% ============================================================
% No Evidence for Large-Scale Parity Violation in Galaxy Morphology:
% A Survey-Scale Chirality Catalog of 8.47 Million Galaxies
%
% Author: Houston Golden (BigBounce Research / Hubify Lab)
% Date: March 2026
% ============================================================
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% Convenience macros
\newcommand{\fnl}{f_{\rm NL}}
\newcommand{\nside}{\texttt{NSIDE}}
\newcommand{\pcw}{P_{\rm CW}}
\newcommand{\pccw}{P_{\rm CCW}}
\newcommand{\pns}{P_{\rm NS}}
\newcommand{\CW}{\textsc{cw}\xspace}
\newcommand{\CCW}{\textsc{ccw}\xspace}
\newcommand{\NS}{\textsc{not\_spiral}\xspace}
\newcommand{\sigmaunit}{\sigma}
\newcommand{\vit}{ViT-Small\xspace}

\begin{document}

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\begin{center}
{\LARGE\bfseries No Evidence for Large-Scale Parity Violation\\[4pt] in Galaxy Morphology}\\[12pt]
{\Large A Survey-Scale Chirality Catalog of 8.47 Million Galaxies}\\[20pt]
{\large Houston Golden}\\[6pt]
{\normalsize BigBounce Research / Hubify Lab}\\[2pt]
{\small \url{https://bigbounce.hubify.app} \quad \texttt{houston@hubify.com}}\\[4pt]
{\small March 2026}
\end{center}

\vspace{16pt}
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% ---- Full-width abstract ----
\noindent\textbf{Abstract.}\quad\begin{minipage}[t]{0.95\textwidth}
We present the largest galaxy chirality catalog to date, comprising
8,474,531 galaxies from the DESI Legacy Imaging Surveys Data Release~8
(Smith42/galaxies). Each galaxy is classified as clockwise~(\CW),
counter-clockwise~(\CCW), or not spiral~(\NS) by a Vision Transformer
(ViT-Small) fine-tuned on 26,626 images from Galaxy Zoo~1 citizen-science
labels, high-confidence CE-ResNet classifications, and synthetic hard
negatives. The classifier achieves 93.7\% three-class validation accuracy.
All eight tests in a purpose-built bias hardening suite pass their
predefined thresholds, including flip-equivariance consistency, rotation
stability, artifact rejection, brightness robustness, and metadata
leakage. A flip-equivariance consistency loss during training and
test-time equivariant averaging at inference eliminate
orientation-dependent classification bias by construction, yielding
a global clockwise fraction of $\CW/(\CW+\CCW) = 0.4974$ after
equivariant post-processing.

We find no significant large-scale chirality dipole: a simple
dipole fit gives $0.43\sigmaunit$, while a full angular power spectrum
analysis yields a marginal $2.75\sigmaunit$ at $\ell = 1$. A
hemisphere asymmetry of $3.05\sigmaunit$ is detected, but its amplitude
of only 0.17\% does not survive a look-elsewhere correction across the
directions tested. This constitutes the most sensitive chirality
measurement ever performed, with a minimum detectable dipole of $0.2\%$
at $3\sigmaunit$ significance.  The $\sim\!3\%$ large-scale chirality
asymmetry reported by \citet{Shamir:2020,Shamir:2022} is definitively
refuted: our maximum regional asymmetry is 0.47\%, a factor of~7
smaller.  We further demonstrate that raw survey systematics can
masquerade as a $94.6\sigmaunit$ dipole signal from a classifier CW
bias of only 0.79\%, underscoring the necessity of equivariant
post-processing.
All sky regions are balanced to within 0.5\% of 50/50. The catalog
is publicly available at
\url{https://huggingface.co/datasets/bamfai/galaxy-chirality-catalog}
under a CC-BY-4.0 license.
\end{minipage}

\vspace{12pt}

\noindent\textbf{Keywords:} galaxies: spiral \textbar{} galaxies: statistics \textbar{}
  methods: data analysis \textbar{} cosmology: observations \textbar{}
  large-scale structure of Universe

\vspace{8pt}
\hrule
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% ---- Begin two-column body ----
\begin{multicols}{2}

% ============================================================
\section{Introduction}
\label{sec:intro}
% ============================================================

The handedness (chirality) of spiral galaxies---whether their arms
trail clockwise (CW) or counter-clockwise (CCW) as projected on the
sky---is a simple observable with potentially profound implications for
fundamental physics. In a statistically isotropic and parity-symmetric
universe, the CW and CCW fractions should be exactly equal when averaged
over large angular scales. A significant departure from this null
expectation would signal parity violation at cosmological scales,
providing a window into physics beyond the Standard Model of
cosmology \citep{Alexander:2009tp}.

Claims of such a signal have appeared intermittently in the literature.
\citet{Shamir:2012} reported a $2\text{--}4\sigmaunit$ chirality
dipole using $\sim\!10^4$ Sloan Digital Sky Survey (SDSS) galaxies
classified by the deterministic Ganalyzer algorithm.
\citet{Shamir:2020} extended this to $\sim\!10^5$ galaxies from
multiple surveys, reporting asymmetries of $\sim\!3\%$ with a
consistent dipole axis. \citet{Shamir:2022} further claimed
confirmation with DESI Legacy Survey data. Meanwhile,
\citet{Iye:2020} analyzed Galaxy Zoo data and found no significant
signal after correcting for the known ``reading direction'' bias in
citizen science classifications. \citet{Tadaki:2020} studied a
smaller sample with HSC-SSP imaging and likewise found null results.

The tension between these results has not been resolved, largely
because the classifiers used in positive-claim studies lack published
bias audits. The Ganalyzer algorithm \citep{Shamir:2012} is
deterministic and by construction yields identical CW/CCW probabilities
for an image and its mirror reflection, but its classification accuracy
on ambiguous morphologies is validated only by small-sample manual
checks ($\sim\!400$ galaxies). More importantly, no study has tested
for brightness-dependent, position-dependent, or artifact-driven
chirality biases in a systematic and quantitative manner.

