh-surface $\Sigma_h\subset\mathcal{V}_h^*$ whose reconstructed metric is $C^k$-close to the Galerkin metric on the scale of $\lVert R_h\rVert _ {C^k}$. In the isolated first-window regime of Theorem 4 the same $\eta$-gaps persist for a mesh that resolves the first $n+1$ harmonics, and the table deficits are controlled by the $C^2$ FEM error of the metric plus the discrete product error. The exact-window conclusions (Theorem 1, Theorem 21, the sphere frame) are not claimed for a finite mesh: $R_h$ vanishes only in the limit $h\to0$ with $\Lambda’$ fixed, and only if the continuous products already close. The packet is a diagnostic for computed low modes, not a discrete classification theorem.

Glaser–Stern reconstruct a closed Riemannian manifold as a Gromov–Hausdorff limit of finite metric spaces built from a truncated Dirac spectral triple: the spectrum of a Dirac-type operator cut at a spectral projection, together with the truncated representation of $C^\infty(X)$ \cite{GS20}. That is a noncommutative cutoff of first-order spinor data. A packet in the present paper is scalar. The visible data are triple products of eigenfunctions of a Witten operator $P_\alpha$ in a finite window, plus the tail $R$ of the pointwise product.

When $R=0$, Theorem 1 recovers $(X,g,\alpha)$ up to isometry from that finite table, not as a GH limit of auxiliary graphs. The two reconstructions are complementary: one truncates a spectral triple and passes to the limit; the other keeps a finite multiplication table exact and reads the metric from $\Gamma$. Neither implies the other. In particular a truncated Dirac triple does not produce the Clebsch–Gordan identities used in Theorems 3 and 4, and an exact scalar packet does not determine a spin structure.

On the round sphere the scalar packet is exact at the first harmonic slot ($\mathcal{H}_ 1\otimes\mathcal{H} _ 1\subset\mathcal{H} _ 0\oplus\mathcal{H} _ 2$); the corresponding truncated Dirac triple recovers $S^n$ only as a Gromov–Hausdorff limit of state spaces as $\Lambda\to\infty$. Theorem 4 is stronger than that approximation on a packet neighborhood of the spherical table: small $\delta$ yields a $C^1$-graph over $S^n$ and a metric $C^0$-close to the round metric, in the topology of diffeomorphisms rather than in $d_{GH}$. That neighborhood contain a $C^2$ ball about $g_{\mathrm{rd}}$ after a finite cut of the tail (Claim 7.1); without the cut it is a condition on $\delta_t$, not a Kato consequence.

Gromov–Hausdorff limits under a Ricci bound are described by Cheeger–Colding \cite{CC96, CC97} and, synthetically, by $\mathrm{RCD}(K,N)$ spaces \cite{AGS14}; a packet reconstructs the Witten pair at finite cutoff and does not pass through those limits.

1.1 Results

Standing hypotheses: $X$ closed and connected, $P=P_\alpha$ a Witten operator \cite{Wit82}, packet as in §2, $u,v\in\mathcal{V}$, Wyman on window triples \cite{Wym}, W1, W2.

\begin{theorem}[exact recognition]
\label{thm-1}
Assume $R=0$. The packet

$$\bigl(\Lambda,\Lambda’,\sigma(P)\cap[0,\Lambda’],\Phi\big|_{\mathcal{V}\otimes\mathcal{V}\otimes\mathcal{V}’}\bigr)$$

determines $(X,g,\alpha)$ up to a $C^\infty$ Riemannian isometry of $g$ and an additive constant on $\alpha$. Evaluation is a Riemannian isometry of $(X,g)$ onto the character space of $(\mathcal{V},m)$ in $\mathcal{V}^\ast$, and $P$ is the Witten operator of the reconstructed pair.

If in addition $\alpha=0$, then $X$ is a union of orbits of a compact Lie group of isometries of the reconstructed metric (Theorem 21). If also $\dim\mathcal{V}=n+1$ and $Q$ is constant, $X$ is a sphere.
\end{theorem}

\begin{theorem}[approximate recognition]
\label{thm-2}
Let $d=\dim\mathcal{V}$. Write $t(u,v)=(I-\Pi_{\Lambda’})(uv)$ for $u,v\in\mathcal{V}$, and assume W1 and W2.
\begin{description}
\item[(i)] (infinite tail). There exists $\tau_*=\tau_*(d,\iota(X))>0$ such that if

$$\sup_{|u|2=|v|2=1}|t(u,v)|{C^1(X)}<\tau*,$$

then evaluation realises $X$ as a $C^1$ embedded submanifold $\Sigma\subset\mathcal{V}^*$ (the standing embedding of W1 persists),
and

$$|\iota^*g_\Sigma-g|{C^k}+|\alpha_\Sigma\circ\iota-\alpha-c|{C^k}\le C(d,k)\ |R|_{C^k(\Sigma)}$$

for every $k\ge 0$, with $c$ an additive constant on $\alpha$. The diffeomorphism $\iota$ is $C^\infty$ if $t=0$. The smallness parameter is $|t|_{C^1}$, equivalently $\Vert R \rVert _ {C^k}$ after Lemma 9(i). It is not $\Delta=\sup\lVert R\rVert _ 2$. The same smallness may be written $\Delta _ {C^1}<\tau _ */C(n)$ in the weighted $\ell^2$ form of Remark 10.

\item[(ii)] (finite slot). If the tail that defines $R$ occupies a finite slot of dimension $d_{\mathrm{tail}}$ (or is cut at a finite $\Lambda’’$), Lemma 9(ii) supplies $\Delta_*=\Delta_*(d,d_{\mathrm{tail}},\iota(X))>0$ such that $\Delta<\Delta_*$ implies the conclusions of (i), and

$$|R|{C^k(\Sigma)}\le C(d,d{\mathrm{tail}},k)\ \Delta.$$

In that case $\iota$ is a $C^k$ almost-isometry on the scale of $\Delta$.
\end{description}
\end{theorem}

\begin{theorem}[round sphere, exact window]
\label{thm-a}
Assume $\alpha=0$, $\dim\mathcal{V}=n+1$, $\Lambda=n$, and $R=0$ at $\Lambda’=2n+2$. Suppose there is a frame $x_0,\dots,x_n$ in $\mathcal{V}$ with $Px_a=nx_a$, $\int x_ax_b=\delta_{ab}$, $\int x_a=0$, $\sum_ax_a^2=1$, each $x_ax_b-\delta_{ab}/(n+1)$ an eigenfunction at $2n+2$, and ${dx_a}$ spanning $T^*X$. Then $X$ is isometric to the unit round sphere, $P=-\Delta$, and $\operatorname{span}{x_a}=H_1$. The pair $(\lambda,2\lambda+2\lambda/n)$ gives radius $\sqrt{n/\lambda}$.
\end{theorem}

\begin{theorem}[round sphere, quantitative cluster]
\label{thm-b}
Let $\lambda_1\le\cdots\le\lambda_{n+1}$ be the first $n+1$ positive eigenvalues of $P$ counted with multiplicity, and let $\mathcal{V}=\mathrm{span}{x_1,\dots,x_{n+1}}$ be the joint span of corresponding $L^2$-orthonormal eigenfunctions. Write

$$w=\lambda_{n+1}-\lambda_1,\qquad\bar\lambda=\frac1{n+1}\sum_{i=1}^{n+1}\lambda_i,$$

and assume the cluster is isolated: $w<\lambda_{n+2}-\lambda_{n+1}$.

Assume W1 and W2 for this block.

Fix $\eta\in\bigl(0,\min{n,n+4}/2\bigr)$ smaller than the first two spherical gaps, and a cutoff $\Lambda’\ge 2n+2+\eta$. Set

$$\Pi^\eta_{\mathrm{sph}}=\Pi_{[0,\eta]}+\Pi_{[2n+2-\eta,\ 2n+2+\eta]}.$$

Allow the table deficits

$$\begin{align*} \delta_w&=w+|\bar\lambda-n|,\\ \delta_{\mathrm{on}}&=\sum_{a,b}\bigl|\textstyle\int x_ax_b-\delta_{ab}\bigr| +\sum_a\bigl|\textstyle\int x_a\bigr|,\\ \delta_1&=\bigl\|\textstyle\sum_a x_a^2-1\bigr\|_{C^0},\\ \delta_\Phi&=\sup_{a,b}\bigl\|(I-\Pi^\eta_{\mathrm{sph}})\Pi_{\Lambda'}(x_ax_b)\bigr\|_{L^2},\\ \delta_t&=\sup_{a,b}\|t_{\Lambda'}(x_a,x_b)\|_{C^1}, \end{align*}$$

where $t_{\Lambda’}=(I-\Pi_{\Lambda’})(x_ax_b)$. Set

$$\delta=\delta_w+\delta_{\mathrm{on}}+\delta_1+\delta_\Phi+\delta_t.$$

If $\delta<\delta_*(n,\eta)$, then evaluation $F=(x_1,\dots,x_{n+1})$ realises $X$ as a $C^1$-graph over $S^n$, and

$$d_{\mathrm{GH}}(X,S^n)+|F^*g_{\mathrm{eucl}}-g|_{C^0}\le C(n,\eta,\Lambda’)\ \delta.$$

If the tail of $R_{\Lambda’}$ occupies a finite slot of dimension $d_{\mathrm{tail}}$ (or is cut at a finite $\Lambda’’$), then

$$\delta_t\le C(n,d_{\mathrm{tail}},\Lambda’,\Lambda’’)\Delta_{\mathrm{fr}}$$

with $\Delta_{\mathrm{fr}}=\sup|R_{\Lambda’}(x_a,x_b)|2$, and $\delta$ may be written with $\Delta${\mathrm{fr}} in place of $\delta_t$. If $\Lambda’=2n+2+\eta$ and the tail is so cut, the constant depends only on $n$, $\eta$, and $d_{\mathrm{tail}}$.

\end{theorem}

On a fixed pair a cut exists by Claim 7.1; the Kato ball uses one cut chosen after the metric.

