==== TABLE OF CONTENTS ====
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//01-introduction.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//02-flat-scales.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//03-normal-inputs.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//04-fitting-frequency.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//05-centered-blowup.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//06-linearized-measures.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//06b-density-centers.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//07-dimension-reduction.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//macros.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//main.tex
A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//references.bib
===========================

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//01-introduction.tex ====
\section{Introduction}\label{sec:introduction}

Stationarity is the weakest variational condition naturally imposed on a
minimal submanifold: the first variation of area vanishes under every
compactly supported ambient deformation. Integral varifolds retain this
condition under weak limits while allowing integer multiplicity,
self-intersections, and singularities. Their general regularity problem
asks how much of the support must nevertheless be a smooth embedded
minimal submanifold.

Let $m,n\ge1$ and let $U\subset\R^{m+n}$ be open. An integral
$m$-varifold $V$ has weight
$\mu=\theta\Hh^m\llcorner E$, where $E$ is countably $m$-rectifiable
and $\theta$ is positive, integer-valued, and locally integrable.
At $\mu$-almost every $x$, write $T_xV$ for the approximate tangent
$m$-plane and $P_{T_xV}$ for its orthogonal projection. Stationarity means
\begin{equation}\label{eq:stationarity}
 \int\tr(P_{T_xV}DX(x))\,\dd\mu(x)=0
 \qquad\text{for every }X\in C_c^1(U;\R^{m+n}).
\end{equation}
A point $p\in\spt\mu\cap U$ is \emph{regular} if some neighborhood $W$
of $p$ contains a smooth embedded minimal $m$-submanifold $\Sigma$,
without boundary in $W$, such that $\spt\mu\cap W=\Sigma$ and
$V\llcorner W=q|\Sigma|$ for a positive integer $q$.
We denote the remaining support points by
$\Sing V$. Thus regularity includes equality of the local support with
one embedded submanifold, not merely a smooth parametrization on a set
of full measure.
For stationary integral varifolds this is equivalent to local $C^1$
embeddedness of the support, by constancy and elliptic regularity.

\begin{theorem}\label{thm:main}
For every stationary integral $m$-varifold $V$ in an open set
$U\subset\R^{m+n}$, where $m,n\ge1$,
\[
 \dim_{\Hh}\Sing V\le m-1.
\]
\end{theorem}

Theorem~\ref{thm:main} proves the Euclidean conjecture recorded as
Conjecture~1.1 by Brena, Decio, and De Lellis \cite{BDD2025}.
There is no stability, minimizing, orientability, density-upper-bound,
or codimension hypothesis. The estimate is sharp: the sum of the
varifolds of two distinct $m$-planes meeting along an $(m-1)$-plane is
stationary, and its singular set is exactly that intersection.

\subsection{Context and antecedents}
Allard's compactness and regularity theory \cite{Allard1972} supplies
the basic passage from weak convergence to geometric information and
gives a relatively open dense regular set. The dimension-reduction
method originating in \cite{Federer1970} leads to the classical
stratification by translation symmetries of tangent cones. For stationary
integral varifolds, the set of points admitting no planar tangent has
Hausdorff dimension at most $m-1$. Naber and Valtorta
\cite{NV2020} proved rectifiability of these classical strata for
integral varifolds with bounded mean curvature. The remaining difficulty
is therefore the singular set at higher-multiplicity planar tangents,
where Allard's multiplicity-one regularity theorem does not apply.
Standard treatments of the compactness, monotonicity, and tangent-cone
theory used here include \cite{Federer1969,Simon1983}.

For area-minimizing integral currents, Almgren's regularity theory
\cite{Almgren2000} gives an interior singular-set bound of dimension $m-2$ in
arbitrary codimension. De Lellis and Spadaro developed a new proof based
on multiple-valued functions, center manifolds, and a frequency analysis
of the blow-up \cite{DLS2011,DLS2016II,DLS2016III}.
These techniques explain the role of subtracting a well-chosen smooth
center before normalizing the height. Their regularity theorems do
not apply to arbitrary stationary integral varifolds; even the crossing
example above prevents the same dimension bound in our setting.

Hirsch and Spolaor proved the $m-1$ bound for two-valued stationary
Lipschitz graphs \cite{HSgraphs}. Their use of generalized-gradient
Young measures retains height and gradient information under weak
convergence and is a methodological antecedent of our measure-valued
blow-up, which does not assume graphicality or two-valuedness.
Hirsch and Spolaor also proved almost-everywhere regularity for stationary
codimension-one integral currents whose density is at most two at every support
point \cite{HS2025}. Krummel, Minter, and Wickramasekera
\cite{KMW2026} obtained singular-set bounds of dimension at most $m-1$ at
controlled densities in classes satisfying an explicit
epsilon-regularity assumption.

A different line of work addresses higher-order rectifiability.
Brakke's work gives $C^{1,\alpha}$ rectifiability, for every
$0<\alpha<1$, of stationary integral varifolds \cite{Brakke1978}.
Menne proved second-order rectifiability under locally bounded first
variation \cite{Menne2013}. Brena, De Lellis, and Franceschini
\cite{BDLF2025} established smooth rectifiability for stationary
integral varifolds; Kolasi\'nski \cite{Kolasinski2026} subsequently
gave an independent alternative proof, following a program he
attributes to Menne. Smooth rectifiability describes
the measure up to a null
set by smooth pieces; it does not assert that the entire support near
almost every point equals one such piece. This distinction is essential
for the singular set defined above.

The ambient geometry is also important. De Lellis, Hirsch, Lihn, and
Spolaor \cite{DLHLS2026} construct stationary integral varifolds with
planar tangents and infinite local topology in smooth, generally
non-Euclidean metrics. Their examples do not contradict the Euclidean
dimension bound proved here.

Brena, Decio, and De Lellis \cite[Conjectures~1.1 and~1.2]{BDD2025}
separate the dimension bound from the weaker assertion
$\Hh^m(\Sing V)=0$. The latter Euclidean assertion is proved in
\emph{Almost-everywhere regularity of stationary integral varifolds}
\cite{AE}, which we henceforth call AE. Our proof uses AE's signed-excess
theorem and local estimates from its fitting and frequency arguments.
The elliptic correction of the averaged graph in
\cite[Sections~6.3--6.4]{BDLF2025} is an antecedent of this fitting
construction. The almost-everywhere
regularity conclusion itself is not an input: a set can have zero
$\Hh^m$ measure and still have Hausdorff dimension greater than $m-1$.
We state the imported measure estimates in
Section~\ref{sec:normal-inputs}. Section~\ref{sec:fitting} specifies
the local fitting estimates and verifies the hypotheses needed for
their use at the prescribed scales.

\subsection{The proof and its new components}
Suppose that $\dim_{\Hh}\Sing V>m-1$ and choose an exponent $s$ strictly
between these two numbers. A first dimension-reduction argument produces
a singular point, placed at the origin, an integer $Q$, and radii
$L_i\downarrow0$ for which the rescaled varifolds converge to the
multiplicity-$Q$ plane. At these same scales there are nonzero measures
$\sigma_i$ carried by density-$Q$ points in a fixed compact ball, with
mass bounded above and away from zero and the estimate
\[
 \sigma_i(\mathbf B_r(x))\le Cr^s.
\]
Here $\mathbf B_r(x)$ is an ambient Euclidean ball. The limiting measure
of centers cannot be carried by an $(m-1)$-plane. This first reduction,
proved in Section~\ref{sec:flat-scales}, is the familiar geometric
starting point of dimension reduction; the challenge is to retain
these centers in a further, height-normalized limit.

The proof has four further stages. First, we construct smooth
reference graphs at the prescribed scales $L_i$. AE's minimal fitting
gives intervals of radii on which a centered height admits frequency
control. Section~\ref{sec:fitting} proves that these intervals cover
every sufficiently small prescribed flat scale, including their
transition regions. Section~\ref{sec:centered-blowup} then normalizes
the height by numbers $a_i\downarrow0$. The normalized second moment
is nonzero, the first moment tends to zero, and the reference graphs'
mean curvature and its first derivative are $o(a_i)$.
Both moment conclusions are needed: subtracting a reference graph
must neither remove the entire limit nor leave an arbitrary translation
in it.

Second, Section~\ref{sec:linearized} passes the first variations and
signed excess to measures on the limiting plane. The resulting data
consist of a nonnegative height measure, a positive-semidefinite tilt
measure, a mixed height--tilt measure, and a signed mass measure.
They satisfy linear first-variation identities, a positive-semidefinite
block inequality, and a one-sided comparison of mass and tilt. This
formulation avoids assuming that the limiting sheets can be globally
labeled or represented by a single-valued function.

Third, the density of the points carrying $\sigma_i$ must still be visible
after division by $a_i^2$. Ordinary uncorrected radial comparison
does not provide that precision on curved reference graphs.
Section~\ref{sec:density-centers} uses geodesic radial fields weighted by
the inverse reference-volume density. The resulting curvature error
can be averaged against $\sigma_i$ at every radius down to zero. For a
fixed $l>0$, its decisive integral is
\[
 \int_0^l t^{s-m}\,\dd t<\infty,
\]
which holds exactly when $s>m-1$. The centered first-moment estimate
then removes the translation terms and yields a density comparison
at every limiting center.

Finally, Section~\ref{sec:dimension-reduction} isolates an abstract
dimension-reduction statement for these measures. Its frequency is
monotone at the retained centers. A second blow-up makes that frequency
constant at enough independent centers. Equality in the
positive-semidefinite inequality then forces the height measure to
vanish, contrary to its normalization. This is the contradiction that
proves Theorem~\ref{thm:main}.

The prescribed-scale construction, the averaged curved radial
comparison, and the final measure-valued reduction are the parts needed
to strengthen AE's almost-everywhere conclusion to a dimension bound.
The center-manifold and frequency viewpoint belongs to the broader
regularity tradition described above; our specific local fitting and
signed-excess inputs are those of AE. The initial and final reductions
use the classical measure-theoretic framework of
\cite{Federer1969,Mattila1995}.

\subsection{Conventions}
All assertions are local in $U$, and supports are relative to $U$ until
a blow-up is taken. If $\eta_r(x)=x/r$, then $V_r=(\eta_r)_\#V$ is the
varifold pushforward, whose weight is $r^{-m}(\eta_r)_\#\mu$.
Translated blow-ups will be specified explicitly. Bold balls $\mathbf B_r$ are ambient, whereas
$B_r$ denotes a base-coordinate ball. The letters $C,c>0$ denote
constants depending only on previously fixed data and may change
between occurrences. Weak convergence of Radon measures is always
tested on continuous compactly supported functions. Positive
semidefiniteness of a symmetric matrix-valued measure means that its contraction
with every constant vector is a nonnegative measure.
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//01-introduction.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//02-flat-scales.tex ====
\section{Flat scales carrying many singular centers}
\label{sec:flat-scales}

The first reduction uses only stationary monotonicity and Hausdorff
measure theory. If the singular set has dimension greater than $m-1$,
we shall find a singular point with a planar tangent cone and retain,
at the same scales, a nonzero measure of nearby points having exactly
the same density. Retaining this measure is essential: a planar
tangent cone at a single singular point gives no contradiction.

For $p\in U$ and $r>0$, write
\[
 \eta_{p,r}(X)=\frac{X-p}{r},\qquad
 V_{p,r}=(\eta_{p,r})_\#V.
\]
Thus the weight of $V_{p,r}$ is $r^{-m}(\eta_{p,r})_\#\mu$.
After translating $p$ to the origin we abbreviate $V_{p,r}$ to $V_r$.
We use $\mathbf B_r$ for ambient balls and, later, $B_r$ for balls in
an $m$-dimensional base plane. The stationary density is
\[
 \Theta_V(p)=\lim_{r\downarrow0}
       \frac{\mu(\mathbf B_r(p))}{\omega_m r^m}.
\]
It exists at every support point, is at least one, and is locally
bounded and upper semicontinuous. Moreover, under locally mass-bounded
convergence of stationary integral varifolds,
\begin{equation}
 \label{eq:density-upper-limit}
 V_i\rightharpoonup V_\infty,\quad p_i\longrightarrow p
 \quad\Longrightarrow\quad
 \Theta_{V_\infty}(p)\ge\limsup_i\Theta_{V_i}(p_i).
\end{equation}
These facts follow from monotonicity and integral compactness
\cite{Allard1972,Simon1983}. For example, to obtain
\eqref{eq:density-upper-limit}, bound each density by its mass ratio
on a fixed ball, pass to the limit at radii whose boundary has zero
limiting mass, and then let the radius decrease to zero.