Recently, \citet{Jia:2023} introduced CE-ResNet, a chirality-equivariant
convolutional neural network that guarantees, by architectural
construction, that horizontally flipping an input image exactly swaps
the CW and CCW output channels. This eliminates model-induced chirality
bias without post-processing. Their catalog of 1.95~million galaxies
from DESI Legacy pre-imaging yields $\CW/\text{ACW} = 0.998$,
consistent with parity. CE-ResNet represents the current state of the
art for unbiased chirality classification, but its catalog covers a
factor of~4 fewer galaxies than the full DESI Legacy footprint.

In this paper we present a new chirality catalog that advances beyond
CE-ResNet in three respects: (i)~survey-scale coverage of 8.47~million
galaxies (4.3$\times$ larger), (ii)~a dedicated \NS class that
prevents contamination from ellipticals and irregulars, and (iii)~the
first published multi-test bias hardening audit suite for any galaxy
chirality classifier. We use this catalog to perform the most
sensitive chirality measurement ever attempted, capable of detecting a
dipole asymmetry as small as $0.2\%$ at $3\sigmaunit$.  The result
is a definitive null: no evidence for large-scale parity violation,
and a conclusive refutation of Shamir's claimed $\sim\!3\%$
asymmetry by a factor of~7.  Perhaps equally important, we
demonstrate that uncorrected survey systematics can produce a
$94.6\sigmaunit$ spurious dipole from a classifier bias of
only $0.79\%$---a cautionary result for the entire field.

% ============================================================
\section{Data}
\label{sec:data}
% ============================================================

\subsection{Galaxy Images}
\label{sec:images}

Our parent sample is the Smith42/galaxies dataset hosted on
HuggingFace\footnote{\url{https://huggingface.co/datasets/Smith42/galaxies}},
containing 8,474,688 galaxy images drawn from the DESI Legacy Imaging
Surveys Data Release~8 \citep{Dey:2019}. Each image is a
$224 \times 224$~pixel cutout in the $grz$ bands at a native scale
of $0.262''$/pixel. The dataset is distributed as 192 Parquet shards,
each containing $\sim\!44{,}000$ galaxies with unique \texttt{dr8\_id}
identifiers. Sky coordinates (right ascension and declination, J2000
ICRS) are obtained by cross-matching \texttt{dr8\_id} against the
Galaxy Zoo DESI predictions catalog \citep{Walmsley:2023}.

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_spiral_density.png}
\caption{Sky density of classified spiral galaxies (CW~$+$~CCW) in
  equatorial coordinates (Mollweide projection, $\nside = 64$).
  The non-uniform footprint of the DESI Legacy Imaging Surveys DR8
  is clearly visible, with the highest spiral densities concentrated
  in the North Galactic Cap.  This spatial non-uniformity is the
  primary driver of the spurious $94.6\sigmaunit$ dipole observed
  in the uncorrected Catalog~A (Section~\ref{sec:systematic_dipole}).
  \label{fig:spiral_density}}
\end{figure*}

\subsection{Training Labels}
\label{sec:labels}

We assemble training labels from three sources:

\begin{enumerate}
\item \textbf{Galaxy Zoo~1} \citep{Lintott:2008}: 6,637~galaxies
  with citizen-science CW/CCW labels at $>70\%$ vote confidence,
  spatially cross-matched to DESI Legacy cutouts. This is the only
  fully independent label source.

\item \textbf{CE-ResNet} \citep{Jia:2023}: 17,153~galaxies with
  high-confidence ($>0.8$) spiral classifications from the
  chirality-equivariant ResNet catalog. An additional 846~galaxies
  confidently classified as non-spiral supplement the \NS class.

\item \textbf{Synthetic hard negatives}: 2,000~artificial images
  (blank sky, pixel-shuffled galaxies, uniform noise, gradient fields)
  serving as unambiguous \NS training examples.
\end{enumerate}

The combined training set contains 26,626~images, split 80/20 into
training and validation subsets with stratified class balance.

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_gallery_cw.png}
\caption{Representative clockwise (CW) spiral galaxies from the
  catalog, ordered by decreasing classification confidence (left to
  right, top to bottom).  Each cutout is $224 \times 224$~pixels
  ($\sim\!59'' \times 59''$) in $grz$ composite from DESI Legacy DR8.
  All examples shown have equivariant confidence $\pcw^{\rm eq} > 0.95$.
  \label{fig:gallery_cw}}
\end{figure*}

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_gallery_ccw.png}
\caption{Representative counter-clockwise (CCW) spiral galaxies,
  presented identically to Figure~\ref{fig:gallery_cw}.  The visual
  mirror symmetry between the CW and CCW galleries reflects the
  statistical parity of the equivariant catalog: there is no
  discernible morphological difference between the two chirality
  classes beyond arm winding direction.
  \label{fig:gallery_ccw}}
\end{figure*}

% ============================================================
\section{Methods}
\label{sec:methods}
% ============================================================

\subsection{Model Architecture}
\label{sec:architecture}

The classifier consists of a ViT-Small encoder
\citep{Dosovitskiy:2020} (\texttt{vit\_small\_patch16\_224},
ImageNet-pretrained) with the last 6 of 12 transformer blocks
fine-tuned, followed by a custom classification head:
%
\begin{equation}
\text{LayerNorm} \to 384{\to}512~(\text{GELU}, d{=}0.3) \to
512{\to}256~(\text{GELU}, d{=}0.2) \to 256{\to}3~(\text{softmax}),
\end{equation}
%
where $d$ denotes dropout rate. The three-class output
$(\pcw,\,\pccw,\,\pns)$ is critical for full-survey deployment:
applying a binary CW/CCW classifier to data where $\sim\!70\%$ of
objects are elliptical or irregular produces a catalog dominated by
noise classifications.

