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@@ -7361,3 +7361,5 @@ SMART/HiddenLayerExperiment/cache/imagenet_resnet50_seed1_vs0.2/test_logits.npy
 DensityAwareCalibration/Tomani[[:space:]]et[[:space:]]al.[[:space:]]-[[:space:]]2023[[:space:]]-[[:space:]]Beyond[[:space:]]In-Domain[[:space:]]Scenarios[[:space:]]Robust[[:space:]]Density-Aware[[:space:]]Calibration.pdf filter=lfs diff=lfs merge=lfs -text
 DensityAwareCalibration/classifier_ckpts/cifar10/resnet18.pth filter=lfs diff=lfs merge=lfs -text
 DensityAwareCalibration/teaser_fig.png filter=lfs diff=lfs merge=lfs -text
+icml2026/example_paper.pdf filter=lfs diff=lfs merge=lfs -text
+icml2026/figs/logitsgap_vs_confidence_change.pdf filter=lfs diff=lfs merge=lfs -text

icml2026/algorithm.sty

@@ -0,0 +1,79 @@
+% ALGORITHM STYLE -- Released 8 April 1996
+%    for LaTeX-2e
+% Copyright -- 1994 Peter Williams
+% E-mail Peter.Williams@dsto.defence.gov.au
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{algorithm}
+\typeout{Document Style `algorithm' - floating environment}
+
+\RequirePackage{float}
+\RequirePackage{ifthen}
+\newcommand{\ALG@within}{nothing}
+\newboolean{ALG@within}
+\setboolean{ALG@within}{false}
+\newcommand{\ALG@floatstyle}{ruled}
+\newcommand{\ALG@name}{Algorithm}
+\newcommand{\listalgorithmname}{List of \ALG@name s}
+
+% Declare Options
+% first appearance
+\DeclareOption{plain}{
+  \renewcommand{\ALG@floatstyle}{plain}
+}
+\DeclareOption{ruled}{
+  \renewcommand{\ALG@floatstyle}{ruled}
+}
+\DeclareOption{boxed}{
+  \renewcommand{\ALG@floatstyle}{boxed}
+}
+% then numbering convention
+\DeclareOption{part}{
+  \renewcommand{\ALG@within}{part}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{chapter}{
+  \renewcommand{\ALG@within}{chapter}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{section}{
+  \renewcommand{\ALG@within}{section}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{subsection}{
+  \renewcommand{\ALG@within}{subsection}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{subsubsection}{
+  \renewcommand{\ALG@within}{subsubsection}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption{nothing}{
+  \renewcommand{\ALG@within}{nothing}
+  \setboolean{ALG@within}{true}
+}
+\DeclareOption*{\edef\ALG@name{\CurrentOption}}
+
+% ALGORITHM
+%
+\ProcessOptions
+\floatstyle{\ALG@floatstyle}
+\ifthenelse{\boolean{ALG@within}}{
+  \ifthenelse{\equal{\ALG@within}{part}}
+     {\newfloat{algorithm}{htbp}{loa}[part]}{}
+  \ifthenelse{\equal{\ALG@within}{chapter}}
+     {\newfloat{algorithm}{htbp}{loa}[chapter]}{}
+  \ifthenelse{\equal{\ALG@within}{section}}
+     {\newfloat{algorithm}{htbp}{loa}[section]}{}
+  \ifthenelse{\equal{\ALG@within}{subsection}}
+     {\newfloat{algorithm}{htbp}{loa}[subsection]}{}
+  \ifthenelse{\equal{\ALG@within}{subsubsection}}
+     {\newfloat{algorithm}{htbp}{loa}[subsubsection]}{}
+  \ifthenelse{\equal{\ALG@within}{nothing}}
+     {\newfloat{algorithm}{htbp}{loa}}{}
+}{
+  \newfloat{algorithm}{htbp}{loa}
+}
+\floatname{algorithm}{\ALG@name}
+
+\newcommand{\listofalgorithms}{\listof{algorithm}{\listalgorithmname}}
+

icml2026/algorithmic.sty

@@ -0,0 +1,201 @@
+% ALGORITHMIC STYLE -- Released 8 APRIL 1996
+%    for LaTeX version 2e
+% Copyright -- 1994 Peter Williams
+% E-mail PeterWilliams@dsto.defence.gov.au
+%
+% Modified by Alex Smola (08/2000)
+% E-mail Alex.Smola@anu.edu.au
+%
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{algorithmic}
+\typeout{Document Style `algorithmic' - environment}
+%
+\RequirePackage{ifthen}
+\RequirePackage{calc}
+\newboolean{ALC@noend}
+\setboolean{ALC@noend}{false}
+\newcounter{ALC@line}
+\newcounter{ALC@rem}
+\newlength{\ALC@tlm}
+%
+\DeclareOption{noend}{\setboolean{ALC@noend}{true}}
+%
+\ProcessOptions
+%
+% ALGORITHMIC
+\newcommand{\algorithmicrequire}{\textbf{Require:}}
+\newcommand{\algorithmicensure}{\textbf{Ensure:}}
+\newcommand{\algorithmiccomment}[1]{\{#1\}}
+\newcommand{\algorithmicend}{\textbf{end}}
+\newcommand{\algorithmicif}{\textbf{if}}
+\newcommand{\algorithmicthen}{\textbf{then}}
+\newcommand{\algorithmicelse}{\textbf{else}}
+\newcommand{\algorithmicelsif}{\algorithmicelse\ \algorithmicif}
+\newcommand{\algorithmicendif}{\algorithmicend\ \algorithmicif}
+\newcommand{\algorithmicfor}{\textbf{for}}
+\newcommand{\algorithmicforall}{\textbf{for all}}
+\newcommand{\algorithmicdo}{\textbf{do}}
+\newcommand{\algorithmicendfor}{\algorithmicend\ \algorithmicfor}
+\newcommand{\algorithmicwhile}{\textbf{while}}
+\newcommand{\algorithmicendwhile}{\algorithmicend\ \algorithmicwhile}
+\newcommand{\algorithmicloop}{\textbf{loop}}
+\newcommand{\algorithmicendloop}{\algorithmicend\ \algorithmicloop}
+\newcommand{\algorithmicrepeat}{\textbf{repeat}}
+\newcommand{\algorithmicuntil}{\textbf{until}}
+
+%changed by alex smola
+\newcommand{\algorithmicinput}{\textbf{input}}
+\newcommand{\algorithmicoutput}{\textbf{output}}
+\newcommand{\algorithmicset}{\textbf{set}}
+\newcommand{\algorithmictrue}{\textbf{true}}
+\newcommand{\algorithmicfalse}{\textbf{false}}
+\newcommand{\algorithmicand}{\textbf{and\ }}
+\newcommand{\algorithmicor}{\textbf{or\ }}
+\newcommand{\algorithmicfunction}{\textbf{function}}
+\newcommand{\algorithmicendfunction}{\algorithmicend\ \algorithmicfunction}
+\newcommand{\algorithmicmain}{\textbf{main}}
+\newcommand{\algorithmicendmain}{\algorithmicend\ \algorithmicmain}
+%end changed by alex smola
+
+\def\ALC@item[#1]{%
+\if@noparitem \@donoparitem
+  \else \if@inlabel \indent \par \fi
+         \ifhmode \unskip\unskip \par \fi
+         \if@newlist \if@nobreak \@nbitem \else
+                        \addpenalty\@beginparpenalty
+                        \addvspace\@topsep \addvspace{-\parskip}\fi
+           \else \addpenalty\@itempenalty \addvspace\itemsep
+          \fi
+    \global\@inlabeltrue
+\fi
+\everypar{\global\@minipagefalse\global\@newlistfalse
+          \if@inlabel\global\@inlabelfalse \hskip -\parindent \box\@labels
+             \penalty\z@ \fi
+          \everypar{}}\global\@nobreakfalse
+\if@noitemarg \@noitemargfalse \if@nmbrlist \refstepcounter{\@listctr}\fi \fi
+\sbox\@tempboxa{\makelabel{#1}}%
+\global\setbox\@labels
+ \hbox{\unhbox\@labels \hskip \itemindent
+       \hskip -\labelwidth \hskip -\ALC@tlm
+       \ifdim \wd\@tempboxa >\labelwidth
+                \box\@tempboxa
+          \else \hbox to\labelwidth {\unhbox\@tempboxa}\fi
+       \hskip \ALC@tlm}\ignorespaces}
+%
+\newenvironment{algorithmic}[1][0]{
+\let\@item\ALC@item
+  \newcommand{\ALC@lno}{%
+\ifthenelse{\equal{\arabic{ALC@rem}}{0}}
+{{\footnotesize \arabic{ALC@line}:}}{}%
+}
+\let\@listii\@listi
+\let\@listiii\@listi
+\let\@listiv\@listi
+\let\@listv\@listi
+\let\@listvi\@listi
+\let\@listvii\@listi
+  \newenvironment{ALC@g}{
+    \begin{list}{\ALC@lno}{ \itemsep\z@ \itemindent\z@
+    \listparindent\z@ \rightmargin\z@ 
+    \topsep\z@ \partopsep\z@ \parskip\z@\parsep\z@
+    \leftmargin 1em
+    \addtolength{\ALC@tlm}{\leftmargin}
+    }
+  }
+  {\end{list}}
+  \newcommand{\ALC@it}{\addtocounter{ALC@line}{1}\addtocounter{ALC@rem}{1}\ifthenelse{\equal{\arabic{ALC@rem}}{#1}}{\setcounter{ALC@rem}{0}}{}\item}
+  \newcommand{\ALC@com}[1]{\ifthenelse{\equal{##1}{default}}%
+{}{\ \algorithmiccomment{##1}}}
+  \newcommand{\REQUIRE}{\item[\algorithmicrequire]}
+  \newcommand{\ENSURE}{\item[\algorithmicensure]}
+  \newcommand{\STATE}{\ALC@it}
+  \newcommand{\COMMENT}[1]{\algorithmiccomment{##1}}
+%changes by alex smola
+  \newcommand{\INPUT}{\item[\algorithmicinput]}
+  \newcommand{\OUTPUT}{\item[\algorithmicoutput]}
+  \newcommand{\SET}{\item[\algorithmicset]}
+%  \newcommand{\TRUE}{\algorithmictrue}
+%  \newcommand{\FALSE}{\algorithmicfalse}
+  \newcommand{\AND}{\algorithmicand}
+  \newcommand{\OR}{\algorithmicor}
+  \newenvironment{ALC@func}{\begin{ALC@g}}{\end{ALC@g}}
+  \newenvironment{ALC@main}{\begin{ALC@g}}{\end{ALC@g}}
+%end changes by alex smola
+  \newenvironment{ALC@if}{\begin{ALC@g}}{\end{ALC@g}}
+  \newenvironment{ALC@for}{\begin{ALC@g}}{\end{ALC@g}}
+  \newenvironment{ALC@whl}{\begin{ALC@g}}{\end{ALC@g}}
+  \newenvironment{ALC@loop}{\begin{ALC@g}}{\end{ALC@g}}
+  \newenvironment{ALC@rpt}{\begin{ALC@g}}{\end{ALC@g}}
+  \renewcommand{\\}{\@centercr}
+  \newcommand{\IF}[2][default]{\ALC@it\algorithmicif\ ##2\ \algorithmicthen%
+\ALC@com{##1}\begin{ALC@if}}
+  \newcommand{\SHORTIF}[2]{\ALC@it\algorithmicif\ ##1\
+    \algorithmicthen\ {##2}}
+  \newcommand{\ELSE}[1][default]{\end{ALC@if}\ALC@it\algorithmicelse%
+\ALC@com{##1}\begin{ALC@if}}
+  \newcommand{\ELSIF}[2][default]%
+{\end{ALC@if}\ALC@it\algorithmicelsif\ ##2\ \algorithmicthen%
+\ALC@com{##1}\begin{ALC@if}}
+  \newcommand{\FOR}[2][default]{\ALC@it\algorithmicfor\ ##2\ \algorithmicdo%
+\ALC@com{##1}\begin{ALC@for}}
+  \newcommand{\FORALL}[2][default]{\ALC@it\algorithmicforall\ ##2\ %
+\algorithmicdo%
+\ALC@com{##1}\begin{ALC@for}}
+  \newcommand{\SHORTFORALL}[2]{\ALC@it\algorithmicforall\ ##1\ %
+    \algorithmicdo\ {##2}}
+  \newcommand{\WHILE}[2][default]{\ALC@it\algorithmicwhile\ ##2\ %
+\algorithmicdo%
+\ALC@com{##1}\begin{ALC@whl}}
+  \newcommand{\LOOP}[1][default]{\ALC@it\algorithmicloop%
+\ALC@com{##1}\begin{ALC@loop}}
+%changed by alex smola
+  \newcommand{\FUNCTION}[2][default]{\ALC@it\algorithmicfunction\ ##2\ %
+    \ALC@com{##1}\begin{ALC@func}}
+  \newcommand{\MAIN}[2][default]{\ALC@it\algorithmicmain\ ##2\ %
+    \ALC@com{##1}\begin{ALC@main}}
+%end changed by alex smola
+  \newcommand{\REPEAT}[1][default]{\ALC@it\algorithmicrepeat%
+    \ALC@com{##1}\begin{ALC@rpt}}
+    \newcommand{\UNTIL}[1]{\end{ALC@rpt}\ALC@it\algorithmicuntil\ ##1}
+  \ifthenelse{\boolean{ALC@noend}}{
+    \newcommand{\ENDIF}{\end{ALC@if}}
+    \newcommand{\ENDFOR}{\end{ALC@for}}
+    \newcommand{\ENDWHILE}{\end{ALC@whl}}
+    \newcommand{\ENDLOOP}{\end{ALC@loop}}
+    \newcommand{\ENDFUNCTION}{\end{ALC@func}}
+    \newcommand{\ENDMAIN}{\end{ALC@main}}
+  }{
+    \newcommand{\ENDIF}{\end{ALC@if}\ALC@it\algorithmicendif}
+    \newcommand{\ENDFOR}{\end{ALC@for}\ALC@it\algorithmicendfor}
+    \newcommand{\ENDWHILE}{\end{ALC@whl}\ALC@it\algorithmicendwhile}
+    \newcommand{\ENDLOOP}{\end{ALC@loop}\ALC@it\algorithmicendloop}
+    \newcommand{\ENDFUNCTION}{\end{ALC@func}\ALC@it\algorithmicendfunction}
+    \newcommand{\ENDMAIN}{\end{ALC@main}\ALC@it\algorithmicendmain}
+  } 
+  \renewcommand{\@toodeep}{}
+  \begin{list}{\ALC@lno}{\setcounter{ALC@line}{0}\setcounter{ALC@rem}{0}%
+      \itemsep\z@ \itemindent\z@ \listparindent\z@%
+      \partopsep\z@ \parskip\z@ \parsep\z@%
+      \labelsep 0.5em \topsep 0.2em%
+      \ifthenelse{\equal{#1}{0}}
+      {\labelwidth 0.5em }
+      {\labelwidth  1.2em }
+      \leftmargin\labelwidth \addtolength{\leftmargin}{\labelsep}
+      \ALC@tlm\labelsep
+      }
+    }
+  {\end{list}}
+
+
+
+
+
+
+
+
+
+
+
+
+
+

icml2026/appendix.tex

@@ -0,0 +1,1227 @@
+\section{Theoretical Proofs}
+\label{Appendix:Theoretical_Proofs}
+
+
+\subsection{Temperature--Confidence Relation and Margin Bounds}
+\label{subsec:confidence-relation}
+
+\textbf{Problem and observation.} To reach a target top-class confidence $\hat p\in(0,1)$, how constrained is $T$? Empirically, fixing only $z_M$ leaves $T$ ill-determined; using the \emph{Margin} $m$ yields tight bounds. We now prove this, step by step.
+
+\paragraph{Target confidence equation.} Requiring $p_{\phi,\max}=\hat p$ is equivalent to
+\begin{align}
+\frac{e^{z_M/T}}{\sum_{j=1}^K e^{z_j/T}}&=\hat p
+&&\Longleftrightarrow&
+\sum_{j\neq M} e^{(z_j-z_M)/T}&=\frac{1}{\hat p}-1\ :=\ S. 
+\label{eq:temp-conf}
+\end{align}
+Because $\hat p$ is the \emph{maximum} softmax probability, $\hat p\ge 1/K$, hence $S\le K{-}1$; moreover $S>0$ since $\hat p<1$. Thus $S\in(0,K{-}1]$, and if we assume a strict top-1 margin $m>0$ (no top-2 ties) then $\hat p>1/K$ and $S\in(0,K{-}1)$.
+
+\paragraph{Unboundedness if only $z_M$ is known.}
+Assume $z_j-z_M=-\delta$ for all $j\neq M$ with $\delta>0$. Then
+\begin{align}
+\sum_{j\neq M} e^{(z_j-z_M)/T}
+&=\sum_{j\neq M} e^{-\delta/T} 
+=(K-1)\,e^{-\delta/T}
+=S, \label{eq:unb-1}\\
+\Rightarrow\quad T&=-\frac{\delta}{\log\!\left(\frac{S}{K-1}\right)}.
+\label{eq:unb-2}
+\end{align}
+When $S<K{-}1$ (equivalently $\hat p>1/K$), $\log(S/(K{-}1))<0$, so as $\delta\in(0,\infty)$ varies, \eqref{eq:unb-2} sweeps $T\in(0,\infty)$. Thus
+\[
+\boxed{\text{Fixing }z_M\text{ alone leaves the feasible }T\text{ unbounded: }(0,\infty).}
+\]
+
+\paragraph{$m$-boundedness.}
+Let $\mu\in\arg\max_{j\neq M} z_j$ denote a runner-up index and define the (nonnegative) margin $m:=z_M-z_\mu$. For any $j\notin\{M,\mu\}$,
+\[
+z_j-z_M\ \le\ z_\mu-z_M\ =\ -m
+\quad\Longrightarrow\quad
+e^{(z_j-z_M)/T}\ \le\ e^{-m/T}.
+\]
+Since $e^{(z_\mu-z_M)/T}=e^{-m/T}$, we have
+\begin{align}
+\sum_{j\neq M} e^{(z_j-z_M)/T}
+&= e^{-m/T} + \sum_{j\notin\{M,\mu\}} e^{(z_j-z_M)/T} \nonumber\\
+&\ge e^{-m/T}, \label{eq:gb-lower}\\
+\sum_{j\neq M} e^{(z_j-z_M)/T}
+&\le e^{-m/T} + (K-2)\,e^{-m/T}
+= (K-1)\,e^{-m/T}. \label{eq:gb-upper}
+\end{align}
+Combining \eqref{eq:temp-conf}, \eqref{eq:gb-lower}, and \eqref{eq:gb-upper} yields
+\begin{equation}
+e^{-m/T}\ \le\ S\ \le\ (K-1)\,e^{-m/T},
+\qquad S=\tfrac{1}{\hat p}-1\in(0,K{-}1].
+\label{eq:gb-ineq-1}
+\end{equation}
+Equivalently,
+\begin{equation}
+-\log\!\Big(\tfrac{S}{K-1}\Big)\ \ge\ \frac{m}{T}\ \ge\ -\log S.
+\label{eq:gb-ineq-5}
+\end{equation}
+Solving \eqref{eq:gb-ineq-5} for $T>0$ gives the $m$-bounded feasible set:
+\[
+\boxed{
+\begin{cases}
+\displaystyle
+T\ \in\ \Bigl[\,\frac{m}{-\log\!\bigl(S/(K-1)\bigr)}\,,\,\frac{m}{-\log S}\,\Bigr],
+& \text{if } 0<S<1\ \ (\hat p>1/2),\\[1.2ex]
+\displaystyle
+T\ \in\ \Bigl[\,\frac{m}{-\log\!\bigl(S/(K-1)\bigr)}\,,\,\infty\Bigr),
+& \text{if } 1\le S<K-1\ \ (1/K<\hat p\le 1/2),\\[1.2ex]
+\text{feasible iff } m=0, & \text{if } S=K-1\ \ (\hat p=1/K).
+\end{cases}
+}
+\]
+\emph{Interpretation.} Knowing the Margin $m$ pins down $T$ tightly. When the target confidence is above $1/2$, the feasible $T$ is a \emph{finite} interval whose width shrinks as $m$ grows. For lower target confidences ($\hat p\le 1/2$), one still gets a nontrivial \emph{lower} bound on $T$; a finite upper bound appears exactly when $S<1$ (i.e., $\hat p>1/2$). In particular, for $K=2$ the bounds coincide and $T=\frac{m}{-\log S}$ is uniquely determined.
+
+
+
+
+\subsection{Charbonnier--SoftECE Upper-Bounds smCE}
+\label{subsec:Charbonnier-soft-controls-smce}
+
+\paragraph{Setup and goal.}
+Let $p(x)\in[0,1]$ denote the predicted probability of correctness (top-class confidence) and
+$a(x):=\mathbb{1}\{\hat y(x)=y\}\in\{0,1\}$ the correctness indicator.
+We measure calibration via the \emph{smooth calibration error} (smCE),
+the worst-case correlation between the residual $a(X)-p(X)$ and any $1$-Lipschitz probe of the prediction $p(X)$ (cf.\ forecasting(\!\cite{kakade2004deterministic}) and the ML calibration view of \cite{blasiok2023does}):
+\begin{align*}
+\mathrm{smCE}(f) &:= \sup_{\varphi\in\mathcal H}\;\Big|\;\mathbb{E}\big[(a(X)-p(X))\,\varphi\!\big(p(X)\big)\big]\;\Big|, \\
+\mathcal H &:= \big\{\varphi:[0,1]\to[-1,1]\ \text{s.t.}\ \mathrm{Lip}(\varphi)\le 1\big\}.
+\end{align*}
+We study the \emph{Charbonnier--SoftECE} objective (a smoothed, Huberized absolute calibration error):
+\[
+\mathcal{H}_{\lambda,\delta}(f)
+\;:=\;
+\E_{X}\!\left[
+  \int_{0}^{1}
+    K_\lambda\!\big(p(X),u\big)\,\phi_\delta\!\big(a(X)-u\big)\;\rho(u)\,du
+\right],
+\qquad
+\phi_\delta(r):=\sqrt{r^2+\delta^2},
+\]
+where $\rho$ is a reference density on $[0,1]$ and
+\begin{equation}
+\label{eq:kernel-normalization-rev}
+K_\lambda(p,u)\;=\;\frac{k_\lambda(p-u)}{\int_0^1 k_\lambda(p-v)\,\rho(v)\,dv},
+\qquad
+k_\lambda(t):=e^{-\lambda t^2},\ \ \lambda>0,
+\end{equation}
+so that $\int_0^1 K_\lambda(p,u)\,\rho(u)\,du=1$ for every $p\in[0,1]$.
+Assume \emph{boundedness and bounded-away-from-zero} of $\rho$: there exist constants $0<\rho_{\min}\le \rho(u)\le\rho_{\max}<\infty$ for all $u\in[0,1]$, and write $\kappa:=\rho_{\max}/\rho_{\min}$.
+
+\paragraph{Main result.}
+\begin{theorem}[Charbonnier--SoftECE upper-bounds smCE]
+\label{thm:Charbonnier-soft-controls-smce-rev}
+Under the assumptions above, for all classifiers $f$ and all $\delta\ge 0$,
+\begin{equation}
+\label{eq:master-bound-rev}
+\mathrm{smCE}(f)\ \le\ \mathcal{H}_{\lambda,\delta}(f)\ +\ 2\,B_\lambda,
+\qquad
+B_\lambda\;:=\;\sup_{p\in[0,1]}\int_0^1 |p-u|\,K_\lambda(p,u)\,\rho(u)\,du.
+\end{equation}
+Moreover, for the Gaussian kernel $k_\lambda(t)=e^{-\lambda t^2}$,
+\begin{equation}
+\label{eq:Blambda-explicit}
+B_\lambda\ \le\ \min\!\left\{\,1,\ \frac{2\,\kappa}{\sqrt{\pi}}\cdot\frac{1}{\sqrt{\lambda}\ \mathrm{erf}(\sqrt{\lambda})}\right\},
+\end{equation}
+and in particular for $\lambda\ge 1$,
+\begin{equation}
+\label{eq:Blambda-simplified}
+B_\lambda\ \le\ \frac{C_\kappa}{\sqrt{\lambda}},
+\qquad
+C_\kappa\;:=\;\frac{2\,\kappa}{\sqrt{\pi}\ \mathrm{erf}(1)}\ \approx\ 1.339\,\kappa.
+\end{equation}
+\end{theorem}
+
+\paragraph{Proof.}
+For brevity write $p:=p(X)$ and $a:=a(X)$.
+Fix any $\varphi\in\mathcal H$ with $\|\varphi\|_\infty\le 1$.
+Introduce the (normalized) kernel smoothing operator
+\[
+(\mathsf T_\lambda\varphi)(p)\ :=\ \int_0^1 K_\lambda(p,u)\,\varphi(u)\,\rho(u)\,du.
+\]
+
+\emph{Decomposition.}
+\begin{align}
+\label{eq:decomp-rev}
+\mathbb{E}[(a-p)\,\varphi(p)]
+\;=\;
+\mathbb{E}[(a-p)\,(\mathsf T_\lambda\varphi)(p)]
+\;+\;
+\mathbb{E}\big[(a-p)\,\{\varphi(p)-(\mathsf T_\lambda\varphi)(p)\}\big].
+\end{align}
+
+\emph{Approximation (mollification) error.}
+By $\mathrm{Lip}(\varphi)\le 1$ and the triangle inequality,
+\begin{align*}
+\big|\varphi(p)-(\mathsf T_\lambda\varphi)(p)\big|
+&=\left|\int_0^1 K_\lambda(p,u)\,\{\varphi(p)-\varphi(u)\}\,\rho(u)\,du\right|
+\ \le\ \int_0^1 K_\lambda(p,u)\,|p-u|\,\rho(u)\,du.
+\end{align*}
+Taking the supremum over $p$ and $\varphi$ yields
+\begin{equation}
+\label{eq:epsLEB}
+\sup_{\varphi\in\mathcal H}\ \sup_{p\in[0,1]}\big|\varphi(p)-(\mathsf T_\lambda\varphi)(p)\big|\ \le\ B_\lambda.
+\end{equation}
+Hence, using $|a-p|\le 1$,
+\[
+\Big|\mathbb{E}\big[(a-p)\,\{\varphi(p)-(\mathsf T_\lambda\varphi)(p)\}\big]\Big|
+\ \le\ B_\lambda.
+\]
+
+\emph{Aligned main term.}
+By Fubini/Tonelli (bounded integrands) and normalization $\int K_\lambda(p,u)\rho(u)\,du=1$,
+\[
+\mathbb{E}[(a-p)\,(\mathsf T_\lambda\varphi)(p)]
+=\int_0^1 \varphi(u)\,\mathbb{E}\big[(a-p)\,K_\lambda(p,u)\big]\ \rho(u)\,du,
+\]
+and since $|\varphi(u)|\le 1$,
+\begin{equation}
+\label{eq:main-aligned}
+\big|\mathbb{E}[(a-p)\,(\mathsf T_\lambda\varphi)(p)]\big|
+\ \le\ \int_0^1 \Big|\,\mathbb{E}\big[(a-p)\,K_\lambda(p,u)\big]\,\Big|\,\rho(u)\,du.
+\end{equation}
+For each $u$, using $|x+y|\le |x|+|y|$ and $|a-p|\le |a-u|+|p-u|$, together with $\phi_\delta(r)\ge |r|$,
+\begin{align}
+\label{eq:triangle-rev}
+\Big|\,\mathbb{E}\big[(a-p)\,K_\lambda(p,u)\big]\,\Big|
+&\le \mathbb{E}\big[\,|a-p|\,K_\lambda(p,u)\big]\notag\\
+&\le\ \mathbb{E}\big[\,\phi_\delta(a-u)\,K_\lambda(p,u)\big]\ +\ \mathbb{E}\big[\,|p-u|\,K_\lambda(p,u)\big].
+\end{align}
+Integrating \eqref{eq:triangle-rev} against $\rho(u)\,du$ and applying Fubini,
+\begin{align*}
+\int_0^1 \Big|\,\mathbb{E}\big[(a-p)\,K_\lambda(p,u)\big]\,\Big|\,\rho(u)\,du
+&\ \le\ \mathbb{E}\!\left[\int_0^1 K_\lambda(p,u)\,\phi_\delta(a-u)\,\rho(u)\,du\right]\\
+&\quad\ +\ \sup_{p}\int_0^1 |p-u|\,K_\lambda(p,u)\,\rho(u)\,du,
+\end{align*}
+i.e.,
+\begin{equation}
+\label{eq:aligned-bdd}
+\big|\mathbb{E}[(a-p)\,(\mathsf T_\lambda\varphi)(p)]\big|
+\ \le\ \mathcal{H}_{\lambda,\delta}(f)\ +\ B_\lambda.
+\end{equation}
+
+\emph{Conclusion.}
+Combining \eqref{eq:epsLEB}, \eqref{eq:aligned-bdd} with \eqref{eq:decomp-rev} and taking the supremum over $\varphi\in\mathcal H$ gives
+$\mathrm{smCE}(f)\le \mathcal{H}_{\lambda,\delta}(f)+2B_\lambda$.
+
+\medskip
+\noindent\emph{Explicit bounds for $B_\lambda$.}
+By definition,
+\[
+B_\lambda=\sup_{p\in[0,1]}\ \frac{\int_0^1 |p-u|\,k_\lambda(p-u)\,\rho(u)\,du}{\int_0^1 k_\lambda(p-v)\,\rho(v)\,dv}.
+\]
+Using $\rho(u)\le \rho_{\max}$ in the numerator and $\rho(v)\ge \rho_{\min}$ in the denominator, and changing variables $t=p-u$ or $t=p-v$, we obtain for all $p\in[0,1]$:
+\[
+B_\lambda
+\ \le\ \kappa\cdot
+\frac{\int_{\mathbb{R}} |t|\,e^{-\lambda t^2}\,dt}
+     {\int_{p-1}^{p} e^{-\lambda t^2}\,dt}.