We shall also use two consequences of these facts. First, supports
converge locally under this varifold convergence. The lower density
bound prevents support points from persisting where the limit has
zero mass; conversely, any neighborhood of a limiting support point
has positive limiting mass and therefore meets the approximating
supports. Second, $\spt\mu$ has locally finite $\Hh^m$ measure.
Indeed the lower density bound gives
$\mu(\mathbf B_r(p))\ge\omega_m r^m$ whenever the ball is interior
and $p\in\spt\mu$. A disjoint-ball covering argument on a compact
subset of $U$ bounds its Hausdorff measure by a dimensional constant
times the mass in a slightly larger compact set. No local
Hausdorff-measure assumption on the support is needed.

\begin{proposition}[Flat scales and a measure of centers]
\label{prop:flat-scales}
Suppose that $V$ is a stationary integral $m$-varifold in an open
subset of $\R^{m+n}$ and that
\[
 m-1<s<\dim_{\Hh}\Sing(V).
\]
After a translation there exist a fixed point $0\in\Sing(V)$, an
integer $Q\ge1$, an $m$-plane $\Pi$ through zero, radii $L_i\downarrow0$,
and positive Radon measures $\sigma_i$ such that
\begin{equation}
 \label{eq:flat-targets}
 \Theta_V(0)=Q,\qquad V_{L_i}\rightharpoonup Q|\Pi|
 \quad\hbox{locally in }\R^{m+n}.
\end{equation}
Each $\sigma_i$ is supported in $\overline{\mathbf B}_1$ on singular
points of $V_{L_i}$ having density exactly $Q$. There are constants
$c,C>0$, independent of $i$, for which
\begin{equation}
 \label{eq:center-measure-bounds}
 \sigma_i(\mathbf B_u(X))\le C u^s
 \quad(X\in\R^{m+n},\ u>0),\qquad
 c\le\sigma_i(\overline{\mathbf B}_1)\le C.
\end{equation}
\end{proposition}

\begin{proof}
The regular set is relatively open in $\spt\mu$, so the singular set
is Borel. The compact-subset theorem for Hausdorff measure
\cite[Theorem~8.13]{Mattila1995} gives a compact subset of
$\Sing(V)$ with finite positive $\Hh^s$ measure.
We recall the two measure-theoretic refinements that we need
\cite{Federer1969,Mattila1995}. On a set of finite positive
$\Hh^s$ measure, the upper $s$-density is positive and finite almost
everywhere. By restricting to a positive-measure subset where its
upper bound is uniform, and then taking a compact subset, we obtain
a compact set $K$ such that
\begin{equation}
 \label{eq:initial-frostman}
 0<\Hh^s(K)<\infty,\qquad
 \Hh^s(K\cap\mathbf B_r(X))\le C r^s.
\end{equation}
An upper bound initially obtained for centers in $K$ extends to
arbitrary centers by doubling the radius. Enlarging $C$ handles the
remaining large radii. Lusin's theorem permits a further compact
restriction of positive measure on which $\Theta_V$ is continuous.
The upper bound in \eqref{eq:initial-frostman} is preserved.

For $\Hh^s$-almost every $p\in K$, positive upper density supplies
radii $r_i\downarrow0$ and a constant $c_p>0$ with
\[
 \Hh^s(K\cap\mathbf B_{r_i}(p))\ge c_p r_i^s.
\]
Pass to a subsequence on which $V_{p,r_i}$ converges to a stationary
integral tangent cone $C_p$. Define measures on the closed unit ball by
\[
 \lambda_i=r_i^{-s}(\eta_{p,r_i})_\#
       \big(\Hh^s\!\llcorner(K\cap\overline{\mathbf B}_{r_i}(p))\big).
\]
They have uniformly bounded mass, mass bounded away from zero, and
the upper ball bound in \eqref{eq:initial-frostman}. A further
subsequence converges to a nonzero measure $\lambda$ with the same
upper ball bound. Its support lies in $\spt C_p$.

Set $q_p=\Theta_V(p)$; at this point $q_p$ is not assumed to be an
integer. If $z\in\spt\lambda$, there are points $p_i\in K$ with
$(p_i-p)/r_i\to z$. Continuity on $K$ and
\eqref{eq:density-upper-limit} imply
\[
 \Theta_{C_p}(z)\ge q_p.
\]
On the other hand, the density at infinity of $C_p$ about $z$ equals
$q_p$: sandwich $\mathbf B_R(z)$ between the balls about zero of
radii $R-|z|$ and $R+|z|$, and use conicality about zero. Monotonicity
about $z$ consequently has equality at every radius.

We spell out the geometric consequence of this equality. The radial
displacement $X-z$ lies in the tangent plane of $C_p$ almost
everywhere. Testing stationarity with $\varphi(X)(X-z)$ gives
\[
 \int\big(m\varphi+(X-z)\cdot\nabla\varphi\big)
       \,\dd\|C_p\|=0.
\]
Differentiating dilated scalar integrals shows that the weight is
$m$-homogeneous about $z$. Since a rectifiable weight determines its
tangent plane almost everywhere, the varifold is conical about $z$
as well. Composing dilations about $z$ and about zero now gives
translation invariance along the whole line $\R z$.

The support of $\lambda$ spans at least $m$ linearly independent
directions. Otherwise it lies in a linear space of dimension at most
$m-1$; covering its bounded support by $O(\varepsilon^{-(m-1)})$
balls of radius $\varepsilon$ would give
$\lambda(\R^{m+n})\le C\varepsilon^{s-m+1}\to0$.
Choose $m$ independent such directions and let $P$ be their span.
The cone is invariant under every translation in $P$. If its support
contained a point outside $P$, subtracting its $P$-component would
give $w\in P^\perp\setminus\{0\}$ in the support. Conicality and
translation invariance would then place every disk
$tw+(P\cap\mathbf B_1)$, $1\le t\le2$, in the support. These are
disjoint disks of equal positive $m$-dimensional measure, all in a
bounded region, contradicting local finiteness of
$\Hh^m(\spt C_p)$. Thus its support is precisely $P$.
Tangential stationarity on this plane gives constant multiplicity,
and integrality gives
\[
 C_p=q_p|P|,\qquad q_p\in\{1,2,\ldots\}.
\]

We have proved that $\Theta_V$ is integer-valued at
$\Hh^s$-almost every point of $K$. One of its countably many integer
level sets therefore has positive $\Hh^s$ measure. Choose a compact
positive-measure subset $K_Q$ of that level set. Repeating the
preceding argument with $K_Q$, choose a fixed point $p\in K_Q$ and
radii $L_i\downarrow0$ on which its normalized $s$-mass is bounded
below. The resulting tangent cone is a $Q$-plane. Translate $p$ to
zero, denote that plane by $\Pi$, and set
\[
 \sigma_i=L_i^{-s}(\eta_{p,L_i})_\#
       \big(\Hh^s\!\llcorner
          (K_Q\cap\overline{\mathbf B}_{L_i}(p))\big).
\]
Compactness of $K_Q$ ensures that the support consists entirely of
points from that integer-density level set. Dilation preserves both
density and singularity. The asserted convergence and bounds follow.
\end{proof}
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//02-flat-scales.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//03-normal-inputs.tex ====
\section{Normal cylinders and signed excess}\label{sec:normal-inputs}

The next construction compares the varifold with a smooth reference disk,
not necessarily a plane. We fix the geometric conventions and state the
two measure estimates from \cite{AE} used throughout the proof. The local
minimal-fitting statements will be introduced when the reference disk is
constructed in Section~\ref{sec:fitting}.

\subsection{Geometry and units}
Let $S$ be a small-slope $C^J$ graph over an $m$-plane, with a fixed normal
tube and uniform bounds through a sufficiently large finite order $J$.
In that tube write
\[
 X=y+z,\qquad y=\pi(X)\in S,\qquad z\in N_yS.
\]
The horizontal and vertical subspaces are $T_yS$ and $N_yS$. Let $P_T$
denote orthogonal projection onto the approximate tangent plane of the
varifold at $X$. For $v\in N_yS$, the shape operator $\mathcal S_v$ is
defined by
\[
 \langle\mathcal S_v a,b\rangle
   =\langle\II_S(a,b),v\rangle,
 \qquad a,b\in T_yS.
\]
Set
\begin{equation}\label{eq:normal-tensors}
 B=I-(P_T)_{hh},\qquad e=\tr B=\tr(P_T)_{vv},
 \qquad J_z=I-\mathcal S_z,\qquad K_z=J_z^{-1}.
\end{equation}
The tube is chosen so that $J_z$ and $K_z$ are uniformly bounded.
The tensor $B$ is positive semidefinite, $0\le e\le m$, and
\begin{equation}\label{eq:projection-blocks}
 (P_T)_{hv}(P_T)_{vh}=B-B^2,
 \qquad |(P_T)_{vh}a|^2\le B[a,a],
 \qquad |(P_T)_{vh}|\le C\sqrt e.
\end{equation}
These follow by taking blocks in $P_T^2=P_T$.

We restrict to an interior cylinder whose support is separated from its
artificial vertical boundary above every compact subset of the base.
Every field has compact base support and is multiplied by a vertical
cutoff equal to one near the relevant support. Thus the restriction has
no vertical boundary term in its first variation. All enlargements below
stay inside this cylinder.

For the positive integer $Q$ being considered, define measures on $S$ by
\begin{equation}\label{eq:normal-measures}
 M=\pi_\#\mu-Q\vol_S,\qquad D=\pi_\#(e\mu).
\end{equation}
Here $M$ is signed and $D$ is nonnegative. In a specified physical system
of Euclidean units, let $z_{\mathrm{phys}}$ denote the unscaled normal
displacement and put
\begin{equation}\label{eq:physical-error}
 \Lambda=\pi_\#(\lambda\mu),\qquad
 \lambda=|z_{\mathrm{phys}}|^2+|\mathcal H_S|^2
                       +|\nabla^\perp\mathcal H_S|^2,
 \qquad \mathcal H_S=\tr\II_S.
\end{equation}
The curvature and its derivative in this formula are evaluated in those
physical units. On scaling a base chart of physical radius $l$ to unit
radius, all measures, including reference volume, are pushed forward and
divided by $l^m$. The factor $\lambda$ is still evaluated physically; it
is not replaced by the squared rescaled height or the rescaled curvature.
For a flat reference plane the estimates of \cite{AE} use
$\Lambda=0$. Charging an additional nonnegative height term in an upper
bound does not weaken their applicability.

Normal frames are transported radially from the chart center, and base
coordinates are normalized to have Euclidean metric at that center.
For an interior subchart using the inherited frames, the small geometric
parameter remains the original
\begin{equation}\label{eq:physical-gamma}
 \gamma=l+\sup|z_{\mathrm{phys}}|.
\end{equation}
This convention matters when the signed-excess theorem is used on much
smaller cells: changing the coordinates alone does not improve a
physical curvature error.

\subsection{The two imported measure estimates}
The first statement identifies an integer using a smooth projected mass
average, then controls the total variation of $M$. The projected count
$N(y)$ counts normal-fiber entries with their integer multiplicities at
almost every base point; exceptional projection fibers are immaterial
for the statement.

\begin{lemma}[Scalar mass and propagation of the integer]\label{lem:scalar-input}
Fix dimensions, a bound for $Q$, bounded chart geometry, a mass bound, and
an interior enlargement factor $c>1$. Suppose
\[
 s^{-m}\pi_\#\mu(B_{cs})\le C_0,
 \qquad s^{-m}(D+\Lambda)(B_{cs})\le\epsilon
\]
with $\epsilon$ sufficiently small. Put $b=(1+c)/2$. On the
intermediate ball $B_{bs}$, the count selects an integer $q\ge0$ with
$\vol_S\{y\in B_{bs}:N(y)\ne q\}\le C\epsilon s^m$.
There is a fixed finite test order $J_{\mathrm{sc}}$ such that, for
every smooth scalar test $\eta$ supported in $B_{bs}$,
\begin{equation}\label{eq:scalar-distribution-before-count}
 \begin{aligned}
 \left|\int\eta\,\dd(\pi_\#\mu-q\vol_S)\right|
 &\le C\epsilon s^m\|\eta\|_{C^{J_{\mathrm{sc}}}_s},\\
 \|\eta\|_{C^{J_{\mathrm{sc}}}_s}
 &=\max_{0\le j\le J_{\mathrm{sc}}}s^j\|D^j\eta\|_\infty.
 \end{aligned}
\end{equation}
The derivatives are in base coordinates; this test order is included
among the finite orders fixed for the argument.
For every nonnegative smooth test $\chi$ supported in $B_{bs}$,
with reference integral one and
$s^m\|\chi\|_{C^{J_{\mathrm{sc}}}_s}\le C_1$ for a fixed constant
$C_1$, one also has
\begin{equation}\label{eq:scalar-average-before-count}
 \left|\int\chi\,\dd(\pi_\#\mu)-q\right|\le C\epsilon.
\end{equation}
Here the constant may depend on $C_1$; the factor $s^m$ accounts for
the size of a test whose integral, rather than its supremum, is one.
In particular, if such an average is within $1/4$ of $Q$, then $q=Q$,
and
\begin{equation}\label{eq:scalar-input}
 |M|(B_s)+\vol_S\{y\in B_s:N(y)\ne Q\}
                     \le C(D+\Lambda)(B_{cs}).
\end{equation}
The same integer propagates along comparable working balls $B_s$
satisfying these bounds, provided their overlaps have fixed positive
relative volume. The smallness threshold depends on that overlap fraction.
\end{lemma}

This is \cite[Lemma~2.5 and its proof]{AE}, including its distributional,
smooth-average, and overlap conclusions. The distributional estimate
first compares the projected measure with a scalar constant; the
integer-count argument places that constant within $C\epsilon$ of $q$.
It therefore gives Equation~\eqref{eq:scalar-distribution-before-count}
before any identification with $Q$. The enlargement $c$ can be any
fixed number greater than one, by a finite interior covering. In
particular we use $c=2$ when the room has already been fixed. The integer
must be identified before applying Equation~\eqref{eq:scalar-input}; a
bound for the tilt alone does not specify which integer to subtract.