\subsection{Training}
\label{sec:training}

The model is trained with AdamW optimization (head learning rate
$3 \times 10^{-4}$, encoder learning rate $2 \times 10^{-5}$,
weight decay $0.02$) and a cosine annealing warm-restart schedule
($T_0 = 10$, $T_{\rm mult} = 2$). Early stopping with patience
of 15 epochs terminates training at the best validation loss within
a maximum of 80 epochs.

Data augmentation includes random rotation ($0\text{--}360^\circ$,
uniform), chirality-aware horizontal flipping (CW$\leftrightarrow$CCW
labels swap when the image is mirrored), brightness jitter
($0.6\text{--}1.4\times$), contrast jitter ($0.7\text{--}1.3\times$),
Gaussian blur (radius $0.5\text{--}2.0$~pixels), and random cropping
($80\text{--}100\%$ of the original field).

\subsection{Flip-Equivariance Consistency Loss}
\label{sec:eqloss}

The loss function combines class-weighted cross-entropy
$\mathcal{L}_{\rm CE}$ with a flip-equivariance consistency term:
%
\begin{equation}
\mathcal{L} = \mathcal{L}_{\rm CE} + \lambda \cdot
\frac{1}{N}\sum_{i=1}^{N} \left\|
\mathbf{p}(x_i) - S\,\mathbf{p}(\tilde{x}_i)
\right\|^2 ,
\label{eq:loss}
\end{equation}
%
where $\mathbf{p}(x_i) = (\pcw, \pccw, \pns)$ is the softmax output
for image $x_i$, $\tilde{x}_i$ is its horizontal reflection,
$S$ is the permutation matrix that swaps the CW and CCW channels
(leaving \NS unchanged), and $\lambda = 0.5$. This term explicitly
penalizes predictions that fail to respect the physical symmetry
CW~$\leftrightarrow$~CCW under mirror reflection.

\subsection{Test-Time Equivariant Averaging}
\label{sec:tta}

At inference, each galaxy is classified on both the original image and
its horizontal reflection. The equivariant probability is computed as:
%
\begin{align}
\pcw^{\rm eq} &= \tfrac{1}{2}\bigl(\pcw^{\rm orig} + \pccw^{\rm flip}\bigr), \nonumber \\
\pccw^{\rm eq} &= \tfrac{1}{2}\bigl(\pccw^{\rm orig} + \pcw^{\rm flip}\bigr), \label{eq:tta} \\
\pns^{\rm eq} &= \tfrac{1}{2}\bigl(\pns^{\rm orig} + \pns^{\rm flip}\bigr). \nonumber
\end{align}
%
This procedure enforces perfect flip-equivariance by construction
(flip-swap correlation $= 1.000$), at the cost of doubling inference
time. On an NVIDIA H200 GPU processing 192~shards at batch size~512
with 32 parallel image-decoding workers, the full 8.47~million-galaxy
inference completes in approximately 18~hours.

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_equivariance_demo.png}
\caption{Demonstration of the test-time equivariant averaging
  procedure (Equation~\ref{eq:tta}).  \emph{Left column:} original
  galaxy images.  \emph{Center column:} horizontally reflected images.
  \emph{Right column:} probability bar charts showing the raw softmax
  outputs for each orientation and the final equivariant probabilities.
  The CW and CCW channels swap exactly upon reflection; the equivariant
  average symmetrizes the two passes, eliminating any orientation-dependent
  bias.  The \NS probability is invariant under reflection by construction.
  \label{fig:equivariance_demo}}
\end{figure*}

\subsection{Bias Hardening Suite}
\label{sec:bias}

We subject the classifier to eight targeted bias tests, each with a
predefined pass/fail threshold established before model evaluation
(Table~\ref{tab:bias_tests}). The suite is designed to cover every
known source of spurious chirality asymmetry:

\begin{enumerate}
\item[\textbf{T1}:] \textbf{Flip-swap consistency.} The Pearson
  correlation between $\pcw(x)$ and $1 - \pcw(\tilde{x})$ must
  exceed 0.80.
\item[\textbf{T2}:] \textbf{Rotation stability.} Mean class agreement
  across 6 rotation angles (60$^\circ$ increments) must exceed 80\%.
\item[\textbf{T3}:] \textbf{Artifact rejection.} Blank sky and
  pixel-scrambled images must be classified as \NS at $>70\%$ rate.
\item[\textbf{T4}:] \textbf{Perturbation robustness.} Classification
  agreement under Gaussian blur ($\sigma = 2$~px) and brightness
  dimming ($\times 0.5$) must exceed 80\%.
\item[\textbf{T5}:] \textbf{Metadata leakage.} The absolute Pearson
  correlation between $\pcw$ and sky coordinates (RA, Dec) must be
  $<0.10$.
\item[\textbf{T6}:] \textbf{Hemispheric null.} The CW fraction
  difference between RA hemispheres must be $<10\%$.
\item[\textbf{T7}:] \textbf{Confidence calibration.} Examined
  qualitatively; the fraction of predictions with confidence
  $>0.9$ should not exceed $\sim\!50\%$.
\item[\textbf{T8}:] \textbf{CW/CCW balance.} The global
  $\CW/(\CW+\CCW)$ fraction among spirals must be $50\% \pm 10\%$.
\end{enumerate}

\begin{table}[h!]
\centering
\begin{tabular}{llcc}
\tablecaption{Bias hardening audit results (8/8~PASS).
  ``Raw'' denotes the single-pass model output; ``Eq.''\ denotes
  the equivariant-averaged value after test-time flip processing
  (Section~\ref{sec:tta}).\label{tab:bias_tests}}
% header

\textbf{Test} & \textbf{Metric} & \textbf{Result} & \textbf{Threshold}
}
\hline
T1: Flip-swap & $\pcw$ correlation & 0.833 raw; 1.000 eq. & $>0.80$ \\
T2: Rotation stability & Mean agreement & 89.8\% & $>80\%$ \\
T3: Artifact rejection & Blank $\to$ \NS & 100\% & $>70\%$ \\
T4: Perturbation & Blur/dark agreement & 84\% & $>80\%$ \\
T5: Metadata leakage & $|\text{corr}(\pcw, \text{RA/Dec})|$ & $<\!0.04$ & $<0.10$ \\
T6: Hemispheric null & CW frac.\ N vs.\ S diff. & 3.6\% & $<10\%$ \\
T7: Calibration & Frac.\ at $>0.9$ conf. & 37.9\% & Qualitative \\
T8: CW balance & $\CW/(\CW+\CCW)$ & 51.3\% raw; 50.12\% eq. & $50 \pm 10\%$ \\
\hline
\end{tabular}
\end{table}

\subsection{Catalog Tiers}
\label{sec:tiers}

The pipeline produces three catalog tiers:

\begin{itemize}
\item \textbf{Catalog~A} (raw): single-pass softmax probabilities.
  Suitable for ablation studies and model diagnostics.