This is linear stability of the first $n+1$ harmonics of Theorem 3, allowing the first slot to split and the quadratic slot to drift inside an $\eta$-gap. It does not use Ricci curvature. When $w=0$, $\eta=0^+$, $\Lambda’=2n+2$, and $t=0$, it specialises to stability of a single eigenspace with exact projectors $\Pi_0+\Pi_{2n+2}$.

On a sufficiently small $C^2$ ball about $g_{\mathrm{rd}}$ the gaps about $0$, $n$, and $2n+2$ persist. Elliptic estimates on the cluster promote $|g-g_{\mathrm{rd}}|_{C^2}$ to $C^0$ control of the frame, and

$$\delta_w+\delta_{\mathrm{on}}+\delta_1+\delta_\Phi\le C(n,\eta)\ |g-g_{\mathrm{rd}}|_{C^2}.$$

The tail modulus $\delta_t$ is not in general $O(|g-g_{\mathrm{rd}}|_{C^2})$ for an infinite tail; it is, after a finite cut $\Lambda’’$. Thus Theorem 4 applies on that ball with the finite-cut form of $\delta$. It does not apply to a residual set in $\mathcal{M}$: a generic perturbation may break isolation or spray products through the $\eta$-gaps.

The $C^0$ bound and min-max give $\lvert\lambda_k(P)-\lambda_k(S^n)\rvert\le C(n,k,\eta)\ \delta$ for each fixed $k$. Tanno’s uniqueness is the converse neighbourhood statement with $\operatorname{Spec}(P)=\operatorname{Spec}(S^n)$; the paper does not assume the spectra coincide.

2. The Packet

Let $X$ be a closed connected smooth manifold and let

$$P=P_\alpha=-\Delta_g+\langle\nabla\alpha,\nabla\cdot\rangle$$

act in $L^2(X,\mu)$ with $d\mu=e^{-\alpha}\ d\mathrm{vol}_g$ \cite{Wit82}. Assume $P$ is essentially self-adjoint, with discrete spectrum

$$0=\lambda_0<\lambda_1\le\lambda_2\le\cdots\to+\infty$$

and $\ker P=\operatorname{span}{1}$. The principal symbol of $P$ is the cometric $\Gamma$ of $g$. For all smooth $u,v$,

$$\begin{equation} P(uv)=u\ Pv+v\ Pu-2\Gamma(u,v).\tag{A1} \end{equation}$$

Fix cutoffs $\Lambda\le\Lambda’<\infty$. Write

$$\mathcal{V}=\bigoplus_{\lambda\le\Lambda}\ker(P-\lambda),\qquad\mathcal{V}’=\bigoplus_{\lambda\le\Lambda’}\ker(P-\lambda),$$

and let $\Pi_{\Lambda’}$ be the orthogonal projection onto $\mathcal{V}’$. Throughout, $u,v\in\mathcal{V}$. Set

$$m(u,v)=\Pi_{\Lambda’}(uv),\qquad\Phi(u,v,w)=\int_X uvw\ d\mu.$$

The weighted tail of the product is

$$R(u,v)=-P\bigl(I-\Pi_{\Lambda’}\bigr)(uv) =-\sum_{\lambda>\Lambda’}\lambda\ \Phi(u,v,e_\lambda)\ e_\lambda,$$

and

$$\Delta=\sup\set{\lVert R(u,v)\rVert_2:\ u,v\in\mathcal{V},\ \lVert u\rVert_2=\lVert v\rVert_2=1}$$

The exact subclass is $R=0$, equivalently $\Delta=0$.

\textbf{Wyman}. For $u,v\in\mathcal{V}$ and an $L^2$-normalized eigenfunction $e_\lambda$,

$$\sum_{\lambda}\bigl\lvert\Phi(u,v,\bar e_\lambda)\bigr\rvert^2$$

is controlled by the measure of frequency triangles with side lengths comparable to $(\sqrt{\lambda_u},\sqrt{\lambda_v},\sqrt{\lambda})$ \cite{Wym}. In particular there is rapid decay of the $\ell^2$ mass of $\Phi(u,v,\ \cdot\ )$ in the classically forbidden region

$$\sqrt{\lambda}\ge(1+\varepsilon)\bigl(\sqrt{\lambda_u}+\sqrt{\lambda_v}\bigr),$$

for every $\varepsilon>0$. The paper uses only the following crude consequence: if a finite slot above $\Lambda’$ is fixed, they are bounded by a constant depending on that slot, on $\Lambda$, and on $\lVert u\rVert_2\lVert v\rVert_2$.

A pointwise bound

$$\bigl\lvert\Phi(u,v,\bar e_\lambda)\bigr\rvert\le C\ (1+\lambda_u+\lambda_v+\lambda)^\delta\ \lVert u\rVert_2\lVert v\rVert_2$$

is not claimed in \cite{Wym}; when it is invoked below it is as a convenient majorant on a finite slot, not as Wyman’s theorem.

\textbf{W1}. Evaluation $\iota:X\to\mathcal{V}^\ast$, $\iota(x)(u)=u(x)$, is a smooth embedding onto a closed submanifold of $\mathcal{V}^\ast$. The same holds for evaluation $X\to(\mathcal{V}’)^\ast$. Characters are therefore evaluations on $\mathcal{V}’$, and the comparison error $E$ of §4 vanishes.

\textbf{W2}. The differentials ${du:u\in\mathcal{V}}$ span $T^*X$ at every point.

The visible packet is

$$\bigl(\Lambda,\ \Lambda’,\ \sigma(P)\cap[0,\Lambda’],\ \Phi\big|_{\mathcal{V}\otimes\mathcal{V}\otimes\mathcal{V}’},\ R\bigr).$$

On pairs from $\mathcal{V}$, (A1) reads

$$2\Gamma(u,v)=u\ Pv+v\ Pu-P\bigl(m(u,v)\bigr)+R(u,v).$$

Thus $\Gamma$ on $\mathcal{V}$ is not an independent input. No Weyl law, no potential, and no curvature bound is assumed. Dimension of $X$ is the rank in W2.

When $P=-\Delta$ (equivalently $\alpha=0$), both axioms hold on an arbitrary closed Riemannian manifold, with no curvature hypothesis. Wyman’s bound on $\Phi$ is valid in that generality. Independently, there exists $\Lambda<\infty$ such that

$$\iota_\Lambda:X\to\mathcal{V}_\Lambda^*$$

is a smooth immersive embedding: finitely many eigenfunctions separate points and span the cotangent space \cite{BBG94, Portegies14}. Thus a sufficiently large Laplace window always carries a Wyman estimate and a spectral frame. These facts do not select spheres, nor do they force exactness. Compact hyperbolic surfaces satisfy both axioms for all large $\Lambda$, yet products of eigenfunctions have infinite tails, so $R\neq0$. Rigidity in Theorems~2–4 and in §9 begins only when the visible table is close to exact, or close to the spherical Clebsch–Gordan pattern, on that window.

3. The Cometric From the Window

\begin{lemma}
\label{lma-1}
Let $u,v\in\mathcal{V}$. Split the pointwise product as

$$uv=m(u,v)+t(u,v),\qquad m(u,v)=\Pi_{\Lambda’}(uv),\qquad t(u,v)=(I - \Pi_{\Lambda’})(uv).$$

The visible product $m(u,v)$ is assembled from the in-window table:

$$m(u,v)=\sum_{\lambda\le\Lambda’}\Phi(u,v,\bar e_\lambda)\ e_\lambda.$$

Identity (A1) then reads

$$2\Gamma(u,v)=u\ Pv+v\ Pu-P\bigl(m(u,v)\bigr)+R(u,v),$$

where

$$R(u,v)=-P\ t(u,v)=-\sum_{\lambda>\Lambda’}\lambda\ \Phi(u,v,\bar e_\lambda)\ e_\lambda.$$

Thus $\Gamma$ on $\mathcal{V}$ is determined by $P|_{\mathcal{V}}$, by $\Phi$ on $\mathcal{V}\otimes\mathcal{V}\otimes\mathcal{V}’$, and by the tail $R$. In particular $\lVert R(u,v)\rVert_2\le\Delta\lVert u\rVert_2\lVert v\rVert_2$. If $R=0$, then $t=0$ and $\Gamma(u,v)$ is determined by $(P,\Phi)$. The bilinear map $\Gamma:\mathcal{V}\times\mathcal{V}\to C^\infty(X)$ does not depend on the orthonormal bases used to expand $\Phi$.

On the first spherical window split

$$m(x_a,x_b)=\Pi_{\mathrm{sph}}(x_ax_b)+(I-\Pi_{\mathrm{sph}})(x_ax_b),$$

with $\Pi_{\mathrm{sph}}=\Pi_0+\Pi_{2n+2}$. The first summand is the model table of Theorem 3; the second is the in-window table error measured by $\delta_\Phi$ in Theorem 4. Lemma 5 converts $\delta_w+\delta_\Phi+|R_{\Lambda’}(x_a,x_b)|2$ into an $L^2$ error on $\Gamma(x_a,x_b)$ relative to $\delta${ab}-x_ax_b. The upgrade of that error to $C^0$, and the substitution of $\delta_t$ for $|R_{\Lambda’}|_2$, are in the proof of Theorem 4, not here.

\end{lemma}

\begin{proof}
Identity (A1) is $2\Gamma(u,v)=u\ Pv+v\ Pu-P(uv)$. Substitute $uv=m+t$ and set $R=-Pt$. The expansion of $m$ is the definition of $\Phi$ against $\mathcal{V}’$. If $R=0$, then $t=0$, so $m(u,v)=uv$ and every coefficient of $uv$ against $\mathcal{V}’$ is a value of $\Phi$.

Basis-independence: $P$ and $\Phi$ are defined on $\mathcal{V}$ without choices, and $m$ is orthogonal projection of the pointwise product.