+\]
+Since $p\in[0,1]$, the denominator integrates over a length-$1$ interval contained in $[-1,1]$; by symmetry and unimodality of $t\mapsto e^{-\lambda t^2}$, the minimum over such intervals is attained at an endpoint, e.g.\ $[0,1]$. Hence
+\[
+B_\lambda
+\ \le\ \kappa\cdot
+\frac{\int_{\mathbb{R}} |t|\,e^{-\lambda t^2}\,dt}
+     {\int_{0}^{1} e^{-\lambda t^2}\,dt}
+\ =\ \kappa\cdot
+\frac{\tfrac{1}{\lambda}}
+     {\tfrac{\sqrt{\pi}}{2\sqrt{\lambda}}\ \mathrm{erf}(\sqrt{\lambda})}
+\ =\ \frac{2\,\kappa}{\sqrt{\pi}}\cdot\frac{1}{\sqrt{\lambda}\ \mathrm{erf}(\sqrt{\lambda})}.
+\]
+Since $|p-u|\le 1$ and $\int K_\lambda\rho=1$, we also have $B_\lambda\le 1$.
+For $\lambda\ge 1$, $\mathrm{erf}(\sqrt{\lambda})\ge \mathrm{erf}(1)$, yielding \eqref{eq:Blambda-simplified}.
+\hfill$\square$
+
+\paragraph{Interpretation and guidance.}
+The guarantee \eqref{eq:master-bound-rev} decomposes into a \emph{model-dependent} term $\mathcal{H}_{\lambda,\delta}(f)$ and a \emph{design-only} kernel bias $B_\lambda$, the average \emph{soft-bin radius} around $p$. The Charbonnier envelope obeys $\phi_\delta(r)\ge |r|$, so replacing $|a-u|$ with $\phi_\delta(a-u)$ never weakens control of smCE and yields smooth gradients near $r=0$. For Gaussian kernels, $B_\lambda=\mathcal{O}(\kappa/\sqrt{\lambda})$ as in \eqref{eq:Blambda-simplified}, so increasing $\lambda$ monotonically tightens the bound; the cap $B_\lambda\le 1$ ensures uniform validity for all $\lambda>0$.
+(Discrete soft-binned implementations—via Riemann-sum quadrature of the $u$-integral—inherit the same inequality up to a standard design-only quadrature error that vanishes as the grid is refined.)
+
+
+
+
+\subsection{Charbonnier--SoftECE vs.\ NLL}
+\label{subsec:Charbonnier-vs-nll}
+
+We compare negative log-likelihood (NLL) with Charbonnier--SoftECE within the SMART family
+$T(x)=h(m(x))$ that scales by the \emph{margin}
+$m(x):=z_{(1)}(x)-z_{(2)}(x)\in\R_{\ge 0}$.
+Throughout, assume
+(i) $T(x)\in[T_{\min},T_{\max}]$ with $0<T_{\min}\le T_{\max}<\infty$,
+(ii) $\mathbb{E}\|z(X)\|_\infty<\infty$, and
+(iii) $\mathbb{P}(z_{(1)}=z_{(2)})=0$ (no top-2 ties a.s.).
+Write $t(x):=z(x)/T(x)$, $q(x):=\softmax(t(x))$, $M(x):=\arg\max_k t_k(x)$,
+$p(x):=q_{M(x)}(x)\in(0,1)$, and $Y^{\top}(x):=\mathbf{1}\{Y(x)=M(x)\}$.
+Define the pointwise top-class probability $r_X(x):=\mathbb{P}\!\big(Y=M(x)\mid X=x\big)$ and the \emph{reliability} curve $r(p):=\mathbb{E}\!\big[r_X(X)\mid p(X)=p\big]$.
+We measure calibration by the smooth calibration error (smCE)
+\[
+\mathrm{smCE}(f):=\sup_{\substack{\varphi:[0,1]\to[-1,1]\\ \mathrm{Lip}(\varphi)\le 1}}
+\big|\mathbb{E}[(Y^{\top}-p)\varphi(p)]\big|.
+\]
+
+\paragraph{Charbonnier--SoftECE and its smCE control.}
+Charbonnier--SoftECE is the objective
+\[
+\mathcal{H}_{\lambda,\delta}(f)
+\ :=\
+\E_{X}\!\left[\int_0^1 K_\lambda\!\big(p(X),u\big)\,
+\phi_\delta\!\big(Y^{\top}(X)-u\big)\,\rho(u)\,du\right],
+\qquad \phi_\delta(r):=\sqrt{r^2+\delta^2},
+\]
+with normalized kernel $K_\lambda$ as in \eqref{eq:kernel-normalization-rev} and a reference density
+$\rho$ on $[0,1]$.
+We \emph{use} (proved in Sec.~\ref{subsec:Charbonnier-soft-controls-smce}) the smCE control
+\begin{equation}
+\label{eq:smce-Charbonnier-master-again-nll}
+\mathrm{smCE}(f)\ \le\ \mathcal{H}_{\lambda,\delta}(f)\ +\ 2\,B_\lambda,
+\qquad
+B_\lambda\ =\ \sup_{p}\int_0^1 |p-u|\,K_\lambda(p,u)\,\rho(u)\,du,
+\end{equation}
+with $B_\lambda=\mathcal{O}(\kappa/\sqrt{\lambda})$ for Gaussian kernels.
+
+\paragraph{A SMART-feasible local scaling path.}
+Fix a Borel margin slice $G\subset\R_{\ge 0}$ and $A:=\{x:\ m(x)\in G\}$.
+For $s>0$, define the local scaling
+\begin{align*}
+T_s(x) &:= \begin{cases}T(x)/s,& x\in A,\\ T(x),& x\notin A,\end{cases} \\
+t_s(x) &:= \frac{z(x)}{T_s(x)} = \begin{cases}s\,t(x),& x\in A,\\ t(x),& x\notin A,\end{cases} \\
+q^{(s)} &:= \softmax(t_s), \quad p_s := q^{(s)}_{M}.
+\end{align*}
+Because uniform multiplication by $s>0$ preserves coordinate ordering, $M$ is unchanged for all $s>0$; (iii) rules out measure-zero ties at the boundary.
+
+\begin{lemma}[Directional derivatives under local margin-dependent scaling]
+\label{lem:dir-derivatives-nll}
+Let $L_{\mathrm{nll}}(h):=\mathbb{E}[-\log q_Y(X)]$.
+For any $C^1$ probe $\psi:[0,1]\to\R$ with $\mathrm{Lip}(\psi)\le 1$ and $\|\psi\|_\infty\le 1$,
+the Gâteaux derivatives at $s=1$ exist and
+\begin{align}
+\left.\frac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1}
+&=\ \mathbb{E}\!\big[\mathbf{1}_A\,(\langle t\rangle_q - t_Y)\big],
+\label{eq:dir-NLL}\\[-0.3em]
+\left.\frac{d}{ds}\,\mathbb{E}\big[(Y^{\top}-p_s)\psi(p_s)\big]\right|_{s=1}
+&=\ \mathbb{E}\!\Big[\mathbf{1}_A\;p\,\big(t_M-\langle t\rangle_q\big)\,
+\big(\psi'(p)\,(r_X-p)-\psi(p)\big)\Big],
+\label{eq:dir-calib-nll}
+\end{align}
+where $\langle t\rangle_q:=\sum_k q_k\,t_k$ and $r_X:=r_X(X)$.
+\end{lemma}
+
+\begin{proof}
+On $A$, $\partial_s q_k^{(s)}=q_k^{(s)}(t_k-\langle t\rangle_{q^{(s)}})$, hence
+$\partial_s(-\log q_Y^{(s)})=\langle t\rangle_{q^{(s)}}-t_Y$. Outside $A$ the derivative vanishes.
+Dominated convergence applies since $\big|\partial_s(-\log q_Y^{(s)})\big|\le 2\|t\|_\infty$ and $\mathbb{E}\|t\|_\infty\le \mathbb{E}\|z\|_\infty/T_{\min}<\infty$, yielding \eqref{eq:dir-NLL}.
+For $F_\psi(s):=\mathbb{E}[(Y^{\top}-p_s)\psi(p_s)]$, with $M$ fixed,
+$\partial_s p_s=\partial_s q_M^{(s)}=q_M^{(s)}(t_M-\langle t\rangle_{q^{(s)}})=p_s(t_M-\langle t\rangle_{q^{(s)}})$.
+Thus
+\[
+\partial_s\big((Y^{\top}-p_s)\psi(p_s)\big)
+=\big(-\psi(p_s)+(Y^{\top}-p_s)\psi'(p_s)\big)\,\partial_s p_s.
+\]
+Conditioning on $X$ replaces $Y^{\top}$ by $r_X(X)$, whence \eqref{eq:dir-calib-nll} at $s=1$ after integration; dominated convergence holds because $p\,|t_M-\langle t\rangle_q|\le 2\|t\|_\infty$ and $|\psi'|\le 1$, $|\psi|\le 1$.
+\end{proof}
+
+\begin{lemma}[Margin lower bound for the top-logit advantage]
+\label{lem:margin-advantage-nll}
+On $\{M=\arg\max t\}$,
+\begin{equation}
+t_M-\langle t\rangle_q \ \ge\ (1-p)\,\big(t_M-t_{(2)}\big)
+\ =\ (1-p)\,\frac{m}{T}.
+\label{eq:adv-lb-margin-nll}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+$\langle t\rangle_q=p\,t_M+\sum_{j\neq M}q_j t_j\le p\,t_M+(1-p)\,t_{(2)}$; rearrange.
+\end{proof}
+
+\paragraph{A correct NLL directional upper bound (multi-class).}
+Define the \emph{runner-up gap} $g(x):=t_{(1)}(x)-t_{(2)}(x)=m(x)/T(x)\ge 0$
+and the \emph{non-top spread} $\Delta(x):=t_{(2)}(x)-t_{(K)}(x)\ge 0$.
+For any $x$ with predicted index $M$ and confidence $p=q_M(x)$,
+\begin{equation}
+\label{eq:NLL-upper-correct}
+\mathbb{E}\!\left[\langle t\rangle_q-t_Y\mid X=x\right]
+\ \le\ \big(p-r_X(x)\big)\,g(x)\ +\ \big(1-r_X(x)\big)\,\Delta(x).
+\end{equation}
+In particular, for binary classification ($K=2$) one has $\Delta\equiv 0$ and \eqref{eq:NLL-upper-correct} reduces to
+$\mathbb{E}[\langle t\rangle_q-t_Y\mid X]\ =\ (p-r_X)\,g$ (exact).
+
+\begin{proof}[Derivation of \eqref{eq:NLL-upper-correct}]
+With $\eta_k(x):=\mathbb{P}(Y{=}k\mid X{=}x)$,
+\[
+\mathbb{E}\!\left[\langle t\rangle_q-t_Y\mid X=x\right]
+=\sum_k (q_k-\eta_k)\,t_k
+=(p-r_X)\,(t_M-t_{(2)})+\sum_{j\neq M}(q_j-\eta_j)\,(t_j-t_{(2)}).
+\]
+Since $t_j-t_{(2)}\le 0$ and $\sum_{j\neq M}(\eta_j-q_j)_+\le \sum_{j\neq M}\eta_j=1-r_X$, the last sum is $\le (1-r_X)\,(t_{(2)}-t_{(K)})=(1-r_X)\,\Delta$.
+\end{proof}
+
+\paragraph{Consequences and a mild spread control.}
+On a slice $A=\{m\in G\}$, assume the empirically checkable \emph{spread control}
+\begin{equation}
+\label{eq:spread-control-nll}
+\Delta(x)\ \le\ \Delta_G\ <\ \infty\qquad \text{for all }x\in A.
+\end{equation}
+Then, combining \eqref{eq:NLL-upper-correct} with Lemma~\ref{lem:dir-derivatives-nll},
+\begin{align}
+\left.\frac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1}
+&=\mathbb{E}\big[\mathbf{1}_A(\langle t\rangle_q-t_Y)\big]
+\ \le\ \mathbb{E}\!\big[\mathbf{1}_A \,(p-r_X)\,g\big]\ +\ \Delta_G\,\mu_A,
+\label{eq:NLL-signed-with-spread}
+\end{align}
+where $\mu_A:=\mathbb{P}\{X\in A\}$.
+In the binary case $\Delta\equiv 0$ and \eqref{eq:NLL-signed-with-spread} holds with equality.
+
+\paragraph{Two-slice \emph{mismatch} under mild, empirically observed heterogeneity.}
+We next give conditions under which a single SMART-feasible local move reduces NLL yet \emph{increases} smCE.
+
+\emph{Assumptions (empirically checkable).}
+Fix a \emph{compact} margin slice $G\subset[m_{\min},m_{\max}]$ and set $A:=\{x:\ m(x)\in G\}$.
+Let $\gamma_{\min}:=\inf_{x\in A}\frac{m(x)}{T(x)}$ and $\gamma_{\max}:=\sup_{x\in A}\frac{m(x)}{T(x)}$ (finite and positive by $G$ compact and $T\in[T_{\min},T_{\max}]$).
+Assume there exist disjoint compact intervals $J_{\mathrm{U}},J_{\mathrm{O}}\subset(p_0,1)$ with gap $\Delta>0$ and constants $\rho_{\mathrm{U}},\rho_{\mathrm{O}}>0$ such that
+\[
+r(p)-p\ \ge\ \rho_{\mathrm{U}}\quad\text{for }p\in J_{\mathrm{U}},\qquad
+r(p)-p\ \le\ -\,\rho_{\mathrm{O}}\quad\text{for }p\in J_{\mathrm{O}}.
+\]
+Write
+$\mu_{\mathrm{U}}:=\mathbb{P}\{\,p\in J_{\mathrm{U}},\ x\in A\,\}$,
+$\mu_{\mathrm{O}}:=\mathbb{P}\{\,p\in J_{\mathrm{O}},\ x\in A\,\}$,
+$\mu_{\mathrm{gap}}:=\mathbb{P}\{\,x\in A,\ p\notin J_{\mathrm{U}}\cup J_{\mathrm{O}}\,\}$,
+and assume additionally:
+\begin{itemize}
+\item[(a)] (\emph{bounded conditional density of $p$ on $A$}) the conditional distribution of $p$ given $X\in A$ has a density $f_{p|A}$ on $(0,1)$ with $\|f_{p|A}\|_\infty\le D_G<\infty$. In particular, for any interval $I\subset(0,1)$, $\mathbb{P}\{p\in I,\,X\in A\}\le D_G\,|I|$.
+\item[(b)] (\emph{slice-bounded advantage}) there exists $C_G<\infty$ with
+$p(x)\big(t_M(x)-\langle t(x)\rangle_{q(x)}\big)\le C_G$ for $x\in A$ (e.g., it holds with $C_G:=2\operatorname*{ess\,sup}_{x\in A}\|t(x)\|_\infty$ whenever $t$ is essentially bounded on $A$).
+\item[(c)] the spread control \eqref{eq:spread-control-nll} holds on $A$ with constant $\Delta_G$.
+\end{itemize}
+
+\begin{proposition}[Two-slice mismatch: NLL $\downarrow$ but smCE $\uparrow$ (multi-class)]
+\label{prop:two-slice-mismatch-nll}
+Consider the \emph{sharpening} direction $s\uparrow 1$ applied on the SMART-feasible set $A=\{m\in G\}$.
+If
+\begin{equation}
+\label{eq:NLL-mismatch-condition}
+\rho_{\mathrm{U}}\,\gamma_{\min}\,\mu_{\mathrm{U}}
+\ >\ \gamma_{\max}\,\big(\rho_{\mathrm{O}}\,\mu_{\mathrm{O}}+\mu_{\mathrm{gap}}\big)\ +\ \Delta_G\,\mu_A,
+\end{equation}
+then $\left.\tfrac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1}<0$ (NLL strictly decreases).
+Moreover, for any $c\in(0,\min\{1,\Delta\})$ there exists a $1$-Lipschitz probe $\psi$ with
+$\psi\equiv 0$ on $J_{\mathrm{U}}$, $\psi\equiv -c$ on $J_{\mathrm{O}}$, and with transitions confined to a band whose $A$-mass is at most $\varepsilon>0$, such that
+\begin{equation}
+\label{eq:calib-mismatch-deriv-nll}
+\left.\frac{d}{ds}\mathbb{E}[(Y^{\top}-p_s)\psi(p_s)]\right|_{s=1}
+\ \ge\ c\,\underline{p}_{\mathrm{O}}\,(1-\overline{p}_{\mathrm{O}})\,\gamma_{\min}\,\mu_{\mathrm{O}}
+\ -\ (1+c)\,C_G\,\varepsilon,
+\end{equation}
+where $\underline{p}_{\mathrm{O}}:=\inf J_{\mathrm{O}}$ and $\overline{p}_{\mathrm{O}}:=\sup J_{\mathrm{O}}$.
+Choosing $\varepsilon<\tfrac{c\,\underline{p}_{\mathrm{O}}(1-\overline{p}_{\mathrm{O}})\gamma_{\min}}{(1+c)\,C_G}\,\mu_{\mathrm{O}}$ makes the right-hand side strictly positive. Because on $J_{\mathrm{O}}$ one has $(r(p)-p)\psi(p)\ge c\,\rho_{\mathrm{O}}$ while $\psi\equiv 0$ on $J_{\mathrm{U}}$, the signed functional at $s=1$ obeys
+\begin{equation}
+\label{eq:baseline-positive-nll}
+\mathbb{E}[(Y^{\top}-p)\psi(p)]
+\ =\ \mathbb{E}[(r(p)-p)\psi(p)]
+\ \ge\ c\,\rho_{\mathrm{O}}\,\mu_{\mathrm{O}}\ -\ c\,\varepsilon\ >\ 0,
+\end{equation}
+so a positive derivative implies a strict increase of its absolute value. Hence smCE strictly increases along $s\uparrow 1$.
+\end{proposition}
+
+\begin{proof}
+By \eqref{eq:NLL-signed-with-spread} and splitting $A$ into the three regions,
+\begin{align*}
+\left.\frac{d}{ds}L_{\mathrm{nll}}(h_s)\right|_{s=1} &\le \mathbb{E}\!\big[\mathbf{1}_{A\cap\{p\in J_{\mathrm{U}}\}}(p-r_X)g\big] \\
+&\quad + \mathbb{E}\!\big[\mathbf{1}_{A\cap\{p\in J_{\mathrm{O}}\}}(p-r_X)g\big] \\
+&\quad + \mathbb{E}\!\big[\mathbf{1}_{A\cap\{p\notin J_{\mathrm{U}}\cup J_{\mathrm{O}}\}}(p-r_X)g\big] \\
+&\quad + \Delta_G\,\mu_A.
+\end{align*}
+On $A\cap\{p\in J_{\mathrm{U}}\}$, $g\ge \gamma_{\min}$ and
+$\mathbb{E}[p-r_X\mid p]=p-r(p)\le -\,\rho_{\mathrm{U}}$,
+hence the contribution is $\le -\,\rho_{\mathrm{U}}\gamma_{\min}\mu_{\mathrm{U}}$.
+On $A\cap\{p\in J_{\mathrm{O}}\}$, $g\le \gamma_{\max}$ and $\mathbb{E}[p-r_X\mid p]\ge \rho_{\mathrm{O}}$, giving at most $\gamma_{\max}\rho_{\mathrm{O}}\mu_{\mathrm{O}}$.
+On the gap region, $|p-r_X|\le 1$ and $g\le \gamma_{\max}$, giving at most $\gamma_{\max}\,\mu_{\mathrm{gap}}$.
+This yields strict negativity under \eqref{eq:NLL-mismatch-condition}.
+For the probe, on $J_{\mathrm{O}}$ we have $\psi'(p)=0$ and $-\psi(p)=c$,
+so by \eqref{eq:dir-calib-nll} and Lemma~\ref{lem:margin-advantage-nll},
+\[
+\left.\frac{d}{ds}\mathbb{E}[(Y^{\top}-p_s)\psi(p_s)]\right|_{s=1,\,p\in J_{\mathrm{O}}}
+\ \ge\ c\,\underline{p}_{\mathrm{O}}\,(1-\overline{p}_{\mathrm{O}})\,\gamma_{\min}.
+\]
+On $J_{\mathrm{U}}$ the contribution is $0$ since $\psi\equiv 0$.
+On the transition band (of $A$-mass $\varepsilon$), $|\psi'|\le 1$ and $|\psi|\le c$,
+hence $|\psi'(p)(r_X-p)-\psi(p)|\le (1+c)$ while $p(t_M-\langle t\rangle_q)\le C_G$ on $A$ by (b).
+Thus the transition contribution is at most $(1+c)\,C_G\,\varepsilon$ in magnitude, giving \eqref{eq:calib-mismatch-deriv-nll}.
+Finally, \eqref{eq:baseline-positive-nll} holds since $\psi$ depends only on $p$ and $\mathbb{E}[Y^{\top}-p\mid p]=r(p)-p$.
+By (a), we can realize the $1$-Lipschitz $\psi$ with linear ramps of total width at most $2c$, whence $\varepsilon\le 2D_G c$; shrinking $c$ if needed makes the stated choice of $\varepsilon$ feasible.
+\end{proof}
+
+\begin{lemma}[Small-$s$ realization for the mismatch]
+\label{lem:small-s-nll}
+Under Proposition~\ref{prop:two-slice-mismatch-nll}, there exists $s^\uparrow>1$ arbitrarily close to $1$ with
+\[
+L_{\mathrm{nll}}(h_{s^\uparrow})\ <\ L_{\mathrm{nll}}(h)
+\qquad\text{and}\qquad
+\mathrm{smCE}(f_{h_{s^\uparrow}})\ >\ \mathrm{smCE}(f_{h}).
+\]
+\end{lemma}
+
+\begin{proof}
+$L_{\mathrm{nll}}(h_s)$ is $C^1$ at $s=1$ by Lemma~\ref{lem:dir-derivatives-nll}, with strictly negative derivative; hence $L_{\mathrm{nll}}(h_{s^\uparrow})<L_{\mathrm{nll}}(h)$ for all $s^\uparrow>1$ sufficiently close to $1$.
+For smCE, fix the $\psi$ from Proposition~\ref{prop:two-slice-mismatch-nll}; then $F_\psi(s):=\mathbb{E}[(Y^{\top}-p_s)\psi(p_s)]$ is $C^1$ with $F_\psi(1)>0$ and $F_\psi'(1)>0$, so $|F_\psi(s^\uparrow)|>|F_\psi(1)|$ for all $s^\uparrow>1$ close enough to $1$. Since $\mathrm{smCE}(f_{h_s})\ge |F_\psi(s)|$, it follows that $\mathrm{smCE}(f_{h_{s^\uparrow}})>\mathrm{smCE}(f_{h})$.
+\end{proof}
+
+\paragraph{Takeaway.}
+Along SMART-feasible local scalings of the temperature map $T(x)=h(m(x))$, Charbonnier--SoftECE continues to control smCE via \eqref{eq:smce-Charbonnier-master-again-nll}, whereas NLL can be \emph{locally improved} (decreased) while smCE \emph{deteriorates} (increases) under mild, empirically checkable heterogeneity of confidence slices (Proposition~\ref{prop:two-slice-mismatch-nll}). The NLL directional formula is exact in binary classification; in multi-class settings the same conclusion holds under a weak spread control on non-top logits.
+
+
+
+\section{The Use of Large Language Models}
+
+During the preparation of this work, we utilized a Large Language Model (LLM) to assist with editorial refinement of the manuscript. The model's application was limited exclusively to improving textual quality and presentation, not for generating substantive research content. The LLM's contributions included:
+
+\begin{itemize}
+    \item Enhancing sentence structure and paragraph organization to improve clarity, brevity, and scholarly tone.
+    \item Identifying and correcting errors in grammar, spelling, and punctuation.
+    \item Strengthening coherence and smoothing transitions throughout the text.
+\end{itemize}
+
+
+\section{Runtime Efficiency}  
+To verify the time efficiency of our method, we compare the runtime with baseline methods. The result is reported in Table~\ref{tab:imagenet_resnet_runtime}. TS optimizes a single scalar temperature via a few gradient steps or closed‐form updates, then applies this same factor to every logit, resulting in a negligible overhead (2.42\,s). SMART yields a small per‐sample inference cost and hence a modest total runtime (23.03\,s). Logits are input into PTS's small neural network for each sample to predict a bespoke temperature, incurring a larger computational cost than SMART. CTS is the most expensive at more than 1 hour with the highest variance, as it conducts an exhaustive grid search for 5 epochs over a dense temperature grid (e.g. 0.1 -10) for each of the 1 000 classes, leading to $O(C\times G\times N)$ evaluations (classes $\times$ grid points $\times$ samples). The spline‐based calibrator precomputes a monotonic mapping on the validation set and then applies a fast piecewise‐linear transform at test time, yielding intermediate overhead. These differences illustrate the trade‐off between expressive power and efficiency: TS is almost instantaneous, SMART adds only a small network‐forward cost per sample, PTS trades per‐sample flexibility for moderate cost, and CTS’s brute‐force search becomes prohibitive at scale.
+
+\begin{table*}[ht]
+\caption{\textbf{Average Runtime (s) on ImageNet} over 10 runs on a ResNet-50 model.}
+\footnotesize
+\label{tab:imagenet_resnet_runtime}
+\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}lccccc@{}}
+\toprule
+\textbf{Method} & TS & Spline & PTS & CTS & SMART \\
+\midrule
+Runtime (s) &
+$2.42 \pm 0.1$ &
+$28.51 \pm 0.9$ &
+$1050.44 \pm 37.8$ &
+$5457.55 \pm 125.5$ &
+$23.03 \pm 0.41$ \\
+\bottomrule
+\end{tabular*}
+% \vspace{-0.2in}
+\end{table*}
+
+
+% \section{margin Perspective on Calibration}
+% \label{sec:logit_gap_perspective}
+
+% Traditional calibration analysis evaluates models from an overall perspective, potentially masking important sample-specific miscalibration patterns. By examining calibration behavior across the continuous spectrum of margin values, we uncover fundamental insights about how neural networks distribute confidence and why existing calibration methods often fall short. This analysis provides crucial validation for our theoretical framework established in Section~\ref{subsec:theoretical_evidence} and design choices made in Section~\ref{subsec:our_scaling}.
+
+% \begin{figure*}[ht]
+% \centering
+% \begin{subfigure}[b]{0.32\textwidth}
+%     \includegraphics[width=\textwidth]{figs/overall_reliability_imagenet_vit_b_16.pdf}
+%     \caption{Overall calibration for ImgNet}
+%     \label{fig:overall_imagenet_vit}
+% \end{subfigure}
+% \hfill
+% \begin{subfigure}[b]{0.32\textwidth}
+%     \includegraphics[width=\textwidth]{figs/group_3_reliability_imagenet_vit_b_16.pdf}
+%     \caption{High margin samples}
+%     \label{fig:high_gap_imagenet_vit}
+% \end{subfigure}
+% \hfill
+% \begin{subfigure}[b]{0.32\textwidth}
+%     \includegraphics[width=\textwidth]{figs/group_1_reliability_imagenet_vit_b_16.pdf}
+%     \caption{Low margin samples}
+%     \label{fig:low_gap_imagenet_vit}
+% \end{subfigure}
+
+% \begin{subfigure}[b]{0.32\textwidth}
+%     \includegraphics[width=\textwidth]{figs/overall_reliability_ciafr100_resnet50.pdf}
+%     \caption{Overall calibration for Cifar-100}
+%     \label{fig:overall_cifar_resnet}
+% \end{subfigure}
+% \hfill
+% \begin{subfigure}[b]{0.32\textwidth}
+%     \includegraphics[width=\textwidth]{figs/group_3_reliability_ciafr100_resnet50.pdf}
+%     \caption{High margin samples Cifar-100}
+%     \label{fig:high_gap_cifar_resnet}
+% \end{subfigure}
+% \hfill
+% \begin{subfigure}[b]{0.32\textwidth}
+%     \includegraphics[width=\textwidth]{figs/logitsgap_vs_confidence_change.pdf}
+%     \caption{margin vs. confidence change}
+%     \label{fig:gap_vs_change}
+% \end{subfigure}
+
+% \caption{\textbf{margin reveals hidden calibration patterns across the confidence spectrum.} Top row: ImageNet ViT-B/16 shows almost perfect but slightly under-confident overall calibration (a) but reveals systematic under-confidence in high margin samples (b) and almost well-calibrated but slightly under-confident low margin samples (c). Bottom row: CIFAR-100 ResNet-50 demonstrates that even with overall over-confidence (d), high margin samples remain under-confident (e). Panel (f) shows how different methods adjust confidence across margin values, with SMART providing targeted adjustments while TS and PTS show suboptimal patterns.}
+% \label{fig:logit_gap_analysis}
+% \end{figure*}
+
+% \paragraph{Revealing Hidden Calibration Heterogeneity Through margin Groupings} Our reliability diagram analysis unveils a crucial insight that traditional overall calibration metrics obscure: models exhibit fundamentally different calibration behaviors across margin groups. This heterogeneity validates our theoretical framework from Section~\ref{subsec:theoretical_evidence} while revealing why sample-specific calibration approaches are essential for achieving truly reliable uncertainty estimates.
+
+% Figure~\ref{fig:logit_gap_analysis} demonstrates this heterogeneity across different scenarios. For ImageNet with ViT-B/16, while overall calibration appears near-perfect (Figure~\ref{fig:overall_imagenet_vit}), decomposing by margin reveals distinct patterns: low gap samples achieve almost perfect but slightly under-confident calibration (Figure~\ref{fig:low_gap_imagenet_vit}), while high gap samples suffer from systematic miscalibration (Figure~\ref{fig:high_gap_imagenet_vit}). This pattern persists even under different overall calibration conditions, as shown in CIFAR-100 with ResNet-50 (Figures~\ref{fig:overall_cifar_resnet} and~\ref{fig:high_gap_cifar_resnet}), indicating that margin-based groupings reveal fundamental calibration characteristics that transcend dataset-specific or architecture-specific behaviors.