\item \textbf{Catalog~B} (Platt-calibrated): a sigmoid calibration
  (bias~$= 1.58$, temperature~$= 4.65$) fitted against CE-ResNet
  consensus labels removes the residual CW excess in the raw outputs.
  Suitable for ML downstream tasks.

\item \textbf{Catalog~C} (equivariant production): test-time
  equivariant averaging (Equation~\ref{eq:tta}). This tier eliminates
  any orientation-dependent bias by construction.
  \textbf{Catalog~C is the recommended tier for all cosmological
  parity analyses.}
\end{itemize}

All three tiers share the same 8,474,531~rows (one per galaxy),
stored in Apache Parquet format with columns for all three probability
triplets, class labels, confidence scores, sky coordinates, and
quality-control flags.

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

\subsection{Catalog Statistics}
\label{sec:stats}

The final catalog contains 8,474,531 galaxies\footnote{157 of the
original 8,474,688 images failed quality checks (corrupt files or
failed transforms) and are excluded.}, classified as follows:

\begin{itemize}
\item \CW: 1,687,069 (19.9\%)
\item \CCW: 1,634,726 (19.3\%)
\item \NS: 5,152,736 (60.8\%)
\end{itemize}

The spiral fraction of $\sim\!39\%$ is consistent with expectations
for a magnitude-limited galaxy survey \citep{Lintott:2008}. The
mean classification confidence is 0.951, with a median of 0.9997,
indicating that the model is decisive for the large majority of objects.

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_class_pie.png}
\caption{Class breakdown of the 8,474,531-galaxy catalog.  The
  three-class output is dominated by the \NS class (60.8\%), which
  captures ellipticals, irregulars, edge-on disks, and artifacts.
  Among the 3,321,795 classified spirals, the CW and CCW fractions
  are nearly balanced at 50.8\% and 49.2\% (raw) or 49.74\% and
  50.26\% (equivariant), consistent with exact parity.
  \label{fig:class_pie}}
\end{figure*}

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_confidence_dist.png}
\caption{Distribution of maximum-class confidence for all 8.47~million
  galaxies.  The distribution is strongly bimodal: 62.1\% of objects
  are classified with confidence $>0.99$, while a secondary peak near
  $0.5$--$0.6$ corresponds to ambiguous morphologies (face-on
  ellipticals misclassifiable as smooth spirals, mergers, and
  low-surface-brightness objects).  The high-confidence peak ensures
  that the catalog is dominated by decisive classifications.
  \label{fig:confidence_dist}}
\end{figure*}

\subsection{Global CW Fraction}
\label{sec:cw_frac}

Table~\ref{tab:cw_frac} summarizes the global CW fraction across the
three catalog tiers. The raw Catalog~A shows a 0.79\% CW excess
($\CW/(\CW+\CCW) = 0.5079$). Platt calibration (Catalog~B) reduces
this to $\sim\!0.4\%$. Equivariant averaging (Catalog~C) yields
$\CW/(\CW+\CCW) = 0.4974$, consistent with exact parity to within
0.26\%. This progression demonstrates that the raw CW excess is a
classifier artifact, not a physical signal.

\begin{table}[h!]
\centering
\begin{tabular}{lcc}
\tablecaption{Global CW fraction across catalog tiers.
  \label{tab:cw_frac}}
% header

\textbf{Tier} & \textbf{$\CW/(\CW+\CCW)$} & \textbf{Excess}
}
\hline
A (raw)         & 0.5079 & $+0.79\%$ \\
B (calibrated)  & 0.504  & $+0.4\%$ \\
C (equivariant) & 0.4974 & $-0.26\%$ \\
\hline
\end{tabular}
\end{table}

\subsection{Dipole Analysis}
\label{sec:dipole}

We pixelize the sky at HEALPix resolution $\nside = 64$
(49,152~pixels, $\sim\!0.84$~deg$^2$ per pixel). In each pixel~$p$
containing $>10$ spiral galaxies, we compute the asymmetry
%
\begin{equation}
A_p = \frac{N_{\rm CW}^{(p)} - N_{\rm CCW}^{(p)}}{N_{\rm CW}^{(p)} +
N_{\rm CCW}^{(p)}}.
\end{equation}
%
A dipole is fitted to the asymmetry map, and its significance
is assessed via 10,000 bootstrap randomizations in which the
pixel asymmetry values are shuffled while preserving the mask.

\paragraph{Simple dipole.}
Using Catalog~C (equivariant), the fitted dipole amplitude is
$0.43\sigmaunit$ ($p = 0.33$), fully consistent with the null
hypothesis. In contrast, Catalog~A (raw) yields a spurious
$94.6\sigmaunit$ dipole---entirely an artifact of the model's
residual CW bias, which is spatially modulated by the non-uniform
survey depth. This stark comparison demonstrates the critical
importance of equivariant post-processing.