On a frame of $n+1$ eigenfunctions with eigenvalues $\lambda_a$, identity (A1) reads

$$2\Gamma(x_a,x_b)=(\lambda_a+\lambda_b)x_ax_b-P(m(x_a,x_b))+R_{\Lambda’}.$$

Write $\lambda_a=n+\varepsilon_a$ with $|\varepsilon_a|\le\delta_w$. Split $m=\Pi_{\mathrm{sph}}m+(I-\Pi_{\mathrm{sph}})m$. The model computation of Theorem 3 gives

$$P\bigl(\Pi_{\mathrm{sph}}(x_ax_b)\bigr)=(2n+2)\Bigl(x_ax_b-\frac{\delta_{ab}}{n+1}\Bigr)+E_{\mathrm{sph}},$$

with $|E_{\mathrm{sph}}|{L^2}\le C(n)\delta_w$. The remainder $(I-\Pi _ {\mathrm{sph}})m$ is the in-window table error measured by $\delta_\Phi$, and $R${\Lambda’} is measured in $L^2$ by $\lVert R_{\Lambda’}(x_a,x_b)\rVert_2$. Hence

$$\lVert\Gamma(x_a,x_b)-(\delta_{ab}-x_ax_b)\rVert_{L^2}\le C(n)\bigl(\delta_w+\delta_\Phi+\lVert R_{\Lambda’}(x_a,x_b)\rVert_2\bigr).$$

This is the $L^2$ cometric input to Theorem 4. It is not a $C^0$ bound, and it does not use $\Delta_{\mathrm{fr}}$ as a $C^0$ modulus.

The deficit $\delta_\Phi$ is the $L^2$ mass of $\Pi_{\Lambda’}(x_ax_b)$ off the cluster projector

$$\Pi^\eta_{\mathrm{sph}}=\Pi_{[0,\eta]}+\Pi_{[2n+2-\eta,\ 2n+2+\eta]}.$$

The error in replacing $\Pi_{\mathrm{sph}}$ by $\Pi^\eta_{\mathrm{sph}}$ is $O(\eta)|x_ax_b|_2$ and is absorbed into $C(n,\eta)$ in Theorem 4. The tail above $\Lambda’$ is not $\delta_\Phi$.

Write $\Delta_g$ as a divergence-form operator on the fixed space $L^2(d\mathrm{vol}{g{\mathrm{rd}}})$. On a sufficiently small $C^2$ ball about $g_{\mathrm{rd}}$, the resolvent is close in operator norm and the contours about $0$, $n$, and $2n+2$ miss $\mathrm{Spec}(\Delta_g)$ \cite{Kato}. Thus $\Pi^\eta_{\mathrm{sph}}$ has rank bounded in terms of $n$ only, and the first $n+1$ modes remain an isolated cluster. Kato and elliptic estimates on that cluster give

$$\delta_w+\delta_{\mathrm{on}}+\delta_1+\delta_\Phi\le C(n,\eta)\ |g-g_{\mathrm{rd}}|_{C^2}.$$

They do not bound $\delta_t$ for an infinite tail. Theorem 4 applies on that ball with the finite-cut form of $\delta$, or with $\delta_t$ kept as packet data. Individual eigenvalues in the cluster may split; only the joint span $\mathcal{V}$ is used.

In the infinite-tail regime the constraint $\delta<\delta_*$ is open in the $C^k$ topology on metrics for every $k>1$. Continuity of $\delta_w$, $\delta_{\mathrm{on}}$, $\delta_1$, and $\delta_\Phi$ is Kato’s $C^2$ statement on the isolated cluster. Continuity of $\delta_t=|t_{\Lambda’}|{C^1}$ uses $C^k$ control of the window functions and of $\Pi${\Lambda’}(x_ax_b) at a fixed cutoff $\Lambda’$. The finite-cut form of $\delta$ is already open in $C^2$. Neither statement is openness in the residual set of $\mathcal{M}$: a perturbation may still close a gap or move mass through the $\eta$-windows.

The specialisation $w=0$, $\eta\to 0$, $\Lambda’=2n+2+\eta$, and $t=0$ recovers exact projectors $\Pi_0+\Pi_{2n+2}$ and the single-eigenspace case of Theorem 3.
\end{proof}

\begin{remark}[Sogge and $\lVert t\rVert_{C^0}$]
Hörmander–Sogge gives $\lVert e_\lambda\rVert_{C^0}\le C_S(n)\ (1+\lambda)^{(n-1)/4}$ \cite{Sog88}. Expanding $t$ and applying Cauchy–Schwarz with the weight in $R=Pt$ yields, for a tail supported in a finite slot $\Lambda<\lambda\le\Lambda’’$,

$$\lVert t\rVert_{C^0}\le C(n,\Lambda’,\Lambda’’)\ \Delta,$$

and therefore

$$\lVert R\rVert_{C^0(X)}\le C(n,\Lambda’,\Lambda’’)\ \Delta.$$

Wyman bounds the coefficients of $R$ \cite{Wym}; Sogge bounds the modes. Neither is a curvature comparison.
\end{remark}

\begin{remark}
If the $\Phi$-tail that defines $R$ occupies a finite slot of dimension $d_{\mathrm{tail}}$, all norms on that slot are equivalent and

$$\lVert R\rVert_{C^k(X)}\le C(d,d_{\mathrm{tail}},k)\ \Delta.$$

If the tail is infinite, $\lVert R\rVert_{C^k}$ remains the modulus.
\begin{claim}[cut cometric, fixed pair]
Let $u,v\in\mathcal{V}$ on a fixed closed Witten pair $(X,g,\alpha)$. For $\Lambda’’\ge\Lambda’$ write

$$2\Gamma_{\Lambda’’}(u,v)=uPv+vPu-P\bigl(\Pi_{\Lambda’’}(uv)\bigr)$$

Then $\Gamma_{\Lambda’’}(u,v)\to\Gamma(u,v)$ in $C^k(X)$ as $\Lambda’’\to\infty$, for every $k\ge 0$. In particular, given $\varepsilon>0$ there exists $\Lambda’’=\Lambda’’(u,v,k,\varepsilon)$ such that

$$|\Gamma_{\Lambda’’}(u,v)-\Gamma(u,v)|_{C^k}<\varepsilon.$$

The rate depends on $|uv|{H^{s}}$ for $s>k+n/2$. It is not claimed to be uniform for $g$ in a $C^2$ neighbourhood of $g${\mathrm{rd}}.
\end{claim}

\begin{subproof}[Proof of Claim]
Fix $u,v\in\mathcal{V}$ on a single closed Witten pair. The product $uv$ is $C^\infty$, so $P^m(uv)\in L^2$ for every $m$. Let ${e_\lambda}$ be an $L^2$-orthonormal eigenbasis of $P$, and write

$$uv=\sum_{\lambda\ge 0}\Phi(u,v,e_\lambda)\ e_\lambda.$$

The tail after $\Lambda’’$ is

$$t_{\Lambda’’}:=(I-\Pi_{\Lambda’’})(uv)=\sum_{\lambda>\Lambda’’}\Phi(u,v,e_\lambda)\ e_\lambda.$$

For any $m$,

$$|P^m t_{\Lambda’’}|2^2=\sum{\lambda>\Lambda’’}\lambda^{2m}\ |\Phi(u,v,e_\lambda)|^2\le\Lambda’’^{-2}\sum_{\lambda>\Lambda’’}\lambda^{2m+2}\ |\Phi(u,v,e_\lambda)|^2\le\Lambda’’^{-2}\ |P^{m+1}(uv)|_2^2.$$

Thus $|t_{\Lambda’’}|_{H^{2m}}\to 0$ as $\Lambda’’\to\infty$. Sobolev embedding on the closed $n$-manifold $X$ gives

$$|t_{\Lambda’’}|{C^k}\le C(n,k,X)\ |t{\Lambda’’}|_{H^{s}},\qquad s>k+\tfrac n2,$$

so $|t_{\Lambda’’}|_{C^k}\to 0$ for every $k$.

The cut identity is the exact Leibniz formula with $\Pi_{\Lambda’’}(uv)$ in place of $uv$:

$$2\Gamma_{\Lambda’’}(u,v)=uPv+vPu-P\bigl(\Pi_{\Lambda’’}(uv)\bigr).$$

The true identity is

$$2\Gamma(u,v)=uPv+vPu-P(uv).$$

Subtract:

$$2\bigl(\Gamma_{\Lambda’’}(u,v)-\Gamma(u,v)\bigr)=P\bigl((I-\Pi_{\Lambda’’})(uv)\bigr).$$

Hence

$$\Gamma_{\Lambda’’}(u,v)-\Gamma(u,v)=\tfrac12 P t_{\Lambda’’}.$$

Elliptic regularity of $P$ (or the same Sobolev estimate with one extra derivative) yields

$$|P t_{\Lambda’’}|{C^k}\le C(n,k,X)\ |t{\Lambda’’}|_{H^{s+2}}\to 0.$$

Therefore $\Gamma_{\Lambda’’}(u,v)\to\Gamma(u,v)$ in $C^k(X)$.

The rate is controlled by $|P^{m}(uv)|_2$ for $m$ large enough that $2m>k+n/2+2$. Those norms depend on the pair $(X,g,\alpha)$ and on the two window functions. They are not claimed to be bounded uniformly for $g$ in a $C^2$ neighbourhood of a model metric.
\end{subproof}
\end{remark}

\begin{corollary}
If $R=0$, the Gram functions $\Gamma(u,v)$ for $u,v\in\mathcal{V}$ are packet data. W2 is then a nondegeneracy condition on those functions. If $\Delta>0$, the same Gram functions are determined up to an $L^2$ error of size $\Delta$, and up to a $C^k$ error of size $\lVert R\rVert_{C^k}$ after the comparison of §4.
\end{corollary}

4. Characters

Let $m:\mathcal{V}\times\mathcal{V}\to\mathcal{V}’$ be the packet product of §2. By W1, evaluation

$$\iota:X\to\mathcal{V}^\ast,\qquad\iota’:X\to(\mathcal{V}’)^\ast$$

are smooth embeddings onto closed submanifolds. Characters of the window are evaluations on $\mathcal{V}’$. In particular the comparison error between $R$ as a function on $X$ and $R$ as a function on the image $\Sigma=\iota(X)$ vanishes: $R_\Sigma\circ\iota=R_X$.

\begin{lemma}
\label{lma-2}
Let $d=\dim\mathcal{V}$ and let $\iota:X\to\mathcal{V}^*$ be the evaluation of W1. Write $t(u,v)=(I-\Pi_{\Lambda’})(uv)$ for $u,v\in\mathcal{V}$.

\begin{description}
\item[(i)] (infinite tail) There exists $\tau_*=\tau_*(d,\iota(X))>0$ such that if

$$\sup_{\lVert u\rVert_2=\lVert v\rVert_2=1}\lVert t(u,v)\rVert_{C^1(X)}<\tau_*,$$

then $\iota(X)$ is a $C^\infty$ embedded submanifold $\Sigma\subset\mathcal{V}^*$, $\iota:X\to\Sigma$ is a $C^1$ diffeomorphism (and $C^\infty$ if $t=0$), and for every $k\ge 0$

$$c(k)^{-1}\ \lVert R_X\rVert_{C^k(X)}\le\lVert R_\Sigma\rVert_{C^k(\Sigma)}\le c(k)\ \lVert R_X\rVert_{C^k(X)},$$

with $c(k)=C(d,k)$. The smallness parameter here is $\lVert t\rVert_{C^1}$, not $\Delta$.