+
+% \paragraph{The Under-Confidence Paradox in High margin Samples} Perhaps the most counterintuitive finding emerges from examining high margin samples across our reliability diagrams. Despite representing "easy" classifications with substantial margins between top predictions, these samples consistently exhibit under-confidence rather than the expected over-confidence. In Figure~\ref{fig:high_gap_imagenet_vit}, high gap samples from ImageNet ViT-B/16 show systematic under-confidence, with predicted confidence consistently lower than empirical accuracy. This pattern holds even when overall model behavior differs dramatically, as demonstrated in Figure~\ref{fig:high_gap_cifar_resnet} where CIFAR-100 ResNet-50 maintains under-confidence in high gap samples despite overall over-confidence.
+
+% This paradox fundamentally challenges conventional intuitions about neural network confidence and validates our theoretical analysis from Section~\ref{subsec:theoretical_evidence}. High margin samples, despite being far from decision boundaries, require confidence enhancement rather than reduction. This phenomenon suggests that modern neural networks, when trained with standard cross-entropy objectives, develop conservative confidence estimates for samples with clear classification margins. The consistency of this pattern across different architectures and datasets indicates that this represents a fundamental characteristic of neural network training dynamics rather than an artifact of specific experimental conditions.
+
+% \paragraph{Method-Specific Failures from the margin Perspective} The confidence adjustment patterns revealed in Figure~\ref{fig:gap_vs_change} expose fundamental limitations in existing calibration approaches when viewed through the margin lens. Temperature Scaling's uniform adjustment completely ignores the heterogeneous calibration needs we identified, applying identical modifications regardless of sample characteristics. This uniform approach proves particularly problematic because it fails to provide the targeted adjustments different gap groups require.
+
+% More critically, PTS demonstrates the exact problems we highlighted in Section~\ref{subsec:our_scaling} regarding high-dimensional parameterization introducing noise. The method makes substantial, often counterproductive adjustments to low margin samples that, in cases like ImageNet ViT-B/16, already achieve good calibration and require minimal intervention. This unnecessary manipulation exemplifies how "the increased dimensionality introduces substantial noise for precise temperature parameterization," leading to degraded performance where careful preservation of existing calibration would be more appropriate.
+
+% In contrast, SMART's learned adjustment pattern directly validates our design principles. For ImageNet ViT-B/16, SMART provides minimal adjustments to low gap samples that are already well-calibrated, while delivering targeted confidence increases to high gap samples suffering from under-confidence. This adaptive behavior emerges naturally from our lightweight scalar-to-scalar mapping, demonstrating how principled architectural choices translate into appropriate calibration strategies without requiring explicit programming of these patterns.
+
+% \paragraph{Theoretical Validation and Design Implications} These empirical findings provide compelling validation for our theoretical framework established in Propositions~\ref{prop:unbound} and~\ref{prop:g-bound}. The systematic under-confidence in high margin samples directly supports our argument that temperature values should be bounded by margins rather than determined by global statistics or high-dimensional features. The fact that SMART automatically learns to provide confidence enhancement for high gap samples while preserving the calibration of well-adjusted low gap samples demonstrates the practical effectiveness of our theoretical bounds.
+
+
+
+
+
+
+
+\section{The proposed SMART framework}
+\label{Appendix:SMART_framework}
+
+This section presents the detailed algorithmic implementation of SMART, providing a step-by-step procedure for applying margin-based temperature scaling with soft-binned ECE optimization.
+
+% \begin{algorithm}[h]
+% \caption{The proposed SMART framework}
+% \label{alg:SMART}
+% \begin{algorithmic}[1]
+% \State \textbf{Input:} Validation logits and labels $\{\mathbf{z}_i, y_i\}_{i=1}^{N}$, temperature parameterization network $h_\phi(\cdot)$.
+
+% \State Compute Margin $m_i = \mathbf{z}_{i,\text{max}} - \mathbf{z}_{i,\text{2nd}}$ for each sample $i$.
+% \State Normalize: $\hat{m}_i = \frac{m_i - \mu_m}{\sigma_m}$
+
+% \For {each epoch} %{$e = 1$ to $\text{epochs}$}
+%     \State Compute sample-wise temperature $T = h_\phi(\hat{m}_i)$ for each sample $i$.
+%     \State Compute calibrated logits $\hat{\mathbf{z}}_i = \mathbf{z}_i / T$ for each sample $i$.
+%     \State Compute loss $\mathcal{L}_i = \text{CharbonnierSmoothedSoftECE}(\hat{\mathbf{z}}_i, y_i)$ for each sample $i$.
+%     \State Update parameters $\phi$ using gradient descent from the overall loss $\mathcal{L}_{calib}=\sum_{i=1}^N\mathcal{L}_i$
+% \EndFor
+
+% \end{algorithmic}
+% \end{algorithm}
+
+\begin{algorithm}[ht]
+\caption{SMART: Sample Margin-Aware Recalibration of Temperature}
+\label{alg:smart}
+\begin{algorithmic}[1]
+\STATE \textbf{Input:} Validation logits and labels $\{\mathbf{z}_i, y_i\}_{i=1}^{N_{\text{val}}}$, temperature network $h_\phi(\cdot)$
+\STATE Compute margins: $m_i = z_{i,\max} - z_{i,\text{2nd}}$ for each $i \in \{1, \ldots, N_{\text{val}}\}$
+\FOR{epoch $= 1, \ldots, N_{\text{epochs}}$}
+    \STATE Predict temperatures: $T_i = h_\phi(m_i)$ for each $i$
+    \STATE Scale logits: $\tilde{\mathbf{z}}_i = \mathbf{z}_i / T_i$ for each $i$
+    \STATE Compute loss: $\mathcal{L}_i = \text{CharbonnierSoftECE}(\tilde{\mathbf{z}}_i, y_i)$ for each $i$ (Equation~\ref{eq:charbonnier_softece})
+    \STATE Update: $\phi \leftarrow \phi - \eta \nabla_\phi \sum_{i=1}^{N_{\text{val}}} \mathcal{L}_i$ via SGD
+\ENDFOR
+\STATE \textbf{Return:} Trained temperature network $h_\phi$
+\end{algorithmic}
+\end{algorithm}
+
+
+\section{Full Calibration Performance}
+
+Full calibration performance for Table \ref{table:comparison_ECE} is in Table \ref{table:comparison_ECE_full}.
+
+\label{Appendix:full_calibration_performance}
+\begin{table*}[ht]
+\caption{\textbf{Comparison of Post-Hoc Calibration Methods Using ECE (\%, $\downarrow$, 15 bins) Across Various Datasets and Models} (mean $\pm$ std across 5 seeds). The best-performing method for each dataset-model combination is in bold, and our method is highlighted.}
+\centering
+\tiny
+\setlength{\tabcolsep}{2pt}
+{\fontsize{5pt}{6pt}\selectfont
+\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccc}
+\toprule
+\textbf{Dataset} & \textbf{Model} & \textbf{Vanilla} & \textbf{TS} & \textbf{PTS} & \textbf{CTS} & \textbf{Spline} & \textbf{GC} & \textbf{ProCal} & \textbf{FC} & \textbf{SMART (ours)} \\
+\midrule
+\multirow{2}{*}{CIFAR-10}
+ & ResNet-50      & $4.34$  & $1.38 \pm 0.26$ & $1.10 \pm 0.21$  & $0.83 \pm 0.15$ & $1.52 \pm 0.03$ & $1.37 \pm 0.08$ & $4.17 \pm 0.12$ & $1.66 \pm 0.09$ & $\cellgray\mathbf{0.76 \pm 0.02}$ \\
+ & Wide-ResNet    & $3.24$  & $0.93 \pm 0.20$ & $0.90 \pm 0.19$  & $0.81 \pm 0.17$ & $1.74 \pm 0.01$ & $0.89 \pm 0.06$ & $2.81 \pm 0.11$ & $1.12 \pm 0.07$ & $\cellgray\mathbf{0.43 \pm 0.05}$ \\
+\midrule
+\multirow{2}{*}{CIFAR-100}
+ & ResNet-50      & $17.53$ & $5.61 \pm 1.39$ & $1.96 \pm 0.48$  & $3.67 \pm 0.88$ & $3.48 \pm 0.00$ & $5.70 \pm 0.15$ & $9.71 \pm 0.18$ & $2.91 \pm 0.12$ & $\cellgray\mathbf{1.37 \pm 0.27}$ \\
+ & Wide-ResNet    & $15.34$ & $4.50 \pm 0.62$ & $1.96 \pm 0.27$  & $3.01 \pm 0.42$ & $3.76 \pm 0.00$ & $4.55 \pm 0.13$ & $9.44 \pm 0.16$ & $4.49 \pm 0.14$ & $\cellgray\mathbf{1.80 \pm 0.10}$ \\
+\midrule
+\multirow{6}{*}{ImageNet-1K}
+ & ResNet-50      & $3.65$  & $2.17 \pm 0.03$  & $0.95 \pm 0.36$  & $2.17 \pm 0.78$ & $0.62 \pm 0.18$ & $2.44 \pm 0.12$ & $1.08 \pm 0.14$ & $1.71 \pm 0.08$ & $\cellgray\mathbf{0.52 \pm 0.12}$ \\
+ & DenseNet-121   & $2.53$  & $1.85 \pm 0.04$  & $1.02 \pm 0.46$  & $1.86 \pm 0.81$ & $0.81 \pm 0.35$ & $2.20 \pm 0.25$ & $1.52 \pm 0.21$ & $1.35 \pm 0.29$ & $\cellgray\mathbf{0.57 \pm 0.03}$ \\
+ & Wide-ResNet    & $5.43$  & $2.89 \pm 0.11$  & $1.14 \pm 0.24$  & $3.27 \pm 0.69$ & $0.66 \pm 0.10$ & $3.66 \pm 0.16$ & $1.57 \pm 0.10$ & $1.62 \pm 0.09$ & $\cellgray\mathbf{0.52 \pm 0.07}$ \\
+ & Swin-B         & $5.05$  & $3.91 \pm 0.07$  & $1.05 \pm 0.05$   & $1.53 \pm 0.08$  & $0.88 \pm 0.14$ & $4.95 \pm 0.17$ & $1.00 \pm 0.15$ & $5.05 \pm 0.06$ & $\cellgray\mathbf{0.46 \pm 0.03}$ \\
+ & ViT-B-16       & $5.62$  & $3.60 \pm 0.19$  & $1.23 \pm 0.29$  & $4.65 \pm 1.02$ & $0.91 \pm 0.31$ & $4.39 \pm 0.25$ & $0.97 \pm 0.30$ & $5.65 \pm 0.06$ & $\cellgray\mathbf{0.48 \pm 0.13}$ \\
+ & ViT-B-32       & $6.39$  & $3.93 \pm 0.02$  & $1.27 \pm 0.97$  & $2.12 \pm 1.59$ & $0.81 \pm 0.12$ & $4.67 \pm 0.13$ & $0.88 \pm 0.32$ & $6.39 \pm 0.06$ & $\mathbf{\cellgray{0.71 \pm 0.18}}$ \\
+\midrule
+\multirow{5}{*}{ImageNet-C}
+ & ResNet-50      & $13.82$ & $1.97 \pm 0.02$  & $1.12 \pm 0.13$  & $1.69 \pm 0.20$ & $5.61 \pm 0.15$ & $2.69 \pm 0.11$ & $5.79 \pm 0.19$ & $2.51 \pm 0.13$ & $\cellgray\mathbf{0.62 \pm 0.03}$ \\
+ & DenseNet-121   & $12.57$ & $1.58 \pm 0.00$  & $1.19 \pm 0.15$  & $1.44 \pm 0.19$ & $5.18 \pm 0.13$ & $2.01 \pm 0.09$ & $9.88 \pm 0.24$ & $9.44 \pm 0.31$ & $\cellgray\mathbf{0.63 \pm 0.01}$ \\
+ & Swin-B         & $12.03$ & $5.82 \pm 0.05$  & $1.53 \pm 0.00$   & $3.05 \pm 0.01$  & $2.58 \pm 0.21$ & $6.92 \pm 0.18$ & $2.53 \pm 0.12$ & $5.18 \pm 0.17$ & $\cellgray\mathbf{1.23 \pm 0.04}$ \\
+ & ViT-B-16       & $8.28$  & $5.24 \pm 0.01$  & $1.27 \pm 0.05$   & $2.76 \pm 0.10$  & $1.71 \pm 0.22$ & $5.95 \pm 0.15$ & $1.96 \pm 0.14$ & $5.37 \pm 0.20$ & $\cellgray\mathbf{1.06 \pm 0.02}$ \\
+ & ViT-B-32       & $7.69$  & $5.10 \pm 0.00$  & $1.07 \pm 0.08$   & $2.97 \pm 0.24$  & $1.43 \pm 0.24$ & $6.40 \pm 0.16$ & $1.55 \pm 0.11$ & $5.50 \pm 0.18$ & $\cellgray\mathbf{0.96 \pm 0.01}$ \\
+\midrule
+\multirow{6}{*}{ImageNet-LT}
+ & ResNet-50      & $3.63$  & $2.01 \pm 0.02$  & $0.99 \pm 0.32$  & $2.17 \pm 0.68$ & $0.56 \pm 0.10$ & $2.20 \pm 0.17$ & $1.12 \pm 0.20$ & $1.80 \pm 0.23$ & $\cellgray\mathbf{0.56 \pm 0.04}$ \\
+ & DenseNet-121   & $2.50$  & $1.80 \pm 0.06$  & $1.20 \pm 0.26$  & $1.88 \pm 0.41$ & $\mathbf{0.79 \pm 0.07}$ & $2.05 \pm 0.11$ & $1.79 \pm 0.09$ & $1.76 \pm 0.50$ & $\cellgray{0.81 \pm 0.01}$ \\
+ & Wide-ResNet    & $5.40$  & $2.99 \pm 0.05$  & $1.21 \pm 0.77$  & $2.87 \pm 1.79$ & $0.81 \pm 0.24$ & $3.59 \pm 0.18$ & $1.28 \pm 0.06$ & $1.68 \pm 0.10$ & $\cellgray\mathbf{0.53 \pm 0.02}$ \\
+ & Swin-B         & $4.69$  & $3.98 \pm 0.12$  & $1.21 \pm 0.45$  & $1.50 \pm 0.56$ & $0.79 \pm 0.17$ & $4.79 \pm 0.27$ & $0.95 \pm 0.16$ & $4.82 \pm 0.10$ & $\cellgray\mathbf{0.58 \pm 0.01}$ \\
+ & ViT-B-16       & $5.58$  & $3.73 \pm 0.13$  & $1.14 \pm 0.47$  & $1.43 \pm 0.58$ & $0.66 \pm 0.05$  & $4.34 \pm 0.14$ & $0.77 \pm 0.14$ & $5.72 \pm 0.08$ & $\cellgray\mathbf{0.56 \pm 0.14}$ \\
+ & ViT-B-32       & $6.28$  & $3.98 \pm 0.06$  & $1.35 \pm 0.41$  & $2.12 \pm 0.63$ & $0.72 \pm 0.23$ & $4.76 \pm 0.08$ & $0.83 \pm 0.12$ & $6.26 \pm 0.03$ & $\mathbf{\cellgray{0.60 \pm 0.11}}$ \\
+\midrule
+\multirow{5}{*}{ImageNet-S}
+ & ResNet-50      & $22.32$ & $2.06 \pm 0.06$  & $1.69 \pm 0.27$  & $1.48 \pm 0.23$ & $9.76 \pm 0.22$  & $1.99 \pm 0.16$ & $9.52 \pm 0.31$ & $12.58 \pm 1.35$ & $\cellgray\mathbf{0.92 \pm 0.09}$ \\
+ & DenseNet-121   & $20.13$ & $1.67 \pm 0.28$ & $1.93 \pm 0.19$   & $1.16 \pm 0.11$  & $9.20 \pm 0.32$  & $1.77 \pm 0.15$ & $12.93 \pm 0.23$ & $22.67 \pm 1.07$ & $\cellgray\mathbf{0.59 \pm 0.25}$ \\
+ & Swin-B         & $24.61$ & $6.50 \pm 0.05$  & $1.53 \pm 0.19$  & $3.62 \pm 0.45$ & $8.66 \pm 0.15$  & $6.92 \pm 0.35$ & $8.05 \pm 0.30$ & $1.70 \pm 0.06$ & $\cellgray\mathbf{1.26 \pm 0.05}$ \\
+ & ViT-B-16       & $16.57$ & $5.75 \pm 0.08$  & $1.33 \pm 0.21$  & $2.84 \pm 0.43$ & $5.70 \pm 0.19$  & $6.36 \pm 0.29$ & $5.67 \pm 0.38$ & $1.93 \pm 0.18$ & $\cellgray\mathbf{0.98 \pm 0.08}$ \\
+ & ViT-B-32       & $14.22$ & $4.99 \pm 0.15$  & $1.67 \pm 0.27$  & $3.25 \pm 0.50$ & $4.07 \pm 0.21$  & $6.23 \pm 0.16$ & $4.44 \pm 0.23$ & $1.56 \pm 0.09$ & $\cellgray\mathbf{0.87 \pm 0.18}$ \\
+\bottomrule
+\end{tabular*}
+}
+\label{table:comparison_ECE_full}
+% \vspace{-0.2in}
+\end{table*}
+
+\section{Calibration Performance on Other Metrics}
+\label{Appendix:Calibration_Performance_on_Other_Metrics}
+
+
+\subsection{Accuracy Performance}
+
+\begin{table*}[ht]
+\caption{\textbf{Comparison of Classification Accuracy (\%) Across Calibration Methods (Seed 1–5 Averaged).}}
+\centering
+\scriptsize
+\setlength{\tabcolsep}{3pt}
+\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}ccccccccc}
+\toprule
+\textbf{Dataset} & \textbf{Model} & \textbf{Vanilla} & \textbf{TS} & \textbf{PTS} & \textbf{CTS} & \textbf{Spline} & \textbf{SMART} \\
+\midrule
+\multirow{2}{*}{CIFAR-10}
+ & ResNet-50      & 95.05\% & 95.05\% & 95.05\% & 94.88\% & 95.05\% & \cellgray{95.05\%} \\
+ & Wide-ResNet    & 96.13\% & 96.13\% & 96.13\% & 96.09\% & 96.13\% & \cellgray{96.13\%} \\
+\midrule
+\multirow{2}{*}{CIFAR-100}
+ & ResNet-50      & 76.69\% & 76.69\% & 76.69\% & 76.38\% & 76.69\% & \cellgray{76.69\%} \\
+ & Wide-ResNet    & 79.29\% & 79.29\% & 79.29\% & 79.28\% & 79.29\% & \cellgray{79.29\%} \\
+\midrule
+\multirow{6}{*}{ImageNet-1K}
+ & ResNet-50      & 76.16\% & 76.16\% & 76.16\% & 75.32\% & 76.17\% & \cellgray{76.16\%} \\
+ & DenseNet-121   & 74.44\% & 74.44\% & 74.44\% & 73.71\% & 74.43\% & \cellgray{74.44\%} \\
+ & Wide-ResNet    & 78.46\% & 78.46\% & 78.46\% & 77.70\% & 78.46\% & \cellgray{78.46\%} \\
+ & Swin-B         & 83.17\% & 83.17\% & 83.17\% & 82.80\% & 83.17\% & \cellgray{83.17\%} \\
+ & ViT-B-16       & 81.12\% & 81.12\% & 81.12\% & 79.64\% & 80.86\% & \cellgray{81.12\%} \\
+ & ViT-B-32       & 75.95\% & 75.95\% & 75.95\% & 75.14\% & 75.94\% & \cellgray{75.95\%} \\
+\midrule
+\multirow{5}{*}{ImageNet-C}
+ & ResNet-50      & 19.16\% & 19.16\% & 19.16\% & 19.34\% & 19.16\% & \cellgray{19.16\%} \\
+ & DenseNet-121   & 21.25\% & 21.25\% & 21.25\% & 21.36\% & 21.25\% & \cellgray{21.25\%} \\
+ & Swin-B         & 40.83\% & 40.83\% & 40.83\% & 41.22\% & 40.83\% & \cellgray{40.83\%} \\
+ & ViT-B-16       & 41.07\% & 41.07\% & 41.07\% & 41.28\% & 41.07\% & \cellgray{41.07\%} \\
+ & ViT-B-32       & 37.82\% & 37.82\% & 24.56\% & 37.96\% & 37.85\% & \cellgray{37.82\%} \\
+\midrule
+\multirow{6}{*}{ImageNet-LT}
+ & ResNet-50      & 76.04\% & 76.04\% & 76.04\% & 75.43\% & 76.04\% & \cellgray{76.04\%} \\
+ & DenseNet-121   & 74.34\% & 74.34\% & 74.34\% & 73.88\% & 74.40\% & \cellgray{74.34\%} \\
+ & Wide-ResNet    & 78.39\% & 78.39\% & 78.39\% & 77.67\% & 78.40\% & \cellgray{78.39\%} \\
+ & Swin-B         & 82.95\% & 82.95\% & 82.95\% & 82.55\% & 82.94\% & \cellgray{82.95\%} \\
+ & ViT-B-16       & 80.95\% & 80.95\% & 80.95\% & 80.58\% & 81.00\% & \cellgray{80.95\%} \\
+ & ViT-B-32       & 75.89\% & 75.89\% & 75.89\% & 75.14\% & 75.92\% & \cellgray{75.89\%} \\
+\midrule
+\multirow{5}{*}{ImageNet-S}
+ & ResNet-50      & 24.09\% & 24.09\% & 24.09\% & 23.88\% & 24.09\% & \cellgray{24.09\%} \\
+ & DenseNet-121   & 24.30\% & 24.30\% & 24.30\% & 23.87\% & 31.55\% & \cellgray{24.30\%} \\
+ & Swin-B         & 31.54\% & 31.54\% & 31.54\% & 31.65\% & 31.55\% & \cellgray{31.54\%} \\
+ & ViT-B-16       & 29.37\% & 29.37\% & 29.37\% & 29.51\% & 29.39\% & \cellgray{29.37\%} \\
+ & ViT-B-32       & 27.77\% & 27.77\% & 27.77\% & 27.76\% & 27.75\% & \cellgray{27.77\%} \\
+\bottomrule
+\end{tabular*}
+\label{table:accuracy_comparison}
+\end{table*}
+
+
+\paragraph{Accuracy Preservation Analysis} Table \ref{table:accuracy_comparison} confirms that SMART achieves superior calibration while perfectly preserving classification accuracy—a fundamental advantage of post-hoc methods. Unlike CTS, which suffers accuracy drops up to 1.48 percentage points due to class-specific boundary alterations, or Spline's variable impacts on transformers, SMART's design ensures zero accuracy loss. By operating exclusively on the margin rather than full logit vectors, SMART focuses solely on confidence scaling without disturbing the relative ordering that determines predictions. This preservation holds even under severe distribution shifts like ImageNet-C and ImageNet-Sketch, where SMART simultaneously maintains base model accuracy while dramatically improving calibration. This dual guarantee makes SMART uniquely suitable for safety-critical applications requiring both correct predictions and reliable uncertainty estimates.
+
+
+
+
+
+\subsection{AdaECE Performance}
+\label{appendix:detailed_adaece_analysis}
+
+
+\begin{table*}[ht]
+\caption{\textbf{Comparison of AdaECE Calibration Methods Using AdaECE($\downarrow$, \%, 15bins) Across Various Datasets and Models (Seed 1--5 Averaged).}}
+\centering
+\scriptsize
+\setlength{\tabcolsep}{3pt}
+\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccc}
+\toprule
+\textbf{Dataset} & \textbf{Model} & \textbf{Vanilla} & \textbf{TS} & \textbf{PTS} & \textbf{CTS} & \textbf{Spline} & \textbf{SMART} \\
+\midrule
+\multirow{2}{*}{CIFAR-10}
+ & ResNet-50      & $4.33\pm0.0\%$  & $2.14\pm0.0\%$ & $0.83\pm28.6\%$ & $1.56\pm26.2\%$ & $2.14\pm1.1\%$  & $\cellgray\mathbf{0.99\pm4.3\%}$ \\
+ & Wide-ResNet    & $3.24\pm0.0\%$  & $1.71\pm0.0\%$ & $0.89\pm21.9\%$ & $1.47\pm19.7\%$ & $2.30\pm0.4\%$  & $\cellgray\mathbf{0.50\pm12.2\%}$ \\
+\midrule
+\multirow{2}{*}{CIFAR-100}
+ & ResNet-50      & $17.53\pm0.0\%$ & $5.66\pm0.0\%$ & $1.91\pm35.3\%$ & $3.43\pm32.0\%$ & $3.55\pm0.0\%$  & $\cellgray\mathbf{2.27\pm25.2\%}$ \\
+ & Wide-ResNet    & $15.34\pm0.0\%$ & $4.41\pm0.0\%$ & $1.69\pm13.0\%$ & $2.95\pm11.6\%$ & $3.95\pm0.1\%$  & $\cellgray\mathbf{1.83\pm2.1\%}$ \\
+\midrule
+\multirow{6}{*}{ImageNet-1K}
+ & ResNet-50      & $3.68\pm1.3\%$  & $2.13\pm0.5\%$ & $0.92\pm44.1\%$ & $2.21\pm39.8\%$ & $0.81\pm28.7\%$ & $\cellgray\mathbf{0.79\pm8.7\%}$ \\
+ & DenseNet-121   & $2.52\pm1.4\%$  & $1.74\pm1.8\%$ & $1.05\pm41.3\%$ & $1.78\pm38.0\%$ & $0.77\pm28.0\%$ & $\cellgray\mathbf{0.65\pm10.2\%}$ \\
+ & Wide-ResNet    & $5.31\pm0.3\%$  & $2.87\pm2.8\%$ & $1.04\pm20.6\%$ & $3.24\pm18.0\%$ & $\mathbf{0.83\pm36.3\%}$ & $\cellgray{0.87\pm14.3\%}$\\
+ & Swin-B         & $4.86\pm0.6\%$  & $4.50\pm1.0\%$ & $1.05\pm4.6\%$  & $1.59\pm5.1\%$  & $1.04\pm5.3\%$  & $\cellgray\mathbf{0.74\pm12.2\%}$\\
+ & ViT-B-16       & $5.57\pm1.2\%$  & $4.10\pm2.3\%$ & $1.09\pm29.7\%$ & $4.85\pm27.4\%$ & $1.07\pm29.2\%$ & $\cellgray\mathbf{0.79\pm15.4\%}$\\
+ & ViT-B-32       & $6.41\pm0.4\%$  & $3.92\pm1.7\%$ & $1.27\pm71.9\%$ & $1.90\pm66.4\%$ & $0.96\pm15.3\%$ & $\cellgray\mathbf{0.78\pm3.6\%}$ \\
+\midrule
+\multirow{5}{*}{ImageNet-C}
+ & ResNet-50      & $13.84\pm0.2\%$ & $2.02\pm1.7\%$ & $1.06\pm0.7\%$  & $1.76\pm0.6\%$ & $5.49\pm2.8\%$  & $\cellgray\mathbf{0.74\pm8.0\%}$ \\
+ & DenseNet-121   & $12.57\pm0.1\%$ & $1.64\pm0.7\%$ & $1.17\pm9.9\%$  & $1.48\pm8.2\%$ & $2.57\pm7.9\%$  & $\cellgray\mathbf{0.70\pm3.6\%}$ \\
+ & Swin-B         & $11.98\pm0.1\%$ & $5.83\pm1.0\%$ & $1.58\pm0.0\%$  & $3.07\pm0.2\%$ & $5.13\pm2.3\%$  & $\cellgray\mathbf{1.31\pm2.9\%}$\\
+ & ViT-B-16       & $8.24\pm0.3\%$  & $5.25\pm0.9\%$ & $1.27\pm5.9\%$  & $2.77\pm5.3\%$ & $2.57\pm7.9\%$  & $\cellgray\mathbf{1.09\pm4.0\%}$\\
+ & ViT-B-32       & $7.66\pm0.2\%$  & $5.11\pm0.0\%$ & $1.07\pm4.3\%$  & $2.97\pm3.7\%$ & $1.45\pm16.8\%$ & $\cellgray\mathbf{1.01\pm4.2\%}$\\
+\midrule
+\multirow{6}{*}{ImageNet-LT}
+ & ResNet-50      & $3.54\pm0.9\%$  & $2.02\pm1.2\%$ & $0.92\pm35.5\%$ & $2.17\pm33.0\%$ & $0.71\pm20.7\%$ & $\cellgray\mathbf{0.67\pm3.3\%}$\\
+ & DenseNet-121   & $2.37\pm3.4\%$  & $1.74\pm2.1\%$ & $1.17\pm23.6\%$ & $1.86\pm21.3\%$ & $\mathbf{0.73\pm26.4\%}$ & $\cellgray{0.76\pm0.7\%}$\\
+ & Wide-ResNet    & $5.22\pm0.4\%$  & $2.98\pm0.9\%$ & $1.22\pm62.4\%$ & $2.83\pm58.1\%$ & $0.79\pm18.1\%$ & $\cellgray\mathbf{0.98\pm4.4\%}$\\
+ & Swin-B         & $4.69\pm0.6\%$  & $4.48\pm1.2\%$ & $1.43\pm19.1\%$ & $1.23\pm18.0\%$ & $0.95\pm6.7\%$  & $\cellgray\mathbf{0.74\pm31.3\%}$\\
+ & ViT-B-16       & $5.57\pm0.8\%$  & $4.18\pm2.9\%$ & $1.13\pm43.4\%$ & $1.06\pm40.1\%$ & $0.95\pm12.9\%$ & $\cellgray\mathbf{0.85\pm15.1\%}$\\
+ & ViT-B-32       & $6.26\pm0.6\%$  & $3.97\pm1.6\%$ & $1.30\pm31.1\%$ & $2.04\pm28.2\%$ & $0.86\pm26.5\%$ & $\cellgray\mathbf{0.84\pm10.1\%}$\\
+\midrule
+\multirow{5}{*}{ImageNet-S}
+ & ResNet-50      & $22.31\pm0.3\%$ & $2.01\pm2.9\%$ & $1.64\pm16.4\%$ & $1.51\pm14.7\%$ & $9.51\pm2.4\%$ & $\cellgray\mathbf{0.90\pm15.8\%}$\\
+ & DenseNet-121   & $20.15\pm0.5\%$ & $1.67\pm17.0\%$& $1.93\pm9.6\%$  & $1.16\pm8.3\%$  & $8.7\pm1.92\%$ & $\cellgray\mathbf{0.76\pm32.3\%}$\\
+ & Swin-B         & $24.62\pm0.0\%$ & $6.40\pm0.5\%$ & $1.53\pm12.2\%$ & $3.57\pm11.1\%$ & $9.06\pm4.2\%$ & $\cellgray\mathbf{1.53\pm3.8\%}$\\
+ & ViT-B-16       & $16.57\pm0.2\%$ & $5.62\pm0.7\%$ & $1.33\pm8.7\%$  & $2.98\pm7.3\%$  & $8.66\pm1.9\%$ & $\cellgray\mathbf{1.08\pm4.3\%}$\\
+ & ViT-B-32       & $14.19\pm0.3\%$ & $4.98\pm2.9\%$ & $1.66\pm16.0\%$ & $3.23\pm14.1\%$ & $5.64\pm3.3\%$ & $\cellgray\mathbf{1.07\pm19.9\%}$\\
+\bottomrule
+\end{tabular*}
+\label{table:adaece_comparison_updated}
+\end{table*}
+
+
+This section provides an in-depth analysis of calibration performance using AdaECE across different datasets and model architectures, complementing the results presented in Section~\ref{Sec:Calibration_performance}. Adaptive-ECE is a measure of calibration performance that addresses the bias of equal-width binning scheme of ECE. It adapts the bin-size to the number of samples and ensures that each bin is evenly distributed with samples. The formula for Adaptive-ECE is as follows:
+\begin{equation}
+\text{Adaptive-ECE} = \sum_{i=1}^{\mathbb{B}} \frac{|B_i|}{N}|I_i - C_i| \text{ s.t. } \forall i, j \cdot |B_i| = |B_j|
+\end{equation}
+AdaECE offers a more rigorous assessment of calibration quality than standard ECE by adapting bin boundaries to ensure uniform sample distribution, preventing calibration errors from being masked in sparsely populated confidence regions. Table \ref{table:adaece_comparison_updated} presents comprehensive AdaECE results across all evaluated datasets and architectures. SMART consistently outperforms competing methods under this metric, achieving the lowest AdaECE on 24 of 26 dataset-architecture combinations.
