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% Title and authors
\title{A Positive Dispersion Optimum for the Rounded Sample Mean:\\
Sufficiency and Necessity of Finite Variance}

\author{%
  Mark R. Diamond\,\orcidlink{0000-0002-0244-8269}%
  \thanks{Corresponding author: \href{mailto:mark.diamond@yahoo.co.uk}%
  {\texttt{mark.diamond@yahoo.co.uk}}}
  \and
  Angela O'Brien-Malone\,\orcidlink{0000-0001-7787-8258}
}

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\begin{document}

\maketitle



\begin{abstract}
A continuous quantity, observable only after nearest-lattice rounding, is
estimated by the mean of $n$ rounded readings of an imperfect instrument.
Averaged over the position of the true value within a rounding cell, the
mean-squared error of this estimator is a function of one dimensionless
parameter: the pre-rounding standard deviation $s$, measured in cell widths.
A perfectly repeatable instrument ($s=0$) returns the same rounded value on
every reading, so averaging provides no additional information; its
cell-averaged error is the rounding floor $1/12$. We show that this floor is
never optimal once more than one reading is taken. For \emph{any} centred
pre-rounding law with finite, non-zero variance and any sample size $n>1$,
some strictly positive dispersion yields a smaller cell-averaged error than
zero, so the global minimizer lies away from the no-scatter endpoint; and each
of the three hypotheses---$n>1$, finite variance, non-degeneracy---is
individually necessary, so existence is \emph{exactly} characterized. The
mechanism is elementary: the autocorrelation of the rounding error has a
\emph{corner} at zero lag, so a little scatter relieves the rounding bias at
first order in $s$ while its variance contribution grows only at second order.
We give the short corner-based proof and then derive the spectral form of the
criterion, using it to compute the optimum for four laws. The Gaussian
optimizer is unique and solves a Jacobi-theta equation. For the Laplace law
the bias term has a hyperbolic closed form, and the optimizer is unique and
characterized by the corresponding derivative equation. For the uniform and
two-point laws the optimizer is explicit. The uniform law cancels every bias
harmonic exactly at its first positive cancellation scale, which saturates
the natural localization bound when $n=2$; and the two-point law, whose
criterion has several local wells once $n\ge4$, shows that finite variance
ensures existence but not a single-well objective.
\end{abstract}

\medskip
\noindent
{\small
{\bf Keywords:}
  quantization error, rounded sample mean, rounding error, characteristic functions,
  mean-squared error, Fourier series.

\medskip\noindent
{\bf 2020 Mathematics Subject Classification:}
  Primary 62F10; Secondary 94A12, 60E10, 42A16.
}

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

From ancient astronomical practice onward, observers have sought both
better instruments and better methods for identifying, reducing, bounding,
and combining observational errors \cite{Sheynin1973,Sheynin2007}. The
advantage of averaging was itself still contested in the eighteenth century:
Simpson gave a probabilistic defence of the mean of repeated astronomical
observations against the claim that a single observation taken with due care
was equally reliable \cite{Simpson1755}. This history supports a natural
limiting intuition: other things equal, greater precision and greater
repeatability should improve a measurement. The question
motivating this paper is whether that intuition remains valid when
repeatability is considered together with finite numerical resolution. If all
random unrepeatability could be removed, would a perfectly repeatable
instrument necessarily be best?

Finite numerical reporting introduces a distinct limitation. An observation
may concern a continuous quantity while being recorded only at fixed
increments. Before this limitation was described in the language of quantization,
closely related problems were treated through the graduation and
interpolation of instrument scales, and through the rounding or grouping
of recorded data \cite{Chapman1995,Hald2001}. Sheppard's analysis of data
arranged according to equidistant divisions of a scale is an early
statistical example \cite{Sheppard1897}.
In
the present paper, finite resolution is modelled by nearest-lattice
quantization: after rescaling, a continuous input is reported as the nearest
point of a fixed lattice. This is a deliberately restricted model of the more
general connection between finite-precision measurement and quantization
\cite[p.~1]{WidrowKollar2008}.

A perfectly repeatable instrument returns the same rounded value on every
reading, so repeated observations cannot recover the unknown position of the
true value within a rounding cell. An instrument with random
unrepeatability may instead return neighbouring lattice values. We ask
whether the average of $n$ such rounded readings can have smaller
cell-averaged squared error than the repeated output of the perfectly
repeatable instrument.

Section~\ref{sec:model} formulates the model and the cell-averaged criterion,
and Section~\ref{sec:existence} reduces the existence question to the local
behaviour of the rounding-error autocorrelation. The resulting theorem
characterizes when a positive minimizing dispersion exists. The spectral
representation in Section~\ref{sec:spectral} is then used to study four
representative scatter laws in Section~\ref{sec:gallery}.
Section~\ref{sec:relation} explains the relation to dithered quantization, and
Section~\ref{sec:discussion} records the scope of the result and the remaining
open questions.

The parameter $s$ describes residual pre-rounding unrepeatability. It is not a
deliberately injected dither or an independently controllable design variable.
Accordingly, the optimum studied here is a property of the model and criterion,
not a recommendation to add noise to an instrument.

% ============================================================
\section{Model and the cell-averaged criterion}
\label{sec:model}
% ============================================================

All lengths are measured in cell widths, so the cell width is $1$. We use
the nearest-lattice rounding map
\begin{equation}
  \Quant(x)=\left\lfloor x+\tfrac12\right\rfloor,
  \label{eq:rounding-map}
\end{equation}
which rounds half-integers upward. Any other fixed convention at
half-integers would give the same cell-averaged quantities considered below.
%
Let $Z$ be a real random variable
with
\begin{equation}
  \E [Z]=0,\qquad \E [Z^2]=1,
  \label{eq:noise-assumptions}
\end{equation}
and let $Z_1,\dots,Z_n$ be independent copies of $Z$.
For an unknown sub-cell
position $u\in[-\tfrac12,\tfrac12]$ and a dispersion $s\ge0$, put
\begin{equation}
  Y_j(u,s)=\Quant(u+sZ_j),\qquad
  \bar Y_n(u,s)=\frac1n\sum_{j=1}^n Y_j(u,s).
  \label{eq:readings}
\end{equation}
The criterion is the squared error of the rounded sample mean about the
underlying position $u$, averaged over one representative cell:
\begin{equation}
  \MSE_n(s)
  =
  \int_{-1/2}^{1/2}\E\!\left[(\bar Y_n(u,s)-u)^2\right]\dd u .
  \label{eq:cell-average-definition}
\end{equation}
For a physical cell width $h$ the model corresponds to observing
$h\,\Quant(\eta/h+sZ)$ with $u=\eta/h$ taken modulo one; the dimensional
cell-averaged error is then $h^2\,\MSE_n(s)$, and $s=\sigma/h$ is the
pre-rounding standard deviation expressed in cell widths.
The magnitudes of $\sigma$ and $h$ thus never enter separately: the cell
width contributes only the overall factor $h^2$, so the shape of the
cell-averaged error, and with it the location of its minimum, is governed
by their ratio $s$ alone.

It is convenient to separate
quantization (\emph{rounding}) from the identity.
The periodic quantization error
\begin{equation}
  q_{\varepsilon}(x)= x - \Quant(x)
  \label{eq:qeps-definition}
\end{equation}
equals $x$ on $(-\tfrac12,\tfrac12)$ and is extended with period one. Values at
half-integers depend on the chosen tie convention, but all identities below are
cell-integral identities and are therefore unaffected by changes on this null
set. Thus, up to null sets, $q_{\varepsilon}$ is an odd sawtooth with
\begin{equation}
  \int_{-1/2}^{1/2}q_{\varepsilon}(u)\dd u=0,\qquad
  \int_{-1/2}^{1/2}q_{\varepsilon}(u)^2\dd u=\frac1{12},
  \label{eq:qeps-moments}
\end{equation}
and a single reading decomposes as
\begin{equation}
  Y_j(u,s)-u=sZ_j-q_{\varepsilon}(u+sZ_j).
  \label{eq:single-error-decomposition}
\end{equation}
Throughout, $\varphi(t)=\E [\,\e^{\iu tZ}\,]$ denotes the characteristic function of
$Z$, $Z'$ an independent copy of $Z$, and $W=Z-Z'$ the symmetrized noise.

% ============================================================
\section{Variance cost \texorpdfstring{\emph{versus}}{versus} bias reduction: the decomposition}
\label{sec:decomposition}
% ============================================================

The variance--bias split of the sample mean turns
\eqref{eq:cell-average-definition} into a contest between two cell averages.

\begin{proposition}[Variance cost / bias-reduction decomposition]
\label{prop:decomposition}
Under \eqref{eq:noise-assumptions}, for every integer $n\ge1$ and every
$s\ge0$,
\begin{equation}
  \MSE_n(s)
  =
  \frac1n\!\left(s^2+\frac1{12}\right)
  +\frac{n-1}{n}\,A(s),
  \qquad
  A(s):=\int_{-1/2}^{1/2} b_s(u)^2\dd u,\quad
  b_s(u):=\E[\,q_{\varepsilon}(u+sZ)\,].
  \label{eq:decomposition}
\end{equation}
\end{proposition}

\begin{proof}
Write $\Delta(u,s)=Y_1(u,s)-u$. For independent identically distributed
readings the mean-squared error of the average splits as
\begin{equation}
  \E\!\left[(\bar Y_n(u,s)-u)^2\right]
  =\frac1n\,\E[\,\Delta(u,s)^2\,]+\frac{n-1}{n}\E[\,\Delta(u,s)\,]^2 .
  \label{eq:sample-mean-identity}
\end{equation}
We average the two terms over the cell separately. For the first, fix a
realization $z$. By \eqref{eq:single-error-decomposition}, the period-one
shift-invariance of $\int_{-1/2}^{1/2}q_{\varepsilon}(\,\cdot+c)$, and
\eqref{eq:qeps-moments},
\[
  \int_{-1/2}^{1/2}\!\bigl\{sz-q_{\varepsilon}(u+sz)\bigr\}^2\dd u
  =s^2z^2-2sz\!\int_{-1/2}^{1/2}\!q_{\varepsilon}(u+sz)\dd u
   +\!\int_{-1/2}^{1/2}\!q_{\varepsilon}(u+sz)^2\dd u
  =s^2z^2+\frac1{12}.
\]
Taking the expectation over $Z$ and using $\E [Z^2]=1$ gives
$\int_{-1/2}^{1/2}\E [ \Delta(u,s)^2 ]\dd u=s^2+\tfrac1{12}$. For the second, since
$\E [Z]=0$ the linear part of \eqref{eq:single-error-decomposition} averages to
zero, so $\E [ \Delta(u,s) ]=-\E[\,q_{\varepsilon}(u+sZ)\,]=-b_s(u)$, and the cell average of its
square is $A(s)$ by definition. Substituting both into the cell average of
\eqref{eq:sample-mean-identity} gives \eqref{eq:decomposition}.
\end{proof}

In \eqref{eq:decomposition}, the $s^2/n$ part is the \emph{variance cost} of scatter;
at $s=0$ the first summand is only $1/(12n)$, and the full rounding floor
$1/12$ is recovered only after adding the $A(0)$ contribution.
The function $A$ is
the cell-averaged squared rounding bias; it carries all the noise-dependence of
the bias-reduction. The two facts we need about it are immediate. First,
$q_{\varepsilon}$ has mean zero on each cell, so $b_0(u)=q_{\varepsilon}(u)$ and, by
\eqref{eq:qeps-moments},
\begin{equation}
  A(0)=\int_{-1/2}^{1/2}q_{\varepsilon}(u)^2\dd u=\frac1{12},
  \qquad\text{whence}\qquad
  \MSE_n(0)=\frac1{12}.
  \label{eq:floor}
\end{equation}
The value $\tfrac1{12}$ is the classical second moment of the rounding error
over a cell \cite{Sheppard1897}.
Second, $|b_s(u)|\le\tfrac12$ always, so $A(s)\le\tfrac14$ and, as we show
below, $A(s)\le A(0)$: zero scatter maximizes the cell-averaged squared
bias, and dispersion cannot increase it. Subtracting \eqref{eq:floor} from \eqref{eq:decomposition}
isolates the comparison,
\begin{equation}
  \MSE_n(s)-\frac1{12}
  =\frac{s^2}{n}-\frac{n-1}{n}\bigl(A(0)-A(s)\bigr),
  \label{eq:contest}
\end{equation}
in which a positive dispersion improves on the no-scatter endpoint exactly
when the bias reduction $A(0)-A(s)$, weighted by $(n-1)/n$, exceeds the
variance cost $s^2/n$. The remaining question is the rate at which
$A(0)-A(s)$ grows as $s$ leaves zero. The next
section answers this without any distributional assumption.