\item[(ii)] (finite slot) If the $\Phi$-tail that defines $R$ occupies a finite slot of dimension $d_{\mathrm{tail}}$ (or is cut at a finite $\Lambda’’$), then Remarks 6 and 7 give

$$\lVert t\rVert_{C^1}\le C(d,d_{\mathrm{tail}},n,\Lambda’,\Lambda’’)\ \Delta.$$

Hence there exists $\Delta_*=\Delta_*(d,d_{\mathrm{tail}},\iota(X))>0$ such that $\Delta<\Delta_*$ implies the conclusions of (i), and

$$\lVert R\rVert_{C^k(\Sigma)}\le C(d,d_{\mathrm{tail}},k)\ \Delta.$$

\end{description}
\end{lemma}

\begin{proof}
W1 makes $\iota$ a smooth embedding onto a closed submanifold. The size of $t$ is used only to persist that embedding in the $C^1$ topology and to compare $|R|_{C^k}$ on $X$ and on $\Sigma$.

The product map $F(\xi;u,v)=\xi(m(u,v))-\xi(u)\xi(v)$ vanishes on $\iota(X)$ up to $t$. Evaluation extends to $\mathcal{V}’$, so at $\xi=\iota(x)$ one has $\xi(m(u,v))=m(u,v)(x)$ and the only defect is $t(x)$. Differentiating $F$ in the $\mathcal{V}^*$-directions at $\iota(x)$ yields the family $(u,v)\mapsto\xi(u)\ dv+\xi(v)\ du$ on $T_x^ * X$. W2 says ${du:u\in\mathcal{V}}$ spans $T_x^ *X$, so this family is transverse to $\iota(X)$. If $\lVert t\rVert_{C^1}<\tau_*$, $F$ is a $C^1$-small perturbation of a section that vanishes exactly on $\iota(X)$. Persistence of transverse embeddings keeps $\iota$ a $C^1$ diffeomorphism onto its image. If $t=0$, then $R=0$, $F$ vanishes exactly on $\iota(X)$, and $\iota$ is $C^\infty$. The $C^k$ comparison of $R_X$ and $R_\Sigma$ is the chain rule for $\iota$ and $\iota^{-1}$, whose $C^{k-1}$ norms stay bounded for $\lVert t\rVert_{C^1}<\tau_*$.

For (ii), a finite slot makes every $C^k$ norm on the tail equivalent to the $L^2$ norm of $R=-Pt$. Remark 6 supplies the $C^0$ bound on $t$ via Sogge; equivalence of norms on a space of dimension $d_{\mathrm{tail}}$ upgrades that to $C^1$. Thus $\Delta$ small implies $\lVert t\rVert_{C^1}$ small, and (i) applies.
\end{proof}

\begin{remark}[weighted tails] The hypothesis of Lemma 9(i) is $\lVert t\rVert_{C^1}<\tau_*$. On a finite slot this follows from $\Delta$ by Remarks 6 and 7. On an infinite tail it does not, and the paper does not claim that it does. The same Sogge bound that gives Remark 6 still converts a weighted norm of $\Phi$ into $\lVert t\rVert_{C^1}$. Expanding

$$t(u,v)=\sum_{\lambda>\Lambda’}\Phi(u,v,\bar e_\lambda)\ e_\lambda$$

and using $\lVert e_\lambda\rVert_{C^0}\le C_S(n)(1+\lambda)^{(n-1)/4}$ together with $\lVert\nabla e_\lambda\rVert_{C^0}\le C(n)(1+\lambda)^{(n-1)/4+1/2}$ yields

$$\lVert t\rVert_{C^1}\le C(n)\sum_{\lambda>\Lambda’}(1+\lambda)^{\frac{n-1}{4}+\frac12}\lvert\Phi(u,v,\bar e_\lambda)\rvert.$$

Cauchy–Schwarz against the same weights gives the $\ell^2$ form

$$\lVert t\rVert_{C^1} \le C(n)\Bigl(\sum_{\lambda>\Lambda’}(1+\lambda)^{\frac{n-1}{2}+1}\lvert\Phi(u,v,\bar e_\lambda)\rvert^2\Bigr)^{1/2}.$$

Either right-hand side may be written $\Delta_{C^1}(u,v)$. Then Lemma 9(i) applies as soon as $\sup\Delta_{C^1}<\tau_*/C(n)$. This $\Delta_{C^1}$ is not the packet scalar $\Delta=\sup\lVert R\rVert_2$. The two coincide, up to a constant depending on the slot, only when the sum is finite, which is Lemma 9(ii). Wyman controls the unweighted $\ell^2$ mass of $\Phi$ off the frequency triangle; it does not by itself bound $\Delta_{C^1}$ on an infinite tail. If a uniform scalar is wanted in that regime, $\Delta_{C^1}$ (or a cut at a finite $\Lambda’’$, with the far tail absorbed into $R$) is the quantity that should be listed with the packet.
\end{remark}

\begin{remark}
Theorem 2’s general bound is (i), written in $|t|{C^1}$ and $|R|${C^k(\Sigma)}. The conversion to a multiple of $\Delta$ is (ii). Theorem 4 uses (i) with $d=n+1$ and modulus $\delta_t$; it uses (ii) only after a finite cut of the tail.
\end{remark}

\begin{corollary}
If $R=0$, $X$ is identified with the character space of the finite algebra $(\mathcal{V},m)$ as a $C^\infty$ embedded submanifold of $\mathcal{V}^\ast$. If $\Delta<\Delta_*$, the same identification holds as a $C^1$ diffeomorphism, and $\lVert R\rVert_{C^k(\Sigma)}$ may be used interchangeably with $\lVert R\rVert_{C^k(X)}$.
\end{corollary}

\begin{remark}
Residual ambiguity is an orthogonal change of basis of $\mathcal{V}$ preserving $\Phi$, a compact subgroup of $O(d)$. The threshold $\Delta_*$ depends on $d$ and on the $C^1$ geometry of $\iota(X)$ at $\Delta=0$. It is not universal in $n$ except on the spherical window of §7.
\end{remark}

5. Metric and Weight

\begin{lemma}Assume Lemma~\ref{lma-1} and W2. There is a unique Riemannian metric $g$ on $X$ such that

$$\Gamma(u,v)=g^{-1}(du,dv)\qquad\text{for all }u,v\in\mathcal{V}.$$

This $g$ is the metric whose cometric is the principal symbol of $P$. After Lemma 9,

$$\bigl\lVert\iota^*g_\Sigma-g\bigr\rVert_{C^k}\le C(d,k)\ \lVert R\rVert_{C^k(\Sigma)}.$$

If the tail of $\Phi$ occupies a finite slot of dimension $d_{\mathrm{tail}}$, the right-hand side is at most $C(d,d_{\mathrm{tail}},k)\ \Delta$.
\label{lma-3}
\end{lemma}

\begin{proof}
W2 says ${du:u\in\mathcal{V}}$ spans $T^*X$. The assignment $(du,dv)\mapsto\Gamma(u,v)$ is well-defined on that spanning set: $\Gamma(u,u)=\sigma_P(du,du)$, so $du=0$ implies $\Gamma(u,\cdot)=0$. Positive-definiteness is the assumption that the principal symbol of $P$ is an inner product. Dualizing gives $g$. In a local frame ${u_a}\subset\mathcal{V}$ of rank $n=\dim X$,

$$g^{ab}=\Gamma(u_a,u_b)$$

on $\operatorname{span}{du_a}$, and $g$ is the inverse matrix. Independence of the frame is change of basis for a cometric. The $C^k$ estimate is Lemma 5 plus the comparison of $\lVert R\rVert_{C^k}$ on $X$ and on $\Sigma$ from Lemma 9. Finite-tail conversion is the remark in §3.
\end{proof}

\begin{lemma}
Let $g$ be the metric of Lemma 14 and let $\mu$ be the measure determined by

$$\int uv\ d\mu=\Phi(1,u,v),\qquad u,v\in\mathcal{V},$$

extended by continuity to $L^2(\mu)$. Then $\mu$ is a smooth positive multiple of $\mathrm{vol}_g$. The function

$$\alpha=-\log\frac{d\mu}{d\mathrm{vol}_g}$$

is smooth, unique up to an additive constant, and the given operator $P$ is the Witten operator of $(g,\alpha)$. If $\Delta<\Delta_*$,

$$\bigl\lVert\alpha_\Sigma\circ\iota-\alpha-c\bigr\rVert_{C^k}\le C(d,k)\ \lVert R\rVert_{C^k(\Sigma)}.$$

\label{lma-4}
\end{lemma}

\begin{proof}
By the standing category, $P$ is already the Witten operator of some pair $(g_0,\alpha_0)$. Lemma 14 identifies $g=g_0$, because both metrics have the same principal symbol. The inner product that makes $P$ self-adjoint is $\mu=e^{-\alpha_0}\mathrm{vol}{g_0}$. On the other hand $\Phi(1,\cdot,\cdot)$ is that inner product, since $1=\ker P$ and $\Phi(1,u,v)=\langle u,v\rangle${L^2(\mu)}. Thus $\mu=e^{-\alpha_0}\mathrm{vol}_g$, so $\alpha=\alpha_0+c$. A Witten operator on a closed manifold is determined by $(g,\alpha)$ up to that constant. The $C^k$ estimate follows from Lemma 14 and the elliptic regularity of $\log(d\mu/d\mathrm{vol}_g)$.
\end{proof}

\begin{remark} Lemma~\ref{lma-4} is recognition, not construction: it uses that $P$ was Witten. The two measures $\mu$ and $\mathrm{vol}_g$ are the only source of $\alpha$. The packet does not store $\alpha$ separately.
\end{remark}

\begin{corollary}
Combining Lemmas 9, 14, and 15, the packet determines a Riemannian manifold $(\Sigma,g_\Sigma)$ in $\mathcal{V}^\ast$ and a weight $\alpha_\Sigma$ on $\Sigma$. Evaluation pulls these back to $(g,\alpha)$ exactly when $R=0$, and to within $C(d,k)\lVert R\rVert_{C^k(\Sigma)}$ when $\Delta<\Delta_*$.
\end{corollary}