+
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=\textwidth]{figs/adaece_cifar_figure.pdf}
+\caption{\textbf{AdaECE comparison on CIFAR datasets.} SMART consistently achieves superior calibration on both CIFAR-10 and CIFAR-100 across multiple architectures. From left to right are Cifar10 ResNet-50/Wide-ResNet, Cifar100 ResNet-50/Wide-ResNet.}
+\label{fig:adaece_cifar}
+\end{figure}
+
+
+\paragraph{CIFAR Performance Analysis.} SMART demonstrates exceptional calibration on CIFAR datasets in Figure~\ref{fig:adaece_cifar}, achieving the lowest AdaECE with notably stable variance compared to competitors. The key insight emerges when comparing CIFAR-10 to CIFAR-100: while global methods like TS suffer dramatic degradation as class count increases, SMART maintains robust performance. PTS shows competitive results but with substantially higher variance, indicating reliability issues. Spline struggles particularly with CIFAR-100's complex confidence landscape, revealing how non-parametric methods become less effective as classification complexity increases.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=\textwidth]{figs/adaece_imagenet_figure.pdf}
+\caption{\textbf{AdaECE($\downarrow$, \%, 15bins) comparison on ImageNet-1K.} SMART delivers consistent calibration across diverse architectures, from CNNs to vision transformers. From left to right are ResNet-50, DenseNet-121, Wide-ResNet, Swin-B, ViT-B-16, ViT-B-32.}
+\label{fig:adaece_imagenet}
+\end{figure}
+
+\paragraph{Large-Scale Classification on ImageNet.} In Figure~\ref{fig:adaece_imagenet}, The ImageNet results reveal a crucial architectural insight: SMART maintains consistent performance across both CNN and transformer designs, while traditional methods like TS and CTS show pronounced degradation on transformers. This architectural robustness highlights SMART's ability to capture fundamental uncertainty signals through the margin regardless of model inductive biases. PTS exhibits extreme variance, confirming that high-dimensional parameterizations struggle with reliability when learning complex temperature mappings, particularly on large-scale datasets.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=\textwidth]{figs/adaece_imagenet_c_figure.pdf}
+\caption{\textbf{AdaECE($\downarrow$, \%, 15bins) comparison on ImageNet-C.} SMART maintains exceptional calibration under corruption, while Spline and TS-based methods demonstrate significant degradation. From left to right are ResNet-50, DenseNet-121, Swin-B, ViT-B-16, ViT-B-32.}
+\label{fig:adaece_imagenet_c}
+\end{figure}
+
+\paragraph{Robustness to Input Corruption.} As shown in Figure~\ref{fig:adaece_imagenet_c}, SMART's resilience under corruption provides compelling evidence for the stability of decision boundary information. While Spline performs competitively on clean ImageNet, it deteriorates dramatically under corruption with values 5-7$\times$ higher than SMART. This collapse reveals a fundamental limitation: non-parametric methods overfit to validation distributions and fail when input characteristics change. SMART's focus on decision boundary uncertainty via the margin remains informative even when input distributions shift substantially.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=\textwidth]{figs/adaece_imagenet_lt_figure.pdf}
+\caption{\textbf{AdaECE($\downarrow$, \%, 15bins) comparison on ImageNet-LT.} SMART maintains strong calibration under long-tailed class distributions, particularly on CNN architectures. From left to right are ResNet-50, DenseNet-121, Wide-ResNet, Swin-B, ViT-B-16, ViT-B-32.}
+\label{fig:adaece_imagenet_lt}
+\end{figure}
+
+\paragraph{Long-Tailed Distribution Calibration.} As shown in Figure~\ref{fig:adaece_imagenet_lt}, The ImageNet-LT results reveal that class imbalance presents a fundamentally different calibration challenge than input corruption. Interestingly, Spline performs competitively here, suggesting non-parametric methods can handle statistical imbalances better than distributional shifts. However, CTS underperforms despite being explicitly designed for per-class variations, demonstrating that simply applying different temperatures per class is insufficient for complex imbalanced scenarios.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=\textwidth]{figs/adaece_imagenet_sketch_figure.pdf}
+\caption{\textbf{AdaECE($\downarrow$, \%, 15bins) comparison on ImageNet-Sketch.} SMART maintains exceptional calibration under extreme domain shift, while Spline struggles significantly. From left to right are ResNet-50, DenseNet-121, Swin-B, ViT-B-16, ViT-B-32.}
+\label{fig:adaece_imagenet_sketch}
+\end{figure}
+
+\paragraph{Extreme Domain Shift Calibration.} The sketch-based domain shift represents the most challenging calibration scenario in Figure~\ref{fig:adaece_imagenet_sketch} , where SMART demonstrates its most dramatic advantage. Spline's collapse here reinforces the brittleness of non-parametric methods under distribution shifts, while SMART's consistent performance across all architectures provides strong evidence that margin information captures robust uncertainty signals that transcend specific input characteristics or domains.
+
+
+
+% \section{In-training Bin Size}
+
+% \begin{figure}[ht]
+% \centering
+% \includegraphics[width=0.7\textwidth]{figs/bin_count_smart.pdf}
+% \caption{\textbf{Ablation Study for in‐training Soft‐ECE bin count}}
+% \captionsetup{justification=raggedright,singlelinecheck=false}
+% \centering
+% \label{fig:bin_count_sensitivity}
+% \end{figure}
+
+
+% \paragraph{Effect of in-training Bin Size on SMART Calibration.}
+% We examine how the choice of histogram bin count during SMART's in-training ECE estimation influences final calibration performance. SMART exhibits good robustness to this hyperparameter across these architectures and datasets. For CIFAR-10 with ResNet-50, the calibration error remains virtually unchanged across all tested bin counts (5-25), demonstrating exceptional stability. ImageNet with ResNet-50 shows modest sensitivity, with performance gradually improving as bin count increases, before stabilizing around 20 bins. ViT-B/16 on ImageNet displays more pronounced variability, exhibiting a performance peak at 10 bins before recovering at 15 bins. This architecture-dependent behavior aligns with our theoretical understanding of the bias-variance trade-off in the Soft-ECE objective: convolutional architectures demonstrate greater stability to binning discretization compared to transformer models, likely due to their inherently different confidence distribution characteristics. The overall bin-count insensitivity of SMART reinforces our method's practical utility—practitioners can generally select a moderate bin count (15-20) without extensive tuning. This robustness stems from the Soft-ECE's smoothing effect through membership functions, which distributes calibration signals across neighboring bins and creates a more continuous optimization landscape that remains stable across different discretization levels.
+
+\section{Comparison Of Various Training-Time Calibration Methods On Other Metrics}
+\label{appendix:training_time_calibration}
+
+This section presents a comprehensive evaluation of SMART when combined with various training-time calibration methods across multiple metrics, extending the ECE analysis provided in Section~\ref{sec:training_time_calibration}. We examine SMART's performance using AdaECE, Classwise ECE (CECE), Negative Log-Likelihood (NLL), and classification accuracy.
+
+\subsection{Accuracy Preservation}
+
+\begin{table*}[ht]
+\centering
+\caption{\textbf{Comparison of Train-time Calibration Methods Using Accuracy($\uparrow$, \%) Across Various Datasets and Models.} Results demonstrate that SMART preserves the original model accuracy across all training methods. Results are from the best run of 5 seeds.}
+\label{table:accuracy_compare_with_training_time_methods}
+\scriptsize
+\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc}
+\toprule
+\textbf{Dataset} & \textbf{Model} &\multicolumn{2}{c}{\textbf{NLL}} &\multicolumn{2}{c}{\textbf{Brier Loss}} &\multicolumn{2}{c}{\textbf{MMCE}} &\multicolumn{2}{c}{\textbf{LS-0.05}} &\multicolumn{2}{c}{\textbf{FLSD-53}} &\multicolumn{2}{c}{\textbf{FL-3}} \\
+ & & base & ours & base & ours & base & ours & base & ours & base & ours & base & ours \\
+\midrule
+\multirow{4}{*}{CIFAR10}
+ & ResNet-50 & 95.1 & \cellgray{95.1} & 95.0 & \cellgray{95.0} & 95.0 & \cellgray{95.0} & 94.7 & \cellgray{94.7} & 95.0 & \cellgray{95.0} & 94.8 & \cellgray{94.8} \\
+ & ResNet-110 & 95.1 & \cellgray{95.1} & 94.5 & \cellgray{94.5} & 94.6 & \cellgray{94.6} & 94.5 & \cellgray{94.5} & 94.6 & \cellgray{94.6} & 94.9 & \cellgray{94.9} \\
+ & DenseNet-121 & 95.0 & \cellgray{95.0} & 94.9 & \cellgray{94.9} & 94.6 & \cellgray{94.6} & 94.9 & \cellgray{94.9} & 94.6 & \cellgray{94.6} & 94.7 & \cellgray{94.7} \\
+ & Wide-ResNet & 96.1 & \cellgray{96.1} & 95.9 & \cellgray{95.9} & 96.1 & \cellgray{96.1} & 95.8 & \cellgray{95.8} & 96.0 & \cellgray{96.0} & 95.9 & \cellgray{95.9} \\
+\midrule
+\multirow{4}{*}{CIFAR100}
+ & ResNet-50 & 76.7 & \cellgray{76.7} & 76.6 & \cellgray{76.6} & 76.8 & \cellgray{76.8} & 76.6 & \cellgray{76.6} & 76.8 & \cellgray{76.8} & 77.3 & \cellgray{77.3} \\
+ & ResNet-110 & 77.3 & \cellgray{77.3} & 74.9 & \cellgray{74.9} & 76.9 & \cellgray{76.9} & 76.6 & \cellgray{76.6} & 77.5 & \cellgray{77.5} & 77.1 & \cellgray{77.1} \\
+ & DenseNet-121 & 75.5 & \cellgray{75.5} & 76.3 & \cellgray{76.3} & 76.0 & \cellgray{76.0} & 75.9 & \cellgray{75.9} & 77.3 & \cellgray{77.3} & 76.8 & \cellgray{76.8} \\
+ & Wide-ResNet & 79.3 & \cellgray{79.3} & 79.4 & \cellgray{79.4} & 79.3 & \cellgray{79.3} & 78.8 & \cellgray{78.8} & 79.9 & \cellgray{79.9} & 80.3 & \cellgray{80.3} \\
+\bottomrule
+\end{tabular*}
+\vspace{-0.1in}
+\end{table*}
+
+
+\paragraph{Accuracy Analysis} As shown in Table~\ref{table:accuracy_compare_with_training_time_methods}, SMART consistently preserves the classification accuracy of all base models across all training-time calibration methods. This is a critical property of post-hoc calibration methods, as improving confidence estimates should not come at the cost of predictive performance. The perfect accuracy preservation is by design, as SMART's temperature scaling mechanism operates solely on the scaling of logits without altering their relative ordering, thus maintaining the same class predictions. This contrasts with some training-time methods that may involve trade-offs between accuracy and calibration quality during the model optimization process. The preservation of accuracy across diverse architectures and datasets further validates SMART's practical utility as a calibration method that can be safely applied in real-world scenarios where maintaining predictive performance is essential.
+
+\subsection{AdaECE Performance}
+
+\begin{table*}[ht]
+\centering
+\caption{\textbf{Comparison of Train-time Calibration Methods Using AdaECE($\downarrow$, \%, 15bins) Across Various Datasets and Models.} The best-performing method for each dataset-model combination is in bold, and our method (SMART) is highlighted. Results are from the best run of 5 seeds.}
+\label{table:adaece_compare_with_training_time_methods}
+\scriptsize
+\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc}
+\toprule
+\textbf{Dataset} & \textbf{Model} &\multicolumn{2}{c}{\textbf{NLL}} &\multicolumn{2}{c}{\textbf{Brier Loss}} &\multicolumn{2}{c}{\textbf{MMCE}} &\multicolumn{2}{c}{\textbf{LS-0.05}} &\multicolumn{2}{c}{\textbf{FLSD-53}} &\multicolumn{2}{c}{\textbf{FL-3}} \\
+ & & base & ours & base & ours & base & ours & base & ours & base & ours & base & ours \\
+\midrule
+\multirow{4}{*}{CIFAR10}
+ & ResNet-50 & 4.33 & \cellgray{\textbf{0.80}} & 1.74 & \cellgray{\textbf{1.01}} & 4.55 & \cellgray{\textbf{0.67}} & 3.88 & \cellgray{\textbf{2.18}} & 1.56 & \cellgray{\textbf{0.45}} & 1.95 & \cellgray{\textbf{0.48}} \\
+ & ResNet-110 & 4.40 & \cellgray{\textbf{1.22}} & 2.61 & \cellgray{\textbf{0.56}} & 5.07 & \cellgray{\textbf{0.93}} & 4.46 & \cellgray{\textbf{3.66}} & 2.07 & \cellgray{\textbf{0.40}} & 1.64 & \cellgray{\textbf{0.52}} \\
+ & DenseNet-121 & 4.49 & \cellgray{\textbf{0.61}} & 2.01 & \cellgray{\textbf{0.51}} & 5.10 & \cellgray{\textbf{0.96}} & 4.40 & \cellgray{\textbf{2.95}} & 1.38 & \cellgray{\textbf{0.62}} & 1.23 & \cellgray{\textbf{0.83}} \\
+ & Wide-ResNet & 3.24 & \cellgray{\textbf{0.44}} & 1.70 & \cellgray{\textbf{0.44}} & 3.29 & \cellgray{\textbf{0.53}} & 4.27 & \cellgray{\textbf{0.97}} & 1.52 & \cellgray{\textbf{0.44}} & 1.84 & \cellgray{\textbf{0.59}} \\
+\midrule
+\multirow{4}{*}{CIFAR100}
+ & ResNet-50 & 17.53 & \cellgray{\textbf{1.00}} & 6.54 & \cellgray{\textbf{1.41}} & 15.31 & \cellgray{\textbf{1.08}} & 7.63 & \cellgray{\textbf{1.75}} & 4.40 & \cellgray{\textbf{1.35}} & 5.08 & \cellgray{\textbf{0.95}} \\
+ & ResNet-110 & 19.06 & \cellgray{\textbf{1.67}} & 7.73 & \cellgray{\textbf{0.93}} & 19.13 & \cellgray{\textbf{1.98}} & 11.07 & \cellgray{\textbf{2.72}} & 8.54 & \cellgray{\textbf{0.93}} & 8.65 & \cellgray{\textbf{1.22}} \\
+ & DenseNet-121 & 20.99 & \cellgray{\textbf{2.23}} & 5.04 & \cellgray{\textbf{1.02}} & 19.10 & \cellgray{\textbf{1.73}} & 12.83 & \cellgray{\textbf{1.96}} & 3.54 & \cellgray{\textbf{0.93}} & 4.14 & \cellgray{\textbf{0.97}} \\
+ & Wide-ResNet & 15.34 & \cellgray{\textbf{1.55}} & 4.28 & \cellgray{\textbf{0.97}} & 13.16 & \cellgray{\textbf{1.12}} & 5.13 & \cellgray{\textbf{2.11}} & 2.77 & \cellgray{\textbf{0.75}} & 2.07 & \cellgray{\textbf{1.15}} \\
+\bottomrule
+\end{tabular*}
+\vspace{-0.1in}
+\end{table*}
+
+\paragraph{AdaECE Analysis} The adaptive ECE results in Table~\ref{table:adaece_compare_with_training_time_methods} provide further validation of SMART's effectiveness when combined with various training-time calibration methods. AdaECE, which uses adaptive binning to ensure equal sample counts in each bin, offers a more robust calibration measure than standard ECE by eliminating potential biases from uneven confidence distributions. SMART consistently improves AdaECE across all training methods, with particularly dramatic improvements for models trained with NLL and MMCE, where we observe reductions of up to 18$\times$ (17.53\% → 1.00\% for CIFAR-100 ResNet-50). 
+
+The most substantial AdaECE improvements occur on CIFAR-100, which has ten times more classes than CIFAR-10 and thus represents a more challenging calibration scenario. This suggests that SMART's effectiveness scales favorably with task complexity. Even for models already trained with calibration-oriented objectives like Focal Loss or FLSD, SMART provides further substantial improvements, indicating that its margin-based temperature adjustment captures complementary information to these training-time approaches. Notably, the combination of SMART with FLSD-53 achieves some of the lowest overall AdaECE values (e.g., 0.40\% on CIFAR-10 ResNet-110), suggesting a particularly effective synergy between these methods.
+
+\subsection{Classwise ECE Performance}
+
+\begin{table*}[ht]
+\centering
+\caption{\textbf{Comparison of Train-time Calibration Methods Using Classwise ECE($\downarrow$, \%, 15bins) Across Various Datasets and Models.} The best-performing method for each dataset-model combination is in bold, and our method (SMART) is highlighted. Results are from the best run of 5 seeds.}
+\label{table:cece_compare_with_training_time_methods}
+\scriptsize
+\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc}
+\toprule
+\textbf{Dataset} & \textbf{Model} &\multicolumn{2}{c}{\textbf{NLL}} &\multicolumn{2}{c}{\textbf{Brier Loss}} &\multicolumn{2}{c}{\textbf{MMCE}} &\multicolumn{2}{c}{\textbf{LS-0.05}} &\multicolumn{2}{c}{\textbf{FLSD-53}} &\multicolumn{2}{c}{\textbf{FL-3}} \\
+ & & base & ours & base & ours & base & ours & base & ours & base & ours & base & ours \\
+\midrule
+\multirow{4}{*}{CIFAR10}
+ & ResNet-50 & 0.91 & \cellgray{\textbf{0.43}} & 0.46 & \cellgray{\textbf{0.40}} & 0.94 & \cellgray{\textbf{0.51}} & 0.71 & \cellgray{\textbf{0.51}} & 0.42 & \cellgray{\textbf{0.37}} & 0.43 & \cellgray{\textbf{0.38}} \\
+ & ResNet-110 & 0.92 & \cellgray{\textbf{0.49}} & 0.59 & \cellgray{\textbf{0.45}} & 1.04 & \cellgray{\textbf{0.54}} & 0.66 & \cellgray{\textbf{0.54}} & 0.47 & \cellgray{\textbf{0.41}} & 0.44 & \cellgray{\textbf{0.38}} \\
+ & DenseNet-121 & 0.92 & \cellgray{\textbf{0.45}} & 0.46 & \cellgray{\textbf{0.41}} & 1.04 & \cellgray{\textbf{0.59}} & 0.60 & \cellgray{\textbf{0.50}} & 0.41 & \cellgray{\textbf{0.38}} & 0.42 & \cellgray{\textbf{0.35}} \\
+ & Wide-ResNet & 0.68 & \cellgray{\textbf{0.37}} & 0.44 & \cellgray{\textbf{0.39}} & 0.70 & \cellgray{\textbf{0.38}} & 0.79 & \cellgray{\textbf{0.40}} & 0.41 & \cellgray{\textbf{0.29}} & 0.44 & \cellgray{\textbf{0.34}} \\
+\midrule
+\multirow{4}{*}{CIFAR100}
+ & ResNet-50 & 0.38 & \cellgray{\textbf{0.21}} & 0.22 & \cellgray{\textbf{0.20}} & 0.34 & \cellgray{\textbf{0.20}} & 0.23 & \cellgray{\textbf{0.21}} & 0.20 & \cellgray{\textbf{0.20}} & 0.20 & \cellgray{\textbf{0.20}} \\
+ & ResNet-110 & 0.41 & \cellgray{\textbf{0.20}} & 0.24 & \cellgray{\textbf{0.21}} & 0.42 & \cellgray{\textbf{0.21}} & 0.26 & \cellgray{\textbf{0.20}} & 0.24 & \cellgray{\textbf{0.20}} & 0.24 & \cellgray{\textbf{0.21}} \\
+ & DenseNet-121 & 0.45 & \cellgray{\textbf{0.23}} & 0.20 & \cellgray{\textbf{0.20}} & 0.42 & \cellgray{\textbf{0.23}} & 0.29 & \cellgray{\textbf{0.21}} & \textbf{0.19} & \cellgray{0.20} & 0.20 & \cellgray{\textbf{0.20}} \\
+ & Wide-ResNet & 0.34 & \cellgray{\textbf{0.19}} & 0.19 & \cellgray{\textbf{0.19}} & 0.30 & \cellgray{\textbf{0.19}} & 0.21 & \cellgray{\textbf{0.20}} & 0.18 & \cellgray{\textbf{0.18}} & 0.18 & \cellgray{\textbf{0.18}} \\
+\bottomrule
+\end{tabular*}
+\vspace{-0.1in}
+\end{table*}
+
+\paragraph{CECE Analysis} Classwise ECE (CECE) provides insights into calibration performance at the individual class level rather than aggregated across all classes. The formula for classwise ECE is:
+
+\begin{equation}
+\text{Classwise-ECE} = \frac{1}{\mathcal{K}} \sum_{i=1}^{B} \sum_{j=1}^{\mathcal{K}} \frac{|B_{i,j}|}{N} |I_{i,j} - C_{i,j}|
+\end{equation}
+
+where the calibration error is computed separately for each class $j$ across all bins $i$, then averaged across all $\mathcal{K}$ classes. This metric is particularly valuable for understanding whether calibration improvements are uniformly distributed across classes or concentrated in specific categories.
+
+Table~\ref{table:cece_compare_with_training_time_methods} demonstrates SMART's ability to improve per-class calibration across almost all training methods and architectures. The improvements are particularly prominent for models trained with NLL and MMCE, where CECE values are typically reduced by 50\% or more after applying SMART (e.g., from 0.91\% to 0.43\% for CIFAR-10 ResNet-50). This substantial improvement suggests that SMART's margin-based temperature scaling effectively addresses class-specific miscalibration patterns that may arise during training with these standard objectives.
+
+Interestingly, CECE values are consistently lower on CIFAR-100 compared to CIFAR-10 despite the higher class count, which contrasts with the pattern observed for ECE and AdaECE. This phenomenon occurs because CECE averages calibration errors across classes, and with 100 classes, individual class miscalibrations tend to average out more effectively than with only 10 classes. Additionally, the higher granularity of class divisions in CIFAR-100 may lead to more balanced per-class confidence distributions, making the averaging effect more pronounced.
+
+For models already trained with calibration-oriented losses like FLSD-53 and FL-3, SMART provides more modest improvements in CECE, and in a few cases maintains the same level of performance. This suggests that these training-time methods are already effective at addressing per-class calibration issues through their specialized loss formulations that inherently consider class-wise balance. However, SMART can still provide complementary benefits in most scenarios, particularly for classes that may remain poorly calibrated even after specialized training procedures.
+
+\subsection{Negative Log-Likelihood Performance}
+
+\begin{table*}[ht]
+\centering
+\caption{\textbf{Comparison of Train-time Calibration Methods Using NLL($\downarrow$, \%) Across Various Datasets and Models.} The best-performing method for each dataset-model combination is in bold, and our method (SMART) is highlighted. Results are from the best run of 5 seeds.}
+\label{table:nll_compare_with_training_time_methods}
+\scriptsize
+\setlength{\tabcolsep}{5pt}
+\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}cccccccccccccc}
+\toprule
+\textbf{Dataset} & \textbf{Model} &
+\multicolumn{2}{c}{\textbf{NLL}} &
+\multicolumn{2}{c}{\textbf{Brier Loss}} &
+\multicolumn{2}{c}{\textbf{MMCE}} &
+\multicolumn{2}{c}{\textbf{LS-0.05}} &
+\multicolumn{2}{c}{\textbf{FLSD-53}} &
+\multicolumn{2}{c}{\textbf{FL-3}} \\
+ & & \textbf{Base} & \textbf{Ours} &
+ \textbf{Base} & \textbf{Ours} &
+ \textbf{Base} & \textbf{Ours} &
+ \textbf{Base} & \textbf{Ours} &
+ \textbf{Base} & \textbf{Ours} &
+ \textbf{Base} & \textbf{Ours} \\
+\midrule
+\multirow{4}{*}{CIFAR-10}
+ & ResNet-50 & 41.2 & \cellgray \textbf{19.7} & 18.7 & \cellgray \textbf{18.4} & 44.8 & \cellgray \textbf{21.0} & 27.7 & \cellgray \textbf{27.7} & 17.6 & \cellgray \textbf{17.1} & 18.4 & \cellgray \textbf{17.9} \\
+ & ResNet-110 & 47.5 & \cellgray \textbf{22.5} & 20.4 & \cellgray \textbf{19.4} & 55.7 & \cellgray \textbf{23.6} & 29.9 & \cellgray \textbf{29.4} & 18.5 & \cellgray \textbf{17.9} & 17.8 & \cellgray \textbf{17.3} \\
+ & DenseNet-121 & 42.9 & \cellgray \textbf{20.8} & 19.1 & \cellgray \textbf{18.6} & 52.1 & \cellgray \textbf{24.1} & 28.7 & \cellgray \textbf{28.7} & 18.4 & \cellgray \textbf{18.1} & 18.0 & \cellgray \textbf{17.9} \\
+ & Wide-ResNet & 26.8 & \cellgray \textbf{14.9} & 15.9 & \cellgray \textbf{15.4} & 28.5 & \cellgray \textbf{15.9} & 21.7 & \cellgray \textbf{19.9} & 14.6 & \cellgray \textbf{13.7} & 15.2 & \cellgray \textbf{14.9} \\
+\midrule
+\multirow{4}{*}{CIFAR-100}
+ & ResNet-50 & 153.7 & \cellgray \textbf{105.3} & 99.6 & \cellgray \textbf{99.5} & 125.3 & \cellgray \textbf{100.7} & 121.0 & \cellgray \textbf{120.1} & 88.0 & \cellgray 88.4 & 87.5 & \cellgray 88.1 \\
+ & ResNet-110 & 179.2 & \cellgray \textbf{104.0} & 110.7 & \cellgray \textbf{110.0} & 180.6 & \cellgray \textbf{106.1} & 133.1 & \cellgray \textbf{128.8} & 89.9 & \cellgray \textbf{88.3} & 90.9 & \cellgray \textbf{90.0} \\
+ & DenseNet-121 & 205.6 & \cellgray \textbf{119.1} & 98.3 & \cellgray 98.9 & 166.6 & \cellgray \textbf{112.6} & 142.0 & \cellgray \textbf{134.3} & 85.5 & \cellgray 86.5 & 87.1 & \cellgray 87.3 \\
+ & Wide-ResNet & 140.1 & \cellgray \textbf{95.2} & 84.6 & \cellgray 84.9 & 119.6 & \cellgray \textbf{94.1} & 108.1 & \cellgray \textbf{106.5} & 76.9 & \cellgray 77.4 & 74.7 & \cellgray 75.8 \\
+\bottomrule
+\end{tabular*}
+\end{table*}
+
+
+
+\paragraph{NLL Analysis} NLL is a probabilistic metric that measures both calibration quality and discriminative power. Table~\ref{table:nll_compare_with_training_time_methods} shows that SMART improves NLL for most models, with the most significant gains observed for NLL, MMCE, and LS-0.05 trained models. The improvements are particularly striking for CIFAR-10, where NLL is reduced by up to 60\% after applying SMART (e.g., 41.22 → 19.70 for ResNet-50 with NLL).
+
+However, a different pattern emerges for models trained with specialized losses like FLSD-53 and FL-3 on CIFAR-100, where SMART sometimes leads to slight increases in NLL despite improvements in calibration metrics like ECE and AdaECE. This suggests that these specialized training losses optimize directly for NLL-like objectives, creating a scenario where SMART's temperature scaling might slightly disturb the carefully optimized probability distributions. Nevertheless, the overall trend across metrics indicates that SMART maintains or improves model performance in the vast majority of cases.