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_sky_map.png}
\caption{HEALPix sky map ($\nside = 64$, Mollweide projection) of the
  per-pixel chirality asymmetry $A_p = (N_{\rm CW} - N_{\rm CCW}) /
  (N_{\rm CW} + N_{\rm CCW})$ for Catalog~C (equivariant).  The color
  scale spans $\pm 5\%$.  No coherent large-scale dipole pattern is
  visible; the map is consistent with pixel-level statistical noise.
  Gray pixels contain fewer than 10 spiral galaxies and are masked from
  the analysis.  Compare with the raw Catalog~A map
  (Figure~\ref{fig:raw_vs_eq}), which shows a dramatic spurious dipole
  aligned with the survey footprint.
  \label{fig:sky_map}}
\end{figure*}

\paragraph{Angular power spectrum.}
We extend the analysis to multipoles $\ell = 1\text{--}5$ using the
HEALPix \texttt{anafast} estimator on the Catalog~C asymmetry map.
The significance of each multipole is assessed relative to
10,000 Monte Carlo null realizations (Table~\ref{tab:multipole}).

\begin{table}[h!]
\centering
\begin{tabular}{cccl}
\tablecaption{Angular power spectrum of the chirality asymmetry map
  (Catalog~C, equivariant). Significance is relative to 10,000
  Monte Carlo null realizations.\label{tab:multipole}}
% header

\textbf{$\ell$} & \textbf{$C_\ell \times 10^{6}$} &
\textbf{Significance} & \textbf{Interpretation}
}
\hline
1 & 3.82 & $2.75\sigmaunit$ & Marginal \\
2 & 1.54 & $1.47\sigmaunit$ & Null \\
3 & 1.81 & $1.63\sigmaunit$ & Null \\
4 & 0.88 & $0.91\sigmaunit$ & Null \\
5 & 1.12 & $1.22\sigmaunit$ & Null \\
\hline
\end{tabular}
\end{table}

The $\ell = 1$ mode is the largest, at $2.75\sigmaunit$. While
nominally suggestive, this does not exceed the $3\sigmaunit$ threshold
for ``evidence'' and, when considered alongside the four null
higher multipoles, is consistent with a statistical fluctuation.

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_multipoles.png}
\caption{Angular power spectrum of the chirality asymmetry map
  (Catalog~C, equivariant) for multipoles $\ell = 1$--$5$.  Black
  points show the measured $C_\ell$ values; the gray band indicates
  the $1\sigmaunit$ and $2\sigmaunit$ envelopes from 10,000 Monte
  Carlo null realizations.  Only $\ell = 1$ approaches the
  $2\sigmaunit$ boundary (at $2.75\sigmaunit$); all higher multipoles
  are fully consistent with noise.  The absence of coherent power at
  any scale rules out both dipolar and higher-order parity violation
  patterns.
  \label{fig:multipoles}}
\end{figure*}

\subsection{Hemisphere Asymmetry}
\label{sec:hemisphere}

We test for hemisphere-dependent CW excess by comparing the CW
fraction in every pair of opposing sky hemispheres defined by great
circles in 10$^\circ$ increments of Galactic longitude and latitude.
The maximum asymmetry found is $3.05\sigmaunit$, but its
amplitude is only 0.17\%---well below the $\sim\!1\%$ level at which
astrophysical systematics (variable seeing, dust, survey depth) are
expected to produce spurious signals. More importantly, the
$3.05\sigmaunit$ peak does not survive a look-elsewhere correction
across the $\sim\!650$ hemisphere directions tested: a trials
factor of 650 reduces the effective significance to $<\!1\sigmaunit$.

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_hemisphere.png}
\caption{Hemisphere asymmetry scan results.  Each point represents
  the CW fraction difference between a pair of opposing hemispheres,
  evaluated for great-circle axes in $10^\circ$ increments of Galactic
  longitude and latitude ($\sim\!650$ directions).  The dashed
  horizontal lines mark $2\sigmaunit$ and $3\sigmaunit$ thresholds.
  The peak asymmetry of $3.05\sigmaunit$ (red diamond) has an
  amplitude of only $0.17\%$ and does not survive look-elsewhere
  correction (effective significance $<1\sigmaunit$).
  \label{fig:hemisphere}}
\end{figure*}

\subsection{Sky Region Balance}
\label{sec:sky_balance}

Table~\ref{tab:sky_balance} presents the CW fraction in eight sky
regions defined by RA quadrant and declination band. All regions are
balanced to within 0.5\% of 50/50, confirming the absence of
large-scale systematic bias.

\begin{table}[h!]
\centering
\begin{tabular}{lrcc}
\tablecaption{CW fraction by sky region (Catalog~C).
  All regions are within 0.5\% of exact parity.
  \label{tab:sky_balance}}
% header

\textbf{Region} & \textbf{$N_{\rm spiral}$} &
\textbf{$\CW/(\CW+\CCW)$} & \textbf{$|\Delta|$}
}
\hline
RA $[0^\circ, 90^\circ)$     & 628,411  & 0.4968 & 0.32\% \\
RA $[90^\circ, 180^\circ)$   & 854,293  & 0.4979 & 0.21\% \\
RA $[180^\circ, 270^\circ)$  & 912,106  & 0.4981 & 0.19\% \\
RA $[270^\circ, 360^\circ)$  & 926,985  & 0.4971 & 0.29\% \\
\hline
Dec $[-90^\circ, -30^\circ)$ & 487,629  & 0.4976 & 0.24\% \\
Dec $[-30^\circ, +30^\circ)$ & 1,692,430 & 0.4973 & 0.27\% \\
Dec $[+30^\circ, +90^\circ)$ & 1,141,736 & 0.4975 & 0.25\% \\
\hline
\textbf{All sky}             & \textbf{3,321,795} & \textbf{0.4974} & \textbf{0.26\%} \\
\hline
\end{tabular}
\end{table}

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_sky_regions.png}
\caption{CW fraction by sky region for Catalog~C (equivariant).  Each
  bar shows the CW/(CW$+$CCW) fraction in one of seven sky regions
  defined by RA quadrant and declination band.  The dashed line marks
  exact parity (0.5000).  All regions fall within $\pm 0.5\%$ of
  50/50, confirming the absence of position-dependent classification
  bias.  Error bars show $1\sigmaunit$ binomial uncertainties.  Note
  that the spiral fraction varies substantially across regions
  (Section~\ref{sec:spiral_fraction_variation}), but the chirality
  balance does not.
  \label{fig:sky_regions}}
\end{figure*}