For a positive integer $j$, write $\exp_j$ and $\log_j$ for $j$-fold
composition. For sufficiently large $T$, let
\begin{equation}\label{eq:signed-rate}
 F_Q(T)=\exp_{2Q}\bigl(\sqrt{\log_{2Q}T}\bigr).
\end{equation}
We will use only the properties
\[
 (\log T)^a=o(F_Q(T))\quad\text{for every fixed }a>0,
 \qquad F_Q(T)=T^{o(1)}.
\]

\begin{theorem}[Signed excess]\label{thm:signed-input}
Fix an integer $Q\ge1$ and $\kappa>0$. In the normal-cylinder setting above, scale a
physical base radius $l$ to one and suppose $B_4$ is available. Set
$\tau=|M|+D+\Lambda$. If
\begin{equation}\label{eq:signed-hypotheses}
 \tau(B_4)\le d=e^{-T},\qquad
 l+\sup|z_{\mathrm{phys}}|\le d^\kappa,
\end{equation}
where the supremum is over the restricted support above the available
chart, then, for $T$ sufficiently large,
\begin{equation}\label{eq:signed-input}
 \int w\,\dd(2M-D-C_Q\Lambda)
       \le d\exp[-F_Q(T)]
\end{equation}
whenever $w$ is supported in $B_1$, $0\le w\le1$, and $\Lip w\le1$.
The finite derivative order $J$, threshold, and constants depend only
on the stated fixed data. For a subchart in inherited frames, the second
condition is imposed on $\gamma$ in Equation~\eqref{eq:physical-gamma}.
For a flat reference, take $\Lambda=0$ and omit the second condition.
\end{theorem}

This is \cite[Theorem~3.1]{AE}. Multiplying the test by a fixed constant
allows any fixed Lipschitz bound, with that constant on the right-hand
side. The theorem does not assume that the multiplicity is bounded by
$Q$: $Q$ is the integer subtracted in $M$.

\subsection{Fixed parameters}
All dimensional constants and interior enlargement factors are fixed
before small scales are sent to zero. Fix an integer kernel exponent $p\ge4$, put
$J_0=m+p+4$, and choose
\begin{equation}\label{eq:parameter-order}
 0<\beta_1<\tfrac14,\qquad
 0<\zeta<\min\{1,1/m\},\qquad \kappa>0,\qquad
 \kappa(2J_0+4)<2\beta_1,\qquad 3\kappa<\zeta.
\end{equation}
We then fix derivative orders large enough for Theorem~\ref{thm:signed-input}
and the compactness arguments below. Finally we choose the fitting order
and its small height coefficient in Section~\ref{sec:fitting}.
This order is noncircular: $J_0$, $\beta_1$, $\zeta$, and $\kappa$ do not
depend on the fitting order. The two inequalities in
Equation~\eqref{eq:parameter-order} will respectively permit the signed
comparison in the frequency estimate and in the height-normalized limit.
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//03-normal-inputs.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//04-fitting-frequency.tex ====
\section{Fitting intervals and frequency control}\label{sec:fitting}

Fix the singular origin, integer $Q$, plane $\Pi$, and target scales
$L_i\downarrow0$ supplied by Proposition~\ref{prop:flat-scales}.
The scales $L_i$ retain the measure of density-$Q$ centers, but were not
chosen by any height or frequency criterion. We shall cover them by
intervals on which a fitted center has uniform height doubling. The
construction of the local fits comes from \cite[Section~7]{AE}; the
selection and comparison of successive fitting intervals are needed here
because the origin need not satisfy a typical-point multiplicity condition.

At each start, local minimal graphs fit the support until their height
test first fails. We glue these graphs to one reference disk. The failure
scales determine the smallest radius on which we estimate its height
and frequency. There are then two possibilities. If this radius is zero,
the frequency estimates apply inward indefinitely. Otherwise we restart
at a smaller scale where the varifold is again close to a plane.
On late intervals carrying the targets, the preceding failure bounds the
frequency at the outer end, and the new failure bounds it at the inner
end. These two endpoint bounds are what make height doubling
uniform. The last subsection proves that the resulting intervals contain
every sufficiently late $L_i$; no coverage of arbitrary small scales is
required.

\subsection{Local fits and their gluing}

A \emph{good start} is a sufficiently small dyadic scale $R$ for which
$V_R$ is within a fixed small weak-varifold distance of a multiplicity-$Q$
plane through the origin in $\mathbf B_{2000}$. Use a metric on varifolds
with bounded mass, and choose a plane whose distance realizes, or
approximates within a fixed factor, the best distance. In particular, if
this distance tends to zero, the corresponding varifolds converge to
their chosen planes after taking a subsequence. Throughout a start we
work in units of $R$ and take its chosen plane as horizontal; $\mu$
denotes the weight of $V_R$ until target units are introduced explicitly.

The goodness threshold will be decreased a finite number of times below.
Weak convergence to a plane, support convergence, and monotonicity then
provide a cylinder with base containing $B_{300}$, fixed vertical room,
and a support-free gap at its vertical boundary. From now on $\mu$
is restricted to this working cylinder, as in
Section~\ref{sec:normal-inputs}; every support condition in the fitting
construction refers to this restriction. On its interior we have
the upper mass bound $\mu(\mathbf B_u(X))\le C u^m$ at all radii used
below, arbitrarily small support height and weak mass error, and small
tilt. The scalar estimate of \cite[Lemma~2.5]{AE} also gives small
projected mass defect on smaller cylinders. These statements concern
all support points in the cylinder, not a selected collection of sheets.
Since the density at the origin is $Q$, we require the start to be small
enough that all needed smaller and fixed-factor larger density ratios
at the origin are uniformly as close to $Q$ as prescribed.

Choose a finite center derivative order $J>m/2+6$, larger than the orders
needed for the signed estimate and subsequent compactness arguments.
Apply the fitting lemma with order at least $J+1$; denote its exponent
$k_0$ by $k$. In particular,
\begin{equation}\label{eq:fitting-order}
 k>4+m/2.
\end{equation}
Next choose its height parameter $m_0>0$ sufficiently small. We recall
precisely the fitting information to be used.

\begin{lemma}[Local fitting inputs]\label{lem:local-fits}
For every $x\in B_5$, there are smooth minimal graphs
$S_{x,\ell}=\operatorname{graph}f_{x,\ell}$ over $B_{80\ell}(x)$,
at descending dyadic scales $\ell=2^{-j}$ starting at $j=0$, through
the first scale at which
\[
 \sup\{|X_v-f_{x,\ell}(X_h)|:
          X\in\spt\mu,\ X_h\in B_{64\ell}(x)\}
                                  \le m_0\ell^k
\]
fails. Every constructed fit, including the first failed one, has
support height at most $Cm_0\ell^k$ on the fixed interior enlargements
of this test ball. On $B_{70\ell}(x)$ its normalized tilt and projected
mass defect are bounded by $Cm_0^2\ell^{2k-2}$.
For a fixed-base smooth vector test $\eta$ supported in
$B_{70\ell}(x)$, put $\eta_{x,\ell}(w)=\eta(x+\ell w)$ on
$B_{70}$. Its first height moment is bounded by
$Cm_0^2\ell^{m+2k-1}\|\eta_{x,\ell}\|_{H^q(B_{70})}$,
for some fixed integer $q$. Thus the test norm is taken after rescaling
the base ball by $\ell^{-1}$, with no additional amplitude factor.
Comparable overlapping fits differ in derivative
order $j\le J+1$ by at most $Cm_0^2\ell^{2k-1-j}$ on smaller common
balls. Their fixed-coordinate derivatives are uniformly bounded, and
their slopes are arbitrarily small when $m_0$ is small.

At a first failure $(x,\ell)$, every fixed sufficiently small comparable
base subball with center in $B_{30\ell}(x)$ contains a set of base
volume at least $c\ell^m$ whose normal fibers contain two entries
separated by at least $cm_0\ell^k$. Both entries have small tangent tilt
relative to the fit and lie at height at most $Cm_0\ell^k$ from it.
The constants may depend on the fixed relative subball radius and tilt
threshold, but not on the start or the failed scale.
\end{lemma}

These are \cite[Lemmas~7.1--7.2 and Equations~(7.7)--(7.8)]{AE}.
Only small initial height and mass error are required for these inputs.
Indeed the scale-zero correction uses the horizontal plane and smooth
mass averages identifying $Q$; subsequent corrections use the preceding
fit and its selected integer. Compatibility and failed-fit separation
use these estimates and the failed height test. The multiplicity-defect
condition imposed later in that source is not used in these arguments.
At every new scale the preceding fit already bounds all support heights
on a horizontal enlargement. Consequently the normal cylinders used
in the estimates retain a support-free vertical gap and sufficient
interior room. The initial cylinder supplies this property at scale zero.

Write $\ell_x$ for the first failed scale, and put $\ell_x=0$ if no
failure occurs. We use the failures in two ways. Their tents prescribe
a spatially varying scale for gluing the local fits; the largest failure
whose $16$-fold base ball contains the origin determines the lower end
of the radial interval. Define these two quantities by
\begin{equation}\label{eq:tents}
 t(w)=\sup_{x\in B_5}(\ell_x-|w-x|)_+,
 \qquad
 b=\max\bigl(\{\ell_x:|x|\le16\ell_x\}\cup\{0\}\bigr).
\end{equation}
The maximum exists because the nonzero scales are dyadic. The function
$t$ is $1$-Lipschitz, $t(0)\le b$, and hence $t(w)\le b+|w|$.
By decreasing the goodness threshold, every failure scale can be made
smaller than any prescribed fixed positive number. Otherwise the
separated entries in Lemma~\ref{lem:local-fits}, whose direction is
nearly vertical, contradict convergence of the support to the start
plane on the initial cylinder.

Glue the fits using the construction in the proof of
\cite[Proposition~7.3]{AE}, before its multiplicity-defect argument.
For clarity, regularize $t$ by $t_\epsilon=\max(t,\epsilon)$, cover by
balls of radii a small fixed multiple of $t_\epsilon$, and choose a
partition of unity with derivatives controlled at those scales. At a
cell center use a constructed dyadic fit of scale comparable to
$t_\epsilon$ and larger than its failure scale. The overlap estimate
in Lemma~\ref{lem:local-fits} controls derivatives of the partitioned
differences. Passing to a $C^J$ limit as $\epsilon\downarrow0$ gives a
graph $S$ with small slope and uniformly bounded $C^J$ geometry, for which
\begin{equation}\label{eq:glued-center}
 |\mathcal H_S|+|\nabla^\perp\mathcal H_S|
       \le Cm_0^2t^{2k-4},
 \qquad |z|\le Cm_0t^k\quad\hbox{on the support}.
\end{equation}
Here $t$ is evaluated at the fixed horizontal argument; in the height
bound one may equivalently use the horizontal argument of the nearest
projection. The same estimate holds for vertical graph-height
differences at fixed horizontal argument. On a cell of size comparable
to $t>0$, the glued graph differs from a constructed fit of comparable
scale by at most $Cm_0^2t^{2k-1}$ in height. These statements follow
from the overlap estimate for the finitely many comparable fits in
each partition sum; they require no density statement about $\{t>0\}$.

We make slopes, normal tubes, and geodesic coordinate neighborhoods
small enough that the relevant parametrizations and nearest projections
have Lipschitz constants below $3/2$. The point of $S$ over horizontal
$0$ is at distance at most $Cm_0b^k$ from ambient $0$. Use this point
as radial center on $S$, and denote geodesic distance from it by $\rho$.
Fix $B_*=100$ and a sufficiently small $r_*>0$. We shall work on
\begin{equation}\label{eq:fitting-radius-range}
 B_*b\le r\le4r_*,\qquad r>0.
\end{equation}
All these radii lie strictly inside one geodesic chart. The maximum
failure scale is taken small enough that the interval has the required
interior room.