% ============================================================
\section{Existence via the corner}
\label{sec:existence}
% ============================================================

The reduction in the bias is governed by a single elementary function, the autocorrelation of
the rounding error, and by one feature of it: a corner.

\subsection{The squared bias as an autocorrelation averaged over a random lag}

\begin{lemma}[Autocorrelation form of the bias]
\label{lem:autocorr}
Let
\begin{equation}
  g(\delta):=\int_{-1/2}^{1/2}q_{\varepsilon}(v)\,q_{\varepsilon}(v+\delta)\dd v .
  \label{eq:g-def}
\end{equation}
Then for every $s\ge0$,
\begin{equation}
  A(s)=\E\bigl[g(sW)\bigr],\qquad W=Z-Z'.
  \label{eq:A-is-Eg}
\end{equation}
\end{lemma}

\begin{proof}
Using $b_s(u)=\E[\,q_{\varepsilon}(u+sZ)\,]$ and an independent copy $Z'$ to write the square
as a product of two expectations,
\[
  b_s(u)^2=\E\bigl[q_{\varepsilon}(u+sZ)\bigr]\,\E\bigl[q_{\varepsilon}(u+sZ')\bigr]
          =\E\bigl[q_{\varepsilon}(u+sZ)\,q_{\varepsilon}(u+sZ')\bigr],
\]
the last step by independence. The integrand is bounded by $\tfrac14$ on a
bounded interval, so Fubini's theorem moves the cell integral inside the
expectation:
\[
  A(s)=\E\!\left[\int_{-1/2}^{1/2}q_{\varepsilon}(u+sZ)\,q_{\varepsilon}(u+sZ')\dd u\right].
\]
For fixed realizations $z,z'$, substitute $v=u+sz$ and use the period-one
shift-invariance of the integral of a period-one integrand; the inner integral
becomes $\int_{-1/2}^{1/2}q_{\varepsilon}(v)\,q_{\varepsilon}\bigl(v+s(z'-z)\bigr)\dd v=g\bigl(s(z'-z)\bigr)$,
which depends on $z,z'$ only through the lag. Since $g$ is even (substitute
$v\mapsto-v$ and use $q_{\varepsilon}(-x)=-q_{\varepsilon}(x)$) and $Z-Z'$ is symmetric, replacing
$Z'-Z$ by $W=Z-Z'$ leaves the expectation unchanged, giving
\eqref{eq:A-is-Eg}.
\end{proof}

\subsection{The autocorrelation and its corner}

\begin{lemma}[Closed form and corner]
\label{lem:g-closed}
With $\nint{\delta}:=\dist(\delta,\Z)\in[0,\tfrac12]$ the distance from
$\delta$ to the nearest integer,
\begin{equation}
  g(\delta)=\frac1{12}-\frac12\nint{\delta}+\frac12\nint{\delta}^2,
  \qquad\text{equivalently}\qquad
  \frac1{12}-g(\delta)=\frac12\,\nint{\delta}\bigl(1-\nint{\delta}\bigr).
  \label{eq:g-closed}
\end{equation}
In particular $g$ is continuous, has period one, attains its maximum
$g(0)=\tfrac1{12}$ at the integers, and has the one-sided slopes
\begin{equation}
  g'(0^+)=-\tfrac12,\qquad g'(0^-)=+\tfrac12 .
  \label{eq:corner-slopes}
\end{equation}
\end{lemma}

\begin{proof}
By evenness and periodicity it suffices to compute $g$ on $[0,\tfrac12]$. Fix
$\delta\in[0,\tfrac12]$. On the window $v\in[-\tfrac12,\tfrac12)$ the first
factor is $q_{\varepsilon}(v)=v$. The shifted argument $v+\delta$ crosses the half-integer
discontinuity at $v=\tfrac12-\delta\in[0,\tfrac12]$, where $q_{\varepsilon}(v+\delta)$ drops by
one: it equals \(v+\delta\)
for \(v<\tfrac12-\delta\) and \(v+\delta-1\) for \(v\ge\tfrac12-\delta\). Splitting \eqref{eq:g-def} at this point,
\[
  g(\delta)=\int_{-1/2}^{1/2-\delta}\!\! v(v+\delta)\dd v
           +\int_{1/2-\delta}^{1/2}\!\! v(v+\delta-1)\dd v
          =\tfrac12\bigl(\tfrac12-\delta\bigr)^2-\tfrac1{24}
          =\frac1{12}-\frac\delta2+\frac{\delta^2}{2},
\]
the cubic endpoint terms cancelling at the split point. On $[0,\tfrac12]$ the
nearest integer is $0$, so $\nint{\delta}=\delta$ and this is
\eqref{eq:g-closed}; evenness and periodicity carry it to all $\delta$ through
$\nint{\delta}$. On $(0,\tfrac12)$, $g(\delta)=\tfrac1{12}-\tfrac\delta2+\tfrac{\delta^2}{2}$
has $g'(\delta)=-\tfrac12+\delta$; letting $\delta\downarrow0$ gives
$g'(0^+)=-\tfrac12$, and evenness gives $g'(0^-)=+\tfrac12$.
\end{proof}

The parabolic term $\tfrac12\nint{\delta}^2$ is flat at the origin, so all of
the behaviour of $g$ near a peak comes from the term $-\tfrac12\nint{\delta}$:
an inverted wedge with slopes $\pm\tfrac12$ (Figure~\ref{fig:g-autocorr}). A
smooth peak could not do this. By Taylor's theorem a differentiable function
with a vanishing derivative at its maximum falls only quadratically; the corner
falls linearly. That gap---linear bias reduction against quadratic variance
cost---is the entire content of the existence theorem.

\begin{figure}[tp]
  \centering
\begin{tikzpicture}[x=2.5cm,y=26cm,>=stealth,line cap=butt,line join=miter]
  % detached y-axis, parked to the left
  \draw[thin,->] (-1.75,-0.058) -- (-1.75,0.103);
  \draw[thin] (-1.75,0.0833333) -- (-1.70,0.0833333);
  \node[left=2pt,font=\small] at (-1.75,0.0833333) {$\tfrac{1}{12}$};
  \draw[thin] (-1.75,-0.0416667) -- (-1.70,-0.0416667);
  \node[left=2pt,font=\small] at (-1.75,-0.0416667) {$-\tfrac{1}{24}$};
  % x-axis at y=0
  \draw[thin,->] (-1.60,0) -- (1.60,0);
  \node[below=1pt,font=\small] at (1.60,0) {lag~$\delta$};
  \foreach \k in {-1,0,1}{
    \draw[thin] (\k,-0.0016) -- (\k,0.0016);
    \node[below=2pt,font=\small] at (\k,-0.0016) {$\k$};
  }
  % the autocorrelation curve g(delta)
  \draw[plotblue,very thick]
     plot[domain=-1.5:1.5,samples=121]
       (\x,{1/12 - 0.5*abs(\x-round(\x)) + 0.5*(\x-round(\x))^2});
  % tangents at the central corner, slopes +1/2 (left) and -1/2 (right)
  \draw[cornerred,thick,dashed] (-0.2083333,-0.0208333) -- (0.0416667,0.1041667);
  \draw[cornerred,thick,dashed] (-0.0416667,0.1041667) -- (0.2083333,-0.0208333);
  \node[rotate=90,anchor=south,font=\small] at (-2.10,0.0208) {$g(\delta)$};
\end{tikzpicture}
  \caption{The rounding-error autocorrelation $g(\delta)$ of
    \eqref{eq:g-closed}, period one with maximum $\tfrac1{12}$ and minimum
    $-\tfrac1{24}$. At each integer lag $g$ has a downward \emph{corner}; the
    red dashed lines are the one-sided tangents of slopes $+\tfrac12$ (left)
    and $-\tfrac12$ (right) at the peak $(0,\tfrac1{12})$. The vertical scale
    is greatly exaggerated.}
  \label{fig:g-autocorr}
\end{figure}

\subsection{Linear relief and the existence theorem}

\begin{lemma}[First-order reduction in the bias]
\label{lem:relief}
If $\E [Z^2]<\infty$ then $\E|W|<\infty$ and
\begin{equation}
  \frac{A(0)-A(s)}{s}\;\longrightarrow\;\tfrac12\,\E|W|=\tfrac12\,\E|Z-Z'|
  \qquad(s\downarrow0).
  \label{eq:relief-slope}
\end{equation}
\end{lemma}

\begin{proof}
By \eqref{eq:A-is-Eg} and $A(0)=\E[g(0)]=\tfrac1{12}$,
\[
  \frac{A(0)-A(s)}{s}
  =\E\!\left[\frac{\tfrac1{12}-g(sW)}{s}\right]
  =\E\!\left[\frac{\tfrac12\,\nint{sW}\bigl(1-\nint{sW}\bigr)}{s}\right],
\]
using \eqref{eq:g-closed}. For fixed $w$, once $s|w|<\tfrac12$ we have
$\nint{sw}=s|w|$, so the bracket equals
$\tfrac12|w|\bigl(1-s|w|\bigr)\to\tfrac12|w|$ as $s\downarrow0$. Moreover
$\nint{x}\bigl(1-\nint{x}\bigr)\le\nint{x}\le|x|$ gives the uniform domination
$0\le\tfrac12\nint{sw}(1-\nint{sw})/s\le\tfrac12|w|$, an integrable bound since
$\E|W|\le2\,\E|Z|<\infty$. Dominated convergence yields
\eqref{eq:relief-slope}.
\end{proof}

\begin{theorem}[A positive optimum under finite variance]
\label{thm:existence}
Let $Z$ satisfy \eqref{eq:noise-assumptions} and let $n>1$. Then $\MSE_n$
attains its global minimum on $[0,\infty)$. Let $s_n^*$ denote any global
minimizer. Then $s_n^*>0$ and
\begin{equation}
  \boxed{\;\MSE_n(s_n^*)<\MSE_n(0)=\frac1{12}.\;}
  \label{eq:strict-improvement}
\end{equation}
\end{theorem}
\begin{proof}
The autocorrelation $g$ is continuous and bounded, so by bounded convergence
$A(s)=\E[g(sW)]$ is continuous, and hence so is $\MSE_n$. From
\eqref{eq:decomposition} and $A\ge0$, $\MSE_n(s)\ge s^2/n\to\infty$, so the
continuous $\MSE_n$ attains a global minimum on $[0,\infty)$.