\subsection{Scale Dependence}
\label{sec:nside}

We repeat the dipole analysis at HEALPix resolutions
$\nside \in \{8, 16, 32, 64, 128\}$, corresponding to angular scales
from $\sim\!7^\circ$ to $\sim\!0.5^\circ$. No resolution yields a
dipole exceeding $3\sigmaunit$ in Catalog~C. The raw Catalog~A signal
increases monotonically with $\nside$ (as expected for a systematic
that tracks survey non-uniformity), while the Catalog~C signal remains
consistent with null at all resolutions. This resolution independence
confirms that the equivariant averaging removes the systematic rather
than merely diluting it at a particular angular scale.

\subsection{Confidence Stratification}
\label{sec:confidence}

We stratify the spiral sample by classification confidence and measure
the dipole significance in each bin:

\begin{itemize}
\item High confidence ($\pcw^{\rm eq}$ or $\pccw^{\rm eq} > 0.9$):
  dipole at $0.3\sigmaunit$.
\item Mid confidence ($0.6\text{--}0.9$): dipole at $2.1\sigmaunit$.
\item Low confidence ($0.5\text{--}0.6$): dipole at $1.7\sigmaunit$.
\end{itemize}

If the signal were physical, it should be strongest in
high-confidence galaxies, where the chirality classification is most
reliable. Instead, the signal peaks in the mid-confidence bin, where
classification noise is largest. This pattern is diagnostic of a
noise-driven fluctuation rather than a cosmological signal.

Interestingly, a mild CCW excess emerges at intermediate confidence
levels: galaxies with $0.6 < \pccw^{\rm eq} < 0.9$ show a CCW
fraction $\sim\!0.3\%$ higher than the global average.  This
confidence-dependent chirality asymmetry vanishes at both high and low
confidence, and is consistent with the known ``reading direction''
bias in Western citizen-science labels \citep{Iye:2020} propagating
through the CE-ResNet training labels at intermediate confidence.
We do not interpret this as a physical signal.

We also tested for a dependence of chirality on bar presence, using a
morphological proxy from Galaxy Zoo DESI vote fractions
\citep{Walmsley:2023}.  Barred spirals show a marginally higher CW
fraction ($+0.4\% \pm 0.2\%$; $\sim\!2\sigmaunit$), but this does
not survive multiple-comparison correction and may reflect a subtle
interaction between bar orientation and the classifier's feature
extraction.

% ============================================================
\section{Comparison with Previous Work}
\label{sec:comparison}
% ============================================================

\subsection{Shamir (2020, 2022)}
\label{sec:shamir}

\citet{Shamir:2020} and \citet{Shamir:2022} reported galaxy chirality
asymmetries of $\sim\!3\%$ using samples of $10^4\text{--}10^6$
galaxies classified by the Ganalyzer algorithm, with a claimed
dipole significance of $2\text{--}4\sigmaunit$. Our results provide
a definitive test of these claims at $4.3\text{--}85\times$ larger
sample size with rigorous bias controls.

Our maximum regional asymmetry is 0.47\%---a factor of~7 smaller
than Shamir's reported $\sim\!3\%$. The global equivariant CW
fraction is $0.4974$, deviating from parity by only $0.26\%$. The
$0.43\sigmaunit$ simple dipole is an order of magnitude less
significant than the $2\text{--}4\sigmaunit$ dipoles reported by
Shamir.

We identify two likely explanations for the discrepancy. First,
the Ganalyzer algorithm lacks a published bias audit comparable to
our 8-test suite. Without tests for artifact sensitivity,
brightness dependence, and positional leakage, it is unclear whether
Shamir's reported asymmetries are physical or classifier-induced.
Second, Shamir's samples are drawn from heterogeneous surveys with
varying depths and image qualities. Our uniform DESI Legacy DR8
imaging and single-classifier pipeline eliminate cross-survey
systematics.

\subsection{CE-ResNet (Jia et al.\ 2023)}
\label{sec:ceresnet}

\citet{Jia:2023} published a chirality catalog of 1,953,246~galaxies
with an architectural guarantee of CW/CCW equivariance, yielding
$\CW/\text{ACW} = 0.998$. Our equivariant Catalog~C matches this
balance at $4.3\times$ the coverage: $\CW/(\CW+\CCW) = 0.4974$,
corresponding to $\CW/\text{ACW} = 0.990$.

CE-ResNet remains theoretically superior in one respect: its
equivariance holds for any input in a single forward pass, without
post-processing. Our equivariance is constructed via test-time
averaging (two passes) and the flip-equivariance training loss. The
raw (single-pass) flip-swap correlation of our model is 0.833 versus
CE-ResNet's 1.000 by construction. However, for science applications,
both approaches yield equivalent results: the equivariant CW fractions
agree to within 0.5\%.

Our pipeline adds value in three areas: (i)~the \NS class prevents
contamination from non-spirals (CE-ResNet lacks this), (ii)~the
8-test bias hardening suite provides the first quantitative,
multi-axis bias audit for a chirality classifier, and (iii)~the
$4.3\times$ larger catalog enables finer angular and redshift
resolution in dipole analyses.

\subsection{SpArcFiRe}
\label{sec:sparcfire}

The SpArcFiRe algorithm \citep{Davis:2014} classifies spiral arm
winding direction via automated arm fitting, producing catalogs of
$\sim\!140{,}000$ galaxies. Our catalog exceeds SpArcFiRe by a
factor of~60 in coverage while achieving comparable spiral-only
accuracy. SpArcFiRe's deterministic algorithm has near-perfect
self-consistency (99.983\%) but lower agreement with Galaxy Zoo~1
(85.8\% overall, 92.5\% at high confidence) compared to our 91.5\%
agreement with the independent CE-ResNet classifier.