+
+% \subsection{Comparative Analysis Across Metrics}
+
+% Examining the results across all metrics provides a comprehensive view of SMART's effectiveness when combined with training-time calibration methods. Several key insights emerge:
+
+% \begin{enumerate}
+%     \item \textbf{Complementary Benefits:} SMART provides substantial improvements across almost all combinations of architectures, datasets, and training methods, indicating that its margin-based temperature scaling captures calibration signals complementary to those targeted by training-time approaches.
+    
+%     \item \textbf{Most Effective Combinations:} The combinations of SMART with FLSD-53 and FL-3 often achieve the best overall calibration performance, suggesting a particularly effective synergy between these focal-loss based training methods and SMART's post-hoc adjustments.
+    
+%     \item \textbf{Architectural Differences:} Wide-ResNet consistently shows the best calibration performance across most metrics and training methods when combined with SMART, possibly due to its architectural properties that produce more informative margins.
+    
+%     \item \textbf{Scale with Task Complexity:} The relative improvements from applying SMART are generally larger on CIFAR-100 than CIFAR-10, suggesting that SMART's benefits scale favorably with task complexity and class count.
+% \end{enumerate}
+
+% These findings further validate SMART as a versatile, effective calibration method that can be applied as a complementary step to various training-time calibration approaches, providing additional performance improvements with minimal computational overhead and without compromising accuracy.
+
+
+
+\section{Calibration Performance Under Specific Corruption Types}
+\label{appendix:corruption_analysis}
+
+To provide deeper insights into SMART's robustness across different corruption scenarios, we examine the calibration error reduction achieved by various methods on individual corruption types in ImageNet-C. We analyze performance across two architectures (ResNet-50 and ViT-B/16) and two metrics (ECE and AdaECE), providing a comprehensive view of how different calibration approaches respond to specific distribution shifts. This granular analysis helps understand which corruption types pose the greatest calibration challenges and how architectural differences influence calibration robustness.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=0.95\textwidth]{figs/resnet50_ece_reduced_error.pdf}
+\caption{\textbf{ECE reduction($\uparrow$, \%, 15bins)  across corruption types for ResNet-50.} SMART consistently achieves superior calibration improvements across diverse corruption scenarios, demonstrating exceptional robustness to distribution shifts.}
+\label{fig:resnet50_ece_corruption_analysis}
+\end{figure}
+
+\paragraph{ResNet-50 ECE Analysis} The corruption-specific analysis reveals that SMART demonstrates remarkable consistency, achieving the highest ECE reduction across most corruption categories with improvements often exceeding 20\%. The inclusion of Spline calibration exposes a critical limitation of non-parametric methods: extreme brittleness under distribution shifts. While Spline achieves competitive results on certain corruptions like Snow, it completely fails on others such as Brightness and Contrast, highlighting how non-parametric approaches overfit to validation characteristics and break down when faced with novel corruptions.
+
+This contrasts sharply with SMART's robust performance across all corruption types. The key insight is that SMART's margin indicator captures decision boundary information that remains meaningful regardless of input degradation type—whether geometric distortions, noise, or digital artifacts. Temperature Scaling and other global methods show predictable limitations on uniform corruptions, while parametric methods like PTS exhibit moderate consistency but still significant variability. SMART's sample-specific adaptation based on decision boundary information provides the most reliable calibration improvements, making it uniquely suitable for real-world scenarios where corruption characteristics are unpredictable.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=0.95\textwidth]{figs/resnet50_adaece_reduced_error.pdf}
+\caption{\textbf{AdaECE reduction($\uparrow$, \%, 15bins) across corruption types for ResNet-50.} SMART maintains consistent superiority across corruption types under adaptive binning, confirming robust calibration improvements independent of evaluation methodology.}
+\label{fig:resnet50_adaece_corruption_analysis}
+\end{figure}
+
+\paragraph{ResNet-50 AdaECE Analysis} The AdaECE results closely mirror the ECE patterns, confirming that SMART's calibration improvements are fundamental rather than evaluation artifacts. SMART achieves the highest reduction rates across most corruptions, with particularly strong performance on geometric distortions approaching 25\% improvement. Spline's brittleness persists under adaptive binning—performing reasonably on weather corruptions but failing on uniform transforms, confirming that its limitations stem from overfitting rather than evaluation methodology. The near-identical performance rankings across both metrics demonstrate that SMART's margin approach captures robust calibration signals regardless of how calibration quality is measured.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=0.95\textwidth]{figs/vit_b_16_ece_reduced_error.pdf}
+\caption{\textbf{ECE reduction($\uparrow$, \%, 15bins) across corruption types for ViT-B/16.} Transformer architectures exhibit distinct calibration challenges under corruption, with global methods often failing while SMART maintains consistent improvements.}
+\label{fig:vit_ece_corruption_analysis}
+\end{figure}
+
+\paragraph{ViT-B/16 ECE Analysis} The transformer results reveal striking architectural differences in calibration behavior under corruption. Most notably, Temperature Scaling frequently worsens calibration, showing negative improvements on multiple corruption types including Shot Noise, Speckle Noise, Snow, Brightness, Pixelate, Jpeg Compression, Saturate and Spatter. This demonstrates that transformers' attention mechanisms and different inductive biases make them fundamentally incompatible with global temperature adjustments under distribution shifts.
+
+SMART maintains consistent positive improvements across all corruption types, though generally more modest than with ResNet-50. This architectural difference suggests that while transformers are inherently better calibrated, they also present unique challenges that require more sophisticated approaches than global scaling. The convergence of all methods on Fog corruption (around 25\% improvement) indicates that certain atmospheric corruptions create calibration conditions where architectural differences become less relevant.
+
+A key insight emerges: the margin's decision boundary information remains meaningful across architectures, while global statistics become unreliable for transformers under corruption. PTS and CTS show more consistent improvements than TS, but SMART's sample-specific adaptation consistently outperforms all alternatives, confirming its architectural robustness.
+
+\begin{figure}[ht]
+\centering
+\includegraphics[width=0.95\textwidth]{figs/vit_b_16_adaece_reduced_error.pdf}
+\caption{\textbf{AdaECE reduction($\uparrow$, \%, 15bins) across corruption types for ViT-B/16.} Transformer calibration patterns remain consistent under adaptive binning, confirming architectural-specific calibration challenges and SMART's robustness.}
+\label{fig:vit_adaece_corruption_analysis}
+\end{figure}
+
+\paragraph{ViT-B/16 AdaECE Analysis} The AdaECE results closely replicate the ECE patterns, confirming that transformer calibration behaviors are fundamental architectural characteristics rather than evaluation artifacts. Temperature Scaling's negative performance persists under adaptive binning, while SMART maintains consistent positive improvements across all corruption types. This metric independence demonstrates that SMART's margin approach captures robust decision boundary information that remains effective regardless of how calibration quality is measured.
+
+\section{Margin Perspective on Calibration}
+\label{sec:logit_gap_perspective}
+
+
+\begin{figure*}[ht]
+\centering
+\begin{subfigure}[b]{0.32\textwidth}
+    \includegraphics[width=\textwidth]{figs/overall_reliability_imagenet_vit_b_16.pdf}
+    \caption{Overall calibration for ImgNet}
+    \label{fig:overall_imagenet_vit}
+\end{subfigure}
+\hfill
+\begin{subfigure}[b]{0.32\textwidth}
+    \includegraphics[width=\textwidth]{figs/group_3_reliability_imagenet_vit_b_16.pdf}
+    \caption{High margin samples}
+    \label{fig:high_gap_imagenet_vit}
+\end{subfigure}
+\hfill
+\begin{subfigure}[b]{0.32\textwidth}
+    \includegraphics[width=\textwidth]{figs/group_1_reliability_imagenet_vit_b_16.pdf}
+    \caption{Low margin samples}
+    \label{fig:low_gap_imagenet_vit}
+\end{subfigure}
+
+\begin{subfigure}[b]{0.32\textwidth}
+    \includegraphics[width=\textwidth]{figs/overall_reliability_ciafr100_resnet50.pdf}
+    \caption{Overall calibration for Cifar-100}
+    \label{fig:overall_cifar_resnet}
+\end{subfigure}
+\hfill
+\begin{subfigure}[b]{0.32\textwidth}
+    \includegraphics[width=\textwidth]{figs/group_3_reliability_ciafr100_resnet50.pdf}
+    \caption{High margin samples Cifar-100}
+    \label{fig:high_gap_cifar_resnet}
+\end{subfigure}
+\hfill
+\begin{subfigure}[b]{0.32\textwidth}
+    \includegraphics[width=\textwidth]{figs/logitsgap_vs_confidence_change.pdf}
+    \caption{Margin vs. confidence change}
+    \label{fig:gap_vs_change}
+\end{subfigure}
+
+\caption{\textbf{Margin reveals hidden calibration patterns across the confidence spectrum.} ImageNet ViT-B/16 shows near-perfect overall calibration (a) but reveals systematic under-confidence in high margin samples (b) and well-calibrated low margin samples (c). CIFAR-100 ResNet-50 demonstrates that even with overall over-confidence (d), high margin samples remain under-confident (e). Panel (f) shows SMART provides targeted adjustments whilst TS and PTS show suboptimal patterns.}
+\label{fig:logit_gap_analysis}
+\end{figure*}
+
+Traditional calibration analysis evaluates models from an overall perspective, potentially masking important sample-specific miscalibration patterns. By examining calibration behaviour across margin values, we uncover fundamental insights about how neural networks distribute confidence and validate our method visually. 
+
+Figure~\ref{fig:logit_gap_analysis} demonstrates heterogeneity across margin groups. For ImageNet with ViT-B/16, whilst overall calibration appears near-perfect (Figure~\ref{fig:overall_imagenet_vit}), decomposing by margin reveals distinct patterns: low margin samples achieve good calibration (Figure~\ref{fig:low_gap_imagenet_vit}), whilst high margin samples show systematic under-confidence (Figure~\ref{fig:high_gap_imagenet_vit}). This pattern persists across different conditions, as shown in CIFAR-100 with ResNet-50 (Figures~\ref{fig:overall_cifar_resnet} and~\ref{fig:high_gap_cifar_resnet}), indicating that margin-based groupings reveal fundamental calibration characteristics transcending dataset-specific or architecture-specific behaviors.
+
+\paragraph{The Under-Confidence Paradox in High Margin Samples} Perhaps the most counterintuitive finding emerges from examining high margin samples. Despite representing easy classifications with substantial separation between top predictions, these samples consistently exhibit under-confidence rather than expected over-confidence. High margin samples from ImageNet ViT-B/16 show systematic under-confidence, with predicted confidence consistently lower than empirical accuracy (Figure~\ref{fig:high_gap_imagenet_vit}). This pattern persists even when overall model behaviour differs dramatically, as CIFAR-100 ResNet-50 maintains under-confidence in high margin samples despite overall over-confidence (Figure~\ref{fig:high_gap_cifar_resnet}).
+
+\paragraph{Method-Specific Failures from the Margin Perspective} The confidence adjustment patterns in Figure~\ref{fig:gap_vs_change} expose fundamental limitations in existing approaches. Temperature Scaling's uniform adjustment completely ignores heterogeneous calibration needs across margin groups, applying identical modifications regardless of sample characteristics. More critically, PTS makes substantial adjustments to low margin samples that already achieve good calibration and require minimal intervention. This unnecessary manipulation exemplifies how increased dimensionality introduces noise for precise temperature parameterisation. In contrast, SMART provides minimal adjustments to low margin samples that are already well-calibrated, whilst delivering targeted confidence increases to high margin samples suffering from under-confidence. This adaptive behavior emerges naturally from our lightweight margin-to-temperature mapping, demonstrating how principled architectural choices translate into appropriate calibration strategies.
+
+
+% \section{Additional Visualization of SmoothSoftECE}
+% \label{app:hubersoeftece-viz}
+
+% \begin{figure}[t]
+%     \centering
+%     \begin{minipage}{0.48\linewidth}
+%         \centering
+%         \includegraphics[width=\linewidth]{figs/test_loss_convergence_vit_b_16.pdf}
+%     \end{minipage}\hfill
+%     \begin{minipage}{0.48\linewidth}
+%         \centering
+%         \includegraphics[width=\linewidth]{figs/gradient_norm_evolution_vit_b_16.pdf}
+%     \end{minipage}
+%     \caption{Post-hoc calibration on ImageNet with ViT-B/16: test loss (left) and gradient norm (right) over training epochs for NLL, SoftECE, and SmoothSoftECE.}
+%     \label{fig:vitb16-hubersoeftece}
+% \end{figure}
+
+% Figure~\ref{fig:vitb16-hubersoeftece} illustrates how the three objectives behave during optimization for a ViT-B/16 calibrator on ImageNet. The left panel shows that NLL rapidly plateaus at a comparatively high value, while SoftECE continues to decrease but remains consistently above SmoothSoftECE. SmoothSoftECE converges faster and reaches the lowest final loss. The right panel plots the corresponding gradient norms: SoftECE exhibits large and persistent oscillations, suggesting noisy and unstable gradients; NLL’s gradient norm decays smoothly but, consistent with the loss curve, does not lead to substantial improvement in the calibration objective. In contrast, SmoothSoftECE shows a more gradual and stable decay of the gradient norm, indicating a better-conditioned optimization process. These curves are representative of typical runs and are meant to provide intuition rather than a separate benchmark.
+
+
+
+
+
+\subsection{Sensitivity to Hyperparameters $\lambda$ and $\delta$}
+\label{subsec:hyperparameter_sensitivity}
+
+We examine the sensitivity of SMART's performance to the bandwidth parameter $\lambda$ and Charbonnier smoothing parameter $\delta$ in Equation~\eqref{eq:charbonnier_softece}. Tables~\ref{tab:hyperparam_resnet50} and~\ref{tab:hyperparam_vitb16} report ECE (15 bins) on ImageNet for ResNet-50 and ViT-B/16 across different $(\lambda, \delta)$ combinations.
+
+\begin{table}[h]
+\centering
+\caption{ECE (\%, $\downarrow$, 15 bins) on ImageNet ResNet-50 for different $(\lambda, \delta)$ combinations.}
+\label{tab:hyperparam_resnet50}
+\small
+\begin{tabular}{@{}lcccc@{}}
+\toprule
+$\lambda \backslash \delta$ & 0.001 & 0.010 & 0.100 & 1.000 \\
+\midrule
+0.01 & 0.66 & 0.83 & 1.02 & 0.67 \\
+\cellgray 0.05 & \cellgray 0.61 & \cellgray 0.66 & \cellgray 0.67 & \cellgray 0.66 \\
+0.10 & 0.66 & 3.11 & 0.71 & 0.72 \\
+0.50 & 0.85 & 0.95 & 1.29 & 1.26 \\
+1.00 & 0.79 & 1.15 & 2.49 & 2.51 \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+\begin{table}[h]
+\centering
+\caption{ECE (\%, $\downarrow$, 15 bins) on ImageNet ViT-B/16 for different $(\lambda, \delta)$ combinations.}
+\label{tab:hyperparam_vitb16}
+\small
+\begin{tabular}{@{}lcccc@{}}
+\toprule
+$\lambda \backslash \delta$ & 0.001 & 0.010 & 0.100 & 1.000 \\
+\midrule
+0.01 & 1.32 & 0.84 & 0.78 & 0.85 \\
+\cellgray 0.05 & \cellgray 0.84 & \cellgray 0.80 & \cellgray 0.89 & \cellgray 0.86 \\
+0.10 & 0.99 & 0.97 & 0.81 & 2.26 \\
+0.50 & 2.09 & 2.02 & 2.06 & 2.05 \\
+1.00 & 2.04 & 2.48 & 2.56 & 2.56 \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+The results demonstrate that performance remains stable within a reasonable range of values. For $\lambda \in \{0.01, 0.05, 0.10\}$, ECE varies by less than 0.2\% across different $\delta$ choices, indicating robustness to the Charbonnier smoothing parameter. Larger values ($\lambda \geq 0.50$) lead to degraded performance due to over-localization of kernel weights, creating high variance in calibration estimates. Our choice of $\lambda=0.05$ and $\delta=0.001$ (highlighted rows) provides consistent performance across both CNN and transformer architectures, though the method is not particularly sensitive to $\delta$ within the range $[0.001, 0.100]$ when $\lambda$ is appropriately chosen.
+
+\section{Additional Analysis of the Margin--Temperature Relationship}
+\label{app:margin_temperature}
+
+\begin{figure}[t]
+    \centering
+    \begin{minipage}{0.48\linewidth}
+        \centering
+        \includegraphics[width=\linewidth]{figs/logitsgap_temperature_density_map_resnet50.pdf}
+    \end{minipage}\hfill
+    \begin{minipage}{0.48\linewidth}
+        \centering
+        \includegraphics[width=\linewidth]{figs/logitsgap_temperature_density_map_vit_b_16.pdf}
+    \end{minipage}
+    \caption{Empirical margin--temperature relationship learned by Left: ImageNet with a ResNet-50 backbone, where the mapping is approximately linear and monotone increasing (Pearson $r=0.94$). Right: ImageNet with a ViT-B/16 backbone, where the mapping becomes non-monotonic with a pronounced U-shaped pattern (Pearson $r=-0.63$ for the best linear fit).}
+    \label{fig:margin_temperature_appendix}
+\end{figure}
+
+Figure~\ref{fig:margin_temperature_appendix} illustrates that the learned margin--temperature mapping is not constrained to be monotonic. For ImageNet ResNet-50, the mapping closely follows an increasing linear trend: samples with larger logit margins receive higher temperatures (softer probabilities), while low-margin samples are assigned temperatures closer to one. In contrast, on ImageNet with ViT-B/16 the mapping is clearly non-monotonic, with an approximately U-shaped dependence on the margin. This behavior indicates that the relationship between margin and miscalibration is architecture- and dataset-dependent; SMART adapts to these differences rather than enforcing a fixed monotone form, and understanding the underlying theoretical reasons is left for future work.

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+      \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth}
+        \fancyhf{}
+        \fancyhead[LE,RO]{\thepage}%
+        \fancyhead[RE]{\slshape\leftmark}%
+        \fancyhead[LO]{\slshape\rightmark}%
+        \def\chaptermark##1{%
+          \markboth{\MakeUppercase{%
+            \ifnum \c@secnumdepth >\m@ne \if@mainmatter
+              \@chapapp\ \thechapter. \ \fi\fi##1}}{}}%
+        \def\sectionmark##1{%
+          \markright {\MakeUppercase{%
+            \ifnum \c@secnumdepth >\z@ \thesection. \ \fi##1}}}%
+      }%
+    \else
+      \def\ps@headings{\ps@f@nch@fancyproto \def\@mkboth{\protect\markboth}
+        \fancyhf{}
+        \fancyhead[LE,RO]{\thepage}%
+        \fancyhead[RE]{\slshape\leftmark}%
+        \fancyhead[LO]{\slshape\rightmark}%
+        \def\chaptermark##1{%
+          \markright{\MakeUppercase{%
+            \ifnum \c@secnumdepth >\m@ne \if@mainmatter
+              \@chapapp\ \thechapter. \ \fi\fi##1}}}%
+        \let\sectionmark\@gobble % Not needed but inserted for safety
+      }%
+    \fi
+  }%
+}
+\ProcessOptions*
+\newcommand{\f@nch@forc}[3]{\expandafter\f@nchf@rc\expandafter#1\expandafter{#2}{#3}}
+\newcommand{\f@nchf@rc}[3]{\def\temp@ty{#2}\ifx\@empty\temp@ty\else
+                                    \f@nch@rc#1#2\f@nch@rc{#3}\fi}
+\long\def\f@nch@rc#1#2#3\f@nch@rc#4{\def#1{#2}#4\f@nchf@rc#1{#3}{#4}}
+\newcommand{\f@nch@for}[3]{\edef\@fortmp{#2}%
+  \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}}
+\newcommand\f@nch@default[3]{%
+  \edef\temp@a{\lowercase{\edef\noexpand\temp@a{#3}}}\temp@a \def#1{}%
+  \f@nch@forc\tmpf@ra{#2}%
+  {\expandafter\f@nch@ifin\tmpf@ra\temp@a{\edef#1{#1\tmpf@ra}}{}}%
+  \ifx\@empty#1\def#1{#2}\fi}
+\newcommand{\f@nch@ifin}[4]{%
+  \edef\temp@a{#2}\def\temp@b##1#1##2\temp@b{\def\temp@b{##1}}%
+  \expandafter\temp@b#2#1\temp@b\ifx\temp@a\temp@b #4\else #3\fi}
+\newcommand{\fancyhead}[2][]{\f@nch@fancyhf\fancyhead h[#1]{#2}}%
+\newcommand{\fancyfoot}[2][]{\f@nch@fancyhf\fancyfoot f[#1]{#2}}%
+\newcommand{\fancyhf}[2][]{\f@nch@fancyhf\fancyhf {}[#1]{#2}}%
+\newcommand{\fancyheadoffset}[2][]{\f@nch@fancyhfoffs\fancyheadoffset h[#1]{#2}}%
+\newcommand{\fancyfootoffset}[2][]{\f@nch@fancyhfoffs\fancyfootoffset f[#1]{#2}}%
+\newcommand{\fancyhfoffset}[2][]{\f@nch@fancyhfoffs\fancyhfoffset {}[#1]{#2}}%
+\def\f@nch@fancyhf@Echeck#1{%
+  \if@twoside\else
+    \iff@nch@twoside\else
+      \if\f@nch@@eo e%
+        \PackageWarning{fancyhdr} {\string#1's `E' option without twoside option is useless.\MessageBreak
+          Please consider using the `twoside' option}%
+  \fi\fi\fi
+}
+\long\def\f@nch@fancyhf#1#2[#3]#4{%
+  \def\temp@c{}%
+  \f@nch@forc\tmpf@ra{#3}%
+  {\expandafter\f@nch@ifin\tmpf@ra{eolcrhf,EOLCRHF}%
+    {}{\edef\temp@c{\temp@c\tmpf@ra}}}%
+  \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in
+    \string#1 argument: [#3]}{}%
+  \fi \f@nch@for\temp@c{#3}%
+  {\f@nch@default\f@nch@@eo{eo}\temp@c
+    \f@nch@fancyhf@Echeck{#1}%
+    \f@nch@default\f@nch@@lcr{lcr}\temp@c
+    \f@nch@default\f@nch@@hf{hf}{#2\temp@c}%
+    \f@nch@forc\f@nch@eo\f@nch@@eo
+        {\f@nch@forc\f@nch@lcr\f@nch@@lcr
+          {\f@nch@forc\f@nch@hf\f@nch@@hf
+            {\expandafter\f@nch@def\csname
+              f@nch@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname {#4}}}}}}
+\def\f@nch@fancyhfoffs#1#2[#3]#4{%
+  \def\temp@c{}%
+  \f@nch@forc\tmpf@ra{#3}%
+  {\expandafter\f@nch@ifin\tmpf@ra{eolrhf,EOLRHF}%
+    {}{\edef\temp@c{\temp@c\tmpf@ra}}}%
+  \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in
+    \string#1 argument: [#3]}{}%
+  \fi \f@nch@for\temp@c{#3}%
+  {\f@nch@default\f@nch@@eo{eo}\temp@c
+    \f@nch@fancyhf@Echeck{#1}%
+    \f@nch@default\f@nch@@lcr{lr}\temp@c
+    \f@nch@default\f@nch@@hf{hf}{#2\temp@c}%
+    \f@nch@forc\f@nch@eo\f@nch@@eo
+        {\f@nch@forc\f@nch@lcr\f@nch@@lcr
+          {\f@nch@forc\f@nch@hf\f@nch@@hf
+            {\expandafter\setlength\csname
+              f@nch@offset@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname {#4}}}}}%
+  \f@nch@setoffs}
+\NewDocumentCommand {\fancyheadwidth}{ s O{} O{} m }
+                      {\f@nch@fancyhfwidth{#1}\fancyheadwidth h[#2][#3]{#4}}%
+\NewDocumentCommand {\fancyfootwidth}{ s O{} O{} m }
+                      {\f@nch@fancyhfwidth{#1}\fancyfootwidth f[#2][#3]{#4}}%
+\NewDocumentCommand {\fancyhfwidth}  { s O{} O{} m }
+                      {\f@nch@fancyhfwidth{#1}\fancyhfwidth  {}[#2][#3]{#4}}%
+\def\f@nch@fancyhfwidth#1#2#3[#4][#5]#6{%
+  \setlength\@tempdima{#6}%
+  \def\temp@c{}%
+  \f@nch@forc\tmpf@ra{#4}%
+  {\expandafter\f@nch@ifin\tmpf@ra{eolcrhf,EOLCRHF}%
+    {}{\edef\temp@c{\temp@c\tmpf@ra}}}%
+  \ifx\@empty\temp@c\else \PackageError{fancyhdr}{Illegal char `\temp@c' in
+    \string#2 argument: [#4]}{}%
+  \fi
+  \f@nch@for\temp@c{#4}%
+  {\f@nch@default\f@nch@@eo{eo}\temp@c
+    \f@nch@fancyhf@Echeck{#2}%
+    \f@nch@default\f@nch@@lcr{lcr}\temp@c
+    \f@nch@default\f@nch@@hf{hf}{#3\temp@c}%
+    \f@nch@forc\f@nch@eo\f@nch@@eo
+        {\f@nch@forc\f@nch@lcr\f@nch@@lcr
+          {\f@nch@forc\f@nch@hf\f@nch@@hf
+            {%
