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\newcommand{\ip}[2]{\langle #1 \,|\, #2 \rangle}
\newcommand{\snr}{\rho}
\newcommand{\rhoR}{\snr_{\mathrm{R}}}
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\newcommand{\Msec}{M_{\mathrm{s}}}
\newcommand{\pair}[2]{(2,2,#1)\text{--}(2,2,#2)}

\begin{document}

% Template rule (IEEE conference template): no math or special symbols
% in the title or abstract.
\title{The Cost of Ignoring Mode Correlations in Black-Hole\\
Quasinormal-Mode Resolvability Forecasts}

% Author block: IEEE conference template (URTC style guide).
% Eligibility route (author's own statement, 2026-08-08): Diwen is a
% high-school student at Port Moody Secondary working directly and
% substantially with a university student (Zelin, UNC) -- the second
% branch of URTC's per-author rule. Zelin qualifies as an enrolled
% undergraduate.
\author{\IEEEauthorblockN{Diwen Huang\textsuperscript{*}%
\thanks{\textsuperscript{*}Equal contribution.}}
\IEEEauthorblockA{\textit{Port Moody Secondary School}\\
Port Moody, BC, Canada\\
diwenhuang42@gmail.com}
\and
\IEEEauthorblockN{Zelin Zhu\textsuperscript{*}}
\IEEEauthorblockA{\textit{University of North Carolina at Chapel Hill}\\
Chapel Hill, NC, USA\\
zelinzhu@ad.unc.edu}}

\maketitle

\begin{abstract}
Forecasts in black-hole spectroscopy decide whether two quasinormal
modes are distinguishable using a Rayleigh-type criterion: the modes
are declared resolvable when their frequency and damping-time
separations exceed the statistical uncertainties of the corresponding
estimates. Those uncertainties can be drawn from the joint two-mode
Fisher matrix that keeps the correlations between the mode templates,
as the forecast literature does, or from a single-mode analysis that
treats each mode as if alone, a natural shortcut when correlations
are presumed small. This paper quantifies what the shortcut costs.
The ratio between the two resulting signal-to-noise thresholds is
computed for damped-sinusoid fits over a finite analysis window,
across the full astrophysical spin range and a wide family of window
choices, under white noise. For the contested fundamental-overtone
pair the correlation-aware threshold exceeds the single-mode one by a
factor of about four to eight everywhere on the map (median over
relative phase): at remnant spin 0.69 the thresholds are 71 against
310. For a well-separated control pair the factor stays near or
below two. The penalty is set almost entirely by the threefold
difference in damping times, not by the two-percent frequency
splitting, and the template overlap behind it has a simple closed
form. Two properties are exact: the ratio is independent of the mode
amplitude ratio, and the overlap is invariant under translation of
the analysis window, which also removes any remnant-mass dependence
under white noise. Individual-phase thresholds spread by tens of
percent about the reported medians, and the noise model is white
throughout; both limitations are stated where they enter. The full
pipeline reproduces every figure with one command.
\end{abstract}

\begin{IEEEkeywords}
black-hole spectroscopy, quasinormal modes, ringdown, Fisher
information, non-normal operators, condition number
\end{IEEEkeywords}

\section{Introduction}
After two black holes merge, the remnant settles down by emitting
gravitational waves at a discrete set of complex frequencies, the
quasinormal modes (QNMs) of the final Kerr black hole. Because general
relativity fixes every QNM frequency by the remnant's mass and spin
alone, detecting two or more modes in the ringdown and checking their
consistency is a direct test of the no-hair theorem, the statement
that an isolated black hole carries no structure beyond those two
numbers \cite{berti2006lisa,isi2019nohair}. Whether current detectors
have already achieved this with the first overtone of GW150914 (a
shorter-lived mode sharing the fundamental's angular harmonic at
nearly the same frequency) remains actively disputed
\cite{isi2019nohair,cotesta2022overtones}.