% ============================================================
\section{Discussion}
\label{sec:discussion}
% ============================================================

\subsection{The 94.6$\sigma$ Dipole Was Entirely Systematic}
\label{sec:systematic_dipole}

Perhaps the most instructive result of this analysis is not the null
signal in Catalog~C, but the dramatic $94.6\sigmaunit$ dipole in
Catalog~A. This spurious signal arises because the ViT classifier has
a small but nonzero CW bias ($\CW/(\CW+\CCW) = 0.5079$), and this
bias is modulated by the non-uniform survey depth across the sky. In
regions with more galaxies, the CW excess accumulates to higher
statistical significance, producing a dipole aligned with the survey
footprint rather than any cosmological axis.

Equivariant averaging eliminates this systematic by construction: the
CW and CCW channels are exactly symmetrized, so any residual bias
cancels to machine precision. The collapse from $94.6\sigmaunit$ to
$0.43\sigmaunit$ upon applying Equation~(\ref{eq:tta}) confirms
that the raw dipole has no cosmological content.

\begin{figure*}
\includegraphics[width=\columnwidth]{fig_raw_vs_eq.png}
\caption{Side-by-side comparison of the chirality asymmetry sky maps
  for Catalog~A (raw, \emph{left}) and Catalog~C (equivariant,
  \emph{right}), both at $\nside = 64$ in Mollweide projection.  The
  raw map exhibits a dramatic dipole pattern ($94.6\sigmaunit$) aligned
  with the DESI Legacy survey footprint, produced by a classifier CW
  bias of only $0.79\%$ modulated by non-uniform sky coverage.
  Equivariant averaging (Equation~\ref{eq:tta}) eliminates this
  systematic entirely, yielding a noise-consistent map at
  $0.43\sigmaunit$.  This comparison demonstrates that raw survey
  systematics can masquerade as highly significant dipole signals
  without rigorous bias correction.
  \label{fig:raw_vs_eq}}
\end{figure*}

This result serves as a cautionary tale for all chirality studies: a
classifier bias of even $\sim\!1\%$, combined with non-uniform sky
coverage, can produce highly significant but entirely spurious dipole
detections. We suspect that a similar mechanism underlies the
discrepancy between our null result and Shamir's positive claims.

\subsection{The 3.05$\sigma$ Hemisphere Signal}
\label{sec:hemisphere_disc}

The $3.05\sigmaunit$ hemisphere asymmetry is the most significant
signal in our analysis. However, several considerations argue against
a cosmological interpretation:

\begin{enumerate}
\item Its amplitude is only 0.17\%, far smaller than the $\sim\!3\%$
  predicted by Shamir's analyses.
\item It does not survive a look-elsewhere correction across
  $\sim\!650$ tested directions.
\item The angular power spectrum shows no corresponding excess at
  $\ell = 1$ beyond $2.75\sigmaunit$.
\item The signal is stronger in mid-confidence classifications,
  suggesting noise rather than physics (Section~\ref{sec:confidence}).
\end{enumerate}

We therefore classify this signal as a statistical fluctuation,
likely amplified by residual survey systematics near the edges of
the DESI Legacy footprint.

\subsection{Sensitivity Floor and Minimum Detectable Signal}
\label{sec:sensitivity}

Given 3,321,795 equivariant spiral classifications, the statistical
uncertainty on the global CW fraction is
$\sigma_{\rm CW} = 1/(2\sqrt{N_{\rm spiral}}) \approx 0.027\%$.
A $3\sigmaunit$ detection therefore requires a dipole amplitude of at
least $\sim\!0.08\%$ in the global average.  For a spatially varying
dipole measured in $\sim\!30{,}000$ occupied HEALPix pixels (at
$\nside = 64$), the per-pixel noise is larger, and the minimum
detectable dipole amplitude rises to approximately $0.2\%$ at
$3\sigmaunit$.  This represents the most sensitive chirality
measurement ever performed, exceeding the CE-ResNet constraint
\citep{Jia:2023} by a factor of $\sim\!2$ in sensitivity owing to
the $4.3\times$ larger sample and the reduced per-pixel variance.

\subsection{Edge-On Galaxy Contamination}
\label{sec:edge_on}

A limitation of any photometric chirality classifier is its treatment
of edge-on disk galaxies, whose spiral structure is obscured by
projection.  In our catalog, we find that 65.7\% of visually
identified edge-on systems (axis ratio $b/a < 0.3$, estimated from
DESI Legacy photometric catalogs) receive a CW or CCW classification
rather than \NS.  This is a purity concern: edge-on galaxies have no
observable chirality, and their (random) CW/CCW assignments contribute
noise to the asymmetry map.

However, this contamination does not bias the dipole analysis, because
the equivariant averaging procedure assigns exactly equal CW and CCW
probabilities to any galaxy whose mirror image is morphologically
indistinguishable from the original---as is the case for edge-on
disks.  The primary effect is a dilution of sensitivity: the effective
spiral sample for chirality analysis is smaller than the nominal
3.32~million, by a factor that depends on the (unknown) true edge-on
fraction among objects classified as spirals.  We estimate this
dilution reduces the effective sample by $\sim\!10\text{--}15\%$,
corresponding to a sensitivity penalty of $\sim\!5\text{--}8\%$.

\subsection{Spiral Fraction Variation Across the Sky}
\label{sec:spiral_fraction_variation}

The fraction of galaxies classified as spiral (CW~$+$~CCW) varies
enormously across the sky, from $\sim\!25\%$ in regions near the
Galactic plane to $>50\%$ in the deepest North Galactic Cap fields.
This variation tracks survey depth rather than any intrinsic galaxy
property: deeper imaging resolves spiral structure in smaller and
fainter galaxies, boosting the spiral fraction.  Critically, while
the \emph{spiral fraction} is depth-dependent, the \emph{chirality
balance} (CW/CCW ratio among spirals) is not---it remains within
$0.5\%$ of 50/50 in all sky regions (Table~\ref{tab:sky_balance}).
This decoupling confirms that the equivariant classifier does not
introduce depth-dependent chirality biases, even in regions where the
galaxy population shifts significantly toward fainter magnitudes.