Combining the contest \eqref{eq:contest} with the rate at which the bias is reduced,
\eqref{eq:relief-slope},
\begin{equation}
  \MSE_n(s)-\frac1{12}
  =\frac{s^2}{n}-\frac{n-1}{2n}\,\E|W|\,s+o(s)
  \qquad(s\downarrow0).
  \label{eq:strict-help}
\end{equation}
Because $\E [Z^2]=1$ the law is non-degenerate, so $\E|W|=\E|Z-Z'|>0$; the
negative linear term dominates the positive quadratic one, and the right-hand
side is negative for all sufficiently small $s>0$. By \eqref{eq:contest} every such $s$ satisfies
$\MSE_n(s)<\MSE_n(0)$. The endpoint $s=0$ is therefore not a minimizer, so every
global minimizer lies in $(0,\infty)$ and takes a value below $1/12$.
\end{proof}


\begin{remark}[Uniqueness is separate from existence]
Uniqueness is not asserted in Theorem~\ref{thm:existence}. It is established
for the two-point and uniform laws in \S\ref{ssec:twopoint} and
\S\ref{ssec:uniform}, and for Gaussian scale
mixtures---including the Gaussian and Laplace laws---in
Proposition~\ref{prop:cm-uniqueness}.
\end{remark}

The proof drew on finite variance twice---for coercivity and, through
$\E|W|<\infty$, for the bias-reduction rate---and on non-degeneracy once. With $n>1$
these three are also necessary, as we now show; existence is then characterized
exactly.

% ============================================================
\section{The characterization}
\label{sec:characterization}
% ============================================================

\begin{proposition}[The hypotheses are necessary]
\label{prop:necessity}
Let $Z$ be centred and $n\ge1$. A strictly improving positive dispersion---a
value $s>0$ with $\MSE_n(s)<\MSE_n(0)$---exists if and only if
\begin{equation}
  n>1\qquad\text{and}\qquad 0<\Var Z<\infty .
  \label{eq:characterization}
\end{equation}
\end{proposition}

\begin{proof}
Sufficiency follows from Theorem~\ref{thm:existence} by scaling.  If
$0<\Var Z=\tau^2<\infty$, put $\widetilde Z=Z/\tau$.  Theorem~\ref{thm:existence}
applied to $\widetilde Z$ gives a $t>0$ such that
$\MSE_n^{\widetilde Z}(t)<\MSE_n^{\widetilde Z}(0)$; since the original law at dispersion
$s=t/\tau$ has the same rounded readings as the normalized law at dispersion
$t$, this gives $\MSE_n^Z(t/\tau)<\MSE_n^Z(0)$.
%
For necessity, suppose one condition fails.

\emph{If $n=1$}, there is no second reading to average: the bias-reduction term drops
out and $\MSE_1(s)=\int_{-1/2}^{1/2}\E[\,\Delta(u,s)^2\,]\dd u$. The first-term
calculation in the proof of Proposition~\ref{prop:decomposition} invokes the
normalization \eqref{eq:noise-assumptions} only at its closing step; carried out
for a centred $Z$ of arbitrary finite variance it gives
$\int_{-1/2}^{1/2}\E[\,\Delta(u,s)^2\,]\dd u=s^2\,\E[Z^2]+\tfrac1{12}$, whence
\[
  \MSE_1(s)=s^2\Var Z+\frac1{12}\qquad(\Var Z<\infty),
\]
so no positive $s$ strictly improves on $s=0$; if $\Var Z=\infty$, the
infinite-variance case below shows that $\MSE_1(s)=\infty$ for every $s>0$.

\emph{If $\Var Z=0$}, then, since $Z$ is centred, $Z=0$ almost surely. Hence
$Y_j(u,s)=\Quant(u)$ for every $j$ and every $s\ge0$, so
\[
  \MSE_n(s)
  =\int_{-1/2}^{1/2}(\Quant(u)-u)^2\dd u
  =\int_{-1/2}^{1/2}q_{\varepsilon}(u)^2\dd u
  =\frac1{12}
\]
for all $s\ge0$. Thus no positive dispersion strictly improves on the endpoint.

\emph{If $\Var Z=\infty$} (equivalently $\E[Z^2]=\infty$), fix $s>0$ and
$u\in[-\tfrac12,\tfrac12]$, and write $\Delta_j(u,s)=Y_j(u,s)-u$. The pointwise
inequality $(a+b)^2\ge\tfrac12 a^2-b^2$, applied with $a=sZ_j$, $b=-q_{\varepsilon}(u+sZ_j)$
and $|q_{\varepsilon}|\le\tfrac12$, gives
\[
  \Delta_j(u,s)^2=\bigl(sZ_j-q_{\varepsilon}(u+sZ_j)\bigr)^2\ge\tfrac12 s^2Z_j^2-\tfrac14,
  \qquad\text{hence}\qquad
  \E[\,\Delta_j(u,s)^2\,]\ge\tfrac12 s^2\,\E[Z^2]-\tfrac14=\infty.
\]
Each $\Delta_j(u,s)$ is almost surely finite, since $sZ_j$ is finite and
$|q_{\varepsilon}|\le\tfrac12$; hence $C:=\sum_{j=2}^{n}\Delta_j(u,s)$ is a.s.\ finite and
independent of $\Delta_1(u,s)$.
%
Applying the same inequality once more, now with $a=\Delta_1(u,s)$ and
$b=C$, and conditioning on the event $C=c$, gives for every finite $c$
\[
  \E\bigl[(\Delta_1(u,s)+c)^2\bigr]
  \ge \tfrac12\,\E[\,\Delta_1(u,s)^2\,]-c^2
  =\infty .
\]
Since $C$ is a.s.\ finite and independent of $\Delta_1(u,s)$, it follows that
\[
  \E\bigl[(\Delta_1(u,s)+C)^2\mid C\bigr]=\infty
  \quad\text{a.s.}
\]

Since $\bar Y_n(u,s)-u=\tfrac1n\bigl(\Delta_1(u,s)+C\bigr)$, taking expectations
gives
\[
  \E\!\left[\left(\bar Y_n(u,s)-u\right)^2\right]
  =\frac1{n^2}\,\E\bigl[(\Delta_1(u,s)+C)^2\bigr]=\infty
\]
for every $u$ and every $s>0$. Therefore $\MSE_n(s)=\infty$ for every $s>0$,
whereas $\MSE_n(0)=\tfrac1{12}$; no positive value improves on the endpoint.

In each case no strictly improving positive
dispersion exists.
\end{proof}

Theorem~\ref{thm:existence} and Proposition~\ref{prop:necessity} together close
the existence question.

\begin{theorem}[Existence, exactly characterized]
\label{thm:iff}
For a centred pre-rounding law and $n\ge1$, there exists a strictly improving
positive dispersion,
\[
  \exists\,s>0\quad\text{such that}\quad \MSE_n(s)<\MSE_n(0),
\]
if and only if $n>1$ and $0<\Var Z<\infty$.
\end{theorem}

Two remarks fix the logical shape of this result. First, the two directions
have different characters. Sufficiency is \emph{local}: it reads the sign of the
one-sided derivative
\begin{equation}
  \left.\frac{\dd}{\dd s}\right|_{s=0^+}\!\MSE_n(s)
  =-\frac{n-1}{n}\cdot\tfrac12\,\E|W| < 0
  \label{eq:right-derivative}
\end{equation}
straight off the corner, with no need to know where the minimum is. Necessity is
\emph{global}: each failure mode is excluded by a statement about the whole
function $\MSE_n$, not its germ at the origin. The asymmetry is worth stating
plainly, because it is what makes the characterization usable: to certify that
scatter helps one differentiates once at the endpoint, whereas to certify that
it cannot one must control the entire curve.

\begin{remark}[Centredness is a normalization, not a hypothesis]
\label{rem:centring}
The assumption $\E[Z]=0$ is a normalization. Write $Z=m+Z^\circ$, where $Z^\circ$ is
centred. The mean error of one rounded reading is
\[
  B_s(u):=\E\left[\,Y_1(u,s)-u\,\right]
        =sm-\E[\,q_{\varepsilon}(u+sm+sZ^\circ)\,],
\]
so, by the period-one shift-invariance of the cell integral and
$\int_{-1/2}^{1/2}\E[\,q_{\varepsilon}(v+sZ^\circ)\,]\dd v=0$,
\[
  \int_{-1/2}^{1/2} B_s(u)^2\dd u
  =\int_{-1/2}^{1/2}\bigl(sm-\E[\,q_{\varepsilon}(v+sZ^\circ)\,]\bigr)^2\dd v
  =s^2m^2+A_{Z^\circ}(s).
\]
The cost term carries the same factor: by the cell-integral calculation in the
proof of Proposition~\ref{prop:decomposition}, without normalising,
$\int_{-1/2}^{1/2}\E[\,\Delta(u,s)^2\,]\dd u=s^2\,\E[Z^2]+\tfrac1{12}
=s^2(m^2+\Var Z^\circ)+\tfrac1{12}$. Assembling the cell average of the
variance--bias split \eqref{eq:sample-mean-identity}, the mean enters both terms
and collects as $\tfrac1n s^2m^2+\tfrac{n-1}{n}s^2m^2=s^2m^2$, leaving
\[
  \MSE_n(s)=s^2m^2+\MSE_n^{Z^\circ}(s),
\]
where $\MSE_n^{Z^\circ}$ is the cell-averaged error \eqref{eq:cell-average-definition}
of the centred law $Z^\circ$. Thus a non-zero mean adds the pure quadratic cost
$s^2m^2$ to the centred problem; since $s^2m^2$ vanishes to second order at
$s=0$, it can shift the optimizer toward zero but cannot create or destroy the
first-order reduction in the bias. The substantive condition is $0<\Var Z<\infty$.
\end{remark}

With existence settled, the rest of the note computes the optimum. For that we
need the spectral form of $A$.