Forecasts of when such tests become possible rest on a resolvability
criterion. The standard choice is Rayleigh-like
\cite{berti2006lisa,ota2022horizons,shi2024spaceqnm}: two modes are
declared distinguishable at a given signal-to-noise ratio (SNR) if
their frequency separation exceeds the statistical uncertainty of the
frequency estimates, and likewise for the damping times. Its users are
not naive about it: Ota and Chirenti found it \emph{overly
restrictive} relative to a Bayes-factor threshold and argued for the
latter \cite{ota2022horizons}. The criterion nonetheless continues to
anchor quantitative statements throughout the forecast literature
\cite{shi2024spaceqnm,berti2025review}.

In parallel, the pseudospectrum programme (which maps an operator's
approximate eigenvalues to ask how far its spectrum moves under small
perturbations) has put a structural fact on firm footing: the
operator governing black-hole ringdown is non-self-adjoint, meaning
it differs from its adjoint and so need not admit an orthogonal
eigenbasis; its spectrum can be unstable under small perturbations;
and its eigenfunctions, the QNMs, are not orthogonal in physically
motivated inner products
\cite{jaramillo2021pseudospectrum,destounis2021rn,gasperin2022scalar}.
Estimation theory is unambiguous about what non-orthogonality does to
a \emph{joint} fit: the covariance of the estimated parameters is
governed by the correlations of the template set, so the achievable
precision is controlled by the conditioning of the template Gram
matrix, not by frequency separation alone.

The forecast literature is not naive about these correlations.
Berti, Cardoso, and Will, who introduced the criterion in the form
used here, already computed its thresholds from a joint two-mode
Fisher matrix, naming the ``cross-terms'' in the inner products as
the reason the same-harmonic overtone case resists closed-form
treatment [Sec.~VII of \cite{berti2006lisa}]; Ota and Chirenti
evaluate an eight-parameter joint matrix [Eq.~(9) of
\cite{ota2022horizons}], and Shi, Zhang, and Mei a twenty-six
parameter one \cite{shi2024spaceqnm}. What none of these works
isolates is how much of the resulting threshold is the correlation
penalty itself: the factor separating the joint-Fisher threshold
from a single-mode baseline in which each mode is analyzed as if
alone. That baseline is a diagnostic, not a position held in the
literature: the two constructions coincide exactly for orthogonal
templates, so their ratio directly measures what mode overlap costs
the criterion, and to the authors' knowledge that ratio has not been
mapped for the astrophysically contested pairs. This paper computes
the map. The working hypothesis (H1) was that the ratio exceeds two
somewhere in parameter space, largest for the $\pair{0}{1}$
fundamental--overtone pair, the same pair contested in GW150914
\cite{isi2019nohair,cotesta2022overtones}. The computation returns
the strong form of H1: the factor is between 4.25 and 8.1
\emph{everywhere} on the map (median over relative phase), and near
or below 2 for a control pair.

The contributions are:
\begin{itemize}
  \item both threshold constructions stated in one Fisher framework,
        pinned to the anchor forecasts' equations
        (\cref{sec:criteria});
  \item a map of the correlation penalty over remnant spin and
        analysis-window start, per mode pair (\cref{sec:results});
  \item a closed form for the template overlap that sets the
        penalty's scale, showing it is governed by the damping-time
        ratio rather than the frequency splitting, plus exact
        invariances with one-line proofs
        (\cref{sec:conditioning,sec:robustness});
  \item a reproducible pipeline (laptop-scale; the full map runs in
        about six minutes; \cref{sec:setup}).
\end{itemize}

Scope is deliberately narrow. This paper analyzes no detector data,
performs no Bayesian inference, and takes no position on whether
GW150914's overtone was real; it audits a \emph{criterion}, not an
event. The inner product is white-noise throughout; detector-shaped
weighting and exact reproduction of the anchors' published horizon
numbers are the next milestones and are not claimed here.