+              \IfBooleanTF{#1}{%
+                \expandafter\edef\csname
+                  f@nch@width@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname{\the\@tempdima}%
+              }%
+              {%
+                \expandafter\def\csname
+                  f@nch@width@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname{#6}%
+              }%
+              \csname f@nchdrwdt@align@v@\f@nch@hf\endcsname
+              \edef\f@nch@align@@h{\f@nch@lcr}%
+              \def\temp@a{#5}%
+              \ifx\temp@a\@empty \else \f@nchdrwdt@align#5\@nil{#2}\fi
+              \expandafter\edef\csname
+                f@nch@align@\f@nch@eo\f@nch@lcr\f@nch@hf\endcsname
+                   {\f@nch@align@@v\f@nch@align@@h}}}}}}
+\def\f@nch@width@elh{\headwidth}
+\def\f@nch@width@ech{\headwidth}
+\def\f@nch@width@erh{\headwidth}
+\def\f@nch@width@olh{\headwidth}
+\def\f@nch@width@och{\headwidth}
+\def\f@nch@width@orh{\headwidth}
+\def\f@nch@width@elf{\headwidth}
+\def\f@nch@width@ecf{\headwidth}
+\def\f@nch@width@erf{\headwidth}
+\def\f@nch@width@olf{\headwidth}
+\def\f@nch@width@ocf{\headwidth}
+\def\f@nch@width@orf{\headwidth}
+\def\f@nch@align@elh{bl}
+\def\f@nch@align@ech{bc}
+\def\f@nch@align@erh{br}
+\def\f@nch@align@olh{bl}
+\def\f@nch@align@och{bc}
+\def\f@nch@align@orh{br}
+\def\f@nch@align@elf{tl}
+\def\f@nch@align@ecf{tc}
+\def\f@nch@align@erf{tr}
+\def\f@nch@align@olf{tl}
+\def\f@nch@align@ocf{tc}
+\def\f@nch@align@orf{tr}
+\def\f@nchdrwdt@align@v@h{\def\f@nch@align@@v{b}}%
+\def\f@nchdrwdt@align@v@f{\def\f@nch@align@@v{t}}%
+\long\def\f@nchdrwdt@align#1#2\@nil#3{%
+  \f@nch@ifin{#1}{TtcbB-}{%
+    \f@nch@ifin{#1}{-}{}{\def\f@nch@align@@v{#1}}%
+    \def\@tempa{#2}%
+    \ifx\@tempa\@empty \else \def\f@nch@align@@h{#2}\fi
+  }%
+  {\def\f@nch@align@@h{#1}}%
+  \expandafter\f@nch@ifin\expandafter{\f@nch@align@@h}{lcrj}{}%
+    {\PackageError{fancyhdr}
+                  {\string#3: Illegal char `\f@nch@align@@h'\MessageBreak
+                              in alignment argument}{}}%
+}
+\newcommand{\lhead}[2][\f@nch@olh]%
+                     {\f@nch@def\f@nch@olh{#2}\f@nch@def\f@nch@elh{#1}}
+\newcommand{\chead}[2][\f@nch@och]%
+                     {\f@nch@def\f@nch@och{#2}\f@nch@def\f@nch@ech{#1}}
+\newcommand{\rhead}[2][\f@nch@orh]%
+                     {\f@nch@def\f@nch@orh{#2}\f@nch@def\f@nch@erh{#1}}
+\newcommand{\lfoot}[2][\f@nch@olf]%
+                     {\f@nch@def\f@nch@olf{#2}\f@nch@def\f@nch@elf{#1}}
+\newcommand{\cfoot}[2][\f@nch@ocf]%
+                     {\f@nch@def\f@nch@ocf{#2}\f@nch@def\f@nch@ecf{#1}}
+\newcommand{\rfoot}[2][\f@nch@orf]%
+                     {\f@nch@def\f@nch@orf{#2}\f@nch@def\f@nch@erf{#1}}
+\newlength{\f@nch@headwidth} \let\headwidth\f@nch@headwidth
+\newlength{\f@nch@offset@elh}
+\newlength{\f@nch@offset@erh}
+\newlength{\f@nch@offset@olh}
+\newlength{\f@nch@offset@orh}
+\newlength{\f@nch@offset@elf}
+\newlength{\f@nch@offset@erf}
+\newlength{\f@nch@offset@olf}
+\newlength{\f@nch@offset@orf}
+\newcommand{\headrulewidth}{0.4pt}
+\newcommand{\footrulewidth}{0pt}
+\@ifundefined{headruleskip}%
+      {\newcommand{\headruleskip}{0pt}}{}
+\@ifundefined{footruleskip}%
+      {\newcommand{\footruleskip}{.3\normalbaselineskip}}{}
+\newcommand{\plainheadrulewidth}{0pt}
+\newcommand{\plainfootrulewidth}{0pt}
+\newif\if@fancyplain \@fancyplainfalse
+\def\fancyplain#1#2{\if@fancyplain#1\else#2\fi}
+\headwidth=-123456789sp
+\let\f@nch@raggedleft\raggedleft
+\let\f@nch@raggedright\raggedright
+\let\f@nch@centering\centering
+\let\f@nch@everypar\everypar
+\ifdefined\ExplSyntaxOn
+  \ExplSyntaxOn
+  \providecommand\IfFormatAtLeastTF{\@ifl@t@r\fmtversion}
+  \IfFormatAtLeastTF{2021-06-01}{
+    \def\f@nch@saveclr@parhook #1{
+      \expandafter\let\csname f@nch@__hook~#1\expandafter\endcsname
+                      \csname __hook~#1\endcsname
+      \expandafter\let\csname f@nch@__hook_toplevel~#1\expandafter\endcsname
+                      \csname __hook_toplevel~#1\endcsname
+      \expandafter\let\csname f@nch@__hook_next~#1\expandafter\endcsname
+                      \csname __hook_next~#1\endcsname
+      \expandafter\let\csname f@nch@g__hook_#1_code_prop\expandafter\endcsname
+                      \csname g__hook_#1_code_prop\endcsname
+      \RemoveFromHook{#1}[*]
+      \ClearHookNext{#1}
+    }
+    \def\f@nch@restore@parhook #1{
+      \global\expandafter\let\csname __hook~#1\expandafter\endcsname
+                             \csname f@nch@__hook~#1\endcsname
+      \global\expandafter\let\csname __hook_toplevel~#1\expandafter\endcsname
+                             \csname f@nch@__hook_toplevel~#1\endcsname
+      \global\expandafter\let\csname __hook_next~#1\expandafter\endcsname
+                             \csname f@nch@__hook_next~#1\endcsname
+      \global\expandafter\let\csname g__hook_#1_code_prop\expandafter\endcsname
+                             \csname f@nch@g__hook_#1_code_prop\endcsname
+    }
+    \def\f@nch@resetpar{
+      \f@nch@everypar{}
+      \f@nch@saveclr@parhook{para/before}
+      \f@nch@saveclr@parhook{para/begin}
+      \f@nch@saveclr@parhook{para/end}
+      \f@nch@saveclr@parhook{para/after}
+    }
+    \def\f@nch@restorepar{
+      \f@nch@restore@parhook{para/before}
+      \f@nch@restore@parhook{para/begin}
+      \f@nch@restore@parhook{para/end}
+      \f@nch@restore@parhook{para/after}
+    }
+  }{
+    \def\f@nch@resetpar{
+      \f@nch@everypar{}
+    }
+    \def\f@nch@restorepar{}
+  }
+  \ExplSyntaxOff
+\else
+  \def\f@nch@resetpar{%
+    \f@nch@everypar{}%
+  }
+  \def\f@nch@restorepar{}
+\fi
+\newcommand\f@nch@noUppercase[2][]{#2}
+\def\f@nch@reset{\f@nch@resetpar\restorecr\endlinechar=13
+  \catcode`\\=0\catcode`\{=1\catcode`\}=2\catcode`\$=3\catcode`\&=4
+  \catcode`\#=6\catcode`\^=7\catcode`\_=8\catcode`\ =10\catcode`\@=11
+  \catcode`\:=11\catcode`\~=13\catcode`\%=14
+  \catcode0=15 %NULL
+  \catcode9=10 %TAB
+  \let\\\@normalcr \let\raggedleft\f@nch@raggedleft
+  \let\raggedright\f@nch@raggedright \let\centering\f@nch@centering
+  \def\baselinestretch{1}%
+  \hsize=\headwidth
+  \def\nouppercase##1{{%
+      \let\uppercase\relax\let\MakeUppercase\f@nch@noUppercase
+      \expandafter\let\csname MakeUppercase \endcsname\relax
+      \expandafter\def\csname MakeUppercase\space\space\space\endcsname
+                                                   [####1]####2{####2}%
+      ##1}}%
+  \@ifundefined{@normalsize} {\normalsize} % for ucthesis.cls
+   {\@normalsize}%
+  }
+\newcommand*{\fancycenter}[1][1em]{%
+  \@ifnextchar[{\f@nch@center{#1}}{\f@nch@center{#1}[3]}%
+}
+\def\f@nch@center#1[#2]#3#4#5{%
+  \def\@tempa{#4}\ifx\@tempa\@empty
+    \hbox to\linewidth{\color@begingroup{#3}\hfil {#5}\color@endgroup}%
+  \else
+    \setlength\@tempdima{#1}%
+    \setlength{\@tempdimb}{#2\@tempdima}%
+    \@tempdimc \@tempdimb \advance\@tempdimc -\@tempdima
+    \setlength\@tempskipa{\@tempdimb \@plus 1fil \@minus \@tempdimc}%
+    \@tempskipb\@tempskipa
+    \def\@tempa{#3}\ifx\@tempa\@empty
+      \addtolength\@tempskipa{\z@ \@minus \@tempdima}%
+    \fi
+    \def\@tempa{#5}\ifx\@tempa\@empty % empty right
+      \addtolength\@tempskipb{\z@ \@minus \@tempdima}%
+    \fi
+    \settowidth{\@tempdimb}{#3}%
+    \settowidth{\@tempdimc}{#5}%
+    \ifdim\@tempdimb>\@tempdimc
+      \advance\@tempdimb -\@tempdimc
+      \addtolength\@tempskipb{\@tempdimb \@minus \@tempdimb}%
+    \else
+      \advance\@tempdimc -\@tempdimb
+      \addtolength\@tempskipa{\@tempdimc \@minus \@tempdimc}%
+    \fi
+    \hbox to\linewidth{\color@begingroup{#3}\hskip \@tempskipa
+                      {#4}\hskip \@tempskipb {#5}\color@endgroup}%
+  \fi
+}
+\newcommand{\f@nch@headinit}{}
+\newcommand{\fancyheadinit}[1]{%
+  \def\f@nch@headinit{#1}%
+}
+\newcommand{\f@nch@footinit}{}
+\newcommand{\fancyfootinit}[1]{%
+  \def\f@nch@footinit{#1}%
+}
+\newcommand{\fancyhfinit}[1]{%
+  \def\f@nch@headinit{#1}%
+  \def\f@nch@footinit{#1}%
+}
+\ifdefined\NewMirroredHookPair
+  \NewMirroredHookPair{fancyhdr/before}{fancyhdr/after}
+  \NewMirroredHookPair{fancyhdr/head/begin}{fancyhdr/head/end}
+  \NewMirroredHookPair{fancyhdr/foot/begin}{fancyhdr/foot/end}
+\fi
+\newlength\f@nch@height
+\newlength\f@nch@footalignment
+\newif\iff@nch@footalign\f@nch@footalignfalse
+\newcommand{\fancyfootalign}[1]{%
+  \def\temp@a{#1}%
+  \ifx\temp@a\@empty
+    \f@nch@footalignfalse
+  \else
+    \f@nch@footaligntrue
+    \setlength\f@nch@footalignment{#1}%
+  \fi
+}
+\newcommand\fancyhdrsettoheight[2]{%
+  \expandafter\ifx\csname f@nch@#2\endcsname\fancyhdrsettoheight
+    \else\PackageError{fancyhdr}{Unknown parameter #2 in \string\fancyhdrsettoheight}{}\fi
+  \setbox\@tempboxa\hbox{{\f@nch@checkfalse\csname @#2\endcsname}}%
+  \setlength{#1}\f@nch@height
+  \setbox\@tempboxa\box\voidb@x
+}
+\let\f@nch@oddhead\fancyhdrsettoheight
+\let\f@nch@evenhead\fancyhdrsettoheight
+\let\f@nch@oddfoot\fancyhdrsettoheight
+\let\f@nch@evenfoot\fancyhdrsettoheight
+\newcommand\f@nch@vbox[2]{%
+  \setbox0\vbox{#2}%
+  \global\f@nch@height=\ht0
+  \ifdim\ht0>#1\relax
+    \iff@nch@check
+      \dimen0=#1\advance\dimen0-\ht0
+      \PackageWarning{fancyhdr}{%
+        \string#1 is too small (\the#1): \MessageBreak
+        Make it at least \the\ht0, for example:\MessageBreak
+        \string\setlength{\string#1}{\the\ht0}%
+        \iff@nch@compatViii .\MessageBreak
+        We now make it that large for the rest of the document.\MessageBreak
+        This may cause the page layout to be inconsistent, however
+        \fi
+        \ifx#1\headheight .\MessageBreak
+          You might also make \topmargin smaller:\MessageBreak
+          \string\addtolength{\string\topmargin}{\the\dimen0}%
+        \fi
+        \@gobble
+      }%
+      \iff@nch@compatViii
+        \dimen0=#1\relax
+        \global#1=\ht0\relax
+        \ht0=\dimen0 %
+      \else
+        \ht0=#1\relax
+      \fi
+    \else
+      \ht0=#1\relax
+    \fi
+  \fi
+  \box0}
+\newcommand\f@nch@head[6]{%
+  \f@nch@reset
+  \ifdefined\UseHook\UseHook{fancyhdr/before}\UseHook{fancyhdr/head/begin}\fi
+  \f@nch@headinit\relax
+  #1%
+  \hbox to\headwidth{%
+    \f@nch@vbox\headheight{%
+      \f@nch@hfbox{#2}{#3}{#4}{#6}{h}%
+      \vskip\headruleskip\relax
+      \headrule
+    }%
+  }%
+  #5%
+  \ifdefined\UseHook\UseHook{fancyhdr/head/end}\UseHook{fancyhdr/after}\fi
+  \f@nch@restorepar
+}
+\newcommand\f@nch@foot[6]{%
+  \f@nch@reset
+  \ifdefined\UseHook\UseHook{fancyhdr/before}\UseHook{fancyhdr/foot/begin}\fi
+  \f@nch@footinit\relax
+  #1%
+  \hbox to\headwidth{%
+    \f@nch@vbox\footskip{%
+      \setbox0=\vbox{\footrule}\unvbox0
+      \vskip\footruleskip
+      \f@nch@hfbox{#2}{#3}{#4}{#6}{f}%
+    \iff@nch@footalign \vskip\f@nch@footalignment \fi
+    }%
+  }%
+  #5%
+  \ifdefined\UseHook\UseHook{fancyhdr/foot/end}\UseHook{fancyhdr/after}\fi
+  \f@nch@restorepar
+}
+\newlength\f@nch@widthL
+\newlength\f@nch@widthC
+\newlength\f@nch@widthR
+\newcommand\f@nch@hfbox[5]{%
+  \setlength\f@nch@widthL{\csname f@nch@width@#4l#5\endcsname}%
+  \setlength\f@nch@widthC{\csname f@nch@width@#4c#5\endcsname}%
+  \setlength\f@nch@widthR{\csname f@nch@width@#4r#5\endcsname}%
+  \let\@tempa\f@nch@hfbox@center
+  \ifdim \dimexpr \f@nch@widthL+\f@nch@widthC+\f@nch@widthR>\headwidth
+  \else
+    \ifdim \dimexpr \f@nch@widthL+0.5\f@nch@widthC>0.5\headwidth
+      \let \@tempa\f@nch@hfbox@fit
+    \fi
+    \ifdim \dimexpr \f@nch@widthR+0.5\f@nch@widthC>0.5\headwidth
+      \let \@tempa\f@nch@hfbox@fit
+    \fi
+  \fi
+  \@tempa{#1}{#2}{#3}#4#5%
+}
+\newcommand\f@nch@hfbox@center[5]{%
+  \hbox to \headwidth{%
+    \rlap{\f@nch@parbox{#1}\f@nch@widthL{#4}l{#5}}%
+    \hfill
+    \f@nch@parbox{#2}\f@nch@widthC{#4}c{#5}%
+    \hfill
+    \llap{\f@nch@parbox{#3}\f@nch@widthR{#4}r{#5}}%
+  }%
+}
+\newcommand\f@nch@hfbox@fit[5]{%
+  \hbox to \headwidth{%
+    \f@nch@parbox{#1}\f@nch@widthL{#4}l{#5}%
+    \hfill
+    \f@nch@parbox{#2}\f@nch@widthC{#4}c{#5}%
+    \hfill
+    \f@nch@parbox{#3}\f@nch@widthR{#4}r{#5}%
+  }%
+}%
+\newcommand\f@nch@parbox[5]{%
+  \expandafter\expandafter\expandafter\f@nch@parbox@align
+                     \csname f@nch@align@#3#4#5\endcsname
+  \parbox[\f@nch@align@@v]{#2}%
+    {%
+      \f@nch@align@@pre
+      \f@nch@align@@h\leavevmode\ignorespaces#1%
+      \f@nch@align@@post
+    }%
+}
+\newcommand\f@nch@parbox@align[2]{%
+  \def\f@nch@align@@pre{}%
+  \def\f@nch@align@@post{}%
+  \csname f@nch@parbox@align@v#1\endcsname
+  \csname f@nch@parbox@align@h#2\endcsname
+}
+\def\f@nch@parbox@align@vT{\def\f@nch@align@@v{t}\def\f@nch@align@@pre{\vspace{0pt}}}
+\def\f@nch@parbox@align@vt{\def\f@nch@align@@v{t}}
+\def\f@nch@parbox@align@vc{\def\f@nch@align@@v{c}}
+\def\f@nch@parbox@align@vb{\def\f@nch@align@@v{b}}
+\def\f@nch@parbox@align@vB{\def\f@nch@align@@v{b}\def\f@nch@align@@post{\vspace{0pt}}}
+\def\f@nch@parbox@align@hl{\def\f@nch@align@@h{\raggedright}}
+\def\f@nch@parbox@align@hc{\def\f@nch@align@@h{\centering}}
+\def\f@nch@parbox@align@hr{\def\f@nch@align@@h{\raggedleft}}
+\def\f@nch@parbox@align@hj{\def\f@nch@align@@h{}}
+\@ifundefined{@chapapp}{\let\@chapapp\chaptername}{}%
+\def\f@nch@initialise{%
+  \@ifundefined{chapter}%
+   {\def\sectionmark##1{\markboth{\MakeUppercase{\ifnum \c@secnumdepth>\z@
+          \thesection\hskip 1em\relax
+        \fi ##1}}{}}%
+    \def\subsectionmark##1{\markright {\ifnum \c@secnumdepth >\@ne
+      \thesubsection\hskip 1em\relax \fi ##1}}}%
+   {\def\chaptermark##1{\markboth {\MakeUppercase{\ifnum
+        \c@secnumdepth>\m@ne \@chapapp\ \thechapter. \ \fi ##1}}{}}%
+    \def\sectionmark##1{\markright{\MakeUppercase{\ifnum \c@secnumdepth >\z@
+        \thesection. \ \fi ##1}}}%
+   }%
+  \def\headrule{{\if@fancyplain\let\headrulewidth\plainheadrulewidth\fi
+      \hrule\@height\headrulewidth\@width\headwidth
+      \vskip-\headrulewidth}}%
+  \def\footrule{{\if@fancyplain\let\footrulewidth\plainfootrulewidth\fi
+      \hrule\@width\headwidth\@height\footrulewidth}}%
+  \def\headrulewidth{0.4pt}%
+  \def\footrulewidth{0pt}%
+  \def\headruleskip{0pt}%
+  \def\footruleskip{0.3\normalbaselineskip}%
+  \fancyhf{}%
+  \if@twoside
+    \fancyhead[el,or]{\fancyplain{}{\slshape\rightmark}}%
+    \fancyhead[er,ol]{\fancyplain{}{\slshape\leftmark}}%
+  \else
+    \fancyhead[l]{\fancyplain{}{\slshape\rightmark}}%
+    \fancyhead[r]{\fancyplain{}{\slshape\leftmark}}%
+  \fi
+  \fancyfoot[c]{\rmfamily\thepage}% page number
+}
+\f@nch@initialise
+\def\ps@f@nch@fancyproto{%
+  \ifdim\headwidth<0sp
+    \global\advance\headwidth123456789sp\global\advance\headwidth\textwidth
+  \fi
+  \gdef\ps@f@nch@fancyproto{\@fancyplainfalse\ps@f@nch@fancycore}%
+  \@fancyplainfalse\ps@f@nch@fancycore
+}%
+\@namedef{f@nch@ps@f@nch@fancyproto-is-fancyhdr}{}
+\def\ps@fancy{\ps@f@nch@fancyproto}
+\@namedef{f@nch@ps@fancy-is-fancyhdr}{}
+\def\ps@fancyplain{\ps@f@nch@fancyproto \let\ps@plain\ps@plain@fancy}
+\def\ps@plain@fancy{\@fancyplaintrue\ps@f@nch@fancycore}
+\let\f@nch@ps@empty\ps@empty
+\def\ps@f@nch@fancycore{%
+  \f@nch@ps@empty
+  \def\@mkboth{\protect\markboth}%
+  \def\f@nch@oddhead{\f@nch@head\f@nch@Oolh\f@nch@olh\f@nch@och\f@nch@orh\f@nch@Oorh{o}}%
+  \def\@oddhead{%
+    \iff@nch@twoside
+      \ifodd\c@page
+        \f@nch@oddhead
+      \else
+        \@evenhead
+      \fi
+    \else
+      \f@nch@oddhead
+    \fi
+  }
+  \def\f@nch@oddfoot{\f@nch@foot\f@nch@Oolf\f@nch@olf\f@nch@ocf\f@nch@orf\f@nch@Oorf{o}}%
+  \def\@oddfoot{%
+    \iff@nch@twoside
+      \ifodd\c@page
+        \f@nch@oddfoot
+      \else
+        \@evenfoot
+      \fi
+    \else
+      \f@nch@oddfoot
+    \fi
+  }
+  \def\@evenhead{\f@nch@head\f@nch@Oelh\f@nch@elh\f@nch@ech\f@nch@erh\f@nch@Oerh{e}}%
+  \def\@evenfoot{\f@nch@foot\f@nch@Oelf\f@nch@elf\f@nch@ecf\f@nch@erf\f@nch@Oerf{e}}%
+}
+\def\f@nch@Oolh{\if@reversemargin\hss\else\relax\fi}
+\def\f@nch@Oorh{\if@reversemargin\relax\else\hss\fi}
+\let\f@nch@Oelh\f@nch@Oorh
+\let\f@nch@Oerh\f@nch@Oolh
+\let\f@nch@Oolf\f@nch@Oolh
+\let\f@nch@Oorf\f@nch@Oorh
+\let\f@nch@Oelf\f@nch@Oelh
+\let\f@nch@Oerf\f@nch@Oerh
+\def\f@nch@offsolh{\headwidth=\textwidth\advance\headwidth\f@nch@offset@olh
+                   \advance\headwidth\f@nch@offset@orh\hskip-\f@nch@offset@olh}
+\def\f@nch@offselh{\headwidth=\textwidth\advance\headwidth\f@nch@offset@elh
+                   \advance\headwidth\f@nch@offset@erh\hskip-\f@nch@offset@elh}
+\def\f@nch@offsolf{\headwidth=\textwidth\advance\headwidth\f@nch@offset@olf
+                   \advance\headwidth\f@nch@offset@orf\hskip-\f@nch@offset@olf}
+\def\f@nch@offself{\headwidth=\textwidth\advance\headwidth\f@nch@offset@elf
+                   \advance\headwidth\f@nch@offset@erf\hskip-\f@nch@offset@elf}
+\def\f@nch@setoffs{%
+  \f@nch@gbl\let\headwidth\f@nch@headwidth
+  \f@nch@gbl\def\f@nch@Oolh{\f@nch@offsolh}%
+  \f@nch@gbl\def\f@nch@Oelh{\f@nch@offselh}%
+  \f@nch@gbl\def\f@nch@Oorh{\hss}%
+  \f@nch@gbl\def\f@nch@Oerh{\hss}%
+  \f@nch@gbl\def\f@nch@Oolf{\f@nch@offsolf}%
+  \f@nch@gbl\def\f@nch@Oelf{\f@nch@offself}%
+  \f@nch@gbl\def\f@nch@Oorf{\hss}%
+  \f@nch@gbl\def\f@nch@Oerf{\hss}%
+}
+\newif\iff@nch@footnote
+\AtBeginDocument{%
+  \let\latex@makecol\@makecol
+  \def\@makecol{\ifvoid\footins\f@nch@footnotefalse\else\f@nch@footnotetrue\fi
+    \let\f@nch@topfloat\@toplist\let\f@nch@botfloat\@botlist\latex@makecol}%
+}
+\newcommand\iftopfloat[2]{\ifx\f@nch@topfloat\@empty #2\else #1\fi}%
+\newcommand\ifbotfloat[2]{\ifx\f@nch@botfloat\@empty #2\else #1\fi}%
+\newcommand\iffloatpage[2]{\if@fcolmade #1\else #2\fi}%
+\newcommand\iffootnote[2]{\iff@nch@footnote #1\else #2\fi}%
+\ifx\@temptokenb\undefined \csname newtoks\endcsname\@temptokenb\fi
+\newif\iff@nch@pagestyle@star
+\newcommand\fancypagestyle{%
+  \@ifstar{\f@nch@pagestyle@startrue\f@nch@pagestyle}%
+          {\f@nch@pagestyle@starfalse\f@nch@pagestyle}%
+}
+\newcommand\f@nch@pagestyle[1]{%
+  \@ifnextchar[{\f@nch@@pagestyle{#1}}{\f@nch@@pagestyle{#1}[f@nch@fancyproto]}%
+}
+\long\def\f@nch@@pagestyle#1[#2]#3{%
+  \@ifundefined{ps@#2}{%
+    \PackageError{fancyhdr}{\string\fancypagestyle: Unknown base page style `#2'}{}%
+  }{%
+    \@ifundefined{f@nch@ps@#2-is-fancyhdr}{%
+      \PackageError{fancyhdr}{\string\fancypagestyle: Base page style `#2' is not fancyhdr-based}{}%
+    }%
+    {%
+      \f@nch@pagestyle@setup
+      \def\temp@b{\@namedef{ps@#1}}%
+      \expandafter\temp@b\expandafter{\the\@temptokenb
+          \let\f@nch@gbl\relax\@nameuse{ps@#2}#3\relax}%
+      \@namedef{f@nch@ps@#1-is-fancyhdr}{}%
+    }%
+  }%
+}
+\newcommand\f@nch@pagestyle@setup{%
+  \iff@nch@pagestyle@star
+    \iff@nch@check\@temptokenb={\f@nch@checktrue}\else\@temptokenb={\f@nch@checkfalse}\fi
+    \@tfor\temp@a:=
+      \f@nch@olh\f@nch@och\f@nch@orh\f@nch@elh\f@nch@ech\f@nch@erh
+      \f@nch@olf\f@nch@ocf\f@nch@orf\f@nch@elf\f@nch@ecf\f@nch@erf
+      \f@nch@width@elh\f@nch@width@ech\f@nch@width@erh\f@nch@width@olh
+      \f@nch@width@och\f@nch@width@orh\f@nch@width@elf\f@nch@width@ecf
+      \f@nch@width@erf\f@nch@width@olf\f@nch@width@ocf\f@nch@width@orf
+      \f@nch@align@elh\f@nch@align@ech\f@nch@align@erh\f@nch@align@olh
+      \f@nch@align@och\f@nch@align@orh\f@nch@align@elf\f@nch@align@ecf
+      \f@nch@align@erf\f@nch@align@olf\f@nch@align@ocf\f@nch@align@orf
+      \f@nch@Oolh\f@nch@Oorh\f@nch@Oelh\f@nch@Oerh
+      \f@nch@Oolf\f@nch@Oorf\f@nch@Oelf\f@nch@Oerf
+      \f@nch@headinit\f@nch@footinit
+      \headrule\headrulewidth\footrule\footrulewidth
+    \do {%
+      \toks@=\expandafter\expandafter\expandafter{\temp@a}%
+      \toks@=\expandafter\expandafter\expandafter{%
+        \expandafter\expandafter\expandafter\def
+        \expandafter\expandafter\temp@a\expandafter{\the\toks@}}%
+      \edef\temp@b{\@temptokenb={\the\@temptokenb\the\toks@}}%
+      \temp@b
+    }%
+    \@tfor\temp@a:=
+      \f@nch@offset@olh\f@nch@offset@orh\f@nch@offset@elh\f@nch@offset@erh
+      \f@nch@offset@olf\f@nch@offset@orf\f@nch@offset@elf\f@nch@offset@erf
+    \do {%
+      \toks@=\expandafter\expandafter\expandafter{\expandafter\the\temp@a}%
+      \toks@=\expandafter\expandafter\expandafter{%
+        \expandafter\expandafter\expandafter\setlength
+        \expandafter\expandafter\temp@a\expandafter{\the\toks@}}%
+      \edef\temp@b{\@temptokenb={\the\@temptokenb\the\toks@}}%
+      \temp@b
+    }%
+  \else
+    \@temptokenb={}%
+  \fi
+}
+\newcommand\fancypagestyleassign[2]{%
+  \@ifundefined{ps@#2}{%
+    \PackageError{fancyhdr}{\string\fancypagestyleassign: Unknown page style `#2'}{}%
+    }{%
+     \expandafter\let
+       \csname ps@#1\expandafter\endcsname
+       \csname ps@#2\endcsname
+     \@ifundefined{f@nch@ps@#2-is-fancyhdr}{%
+       \expandafter\let\csname f@nch@ps@#1-is-fancyhdr\endcsname\@undefined
+     }{%
+       \@namedef{f@nch@ps@#1-is-fancyhdr}{}%
+     }%
+   }%
+}
+\fancypagestyle*{fancydefault}{\f@nch@initialise}
+\def\f@nchdrbox@topstrut{\vrule height\ht\strutbox width\z@}
+\def\f@nchdrbox@botstrut{\vrule depth\dp\strutbox width\z@}
+\def\f@nchdrbox@nostrut{\noalign{\vspace{0pt}}\let\f@nchdrbox@@crstrut\f@nchdrbox@botstrut}
+\NewDocumentCommand{\fancyhdrbox}{ O{cl} o m }{%
+\begingroup
+  \let\f@nchdrbox@@pre\f@nchdrbox@topstrut
+  \let\f@nchdrbox@@postx\f@nchdrbox@botstrut
+  \let\f@nchdrbox@@posty\relax
+  \let\f@nchdrbox@@crstrut\strut
+  \IfNoValueTF{#2}%
+    {\let\f@nchdrbox@@halignto\@empty}%
+    {\setlength\@tempdima{#2}%
+      \def\f@nchdrbox@@halignto{to\@tempdima}}%
+  \def\@tempa{#1}%
+  \ifx\@tempa\@empty
+    \f@nchdrbox@align cl\@nil{#3}%
+  \else
+    \f@nchdrbox@align #1\@nil{#3}%
+  \fi
+\endgroup
+}
+\protected\def\f@nchdrbox@cr{%
+  {\ifnum0=`}\fi\@ifstar\@f@nchdrbox@xcr\@f@nchdrbox@xcr}
+
+\def\@f@nchdrbox@xcr{%
+  \unskip\f@nchdrbox@@crstrut
+  \@ifnextchar[\@f@nchdrbox@argc{\ifnum0=`{\fi}\cr}%
+}
+
+\def\@f@nchdrbox@argc[#1]{%
+  \ifnum0=`{\fi}%
+    \ifdim #1>\z@
+      \unskip\@f@nchdrbox@xargc{#1}%
+    \else
+      \@f@nchdrbox@yargc{#1}%
+    \fi}
+
+\def\@f@nchdrbox@xargc#1{\@tempdima #1\advance\@tempdima \dp \strutbox
+   \vrule \@height\z@ \@depth\@tempdima \@width\z@ \cr}
+
+\def\@f@nchdrbox@yargc#1{\cr\noalign{\setlength\@tempdima{#1}\vskip\@tempdima}}
+\def\f@nchdrbox@T{\let\f@nchdrbox@@pre\f@nchdrbox@nostrut
+                  \f@nchdrbox@t}
+\def\f@nchdrbox@t{\def\f@nchdrbox@@v{t}\def\f@nchdrbox@@h{l}}
+\def\f@nchdrbox@c{\def\f@nchdrbox@@v{c}\def\f@nchdrbox@@h{c}}
+\def\f@nchdrbox@b{\def\f@nchdrbox@@v{b}\def\f@nchdrbox@@h{l}}
+\def\f@nchdrbox@B{\let\f@nchdrbox@@postx\relax
+                  \def\f@nchdrbox@@posty{\vspace{0pt}}%
+                  \f@nchdrbox@b}
+\long\def\f@nchdrbox@align#1#2\@nil#3{%
+  \f@nch@ifin{#1}{TtcbB}{%
+    \@nameuse{f@nchdrbox@#1}%
+    \def\@tempa{#2}%
+    \ifx\@tempa\@empty\else \def\f@nchdrbox@@h{#2}\fi
+  }%
+  {\def\f@nchdrbox@@v{c}\def\f@nchdrbox@@h{#1}}%
+  \expandafter\f@nch@ifin\expandafter{\f@nchdrbox@@h}{lcr}{}%
+  {\PackageError{fancyhdr}{\string\fancyhdrbox: Illegal char `\f@nchdrbox@@h'\MessageBreak
+                            in alignment argument}{}}%
+  \let\\\f@nchdrbox@cr
+  \setbox0=\if \f@nchdrbox@@v t\vtop
+  \else \vbox
+  \fi
+  {%
+     \ialign \f@nchdrbox@@halignto
+     \bgroup \relax
+     {\if \f@nchdrbox@@h l\hskip 1sp\else \hfil \fi
+       \ignorespaces ##\unskip
+       \if\f@nchdrbox@@h r\else \hfil \fi
+     }%
+     \tabskip\z@skip \cr
+     \f@nchdrbox@@pre
+     #3\unskip \f@nchdrbox@@postx
+     \crcr
+     \egroup
+     \f@nchdrbox@@posty
+  }%
+  \if\f@nchdrbox@@v c\@tempdima=\ht0\advance\@tempdima\dp0%
+    \ht0=0.5\@tempdima\dp0=0.5\@tempdima\fi
+  \leavevmode \box0
+}
+\@ifclassloaded{newlfm}
+{
+  \let\ps@@empty\f@nch@ps@empty
+  \AtBeginDocument{%
+    \renewcommand{\@zfancyhead}[5]{\relax\hbox to\headwidth{\f@nch@reset
+      \@zfancyvbox\headheight{\hbox
+        {\rlap{\parbox[b]{\headwidth}{\raggedright\f@nch@olh}}\hfill
+          \parbox[b]{\headwidth}{\centering\f@nch@olh}\hfill
+          \llap{\parbox[b]{\headwidth}{\raggedleft\f@nch@orh}}}%
+        \zheadrule}}\relax}%
+  }
+}
+{}
+\endinput
+%%
+%% End of file `fancyhdr.sty'.