% ============================================================
\section{The spectral form and a localization bound}
\label{sec:spectral}
% ============================================================

The bridge from the corner to closed-form optima is the Fourier series of $g$.

\begin{lemma}[Spectral form of the bias functional]
\label{lem:spectral}
For every $s\ge0$,
\begin{equation}
  A(s)=\frac1{2\pi^2}\sum_{k=1}^{\infty}\frac{|\varphi(2\pi ks)|^2}{k^2},
  \qquad A(0)=\frac1{2\pi^2}\sum_{k\ge1}\frac1{k^2}=\frac1{12}.
  \label{eq:spectral}
\end{equation}
Moreover, writing $|\varphi(t)|^2=\E[\,\cos(tW)\,]$,
\begin{equation}
  A(0)-A(s)=\frac1{2\pi^2}\sum_{k\ge1}\frac{1-\E[\,\cos(2\pi ksW)\,]}{k^2}
           =\E\!\left[\tfrac12\,\nint{sW}\bigl(1-\nint{sW}\bigr)\right].
  \label{eq:deficit}
\end{equation}
\end{lemma}

\begin{proof}
In $L^2[-\tfrac12,\tfrac12]$ the sawtooth has the sine series
$q_{\varepsilon}(x)=\sum_{k\ge1}\beta_k\sin(2\pi kx)$ with
$\beta_k=(-1)^{k+1}/(\pi k)$, so $\beta_k^2=1/(\pi^2k^2)$. Sine orthogonality
gives $\int_{-1/2}^{1/2}\sin(2\pi kv)\sin(2\pi j(v+\delta))\dd v
=\tfrac12\delta_{jk}\cos(2\pi k\delta)$, hence
\begin{equation}
  g(\delta)=\sum_{k\ge1}\frac{\beta_k^2}{2}\cos(2\pi k\delta)
           =\frac1{2\pi^2}\sum_{k\ge1}\frac{\cos(2\pi k\delta)}{k^2}.
  \label{eq:g-fourier}
\end{equation}
This series converges absolutely and uniformly (dominated by
$\sum k^{-2}$), so expectation passes through the sum in
$A(s)=\E[g(sW)]$; with $\E[\,\cos(2\pi ksW)\,]=|\varphi(2\pi ks)|^2$ from
$|\varphi(t)|^2=\E[\,\e^{\iu tW}\,]$ this gives \eqref{eq:spectral}.
Subtracting from $A(0)=g(0)=\tfrac1{12}$ gives the first equality of
\eqref{eq:deficit}. Applying the classical evaluation
\begin{equation}
  \sum_{k\ge1}\frac{1-\cos(2\pi kx)}{k^2}=\pi^2\,\nint{x}\bigl(1-\nint{x}\bigr)
  \label{eq:bernoulli}
\end{equation}
inside the expectation (legitimate by Tonelli, the summand being non-negative)
gives the second equality, which is the deficit form of \eqref{eq:g-closed}
and recovers Lemma~\ref{lem:relief} on dividing by $s$.
\end{proof}

Identity \eqref{eq:bernoulli} is the single welded joint between the two routes:
read left to right it is the spectral series; read right to left it is the
corner term $\tfrac12\nint{x}(1-\nint{x})$ whose linear cusp drove the existence
proof. The same fact does both jobs.

The spectral form makes one universal bound on the optimizer immediate.

\begin{corollary}[Localization]
\label{cor:localization}
Under the hypotheses of Theorem~\ref{thm:existence}, every strictly improving
positive dispersion---in particular every global minimizer $s_n^*$---satisfies
\begin{equation}
  (s_n^*)^2<\frac{n-1}{12}.
  \label{eq:localization}
\end{equation}
\end{corollary}

\begin{proof}
Improvement requires that the reduction in the bias outweighs the variance
cost in \eqref{eq:contest}:
$s^2/n<\tfrac{n-1}{n}\bigl(A(0)-A(s)\bigr)$. But
$A(0)-A(s)\le A(0)=\tfrac1{12}$, so $s^2<(n-1)/12$.
\end{proof}

The bound \eqref{eq:localization} is honest but loose: its scale
$\sqrt{(n-1)/12}$ grows like $\sqrt n$, while the optimizers we compute below
grow far more slowly. For the Gaussian law the optimizer grows like
$\sqrt{\log n}$ (\S\ref{sec:gallery}), whereas the uniform and two-point
optimizers converge to finite limits. Rapid decay of $\varphi$ alone does not
force the Gaussian rate. If $\varphi$ vanishes outside a bounded interval
$[-T,T]$---as it does, after rescaling to unit variance, for the centred
finite-variance law with density proportional to
$\bigl(\sin(x/2)/(x/2)\bigr)^4$, whose characteristic function is the
self-convolution of a triangle---then every term of \eqref{eq:spectral}
vanishes for $2\pi s\ge T$, so $A(s)=0$ and $\MSE_n$ is strictly increasing
there; hence $s_n^*\le T/(2\pi)$ for every $n$. The localization scale is nonetheless the correct endpoint scale:
for the uniform law, the exact-debiasing point reaches the boundary when
$n=2$, although there it ties the floor rather than strictly improving.

% ============================================================
\section{A gallery of laws}
\label{sec:gallery}
% ============================================================

We compute the optimum for four centred, unit-variance laws. They are arranged
to do four different jobs: the two-point law furnishes the cheapest witness, an
exact rational optimizer, and a counterexample to single-well behaviour all at
once; the Gaussian is the canonical smooth, uniquely solvable case; the Laplace
shows that a heavier tail keeps uniqueness while changing the special function;
and the uniform cancels the bias exactly and marks the boundary by saturating
the localization bound. Throughout,
the \emph{bias-reduction rate} is $\tfrac12\E|W|$, the slope from
Lemma~\ref{lem:relief}. Optimizers in Table~\ref{tab:gallery} were obtained by
minimizing \eqref{eq:decomposition} numerically and agree with the closed forms
below to the digits shown.

When the symmetrized noise is bounded, the criterion is exactly quadratic on
an initial interval, and the two bounded laws of the gallery share a single
closed-form optimizer.

\begin{lemma}[Bounded symmetrized noise]
\label{lem:bounded}
Suppose $|W|\le w$ almost surely for some $w>0$, and put
\[
  c_n:=\frac{(n-1)\,\E|W|}{4n}.
\]
Then for $0\le s\le 1/w$,
\begin{equation}
  \MSE_n(s)-\frac1{12}=s^2-2c_n\,s .
  \label{eq:bounded-quadratic}
\end{equation}
Consequently, if $c_n\le1/w$, the unique minimizer of $\MSE_n$ over
$[0,1/w]$ is $c_n$, with $\MSE_n(c_n)=\tfrac1{12}-c_n^2$.
\end{lemma}

\begin{proof}
The function $x\mapsto x(1-x)$ is symmetric about $x=\tfrac12$, so for
$|\delta|\le1$ one has $\nint{\delta}\bigl(1-\nint{\delta}\bigr)=|\delta|\bigl(1-|\delta|\bigr)$
whether $\nint{\delta}=|\delta|$ or $\nint{\delta}=1-|\delta|$. For
$0\le s\le1/w$ we have $|sW|\le1$, so by \eqref{eq:deficit} and
$\E[W^2]=2$,
\[
  A(0)-A(s)=\E\!\left[\tfrac12\,s|W|\bigl(1-s|W|\bigr)\right]
           =\frac s2\,\E|W|-s^2 .
\]
Substitution into \eqref{eq:contest} gives
$\MSE_n(s)-\tfrac1{12}=\tfrac{s^2}{n}-\tfrac{n-1}{n}\bigl(\tfrac s2\E|W|-s^2\bigr)
=s^2-2c_ns$, which is \eqref{eq:bounded-quadratic}. This parabola has its
vertex at $s=c_n$ and value $-c_n^2$ there.
\end{proof}

\subsection{Two points: the minimal witness, exactly}
\label{ssec:twopoint}

Let $Z=\pm1$ with probability $\tfrac12$ each---the cheapest non-degenerate
centred unit-variance law. Then $\varphi(t)=\cos t$, $W=Z-Z'\in\{-2,0,2\}$ with
$\Pr(W=0)=\tfrac12$ and $\Pr(W=\pm2)=\tfrac14$, and $\E|W|=1$, so the relief
rate is exactly $\tfrac12$. Because $W$ is supported on $\{0,\pm2\}$,
\eqref{eq:A-is-Eg} gives $A$ in closed form with no series at all:
$A(s)=\tfrac12 g(0)+\tfrac12 g(2s)$. For $s\in[0,\tfrac14]$ one has
$2s\in[0,\tfrac12]$, so $\nint{2s}=2s$ and
\begin{equation}
  A(s)=\frac12\cdot\frac1{12}+\frac12\Bigl(\frac1{12}-s+2s^2\Bigr)
      =s^2-\frac s2+\frac1{12},
  \qquad s\in[0,\tfrac14].
  \label{eq:twopoint-A}
\end{equation}
Substituting into \eqref{eq:contest}, the contest collapses to a single
parabola,
\[
  \MSE_n(s)-\frac1{12}
  =s^2-\frac{n-1}{2n}\,s,
\]
which is \eqref{eq:bounded-quadratic} of Lemma~\ref{lem:bounded} with $w=2$
and $c_n=(n-1)/(4n)$. Its minimum over $[0,\tfrac14]$ is attained at
\begin{equation}
  s_n^*=\frac{n-1}{4n}\in\bigl(0,\tfrac14\bigr),
  \qquad
  \MSE_n(s_n^*)=\frac1{12}-\frac{(n-1)^2}{16\,n^2}.
  \label{eq:twopoint-opt}
\end{equation}
This is a fully elementary, rational optimum: no special function is
involved, it is valid for every $n>1$, and, since $s_n^*<\tfrac14$, it
always lies inside $[0,\tfrac14)$.

It is also the global minimizer. With
\[
  r(s)=\dist(2s,\Z),
\]
one has $A(0)-A(s)=\tfrac14 r(s)\bigl(1-r(s)\bigr)$, so by \eqref{eq:contest}
\begin{equation}
  n\Bigl(\MSE_n(s)-\frac1{12}\Bigr)
  =s^2-\frac{n-1}{4}\,r(s)\bigl(1-r(s)\bigr)
  \ge s^2-\frac{n-1}{16},
  \label{eq:twopoint-global}
\end{equation}
because $r(1-r)\le\tfrac14$. By \eqref{eq:twopoint-opt} the minimum over
$[0,\tfrac14]$ has $n\bigl(\MSE_n(s_n^*)-\tfrac1{12}\bigr)=-(n-1)^2/(16n)$, and the
lower bound in \eqref{eq:twopoint-global} exceeds this whenever
$s^2>(n-1)/(16n)$. Since $(n-1)/(16n)<\tfrac1{16}$, every $s\ge\tfrac14$
gives a strictly larger value than $s_n^*$. On the remaining interval
$[0,\tfrac14)$ the criterion is the parabola above, so $s_n^*$ is the unique
global minimizer.