\section{Signal Model and Estimation Framework}
\label{sec:model}

\subsection{Ringdown model}
The observed ringdown is modeled over a finite analysis window
$t \in [t_0,\, t_0+T]$ as a sum of damped sinusoids referenced to the
waveform peak at $t=0$,
\begin{equation}
  h(t) \;=\; \sum_{k} A_k\, e^{-t/\tau_k}
  \cos\!\bigl(2\pi f_k t + \phi_k\bigr),
  \label{eq:model}
\end{equation}
where $k$ runs over the mode pair under test and $R = A_1/A_0$ is the
amplitude ratio at the peak. The frequencies and damping times are
those of the Kerr QNMs,
\begin{equation}
  2\pi f_k - \frac{i}{\tau_k} \;=\;
  \frac{\hat\omega_{\ell m n}(\chi)}{\Msec},
  \qquad \Msec \equiv \frac{GM}{c^3},
  \label{eq:kerr}
\end{equation}
with $\hat\omega_{\ell m n}(\chi)$ the dimensionless Kerr QNM
frequency at spin $\chi$, obtained by table lookup from the
\texttt{qnm} package \cite{stein2019qnm}; no QNM frequencies are
computed in this work. Here $(\ell, m)$ label the angular harmonic of
the perturbation and $n$ is the overtone index, which orders the modes
of a given $(\ell, m)$ by decreasing damping time: $n=0$ is the
fundamental and $n=1$ its first overtone, the shorter-lived mode at
nearly the same frequency. The dimensionless spin is
$\chi = cJ / (GM^2) \in [0,1)$ for remnant angular momentum $J$, and
the map axes run over it. Time dependence is $e^{-i\omega t}$
throughout, so $\operatorname{Re}\hat\omega > 0$ and
$\operatorname{Im}\hat\omega = -\Msec/\tau < 0$; the code takes
$f = \operatorname{Re}\hat\omega / 2\pi$ and
$\tau = -1/\operatorname{Im}\hat\omega$ in units of $M$.
At $\chi=0.69$ the pair of interest has
$\hat f_{220}=0.0841$, $\hat\tau_{220}=12.3M$ and
$\hat f_{221}=0.0822$, $\hat\tau_{221}=4.07M$: a frequency splitting
of only $2.2\%$ against damping times differing by a factor of three.

\subsection{Inner product, SNR, and Fisher matrix}
For white noise of one-sided power spectral density $S_0$, the
noise-weighted inner product over the window is
\begin{equation}
  \ip{a}{b} \;=\; \frac{2}{S_0}\int_{t_0}^{t_0+T} a(t)\, b(t)\, dt,
  \label{eq:innerprod}
\end{equation}
and the total SNR is $\snr^2 = \ip{h}{h}$. With time measured in
units of $M$, every quantity entering the analysis is dimensionless,
so the results below are exactly independent of the remnant mass; the
map axes are spin and window start in units of $M$.

For parameters $\theta = \{A_k, \phi_k, f_k, \tau_k\}$, the Fisher
matrix and the Cram\'er--Rao bound on any unbiased estimator are
\begin{equation}
  \Gamma_{ij} = \ip{\partial_i h}{\partial_j h},
  \qquad
  \sigma_{\theta_i} \ge \sqrt{(\Gamma^{-1})_{ii}} .
  \label{eq:fisher}
\end{equation}
All statistical uncertainties below are Fisher-level; this matches the
anchor forecasts and is the regime where the Rayleigh criterion is
stated.