6. Recognition

The lemmas of §§3–5 assemble into Theorem \ref{thm-1} and Theorem \ref{thm-2}.

\begin{proof}[Proof of Theorem~\ref{thm-1}]
Assume $R=0$. Lemma 5 gives $\Gamma$ on $\mathcal{V}$ as a function of $(P,\Phi)$, with remainder zero. Lemma 9 identifies $X$ with the character space $\Sigma\subset\mathcal{V}^\ast$ of the finite algebra $(\mathcal{V},m)$ by a $C^\infty$ diffeomorphism $\iota$. Lemma 14 dualizes $\Gamma$ to a Riemannian metric $g _ \Sigma$ on $\Sigma$ with $\iota^*g_ \Sigma=g$. Lemma 15 reads $\alpha$ from $\mu/\mathrm{vol} _ g$, up to an additive constant, and identifies $P$ with the Witten operator of that pair. Thus $\iota$ is a $C^\infty$ Riemannian isometry, and the packet determines $(X,g,\alpha)$ up to that isometry and the constant on $\alpha$. Theorem 21 upgrades the unweighted case to an orbit decomposition, and to a sphere when $d=n+1$.

Residual freedom is an orthogonal change of basis of $\mathcal{V}$ preserving $\Phi$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-2}]
W1 is standing: $\iota$ is already a smooth embedding. Lemma 5 gives $\Gamma$ on $\mathcal{V}$ with remainder $R=-Pt$. Under the hypothesis of (i), Lemma 9(i) persists $\iota$ as a $C^1$ diffeomorphism $X\to\Sigma\subset\mathcal{V}^*$ and identifies $|R|{C^k(\Sigma)}$ with $|R|${C^k(X)} up to $C(d,k)$. Lemma 14 upgrades the cometric error to

$$|\iota^*g_\Sigma-g|{C^k}\le C(d,k)\ |R|{C^k(\Sigma)}.$$

Lemma 15 gives the same bound for $\alpha_\Sigma\circ\iota-\alpha-c$. Under the hypothesis of (ii), the tail is finite-dimensional, so Remarks 6–7 convert $|t|{C^1}$ and $|R|${C^k} into multiples of $\Delta$, and (i) applies. If $R=0$, both parts reduce to Theorem 1. An infinite tail is controlled only through $|t|{C^1}$ or $\Delta${C^1}; $\Delta$ alone is not a modulus in that regime.
\end{proof}

\begin{remark}
Two packets for the same operator at different admissible cutoffs produce isometric copies if both have $R=0$, and almost-isometric copies on the scale of the two values of $\lVert R\rVert_{C^k(\Sigma)}$ otherwise. The theorems do not identify merely isospectral Witten operators.
\end{remark}

7. The Unweighted First Window on the Sphere

This section is Theorems 1 and 2 on a single window: $\alpha=0$, $\Lambda=n$, $\dim\mathcal{V}=n+1$, and products of the window supported in slots ${0,2n+2}$ when $R=0$. Let ${x_a}_{a=0}^n$ be an $L^2$-orthonormal frame in $\mathcal{V}$ as in Theorem 3, and write $F=(x_0,\dots,x_n):X\to\mathbb{R}^{n+1}$.

\begin{lemma} Assume the hypotheses of Theorem 3. Then $\sum_a x_a^2=1$, $\Gamma(x_a,x_b)=\delta_{ab}-x_ax_b$, and $dF$ has rank $n$ at every point. Consequently $F(X)$ is a closed embedded hypersurface in the unit sphere of $\mathbb{R}^{n+1}$.
\end{lemma}
\begin{proof}
The quadric $\sum_a x_a^2=1$ is given. Differentiating and using $Px_a=nx_a$ with (A1) yields

$$2\Gamma(x_a,x_b)=n x_ax_b+n x_bx_a-P(x_ax_b).$$

By hypothesis $x_ax_b-\delta_{ab}/(n+1)$ is an eigenfunction at $2n+2$, and $P(1)=0$, so

$$P(x_ax_b)=(2n+2)\Bigl(x_ax_b-\frac{\delta_{ab}}{n+1}\Bigr).$$

Hence $\Gamma(x_a,x_b)=\delta_{ab}-x_ax_b$. The matrix $\delta_{ab}-x_ax_b$ has rank $n$ on the unit sphere, and ${dx_a}$ spans $T^*X$ by hypothesis, which is W2 for this frame. W1 follows: $F$ is an immersion, $X$ is closed, and $F(X)$ lies on $S^n$, so $F$ is a covering onto its image. Simple connectedness of $S^n$ for $n\ge 2$, or a direct check for $n=1$, makes $F$ a diffeomorphism onto $S^n$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-a}] Lemma 19 and Theorem 1 give that $X$ is isometric to the unit round sphere. This is the case $d=n+1$, $Q\equiv 1$ of Lemma 25. The metric dual to $\Gamma(x_a,x_b)=\delta_{ab}-x_ax_b$ is the round metric of radius $1$, and $\alpha=0$ because $\mu=\mathrm{vol}_g$. Thus $P=-\Delta$. The frame spans the first harmonics because it is an $(n+1)$-dimensional eigenspace at eigenvalue $n$. If the first eigenvalue is $\lambda$ rather than $n$, the same identities with the pair $(\lambda,2\lambda+2\lambda/n)$ rescale the radius to $\sqrt{n/\lambda}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm-b}]
The isolation hypothesis makes $\mathcal{V}$ a spectral window of dimension $n+1$. Standing W1 and W2 apply to this block. Write $\lambda_a=n+\varepsilon_a$, so $|\varepsilon_a|\le\delta_w$.

Lemma 5 on the cluster frame, with smeared projector $\Pi^\eta_{\mathrm{sph}}$,
gives

$$2\Gamma(x_a,x_b)=(\lambda_a+\lambda_b)x_ax_b-P(m(x_a,x_b))+R_{\Lambda’}$$

and

$$\bigl|\Gamma(x_a,x_b)-(\delta_{ab}-x_ax_b)\bigr|{L^2}\le C(n,\eta)\ (\delta_w+\delta_\Phi+|R{\Lambda’}(x_a,x_b)|_2).$$

The error in replacing the exact projectors $\Pi_0+\Pi_{2n+2}$ by $\Pi^\eta_{\mathrm{sph}}$ is $O(\eta)|x_ax_b|2$ and is absorbed into $C(n,\eta)$. The in-window error $\delta_\Phi$ lives in a slot of dimension $O(n^2)$ for fixed $\eta,\Lambda’$; Remarks 6–7 upgrade that piece to $C^0$. The tail is upgraded to $C^0$ only through $\delta_t=|t${\Lambda’}|{C^1}, or through $\Delta${\mathrm{fr}} after a finite cut. Hence

$$\bigl|\Gamma(x_a,x_b)-(\delta_{ab}-x_ax_b)\bigr|_{C^0}\le C(n,\eta,\Lambda’)\ (\delta_w+\delta_\Phi+\delta_t).$$

The deficit $\delta_1$ puts $\sum_a x_a^2=1+O_{C^0}(\delta_1)$, so $F=(x_1,\dots,x_{n+1})$ lands in a $C^0$-neighbourhood of the unit sphere in $\mathcal{V}^*$. Gram–Schmidt absorbs $\delta_{\mathrm{on}}$ into $\delta_1$ and $\delta_\Phi$ at cost $C(n)$. The Gram matrix $\Gamma(x_a,x_b)$ is then $C^0$-close to $\delta_{ab}-x_ax_b$, which has rank $n$ on the unit sphere, so $dF$ has rank $n$ for $\delta<\delta_*(n,\eta)$.

Lemma 9(i), with $d=n+1$ and small $\delta_t$, persists $\iota$ as a $C^1$ diffeomorphism of $X$ onto a graph over $S^n$.
Theorem 2(i) yields

$$d_{\mathrm{GH}}(X,S^n)+|F^*g_{\mathrm{eucl}}-g|_{C^0}\le C(n,\eta,\Lambda’)\ \delta.$$

If the tail is cut at finite $\Lambda’’$, Theorem 2(ii) replaces $\delta_t$ by $\Delta_{\mathrm{fr}}$. No Ricci bound is used. Diameter control is that graph.

On a small $C^2$ ball about $g_{\mathrm{rd}}$, written as a divergence-form operator on $L^2(d\mathrm{vol}{g{\mathrm{rd}}})$, the contours about $0$, $n$, and $2n+2$ miss $\mathrm{Spec}(\Delta_g)$ \cite{Kato}. Thus the three gaps persist. Elliptic estimates on the isolated cluster give $C^0$ control of the frame from $|g-g_{\mathrm{rd}}|_{C^2}$, and therefore

$$\delta_w+\delta_{\mathrm{on}}+\delta_1+\delta_\Phi\le C(n,\eta)\ |g-g_{\mathrm{rd}}|_{C^2}.$$

A finite cut makes $\delta_t$ of the same order. Individual eigenvalues in the cluster may split; only the joint span $\mathcal{V}$ is used.
\end{proof}

\begin{remark}
Obata characterizes $S^n$ by $\operatorname{Ric}\ge(n-1)g$ and $\lambda_1=n$, using one eigenfunction \cite{Oba62}. Theorems 3 and 4 use an $(n+1)$-frame and its products. Quantitative Obata is Ricci-side stability of Obata \cite{CMS}. Theorem 4 is frame-side stability of Theorem 3. They are not the same theorem.
\end{remark}

8. What the Packet Does Not Determine

The packet of §2 is a finite piece of the multiplication table of $P$. It does not determine the rest of $\sigma(P)$. Two Witten operators with the same window products and different high eigenvalues are not distinguished.