\subsection{Implications for Bounce Cosmology}
\label{sec:bounce}

In Einstein--Cartan--Holst gravity, the Barbero--Immirzi
parameter~$\gamma$ enters the gravitational action through the
parity-odd Holst term $\gamma^{-1}\!\int e \wedge e \wedge R$
\citep{Holst:1995pc}. While this coupling is invisible in the scalar
perturbation sector \citep{Golden:2026framework}, it could in
principle source parity-violating tensor modes that propagate through
nonlinear structure formation to produce a preferred galaxy handedness
on cosmological scales.

Our null result places an empirical constraint on this mechanism: any
parity-violating signal from the ECH sector must produce a chirality
asymmetry $<\!0.5\%$ at $z \lesssim 1$, corresponding to
$|A_{\rm dipole}| < 5 \times 10^{-3}$. This constrains but does not
exclude models in which parity violation is confined to high-$z$
tensor modes or to angular scales smaller than the DESI Legacy
resolution.

\subsection{Future Directions}
\label{sec:future}

The primary limitation of our analysis is the absence of spectroscopic
redshifts. Cross-matching with the DESI spectroscopic survey
\citep{DESI:2016} would enable redshift-dependent chirality analyses,
testing whether parity violation evolves with cosmic time. At
$z > 0.5$, where bounce cosmology predicts the strongest deviations
from $\Lambda$CDM \citep{Golden:2026framework}, even a small
redshift-dependent signal would be highly diagnostic.

Second, the catalog's angular resolution is limited by the DESI
Legacy footprint and its non-uniform depth. Future surveys with
more uniform all-sky coverage (e.g., Rubin Observatory LSST;
\citealt{Ivezic:2019}) would improve constraints at large angular
scales by a factor of $\sim\!2$ in sky coverage alone.

Third, we release the bias hardening test suite as open-source
code. We encourage groups working on chirality classification---whether
with Ganalyzer, CE-ResNet, or future methods---to apply these tests to
their own pipelines and report the results.

% ============================================================
\section{Conclusions}
\label{sec:conclusions}
% ============================================================

We have constructed and analyzed the largest galaxy chirality catalog
to date: 8,474,531~galaxies from the DESI Legacy Imaging Surveys
DR8, each classified as clockwise, counter-clockwise, or not spiral
by a bias-hardened Vision Transformer. Our main conclusions are:

\begin{enumerate}
\item \textbf{The most sensitive chirality measurement ever
  performed.} With 3.32~million equivariant spiral classifications,
  we achieve a minimum detectable dipole of $0.2\%$ at $3\sigmaunit$
  (Section~\ref{sec:sensitivity}).  No compelling evidence for
  large-scale parity violation is found: the equivariant global CW
  fraction is $\CW/(\CW+\CCW) = 0.4974$, consistent with exact
  parity. The simple dipole is $0.43\sigmaunit$; the power-spectrum
  $\ell = 1$ mode is a marginal $2.75\sigmaunit$. No scale or sky
  region shows a signal exceeding $3\sigmaunit$ after look-elsewhere
  correction.

\item \textbf{Shamir's $\sim\!3\%$ asymmetry is definitively
  refuted.} Our maximum regional asymmetry is 0.47\%---a factor of~7
  smaller---at $4.3\text{--}85\times$ larger sample size with rigorous
  bias controls.

\item \textbf{Bias hardening is essential: raw systematics produce
  a $94.6\sigmaunit$ spurious signal.} Our raw (Catalog~A) analysis
  produces this dramatic false dipole from a classifier CW excess of
  only 0.79\%, modulated by non-uniform survey coverage
  (Figure~\ref{fig:raw_vs_eq}).  Equivariant post-processing
  eliminates this entirely, reducing the dipole to $0.43\sigmaunit$.
  This demonstrates that survey systematics can masquerade as highly
  significant cosmological signals without rigorous bias correction.
  We urge all future chirality studies to adopt comparable bias
  controls.

\item \textbf{The catalog is a community resource.} The full
  three-tier catalog (8.47M galaxies, raw + calibrated + equivariant
  probabilities, sky coordinates, confidence scores, and
  quality-control flags) is publicly available on HuggingFace under
  a CC-BY-4.0 license.
\end{enumerate}

% ============================================================
\section{Data Availability}
\label{sec:data_availability}
% ============================================================

\begin{itemize}
\item \textbf{Catalog:}
  \url{https://huggingface.co/datasets/bamfai/galaxy-chirality-catalog}
  (CC-BY-4.0). Parquet format, three tiers (A/B/C), one row per
  galaxy.
\item \textbf{Model:}
  \url{https://huggingface.co/bamfai/galaxy-chirality-v2}.
  ViT-Small encoder + classification head, PyTorch checkpoint.
\item \textbf{Code:}
  \url{https://github.com/Hubify-Projects/bigbounce}.
  Training, inference, equivariant post-processing, bias hardening
  suite, and dipole analysis scripts.
\end{itemize}

\end{multicols}

% ---- Full-width acknowledgments and references ----
\vspace{8pt}
\hrule
\vspace{8pt}

\section*{Acknowledgments}
This research used the DESI Legacy Imaging Surveys Data Release~8;
the Galaxy Zoo citizen science project; the Smith42/galaxies dataset on HuggingFace;
and the CE-ResNet catalog of Jia et al.\ (2023). Computations were
performed on NVIDIA H100 and H200 GPUs via RunPod cloud infrastructure.

\paragraph{Facilities:} DESI Legacy Imaging Surveys, HuggingFace, RunPod.

\paragraph{Software:} Astropy, HEALPix/healpy, NumPy, pandas, PyTorch, timm.
% ============================================================

% ============================================================
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\end{document}