\section{Two Constructions of One Criterion}
\label{sec:criteria}

\subsection{The Rayleigh criterion}
\label{sec:rayleigh}
Following Berti, Cardoso, and Will [Eqs.~(57)--(59) of
\cite{berti2006lisa}], as adopted by Ota and Chirenti
[Eqs.~(6a)--(6b) of \cite{ota2022horizons}], the pair is resolvable
at SNR $\snr$ when
both separations clear the larger of the corresponding uncertainties,
\begin{equation}
  |f_1{-}f_0| > \max(\sigma_{f_0}, \sigma_{f_1})
  \;\;\text{and}\;\;
  |\tau_1{-}\tau_0| > \max(\sigma_{\tau_0}, \sigma_{\tau_1}).
  \label{eq:rayleigh}
\end{equation}
Every $\sigma$ scales as $1/\snr$ at fixed signal shape, so
\cref{eq:rayleigh} defines a critical SNR: the smallest $\snr$ at
which both conditions hold. The two constructions differ only in the
Fisher matrix that supplies the $\sigma$'s:
\begin{itemize}
  \item \emph{single-mode baseline}, critical SNR $\rhoR$: each
        mode's $(A,\phi,f,\tau)$ errors from its own $4\times4$
        Fisher matrix, the other mode absent: the correlation-free
        reference point, not a procedure attributed to any cited
        forecast;
  \item \emph{correlation-aware}, critical SNR $\rhoC$: all errors
        from the joint $8\times8$ Fisher matrix of
        \cref{eq:model}, as the anchor forecasts implement
        [Eq.~(9) of \cite{ota2022horizons}].
\end{itemize}
The quantity mapped here is the ratio
\begin{equation}
  D(\chi, t_0;\ \text{pair}) \;=\; \rhoC / \rhoR \;\ge\; 1,
  \label{eq:ratio}
\end{equation}
the factor by which mode correlations raise the SNR the criterion
requires over the orthogonal-template reference. Both thresholds are
referred to the same total two-mode SNR of \cref{eq:innerprod}, so
$D$ isolates the effect of the off-diagonal Fisher content at fixed
signal strength.

\subsection{Why the readings must differ: conditioning}
\label{sec:conditioning}
Write \cref{eq:model} in quadrature form: each mode contributes
$e^{-t/\tau_k}\cos 2\pi f_k t$ and $e^{-t/\tau_k}\sin 2\pi f_k t$, so
a $k$-mode fit is linear in $2k$ amplitudes with Gram matrix
$G_{ab} = \ip{\hat u_a}{\hat u_b}$ of the normalized basis. The
joint covariance is proportional to $G^{-1}$; the variance of
amplitude $a$ inflates over the orthogonal case by
$(G^{-1})_{aa} \ge 1$, bounded by the condition number
$\kappa(G) = \lambda_{\max}/\lambda_{\min}$, with $\lambda_{\max}$ and
$\lambda_{\min}$ the largest and smallest eigenvalues of $G$. The
cross-mode content
is captured by the overlap of the normalized complex templates
$z_k(t) = \exp[(2\pi i f_k - 1/\tau_k)\,t]$ on the window,
$\mu = |\ip{\hat z_0}{\hat z_1}|$, for which the $2\times2$ case is
fully analytic:
\begin{equation}
  (G^{-1})_{aa} = \frac{1}{1-\mu^2},
  \qquad
  \kappa = \frac{1+\mu}{1-\mu},
  \label{eq:twobytwo}
\end{equation}
giving an SNR penalty of at least $(1-\mu^2)^{-1/2}$ for amplitude
estimation alone. For orthogonal modes $\mu \to 0$, the penalty
disappears, and the two constructions of \cref{eq:rayleigh} coincide;
\cref{eq:twobytwo} is also the pipeline's unit test.

For a window long compared to the damping times the overlap itself
has a closed form,
\begin{equation}
  \mu \;=\; \frac{2}{\sqrt{\tau_0\tau_1}\,
    \bigl|\,\tau_0^{-1}+\tau_1^{-1}-2\pi i\,(f_1{-}f_0)\bigr|}
  \;\;\xrightarrow{\,f_1\to f_0\,}\;\;
  \frac{2\sqrt{\tau_0\tau_1}}{\tau_0+\tau_1},
  \label{eq:muclosed}
\end{equation}
whose equal-frequency limit is the ratio of the geometric to the
arithmetic mean of the two damping times. The overlap that drives
the penalty is therefore governed by the damping-time ratio; the
frequency splitting enters only through the second term in the
modulus, which is small for this pair.