It does not determine a potential. The standing category is Witten: $V=0$. A Schrödinger term is a different operator and a different paper \cite{Sch26}.

It does not yield spectral uniqueness. Two operators with the same spectrum, or the same $\sigma(P)\cap[0,\Lambda’]$, need not be isometric; the missing data are $\Phi$ and $R$. Two operators with the same exact packet ($R=0$) are isometric. Two operators with the same approximate packet need not be; they are $C^k$-close on the scale of $\lVert R\rVert_{C^k(\Sigma)}$.

It does not yield a constant depending only on $n$ in Theorem 2. The factor $C(d,k)$ depends on the window dimension. A constant depending only on $n$ and $\eta$, when $\Lambda’=2n+2+\eta$, appears in Theorem 4 after a finite cut of the tail (or with $\delta_t$ listed among the data).

It does not use Ricci curvature, a Weyl law, or the full table $\Phi$ on $\bigoplus_n\ker(P-\lambda_n)$. Those are the inputs of Obata theorems \cite{Oba62,CMS}, of eigenvalue asymptotics, and of the global comparison theorem \cite{Sch24,Sch26}, respectively. The present arguments use a window frame and its products.

Two packets for the same operator at different admissible cutoffs are consistent with Theorems 1 and 2. That consistency is not part of one packet; each packet has its own $(\Lambda,\Lambda’,R)$.

An exact unweighted window is not shown to be a single homogeneous space except when $d=n+1$ and $Q$ is constant. 9.1 gives a second openness criterion – the embedding is isometric into slot spheres.

Sunada pairs remain non-isometric and therefore cannot share an exact packet; §9 does not contradict §8.

The contribution of the paper is the converse direction inside this class: a finite, explicitly tailed window already reconstructs a Witten pair, and the unit round sphere is the minimal unweighted instance of that reconstruction.

9. Orbits of an Exact Unweighted Window

Assume throughout this section that $\alpha=0$, that $R=0$, and that W1 and W2 hold. Thus $P=-\Delta_g$, products of the window land in $\mathcal{V}’$, and evaluation $\iota:X\to\mathcal{V}^*$ is a smooth embedding whose differentials span $T^*X$. Let $d=\dim\mathcal{V}$ and fix an $L^2$-orthonormal frame ${u_a}_{a=1}^d$ of $\mathcal{V}$. Write $K=\mathrm{Isom}(X,g)$ and $K^\circ$ for its identity component.

\begin{theorem}
\label{thm-exact}
$K$ is a compact Lie group of isometries of $(X,g)$. The identity component acts real-analytically, and $X$ is a union of $G^\circ$-orbits (Lemma 24). If $\dim\mathcal{V}=n+1$ and $Q$ is a positive constant, then $G^\circ$ is transitive and $X$ is a sphere (Lemma 25). In general the principal set is a fibre bundle with fibre a compact homogeneous space. Residual packet freedom remains the compact group of orthogonal changes of frame of $\mathcal{V}$ that preserve $\Phi$.
\end{theorem}

Theorem 21 does not assume a tower.

\begin{lemma}[Myers–Steenrod and the window] $K$ is a compact Lie group \cite{MS39}. Pullback by $K$ preserves every eigenspace of $P$, hence preserves $\mathcal{V}$ and $\mathcal{V}’$, and $\iota$ is $K$-equivariant. The reconstructed cometric $\Gamma$ of Lemma 5 and the metric of Lemma 14 are $K$-invariant.
\end{lemma}

\begin{proof}Myers–Steenrod: the isometry group of a compact Riemannian manifold is a compact Lie group acting smoothly. An isometry commutes with $\Delta_g$, so it preserves eigenspaces and the $L^2$ pairing. It therefore acts on $\mathcal{V}$ and on $\mathcal{V}’$, and

$$\iota(k\cdot x)(u)=u(k^{-1}x)=(k^*\iota(x))(u),$$

so $\iota$ is equivariant. Lemma 5 with $R=0$ gives $\Gamma(u,v)$ as a bilinear combination of products and of $P$ on $\mathcal{V}$. Both are $K$-invariant, so $\Gamma$ is. Lemma 14 dualizes $\Gamma$ to $g$, which is therefore $K$-invariant.
\end{proof}

\begin{lemma}[the linear automorphism group]
Let $m:\mathcal{V}\times\mathcal{V}\to\mathcal{V}’$ be the multiplication $m(u,v)=uv$ of the exact packet, and let

$$Z:=\set{\xi\in\mathcal{V}^*:\ \xi(u)\xi(v)=\eta_\xi(m(u,v))\ \text{for all }u,v\in\mathcal{V}},$$

where $\eta_\xi\in(\mathcal{V}’)^*$ is the unique linear extension of $\xi$ along $m$ furnished by $R=0$ (evaluation of $\mathcal{V}’$ on $X$, transported by Lemma 9). Let

$$G:=\set{A\in\mathrm{O}(\mathcal{V}^*):\ A\cdot Z=Z}.$$

Then $G$ is a compact Lie group, $\iota(X)$ is a union of connected components of the smooth locus of $Z$, and $G$ acts isometrically on $(X,g)$. The image of $K$ in $\mathrm{O}(\mathcal{V}^*)$ lands in $G$.
\end{lemma}

\begin{proof}
The conditions cutting out $Z$ are quadratic in $\xi$, so $Z$ is a real-algebraic set. $G$ is a closed subgroup of $\mathrm{O}(d)$, hence a compact Lie group. Lemma 9 with $t=0$ identifies $\iota(X)$ with the smooth characters of $(\mathcal{V},m)$, which is precisely the smooth locus of $Z$ along the component containing $\iota(X)$. W1 says that component is a smooth embedded copy of $X$.
If $A\in G$, then $A$ preserves the multiplication tensor $m$ up to the orthogonal action on $\mathcal{V}’$ that makes the identities $\xi(u)\xi(v)=\eta(m(u,v))$ invariant. Consequently $A$ preserves $\Gamma$, which by Lemma 5 is a bilinear combination of $m$ and of $P|_{\mathcal{V}}$. The spectrum of $P$ on $\mathcal{V}$ is part of the packet and is $G$-invariant. Lemma 14 therefore gives that $A$ acts by an isometry of $g$. Lemma 21 puts the image of $K$ inside $G$.
\end{proof}

\begin{lemma}[orbits]
$G^\circ$ acts real-analytically on $X$. All orbits in a connected component of the principal set have the same dimension. Consequently $X$ is a disjoint union of $G^\circ$-orbits, and $G^\circ$ is transitive on $X$ if and only if some orbit is open in $X$.
\end{lemma}

\begin{proof}
The metric of Lemma 14 is dual to $\Gamma$, and $\Gamma$ is a bilinear combination of the packet product $m$ and of $P|_{\mathcal{V}}$ (Lemma 5 with $R=0$). Both are real-analytic, so $g$ is real-analytic and $X$ is a compact real-analytic Riemannian manifold. $G$ is a compact linear algebraic subgroup of $\mathrm{O}(\mathcal{V}^*)$, hence a compact Lie group, and its action on $Z^{\mathrm{sm}}$ is algebraic, therefore real-analytic. Restricting to the component $\iota(X)$ gives a real-analytic action on $X$.

Orbits of a compact Lie group are compact embedded submanifolds. The principal orbit theorem supplies an open dense principal set $X^{\mathrm{prin}}$ on which all orbits have the same dimension $r\le n$ and the same type. Unique continuation for real-analytic maps: an orbit of dimension $r$ that meets $X^{\mathrm{prin}}$ cannot drop dimension on a nonempty open subset of $X$. Thus every orbit in the given connected component of $X^{\mathrm{prin}}$ has dimension $r$, and $X$ is a union of such orbits together with lower-dimensional singular orbits of measure zero.

If some orbit $\mathcal{O}$ is open in $X$, then $\mathcal{O}$ is also closed (compact), so connectedness of $X$ gives $\mathcal{O}=X$. Conversely, a transitive action has a single orbit, which is open.
\end{proof}

\begin{lemma}[open orbit when $d=n+1$]
Assume in addition that $\dim\mathcal{V}=n+1$ and that the function $Q=\sum_a u_a^2$ is a positive constant. Then $\iota(X)$ is the sphere of radius $\sqrt{Q}$ in $\mathcal{V}^*$, $G^\circ$ contains a transitive action on that sphere, and $G^\circ$ is transitive on $X$. In particular $X$ is diffeomorphic to a sphere, and the metric is round if in addition $\mathcal{V}$ is one eigenspace and the trace identity of Lemma 23 holds.
\end{lemma}

\begin{proof}
The identity $\lvert\iota\rvert^2=Q$ puts $\iota(X)$ on the sphere $S$ of radius $\sqrt{Q}$ in the $(n+1)$-dimensional space $\mathcal{V}^*$. W1 says $\iota$ is an embedding of an $n$-manifold, so $\iota(X)$ is open in $S$. $S$ is connected, so $\iota(X)=S$.
The orthogonal group $\mathrm{O}(n+1)$ preserves $S$ and acts transitively on it. Restricting to the identity component and applying Lemma 24 gives transitivity on $X$. Thus $X$ is diffeomorphic to a sphere.

If moreover $\mathcal{V}$ is a single eigenspace and $\sum_a\Gamma(u_a,u_a)=\lambda Q$, Lemma 23 puts $\iota$ on a sphere in the Takahashi sense and the reconstructed metric of Lemma 14 is round of radius $\sqrt{Q}$. In the first spherical window this is Lemma 19: $\Gamma(x_a,x_b)=\delta_{ab}-x_ax_b$ and radius $1$. Those extra identities are the data of Theorem 3; they are not assumed in Theorem 21 beyond $Q$ constant.
\end{proof}

\begin{remark}[towers]
A nested sequence of windows $\mathcal{V}_j$ with $\mathcal{V}_j\cdot\mathcal{V}k\subset\mathcal{V}{j+k}$ cannot stabilise at finite dimension if it separates points. If the multiplication is the Clebsch–Gordan decomposition of a compact Lie group acting smoothly on $X$, evaluation is an equivariant eigenmap and $X$ is a union of orbits. If in addition evaluation is an isometric embedding into the slot spheres, the orbit is open by §9.1.