Yet
\[
  |\varphi(2\pi ks)|^2=\cos^2(2\pi ks)
\]
does not decay. Each harmonic revives with period $\tfrac12$ in $s$, and for
larger $n$ the criterion develops later local wells. For each
$\ell\in\mathbb Z_{\ge0}$ parameterize the $\ell$-th arch by
\[
  s=\frac{\ell}{2}+x,
  \qquad
  0\le x\le\frac12 .
\]
On the whole arch $r(s)\bigl(1-r(s)\bigr)=2x(1-2x)$, whether $r=2x$ or
$r=1-2x$, so
\[
  n\Bigl(\MSE_n(s)-\frac1{12}\Bigr)
  =n x^2+\left(\ell-\frac{n-1}{2}\right)x+\frac{\ell^2}{4},
  \qquad 0\le x\le\tfrac12 .
\]
This parabola has an interior minimum, at
$x=\bigl(\tfrac{n-1}{2}-\ell\bigr)/(2n)\in(0,\tfrac14)$, exactly when
$\ell<(n-1)/2$; otherwise it is increasing on the arch. At a junction
$s=\ell/2$ with $\ell\ge1$ the left derivative, taken from arch $\ell-1$ at
$x=\tfrac12$, is $\tfrac{n-1}2+\ell>0$, so a junction is never a local
minimum. Hence $\MSE_n$ has exactly $\lfloor n/2\rfloor$ local minima on
$(0,\infty)$, one on each arch $\ell=0,1,\ldots,\lfloor n/2\rfloor-1$. For
$n=2,3$ the criterion is single-welled; for $n\ge4$ it has several wells, of
which the first, \eqref{eq:twopoint-opt}, contains the unique global minimizer.
The two-point law therefore shows that finite variance ensures existence but
does not imply a globally simple objective.
%
For plots of $\MSE_n$ against \(s\) for two-point scatter,
see figure~\ref{fig:mse-curves-twopoint}.
%
% =====================================================================
% FIGURE -- TWO-POINT  (Z = +/-1, equal mass)
% =====================================================================
\begin{figure}[p]
\centering
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  \node[anchor=west,font=\scriptsize] at (axis cs:0.30,0.038) {$s_n^*\!\approx\!0.125$};

\nextgroupplot[title={$n=3$}]
  \addplot[plotblue] table[x=s,y=I3]{\msedatatp};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.16667,0.05556)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.30,0.036) {$s_n^*\!\approx\!0.167$};

\nextgroupplot[title={$n=4$}]
  \addplot[plotblue] table[x=s,y=I4]{\msedatatp};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.18750,0.04818)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.31,0.030) {$s_n^*\!\approx\!0.188$};

\nextgroupplot[title={$n=8$}]
  \addplot[plotblue] table[x=s,y=I8]{\msedatatp};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.21875,0.03548)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.31,0.022) {$s_n^*\!\approx\!0.219$};

\nextgroupplot[title={$n=20$}]
  \addplot[plotblue] table[x=s,y=I20]{\msedatatp};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.23750,0.02693)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.32,0.018) {$s_n^*\!\approx\!0.238$};
\end{groupplot}
\end{tikzpicture}
\caption{\textbf{Two-point noise} ($Z=\pm1$ with equal probability).
Cell-averaged $\MSE_n(s)$ as a
function of the scatter level $s$ measured in cell widths, for
$n\in\{1,2,3,4,8,20\}$.
The dashed line marks the zero-scatter value
$\MSE_n(0)=1/12$; the dot marks the global minimizer.
%
The minimizer follows the exact rational form
$s_n^*=(n-1)/(4n)$ with value $1/12-(n-1)^2/(16n^2)$. Because
$|\varphi(2\pi ks)|^2=\cos^2(2\pi ks)$ does not decay, the harmonics
revive with period $\tfrac12$ in $s$; the curve has exactly
$\lfloor n/2\rfloor$ local wells, so several wells appear from $n=4$ onward
(here $n=4,8,20$), and the first well always contains the global minimizer.
Vertical scales differ between panels.}
\label{fig:mse-curves-twopoint}
\end{figure}

\subsection{Gaussian: the theta optimizer}
\label{ssec:gaussian}
%
For $Z\sim\mathcal N(0,1)$, $\varphi(t)=\e^{-t^2/2}$ and
$|\varphi(2\pi ks)|^2=\e^{-4\pi^2k^2s^2}$, so \eqref{eq:spectral} reads
\begin{equation}
  A(s)=\frac1{2\pi^2}\sum_{k\ge1}\frac{\e^{-4\pi^2k^2s^2}}{k^2}.
  \label{eq:gaussian-A}
\end{equation}
The damping is faster than any power of $k$, so $A$ is smooth on $(0,\infty)$
and may be differentiated term by term. Writing
$H(s):=\sum_{k\ge1}\e^{-4\pi^2k^2s^2}$, one finds $A'(s)=-4s\,H(s)$ and hence,
from \eqref{eq:decomposition}, $\MSE_n'(s)=\tfrac{2s}{n}\bigl(1-2(n-1)H(s)\bigr)$.
Since $H$ is strictly decreasing from $+\infty$ to $0$, the bracket increases
strictly from $-\infty$ to $1$, vanishing once: the optimizer is \emph{unique}
and solves
\begin{equation}
  H(s_n^*)=\frac1{2(n-1)},
  \qquad\text{equivalently}\qquad
  \vartheta_3\!\bigl(0,\,\e^{-4\pi^2 (s_n^*)^2}\bigr)=\frac{n}{n-1},
  \label{eq:gaussian-theta}
\end{equation}
where $\vartheta_3(0,r)=1+2\sum_{j\ge1}r^{j^2}$ and we used
$\vartheta_3(0,r)=1+2H$.

% =====================================================================
% FIGURE -- GAUSSIAN  (reproduced verbatim from the exposition)
% =====================================================================
\begin{figure}[p]
\centering
\pgfplotstableread[col sep=comma]{
s,I1,I2,I3,I4,I8,I20
0.00,0.08333,0.08333,0.08333,0.08333,0.08333,0.08333
0.03,0.08423,0.07577,0.07295,0.07154,0.06942,0.06815
0.06,0.08693,0.07001,0.06437,0.06154,0.05731,0.05477
0.09,0.09143,0.06604,0.05758,0.05335,0.04700,0.04320
0.12,0.09773,0.06388,0.05260,0.04696,0.03849,0.03342
0.15,0.10583,0.06352,0.04941,0.04236,0.03178,0.02544
0.18,0.11573,0.06495,0.04803,0.03956,0.02687,0.01925
0.21,0.12743,0.06816,0.04841,0.03853,0.02371,0.01482
0.24,0.14093,0.07307,0.05045,0.03914,0.02218,0.01200
0.27,0.15623,0.07954,0.05398,0.04120,0.02202,0.01052
0.30,0.17333,0.08739,0.05874,0.04442,0.02294,0.01004
0.33,0.19223,0.09646,0.06454,0.04857,0.02463,0.01027
0.36,0.21293,0.10662,0.07118,0.05346,0.02688,0.01094
0.39,0.23543,0.11778,0.07856,0.05895,0.02954,0.01189
0.42,0.25973,0.12989,0.08661,0.06497,0.03251,0.01303
0.45,0.28583,0.14293,0.09529,0.07147,0.03574,0.01431
0.48,0.31373,0.15687,0.10458,0.07844,0.03922,0.01569
0.51,0.34343,0.17172,0.11448,0.08586,0.04293,0.01717
0.54,0.37493,0.18747,0.12498,0.09373,0.04687,0.01875
0.57,0.40823,0.20412,0.13608,0.10206,0.05103,0.02041
0.60,0.44333,0.22167,0.14778,0.11083,0.05542,0.02217
0.63,0.48023,0.24012,0.16008,0.12006,0.06003,0.02401
0.66,0.51893,0.25947,0.17298,0.12973,0.06487,0.02595
0.69,0.55943,0.27972,0.18648,0.13986,0.06993,0.02797
0.72,0.60173,0.30087,0.20058,0.15043,0.07522,0.03009
0.75,0.64583,0.32292,0.21528,0.16146,0.08073,0.03229
0.78,0.69173,0.34587,0.23058,0.17293,0.08647,0.03459
0.81,0.73943,0.36972,0.24648,0.18486,0.09243,0.03697
0.84,0.78893,0.39447,0.26298,0.19723,0.09862,0.03945
0.87,0.84023,0.42012,0.28008,0.21006,0.10503,0.04201
0.90,0.89333,0.44667,0.29778,0.22333,0.11167,0.04467
0.93,0.94823,0.47412,0.31608,0.23706,0.11853,0.04741
0.96,1.00493,0.50247,0.33498,0.25123,0.12562,0.05025
0.99,1.06343,0.53172,0.35448,0.26586,0.13293,0.05317
1.02,1.12373,0.56187,0.37458,0.28093,0.14047,0.05619
1.05,1.18583,0.59292,0.39528,0.29646,0.14823,0.05929
1.08,1.24973,0.62487,0.41658,0.31243,0.15622,0.06249
1.11,1.31543,0.65772,0.43848,0.32886,0.16443,0.06577
1.14,1.38293,0.69147,0.46098,0.34573,0.17287,0.06915
1.17,1.45223,0.72612,0.48408,0.36306,0.18153,0.07261
1.20,1.52333,0.76167,0.50778,0.38083,0.19042,0.07617
}\msedata

\begin{tikzpicture}
\begin{groupplot}[
  group style={group size=3 by 2, horizontal sep=1.2cm, vertical sep=1.5cm},
  msepanel,
]
\nextgroupplot[title={$n=1$}]
  \addplot[plotblue] table[x=s,y=I1]{\msedata};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0,0.08333)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.06,0.30) {min at $s=0$};

\nextgroupplot[title={$n=2$}]
  \addplot[plotblue] table[x=s,y=I2]{\msedata};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.14105,0.06344)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.34,0.035) {$s_n^*\!\approx\!0.141$};

\nextgroupplot[title={$n=3$}]
  \addplot[plotblue] table[x=s,y=I3]{\msedata};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.18839,0.04796)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.33,0.033) {$s_n^*\!\approx\!0.188$};

\nextgroupplot[title={$n=4$}]
  \addplot[plotblue] table[x=s,y=I4]{\msedata};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.21331,0.03852)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.25,0.03852) {$s_n^*\!\approx\!0.213$};

\nextgroupplot[title={$n=8$}]
  \addplot[plotblue] table[x=s,y=I8]{\msedata};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.25857,0.02194)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.30,0.02194) {$s_n^*\!\approx\!0.259$};

\nextgroupplot[title={$n=20$}]
  \addplot[plotblue] table[x=s,y=I20]{\msedata};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.30355,0.01004)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.35,0.01004) {$s_n^*\!\approx\!0.304$};
\end{groupplot}
\end{tikzpicture}
\caption{\textbf{Gaussian noise.}
Cell-averaged $\MSE_n(s)$ as a
function of the scatter level $s$ measured in cell widths, for
$n\in\{1,2,3,4,8,20\}$.
The dashed line marks the zero-scatter value
$\MSE_n(0)=1/12$; the dot marks the global minimizer.
%
For $n=1$ the minimum is at
$s=0$;
for every $n>1$ it is strictly interior,
and the minimum value lies
strictly below $1/12$.
Vertical scales differ between panels.}
\label{fig:mse-curves-gaussian}
\end{figure}
%
Keeping only the dominant $k=1$ term gives the
large-$n$ asymptotic
\begin{equation}
  s_n^*\sim\frac{\sqrt{\log\bigl(2(n-1)\bigr)}}{2\pi}\qquad(n\to\infty),
  \label{eq:gaussian-asymptotic}
\end{equation}
the slow $\sqrt{\log n}$ growth promised in \S\ref{sec:spectral}; it already
agrees with the exact root to three decimals by $n=10$
(Table~\ref{tab:gaussian}). The bias-reduction rate is
$\tfrac12\E|W|=\tfrac12\cdot 2/\sqrt\pi=1/\sqrt\pi\approx0.5642$, since
$W\sim\mathcal N(0,2)$.