\section{Numerical Setup}
\label{sec:setup}
The grid covers $\chi \in [0, 0.99]$ in 100 points and window starts
$t_0 \in [0, 30]M$ in steps of $1M$, with fiducial window length
$T=60M$ and amplitude ratio $R=1$; robustness variations use
$T \in \{30, 100\}M$, $R \in \{0.3, 3\}$, and 24 relative phases
$\phi_1{-}\phi_0$ spanning $[0,\pi)$ (with $\phi_0 = 0$; $D$ has
period $\pi$ in the relative phase), with the median over phase
reported. Individual-phase values spread by tens of percent about
the median (\cref{sec:robustness}). An earlier freeze (results v1)
sampled only four phases, which this symmetry collapses to two
distinct values; its medians agree with the current 24-phase freeze
(results v2) at the tabulated point to five digits, and v1 is
archived alongside the pipeline. The mode pairs are the contested
$\pair{0}{1}$ pair and a $(2,2,0)$--$(3,3,0)$ control. Integrals use
a $0.01M$ step; Fisher inversions are correlation-scaled, with a
conditioning ceiling of $10^{13}$ that is never reached on this grid
(the largest correlation-scaled condition number encountered is
$3.4 \times 10^{5}$).
In the analytic orthogonal limit (narrow linewidths, well-separated
tones) the pipeline returns $D = 1.0000$ and $\kappa = 1.011$,
matching \cref{eq:twobytwo}. As an external order-of-magnitude
anchor: at the amplitude ratio of the anchor forecast's overtone
figures ($R = 0.1$), the joint threshold here is a few times $10^3$
over most of the spin range at $t_0 = 0$ (median $3.1 \times 10^3$;
$3.8 \times 10^3$ at $\chi = 0.69$), consistent with that forecast's
statement that resolving both frequencies and damping times of this
pair ``typically requires a SNR greater than about $10^3$''
[Figs.~3--4 of \cite{berti2006lisa}]. Their detector noise and
infinite window differ from the white-noise finite window used here,
so this is a consistency check rather than a reproduction; exact
reproduction of the anchors' published horizon numbers is the next
milestone and is not claimed here.

% Artifact locator: intentionally none printed for now. The pipeline
% is packaged (LaTeX + scripts/ + both frozen result sets in one zip);
% pick a neutral archival home (Zenodo DOI or a GitHub repo) before
% submission and print that here -- reviewers flagged "released with
% no pointer" as a blocking issue, so this sentence should carry a
% locator in the submitted version.
The full analysis is a two-script pipeline shipped with the paper's
source: \texttt{make figures} regenerates every figure from the
frozen map, and \texttt{make map} regenerates the map itself in
about six minutes on a laptop.

\section{Results}
\label{sec:results}

\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth]{figures/fig_map.pdf}
  \caption{Disagreement ratio $D = \rhoC/\rhoR$ of \cref{eq:ratio}
  for the $\pair{0}{1}$ pair over remnant spin and window start
  ($T=60M$, $R=1$, median over 24 phases). $D$ exceeds 4.25
  everywhere and peaks at 8.1 as $\chi \to 0.99$, where the pair
  degenerates; the field is flat in window start to better than
  $\pm 1\%$ at every spin (\cref{sec:robustness}). The star marks
  $(\chi, t_0) = (0.69, 0)$, the GW150914-like point of
  \cref{tab:audit}. The colour scale is banded, not
  linear: $D$ is strongly right-skewed, with $97\%$ of grid points below
  $D = 5$, so the band edges printed on the colourbar are near-quantiles
  of the field at the low end and widen over the high-spin tail. Bands
  are drawn at uniform width; the tick values are the true $D$ at each
  edge.}
  \label{fig:map}
\end{figure}

\begin{table}[t]
  \caption{Critical SNR under the two readings at $\chi = 0.69$,
  $T = 60M$, $R = 1$ (median over phase). $\kappa$ is the
  Gram condition number of \cref{eq:twobytwo}; its window-start
  invariance is exact (\cref{sec:robustness}).}
  \label{tab:audit}
  \centering
  \begin{tabular}{lcccc}
    \toprule
    Window start & $\rhoR$ (clean) & $\rhoC$ (joint) & $D$ & $\kappa$ \\
    \midrule
    $t_0 = 0$    & 71   & 310  & 4.29 & 13.7 \\
    $t_0 = 10M$  & 519  & 2204 & 4.30 & 13.7 \\
    $t_0 = 20M$  & 2217 & 9635 & 4.29 & 13.7 \\
    \bottomrule
  \end{tabular}
\end{table}