icml2026/figs/logitsgap_vs_confidence_change.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:761bf2e01ccfe34bc0fe0026e0541276d3011c3293463727f89ad03bdd844fef
+size 1905210

icml2026/icml2026.bib

@@ -0,0 +1,1223 @@
+@misc{zwaan2019bridging,
+  title={Bridging the gap between uncertainty, confidence and diagnostic accuracy: calibration is key},
+  author={Zwaan, Laura and Hautz, Wolf E},
+  journal={BMJ quality \& safety},
+  volume={28},
+  number={5},
+  pages={352--355},
+  year={2019},
+  publisher={BMJ Publishing Group Ltd}
+}
+
+
+
+
+
+@inproceedings{luo2025beyond,
+  added-at = {2025-12-05T00:00:00.000+0100},
+  author = {Luo, Haoyang and Tao, Linwei and Dong, Minjing and Xu, Chang},
+  biburl = {https://www.bibsonomy.org/bibtex/2ff6f9571f60a72e2d6b71dcae9c0d0f9/dblp},
+  booktitle = {ICML},
+  ee = {https://openreview.net/forum?id=zDZkyvZqFR},
+  interhash = {58681d87ddf6d793599ce25fe4bdffed},
+  intrahash = {ff6f9571f60a72e2d6b71dcae9c0d0f9},
+  keywords = {dblp},
+  publisher = {OpenReview.net},
+  timestamp = {2025-12-08T09:10:01.000+0100},
+  title = {Beyond One-Hot Labels: Semantic Mixing for Model Calibration.},
+  url = {http://dblp.uni-trier.de/db/conf/icml/icml2025.html#LuoTD025},
+  year = 2025
+}
+
+
+@inproceedings{tao2023dual,
+  title={Dual focal loss for calibration},
+  author={Tao, Linwei and Dong, Minjing and Xu, Chang},
+  booktitle={International Conference on Machine Learning},
+  pages={33833--33849},
+  year={2023},
+  organization={PMLR}
+}
+
+@inproceedings{gupta2021calibration,
+  title={Calibration of neural networks using splines},
+  author={Gupta, Kartik and Rahimi, Amir and Ajanthan, Thalaiyasingam and Mensink, Thomas and Sminchisescu, Cristian and Hartley, Richard},
+  booktitle={International Conference on Learning Representations},
+  year={2021}
+}
+
+@article{yang2023beyond,
+  title={Beyond probability partitions: Calibrating neural networks with semantic aware grouping},
+  author={Yang, Jia-Qi and Zhan, De-Chuan and Gan, Le},
+  journal={Advances in Neural Information Processing Systems},
+  volume={36},
+  pages={58448--58460},
+  year={2023}
+}
+@inproceedings{hendrycks2019benchmarking,
+  title={Benchmarking Neural Network Robustness to Common Corruptions and Perturbations},
+  author={Hendrycks, Dan and Dietterich, Thomas},
+  booktitle={International Conference on Learning Representations (ICLR)},
+  year={2019}
+}
+@inproceedings{kakade2004deterministic,
+  title={Deterministic calibration and Nash equilibrium},
+  author={Kakade, Sham M and Foster, Dean P},
+  booktitle={International Conference on Computational Learning Theory},
+  pages={33--48},
+  year={2004},
+  organization={Springer}
+}
+@inproceedings{dosovitskiy2021an,
+  title={An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale},
+  author={Dosovitskiy, Alexey and Beyer, Lucas and Kolesnikov, Alexander and Weissenborn, Dirk and Zhai, Xiaohua and Unterthiner, Thomas and Dehghani, Mostafa and Minderer, Matthias and Heigold, Georg and Gelly, Sylvain and Uszkoreit, Jakob and Houlsby, Neil},
+  booktitle={International Conference on Learning Representations},
+  year={2021}
+}
+@article{blasiok2023does,
+  title={When does optimizing a proper loss yield calibration?},
+  author={Blasiok, Jaroslaw and Gopalan, Parikshit and Hu, Lunjia and Nakkiran, Preetum},
+  journal={Advances in Neural Information Processing Systems},
+  volume={36},
+  pages={72071--72095},
+  year={2023}
+}
+@inproceedings{liu2019large,
+  title={Large-scale long-tailed recognition in an open world},
+  author={Liu, Ziwei and Miao, Zhongqi and Zhan, Xiaohang and Wang, Jiayun and Gong, Boqing and Yu, Stella X},
+  booktitle={Proceedings of the IEEE/CVF conference on computer vision and pattern recognition},
+  pages={2537--2546},
+  year={2019}
+}
+@inproceedings{liu2021swin,
+  title={Swin transformer: Hierarchical vision transformer using shifted windows},
+  author={Liu, Ze and Lin, Yutong and Cao, Yue and Hu, Han and Wei, Yixuan and Zhang, Zheng and Lin, Stephen and Guo, Baining},
+  booktitle={Proceedings of the IEEE/CVF international conference on computer vision},
+  pages={10012--10022},
+  year={2021}
+}
+@article{paszke2019pytorch,
+  title={Pytorch: An imperative style, high-performance deep learning library},
+  author={Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and others},
+  journal={Advances in neural information processing systems},
+  volume={32},
+  year={2019}
+}
+@article{wang2019learning,
+  title={Learning robust global representations by penalizing local predictive power},
+  author={Wang, Haohan and Ge, Songwei and Lipton, Zachary and Xing, Eric P},
+  journal={Advances in neural information processing systems},
+  volume={32},
+  year={2019}
+}
+@article{xiong2024proximity,
+  title={Proximity-informed calibration for deep neural networks},
+  author={Xiong, Miao and Deng, Ailin and Koh, Pang Wei W and Wu, Jiaying and Li, Shen and Xu, Jianqing and Hooi, Bryan},
+  journal={Advances in Neural Information Processing Systems},
+  volume={36},
+  pages={68511--68538},
+  year={2024}
+}
+@inproceedings{tao2025feature,
+  title={Feature clipping for uncertainty calibration},
+  author={Tao, Linwei and Dong, Minjing and Xu, Chang},
+  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
+  volume={39},
+  number={19},
+  pages={20841--20849},
+  year={2025}
+}
+
+@inproceedings{frenkel2021network,
+  title={Network calibration by class-based temperature scaling},
+  author={Frenkel, Lior and Goldberger, Jacob},
+  booktitle={2021 29th European Signal Processing Conference (EUSIPCO)},
+  pages={1486--1490},
+  year={2021},
+  organization={IEEE}
+}
+@inproceedings{tomani2022parameterized,
+  title={Parameterized temperature scaling for boosting the expressive power in post-hoc uncertainty calibration},
+  author={Tomani, Christian and Cremers, Daniel and Buettner, Florian},
+  booktitle={European conference on computer vision},
+  pages={555--569},
+  year={2022},
+  organization={Springer}
+}
+@article{goodfellow2014explaining,
+  title={Explaining and harnessing adversarial examples},
+  author={Goodfellow, Ian J and Shlens, Jonathon and Szegedy, Christian},
+  journal={arXiv preprint arXiv:1412.6572},
+  year={2014}
+}
+@inproceedings{ding2021local,
+  title={Local Temperature Scaling for Probability Calibration},
+  author={Ding, Zhipeng and Han, Xu and Liu, Peirong and Niethammer, Marc},
+  booktitle={International Conference on Computer Vision},
+  year={2021}
+}
+@inproceedings{hendrycksaugmix,
+  title={AugMix: A Simple Data Processing Method to Improve Robustness and Uncertainty},
+  author={Hendrycks, Dan and Mu, Norman and Cubuk, Ekin Dogus and Zoph, Barret and Gilmer, Justin and Lakshminarayanan, Balaji},
+  booktitle={International Conference on Learning Representations},
+  year={2019}
+}
+@inproceedings{wei2022mitigating,
+  title={Mitigating neural network overconfidence with logit normalization},
+  author={Wei, Hongxin and Xie, Renchunzi and Cheng, Hao and Feng, Lei and An, Bo and Li, Yixuan},
+  booktitle={International conference on machine learning},
+  pages={23631--23644},
+  year={2022},
+  organization={PMLR}
+}
+
+@article{feng2019trust,
+  title={Can We Trust You? On Calibration of a Probabilistic Object Detector for Autonomous Driving},
+  author={Di Feng and Lars Rosenbaum and Claudius Glaeser and Fabian Timm and Klaus C. J. Dietmayer},
+  journal={ArXiv},
+  year={2019},
+  volume={abs/1909.12358},
+  url={https://api.semanticscholar.org/CorpusID:203591605}
+}
+
+
+@inproceedings{caruana2015intelligible,
+  title={Intelligible models for healthcare: Predicting pneumonia risk and hospital 30-day readmission},
+  author={Caruana, Rich and Lou, Yin and Gehrke, Johannes and Koch, Paul and Sturm, Marc and Elhadad, Noemie},
+  booktitle={Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining},
+  pages={1721--1730},
+  year={2015}
+}
+@article{cortes1995support,
+  title={Support-vector networks},
+  author={Cortes, Corinna and Vapnik, Vladimir},
+  journal={Machine learning},
+  volume={20},
+  number={3},
+  pages={273--297},
+  year={1995},
+  publisher={Springer}
+}
+@article{hadsell2006dimensionality,
+  title={Dimensionality reduction by learning an invariant mapping},
+  author={Hadsell, Raia and Chopra, Sumit and LeCun, Yann},
+  journal={IEEE transactions on pattern analysis and machine intelligence},
+  volume={28},
+  number={11},
+  pages={185--190},
+  year={2006},
+  publisher={IEEE}
+}
+@inproceedings{schroff2015facenet,
+  title={FaceNet: A unified embedding for face recognition and clustering},
+  author={Schroff, Florian and Kalenichenko, Dmitry and Philbin, James},
+  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
+  pages={815--823},
+  year={2015}
+}
+@article{weston2011wsabie,
+  title={Wsabie: Scaling up to large vocabulary image annotation},
+  author={Weston, Jason and Bengio, Samy},
+  journal={Journal of machine learning research},
+  volume={12},
+  number={Feb},
+  pages={2177--2204},
+  year={2011}
+}
+@article{gruber2022better,
+  title={Better uncertainty calibration via proper scores for classification and beyond},
+  author={Gruber, Sebastian and Buettner, Florian},
+  journal={Advances in Neural Information Processing Systems},
+  volume={35},
+  pages={8618--8632},
+  year={2022}
+}
+@article{popordanoska2022consistent,
+  title={A consistent and differentiable lp canonical calibration error estimator},
+  author={Popordanoska, Teodora and Sayer, Raphael and Blaschko, Matthew},
+  journal={Advances in Neural Information Processing Systems},
+  volume={35},
+  pages={7933--7946},
+  year={2022}
+}
+@article{lin2013network,
+  title={Network in network},
+  author={Lin, Min and Chen, Qiang and Yan, Shuicheng},
+  journal={arXiv preprint arXiv:1312.4400},
+  year={2013}
+}
+@techreport{joachims1996probabilistic,
+  title={A Probabilistic Analysis of the Rocchio Algorithm with TFIDF for Text Categorization.},
+  author={Joachims, Thorsten},
+  year={1996},
+  institution={Carnegie-mellon univ pittsburgh pa dept of computer science}
+}
+@misc{
+ghosh2022adafocal,
+title={AdaFocal: Calibration-aware Adaptive Focal Loss},
+author={Arindam Ghosh and Thomas Schaaf and Matthew R. Gormley},
+year={2022},
+url={https://openreview.net/forum?id=CoMOKHYWf2}
+}
+@incollection{prechelt1998early,
+  title={Early stopping-but when?},
+  author={Prechelt, Lutz},
+  booktitle={Neural Networks: Tricks of the trade},
+  pages={55--69},
+  year={1998},
+  publisher={Springer}
+}
+
+@inproceedings{deng2009imagenet,
+  title={Imagenet: A large-scale hierarchical image database},
+  author={Deng, Jia and Dong, Wei and Socher, Richard and Li, Li-Jia and Li, Kai and Fei-Fei, Li},
+  booktitle={2009 IEEE conference on computer vision and pattern recognition},
+  pages={248--255},
+  year={2009},
+  organization={Ieee}
+}
+@inproceedings{huang2017densely,
+  title={Densely connected convolutional networks},
+  author={Huang, Gao and Liu, Zhuang and Van Der Maaten, Laurens and Weinberger, Kilian Q},
+  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
+  pages={4700--4708},
+  year={2017}
+}
+@article{muller2019does,
+  title={When does label smoothing help?},
+  author={M{\"u}ller, Rafael and Kornblith, Simon and Hinton, Geoffrey E},
+  journal={Advances in neural information processing systems},
+  volume={32},
+  year={2019}
+}
+
+
+@article{brier1950verification,
+  title={VERIFICATION OF FORECASTS EXPRESSED IN TERMS OF PROBABILITY},
+  author={Glenn W. Brier},
+  journal={Monthly Weather Review},
+  year={1950},
+  volume={78},
+  pages={1-3},
+  url={https://api.semanticscholar.org/CorpusID:122906757}
+}
+
+
+@InProceedings{kumar2018trainable,
+  title = 	 {Trainable Calibration Measures for Neural Networks from Kernel Mean Embeddings},
+  author =       {Kumar, Aviral and Sarawagi, Sunita and Jain, Ujjwal},
+  booktitle = 	 {Proceedings of the 35th International Conference on Machine Learning},
+  pages = 	 {2805--2814},
+  year = 	 {2018},
+  editor = 	 {Dy, Jennifer and Krause, Andreas},
+  volume = 	 {80},
+  series = 	 {Proceedings of Machine Learning Research},
+  month = 	 {10--15 Jul},
+  publisher =    {PMLR},
+  pdf = 	 {http://proceedings.mlr.press/v80/kumar18a/kumar18a.pdf},
+  url = 	 {https://proceedings.mlr.press/v80/kumar18a.html},
+  abstract = 	 {Modern neural networks have recently been found to be poorly calibrated, primarily in the direction of over-confidence. Methods like entropy penalty and temperature smoothing improve calibration by clamping confidence, but in doing so compromise the many legitimately confident predictions. We propose a more principled fix that minimizes an explicit calibration error during training. We present MMCE, a RKHS kernel based measure of calibration that is efficiently trainable alongside the negative likelihood loss without careful hyper-parameter tuning. Theoretically too, MMCE is a sound measure of calibration that is minimized at perfect calibration, and whose finite sample estimates are consistent and enjoy fast convergence rates. Extensive experiments on several network architectures demonstrate that MMCE is a fast, stable, and accurate method to minimize calibration error while maximally preserving the number of high confidence predictions.}
+}
+
+@article{zagoruyko2016wide,
+  title={Wide residual networks},
+  author={Zagoruyko, Sergey and Komodakis, Nikos},
+  journal={arXiv preprint arXiv:1605.07146},
+  year={2016}
+}
+@inproceedings{he2016deep,
+  title={Deep residual learning for image recognition},
+  author={He, Kaiming and Zhang, Xiangyu and Ren, Shaoqing and Sun, Jian},
+  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
+  pages={770--778},
+  year={2016}
+}
+@article{krizhevsky2009learning,
+  title={Learning multiple layers of features from tiny images},
+  author={Krizhevsky, Alex and Hinton, Geoffrey and others},
+  year={2009},
+  publisher={Citeseer}
+}
+@inproceedings{szegedy2016rethinking,
+  title={Rethinking the inception architecture for computer vision},
+  author={Szegedy, Christian and Vanhoucke, Vincent and Ioffe, Sergey and Shlens, Jon and Wojna, Zbigniew},
+  booktitle={Proceedings of the IEEE conference on computer vision and pattern recognition},
+  pages={2818--2826},
+  year={2016}
+}
+
+@article{jang2016categorical,
+  title={Categorical reparameterization with gumbel-softmax},
+  author={Jang, Eric and Gu, Shixiang and Poole, Ben},
+  journal={arXiv preprint arXiv:1611.01144},
+  year={2016}
+}
+@incollection{miikkulainen2019evolving,
+  title={Evolving deep neural networks},
+  author={Miikkulainen, Risto and Liang, Jason and Meyerson, Elliot and Rawal, Aditya and Fink, Daniel and Francon, Olivier and Raju, Bala and Shahrzad, Hormoz and Navruzyan, Arshak and Duffy, Nigel and others},
+  booktitle={Artificial intelligence in the age of neural networks and brain computing},
+  pages={293--312},
+  year={2019},
+  publisher={Elsevier}
+}
+@inproceedings{naeini2015obtaining,
+  title={Obtaining well calibrated probabilities using bayesian binning},
+  author={Naeini, Mahdi Pakdaman and Cooper, Gregory and Hauskrecht, Milos},
+  booktitle={Twenty-Ninth AAAI Conference on Artificial Intelligence},
+  year={2015}
+}
+@InProceedings{pmlr-v119-li20c,
+  title = 	 {Neural Architecture Search in A Proxy Validation Loss Landscape},
+  author =       {Li, Yanxi and Dong, Minjing and Wang, Yunhe and Xu, Chang},
+  booktitle = 	 {Proceedings of the 37th International Conference on Machine Learning},
+  pages = 	 {5853--5862},
+  year = 	 {2020},
+  editor = 	 {III, Hal Daumé and Singh, Aarti},
+  volume = 	 {119},
+  series = 	 {Proceedings of Machine Learning Research},
+  month = 	 {13--18 Jul},
+  publisher =    {PMLR},
+  pdf = 	 {http://proceedings.mlr.press/v119/li20c/li20c.pdf},
+  url = 	 {https://proceedings.mlr.press/v119/li20c.html},
+  abstract = 	 {This paper searches for the optimal neural architecture by minimizing a proxy of validation loss. Existing neural architecture search (NAS) methods used to discover the optimal neural architecture that best fits the validation examples given the up-to-date network weights. However, back propagation with a number of validation examples could be time consuming, especially when it needs to be repeated many times in NAS. Though these intermediate validation results are invaluable, they would be wasted if we cannot use them to predict the future from the past. In this paper, we propose to approximate the validation loss landscape by learning a mapping from neural architectures to their corresponding validate losses. The optimal neural architecture thus can be easily identified as the minimum of this proxy validation loss landscape. A novel sampling strategy is further developed for an efficient approximation of the loss landscape. Theoretical analysis indicates that the validation loss estimator learnt with our sampling strategy can reach a lower error rate and a lower label complexity compared with a uniform sampling. Experimental results on benchmarks demonstrate that the architecture searched by the proposed algorithm can achieve a satisfactory accuracy with less time cost.}
+}
+
+@inproceedings{guo2017calibration,
+  title={On calibration of modern neural networks},
+  author={Guo, Chuan and Pleiss, Geoff and Sun, Yu and Weinberger, Kilian Q},
+  booktitle={International conference on machine learning},
+  pages={1321--1330},
+  year={2017},
+  organization={PMLR}
+}
+@article{minderer2021revisiting,
+  title={Revisiting the calibration of modern neural networks},
+  author={Minderer, Matthias and Djolonga, Josip and Romijnders, Rob and Hubis, Frances and Zhai, Xiaohua and Houlsby, Neil and Tran, Dustin and Lucic, Mario},
+  journal={Advances in Neural Information Processing Systems},
+  volume={34},
+  pages={15682--15694},
+  year={2021}
+}
+@article{bojarski2016end,
+  title={End to end learning for self-driving cars},
+  author={Bojarski, Mariusz and Del Testa, Davide and Dworakowski, Daniel and Firner, Bernhard and Flepp, Beat and Goyal, Prasoon and Jackel, Lawrence D and Monfort, Mathew and Muller, Urs and Zhang, Jiakai and others},
+  journal={arXiv preprint arXiv:1604.07316},
+  year={2016}
+}
+@article{esteva2017dermatologist,
+  title={Dermatologist-level classification of skin cancer with deep neural networks},
+  author={Esteva, Andre and Kuprel, Brett and Novoa, Roberto A and Ko, Justin and Swetter, Susan M and Blau, Helen M and Thrun, Sebastian},
+  journal={nature},
+  volume={542},
+  number={7639},
+  pages={115--118},
+  year={2017},
+  publisher={Nature Publishing Group}
+}
+@article{sonderby2020metnet,
+  title={Metnet: A neural weather model for precipitation forecasting},
+  author={S{\o}nderby, Casper Kaae and Espeholt, Lasse and Heek, Jonathan and Dehghani, Mostafa and Oliver, Avital and Salimans, Tim and Agrawal, Shreya and Hickey, Jason and Kalchbrenner, Nal},
+  journal={arXiv preprint arXiv:2003.12140},
+  year={2020}
+}
+@article{degroot1983comparison,
+  title={The comparison and evaluation of forecasters},
+  author={DeGroot, Morris H and Fienberg, Stephen E},
+  journal={Journal of the Royal Statistical Society: Series D (The Statistician)},
+  volume={32},
+  number={1-2},
+  pages={12--22},
+  year={1983},
+  publisher={Wiley Online Library}
+}
+
+@article{goodfellow2013multi,
+  title={Multi-digit number recognition from street view imagery using deep convolutional neural networks},
+  author={Goodfellow, Ian J and Bulatov, Yaroslav and Ibarz, Julian and Arnoud, Sacha and Shet, Vinay},
+  journal={arXiv preprint arXiv:1312.6082},
+  year={2013}
+}
+@inproceedings{ding2020revisiting,
+  title={Revisiting the evaluation of uncertainty estimation and its application to explore model complexity-uncertainty trade-off},
+  author={Ding, Yukun and Liu, Jinglan and Xiong, Jinjun and Shi, Yiyu},
+  booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops},
+  pages={4--5},
+  year={2020}
+}
+@article{kumar2019verified,
+  title={Verified uncertainty calibration},
+  author={Kumar, Ananya and Liang, Percy S and Ma, Tengyu},
+  journal={Advances in Neural Information Processing Systems},
+  volume={32},
+  year={2019}
+}
+@inproceedings{nixon2019measuring,
+  title={Measuring Calibration in Deep Learning.},
+  author={Nixon, Jeremy and Dusenberry, Michael W and Zhang, Linchuan and Jerfel, Ghassen and Tran, Dustin},
+  booktitle={CVPR Workshops},
+  volume={2},
+  year={2019}
+}
+@article{hendrycks2018benchmarking,
+  title={Benchmarking neural network robustness to common corruptions and surface variations},
+  author={Hendrycks, Dan and Dietterich, Thomas G},
+  journal={arXiv preprint arXiv:1807.01697},
+  year={2018}
+}
+@article{ovadia2019can,
+  title={Can you trust your model's uncertainty? evaluating predictive uncertainty under dataset shift},
+  author={Ovadia, Yaniv and Fertig, Emily and Ren, Jie and Nado, Zachary and Sculley, David and Nowozin, Sebastian and Dillon, Joshua and Lakshminarayanan, Balaji and Snoek, Jasper},
+  journal={Advances in neural information processing systems},
+  volume={32},
+  year={2019}
+}
+@inproceedings{wenger2020non,
+  title={Non-parametric calibration for classification},
+  author={Wenger, Jonathan and Kjellstr{\"o}m, Hedvig and Triebel, Rudolph},
+  booktitle={International Conference on Artificial Intelligence and Statistics},
+  pages={178--190},
+  year={2020},
+  organization={PMLR}
+}
+@inproceedings{kull2017beta,
+  title={Beta calibration: a well-founded and easily implemented improvement on logistic calibration for binary classifiers},
+  author={Kull, Meelis and Silva Filho, Telmo and Flach, Peter},
+  booktitle={Artificial Intelligence and Statistics},
+  pages={623--631},
+  year={2017},
+  organization={PMLR}
+}
+
+@article{karandikar2021soft,
+  title={Soft calibration objectives for neural networks},
+  author={Karandikar, Archit and Cain, Nicholas and Tran, Dustin and Lakshminarayanan, Balaji and Shlens, Jonathon and Mozer, Michael C and Roelofs, Becca},
+  journal={Advances in Neural Information Processing Systems},
+  volume={34},
+  pages={29768--29779},
+  year={2021}
+}
+@article{hui2020evaluation,
+  title={Evaluation of neural architectures trained with square loss vs cross-entropy in classification tasks},
+  author={Hui, Like and Belkin, Mikhail},
+  journal={arXiv preprint arXiv:2006.07322},
+  year={2020}
+}
+@inproceedings{platt1999probabilistic,
+  abstract = {The output of a classifier should be a calibrated posterior probability to enable post-processing. Standard SVMs do not provide such probabilities. One method to create probabilities is to directly train a kernel classifier with a logit link function and a regularized maximum likelihood score. However, training with a maximum likelihood score will produce non-sparse kernel machines. Instead, we train an SVM, then train the parameters of an additional sigmoid function to map the SVM outputs into probabilities. This chapter compares classification error rate and likelihood scores for an SVM plus sigmoid versus a kernel method trained with a regularized likelihood error function. These methods are tested on three data-mining-style data sets. The SVM+sigmoid yields probabilities of comparable quality to the regularized maximum likelihood kernel method, while still retaining the sparseness of the SVM.},
+  added-at = {2012-07-13T11:59:27.000+0200},
+  author = {Platt, John C.},
+  biburl = {https://www.bibsonomy.org/bibtex/21a15c25b220dac6b6f222c18fc7e91d2/jabreftest},
+  booktitle = {Advances in Large Margin Classifiers},
+  citeseerurl = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.1639},
+  file = {Platt1999.pdf:1999/Platt1999.pdf:PDF},
+  groups = {public},
+  interhash = {c5df9f9137085cad9cafce3c347b2508},
+  intrahash = {1a15c25b220dac6b6f222c18fc7e91d2},
+  keywords = {},
+  pages = {61--74},
+  publisher = {MIT Press},
+  timestamp = {2012-07-13T11:59:27.000+0200},
+  title = {Probabilistic Outputs for Support Vector Machines and Comparisons to Regularized Likelihood Methods},
+  username = {jabreftest},
+  year = 1999
+}
+
+
+@inproceedings{chen2020simple,
+  title={A simple framework for contrastive learning of visual representations},
+  author={Chen, Ting and Kornblith, Simon and Norouzi, Mohammad and Hinton, Geoffrey},
+  booktitle={International conference on machine learning},
+  pages={1597--1607},
+  year={2020},
+  organization={PMLR}
+}
+@article{hinton2015distilling,
+  title={Distilling the knowledge in a neural network},
+  author={Hinton, Geoffrey and Vinyals, Oriol and Dean, Jeff and others},
+  journal={arXiv preprint arXiv:1503.02531},
+  volume={2},
+  number={7},
+  year={2015}
+}
+
+@article{zadrozny2001obtaining,
+  title={Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers},
+  author={Zadrozny, Bianca and Elkan, Charles},
+  journal={18th International Conferrence on Machine Learning},
+  year={2001}
+}
+
+@inproceedings{zadrozny2002transforming,
+  title={Transforming classifier scores into accurate multiclass probability estimates},
+  author={Zadrozny, Bianca and Elkan, Charles},
+  booktitle={Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining},
+  pages={694--699},
+  year={2002}
+}
+@article{nguyen2015posterior,
+  title={Posterior calibration and exploratory analysis for natural language processing models},
+  author={Nguyen, Khanh and O'Connor, Brendan},
+  journal={arXiv preprint arXiv:1508.05154},
+  year={2015}
+}
+@article{kull2019beyond,
+  title={Beyond temperature scaling: Obtaining well-calibrated multi-class probabilities with dirichlet calibration},
+  author={Kull, Meelis and Perello Nieto, Miquel and K{\"a}ngsepp, Markus and Silva Filho, Telmo and Song, Hao and Flach, Peter},
+  journal={Advances in neural information processing systems},
+  volume={32},
+  year={2019}
+}
+@inproceedings{vaicenavicius2019evaluating,
+  title={Evaluating model calibration in classification},
+  author={Vaicenavicius, Juozas and Widmann, David and Andersson, Carl and Lindsten, Fredrik and Roll, Jacob and Sch{\"o}n, Thomas},
+  booktitle={The 22nd International Conference on Artificial Intelligence and Statistics},
+  pages={3459--3467},
+  year={2019},
+  organization={PMLR}
+}
+@article{widmann2019calibration,
+  title={Calibration tests in multi-class classification: A unifying framework},
+  author={Widmann, David and Lindsten, Fredrik and Zachariah, Dave},
+  journal={Advances in Neural Information Processing Systems},
+  volume={32},
+  year={2019}
+}
+
+@inproceedings{roelofs2022mitigating,
+  title={Mitigating bias in calibration error estimation},
+  author={Roelofs, Rebecca and Cain, Nicholas and Shlens, Jonathon and Mozer, Michael C},
+  booktitle={International Conference on Artificial Intelligence and Statistics},
+  pages={4036--4054},
+  year={2022},
+  organization={PMLR}
+}
+@article{wang2022yolov7,
+  title={YOLOv7: Trainable bag-of-freebies sets new state-of-the-art for real-time object detectors},
+  author={Wang, Chien-Yao and Bochkovskiy, Alexey and Liao, Hong-Yuan Mark},
+  journal={arXiv preprint arXiv:2207.02696},
+  year={2022}
+}
+@inproceedings{wortsman2022model,
+  title={Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time},
+  author={Wortsman, Mitchell and Ilharco, Gabriel and Gadre, Samir Ya and Roelofs, Rebecca and Gontijo-Lopes, Raphael and Morcos, Ari S and Namkoong, Hongseok and Farhadi, Ali and Carmon, Yair and Kornblith, Simon and others},
+  booktitle={International Conference on Machine Learning},
+  pages={23965--23998},
+  year={2022},
+  organization={PMLR}
+}
+@article{yu2022coca,
+  title={Coca: Contrastive captioners are image-text foundation models},
+  author={Yu, Jiahui and Wang, Zirui and Vasudevan, Vijay and Yeung, Legg and Seyedhosseini, Mojtaba and Wu, Yonghui},
+  journal={arXiv preprint arXiv:2205.01917},
+  year={2022}
+}
+@inproceedings{brock2021high,
+  title={High-performance large-scale image recognition without normalization},
+  author={Brock, Andy and De, Soham and Smith, Samuel L and Simonyan, Karen},
+  booktitle={International Conference on Machine Learning},
+  pages={1059--1071},
+  year={2021},
+  organization={PMLR}
+}
+@inproceedings{pham2021meta,
+  title={Meta pseudo labels},
+  author={Pham, Hieu and Dai, Zihang and Xie, Qizhe and Le, Quoc V},
+  booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition},
+  pages={11557--11568},
+  year={2021}
+}
+@article{wang2021you,
+  title={You only learn one representation: Unified network for multiple tasks},
+  author={Wang, Chien-Yao and Yeh, I-Hau and Liao, Hong-Yuan Mark},
+  journal={arXiv preprint arXiv:2105.04206},
+  year={2021}
+}
+@inproceedings{qiao2021detectors,
+  title={Detectors: Detecting objects with recursive feature pyramid and switchable atrous convolution},
+  author={Qiao, Siyuan and Chen, Liang-Chieh and Yuille, Alan},
+  booktitle={Proceedings of the IEEE/CVF conference on computer vision and pattern recognition},
+  pages={10213--10224},
+  year={2021}
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+@article{zhou2021probabilistic,
+  title={Probabilistic two-stage detection},
+  author={Zhou, Xingyi and Koltun, Vladlen and Kr{\"a}henb{\"u}hl, Philipp},
+  journal={arXiv preprint arXiv:2103.07461},
+  year={2021}
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+@article{gupta2020calibration,
+  title={Calibration of neural networks using splines},
+  author={Gupta, Kartik and Rahimi, Amir and Ajanthan, Thalaiyasingam and Mensink, Thomas and Sminchisescu, Cristian and Hartley, Richard},
+  journal={arXiv preprint arXiv:2006.12800},
+  year={2020}
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+@article{krishnan2020improving,
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+  volume={33},
+  pages={18237--18248},
+  year={2020}
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+@article{mukhoti2020calibrating,
+  title={Calibrating deep neural networks using focal loss},
+  author={Mukhoti, Jishnu and Kulharia, Viveka and Sanyal, Amartya and Golodetz, Stuart and Torr, Philip and Dokania, Puneet},
+  journal={Advances in Neural Information Processing Systems},
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+  year={2020}
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+@inproceedings{lin2017focal,
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+  year={2017}
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+@article{thulasidasan2019mixup,
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+  year={2019}
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icml2026/icml2026.bst

@@ -0,0 +1,1443 @@
+%% File: `icml2025.bst'
+%% A modification of `plainnl.bst' for use with natbib package 
+%%
+%% Copyright 2010 Hal Daum\'e III
+%% Modified by J. Fürnkranz
+%% - Changed labels from (X and Y, 2000) to (X & Y, 2000)
+%% - Changed References to last name first and abbreviated first names.
