%============================================================
Appendix D — Scale–cut robustness and failure localization
%============================================================
\section{Scale--cut robustness and failure localization}
\label{app:scalecuts}
%============================================================

This appendix investigates the robustness of the Stage $\Sigma$ closure failure
to conservative angular scale cuts and localizes the dominant contributors to
the residual $\chi^2$ at the level of individual shear correlation points.

The purpose is twofold:
(i) to assess whether the closure failure is driven primarily by small--scale
modeling uncertainties, and
(ii) to identify the angular and tomographic origin of the dominant residuals.

%------------------------------------------------------------
\subsection{Motivation}
%------------------------------------------------------------

The baseline Stage $\Sigma$ shear--only result yields
$\Sigma_{\rm best}=8.5538\pm0.4538$ with
$\chi^2_{\min}/\nu=1644.85/399$ and
$\Delta\chi^2(\Sigma=1)=277$,
indicating a highly significant failure of internal closure.

Since cosmic shear modeling at small angular scales is affected by nonlinear
structure formation, baryonic feedback, and intrinsic--alignment uncertainties,
it is essential to verify whether the observed failure is driven primarily by
poorly modeled small--scale data or whether it persists on conservative scales
where theoretical predictions are robust.

%------------------------------------------------------------
\subsection{Methodology}
%------------------------------------------------------------

We repeat the shear--only Stage $\Sigma$ estimator described in
Appendix~\ref{app:stage_sigma}, restricting the data vector and covariance matrix
to progressively more conservative angular ranges.

The estimator remains
%
\begin{equation}
\bm m(\Sigma) = \Sigma\,\bm t ,
\end{equation}
%
with
%
\begin{equation}
\Sigma_{\rm best} =
\frac{\bm t^{\top}\bm C^{-1}\bm d}{\bm t^{\top}\bm C^{-1}\bm t},
\qquad
\sigma_\Sigma^2 =
\left(\bm t^{\top}\bm C^{-1}\bm t\right)^{-1},
\end{equation}
%
and
%
\begin{equation}
\chi^2(\Sigma) =
(\bm d-\Sigma\bm t)^{\top}\bm C^{-1}(\bm d-\Sigma\bm t),
\qquad
\nu = N_{\rm cut}-1 .
\end{equation}

Scale cuts are implemented by selecting a subset of the shear data vector
according to angular thresholds applied separately to $\xi_+$ and $\xi_-$.
Because $\xi_-$ is significantly more sensitive to nonlinear and baryonic
effects, we adopt asymmetric cuts of the form
%
\begin{equation}
\xi_+ : \theta \ge \theta_{\min}^{+}, 
\qquad
\xi_- : \theta \ge \theta_{\min}^{-}.
\end{equation}

For each cut we recompute
$N_{\rm cut}$, $\Sigma_{\rm best}$, $\sigma_\Sigma$,
$\chi^2_{\min}/\nu$, and $\Delta\chi^2(\Sigma=1)$.

To localize the origin of the failure we evaluate the approximate per--point
contribution to the total chi--square,
%
\begin{equation}
\Delta\chi^2_i \simeq r_i \left(\bm C^{-1}\bm r\right)_i ,
\qquad
\bm r = \bm d - \Sigma_{\rm best}\bm t ,
\end{equation}
%
and rank the dominant contributions by angular scale, correlation type, and
tomographic bin pair.

%------------------------------------------------------------
\subsection{Conservative safe--scale cut}
%------------------------------------------------------------

After scanning several angular thresholds, we adopt as conservative
\emph{safe--scale cut}:
%
\begin{equation}
\xi_+ : \theta \ge 8~{\rm arcmin},
\qquad
\xi_- : \theta \ge 30~{\rm arcmin}.
\end{equation}

This choice removes all strongly nonlinear small--scale regimes while preserving
a large fraction of the cosmological signal.