\Cref{fig:map} maps $D$ for the fundamental--overtone pair. Ignoring
the correlations understates the required SNR by a factor of
$4.25$ to $8.1$ over the \emph{entire} physical range: there is no
corner of spin or window choice where the shortcut is safe for this
pair. At the GW150914-like point $(\chi, t_0) = (0.69, 0)$ the
single-mode baseline declares the pair resolvable at $\snr \approx
71$, while the correlation-aware construction requires $\snr \approx
310$ (\cref{tab:audit}). Both absolute thresholds grow steeply as the
window start moves away from the peak and the overtone decays, but
their ratio stays near $4.3$; the penalty is a property of the
pair, not of the window. Across the sampled grid the frequency
condition of \cref{eq:rayleigh} always sets the threshold (the
damping-time condition never binds for either construction), so the
thresholds reported here are frequency-resolution limited.

The structural driver is the overlap, and \cref{eq:muclosed} shows
what sets its size. At $\chi = 0.69$ the closed form gives
$\mu = 0.8638$, matching the pipeline to four digits
($\kappa = 13.7$); its equal-frequency limit is $0.8643$, so the
$2.2\%$ frequency splitting shifts the overlap by well under
$0.1\%$. So $\mu$, and with it the penalty, is set by the threefold
damping-time difference almost alone, not by the frequency proximity
the criterion is named for. At $\chi = 0.99$, $\mu = 0.878$
($\kappa = 15.4$). The analytic amplitude-only
penalty of \cref{eq:twobytwo}, $(1-\mu^2)^{-1/2} = 1.98$, accounts
for about half of the measured $D = 4.29$; the rest comes from
cross-parameter degeneracies that only the full $8\times8$ analysis
sees, and a per-parameter attribution of it has not been computed
here. The $(2,2,0)$--$(3,3,0)$ control pair
confirms pair-specificity: its splitting is large, $\mu$ is small,
and the phase-median $D$ stays between $1.0$ and $2.0$ (median
$1.4$; individual phases reach at most $2.1$), approaching
unity at high spin (\cref{fig:robust}a).

Two consequences follow. First, the single-mode shortcut is
tempting for the same reason the joint calculation is hard: the
cross-terms \cite{berti2006lisa}. Wherever it is used, it overstates
the prospects of fundamental--overtone spectroscopy by a factor of
about four in required SNR; forecasts built on the joint Fisher matrix,
including the anchors here
\cite{berti2006lisa,ota2022horizons,shi2024spaceqnm}, are on
the correct side of this gap. Second, both constructions place
both-parameter resolvability of the $\pair{0}{1}$ pair at SNRs well
beyond those of the events analyzed in the overtone dispute,
consistent with that dispute remaining open
\cite{isi2019nohair,cotesta2022overtones};
this work takes no position on that event.

\section{Robustness}
\label{sec:robustness}

\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth]{figures/fig_robust.pdf}
  \caption{(a) $D$ at $t_0 = 0$ versus spin for the
  fundamental--overtone pair and the $(2,2,0)$--$(3,3,0)$ control.
  (b) Window-length dependence at $\chi = 0.69$: shortening the
  window to $T = 30M$ raises $D$ to about $6.4$; $T = 100M$ differs
  from $T = 60M$ by under $2\%$. Curves are medians over the
  24-phase grid.}
  \label{fig:robust}
\end{figure}