\begin{table}[!ht]
\centering
\begin{tabular}{rccc}
\toprule
$n$ & $s_n^*$ (exact root) & $\MSE_n(s_n^*)$ & asymptotic \eqref{eq:gaussian-asymptotic} \\
\midrule
$2$    & $0.14105$ & $0.06344$ & $0.13251$ \\
$3$    & $0.18839$ & $0.04796$ & $0.18739$ \\
$5$    & $0.22961$ & $0.03227$ & $0.22951$ \\
$10$   & $0.27059$ & $0.01819$ & $0.27058$ \\
$100$  & $0.36600$ & $0.00243$ & $0.36600$ \\
$1000$ & $0.43876$ & $0.00030$ & $0.43876$ \\
\bottomrule
\end{tabular}
\caption{Gaussian noise: the unique optimizer $s_n^*$ solving the theta
equation \eqref{eq:gaussian-theta}, the cell-averaged error there, and the
asymptotic \eqref{eq:gaussian-asymptotic}. The optimizer grows like
$\sqrt{\log n}$; the asymptotic is accurate to three decimals from $n=10$.}
\label{tab:gaussian}
\end{table}
%
For plots of $\MSE_n$ against \(s\) for Gaussian scatter,
see figure~\ref{fig:mse-curves-gaussian}.
%
\subsection{Laplace: a hyperbolic bias term, still a unique optimizer}
\label{ssec:laplace}

The unit-variance Laplace law has $\varphi(t)=(1+t^2/2)^{-1}$, so
$|\varphi(2\pi ks)|^2=(1+2\pi^2k^2s^2)^{-2}$ and, for $s>0$, the series
\eqref{eq:spectral} sums to a hyperbolic closed form: with $a=1/(s\sqrt2)$,
\begin{equation}
  A(s)=s^2+\frac1{12}-\frac{3}{8a}\coth a-\frac18\operatorname{csch}^2 a  .
  \label{eq:laplace-A}
\end{equation}
(The right-hand side is defined for $s>0$, where $a=1/(s\sqrt2)$ is finite; as
$s\downarrow0$ one has $a\to\infty$, $\tfrac{3}{8a}\coth a\to0$ and
$\tfrac18\operatorname{csch}^2 a\to0$, so it extends continuously to
$A(0)=\tfrac1{12}$, agreeing with the series. We verified \eqref{eq:laplace-A}
against the defining series to better than $10^{-25}$.).
Where the Gaussian produces a theta function, the heavier-tailed
Laplace produces $\coth$; the contrast is a feature, not an accident, and traces
to the analytic class of $|\varphi|^2$. For Laplace noise $|\varphi(t)|^2$ is a
completely monotone function of $t^2$---equivalently $W=Z-Z'$ is a scale mixture
of Gaussians---and that single structural property forces uniqueness, by the
following general mechanism.

\begin{proposition}[Uniqueness for Gaussian scale mixtures]
\label{prop:cm-uniqueness}
Suppose $W=Z-Z'$ is a Gaussian scale mixture; that is,
$|\varphi(t)|^2=\int_0^\infty \e^{-\lambda t^2}\dd\mu(\lambda )$ for a finite measure $\mu$
with $\int_0^\infty \lambda \dd\mu(\lambda )<\infty$. Then for every $n>1$ the optimizer of
$\MSE_n$ is unique.
\end{proposition}

\begin{proof}
By \eqref{eq:spectral} and the representation of $|\varphi|^2$,
\[
  A(s)=\int_0^\infty A_{\mathrm G}(s\sqrt \lambda )\dd\mu(\lambda ),
\]
where $A_{\mathrm G}$ is the Gaussian functional \eqref{eq:gaussian-A}
(legitimate by Tonelli, the summand being non-negative). Fix a compact interval
$[s_1,s_2]\subset(0,\infty)$. For $s\in[s_1,s_2]$ the $s$-derivative of the
integrand is $-4s\,\lambda\,H(s\sqrt\lambda)$, and since $H$ is decreasing,
$0\le \lambda\,H(s\sqrt\lambda)\le \lambda\,H(s_1\sqrt\lambda)=:G(\lambda)$. The
bound $G$ is bounded on $(0,\infty)$---it tends to $0$ as $\lambda\downarrow0$
(since $H(r)\sim 1/(4\sqrt\pi\,r)$, so $\lambda H(s_1\sqrt\lambda)\sim
\sqrt\lambda/(4\sqrt\pi\,s_1)$) and as $\lambda\to\infty$ (where $H$ decays
super-polynomially)---hence $\mu$-integrable, $\mu$ being finite. One may
therefore differentiate under the integral; with $A_{\mathrm G}'(r)=-4r\,H(r)$
this gives, for $s>0$,
\[
  \MSE_n'(s)=\frac{2s}{n}\Bigl(1-2(n-1)\,H_{\mu}(s)\Bigr),
  \qquad
  H_{\mu}(s):=\int_0^\infty \lambda \,H(s\sqrt \lambda )\dd\mu(\lambda ).
\]
For each $\lambda>0$ the map $s\mapsto H(s\sqrt\lambda)$ is strictly decreasing
from $+\infty$ to $0$; the fibre $\lambda=0$ contributes
$\lambda\,H(s\sqrt\lambda)\equiv0$ and is inert. Because $Z$ is non-degenerate,
$\mu\bigl((0,\infty)\bigr)>0$, so $H_{\mu}$ is strictly decreasing on $(0,\infty)$,
with $H_{\mu}(s)\to+\infty$ as $s\downarrow0$ (monotone convergence) and
$H_{\mu}(s)\to0$ as $s\to\infty$ (dominated convergence, via $G$). Hence the bracket
$1-2(n-1)H_{\mu}(s)$ is strictly increasing from $-\infty$ to $1$ and vanishes
exactly once; $\MSE_n'$ changes sign once on $(0,\infty)$, from $-$ to $+$, so
the minimizer is unique.
\end{proof}

Proposition~\ref{prop:cm-uniqueness} covers both the Gaussian ($\mu=\delta_1$)
and the Laplace; for the latter the optimizer is the unique root of
$\MSE_n'=0$ with $A$ given by \eqref{eq:laplace-A}. The bias-reduction rate is
$\tfrac12\E|W|=\tfrac{3}{4\sqrt2}\approx0.530$, slightly below the Gaussian's
$1/\sqrt\pi\approx0.564$. The two optimizers track closely but do not nest: the
smaller bias-reduction rate places the Laplace optimizer marginally \emph{below} the
Gaussian one for small $n$ (at $n=2,3,4$; Table~\ref{tab:gallery}), while the
polynomial---rather than Gaussian---decay of $|\varphi(2\pi ks)|^2$ keeps relief
alive at larger scales, so the Laplace optimizer overtakes the Gaussian near
$n=5$ and thereafter sits higher and grows faster. The crossover is a
spectral-tail effect, not a relief-rate effect. A clean
necessary-and-sufficient condition for uniqueness---a statement about the shape
of $A$ alone, neither requiring complete monotonicity nor excluded by the
two-point oscillation---remains open.
%
For plots of $\MSE_n$ against \(s\) for Laplacian scatter,
see figure~\ref{fig:mse-curves-laplace}.

% =====================================================================
% FIGURE -- LAPLACE  (unit-variance, varphi = (1+t^2/2)^{-1})
% =====================================================================
\begin{figure}[p]
\centering
\pgfplotstableread[col sep=comma]{
s,I1,I2,I3,I4,I8,I20
0.00,0.08333,0.08333,0.08333,0.08333,0.08333,0.08333
0.03,0.08423,0.07628,0.07363,0.07230,0.07031,0.06912
0.06,0.08693,0.07102,0.06572,0.06307,0.05909,0.05670
0.09,0.09143,0.06757,0.05961,0.05564,0.04967,0.04609
0.12,0.09773,0.06591,0.05530,0.05000,0.04204,0.03727
0.15,0.10583,0.06603,0.05276,0.04613,0.03618,0.03021
0.18,0.11573,0.06787,0.05192,0.04394,0.03197,0.02479
0.21,0.12743,0.07132,0.05261,0.04326,0.02923,0.02081
0.24,0.14093,0.07625,0.05469,0.04390,0.02773,0.01803
0.27,0.15623,0.08253,0.05796,0.04568,0.02726,0.01620
0.30,0.17333,0.09006,0.06230,0.04843,0.02761,0.01512
0.33,0.19223,0.09875,0.06759,0.05200,0.02863,0.01461
0.36,0.21293,0.10852,0.07372,0.05632,0.03022,0.01456
0.39,0.23543,0.11934,0.08064,0.06129,0.03227,0.01486
0.42,0.25973,0.13116,0.08830,0.06687,0.03473,0.01544
0.45,0.28583,0.14396,0.09666,0.07302,0.03755,0.01627
0.48,0.31373,0.15771,0.10570,0.07970,0.04069,0.01729
0.51,0.34343,0.17241,0.11540,0.08689,0.04413,0.01848
0.54,0.37493,0.18803,0.12573,0.09458,0.04786,0.01982
0.57,0.40823,0.20459,0.13671,0.10276,0.05185,0.02131
0.60,0.44333,0.22206,0.14830,0.11142,0.05611,0.02291
0.63,0.48023,0.24045,0.16052,0.12056,0.06061,0.02464
0.66,0.51893,0.25975,0.17335,0.13015,0.06536,0.02648
0.69,0.55943,0.27996,0.18680,0.14022,0.07035,0.02843
0.72,0.60173,0.30107,0.20085,0.15074,0.07558,0.03048
0.75,0.64583,0.32309,0.21551,0.16172,0.08104,0.03263
0.78,0.69173,0.34602,0.23078,0.17316,0.08673,0.03488
0.81,0.73943,0.36985,0.24665,0.18506,0.09266,0.03722
0.84,0.78893,0.39458,0.26313,0.19741,0.09882,0.03967
0.87,0.84023,0.42022,0.28021,0.21021,0.10521,0.04220
0.90,0.89333,0.44676,0.29790,0.22347,0.11182,0.04484
0.93,0.94823,0.47420,0.31618,0.23718,0.11867,0.04756
0.96,1.00493,0.50254,0.33507,0.25134,0.12574,0.05038
0.99,1.06343,0.53178,0.35456,0.26595,0.13304,0.05329
1.02,1.12373,0.56192,0.37465,0.28102,0.14056,0.05629
1.05,1.18583,0.59297,0.39534,0.29653,0.14832,0.05939
1.08,1.24973,0.62491,0.41664,0.31250,0.15629,0.06257
1.11,1.31543,0.65776,0.43853,0.32892,0.16450,0.06585
1.14,1.38293,0.69150,0.46103,0.34579,0.17293,0.06922
1.17,1.45223,0.72615,0.48412,0.36311,0.18159,0.07267
1.20,1.52333,0.76170,0.50782,0.38088,0.19047,0.07622
}\msedatalap

\begin{tikzpicture}
\begin{groupplot}[
  group style={group size=3 by 2, horizontal sep=1.2cm, vertical sep=1.5cm},
  msepanel,
]
\nextgroupplot[title={$n=1$}]
  \addplot[plotblue] table[x=s,y=I1]{\msedatalap};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0,0.08333)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.06,0.30) {min at $s=0$};