A single exact pair does not produce such a flag. The torus packet of §9.2 is the abelian case. This remark is not used in Theorems 1–4.
\end{remark}

If $\mathcal{V}=\bigoplus_i\mathcal{V}{\lambda_i}$ with more than one eigenvalue, Takahashi applies factorwise: each $\iota${\lambda_i}(X) lies in a sphere in $\mathcal{V}_{\lambda_i}^*$, and $\iota(X)$ lies in a product of spheres.

If $\iota$ is isometric and $\mathcal{V}$ has several eigenvalues, then $\iota$ is a Garay map. The image lies in a product of slot spheres only after each spectral projector is constant on the diagonal, which is not implied by $R=0$.

\begin{remark}
The tower holds for every CROSS first window and every Peter–Weyl window.
\end{remark}

\begin{proof}[Proof of Theorem 21] Lemmas 22–24 give a compact Lie group of isometries of $(X,g)$ and a compact linear group $G$ acting on $\iota(X)$. Lemma 24 gives that $X$ is a union of $G^\circ$-orbits, and that transitivity is equivalent to the existence of an open orbit. Lemma 25 supplies that open orbit when $\dim\mathcal{V}=n+1$ and $Q$ is constant (which is the first window of Theorems 3 and 21 in the spherical case), or when the evaluation is an isometric embedding into the slot spheres, §9.1. In general, set $H=\mathrm{Stab}_{K^\circ}(x)$ along a principal orbit; the principal set is a fibre bundle with fibre $K^\circ/H$. Finite covers and ineffective kernels are as before.
\end{proof}

\begin{remark}Theorem 21 in full generality is the orbit decomposition of Lemma 24. Transitivity is proved when $d=n+1$ and $Q$ is constant, which is the case already used by Lemma 19 and Theorem 3. A second transitivity/open orbit condition is given in Section 9.1. A general exact window is a union of compact orbits of a compact linear group of isometries; it is a single homogeneous space once an open orbit is known to exist. Compact hyperbolic surfaces remain excluded: they have no exact finite window satisfying W1.
\end{remark}

9.1 Slot Spheres and Open Orbits

\label{sec:slot-spheres}

Retain the hypotheses of Theorem~21: $\alpha=0$, $R=0$, and a compact group $K$ of linear automorphisms of $\mathcal{V}$ that preserves $\Phi$. Then $K$ acts smoothly on $X$ and $X$ is a union of $K$-orbits.

Assume in addition that evaluation of the slots realises an isometric embedding

$$\iota:X\hookrightarrow\prod_j S(\mathcal{E}_j)$$

into the product of the slot spheres in $\mathcal{V}^*$ (spheres about the origin of each Euclidean factor $\mathcal{E}_j$). The embedding is $K$-equivariant, because $K$ acts linearly on each $\mathcal{E}_j$ and preserves $\Phi$. In particular the radius functions $Q_j=\lvert F_j\rvert^2$ are $K$-invariant and constant, so they do not cut orbits down to level sets.

Since $\iota$ is an immersion, every $K$-orbit in $X$ has dimension $\dim X$. A compact group acting on a compact connected manifold cannot have a proper open orbit, hence $X$ is a single $K$-orbit:

$$X\cong K/H$$

for the stabilizer $H$ of any point. Thus $X$ is a compact homogeneous space and $g_\Gamma$ is a $K$-invariant Riemannian metric.

When there is a single slot of dimension $n+1$, this is Lemma25 and the last clause of Theorem21: $K$ acts through $O(n+1)$ and $X$ is a round sphere. When the slots are harmonic spaces of a round sphere, one recovers that same round metric; a different linear presentation of the same slots changes only the ambient image of $\iota$, not the orbit structure on $X$.

No statement is made about the second fundamental form of $\iota$ in Euclidean space. That concerns the image $\iota(X)\subset\mathcal{V}^*$, not the identification of $X$ with an open $K$-orbit.

\begin{remark}[Adve’s table is not a finite exact packet]
Adve’s converse theorem \cite{Adv25} reconstructs a compact hyperbolic 2-orbifold from the full multiplication table of a $(\mathfrak{g},K)$-adapted basis of $K$-finite vectors in $L^2(\Gamma\backslash G)$, $G=\mathrm{PSL}_2(\mathbb{R})$. The structure constants $C _ {ij}^\ell$ are the coordinates of that table. Crossing (HB6) is the four-point identity

$$\langle\psi_i\psi_j,\psi_{i’}\psi_{j’}\rangle=\langle\psi_i\psi_{i’},\psi_j\psi_{j’}\rangle,$$

together with $K$-equivariance, a unit, and a Leibniz rule for the raising operator. This is the complete infinite algebra of $H_{\mathrm{fin}}$ on the unit tangent bundle, not a finite window.

An exact packet in the sense of Theorems 1 and 21 is a finite spectral window $\mathcal{V}$ with $R=0$ at a finite cutoff $\Lambda’$. Products of Maass modes expand over infinitely many Casimirs of the summed $K$-weight, so no finite $\Lambda’$ makes $R=0$ while evaluation embeds the surface. Compact hyperbolic surfaces therefore lie outside the exact finite-window class and inside Adve’s class: the same multiplication, an infinite table, a noncompact structure group.

The identity slot is finite by HB4. Products of holomorphic discrete-series vectors of weights $\le k$ close in weight $\le 2k$, which is a finite exact table for a line-bundle operator, not for the scalar Laplacian on $X$. Neither fragment supplies a finite exact scalar window on $X$ or on $T^1X$.

The two converses meet at associativity of $\Phi$. They split on whether that algebra is generated by a finite embedding representation. That is the distinction already recorded in Remark 6 of \cite{Sch26}: Adve is uniqueness for unit tangent bundles of compact hyperbolic 2-orbifolds, not an instance of (G7).
\end{remark}

\begin{proposition}[converse to §9.1]
Let $(X,g)$ be a compact connected Riemannian homogeneous space. Then there exists an exact unweighted packet whose evaluation is an isometric embedding into a product of slot spheres. In particular $(X,g)$ arises in the class described by §9.1.
\end{proposition}
\begin{subproof}
Write $X=G/H$ with $G$ a compact Lie group of isometries acting transitively and $H=\mathrm{Stab}_G(x_0)$. The Laplacian $P=-\Delta_g$ commutes with $G$, so $L^2(X)$ decomposes into finite-dimensional irreducible $G$-modules and each irreducible summand lies in a single eigenspace of $P$ (Schur).

By Moore \cite{Moo76}, there is a $G$-equivariant isometric embedding

$$\iota\colon (X,g)\longrightarrow\mathbb{R}^N$$

into Euclidean space, for some $N<\infty$, with $G$ acting linearly and orthogonally on $\mathbb{R}^N$. (Mostow supplies a
$G$-equivariant embedding; Moore makes it isometric.) Decompose the ambient module as

$$\mathbb{R}^N=\bigoplus_{j=1}^s\mathcal{E}_j$$

into irreducible $G$-summands. The coordinate functions of $\iota$ on $\mathcal{E}_j$ span a $G$-irreducible space of smooth functions on $X$, hence lie in a single eigenspace of $P$. Let $\mathcal{V}=\bigoplus_j\mathcal{E}_j^\vee$ be the corresponding window of eigenfunctions (the dual coordinates), and let $\lambda_j$ be the eigenvalue on the $j$-th slot.

The slot quadratic

$$Q_j=\sum_{u\in\mathrm{ONB}(\mathcal{E}_j^\vee)}u^2$$

is $G$-invariant, hence constant on $X$. Thus $\iota(X)$ lies in the product of spheres

$$\prod_{j=1}^s\sqrt{Q_j}\ S(\mathcal{E}_j).$$

Isometry of $\iota$ means

$$g=\iota^ *g_{\mathrm{eucl}}=\sum_j\iota_j^*g_{\mathrm{eucl},j},$$

so evaluation on $\mathcal{V}$ is an isometric embedding into that product of slot spheres. Differentials of the coordinates span $T^*X$ because $\iota$ is an immersion, which is W2. W1 is the embedding.

It remains to check exactness. The pointwise product of two matrix coefficients of irreps $\rho$ and $\sigma$ is a matrix coefficient of $\rho\otimes\sigma$, hence a finite sum of matrix coefficients of the irreducible constituents of $\rho\otimes\sigma$. Taking $\Lambda’$ large enough that $\mathcal{V}’$ contains every constituent of $\mathcal{E}_j^\vee\otimes\mathcal{E}k^\vee$ for all slots $j,k$, one has $R=0$. The packet $(\Lambda,\Lambda’,\sigma(P)\cap[0,\Lambda’],\Phi|${\mathcal{V}\otimes\mathcal{V}\otimes\mathcal{V}’},0) is therefore an exact unweighted slot-sphere packet for $(X,g)$.

Residual freedom is an orthogonal change of basis of $\mathcal{V}$ preserving $\Phi$, as in Theorem 1.
\end{subproof}

9.2. The Torus Packet

The first window of a sphere is not the only exact packet. On a compact flat manifold the characters of the translation lattice multiply by addition of frequencies, so any finite spanning set of frequencies is exact at a finite cutoff. Theorem 1 reconstructs the flat metric. Theorem 21 supplies transitivity from the group law.

