beta_table.tex

% Table rows generated programmatically from the analysis results used in this
% work. Do not hand-edit numeric values; notemarks a--d are annotation only.
% NOTE(2026-07-07 trust reset): this campaign-era table shell is not currently
% input by sections/results.tex. Reconcile rail/alpha-limit wording against
% CONTEXT.md before restoring it.
\begin{deluxetable*}{lccccc}
\tablecaption{Turbulence spectral index $\beta$ from the uniform $\beta$-based
joint re-fit (Section~\ref{sec:jointfit}) of the co-detected sample. Eleven of
the twelve sightlines are listed; FRB~20240203A is excluded by the joint-fit
quality gate (catastrophic per-band misfit, $\chi^2_\nu=11.6/9.3$). The PBF
shape and the frequency-scaling index $\alpha=2\beta/(\beta-2)$ are both
derived from $\beta$ at each likelihood evaluation (thin-screen co-model).
$\tau_{1\,\mathrm{GHz}}$ is the shared pulse-broadening time referred to
$1\,\mathrm{GHz}$; $C\times D$ is the number of temporal components modeled in
the CHIME and DSA-110 bands. The Kolmogorov value is $\beta=11/3$
($\alpha\approx4.4$); the square-law limit is $\beta=4$ ($\alpha=4$,
exponential PBF). Two sightlines return interior posteriors (a measured
$\beta$ with 16th/84th-percentile uncertainties); the remaining nine rail at
the $\beta=4$ bound\tablenotemark{d}. All fits are verified against their
posterior-predictive checks; rows with an elevated single-band $\chi^2_\nu$
(FRB~20221203A, FRB~20230913A, FRB~20220207C) are provisional pending a
per-band systematics pass.
\label{tab:beta}}
\tablehead{\colhead{Burst} & \colhead{$C\times D$} & \colhead{$\beta$} &
  \colhead{$\alpha$ (derived)} & \colhead{$\tau_{1\,\mathrm{GHz}}$ [ms]} &
  \colhead{$\chi^2_\nu$ (C/D)}}
\startdata
FRB 20230325A\tablenotemark{a} & $1\times1$ & $3.722^{+0.014}_{-0.015}$ & $4.32^{+0.02}_{-0.02}$ & $0.119$ & 1.29/1.03 \\
FRB 20240229A & $1\times1$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $0.0186$ & 1.57/1.02 \\
FRB 20221203A\tablenotemark{b} & $1\times1$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $0.269$ & 1.57/6.73 \\
FRB 20230913A & $1\times1$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $0.0245$ & 3.96/1.00 \\
FRB 20240122A & $1\times1$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $0.219$ & 1.04/0.90 \\
FRB 20220207C & $1\times1$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $0.294$ & 2.51/1.31 \\
FRB 20220506D & $2\times1$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $0.843$ & 1.02/1.22 \\
FRB 20221113A & $2\times1$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $0.314$ & 1.05/0.91 \\
FRB 20230814B & $2\times2$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $2.22$ & 1.09/1.23 \\
FRB 20220310F\tablenotemark{c} & $2\times2$ & $\to 4$\tablenotemark{d} & $4$ (limit) & $1.18$ & 1.09/1.42 \\
FRB 20230307A & $3\times3$ & $3.228^{+0.020}_{-0.018}$ & $5.26^{+0.05}_{-0.05}$ & $0.469$ & 1.06/1.24 \\
% excluded (not citable): FRB 20240203A (gate FAIL: L2 catastrophic chi2_C=11.59 chi2_D=9.25)
\enddata
\tablenotetext{a}{Interior posterior near the Kolmogorov point; well fit in
both bands, so the measurement is a constraint rather than a prior-rail
artifact.}
\tablenotetext{b}{The elevated DSA-110 $\chi^2_\nu$ is a coherent residual
localized at the bright pulse peak---off-peak residuals are white (RMS $\approx1$)
while the on-peak frequency-summed whitened residual reaches $\pm15\sigma$.
This is not evidence against the $\beta=4$ exponential/EMG branch; it means
the adopted component model leaves bright-pulse profile structure within that
preferred exponential-tail description. The rail classification is nonetheless
robust to the intrinsic-width parametrization.}
\tablenotetext{c}{Multiplicity exemplar: C2D2 fit after local re-prep
(Section~\ref{sec:results-alpha}, Figure~\ref{fig:whitney_mult}). Fixed-s²
evidence confirms the second DSA-110 component.}
\tablenotetext{d}{Posterior railed at the $\beta=4$ prior bound (by the
$3\sigma$-proximity or $\ge30\%$ edge-mass criterion): the sightline is
consistent with square-law statistics and an exponential PBF, and $\alpha=4$
is quotable only as a geometry-conditioned limit under the thin-screen
closure, not as a measurement. An extended-medium PBF family may fit these
sightlines without railing; that re-analysis is deferred.}
\end{deluxetable*}