\nextgroupplot[title={$n=2$}]
  \addplot[plotblue] table[x=s,y=I2]{\msedatalap};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.13290,0.06575)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.33,0.037) {$s_n^*\!\approx\!0.133$};

\nextgroupplot[title={$n=3$}]
  \addplot[plotblue] table[x=s,y=I3]{\msedatalap};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.18094,0.05191)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.32,0.035) {$s_n^*\!\approx\!0.181$};

\nextgroupplot[title={$n=4$}]
  \addplot[plotblue] table[x=s,y=I4]{\msedatalap};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.20963,0.04326)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.27,0.04326) {$s_n^*\!\approx\!0.210$};

\nextgroupplot[title={$n=8$}]
  \addplot[plotblue] table[x=s,y=I8]{\msedatalap};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.27125,0.02726)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.32,0.02726) {$s_n^*\!\approx\!0.271$};

\nextgroupplot[title={$n=20$}]
  \addplot[plotblue] table[x=s,y=I20]{\msedatalap};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.34893,0.01453)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.40,0.01453) {$s_n^*\!\approx\!0.349$};
\end{groupplot}
\end{tikzpicture}
\caption{\textbf{Laplace noise} (unit variance,
$\varphi(t)=(1+t^2/2)^{-1}$).
Cell-averaged $\MSE_n(s)$ as a
function of the scatter level $s$ measured in cell widths, for
$n\in\{1,2,3,4,8,20\}$.
The dashed line marks the zero-scatter value
$\MSE_n(0)=1/12$; the dot marks the global minimizer.
%
The slightly smaller bias-reduction rate
places the optimizer marginally below the Gaussian for small $n$ (visible for
$n=2,3,4$), while the heavier, polynomially decaying spectral tail makes it
overtake the Gaussian near $n=5$ and grow somewhat faster thereafter. Vertical
scales differ between panels.}
\label{fig:mse-curves-laplace}
\end{figure}

\subsection{Uniform: exact debiasing and the localization scale}
\label{ssec:uniform}

For $Z\sim\Unif[-\sqrt3,\sqrt3]$ (variance one),
$\varphi(t)=\sin(\sqrt3\,t)/(\sqrt3\,t)$, so
\[
  |\varphi(2\pi ks)|^2=\left(\frac{\sin(2\sqrt3\,\pi ks)}{2\sqrt3\,\pi ks}\right)^{\!2},
\]
which does not decay monotonically but \emph{vanishes} whenever
$2\sqrt3\,s\in\Z$. Consequently every harmonic is annihilated at each
positive integer multiple $s=ms_0$ ($m=1,2,\ldots$) of the first positive
cancellation scale
\begin{equation}
  s_0=\frac1{2\sqrt3},\qquad s_0^2=\frac1{12}.
  \label{eq:uniform-null}
\end{equation}
At $s_0$, $|\varphi(2\pi ks_0)|^2=\sin^2(\pi k)/(\pi k)^2=0$ for every $k\ge1$,
so $A(s_0)=0$: the cell-averaged rounding bias is cancelled exactly. There
\begin{equation}
  \MSE_n(s_0)=\frac1n\Bigl(\frac1{12}+\frac1{12}\Bigr)=\frac1{6n},
  \label{eq:uniform-value}
\end{equation}
equal to the floor $\tfrac1{12}$ at $n=2$ and strictly below it for every
$n>2$. Two things are worth noting. First, the tie at $n=2$ occurs at
$s_0^2=\tfrac1{12}=(n-1)/12$: the uniform law saturates the localization bound
\eqref{eq:localization}, so its scale $\sqrt{(n-1)/12}$ is realized, not merely
an artefact of the proof.
Second, $s_0$ is a \emph{debiasing} point, not the global optimizer, which
is explicit. Here $|W|\le2\sqrt3$ and $\E|W|=2/\sqrt3$, so
Lemma~\ref{lem:bounded} applies with $w=2\sqrt3$, $1/w=s_0$ and
$c_n=(n-1)/(2\sqrt3\,n)<s_0$: on $[0,s_0]$,
\[
  \MSE_n(s)-\frac1{12}=s^2-\frac{n-1}{\sqrt3\,n}\,s ,
\]
uniquely minimized at
\begin{equation}
  s_n^*=\frac{n-1}{2\sqrt3\,n}=\frac{n-1}{n}\,s_0,
  \qquad
  \MSE_n(s_n^*)=\frac1{12}-\frac{(n-1)^2}{12\,n^2}=\frac{2n-1}{12\,n^2}.
  \label{eq:uniform-opt}
\end{equation}
On the other side, for every $s\ge s_0$,
\[
  \MSE_n(s)=\frac{s^2}{n}+\frac1{12n}+\frac{n-1}{n}A(s)
           \ge\frac{s^2}{n}+\frac1{12n}
           \ge\frac{s_0^2}{n}+\frac1{12n}=\frac1{6n}
           >\frac{2n-1}{12\,n^2},
\]
so no $s\ge s_0$ competes, and \eqref{eq:uniform-opt} is the unique global
minimizer. It lies strictly below the debiasing scale for every $n>1$, rising
to $s_0$ as $n\to\infty$; in particular the localization bound
\eqref{eq:localization} is strict for the optimizer even when $n=2$, where
only the debiasing point attains it (Table~\ref{tab:gallery}).
The bias-reduction rate is $\tfrac12\E|W|=1/\sqrt3\approx0.5774$.

The simultaneous cancellation of all bias harmonics is the familiar uniform-
dither condition of Schuchman \cite{Schuchman1964}; a modern survey
appears in \cite{LWV1992}.
In the present calculation it follows directly from the uniform
characteristic function and is the saturation endpoint of
Corollary~\ref{cor:localization}. Section~\ref{sec:relation} distinguishes this
mathematical overlap from the estimation question studied here.
%
For plots of $\MSE_n$ against \(s\) with uniform scatter,
see figure~\ref{fig:mse-curves-uniform}.
%

\begin{table}[ht]
\centering
\begin{tabular}{lccc}
\toprule
 & $n=2$ & $n=5$ & $n=10$ \\
\midrule
Two-point\quad $s_n^*$            & $0.1250$ & $0.2000$ & $0.2250$ \\
\phantom{Two-point}\quad $\MSE_n(s_n^*)$ & $0.0677$ & $0.0433$ & $0.0327$ \\[2pt]
Gaussian\quad $s_n^*$             & $0.1410$ & $0.2296$ & $0.2706$ \\
\phantom{Gaussian}\quad $\MSE_n(s_n^*)$  & $0.0634$ & $0.0323$ & $0.0182$ \\[2pt]
Laplace\quad $s_n^*$              & $0.1329$ & $0.2302$ & $0.2902$ \\
\phantom{Laplace}\quad $\MSE_n(s_n^*)$   & $0.0657$ & $0.0374$ & $0.0234$ \\[2pt]
Uniform\quad $s_n^*$              & $0.1443$ & $0.2309$ & $0.2598$ \\
\phantom{Uniform}\quad $\MSE_n(s_n^*)$   & $0.0625$ & $0.0300$ & $0.0158$ \\
\bottomrule
\end{tabular}
\caption{The global minimizer $s_n^*$ (in cell widths) and the cell-averaged
error there, for four unit-variance laws and three sample sizes. Every value
lies strictly below the zero-dispersion floor
$\MSE_n(0)=1/12\approx0.0833$, as Theorem~\ref{thm:existence} requires. The
two-point and uniform entries match the exact forms \eqref{eq:twopoint-opt}
and \eqref{eq:uniform-opt}; the
optimizers are of comparable size across laws and grow only slowly with $n$.}
\label{tab:gallery}
\end{table}

% =====================================================================
% FIGURE -- UNIFORM  (Z ~ Unif[-sqrt3, sqrt3], variance one)
% =====================================================================
\begin{figure}[p]
\centering
\pgfplotstableread[col sep=comma]{
s,I1,I2,I3,I4,I8,I20
0.00,0.08333,0.08333,0.08333,0.08333,0.08333,0.08333
0.03,0.08423,0.07557,0.07269,0.07124,0.06908,0.06778
0.06,0.08693,0.06961,0.06384,0.06095,0.05662,0.05402
0.09,0.09143,0.06545,0.05679,0.05246,0.04597,0.04207
0.12,0.09773,0.06309,0.05155,0.04577,0.03711,0.03192
0.15,0.10583,0.06253,0.04810,0.04088,0.03006,0.02356
0.18,0.11573,0.06377,0.04645,0.03779,0.02480,0.01701
0.21,0.12743,0.06681,0.04660,0.03650,0.02135,0.01225
0.24,0.14093,0.07165,0.04856,0.03701,0.01969,0.00930
0.27,0.15623,0.07829,0.05231,0.03932,0.01983,0.00814
0.28868,0.16667,0.08333,0.05556,0.04167,0.02083,0.00833
0.30,0.17333,0.08672,0.05785,0.04342,0.02176,0.00877
0.33,0.19223,0.09660,0.06472,0.04878,0.02487,0.01052
0.36,0.21293,0.10739,0.07221,0.05462,0.02824,0.01241
0.39,0.23543,0.11890,0.08006,0.06064,0.03150,0.01402
0.42,0.25973,0.13108,0.08819,0.06675,0.03458,0.01529
0.45,0.28583,0.14396,0.09667,0.07302,0.03755,0.01627
0.48,0.31373,0.15762,0.10558,0.07956,0.04053,0.01712
0.51,0.34343,0.17214,0.11505,0.08650,0.04368,0.01798
0.54,0.37493,0.18762,0.12518,0.09396,0.04713,0.01903
0.57,0.40823,0.20412,0.13609,0.10207,0.05104,0.02042
0.57735,0.41667,0.20833,0.13889,0.10417,0.05208,0.02083
0.60,0.44333,0.22172,0.14785,0.11091,0.05550,0.02226
0.63,0.48023,0.24031,0.16034,0.12035,0.06037,0.02438
0.66,0.51893,0.25980,0.17342,0.13023,0.06545,0.02658
0.69,0.55943,0.28013,0.18703,0.14048,0.07065,0.02876
0.72,0.60173,0.30129,0.20114,0.15106,0.07595,0.03088
0.75,0.64583,0.32327,0.21575,0.16199,0.08135,0.03297
0.78,0.69173,0.34612,0.23091,0.17331,0.08690,0.03506
0.81,0.73943,0.36985,0.24665,0.18505,0.09266,0.03722
0.84,0.78893,0.39450,0.26302,0.19728,0.09867,0.03951
0.86603,0.83333,0.41667,0.27778,0.20833,0.10417,0.04167
0.87,0.84023,0.42012,0.28008,0.21006,0.10503,0.04201
0.90,0.89333,0.44671,0.29784,0.22340,0.11175,0.04475
0.93,0.94823,0.47424,0.31624,0.23724,0.11874,0.04764
0.96,1.00493,0.50265,0.33522,0.25151,0.12593,0.05059
0.99,1.06343,0.53193,0.35476,0.26618,0.13330,0.05358
1.02,1.12373,0.56207,0.37485,0.28124,0.14083,0.05658
1.05,1.18583,0.59308,0.39550,0.29671,0.14852,0.05961
1.08,1.24973,0.62498,0.41672,0.31260,0.15641,0.06269
1.11,1.31543,0.65776,0.43854,0.32893,0.16451,0.06586
1.14,1.38293,0.69147,0.46099,0.34574,0.17288,0.06916
1.15470,1.41667,0.70833,0.47222,0.35417,0.17708,0.07083
1.17,1.45223,0.72612,0.48409,0.36307,0.18154,0.07262
1.20,1.52333,0.76171,0.50783,0.38090,0.19049,0.07625
}\msedataunif

\begin{tikzpicture}
\begin{groupplot}[
  group style={group size=3 by 2, horizontal sep=1.2cm, vertical sep=1.5cm},
  msepanel,
]
\nextgroupplot[title={$n=1$}]
  \addplot[plotblue] table[x=s,y=I1]{\msedataunif};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0,0.08333)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.06,0.30) {min at $s=0$};