For this cut we obtain:
%
\begin{align}
N_{\rm cut} &= 240, \\
\Sigma_{\rm best} &= 4.8846 \pm 0.6000, \\
\chi^2_{\min} &= 1099.5, \qquad \nu = 239, \qquad \chi^2_{\min}/\nu = 4.59, \\
\Delta\chi^2(\Sigma=1) &= 41.9 .
\end{align}

Several robust features follow.

First, the best--fit amplitude decreases substantially relative to the full data
result, demonstrating that small angular scales significantly amplify the
apparent deviation.

Second, the closure failure remains highly significant even on conservative
scales, with $\Delta\chi^2(\Sigma=1)\simeq 42$, corresponding to a rejection of the
GR closure hypothesis at $\sim6.5\sigma$.

Third, the reduced chi--square remains very large, $\chi^2/\nu\simeq4.6$,
demonstrating that a single amplitude rescaling fails to describe the residuals
even in the safe--scale regime.

We therefore conclude that small--scale modeling uncertainties are not the
primary origin of the closure failure.

%------------------------------------------------------------
\subsection{Localization of dominant contributions}
%------------------------------------------------------------

Ranking the individual contributions $\Delta\chi^2_i$ reveals a highly coherent
pattern.

The dominant contributions arise almost exclusively from:

\begin{itemize}
\item correlation type: $\xi_-$,
\item tomographic pairs: high--redshift bins, primarily $(3,4)$ and $(4,4)$,
\item angular scales: intermediate scales $30' \lesssim \theta \lesssim 60'$.
\end{itemize}

The ten largest contributors are concentrated in the subsets
%
\begin{equation}
(\xi_-,\, z_3\!-\!z_4), \qquad
(\xi_-,\, z_4\!-\!z_4), \qquad
\theta \simeq 30' - 60'.
\end{equation}

Importantly, these points are not located in the smallest angular bins and lie in
a regime where baryonic feedback and extreme nonlinearities are already
subdominant.
The closure failure is therefore driven by intermediate--scale, high--redshift
shear correlations rather than by poorly modeled small--scale physics.

%------------------------------------------------------------
\subsection{Interpretation}
%------------------------------------------------------------

The scale--cut and localization analysis leads to three robust conclusions.

First, the closure failure is not an artifact of nonlinear small--scale modeling.
Removing all small--scale data reduces but does not eliminate the failure and
does not improve the overall goodness of fit.

Second, the failure is dominated by $\xi_-$ correlations in high--redshift
tomographic bins at intermediate angular scales.
This pattern is suggestive of a structural inconsistency affecting the lensing
kernel normalization, tomographic weighting, or residual intrinsic--alignment
contributions, rather than a simple global amplitude mismatch.

Third, the persistence of large $\chi^2/\nu$ even after aggressive scale cuts
demonstrates that the hypothesis of a single multiplicative Weyl--sector
rescaling,
%
\begin{equation}
\bm m(\Sigma) = \Sigma\,\bm t ,
\end{equation}
%
is insufficient to close the shear sector.
The residuals exhibit scale and tomographic structure incompatible with a global
amplitude modification.

These results show that the Stage $\Sigma$ failure cannot be consistently
interpreted as evidence for modified gravity.
Instead, it reflects a structural internal inconsistency of the DES~Y3 shear
sector under the rigid frozen--baseline closure hypothesis.

%------------------------------------------------------------
\subsection{Implications for subsequent stages}
%------------------------------------------------------------

This analysis establishes that:

\begin{itemize}
\item the Stage $\Sigma$ failure is not driven by small--scale systematics,
\item the dominant tension arises in high--redshift $\xi_-$ correlations on
intermediate angular scales,
\item a single--parameter amplitude model is insufficient to restore closure.
\end{itemize}

These findings motivate subsequent stages of the Criterion~C2 program based on:
(i) tomographic--dependent amplitude diagnostics,
(ii) scale--dependent consistency tests, and
(iii) cross--sector comparisons with growth and clustering observables.

Only after such refinements achieve internal closure in both the Weyl and growth
sectors does a full relativistic consistency test become physically meaningful.