Three invariances are exact, not approximate. First, $D$ is
independent of the amplitude ratio $R$: rescaling one mode's
amplitude rescales the corresponding rows and columns of both Fisher
matrices, leaving every correlation coefficient unchanged, so the
$1/\snr$-scaled $\sigma$'s shift identically in numerator and
denominator of \cref{eq:ratio}. The computed maps at $R = 0.3$, $1$,
and $3$ agree to better than ten significant digits everywhere on
the grid. Second, the normalized template
overlap $\mu$, and hence $\kappa$, is exactly invariant under window
translation: shifting $t_0$ multiplies each template by a constant
factor that cancels in the normalization. This is why
\cref{tab:audit} shows the same $\kappa$ at every $t_0$, and why the
phase-median $D$ is nearly flat in $t_0$: its variation over the
full window-start range is below $\pm 1\%$ at every spin. Third,
with the white-noise
product and windows scaled by $M$, the entire analysis is
dimensionless and $D$ carries no mass dependence at all.

The relative phase is the one handle whose effect is not small
pointwise, and it deserves stating as a limitation:
individual-phase values of $D$ spread by about $22\%$ at the
tabulated point and by up to $46\%$ at low spin about the reported
medians. What is robust is the median itself: at the tabulated
point the superseded four-sample grid (results v1) and the current
24-phase grid agree to five digits. Every threshold quoted here
should therefore be read as a phase-median carrying a
tens-of-percent phase spread, a property of the damped-sinusoid
criterion itself rather than of this implementation. The window length behaves smoothly
(\cref{fig:robust}): shortening to $T = 30M$, which curtails the
evolution that separates the
two damping times, raises the penalty to $D \approx 6.4$, while
lengthening to $T = 100M$ shifts $D$ by under $2\%$. The one
weighting not yet swept is a detector-shaped power spectral density,
which breaks the mass invariance by introducing a physical frequency
scale; whether $D$ itself moves qualitatively under a detector-shaped
weighting is not answered here. That arm, and the anchor-exact
horizon reproduction, are the declared next steps.

\section{Conclusions}
\label{sec:conclusions}
\begin{enumerate}
  \item For the contested $\pair{0}{1}$ pair, ignoring the mode
        correlations in the Rayleigh resolvability
        criterion understates the required SNR by a factor of 4.25 to
        8.1 (median over relative phase) across the entire
        astrophysical spin range and all
        analysis windows studied; at the GW150914-like point the
        thresholds are 71 versus 310. A well-separated control pair
        stays near or below a factor of 2.
  \item The penalty is structural, not tuned: it is exactly
        independent of the mode amplitude ratio, exactly
        translation-invariant in its Gram diagnostic, and
        mass-invariant under white noise. Its scale is set by the
        template overlap, $\mu \approx 0.86$ to $0.88$, which the
        closed form of \cref{eq:muclosed} ties to the damping-time
        ratio almost alone (the frequency splitting the criterion
        is named for contributes under one part in a thousand of
        the overlap), with cross-parameter degeneracies roughly
        doubling the analytic amplitude-only penalty of
        $(1-\mu^2)^{-1/2} \approx 2$. The medians are stable under
        phase-grid refinement, but individual-phase thresholds
        spread by tens of percent about them, and the white-noise
        inner product remains the analysis's main untested
        assumption.
  \item Practical recommendation: resolvability forecasts for
        overtone spectroscopy must draw their uncertainties from the
        joint Fisher matrix, as the anchor forecasts
        \cite{berti2006lisa,ota2022horizons,shi2024spaceqnm} do;
        assembling the same criterion from single-mode error
        formulas is unsafe
        for this pair by half an order of magnitude. The pipeline
        evaluates the correction in milliseconds per grid
        point; detector-PSD weighting and anchor-exact threshold
        reproduction are its next milestones.
\end{enumerate}

\begin{thebibliography}{10}
% Every entry below was verified against its arXiv abstract page or
% publisher record on 2026-08-08. Do not add references without the
% same check. Entries are ordered by first citation in the text.

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\bibitem{jaramillo2021pseudospectrum}
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\end{thebibliography}

\end{document}