\subsection{Weighted height on a fitting interval}

Using the integer $p\ge4$ fixed in Section~\ref{sec:normal-inputs}, define
\[
 \phi=(1-\rho^2/r^2)_+^p,
 \qquad \psi=2p(1-\rho^2/r^2)_+^{p-1}.
\]
Use the normal-cylinder notation of the preceding section. In particular,
$B=I-(P_T)_{hh}$, $e=\tr B$, $J_z=I-\mathcal S_z$, and $K_z=J_z^{-1}$.
Put $u=\nabla_S\rho$, $v_z=K_zu$, and
\[
 b_z=B[v_z,v_z],\qquad h=1-b_z/|v_z|^2.
\]
Then $0\le h\le1$. On the central fiber choose any horizontal unit
vector in place of $u$. If this fiber has positive mass, its tangent is
purely vertical almost everywhere on the fiber by rectifiability, so
$h=0$ there almost everywhere. The reason for this factor is the sharp
projection inequality
\[
 |(P_T)_{vh}v_z|^2
   =B[v_z,v_z]-|Bv_z|^2
   \le b_z-\frac{b_z^2}{|v_z|^2}=b_zh.
\]
Weighting squared height by $h$ will therefore pair it directly with
the radial tilt $b_z$ in Cauchy--Schwarz. No positive lower bound for
$h$ is assumed. Define
\begin{equation}\label{eq:interval-quantities}
 \begin{aligned}
 H&=r^{-m}\int\psi|z|^2h\,\dd\mu,
 &\delta&=r^{-1}\sup_{\rho<r,\ X\in\spt\mu}|z|,\\
 A&=r^{2-m}\int\phi\,\dd M,
 &E_*&=r^{2-m}\int\phi\,\dd D,\\
 R_*&=r^{-m}\int\rho^2\psi b_z\,\dd\mu.
 \end{aligned}
\end{equation}
The radius in the supremum refers to the projected point. These
quantities will compare neighboring radii without requiring that the
center be minimal.

\begin{lemma}[Weighted bounds]\label{lem:weighted-bounds}
On \eqref{eq:fitting-radius-range}, $\delta>0$, $\delta\le Cr^{k-1}$,
and normalized mass is bounded. For $J_0=m+p+4$ as fixed in
Section~\ref{sec:normal-inputs},
\begin{align}
 c\delta^{J_0}&\le H/r^2\le C\delta^2,
                                  \label{eq:height-power-bounds}\\
 |\mathcal H_S|+|\nabla^\perp\mathcal H_S|
   &\le C\delta r\quad(\rho<r),\label{eq:point-curvature-bound}\\
 \int\phi\bigl(|\mathcal H_S|^2+
                    |\nabla^\perp\mathcal H_S|^2\bigr)\,\dd\mu
   &\le C\int\phi|z|^2\,\dd\mu.\label{eq:weighted-curvature-bound}
\end{align}
Moreover a nonnegative smooth test $\chi_r$, with reference integral
one and support in $\{\rho<r/4\}$, can be chosen so that its
projected mass average is within $1/8$ of $Q$ and
$r^m\|\chi_r\|_{C^{J_{\mathrm{sc}}}_r}\le C$.
All constants are uniform among good starts.
\end{lemma}

\begin{proof}
If $t(w)>0$ and $\rho(w)<r$, choose a failure $(x,\ell)$ for which
$\ell-|w-x|>t(w)/2$. Since $|w|\le r$, either $|x|\le16\ell$ or
$16\ell<r+\ell$. Thus
\begin{equation}\label{eq:witness-size}
 \ell\le\max(b,r/15)\le r/15.
\end{equation}
Equation~\eqref{eq:glued-center} gives $\delta\le Cr^{k-1}$.
The projected cylinder is contained in an ambient ball of comparable
radius, so monotonicity gives its normalized mass bound.

If $\delta=0$, the support lies on $S$ in a neighborhood of ambient
$0$: its projection lies inside $\{\rho<r\}$ because its distance
from the radial center is $O(m_0b^k)$.
Tangential stationarity then makes the multiplicity, extended by zero
on the disk, constant. It is positive because $0$ belongs to the
support. Normal stationarity and elliptic regularity make $S$ smooth
and minimal there. This contradicts the singularity of $0$.

We next locate the height that pays for a witness. The argument is the
weighted construction of \cite[Proposition~7.4]{AE}, with
\eqref{eq:witness-size} replacing its typical-point estimate.
If the witness is near the kernel boundary, move inward by $\ell$
along the radial geodesic and take a ball of radius a sufficiently
small fixed multiple of $\ell$; otherwise take a nearby such ball.
This ball lies within the kernel support, and its center remains within
$B_{30\ell}(x)$ in the fitting coordinates. On it $\phi$ and $\psi$
dominate, up to fixed constants, their suprema on a small cell containing
the original point, and
$\psi\ge c(\ell/r)^{p-1}$.
Lemma~\ref{lem:local-fits} supplies separated entries above a set of
base volume $c\ell^m$. Two entries separated in a nearly vertical
direction cannot both lie within a sufficiently small multiple of their
separation from a small-slope graph. After transferring to $S$, at
least one entry therefore has height at least $cm_0\ell^k$ and
$h\ge3/4$. Small slopes and the height bounds make changes of horizontal
argument smaller than the available subball margin. The projection
Jacobian bound converts base volume into the same lower bound for mass.

In particular $\delta r\ge cm_0\ell^k$. At a point approaching the
height supremum, the witness also satisfies
$\delta r\le Cm_0\ell^k$, by \eqref{eq:glued-center}. Hence
\[
 H/r^2\ge c\delta^2(\ell/r)^{m+p-1},
 \qquad \ell/r\ge c\delta^{1/k},
\]
where the last inequality uses $r\le1$. This proves the lower bound
in \eqref{eq:height-power-bounds}, for example with
$J_0\ge2+(m+p-1)/k$; its upper bound follows from the mass bound.
The comparison $\delta r\ge cm_0\ell^k$, together with
$k>4$, proves \eqref{eq:point-curvature-bound}.

To prove \eqref{eq:weighted-curvature-bound}, group witnesses by dyadic
scale and use each covering grid cell once at its scale. By
\eqref{eq:glued-center}, the curvature-square integral on a cell of
size $\ell$ is at most
$Cm_0^4\ell^{4k-8+m}\sup\phi$.
Its paying ball contributes at least
$cm_0^2\ell^{2k+m}\sup\phi$ to the height-square integral.
At each scale the paying balls have bounded overlap, since they are
moved by only $O(\ell)$. Summing the factors
$Cm_0^2\ell^{2k-8}$ proves the assertion. The cells may be used as an
overcover, so no measurable choice of witnesses is necessary.

Finally choose a smooth decreasing radial mass test on a small fixed
fraction of $r$. Compare it with the corresponding ambient radial test
at $0$. The displacement of the center is $O(m_0b^k)$, the support
height is $O(r^k)$, and chord and geodesic radii on $S$ differ by
$O(r^3)$. The radius replacement has error $O(r^2)$, so the normalized
averages differ by $O(r)$, including their reference normalizations.
All support encountered by the ambient test lies in the cylinder and
has projected radius below $r/2$. The ambient average is close to $Q$
by the density-ratio requirement at the start. Decreasing the fixed
parameters gives the stated $1/8$ bound.
\end{proof}

\subsection{Frequency estimates and their inward consequences}

The preceding lemma gives positive height and the single-radius
hypotheses for the frequency calculation. Set
\[
 P=A/H,\qquad n_*=E_*/H,\qquad q_*=R_*/H,
 \qquad T=\log(r^2/H),\qquad g=T^{-20}.
\]
Then $T$ is comparable to $|\log\delta|$ and
$T\ge2|\log r|-C$. A dot denotes differentiation with respect to
$\log r$.

The use of signed mass in $P=A/H$ is dictated by horizontal stationarity,
which controls $\dot A$. To see the intended cancellation, temporarily
omit the geometric errors in the estimates below. Signed excess gives
$2P\le n_*$, the sharp flux inequality gives $n_*^2\le q_*$,
and the variation identities give
$\dot A/H=2P-n_*+q_*$ and $\dot H/H\le2n_*$.
For $P\ge0$ these imply
\[
 \dot P\ge2P-n_*+n_*^2-2Pn_*
             =(n_*-1)(n_*-2P).
\]
This expression is nonnegative when $P\ge1/2$.
The next two lemmas quantify the errors and recover the inward bounds
and height comparison needed on a fitting interval.

\begin{lemma}[Frequency on a fitting interval]\label{lem:interval-frequency}
After fixing $r_*$ sufficiently small, on
\eqref{eq:fitting-radius-range} the following inequalities hold:
\begin{align}
 |P|&\le C(n_*+r^2)+\eta q_* ,\label{eq:freq-absolute}\\
 2P&\le n_*+Cg(n_*+1)+Cr^2+\eta q_* ,\label{eq:freq-signed}\\
 n_*&\le\sqrt{q_*}+Cr^2+\eta n_* ,\label{eq:freq-flux}\\
 \left|\dot A/H-(2P-n_*+q_*)\right|
 &\le C\{r\sqrt{q_*}+r^2(1+n_*)\}+\eta(n_*+q_*),
                                      \label{eq:freq-horizontal}\\
 -m\le\dot H/H&\le2n_*+Cr+\eta(n_*+q_*).
                                      \label{eq:freq-height}
\end{align}
Here $\eta=C\delta^\alpha\le C'e^{-cT}$ for fixed positive
$\alpha,c$, and constants are uniform among intervals. The functions
whose derivatives occur are locally absolutely continuous in $\log r$.
\end{lemma}

\begin{proof}
We verify the transfer of the calculation in
\cite[Lemmas~8.1--8.2]{AE}; this also identifies the precise
single-radius inputs. The definitions and positivity of $B$ give
\begin{equation}\label{eq:weighted-height-comparisons}
 \begin{aligned}
 r^{-m}\int\phi|z|^2\,\dd\mu&\le C(H+\delta^2E_*),\\
 r^{-m}\int\psi|z|^2\,\dd\mu
     &\le C\{H+\delta^2(E_*+R_*)\},\\
 r^{-m}\int\phi\,\dd\Lambda&\le C(H+\delta^2E_*).
 \end{aligned}
\end{equation}
For the first bound use $1-h\le Ce$; for the second split at $r/2$
and use $1-h\le Cb_z$ on the outer part. The last bound also uses
\eqref{eq:weighted-curvature-bound}, with physical units equal to the
current start units.

Here are the details of the mass comparison, where preserving the
coefficient one in \eqref{eq:freq-signed} is essential. For the fixed
$0<\beta_1<1/4$ put $\gamma_1=\delta^{\beta_1}$. Away from the
outer collar we use signed excess; on the collar we use a radial
first-variation test. Choose a radial cutoff $\chi$ equal to one at
relative depth $1-\rho/r\le\gamma_1^2$ and zero at depth at least
$2\gamma_1^2$, and put
\[
 \phi_{\rm thin}=\chi\phi,\qquad
 \phi_{\rm bulk}=(1-\chi)\phi.
\]
Cover the support of $\phi_{\rm bulk}$ by cells centered at $x_j$ of
radius $\lambda_1=c_1\gamma_1^2r$, with a smooth partition of unity
$\vartheta_j$ and bounded overlap of fixed enlargements. Choose $c_1$
small enough that these enlargements stay inside $\{\rho<r\}$ and
$\phi$ is comparable there to $\phi_j=\phi(x_j)$. Their mass is
$O(\lambda_1^m)$ because
$\delta r=o(\lambda_1)$. Testing normal stationarity with a cutoff
times $z$ and using \eqref{eq:point-curvature-bound} yields
\[
 \lambda_1^{-m}(D+\Lambda)(B_{c\lambda_1})
                    \le C\delta^{2-4\beta_1}.
\]
This is the local estimate in \cite[Equation~(8.6)]{AE}.
The scalar estimate selects an integer on each cell; the small
exceptional projection sets cannot fill the prescribed overlaps, so
the integers agree throughout the connected bulk; call their value $q$.
Put $\epsilon=C\delta^{2-4\beta_1}$.
Partition the smooth mass test from Lemma~\ref{lem:weighted-bounds}
over the cells. Each piece has the scaled derivative norm in
Equation~\eqref{eq:scalar-distribution-before-count} bounded by
$Cr^{-m}$. That homogeneous estimate bounds its error relative to $q$
by $C\epsilon\lambda_1^m r^{-m}$, without dividing by the piece's
possibly small integral. Bounded overlap gives
$\sum\lambda_1^m\le Cr^m$, so the total error is $O(\epsilon)$.
Since the test average is within $1/8$ of $Q$, integrality gives $q=Q$.

The scalar estimate now bounds $|M|(B_{4\lambda_1}(x_j))$ by
$C(D+\Lambda)(B_{8\lambda_1}(x_j))$. Summing with comparable
weights gives the absolute bulk estimate
\begin{equation}\label{eq:bulk-absolute}
 r^{2-m}\int\phi_{\rm bulk}\,\dd|M|
 \le Cr^{2-m}\int\phi\,\dd(D+\Lambda)
 \le C(E_*+r^2H).
\end{equation}
In the last inequality a small multiple of $E_*$ is absorbed using
\eqref{eq:weighted-height-comparisons}.