\nextgroupplot[title={$n=2$}]
  \addplot[plotblue] table[x=s,y=I2]{\msedataunif};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.14434,0.06250)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.34,0.035) {$s_n^*\!\approx\!0.144$};

\nextgroupplot[title={$n=3$}]
  \addplot[plotblue] table[x=s,y=I3]{\msedataunif};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.19245,0.04630)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.33,0.032) {$s_n^*\!\approx\!0.192$};

\nextgroupplot[title={$n=4$}]
  \addplot[plotblue] table[x=s,y=I4]{\msedataunif};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.21651,0.03646)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.27,0.03646) {$s_n^*\!\approx\!0.217$};

\nextgroupplot[title={$n=8$}]
  \addplot[plotblue] table[x=s,y=I8]{\msedataunif};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.25259,0.01953)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.30,0.01953) {$s_n^*\!\approx\!0.253$};

\nextgroupplot[title={$n=20$}]
  \addplot[plotblue] table[x=s,y=I20]{\msedataunif};
  \addplot[dashed,gray] coordinates {(0,0.08333) (1.2,0.08333)};
  \addplot[optdot] coordinates {(0.27424,0.00812)};
  \node[anchor=west,font=\scriptsize] at (axis cs:0.33,0.00812) {$s_n^*\!\approx\!0.274$};
\end{groupplot}
\end{tikzpicture}
\caption{\textbf{Uniform noise} ($Z\sim\Unif[-\sqrt3,\sqrt3]$, variance
one).
Cell-averaged $\MSE_n(s)$ as a
function of the scatter level $s$ measured in cell widths, for
$n\in\{1,2,3,4,8,20\}$.
The dashed line marks the zero-scatter value
$\MSE_n(0)=1/12$; the dot marks the global minimizer.
%
The spectrum $|\varphi(2\pi ks)|^2$ has nulls, so every
bias harmonic is annihilated at $s_0=1/(2\sqrt3)\approx0.289$ (where the
bias vanishes and $\MSE_n=1/(6n)$), and again at every integer multiple of
$s_0$. The global optimizer is $s_n^*=(n-1)s_0/n$, strictly below $s_0$, with
value $(2n-1)/(12n^2)$. The optimizer barely grows past $n=8$, reflecting that
the bias is already nearly cancelled. Vertical scales differ between panels.}
\label{fig:mse-curves-uniform}
\end{figure}

% ============================================================
\section{Relation to dithered quantization}
\label{sec:relation}
% ============================================================

The same integrals and lattice-frequency sums occur in dither theory, but the
role of the randomness and the inferential objective are different. In the
present model, $sZ$ represents pre-existing measurement unrepeatability. The
criterion is the cell-averaged mean-squared error of the average of $n$ rounded
readings about the unknown sub-cell value. No random signal is introduced by
design, and no realization of the scatter is available for subtraction.

In subtractive dithering, a known random signal is added before quantization and
subtracted after quantization in order to control the statistics of the
quantization error. Schuchman's condition states that this error is independent
of the signal precisely when the dither characteristic function vanishes at
every nonzero lattice frequency; the uniform law is the canonical example
\cite{Schuchman1964}. See \cite{LWV1992} for a survey and
\cite{WidrowKollar2008} for a broader treatment. The exact-cancellation scale in
Section~\ref{ssec:uniform} satisfies the same harmonic condition, which explains
the mathematical overlap.

The questions are nevertheless distinct. Dither theory concerns a designed
randomization and properties of the resulting quantization error; the present
paper concerns estimation by repeated rounded measurements subject to
uncontrolled scatter. Schuchman also observed that saturation prevents full
independence for a finite-level quantizer. We assume an infinite lattice, so
that boundary does not arise. To our knowledge, the specific existence
characterization in Theorem~\ref{thm:iff} has not previously been given.

% ============================================================
\section{Scope and open questions}
\label{sec:discussion}
% ============================================================

For the model of Section~\ref{sec:model}, Theorem~\ref{thm:iff} settles the
existence question. The cell-averaged error of the rounded sample mean is
strictly below its zero-dispersion value $\tfrac1{12}$ for some positive
dispersion---equivalently, every global minimizer lies away from $s=0$---if and
only if $n>1$ and the pre-rounding law is centred with finite, non-zero
variance. No density, smoothness, or additional tail condition is required.
The strict local improvement follows from the corner of the rounding-error
autocorrelation: the bias is reduced at first order in $s$, whereas the
variance contribution is second order. Continuity and the quadratic variance
term then give attainment of the global minimum.

Finite variance alone does not imply uniqueness. The local bias-reduction rate
$\tfrac12\E|W|$ determines the behaviour near zero, whereas uniqueness depends
on the global shape of $A$. When $|\varphi|^2$ is completely monotone in
$t^2$---equivalently, when $W$ is a Gaussian scale mixture, as in the Gaussian
and Laplace examples---the optimizer is unique
(Proposition~\ref{prop:cm-uniqueness}). When $|\varphi|^2$ oscillates, as for the
two-point law, the criterion can have several local wells (for that law,
once $n\ge4$), although its first well contains the unique global minimizer.
A useful open problem is to
identify verifiable structural conditions on $A$, or equivalently on
$|\varphi|^2$, that are necessary and sufficient for uniqueness. The growth
of the optimizer with $n$ is governed by the decay of $|\varphi|^2$ rather than
by the tails of the law itself: the bounded uniform and two-point laws have
optimizers converging to finite limits, the Gaussian optimizer grows like
$\sqrt{\log n}$, and a characteristic function of bounded support keeps the
optimizer bounded (\S\ref{sec:spectral}). It remains to identify spectral-decay
conditions under which the Gaussian $\sqrt{\log n}$ rate holds; decay of
$\varphi$ faster than every inverse power is not sufficient.

Several assumptions delimit the result. The comparison is based on the cell
\emph{average}; it does not assert improvement at every fixed sub-cell
position. The quantizer is an infinite nearest-integer lattice, so saturation
and finite-range effects are excluded; truncation can be reduced to a shifted
version of the same calculation. The readings are independent and identically
distributed within a centred scale family, and the estimator is their sample
mean. Other dependence structures, non-scale families, and estimators tailored
to the rounded data may alter or remove the optimum. Finally, $s$ represents
residual unrepeatability rather than a controllable intervention. The result is
therefore not a prescription to add noise to a measuring instrument. Within
this model, it establishes a non-monotone relation between repeatability and
cell-averaged estimation error.

\section*{Historical note and author contributions}

The problem considered in this paper has its origin
in Lumsden's~\cite{Lumsden1976}
``flogging wall'' thought experiment,
first encountered by MRD as an
undergraduate student in 1979.  Lumsden used the
flogging wall to test the
counterintuitive possibility that, in some circumstances, an unreliable
psychometric instrument might outperform a reliable one.
Between
1989 and 2014, Daniel D. Reidpath (ORCID:
\href{https://orcid.org/0000-0002-8796-0420}{0000-0002-8796-0420}) and MRD intermittently explored
a variety of formalized simulation models of
the flogging wall which strongly suggested that
instrument unreliability could be advantageous.
Two unpublished papers were developed on the basis of the simulation results.

In 2023 MRD reformulated the problem in terms of a weighing scale that
\emph{truncates} to finite precision,
as opposed to the simpler rounding formulation used here.
AOBM observed that this finite
precision formulation could be expressed naturally as a
question about quantization;
see for example \cite[p.~1]{WidrowKollar2008}.
Our first formalization was restricted to Gaussian
scatter using,
as a starting point,
the treatment of quantized distributions
given by
Schwartz \cite{Schwartz1980},
and drawing also on the quantization framework of
Widrow~\cite{Widrow1961}.
This led to a moment-based derivation of a formula for the mean-squared error
\(\MSE_n(u,s)\) around a specific $u$, followed by integration over
\(u\in[-\tfrac12,\tfrac12)\), differentiation with respect to \(s\), and
a Jacobi theta-function expression for the minimizing scale.  That work
was developed into a short unpublished manuscript.

The present paper arose from the attempt to identify which features of
the Gaussian calculation were essential.  In particular, MRD conjectured
from the moment-based model that positive finite variance might be the
necessary and sufficient condition for the existence of a
strictly positive minimizer of
the cell-averaged mean-squared error, for \(n>1\).
The central simplification in the present paper
is the replacement of the Gaussian moment calculation by the
autocorrelation formulation used above.  This change separates the
existence argument from the special analytic form of the Gaussian
density, and is what allows the result to be stated for the more general
class of distributions considered here.

Figures were initially developed in Wolfram Mathematica and then
translated into TikZ for inclusion in the manuscript.
The mathematical
arguments, exposition, and final form of the paper are the responsibility
of the authors.

\section*{Acknowledgements and AI tool disclosure}

The authors used several generative AI tools during the development of
this paper, including OpenAI ChatGPT, Anthropic Claude, Google Gemini,
and Moonshot AI Kimi.
The tools were used to check algebraic derivations
from the quantization literature, to test and expand justifications
involving Fourier series, convergence, and exchanges of limits, sums, and
integrals, and to help identify simpler proof strategies from earlier
moment-based formulations.
In particular, ChatGPT and Claude independently
suggested that it might be possible to simplify
the existence argument by working with
an autocorrelation function rather than with the moment formula derived
from the Gaussian model.

AI tools were also used for proofreading and for checks of notational
consistency.
Google Gemini and Moonshot Kimi were particularly helpful with
LaTeX assistance, and the translation of Wolfram Mathematica plots into TikZ.
All AI-assisted mathematical suggestions were reviewed, revised, and
checked by the authors.
The text,
mathematical claims, and final editorial decisions are those of the
authors, who take responsibility for any remaining errors.


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\end{document}