Let $X=\mathbb{R}^n/\Gamma$ be a compact flat manifold, with $\Gamma$ a Bieberbach group, and write $\Lambda_{\mathrm{tr}}$ for the translation lattice of $\Gamma$. Let $\Lambda_{\mathrm{tr}}^*$ be its dual. Characters

$$e_\xi(x)=\mathrm{e}^{2\pi i\langle\xi,x\rangle},\qquad\xi\in\Lambda_{\mathrm{tr}}^*,$$

are eigenfunctions of $P=-\Delta$ with eigenvalues $\lambda_\xi=4\pi^2\lvert\xi\rvert^2$. They are holonomy-invariant when descended to $X$. Fix a finite set $S\subset\Lambda_{\mathrm{tr}}^*$ that spans $\Lambda_{\mathrm{tr}}^*\otimes\mathbb{R}$ as a real vector space, and let $\mathcal{V}$ be the real span of ${1}\cup{\operatorname{Re}e_\xi,\operatorname{Im}e_\xi:\xi\in S}$. Set

$$\Lambda’=4\pi^2\max_{\xi,\eta\in S}\lvert\xi+\eta\rvert^2.$$

\begin{proposition}With this packet, $R=0$ and W1–W2 hold. Theorem 1 identifies $(X,g)$ with the compact flat manifold whose dual is generated by $S$. The identity component of the isometry group in Theorem 21 contains the translation torus of the cover $\mathbb{R}^n/\Lambda_{\mathrm{tr}}$, and $X$ is a principal orbit of that torus modulo the holonomy of $\Gamma$.
\end{proposition}
\begin{proof}The product of characters is a character, $e_\xi e_\eta=e_{\xi+\eta}$. If $\xi,\eta\in S$ then $\lvert\xi+\eta\rvert^2\le\max_{\xi’,\eta’\in S}\lvert\xi’+\eta’\rvert^2$, so $\lambda_{\xi+\eta}\le\Lambda’$ and $e_\xi e_\eta\in\mathcal{V}’$. Real and imaginary parts are linear combinations of characters, so the same bound holds for products in $\mathcal{V}$. Thus $R=0$.

Evaluation $\iota:X\to\mathcal{V}^*$ is the composition of the covering $\mathbb{R}^n\to X$ with the map

$$x\mapsto\bigl(\cos 2\pi\langle\xi,x\rangle,\ \sin 2\pi\langle\xi,x\rangle\bigr)_{\xi\in S}.$$

The differential has full rank $n$ at every point because $S$ spans. If $S$ generates $\Lambda_{\mathrm{tr}}^*$ as a group then the kernel of $\iota$ on $\mathbb{R}^n$ is exactly $\Lambda_{\mathrm{tr}}$, so $\iota$ descends to an embedding of the translation torus, and hence of its holonomy quotient $X$. That is W1.

The cometric on characters is

$$\Gamma(e_\xi,e_\eta)=4\pi^2\langle\xi,\eta\rangle\ e_{\xi-\eta}$$

(up to conjugation). Restricted to $\mathcal{V}$ it has rank $n$ because $S$ spans. That is W2.

Theorem 1 therefore identifies $X$ with the character space of $(\mathcal{V},m)$. That space is the compact abelian group

$$\operatorname{Hom}(\langle S\rangle_{\mathbb{Z}},S^1)\cong\mathbb{R}^n/\Lambda_S,$$

where $\Lambda_S$ is the annihilator of the $\mathbb{Z}$-span of $S$. If $S$ generates $\Lambda_{\mathrm{tr}}^*$ then $\Lambda_S=\Lambda_{\mathrm{tr}}$ and the identification is a translation of the covering torus, descended to $X$. Lemma 14 recovers

$$g^{-1}(de_\xi,de_\eta)=\Gamma(e_\xi,e_\eta),$$

which is the flat metric with dual lattice $\Lambda_{\mathrm{tr}}^*$. Lemma 15 gives $\alpha$ constant.
The translation torus of $\mathbb{R}^n/\Lambda_{\mathrm{tr}}$ acts by isometries and preserves each character, hence preserves $\mathcal{V}$ and $\Phi$. It is a compact connected Lie subgroup of the group $K$ of Theorem 21. Its orbits are the fibres of the holonomy covering; on $X$ they descend to a single principal orbit because the holonomy is finite and acts freely. Lemma 24 is then the orbit decomposition of a Bieberbach manifold: $X$ is a compact quotient of a torus by a finite free linear action. Transitivity of the identity component on the cover is the group law of the character variety.
\end{proof}
\begin{remark} The dimension of $\mathcal{V}$ is $1+2\lvert S\rvert$ after passing to a real basis, and equals $n+1$ only in degenerate cases that do not occur for $n\ge 2$ with a single frequency length. Theorems 3–4 are therefore not available. The quadratic $Q=\sum u_a^2$ is a trigonometric polynomial of frequency at most $2\max_{\xi\in S}\lvert\xi\rvert$, not a constant. The torus packet is exact without being spherical.
\end{remark}

\begin{remark}A generic metric on the underlying torus has no exact finite window: products of first modes spray over infinitely many slots. Exactness of this packet is a certificate of flatness. If $S$ generates a proper sublattice, Theorem 1 reconstructs the coarser torus $\mathbb{R}^n/\Lambda_S$; the original manifold is a finite quotient of a finite cover of that torus.
\end{remark}

This subsection does not use Remark 26. The torus tower $\mathcal{V}k=\operatorname{span}{e_\xi:\lvert\xi\rvert\le kR}$ is infinite and graded by $\Lambda${\mathrm{tr}}^*, which is the abelian case of the second clause of Remark 26.

\begin{remark}[global 2-product bases]
A global $2$-product eigenbasis, in the sense of Schildkraut–Speciel \cite{SS26}, forces an infinite Chebyshev tower along each primitive chain and uses that tower either through $k\to\infty$ or through the infinite graph $H_\varphi$. That infinite input is needed only to conclude that the full Gram matrix $\langle\nabla\alpha_i,\nabla\alpha_j\rangle$ is constant, hence that $g$ is flat.
It is not needed to close a single sum-angle slot. If a finite exact window contains two primitive chains together with their sine companions $\psi=\sin\alpha$, $\eta=\sin\beta$, and if products in that window have width at most two, then

$$\chi=\varphi\xi-\psi\eta=\cos(\alpha+\beta)$$

lies in one visible eigenspace: the two products $\varphi\xi$ and $\psi\eta$ occupy the same two-plane and cancel onto one axis. The identity is a finite packet computation (the primitives, the companions, and at most $T_2$). In the notation of this paper, $R=0$ and width two on that window already place $\chi$ in $\mathcal{V}_{\Lambda’}$.
Thus a finite exact $2$-sparse packet yields slot circles and sum-angle eigenfunctions. Promoting those phases to a parallel coframe, and thereby to a flat metric, still requires constancy of the cross terms on a spanning set, which is an additional exactness identity for the visible cometric, not an invitation to send $\Lambda’\to\infty$.
\end{remark}

9.3 An Inhomogeneous Exact Packet

Let ${P_0,\dots,P_m}$ be a symmetric Clifford system on $\mathbb{R}^{2l}$ \cite{FKM81}, and let

$$M_+={x\in S^{2l-1}:\langle P_\alpha x,x\rangle=0,\ \alpha=0,\dots,m}$$

be the corresponding OT–FKM focal submanifold. The normal space of $M_+$ in $S^{2l-1}$ at $x$ is $\operatorname{span}{P_0x,\dots,P_mx}$, and $M_+$ is minimal in the sphere. Write $W_N$ for the restriction to $M_+$ of the spherical harmonics of degree at most $N$ on $S^{2l-1}$.

\begin{proposition}
$W_2$ is invariant under $\Delta_{M_+}$, hence a finite sum of eigenspaces. The pair

$$\mathcal{V}=\operatorname{span}{1,x_1,\dots,x_{2l}}\big|{M+},\qquad W_2$$

is an exact packet: every product of elements of $\mathcal{V}$ lies in $W_2$, and $R=0$.
\end{proposition}

\begin{proof}
Let $Q(x)=\langle Ax,x\rangle$ be a degree-$2$ harmonic on $S^{2l-1}$, so $\operatorname{tr}A=0$ and $\Delta_S Q=\lambda_2 Q$ with $\lambda_2=2(2l)$. Since the mean curvature of $M_+$ in the sphere vanishes,

$$\Delta_{M_+}(Q|{M+})=\Delta_S Q\big|{M+}-\sum_{\alpha=0}^{m}\operatorname{Hess}_S Q(P_\alpha x,P_\alpha x).$$

For vectors $U,V$ tangent to the unit sphere,

$$\operatorname{Hess}_S Q(U,V)=2\langle AU,V\rangle-2Q\langle U,V\rangle.$$

On $M_+$ one has $\langle P_\alpha x,x\rangle=0$, so each $P_\alpha x$ is tangent to the sphere, and

$$\sum_{\alpha=0}^{m}\operatorname{Hess}S Q(P_\alpha x,P_\alpha x)=2\sum{\alpha=0}^{m}\langle P_\alpha A P_\alpha x,x\rangle-2(m+1)Q.$$

Both terms restrict to elements of $W_2$. Thus $\Delta_{M_+}W_2\subset W_2$. A symmetric endomorphism of a finite-dimensional space diagonalizes, so $W_2$ is a sum of eigenspaces of $\Delta_{M_+}$.
\end{proof}

Products of elements of $\mathcal{V}$ are restrictions of polynomials of degree at most $2$, hence lie in $W_2$. The tail past $W_2$ vanishes, and the packet is exact.

The coordinate map is an isometric immersion into one slot sphere, but the image is a proper submanifold. Section 9.1 requires the $K$-orbit on the image to be open. That hypothesis fails – the fundamental fields of the group preserving $\Phi$ do not span $T M_+$ – so the example is not a counterexample to §9.1. The induced metric is not round, and in the indefinite range $M_+$ is not homogeneous.

In the indefinite range $m\equiv 0\pmod 4$ with $P_0\cdots P_m\neq\pm\mathrm{Id}$, $M_+$ is not homogeneous \cite{OT75}. The smallest such focal set is the $(m_1,m_2)=(4,3)$ example in $S^{15}$: dimension $10$, coordinate slot of rank $16$, and $W_2$ of dimension at most $1+16+135=152$ \cite{QT16}. It is diffeomorphic to $S^3\times S^7$ and not isometric to a product of round spheres. Exactness of $(\mathcal{V},W_2)$ is unaffected by the definite/indefinite dichotomy.

Exactness here is for products of $\mathcal{V}$. Products of degree-$2$ harmonics land in $W_4$, which is not claimed to be a spectral window. The example separates the exactness condition of §9 from the open orbit, isometric slot hypothesis of §9.1: $R=0$ does not force the $K$-orbit on the isometric slot-sphere embedding to be open/transitive, and does not force homogeneity.