For signed excess, use fresh charts on the cells and set
\[
 d_j=\max\{C\lambda_1^{-m}\tau(B_{4\lambda_1}(x_j)),
                          \delta^{2J_0+4}\},
 \qquad T_j=-\log d_j,\qquad \tau=|M|+D+\Lambda.
\]
The local bounds and the floor give
$c|\log\delta|\le T_j\le(2J_0+4)|\log\delta|$.
The physical smallness hypothesis follows from
\[
 \lambda_1+\sup|z|\le C\delta^{2\beta_1},
 \qquad \kappa(2J_0+4)<2\beta_1.
\]
These are precisely the fixed parameter choices in
Equation~\eqref{eq:parameter-order}; the required derivative orders
were chosen after $J_0$ and $\kappa$, and before the fitting exponent.
The partitioned tests $\vartheta_j\phi_{\rm bulk}/(C\phi_j)$ have
size and Lipschitz constant at most one in cell units. Applying signed
excess to these tests and restoring the measure factors gives
\begin{align*}
 2r^{2-m}\int\phi_{\rm bulk}\,\dd M
 &\le r^{2-m}\int\phi_{\rm bulk}\,\dd D
     +C_Qr^{2-m}\int\phi_{\rm bulk}\,\dd\Lambda\\
 &\quad+Cr^{2-m}\sum_j\phi_j\lambda_1^m d_j e^{-F_Q(T_j)}.
\end{align*}
The weighted budgets satisfy
\[
 \sum_j\phi_j\lambda_1^m d_j
 \le C\int\phi\,\dd(D+\Lambda)+Cr^m\delta^{2J_0+4},
\]
by scalar control on the enlarged cells and bounded overlap. The
exponential factor is smaller than any fixed negative power of $T$,
and the floor costs
$r^2\delta^{2J_0+4}\le C\delta^{J_0+4}H$ by
\eqref{eq:height-power-bounds}. The signed-excess remainder is therefore at most
$Cg(E_*+H)$. The separate $C_Q\Lambda$ term is bounded by
$Cr^2H+Cr^2\delta^2E_*$; since $r^2\delta^2=o(g)$, it contributes
only $Cr^2H+CgE_*$. This proves the bulk signed comparison with
leading tilt coefficient one.

For the thin part, let $a$ be a multiplier with
$\sup(|a|+r|\nabla_Sa|)\le1$. The radial primitive constructed in
\cite[Lemma~8.1, Equation~(8.11)]{AE}, cut off at relative depths
between $\gamma_1$ and $2\gamma_1$, gives a compactly supported
tangent field $X$ with
\[
 \diver_SX=a\phi_{\rm thin}+q_c,\qquad
 |X|/r+|\nabla^SX-a\phi_{\rm thin}u\otimes u|
                                    \le C\gamma_1^2\phi.
\]
Here $q_c$ is supported on that thicker cutoff collar and
$|q_c|\le C\gamma_1^{p+1}\phi$. Its mass contribution is controlled
by the absolute bulk estimate \eqref{eq:bulk-absolute}. In horizontal
stationarity, the principal derivative of $X$ pairs with radial tilt
$B[u,u]$; the remaining derivative has the small factor $\gamma_1^2$.
The resulting bound is
\[
 \begin{aligned}
 r^{2-m}\left|\int a\phi_{\rm thin}\,\dd M\right|
 &\le Cr^{2-m}\int\phi_{\rm thin}B[u,u]\,\dd\mu
                  +C\delta^\alpha E_*+Cr^2H
 \\
 &\le C\delta^{\alpha'}(E_*+R_*)+Cr^2H.
 \end{aligned}
\]
For the last inequality use, on the thin collar,
$\phi_{\rm thin}\le C\gamma_1^2(\rho/r)^2\psi$ and
$B[u,u]\le2b_z+C|z|^2e$.
The errors use only horizontal stationarity, fixed geodesic
geometry, and \eqref{eq:weighted-height-comparisons}.
Combining the absolute bulk and thin estimates proves
\eqref{eq:freq-absolute}, including its version with any bounded
scaled $C^1$ multiplier. For the signed estimate take $a=1$ and
combine with the signed bulk bound. The small tilt error is absorbed
in $CgE_*$, since every positive power of $\delta$ is $o(g)$; the
radial error remains $\eta R_*$. This proves \eqref{eq:freq-signed}.

Normal stationarity gives the flux estimate
\[
 \left|E_*-r^{-m}\int\rho\psi\,
                    z\cdot(P_T)_{vh}v_z\,\dd\mu\right|
                       \le Cr^2H+\eta E_*.
\]
Projection algebra gives the exact inequality
$|(P_T)_{vh}v_z|^2\le b_zh$; Cauchy--Schwarz proves
\eqref{eq:freq-flux}. Horizontal stationarity with the lift of
$\rho u$ gives \eqref{eq:freq-horizontal}, as in
\cite[Lemma~8.2]{AE}. Its geodesic volume-distortion multiplier has
bounded scaled $C^1$ norm, so the preceding absolute estimate applies
with a uniform constant.

One modification is needed for the height identity because our center
need not pass through ambient $0$. Off the central fiber put
\[
 Y=\frac{\rho K_zu}{|K_zu|^2}.
\]
Then $DY=P_h+O(\rho+|z|)$, where $P_h$ is horizontal projection
at the nearest point of $S$. To see this, the pulled-back radial field
$\rho u$ has derivative $P_h+O(\rho+|z|)$. Its correction in $Y$
is $\rho$ times a function of the tube coordinates and $u$ that is
$O(|z|)$, including its angular derivatives. Angular differentiation
therefore costs $\rho\cdot O(|z|)/\rho=O(|z|)$, and the remaining
derivative costs $O(\rho)$.
Testing with $\psi|z|^2Y$ gives the following identity, whose integrand
is extended by zero on the central fiber:
\begin{equation}\label{eq:interval-height-identity}
 \dot H=r^{-m}\int\psi\left\{
 |z|^2(\tr(P_TDY)-mh)
 +\frac{2\rho z\cdot(P_T)_{vh}K_zu}{|K_zu|^2}
 \right\}\,\dd\mu .
\end{equation}
This is \cite[Equation~(8.18)]{AE}. First omit the central fiber
using a radial cutoff. The transition term tends to zero because
$Y=O(\rho)$ and the mass of the open shrinking transition annulus
tends to zero. On the fiber $h=0$ almost everywhere, so the left-hand
side is unchanged; the flux term has a factor $\rho$. The error in
$DY$ is $O(r)$ on the support, and
\eqref{eq:weighted-height-comparisons} and the normal flux estimate
give the upper bound in \eqref{eq:freq-height}. The lower bound follows
by direct differentiation of the nonnegative weight $r^{-m}\psi$.
All parameter choices have been fixed independently of the start.
\end{proof}

We now extract the ODE consequences before selecting successive starts.
The proof adapts \cite[Proposition~8.3]{AE} to finite intervals with
controlled endpoint values.

\begin{lemma}[Inward control]\label{lem:inward-control}
There is a fixed large $K$ such that, writing $v_0=\dot A/H$,
\begin{equation}\label{eq:large-frequency}
 |P|\ge K\quad\Longrightarrow\quad
 q_*/|P|\ge c\min(|P|,\eta^{-1}),\qquad v_0\ge cq_*.
\end{equation}
When $K\le P\le T^4$,
\begin{equation}\label{eq:moderate-frequency}
 \dot P\ge-C(rP+gP^2).
\end{equation}
A lower bound for $P$ at the smaller-radius endpoint of a finite
interval gives a lower bound throughout that interval. On an interval
extending to zero, $P$ is bounded below for all sufficiently small radii.
If at a sufficiently small starting radius $r_0$ one has $P_0\le T_0^2$,
then inward throughout the interval
\begin{equation}\label{eq:inward-upper}
 P\le \exp(1)\max(K,P_0).
\end{equation}
Such a starting radius exists on every interval extending to zero.
Finally, wherever $|P|\le C_0$ and $\eta$ is sufficiently small
depending on $C_0$, every fixed bounded-ratio subinterval satisfies
\begin{equation}\label{eq:height-doubling}
 H(r')\asymp H(r''),\qquad
 \int q_*\,\dd\log r\le C.
\end{equation}
The constants in the last conclusion depend on $C_0$ and the radius
ratio, not on the position of the interval.
\end{lemma}

\begin{proof}
Equations~\eqref{eq:freq-absolute} and \eqref{eq:freq-flux} imply
$|P|\le C(\sqrt{q_*}+r^2)+C\eta q_*$. For $|P|$ large, either
the square-root term or the last term controls a fixed fraction of
$|P|$, proving the first assertion of \eqref{eq:large-frequency}.
Equation~\eqref{eq:freq-horizontal} then gives its second assertion,
after increasing $K$ and decreasing $r_*$.

For positive $P$, quotient differentiation and the upper height bound
give the principal expression
$2P-n_*+q_*-2Pn_*$, with the errors displayed in
Lemma~\ref{lem:interval-frequency}. If $q_*$ is larger than a
sufficiently large fixed multiple of $P^2$, it absorbs all negative
terms, since $n_*\le C\sqrt{q_*}+Cr^2$ and $\eta T^4=o(1)$.
Otherwise $n_*\le CP$ and
\[
 q_*\ge n_*^2-Cr^2P-C\eta P^2,
 \qquad n_*\ge2P-C(gP+r^2+\eta P^2).
\]
For large $K$ the principal terms factor as
$ (n_*-1)(n_*-2P)$, bounded below by
$-CP(gP+r^2+\eta P^2)$. Since $\eta T^4\le g$ at small radii,
the term $\eta P^3$ is absorbed by $gP^2$. This proves
\eqref{eq:moderate-frequency}.

If $P\le-K$, the lower height bound gives
$\dot P=v_0-P\dot H/H\ge cq_*-m|P|>0$.
Thus $P$ decreases in inward time $-\log r$ whenever it is below
$-K$. This proves the finite-interval lower bound from the inner
endpoint. On an infinite interval, such an excursion would also make
$A<0$ decrease inward and remain below a fixed negative value. That is
impossible because $A\to0$: its normalized mass integral is bounded
and it has a factor $r^2$.

To prove the upper bound, work on a component where $P\ge K$ before
any exit through $P=T^4$. In inward time, \eqref{eq:moderate-frequency}
gives
\[
 (\log P)'\le C(r+T^{-16}).
\]
The integral of this right-hand side from a sufficiently small radius
to zero is less than one, since $T\ge2|\log r|-C$. Moreover
$Z=T-\log P=\log(r^2/A)$ satisfies
$Z'=v_0/P-2>0$ by \eqref{eq:large-frequency}.
If the component starts at $r_0$ with $P_0\le T_0^2$, then
$P\le\exp(1)P_0$ and $T\ge T_0-\log P_0+\log K\ge T_0/2$.
For large $T_0$ these inequalities exclude $P=T^4$.
Components starting later at $P=K$ satisfy the same argument, giving
\eqref{eq:inward-upper}.

On an infinite interval, suppose no sufficiently small radius satisfied
$P\le T^2$. Then eventually $P>T^2$, and
\eqref{eq:large-frequency} implies $Z'\ge cT^2-2$.
The lower bound for $T$ makes $Z$ positive eventually. Since $Z\le T$
in this regime, one then has $Z'\ge c'Z^2$, which forces blow-up in
finite inward time. The continuous positive quantities $A,H$ are
defined at every finite time in this regime, a contradiction.

Finally, for $|P|\le C_0$, use the upper height derivative bound if
$P\ge0$ and the lower bound if $P<0$. Absorbing $\eta q_*$ gives
\[
 \dot P\ge cq_*-C_{C_0}.
\]
Integration bounds $\int q_*\,\dd\log r$ on every bounded-ratio
interval. Equation~\eqref{eq:freq-flux} and Cauchy--Schwarz bound
$\int n_*\,\dd\log r$, and \eqref{eq:freq-height} gives both
height comparisons in \eqref{eq:height-doubling}.
\end{proof}

\subsection{Covering the prescribed target scales}

We have obtained height comparison whenever the frequency can be
bounded above and below on a radial interval. For a finite interval,
Lemma~\ref{lem:inward-control} uses an upper bound at its outer endpoint
and a lower bound at its inner endpoint. The restart construction supplies
them from different failures: the preceding start's failure supplies the
outer bound, and the current start's failure supplies the inner bound.
If no failure occurs near the origin, the interval extends to zero and
the infinite-interval conclusions of that lemma apply instead.

We first define the starts and prove that their intervals contain the
prescribed targets. Fix $N_*=1000$, and choose a dyadic factor
\begin{equation}\label{eq:restart-factor}
 F>10N_*B_*/r_*.
\end{equation}
Decrease the maximum failure scale so that
$Fb<1/4$ and $8N_*B_*b<r_*$ for every good start.
Its target interval is
\begin{equation}\label{eq:target-interval}
 B_*bR\le L\le(r_*/N_*)R,\qquad L>0.
\end{equation}
Starting at one good dyadic $R$, if $b>0$ choose the next start to be
the largest good dyadic scale at most $FbR$; if $b=0$, stop. Such a
next start always exists: every dyadic scale comparable to a sufficiently
late $L_i$ by fixed factors is good, because $V_{L_i}\to Q|\Pi|$
locally.

\begin{proposition}[Uniform comparison at the targets]\label{prop:target-frequency}
All sufficiently late $L_i$ belong to intervals
\eqref{eq:target-interval}. After passing to a subsequence, either one
interval extends to zero and contains all these targets, or the indices
of their covering intervals tend to infinity. In the latter case,
on each sufficiently late covering interval,
\begin{equation}\label{eq:uniform-frequency-range}
 |P(r)|\le C\qquad(B_*b\le r\le r_*),
\end{equation}
and \eqref{eq:height-doubling} holds there with uniform constants.
In the former case the same conclusions hold on the sufficiently
small-radius part of the fixed interval.
\end{proposition}

\begin{proof}
\emph{Coverage of the targets.}
The top of the interval belonging to $R$ is $(r_*/N_*)R$.
Successive tops decrease to zero unless the construction stops, since
the next start is at most $FbR<R/4$. Suppose a target $L_i$ lies below
one top and above the next, but misses the first interval. It must then
lie below its lower endpoint, so $L_i<B_*bR$ and
$FbR>10N_*L_i/r_*$. A dyadic scale between
$2N_*L_i/r_*$ and $4N_*L_i/r_*$ is therefore eligible for the next
start. For all sufficiently late $i$ it is good, by convergence at
$L_i$ to a plane. Maximality of the next start would make its top
larger than $L_i$, a contradiction. Hence all sufficiently late targets
are covered.