+%% Modified by Iain Murray 2018 (who suggests adopting a standard .bst in future...)
+%% - Made it actually use abbreviated first names
+%%
+%% Copyright 1993-2007 Patrick W Daly
+%% Max-Planck-Institut f\"ur Sonnensystemforschung
+%% Max-Planck-Str. 2
+%% D-37191 Katlenburg-Lindau
+%% Germany
+%% E-mail: daly@mps.mpg.de
+%%
+%% This program can be redistributed and/or modified under the terms
+%% of the LaTeX Project Public License Distributed from CTAN
+%% archives in directory macros/latex/base/lppl.txt; either
+%% version 1 of the License, or any later version.
+%%
+ % Version and source file information:
+ % \ProvidesFile{icml2010.mbs}[2007/11/26 1.93 (PWD)]
+ %
+ % BibTeX `plainnat' family
+ %   version 0.99b for BibTeX versions 0.99a or later,
+ %   for LaTeX versions 2.09 and 2e.
+ %
+ % For use with the `natbib.sty' package; emulates the corresponding
+ %   member of the `plain' family, but with author-year citations.
+ %
+ % With version 6.0 of `natbib.sty', it may also be used for numerical
+ %   citations, while retaining the commands \citeauthor, \citefullauthor,
+ %   and \citeyear to print the corresponding information.
+ %
+ % For version 7.0 of `natbib.sty', the KEY field replaces missing
+ %   authors/editors, and the date is left blank in \bibitem.
+ %
+ % Includes field EID for the sequence/citation number of electronic journals
+ %  which is used instead of page numbers.
+ %
+ % Includes fields ISBN and ISSN.
+ %
+ % Includes field URL for Internet addresses.
+ %
+ % Includes field DOI for Digital Object Idenfifiers.
+ %
+ % Works best with the url.sty package of Donald Arseneau.
+ %
+ % Works with identical authors and year are further sorted by
+ %   citation key, to preserve any natural sequence.
+ %
+ENTRY
+  { address
+    author
+    booktitle
+    chapter
+    doi
+    eid
+    edition
+    editor
+    howpublished
+    institution
+    isbn
+    issn
+    journal
+    key
+    month
+    note
+    number
+    organization
+    pages
+    publisher
+    school
+    series
+    title
+    type
+    url
+    volume
+    year
+  }
+  {}
+  { label extra.label sort.label short.list }
+
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+  #1 'mid.sentence :=
+  #2 'after.sentence :=
+  #3 'after.block :=
+}
+
+STRINGS { s t }
+
+FUNCTION {output.nonnull}
+{ 's :=
+  output.state mid.sentence =
+    { ", " * write$ }
+    { output.state after.block =
+        { add.period$ write$
+          newline$
+          "\newblock " write$
+        }
+        { output.state before.all =
+            'write$
+            { add.period$ " " * write$ }
+          if$
+        }
+      if$
+      mid.sentence 'output.state :=
+    }
+  if$
+  s
+}
+
+FUNCTION {output}
+{ duplicate$ empty$
+    'pop$
+    'output.nonnull
+  if$
+}
+
+FUNCTION {output.check}
+{ 't :=
+  duplicate$ empty$
+    { pop$ "empty " t * " in " * cite$ * warning$ }
+    'output.nonnull
+  if$
+}
+
+FUNCTION {fin.entry}
+{ add.period$
+  write$
+  newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+    'skip$
+    { after.block 'output.state := }
+  if$
+}
+
+FUNCTION {new.sentence}
+{ output.state after.block =
+    'skip$
+    { output.state before.all =
+        'skip$
+        { after.sentence 'output.state := }
+      if$
+    }
+  if$
+}
+
+FUNCTION {not}
+{   { #0 }
+    { #1 }
+  if$
+}
+
+FUNCTION {and}
+{   'skip$
+    { pop$ #0 }
+  if$
+}
+
+FUNCTION {or}
+{   { pop$ #1 }
+    'skip$
+  if$
+}
+
+FUNCTION {new.block.checka}
+{ empty$
+    'skip$
+    'new.block
+  if$
+}
+
+FUNCTION {new.block.checkb}
+{ empty$
+  swap$ empty$
+  and
+    'skip$
+    'new.block
+  if$
+}
+
+FUNCTION {new.sentence.checka}
+{ empty$
+    'skip$
+    'new.sentence
+  if$
+}
+
+FUNCTION {new.sentence.checkb}
+{ empty$
+  swap$ empty$
+  and
+    'skip$
+    'new.sentence
+  if$
+}
+
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+    { pop$ "" }
+    'skip$
+  if$
+}
+
+FUNCTION {emphasize}
+{ duplicate$ empty$
+    { pop$ "" }
+    { "\emph{" swap$ * "}" * }
+  if$
+}
+
+INTEGERS { nameptr namesleft numnames }
+
+FUNCTION {format.names}
+{ 's :=
+  #1 'nameptr :=
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't :=
+      nameptr #1 >
+        { namesleft #1 >
+            { ", " * t * }
+            { numnames #2 >
+                { "," * }
+                'skip$
+              if$
+              t "others" =
+                { " et~al." * }
+                { " and " * t * }
+              if$
+            }
+          if$
+        }
+        't
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {format.key}
+{ empty$
+    { key field.or.null }
+    { "" }
+  if$
+}
+
+FUNCTION {format.authors}
+{ author empty$
+    { "" }
+    { author format.names }
+  if$
+}
+
+FUNCTION {format.editors}
+{ editor empty$
+    { "" }
+    { editor format.names
+      editor num.names$ #1 >
+        { " (eds.)" * }
+        { " (ed.)" * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.isbn}
+{ isbn empty$
+    { "" }
+    { new.block "ISBN " isbn * }
+  if$
+}
+
+FUNCTION {format.issn}
+{ issn empty$
+    { "" }
+    { new.block "ISSN " issn * }
+  if$
+}
+
+FUNCTION {format.url}
+{ url empty$
+    { "" }
+    { new.block "URL \url{" url * "}" * }
+  if$
+}
+
+FUNCTION {format.doi}
+{ doi empty$
+    { "" }
+    { new.block "\doi{" doi * "}" * }
+  if$
+}
+
+FUNCTION {format.title}
+{ title empty$
+    { "" }
+    { title "t" change.case$ }
+  if$
+}
+
+FUNCTION {format.full.names}
+{'s :=
+  #1 'nameptr :=
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      "{vv~}{ll}" format.name$ 't :=
+      nameptr #1 >
+        {
+          namesleft #1 >
+            { ", " * t * }
+            {
+              numnames #2 >
+                { "," * }
+                'skip$
+              if$
+              t "others" =
+                { " et~al." * }
+                { " and " * t * }
+              if$
+            }
+          if$
+        }
+        't
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {author.editor.full}
+{ author empty$
+    { editor empty$
+        { "" }
+        { editor format.full.names }
+      if$
+    }
+    { author format.full.names }
+  if$
+}
+
+FUNCTION {author.full}
+{ author empty$
+    { "" }
+    { author format.full.names }
+  if$
+}
+
+FUNCTION {editor.full}
+{ editor empty$
+    { "" }
+    { editor format.full.names }
+  if$
+}
+
+FUNCTION {make.full.names}
+{ type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.full
+    { type$ "proceedings" =
+        'editor.full
+        'author.full
+      if$
+    }
+  if$
+}
+
+FUNCTION {output.bibitem}
+{ newline$
+  "\bibitem[" write$
+  label write$
+  ")" make.full.names duplicate$ short.list =
+     { pop$ }
+     { * }
+   if$
+  "]{" * write$
+  cite$ write$
+  "}" write$
+  newline$
+  ""
+  before.all 'output.state :=
+}
+
+FUNCTION {n.dashify}
+{ 't :=
+  ""
+    { t empty$ not }
+    { t #1 #1 substring$ "-" =
+        { t #1 #2 substring$ "--" = not
+            { "--" *
+              t #2 global.max$ substring$ 't :=
+            }
+            {   { t #1 #1 substring$ "-" = }
+                { "-" *
+                  t #2 global.max$ substring$ 't :=
+                }
+              while$
+            }
+          if$
+        }
+        { t #1 #1 substring$ *
+          t #2 global.max$ substring$ 't :=
+        }
+      if$
+    }
+  while$
+}
+
+FUNCTION {format.date}
+{ year duplicate$ empty$
+    { "empty year in " cite$ * warning$
+       pop$ "" }
+    'skip$
+  if$
+  month empty$
+    'skip$
+    { month
+      " " * swap$ *
+    }
+  if$
+  extra.label *
+}
+
+FUNCTION {format.btitle}
+{ title emphasize
+}
+
+FUNCTION {tie.or.space.connect}
+{ duplicate$ text.length$ #3 <
+    { "~" }
+    { " " }
+  if$
+  swap$ * *
+}
+
+FUNCTION {either.or.check}
+{ empty$
+    'pop$
+    { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+  if$
+}
+
+FUNCTION {format.bvolume}
+{ volume empty$
+    { "" }
+    { "volume" volume tie.or.space.connect
+      series empty$
+        'skip$
+        { " of " * series emphasize * }
+      if$
+      "volume and number" number either.or.check
+    }
+  if$
+}
+
+FUNCTION {format.number.series}
+{ volume empty$
+    { number empty$
+        { series field.or.null }
+        { output.state mid.sentence =
+            { "number" }
+            { "Number" }
+          if$
+          number tie.or.space.connect
+          series empty$
+            { "there's a number but no series in " cite$ * warning$ }
+            { " in " * series * }
+          if$
+        }
+      if$
+    }
+    { "" }
+  if$
+}
+
+FUNCTION {format.edition}
+{ edition empty$
+    { "" }
+    { output.state mid.sentence =
+        { edition "l" change.case$ " edition" * }
+        { edition "t" change.case$ " edition" * }
+      if$
+    }
+  if$
+}
+
+INTEGERS { multiresult }
+
+FUNCTION {multi.page.check}
+{ 't :=
+  #0 'multiresult :=
+    { multiresult not
+      t empty$ not
+      and
+    }
+    { t #1 #1 substring$
+      duplicate$ "-" =
+      swap$ duplicate$ "," =
+      swap$ "+" =
+      or or
+        { #1 'multiresult := }
+        { t #2 global.max$ substring$ 't := }
+      if$
+    }
+  while$
+  multiresult
+}
+
+FUNCTION {format.pages}
+{ pages empty$
+    { "" }
+    { pages multi.page.check
+        { "pp.\ " pages n.dashify tie.or.space.connect }
+        { "pp.\ " pages tie.or.space.connect }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.eid}
+{ eid empty$
+    { "" }
+    { "art." eid tie.or.space.connect }
+  if$
+}
+
+FUNCTION {format.vol.num.pages}
+{ volume field.or.null
+  number empty$
+    'skip$
+    { "\penalty0 (" number * ")" * *
+      volume empty$
+        { "there's a number but no volume in " cite$ * warning$ }
+        'skip$
+      if$
+    }
+  if$
+  pages empty$
+    'skip$
+    { duplicate$ empty$
+        { pop$ format.pages }
+        { ":\penalty0 " * pages n.dashify * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.vol.num.eid}
+{ volume field.or.null
+  number empty$
+    'skip$
+    { "\penalty0 (" number * ")" * *
+      volume empty$
+        { "there's a number but no volume in " cite$ * warning$ }
+        'skip$
+      if$
+    }
+  if$
+  eid empty$
+    'skip$
+    { duplicate$ empty$
+        { pop$ format.eid }
+        { ":\penalty0 " * eid * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+    'format.pages
+    { type empty$
+        { "chapter" }
+        { type "l" change.case$ }
+      if$
+      chapter tie.or.space.connect
+      pages empty$
+        'skip$
+        { ", " * format.pages * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.in.ed.booktitle}
+{ booktitle empty$
+    { "" }
+    { editor empty$
+        { "In " booktitle emphasize * }
+        { "In " format.editors * ", " * booktitle emphasize * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {empty.misc.check}
+{ author empty$ title empty$ howpublished empty$
+  month empty$ year empty$ note empty$
+  and and and and and
+  key empty$ not and
+    { "all relevant fields are empty in " cite$ * warning$ }
+    'skip$
+  if$
+}
+
+FUNCTION {format.thesis.type}
+{ type empty$
+    'skip$
+    { pop$
+      type "t" change.case$
+    }
+  if$
+}
+
+FUNCTION {format.tr.number}
+{ type empty$
+    { "Technical Report" }
+    'type
+  if$
+  number empty$
+    { "t" change.case$ }
+    { number tie.or.space.connect }
+  if$
+}
+
+FUNCTION {format.article.crossref}
+{ key empty$
+    { journal empty$
+        { "need key or journal for " cite$ * " to crossref " * crossref *
+          warning$
+          ""
+        }
+        { "In \emph{" journal * "}" * }
+      if$
+    }
+    { "In " }
+  if$
+  " \citet{" * crossref * "}" *
+}
+
+FUNCTION {format.book.crossref}
+{ volume empty$
+    { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+      "In "
+    }
+    { "Volume" volume tie.or.space.connect
+      " of " *
+    }
+  if$
+  editor empty$
+  editor field.or.null author field.or.null =
+  or
+    { key empty$
+        { series empty$
+            { "need editor, key, or series for " cite$ * " to crossref " *
+              crossref * warning$
+              "" *
+            }
+            { "\emph{" * series * "}" * }
+          if$
+        }
+        'skip$
+      if$
+    }
+    'skip$
+  if$
+  " \citet{" * crossref * "}" *
+}
+
+FUNCTION {format.incoll.inproc.crossref}
+{ editor empty$
+  editor field.or.null author field.or.null =
+  or
+    { key empty$
+        { booktitle empty$
+            { "need editor, key, or booktitle for " cite$ * " to crossref " *
+              crossref * warning$
+              ""
+            }
+            { "In \emph{" booktitle * "}" * }
+          if$
+        }
+        { "In " }
+      if$
+    }
+    { "In " }
+  if$
+  " \citet{" * crossref * "}" *
+}
+
+FUNCTION {article}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  new.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    { journal emphasize "journal" output.check
+      eid empty$
+        { format.vol.num.pages output }
+        { format.vol.num.eid output }
+      if$
+      format.date "year" output.check
+    }
+    { format.article.crossref output.nonnull
+      eid empty$
+        { format.pages output }
+        { format.eid output }
+      if$
+    }
+  if$
+  format.issn output
+  format.doi output
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {book}
+{ output.bibitem
+  author empty$
+    { format.editors "author and editor" output.check
+      editor format.key output
+    }
+    { format.authors output.nonnull
+      crossref missing$
+        { "author and editor" editor either.or.check }
+        'skip$
+      if$
+    }
+  if$
+  new.block
+  format.btitle "title" output.check
+  crossref missing$
+    { format.bvolume output
+      new.block
+      format.number.series output
+      new.sentence
+      publisher "publisher" output.check
+      address output
+    }
+    { new.block
+      format.book.crossref output.nonnull
+    }
+  if$
+  format.edition output
+  format.date "year" output.check
+  format.isbn output
+  format.doi output
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {booklet}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  new.block
+  format.title "title" output.check
+  howpublished address new.block.checkb
+  howpublished output
+  address output
+  format.date output
+  format.isbn output
+  format.doi output
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+  author empty$
+    { format.editors "author and editor" output.check
+      editor format.key output
+    }
+    { format.authors output.nonnull
+      crossref missing$
+        { "author and editor" editor either.or.check }
+        'skip$
+      if$
+    }
+  if$
+  new.block
+  format.btitle "title" output.check
+  crossref missing$
+    { format.bvolume output
+      format.chapter.pages "chapter and pages" output.check
+      new.block
+      format.number.series output
+      new.sentence
+      publisher "publisher" output.check
+      address output
+    }
+    { format.chapter.pages "chapter and pages" output.check
+      new.block
+      format.book.crossref output.nonnull
+    }
+  if$
+  format.edition output
+  format.date "year" output.check
+  format.isbn output
+  format.doi output
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  new.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    { format.in.ed.booktitle "booktitle" output.check
+      format.bvolume output
+      format.number.series output
+      format.chapter.pages output
+      new.sentence
+      publisher "publisher" output.check
+      address output
+      format.edition output
+      format.date "year" output.check
+    }
+    { format.incoll.inproc.crossref output.nonnull
+      format.chapter.pages output
+    }
+  if$
+  format.isbn output
+  format.doi output
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {inproceedings}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  new.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    { format.in.ed.booktitle "booktitle" output.check
+      format.bvolume output
+      format.number.series output
+      format.pages output
+      address empty$
+        { organization publisher new.sentence.checkb
+          organization output
+          publisher output
+          format.date "year" output.check
+        }
+        { address output.nonnull
+          format.date "year" output.check
+          new.sentence
+          organization output
+          publisher output
+        }
+      if$
+    }
+    { format.incoll.inproc.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  format.isbn output
+  format.doi output
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {conference} { inproceedings }
+
+FUNCTION {manual}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  new.block
+  format.btitle "title" output.check
+  organization address new.block.checkb
+  organization output
+  address output
+  format.edition output
+  format.date output
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  new.block
+  format.title "title" output.check
+  new.block
+  "Master's thesis" format.thesis.type output.nonnull
+  school "school" output.check
+  address output
+  format.date "year" output.check
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  title howpublished new.block.checkb
+  format.title output
+  howpublished new.block.checka
+  howpublished output
+  format.date output
+  format.issn output
+  format.url output
+  new.block
+  note output
+  fin.entry
+  empty.misc.check
+}
+
+FUNCTION {phdthesis}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  new.block
+  format.btitle "title" output.check
+  new.block
+  "PhD thesis" format.thesis.type output.nonnull
+  school "school" output.check
+  address output
+  format.date "year" output.check
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+  format.editors output
+  editor format.key output
+  new.block
+  format.btitle "title" output.check
+  format.bvolume output
+  format.number.series output
+  address output
+  format.date "year" output.check
+  new.sentence
+  organization output
+  publisher output
+  format.isbn output
+  format.doi output
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  new.block
+  format.title "title" output.check
+  new.block
+  format.tr.number output.nonnull
+  institution "institution" output.check
+  address output
+  format.date "year" output.check
+  format.url output
+  new.block
+  note output
+  fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  new.block
+  format.title "title" output.check
+  new.block
+  note "note" output.check
+  format.date output
+  format.url output
+  fin.entry
+}
+
+FUNCTION {default.type} { misc }
+
+
+MACRO {jan} {"January"}
+
+MACRO {feb} {"February"}
+
+MACRO {mar} {"March"}
+
+MACRO {apr} {"April"}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"June"}
+
+MACRO {jul} {"July"}
+
+MACRO {aug} {"August"}
+
+MACRO {sep} {"September"}
+
+MACRO {oct} {"October"}
+
+MACRO {nov} {"November"}
+
+MACRO {dec} {"December"}
+
+
+
+MACRO {acmcs} {"ACM Computing Surveys"}
+
+MACRO {acta} {"Acta Informatica"}
+
+MACRO {cacm} {"Communications of the ACM"}
+
+MACRO {ibmjrd} {"IBM Journal of Research and Development"}
+
+MACRO {ibmsj} {"IBM Systems Journal"}
+
+MACRO {ieeese} {"IEEE Transactions on Software Engineering"}
+
+MACRO {ieeetc} {"IEEE Transactions on Computers"}
+
+MACRO {ieeetcad}
+ {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"}
+
+MACRO {ipl} {"Information Processing Letters"}
+
+MACRO {jacm} {"Journal of the ACM"}
+
+MACRO {jcss} {"Journal of Computer and System Sciences"}
+
+MACRO {scp} {"Science of Computer Programming"}
+
+MACRO {sicomp} {"SIAM Journal on Computing"}
+
+MACRO {tocs} {"ACM Transactions on Computer Systems"}
+
+MACRO {tods} {"ACM Transactions on Database Systems"}
+
+MACRO {tog} {"ACM Transactions on Graphics"}
+
+MACRO {toms} {"ACM Transactions on Mathematical Software"}
+
+MACRO {toois} {"ACM Transactions on Office Information Systems"}
+
+MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"}
+
+MACRO {tcs} {"Theoretical Computer Science"}
+
+
+READ
+
+FUNCTION {sortify}
+{ purify$
+  "l" change.case$
+}
+
+INTEGERS { len }
+
+FUNCTION {chop.word}
+{ 's :=
+  'len :=
+  s #1 len substring$ =
+    { s len #1 + global.max$ substring$ }
+    's
+  if$
+}
+
+FUNCTION {format.lab.names}
+{ 's :=
+  s #1 "{vv~}{ll}" format.name$
+  s num.names$ duplicate$
+  #2 >
+    { pop$ " et~al." * }
+    { #2 <
+        'skip$
+        { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+            { " et~al." * }
+            { " \& " * s #2 "{vv~}{ll}" format.name$ * }
+          if$
+        }
+      if$
+    }
+  if$
+}
+
+FUNCTION {author.key.label}
+{ author empty$
+    { key empty$
+        { cite$ #1 #3 substring$ }
+        'key
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {author.editor.key.label}
+{ author empty$
+    { editor empty$
+        { key empty$
+            { cite$ #1 #3 substring$ }
+            'key
+          if$
+        }
+        { editor format.lab.names }
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {author.key.organization.label}
+{ author empty$
+    { key empty$
+        { organization empty$
+            { cite$ #1 #3 substring$ }
+            { "The " #4 organization chop.word #3 text.prefix$ }
+          if$
+        }
+        'key
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {editor.key.organization.label}
+{ editor empty$
+    { key empty$
+        { organization empty$
+            { cite$ #1 #3 substring$ }
+            { "The " #4 organization chop.word #3 text.prefix$ }
+          if$
+        }
+        'key
+      if$
+    }
+    { editor format.lab.names }
+  if$
+}
+
+FUNCTION {calc.short.authors}
+{ type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.key.label
+    { type$ "proceedings" =
+        'editor.key.organization.label
+        { type$ "manual" =
+            'author.key.organization.label
+            'author.key.label
+          if$
+        }
+      if$
+    }
+  if$
+  'short.list :=
+}
+
+FUNCTION {calc.label}
+{ calc.short.authors
+  short.list
+  "("
+  *
+  year duplicate$ empty$
+  short.list key field.or.null = or
+     { pop$ "" }
+     'skip$
+  if$
+  *
+  'label :=
+}
+
+FUNCTION {sort.format.names}
+{ 's :=
+  #1 'nameptr :=
+  ""
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    {
+      s nameptr "{vv{ } }{ll{ }}{  f{ }}{  jj{ }}" format.name$ 't :=
+      nameptr #1 >
+        {
+          "   "  *
+          namesleft #1 = t "others" = and
+            { "zzzzz" * }
+            { numnames #2 > nameptr #2 = and
+                { "zz" * year field.or.null * "   " * }
+                'skip$
+              if$
+              t sortify *
+            }
+          if$
+        }
+        { t sortify * }
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {sort.format.title}
+{ 't :=
+  "A " #2
+    "An " #3
+      "The " #4 t chop.word
+    chop.word
+  chop.word
+  sortify
+  #1 global.max$ substring$
+}
+
+FUNCTION {author.sort}
+{ author empty$
+    { key empty$
+        { "to sort, need author or key in " cite$ * warning$
+          ""
+        }
+        { key sortify }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+
+FUNCTION {author.editor.sort}
+{ author empty$
+    { editor empty$
+        { key empty$
+            { "to sort, need author, editor, or key in " cite$ * warning$
+              ""
+            }
+            { key sortify }
+          if$
+        }
+        { editor sort.format.names }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+
+FUNCTION {author.organization.sort}
+{ author empty$
+    { organization empty$
+        { key empty$
+            { "to sort, need author, organization, or key in " cite$ * warning$
+              ""
+            }
+            { key sortify }
+          if$
+        }
+        { "The " #4 organization chop.word sortify }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+
+FUNCTION {editor.organization.sort}
+{ editor empty$
+    { organization empty$
+        { key empty$
+            { "to sort, need editor, organization, or key in " cite$ * warning$
+              ""
+            }
+            { key sortify }
+          if$
+        }
+        { "The " #4 organization chop.word sortify }
+      if$
+    }
+    { editor sort.format.names }
+  if$
+}
+
+
+FUNCTION {presort}
+{ calc.label
+  label sortify
+  "    "
+  *
+  type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.sort
+    { type$ "proceedings" =
+        'editor.organization.sort
+        { type$ "manual" =
+            'author.organization.sort
+            'author.sort
+          if$
+        }
+      if$
+    }
+  if$
+  "    "
+  *
+  year field.or.null sortify
+  *
+  "    "
+  *
+  cite$
+  *
+  #1 entry.max$ substring$
+  'sort.label :=
+  sort.label *
+  #1 entry.max$ substring$
+  'sort.key$ :=
+}
+
+ITERATE {presort}
+
+SORT
+
+STRINGS { longest.label last.label next.extra }
+
+INTEGERS { longest.label.width last.extra.num number.label }
+
+FUNCTION {initialize.longest.label}
+{ "" 'longest.label :=
+  #0 int.to.chr$ 'last.label :=
+  "" 'next.extra :=
+  #0 'longest.label.width :=
+  #0 'last.extra.num :=
+  #0 'number.label :=
+}
+
+FUNCTION {forward.pass}
+{ last.label label =
+    { last.extra.num #1 + 'last.extra.num :=
+      last.extra.num int.to.chr$ 'extra.label :=
+    }
+    { "a" chr.to.int$ 'last.extra.num :=
+      "" 'extra.label :=
+      label 'last.label :=
+    }
+  if$
+  number.label #1 + 'number.label :=
+}
+
+FUNCTION {reverse.pass}
+{ next.extra "b" =
+    { "a" 'extra.label := }
+    'skip$
+  if$
+  extra.label 'next.extra :=
+  extra.label
+  duplicate$ empty$
+    'skip$
+    { "{\natexlab{" swap$ * "}}" * }
+  if$
+  'extra.label :=
+  label extra.label * 'label :=
+}
+
+EXECUTE {initialize.longest.label}
+
+ITERATE {forward.pass}
+
+REVERSE {reverse.pass}
+
+FUNCTION {bib.sort.order}
+{ sort.label  'sort.key$ :=
+}
+
+ITERATE {bib.sort.order}
+
+SORT
+
+FUNCTION {begin.bib}
+{   preamble$ empty$
+    'skip$
+    { preamble$ write$ newline$ }
+  if$
+  "\begin{thebibliography}{" number.label int.to.str$ * "}" *
+  write$ newline$
+  "\providecommand{\natexlab}[1]{#1}"
+  write$ newline$
+  "\providecommand{\url}[1]{\texttt{#1}}"
+  write$ newline$
+  "\expandafter\ifx\csname urlstyle\endcsname\relax"
+  write$ newline$
+  "  \providecommand{\doi}[1]{doi: #1}\else"
+  write$ newline$
+  "  \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi"
+  write$ newline$
+}
+
+EXECUTE {begin.bib}
+
+EXECUTE {init.state.consts}
+
+ITERATE {call.type$}
+
+FUNCTION {end.bib}
+{ newline$
+  "\end{thebibliography}" write$ newline$
+}
+
+EXECUTE {end.bib}