A fixed interval containing targets tending to zero necessarily has
$b=0$. On that interval Lemma~\ref{lem:inward-control} first supplies
a lower bound and a sufficiently small radius with $P\le T^2$, then
an upper bound and height doubling inward. This proves the assertion
in the fixed-interval case. After taking a subsequence, we may therefore
restrict to targets whose covering interval indices tend to infinity.
There are then infinitely many starts, all with positive failure
parameter $b$.

\emph{Geometry of the late starts.}
We first show that the $b$ values of these covering intervals tend to
zero. Otherwise their start scales would be comparable to their target
scales, since
$B_*b\le L_i/R\le r_*/N_*$. The start varifolds would then converge
to planes, contradicting failed-fit separation at a scale bounded
away from zero.

In fact the start varifolds converge to multiplicity-$Q$ planes even
without comparability to the targets. Any subsequential limit is a
stationary tangent cone at $0$, of density $Q$. Pass also to limits of
the start coordinates and their uniformly controlled center graphs.
Because $t(w)\le b+|w|$ and $b\to0$, the limiting cone lies within
$C|w|^k$ of a $C^1$ graph through $0$. Every conical ray is therefore
in the tangent plane of that graph at $0$. Constancy gives a
multiplicity-$Q$ plane. Thus, if a minus sign denotes the preceding
interval, eventually
\begin{equation}\label{eq:exact-restart}
 R=Fb_-R_-.
\end{equation}
Indeed the right-hand side is dyadic. If it exceeded the chosen $R$,
then $2R$ would still be eligible; planar convergence of $V_R$ on
every fixed ball makes $2R$ good, contradicting maximality. It follows
also that $b_-\to0$. If not, the preceding starts would be comparable
to the current planar starts, again contradicting separation.

\emph{The outer endpoint: the preceding failure.}
We now work entirely in the current start's units. The preceding failure
will give an upper support-height bound and a matching lower height
integral on fixed room. Together these bounds will control $P(r_*)$.
Define the preceding failure height in current units by
\[
 h_0=\frac{m_0b_-^k}{Fb_-}.
\]
In every required fixed bounded region of the current start, all
support points are within $Ch_0$ of the preceding center, expressed
in current units: in preceding units the region has radius $O(Fb_-)$,
and \eqref{eq:glued-center} applies with $t\le b_-+|w|$.
The preceding and current start planes make a small angle. To verify
this, take a subsequence: the preceding centers in current units have
small slope and pass within $Ch_0$ of $0$, and the current planar
support limit lies in their limiting graphs. The current chosen
planes converge to this support plane. This proves the asserted angle
bound, allowing a fixed enlargement of the small slope threshold.

The current center consequently satisfies
\begin{equation}\label{eq:restart-height-upper}
 |z|\le Ch_0
\end{equation}
on the region containing $\{\rho<4r_*\}$. Indeed a current witness
of scale $\ell$ supplies two entries separated by $cm_0\ell^k$ in
a direction nearly vertical also for the preceding plane. Both lie
within $Ch_0$ of its small-slope center, so $cm_0\ell^k\le Ch_0$.
Equation~\eqref{eq:glued-center} now proves
\eqref{eq:restart-height-upper}. By Lemma~\ref{lem:weighted-bounds}
at radius $4r_*$, the mean curvature and its first normal derivative
are at most $Ch_0$ there as well. Every comparison is internal to both
cylinders: the current cylinder has fixed room, whereas its required
region in preceding units shrinks with $Fb_-\to0$.

A preceding failure attaining $b_-$ supplies the complementary bound
\begin{equation}\label{eq:restart-height-lower}
 H(r_*)\ge ch_0^2.
\end{equation}
Its center is within $16b_-$ of horizontal $0$, so the failed-fit
lemma may be used on a subball centered near horizontal $0$.
The entries lie within ambient radius $Cb_-$ in preceding units:
their heights over the fit are $O(m_0b_-^k)$, and the glued center
has the same bound at $0$. In current units they are well inside
$\{\rho<r_*\}$ by \eqref{eq:restart-factor}; their mass is bounded
below by a constant times $F^{-m}$. Their tilt remains small relative
to the current center. At least one entry of each separated pair has
height at least $ch_0$ over that center, proving
\eqref{eq:restart-height-lower}.

The cutoff normal test with $z$ on fixed room around $r_*$, using
\eqref{eq:restart-height-upper} and its curvature bound, gives
$E_*(r_*)+R_*(r_*)\le Ch_0^2$. Hence
\eqref{eq:freq-absolute} and \eqref{eq:restart-height-lower} bound
$|P(r_*)|$ uniformly, while $T(r_*)\to\infty$.
In particular $P(r_*)\le T(r_*)^2$ for all sufficiently late covering
intervals, as required for the inward upper bound.

\emph{The inner endpoint: the current failure.}
For $b>0$ we use the same height-versus-tilt comparison at radius
$B_*b$, now supplied by the current failure. On a fixed enlargement,
$|z|\le Cm_0b^k$ follows from $t\le b+|w|$; the curvature bounds of
Lemma~\ref{lem:weighted-bounds} apply there. The cutoff tilt estimate
on room of scale $b$ gives
\[
 E_*(B_*b)+R_*(B_*b)\le Cm_0^2b^{2k}.
\]
The inverse-square scale in that estimate is canceled by the factors
in \eqref{eq:interval-quantities}. A failure attaining $b$, tested on
a subball of radius $10^{-2}b$ near horizontal $0$, supplies low-tilt
entries well inside the kernel and proves
$H(B_*b)\ge cm_0^2b^{2k}$. Thus $|P(B_*b)|$ is bounded as well.

The outer bound controls $P$ from above as the radius decreases.
The inner bound controls it from below: once $P<-K$, it decreases
inward, so a value below the inner-endpoint bound could not recover
before that endpoint. Lemma~\ref{lem:inward-control} therefore gives
\eqref{eq:uniform-frequency-range}. Its final smallness requirement
on $\eta$ is uniform on late intervals: use $\delta\le Cr$ on
the sufficiently small radii and
\eqref{eq:restart-height-upper} on the remaining radii.
This proves the uniform height comparison.
\end{proof}
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//04-fitting-frequency.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//05-centered-blowup.tex ====
\section{A centered height blow-up}\label{sec:centered-blowup}

The interval construction has supplied height comparison at the target
scales without changing the sequence that carries the density centers.
We now normalize height at those scales. The resulting centers have
mean curvature negligible at the height scale, and their first height
moments vanish. These two properties will allow both first variation
and the density comparison to pass to the limit.

\begin{proposition}[Centered blow-up at the target scales]
\label{prop:centered-blowup}
Let the stationary integral varifold, singular origin, integer $Q$,
plane $\Pi$, and sequence $L_i\downarrow0$ be as in
Proposition~\ref{prop:flat-scales}. After taking a subsequence and
identifying $\Pi$ with the horizontal plane, there are positive
$a_i\to0$ and $C^J$ graph disks $S_i$ over a fixed enlargement of
$B_8\subset\Pi$, converging to $\Pi$ through all derivative orders
needed below, with the following properties. In target units let
$\mu_i$ be the weight of $V_{L_i}$, let $\pi_i$ be nearest projection
to $S_i$, and set
\[
 z_i=X-\pi_iX,\qquad
 M_i=(\pi_i)_\#\mu_i-Q\vol_{S_i},\qquad
 D_i=(\pi_i)_\#(e_i\mu_i).
\]
All measures are restricted to one fixed interior normal cylinder
containing the support in question, with a support-free gap at its
vertical boundary. On the part over $B_8$,
\begin{equation}\label{eq:centered-interface}
 \begin{gathered}
 D_i+|M_i|=O(a_i^2)\quad\hbox{in total mass},
 \qquad |z_i|\le Ca_i\quad\hbox{on the support},\\
 \sup\bigl(|\mathcal H_{S_i}|+
                  |\nabla^\perp\mathcal H_{S_i}|\bigr)=o(a_i),
 \qquad
 a_i^{-1}(\pi_i)_\#(z_i\mu_i)\rightharpoonup0.
 \end{gathered}
\end{equation}
Moreover the normalized height-square measures have uniformly bounded
mass there and are quantitatively nonzero on a fixed compact subset
of $B_4$; in particular, for all sufficiently large $i$,
\begin{equation}\label{eq:nonzero-second-moment}
 c\le a_i^{-2}\int_{\pi_iX\in B_3}|z_i|^2\,\dd\mu_i
 \le a_i^{-2}\int_{\pi_iX\in B_8}|z_i|^2\,\dd\mu_i\le C.
\end{equation}
Balls in this statement are understood through the graph coordinates
over $\Pi$. The weak convergence in \eqref{eq:centered-interface}
holds against compactly supported tests in $B_8$.
\end{proposition}

\begin{proof}
We prove successively the integral normalization, negligible curvature,
uniform support height, and first-moment centering. In all intermediate
estimates we retain larger fixed interior regions than those in the
statement.

\subsubsection*{Integral normalization and convergence of the centers}
For the interval covering $L_i$, let $R$ be its start, use its center
$S$, and put
\begin{equation}\label{eq:height-normalization}
 r_i=L_i/R,\qquad a_i=\frac{\sqrt{H(r_i)}}{r_i}>0.
\end{equation}
The notation $R$ and $S$ here may depend on $i$. If $r_i\to0$,
then $a_i\to0$ by \eqref{eq:height-power-bounds} and
$\delta\le Cr_i^{k-1}$. If $r_i$ stays bounded away from zero,
we are in the multiple-interval case, and
\eqref{eq:restart-height-upper}, with $h_0\to0$, gives the same
conclusion. These cases also show that $\delta(cr_i)\to0$ for every
fixed enlargement used below. Subsequences suffice to handle them.

Proposition~\ref{prop:target-frequency} yields
\begin{equation}\label{eq:target-energy}
 \begin{aligned}
 r_i^{-m}\int_{\rho<200r_i}|z|^2\,\dd\mu
        &\le Ca_i^2r_i^2,\\
 r_i^{-m}D(B^S_{200r_i})&\le Ca_i^2,
 \end{aligned}
\end{equation}
where these expressions remain in start units. Indeed
\eqref{eq:height-doubling} supplies a radius in
$[400r_i,600r_i]$ at which $q_*$, and hence $n_*$, is bounded.
These radii lie below $r_*$ because $r_i\le r_*/1000$.
The weights at that radius are bounded below on
$\{\rho<200r_i\}$, so
\eqref{eq:weighted-height-comparisons} gives the height bound, and
the definition of $E_*$ gives the tilt bound. There is also a converse
height lower bound $ca_i^2r_i^2$ already on $\{\rho<2r_i\}$,
directly from $H(r_i)=a_i^2r_i^2$.

Dilate the center into target units and denote it by $S_i$.
Its graphs have uniform high derivative bounds on fixed balls: their
start derivatives are uniformly bounded and the dilation introduces
factors $r_i^{j-1}$ in derivative order $j$. Their heights at
horizontal $0$ before dilation are at most $Cm_0b^k$, and their slopes
are small. Since $b\le r_i/B_*$, the center positions remain bounded
after dilation. The sup height estimates show that support points on
each needed fixed region are at distance tending to zero from these
graphs in target units. For example, support points in an ambient
ball of radius $60$ have projected geodesic radius below $180$,
by the fixed projection bounds and the position of the radial center.
Support convergence to $Q|\Pi|$ therefore places a corresponding
patch of $\Pi$ in every subsequential graph limit. A small-slope graph
containing this patch must equal the plane there. Compactness with
extra derivative orders gives the asserted smooth-enough convergence.
Passing to a subsequence also fixes the limiting rotations of the start
coordinates. We may now regraph over $\Pi$ on smaller fixed room.

\subsubsection*{Curvature below the height scale}
We claim that, on a fixed region larger than the one in the statement,
\begin{equation}\label{eq:center-curvature}
 \sup\bigl(|\mathcal H_{S_i}|+
                 |\nabla^\perp\mathcal H_{S_i}|\bigr)=o(a_i).
\end{equation}
Temporarily return to start units. At a point with horizontal argument
of norm at most $40r_i$ and $t>0$, choose a witness $(x,\ell)$.
The argument for \eqref{eq:witness-size} gives
$\ell\le\max(b,40r_i/15)\le3r_i$.
Apply failed-fit separation on a small subball near that point; its
center lies within the allowed $30\ell$ of $x$. The subball, and
the changes between its fitting and normal projections, remain inside
the upper-bound region in \eqref{eq:target-energy}. Separation and
small slopes give height at least $cm_0\ell^k$ on mass at least
$c\ell^m$. Consequently
\begin{equation}\label{eq:witness-normalization}
 a_i r_i\ge cm_0\ell^k(\ell/r_i)^{m/2}.
\end{equation}
All such witness scales tend to zero uniformly. This is immediate
when $r_i\to0$, and otherwise follows from $a_i\to0$ in
\eqref{eq:witness-normalization}.

The pointwise start-unit curvature estimate is
$Cm_0^2\ell^{2k-4}$, by \eqref{eq:glued-center} and $t\le2\ell$.
Mean curvature and its first normal derivative acquire factors
$r_i$ and $r_i^2$, respectively, in target units. Thus, for mean
curvature, \eqref{eq:witness-normalization} gives
\[
 \frac{r_i m_0^2\ell^{2k-4}}{a_i}
       \le Cm_0r_i^{2+m/2}\ell^{k-4-m/2}\longrightarrow0.
\]
The derivative has one additional factor $r_i\le1$. The exponent of
$\ell$ is positive by \eqref{eq:fitting-order}. At $t=0$ both
curvature quantities vanish by \eqref{eq:glued-center}. This proves
\eqref{eq:center-curvature}.

\subsubsection*{A uniform bound for support height}
Integral control alone would not suffice for all later expansions.
We strengthen it to
\begin{equation}\label{eq:target-height}
 |z_i|\le Ca_i
\end{equation}
on fixed inner room. The normal-displacement trace formula
\cite[Equation~(7.9)]{AE} is
\begin{equation}\label{eq:normal-displacement-trace}
 P_T:Dz_i=e_i-\mathcal H_{S_i}\cdot z_i
       +O(|\II_{S_i}|^2|z_i|^2+|\II_{S_i}|e_i|z_i|).
\end{equation}
Since $\nabla|z_i|^2=2z_i$, the nonnegative ambient function
$G=|z_i|^2+a_i^2$ satisfies, on the support,
\begin{equation}\label{eq:height-subsolution}
 \tr(P_TD^2G)\ge-CG.
\end{equation}
Here the positive $2e_i$ term absorbs its small multiple in
\eqref{eq:normal-displacement-trace}, and
\eqref{eq:center-curvature} bounds the term linear in height by
$C(|z_i|^2+a_i^2)$.

For completeness, \eqref{eq:height-subsolution} has the requisite
mean-value consequence on a stationary varifold. Fix a support point
$X_*$, put $d=|X-X_*|$, and take a nonnegative decreasing smooth
radial kernel $f(d/l)$, supported in $d<l$ and positive at $d=0$.
Define
\[
 I(l)=l^{-m}\int f(d/l)G\,\dd\mu_i.
\]
Radial stationarity gives
\[
 \frac{\dd I}{\dd\log l}
 \ge l^{-m}\int f(d/l)\nabla_TG\cdot(X-X_*)\,\dd\mu_i.
\]
The discarded term is nonnegative because $f$ is decreasing.
Use the primitive
$F_l(d)=\int_d^\infty t f(t/l)\,\dd t$, whose gradient is
$-f(d/l)(X-X_*)$ and which satisfies
$F_l\le Cl^2f(d/l)$. Stationarity and
\eqref{eq:height-subsolution} bound the right-hand side below by
$-Cl^2I(l)$. Integration of this differential inequality down to zero, using
density at least one, bounds $G(X_*)$ by a fixed-radius integral
of $G$ around $X_*$. The integral is $O(a_i^2)$ by
\eqref{eq:target-energy} and the mass bound, proving
\eqref{eq:target-height}.

These tests may be made for all points with horizontal arguments in a
fixed ball of radius $25$ in target units, using small fixed ambient
balls. Their heights already tend to zero by the earlier sup height
estimate. To check that the normal coordinates include all relevant
support, in start units use $t(w)\le b+|w|$ and
\eqref{eq:glued-center}: on these arguments the heights are small
compared with $r_i$, and changes from horizontal to projected arguments
are smaller still by the small slope. Decreasing $r_*$ once fixes all
needed tube and chart margins. Regraphing over $\Pi$ preserves smaller
fixed interior regions.

\subsubsection*{Vanishing of the first height moment}
It remains to prove
\begin{equation}\label{eq:centered-first-moment}
 a_i^{-1}(\pi_i)_\#(z_i\mu_i)\rightharpoonup0.
\end{equation}
We first use horizontal tests and vertical graph-height differences in
the start coordinates. Take an arbitrary fixed smooth test of $w/r_i$,
supported in horizontal $B_{12r_i}$, so that its derivatives are bounded
at that scale. We do not yet pull tests back through the changing
centers. By convergence of their coordinate changes, this region
contains the supports of target tests on $B_8\subset\Pi$ for all
large $i$; coordinate conversion will be made after proving the
fixed-coordinate limit.
Cover $\{t>0\}$ there by balls of radii a small fixed multiple of
$t$ at their centers, selecting a disjoint smaller family and a
bounded-overlap enlargement. The Lipschitz property of $t$ gives a
partition of unity whose derivatives are bounded at the corresponding
cell scales. Height is zero off $\{t>0\}$.

On a cell choose a constructed fit of dyadic scale $L$ comparable to
$t$ and sufficiently large compared with it, as in the gluing
construction. Cells meeting the test support have centers in an
$O(r_i)$ region. More explicitly, choose the enlarged cell radius
at most $ct(w)$ with $c<1/10$. A cell meeting $B_{12r_i}$ has
\[
 |w|\le12r_i+ct(w)\le12r_i+c(b+|w|),
 \qquad |w|<14r_i,
\]
because $b\le r_i/100$. In particular $L\le Cr_i$ and these centers
lie strictly inside the $40r_i$ region used for
\eqref{eq:witness-normalization}. Applying the
first-moment estimate in Lemma~\ref{lem:local-fits} to the partitioned
test, whose required scaled derivatives are bounded, gives an error
at most
\begin{equation}\label{eq:cell-moment-error}
 CL^m m_0^2L^{2k-1}.
\end{equation}
This controls the height over the local fit. Its difference from the
glued center has height $Cm_0^2L^{2k-1}$, so the same bound controls
that remaining contribution, using the cell mass bound.

To compare \eqref{eq:cell-moment-error} with the target height scale,
choose a witness at the cell center. It satisfies $\ell\ge t/2$,
hence $L\le C\ell$, though no converse comparison is required.
Equation~\eqref{eq:witness-normalization} implies
\[
 \frac{m_0^2L^{2k-1}}{a_i r_i}
    \le Cm_0r_i^{m/2}\ell^{k-1-m/2}\longrightarrow0
\]
uniformly over the cells. Positivity of the exponent follows from
\eqref{eq:fitting-order}, and the witness scales tend uniformly to
zero as proved above. Independently, bounded overlap of the cells
and $L\asymp t$ give $\sum L^m\le Cr_i^m$. Summing
\eqref{eq:cell-moment-error} therefore gives an error
$o(a_i r_i^{m+1})$. This is exactly the vanishing first moment after
mass and height are expressed in target units.

Finally convert from vertical graph-height differences to normal
displacements. At first order this conversion is the normal projection
at the graph point, with uniformly bounded quadratic error. By
\eqref{eq:target-height} the vertical differences also have size
$O(a_i)$. The quadratic error is therefore $O(a_i^2)$, and test
arguments change by $O(a_i)$, which has the same harmless effect.
The coefficient matrices converge uniformly on the fixed chart.
The normalized first-moment measures have bounded total variation,
so their convergence against fixed smooth tests extends to these
converging tests and matrices. This proves
\eqref{eq:centered-first-moment}.

We can now collect the conclusions in target units. The tilt estimate
in \eqref{eq:target-energy} gives $D_i=O(a_i^2)$ on the required
room. The scalar estimate, Lemma~\ref{lem:scalar-input}, with
\eqref{eq:center-curvature} and \eqref{eq:target-height}, gives
$|M_i|=O(a_i^2)$ there as well. Its integer is $Q$, identified by
smooth averages and convergence to $Q|\Pi|$. If a fixed small physical
radius is needed for that estimate, a fixed dilation changes only
constants. The lower height bound after
\eqref{eq:target-energy}, originally in geodesic radius $2r_i$,
lies in graph-coordinate $B_3$ for all large $i$, because the centers
converge to $\Pi$ and their radial origins tend to $0$.
This proves \eqref{eq:nonzero-second-moment}. All calculations retained
enough room to select the fixed cylinder over $B_8$, with a vertical
cutoff in a support-free gap. Hence they concern the full local
varifold measure required in the statement, and the proposition follows.
\end{proof}
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//05-centered-blowup.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//06-linearized-measures.tex ====
\section{Linearized measures and signed excess}\label{sec:linearized}

The centered blow-up of Proposition~\ref{prop:centered-blowup} has small
height, small tilt, and negligible mean curvature at the height scale.
We now retain its second-order information as measures on the limiting
plane. This formulation allows tilt to concentrate: no compactness theorem
for a limiting collection of graphs is required.
A related use of measure-valued limits for stationary two-valued graphs
appears in the generalized-gradient Young measures and linear measure
solutions of Hirsch and Spolaor \cite[Sections~2--3]{HSgraphs}.
The measure equations needed here are derived directly below.

For a concrete model, let $u:\Omega\to\R^n$ be a smooth harmonic
map on an open subset of $\R^m$. Its squared height, height flux,
and gradient energy define the measures
\[
 \begin{aligned}
 \nu_u&=|u|^2\,\dd y,
 & (f_u)_a&=u\cdot\partial_a u\,\dd y,\\
 (\beta_u)_{ab}&=\partial_a u\cdot\partial_b u\,\dd y,
 & \mathfrak m_u&=\tfrac12|Du|^2\,\dd y,
 \end{aligned}
\]
where $1\le a,b\le m$. Their block matrix is a Gram matrix, and
harmonicity gives
$\nabla\nu_u=2f_u$, $\diver f_u=\tr\beta_u$, and
$\diver\beta_u=\nabla\mathfrak m_u$ in distributions.
Moreover $2\mathfrak m_u=\tr\beta_u$.
The proposition below retains these identities and block positivity,
with the last equality replaced by a one-sided inequality. The model
explains the roles of the measures; the actual limit is not assumed
to admit such a harmonic-map representation.

\begin{proposition}[The limiting measure equations]\label{prop:linearized}
Let $V_i$ be stationary integral $m$-varifolds converging locally to
$Q|\Pi|$, where $Q$ is a positive integer and $\Pi$ is an $m$-plane.
Let $S_i$ be reference graphs converging to $\Pi$ in every fixed finite
derivative order used below, on a normal cylinder with fixed interior
room over a connected disk $\Omega\subset\Pi$. The cylinders contain all
support used below and have a support-free gap at their artificial vertical
boundaries above each compact base subset. Let $\mu_i$ denote the weight
of $V_i$ restricted to this cylinder, $\pi_i$ nearest projection onto
$S_i$, and $z_i=X-\pi_iX$. All pushforwards below use this restriction.
Use the horizontal and vertical tangent blocks relative to $S_i$ to set
$B_i=I-(P_T)_{hh}$, $e_i=\tr B_i$,
$D_i=(\pi_i)_\#(e_i\mu_i)$, and
$M_i=(\pi_i)_\#\mu_i-Q\vol_{S_i}$.
Suppose $a_i\downarrow0$ and, on each compact subdisk with interior room,
\begin{gather*}
 |M_i|+D_i=O(a_i^2),\qquad |z_i|\le Ca_i\quad\hbox{on the support},\\
 |\mathcal H_{S_i}|+|\nabla^\perp\mathcal H_{S_i}|=o(a_i),
 \qquad a_i^{-1}(\pi_i)_\#(z_i\mu_i)\rightharpoonup0.
\end{gather*}
Assume also that $a_i^{-2}(\pi_i)_\#(|z_i|^2\mu_i)$ has a positive
uniform lower mass bound on a fixed compact subset of $\Omega$.
Identify the bases by their converging graph coordinates. After passage
to a subsequence there are Radon measures satisfying
\begin{equation}\label{eq:limit-measures}
\begin{aligned}
 a_i^{-2}(\pi_i)_\#(|z_i|^2\mu_i)&\rightharpoonup\nu,&
 a_i^{-2}(\pi_i)_\#(B_i\mu_i)&\rightharpoonup\beta,\\
 a_i^{-2}D_i&\rightharpoonup d_0=\tr\beta,&
 a_i^{-2}M_i&\rightharpoonup\mathfrak m,\\
 a_i^{-2}(\pi_i)_\#\bigl(z_i\cdot(P_T)_{vh}\,\mu_i\bigr)
   &\rightharpoonup f.
\end{aligned}
\end{equation}
Here $\nu\ne0$ and $d_0$ are nonnegative measures, $f$ is vector-valued,
$\beta$ is symmetric matrix-valued, and $\mathfrak m$ is signed. They obey
\begin{equation}\label{eq:limit-psd}
 \begin{pmatrix}\nu&f^{\mathsf t}\\ f&\beta\end{pmatrix}\ge0
\end{equation}
as a matrix-valued measure, and
\begin{equation}\label{eq:limit-identities}
 \nabla\nu=2f,\qquad \diver f=d_0,\qquad
 \int_\Omega\diver X\,\dd\mathfrak m
   =\int_\Omega DX:\dd\beta
 \quad (X\in C_c^\infty(\Omega;\R^m)).
\end{equation}
Finally, locally on $\Omega$,
\begin{equation}\label{eq:limit-sign}
             2\mathfrak m\le d_0,\qquad
             |\mathfrak m|\le C d_0.
\end{equation}
The data supplied by Proposition~\ref{prop:centered-blowup} satisfy these
hypotheses, with a fixed disk containing the closed unit ball of density
centers.
\end{proposition}

\begin{proof}
All estimates are on compact subdisks with a fixed enlargement. Graph
volume factors are included when measures are transported to $\Omega$,
and tensor components are taken in smoothly converging orthonormal frames.
The assumed mass bounds give weak compactness. The projection identity
\[
 (P_T)_{hv}(P_T)_{vh}=B_i-B_i^2\le B_i
\]
shows both that the flux measures have bounded mass and that the block
matrix before passage to the limit is positive semidefinite. This proves
\eqref{eq:limit-psd}; the lower second-moment bound gives $\nu\ne0$.

\subsection*{Vanishing geometric errors}

We record the horizontal first variation carefully, since the height
normalization may be much smaller than the curvature of the reference
graph. Suppress $i$ temporarily. For a tangent field $Y$ on $S$, let
$J_z=I-\mathcal S_z$, where $\mathcal S_z$ is the shape operator, and
lift $Y$ to the ambient field $J_zY$. The curved first-variation calculation
in \cite[Lemma~2.2]{AE} gives
\begin{equation}\label{eq:horizontal-error}
 P_T:D(J_zY)=\diver_S Y-B:\nabla^S Y+\mathcal E_Y,
\end{equation}
with the following sharper error in the present setting. Choose
$\varepsilon_i\to0$ dominating the fixed geometric derivatives of $S_i$
that tend to zero and such that
$|\mathcal H_{S_i}|+|\nabla^\perp\mathcal H_{S_i}|\le\varepsilon_i a_i$.
Then
\begin{equation}\label{eq:horizontal-error-bound}
 |\mathcal E_Y|\le C\varepsilon_i\left\{
 |z_i|e_i|\nabla^{S_i}Y|
 +(a_i|z_i|+|z_i|^2+\sqrt{e_i}\,|z_i|)|Y|\right\}.
\end{equation}
Indeed, in frames whose tangent and normal connections vanish at the
point of calculation, the horizontal derivative has blocks
\[
 D(J_zY)_{hh}=J_z(\nabla^SY)J_z^{-1}-\mathcal K_zJ_z^{-1},
 \qquad \mathcal K_z(a)=(\nabla_a\mathcal S)_zY.
\]
Conjugation preserves the trace of $\nabla^SY$, and its pairing with $B$
has error $O(\varepsilon_i|z|e|\nabla^SY|)$. Codazzi gives
$\tr\mathcal K_z=\langle\nabla_Y^\perp\mathcal H_S,z\rangle$.
The remaining horizontal terms are bounded by
$C\varepsilon_i(a_i|z|+|z|^2+e|z|)|Y|$.
The symmetric sum of the off-diagonal blocks is, when paired with a
tangent vector $a$ and a normal vector $v$,
\[
 \big\langle a,J_z^{-1}[\mathcal S_z,\mathcal S_v]Y\big\rangle.
\]
It is $O(\varepsilon_i^2|z||a||v||Y|)$; pairing with $(P_T)_{vh}$
costs at most $C\sqrt e$. This proves
\eqref{eq:horizontal-error-bound}, including in arbitrary codimension.

The corresponding normal formula, obtained from $D\pi=J_z^{-1}P_h$, is
\begin{equation}\label{eq:normal-trace}
 P_T:Dz=e-\mathcal H_S\cdot z
       +O(|\mathrm{II}_S|^2|z|^2+|\mathrm{II}_S|e|z|).
\end{equation}
This is the normal trace computation used in the minimal-fitting
argument of \cite[Equation~(7.9)]{AE}. At normalization $a_i^{-2}$, every
error in \eqref{eq:horizontal-error-bound} and
\eqref{eq:normal-trace} tends to zero in mass: use $|z_i|\le Ca_i$,
$\int e_i\,\dd\mu_i=O(a_i^2)$, and Cauchy--Schwarz for
$\int |z_i|\sqrt{e_i}\,\dd\mu_i$.

Stationarity tested with $J_{z_i}Y$ now yields the last identity in
\eqref{eq:limit-identities}; the reference-volume integral of a divergence
is zero. Testing instead with $|z_i|^2J_{z_i}Y$ gives
$\nabla\nu=2f$. Terms containing both $B_i$ and $|z_i|^2$ disappear,
and the differentiated height contributes
$2z_i\cdot(P_T)_{vh}J_{z_i}Y$; replacing $J_{z_i}$ by the identity has
vanishing normalized error. Finally, the normal test $\varphi z_i$,
with $\varphi$ a smooth base function, gives
$\int\varphi\,\dd d_0+\int\nabla\varphi\cdot\dd f=0$ by
\eqref{eq:normal-trace}. Thus all equations in
\eqref{eq:limit-identities} hold.

\subsection*{Applying signed excess at the height normalization}

It remains to prove \eqref{eq:limit-sign}. There are two requirements
for applying Theorem~\ref{thm:signed-input} at this normalization.
First, its geometric error $\Lambda_i$ must be $o(a_i^2)$; in target
units its height-square term is only $O(a_i^2)$. We achieve this by
changing physical units. Second, the physical chart radius and support
height must be small relative to the local excess budget. We meet that
condition by dividing the base into cells whose radii are powers of
$a_i$. These choices do not require a rate relating $a_i$ to the
original target scales.

Fix a compact test support and an interior enlargement. Dilate physical
space by $d_i'\downarrow0$ sufficiently slowly that all required finite
geometric derivatives remain bounded and
\[
 \frac{|\mathcal H_{S_i}|}{d_i'a_i}\longrightarrow0,
 \qquad
 \frac{|\nabla^\perp\mathcal H_{S_i}|}{(d_i')^2a_i}\longrightarrow0.
\]
Such a choice exists because each numerator divided by $a_i$ tends to
zero, and all the finitely many graph derivatives tend to their flat
values. To obtain a uniform normal tube after dilation, write the graph
as $g_i$ and extend it using a fixed smooth cutoff $\chi$ equal to one
on the enlargement. The new physical graph and its derivatives are
\[
 g_i^{\rm phys}(v)=d_i'(\chi g_i)(v/d_i'),\qquad
 D^j g_i^{\rm phys}(v)
   =(d_i')^{1-j}D^j(\chi g_i)(v/d_i').
\]
The cutoff graph tends to zero through all the required finite orders,
including order zero. Choosing $d_i'$ still more slowly keeps these
derivatives bounded and the second derivatives small, which provides
the fixed tube. The mean-curvature bounds are needed only on the
unchanged inner region. The cylinder restriction has a support gap at
its artificial vertical boundary, so this operation introduces no
first-variation term.

In the new physical units, the geometric measure $\Lambda_i$ charges
\[
 (d_i')^2|z_i|^2+(d_i')^{-2}|\mathcal H_{S_i}|^2
                  +(d_i')^{-4}|\nabla^\perp\mathcal H_{S_i}|^2.
\]
Expressed with mass returned to target units, it therefore satisfies
$\Lambda_i=o(a_i^2)$. The measures $M_i,D_i$ in target normalization
are unchanged. Thus the geometric error will disappear after division
by $a_i^2$; we now arrange the local hypotheses for signed excess.

Use $\zeta$ and $\kappa$ fixed in Equation~\eqref{eq:parameter-order},
and cover the test support by cells of
target radius $\lambda_i=a_i^\zeta$, with bounded overlap of their fixed
enlargements. In particular $3\kappa<\zeta$. On each enlarged cell use a normalized budget
$b_{i,j}$ for $|M_i|+D_i+\Lambda_i$, enlarged by fixed constants and
with floor $a_i^3$. Thus, uniformly in the cells,
\begin{equation}\label{eq:signed-cell-budget}
 a_i^3\le b_{i,j}\le C a_i^2\lambda_i^{-m}+a_i^3=o(1),
 \qquad \sum_j\lambda_i^m b_{i,j}=O(a_i^2).
\end{equation}
Fresh normal charts on these cells have physical radius
$O(d_i'\lambda_i)$ and physical support height $O(d_i'a_i)$.
Since $3\kappa<\zeta<1$, these are at most $b_{i,j}^{\kappa}$ for
all large $i$, with the fixed factors absorbed by strictness of the
inequality. All physical hypotheses of signed excess are satisfied.

Partition a nonnegative smooth test by a smooth partition of unity at
scale $\lambda_i$. After the fixed normalization of its size and
Lipschitz constant, the signed-excess error on a cell is
$o(1)b_{i,j}$, uniformly because $b_{i,j}\to0$ uniformly. Multiplying
by $\lambda_i^m$, summing \eqref{eq:signed-cell-budget}, and dividing
by $a_i^2$ proves $2\mathfrak m\le d_0$.

For the absolute bound, apply Lemma~\ref{lem:scalar-input}, from
\cite[Lemma~2.5]{AE}, in the same physical units on any fixed interior
ball and its enlargement. Smooth mass averages identify its integer as
$Q$ for all large $i$. Since $\Lambda_i=o(a_i^2)$, passage to the
limit gives, after harmless fixed changes of radii,
\[
       |\mathfrak m|(B_u(y))\le C d_0(B_{3u}(y)).
\]
This inflated-ball estimate implies domination as measures, without a
doubling assumption. For a compact $K\subset O\Subset\Omega$, cover
$K$ by sufficiently fine equal-radius balls whose triple enlargements
lie in $O$ and have bounded overlap. Summing gives
$|\mathfrak m|(K)\le C d_0(O)$. Outer regularity as $O$ decreases to
$K$, followed by inner regularity, proves $|\mathfrak m|\le C d_0$.
This completes the proof.
\end{proof}
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//06-linearized-measures.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//06b-density-centers.tex ====
\section{Density centers survive the height blow-up}\label{sec:density-centers}

The measures of Section~\ref{sec:linearized} retain height and tilt, but
the final dimension argument also needs the large set of density-$Q$
centers. Ordinary monotonicity about those centers is not accurate enough
at the height-square scale: a curvature error depending only on the
reference graph could dominate $a_i^2$. We remove that error by measuring
radius along the reference graph and compensating its volume density.

Fix the integer $p\ge4$ chosen above. For $x,y\in\Pi$ and $l>0$, set
\begin{equation}\label{eq:limit-kernels}
 \phi_{x,l}(y)=l^{-m}\left(1-\frac{|y-x|^2}{l^2}\right)_+^p,
 \qquad
 \psi_{x,l}(y)=2p\,l^{-m}
                  \left(1-\frac{|y-x|^2}{l^2}\right)_+^{p-1}.
\end{equation}

\begin{proposition}[Comparison at the density centers]
\label{prop:density-comparison}
In the setting of Proposition~\ref{prop:linearized}, suppose that
$\Omega$ contains a fixed neighborhood of $\overline B_1\subset\Pi$.
Let $\sigma_i$ be positive measures supported on density-$Q$ points of
$V_i$ in the ambient closed unit ball, with
\[
 c\le\sigma_i(\overline{\mathbf B}_1)\le C,
 \qquad \sigma_i(\mathbf B_r(X))\le Cr^s
 \quad(X\in\R^{m+n},\ r>0),\qquad s>m-1.
\]
Pass to a weak limit $\sigma$ along the subsequence defining the
linearized measures. Then $\sigma\ne0$, its support lies in
$\overline B_1\subset\Pi$, and it has the same upper ball bound.
There is $l_*>0$ such that
\begin{equation}\label{eq:center-comparison}
 2l^2\int\phi_{x,l}\,\dd\mathfrak m
       \ge\int\psi_{x,l}\,\dd\nu
 \qquad(x\in\spt\sigma, 0<l<l_*).
\end{equation}
\end{proposition}

\begin{proof}
Weak compactness and the stated conclusions about $\sigma$ follow from
the mass and ball bounds and from convergence of the varifold supports
to $\Pi$. We prove \eqref{eq:center-comparison} for each fixed
$0<l<l_*$, where $l_*$ is chosen so all the following constructions
have common interior room. All estimates below are uniform over
$X_0\in\spt\sigma_i$.

\subsection*{A radial test adapted to the reference graph}

Write $w=X_0-\pi_iX_0$ and translate the reference graph to
$S=S_i+w$, which passes through $X_0$. By the height bound,
$|w|\le Ca_i$. In this part of the proof, $\pi,z,B,e,J_z$ refer to this
translated graph, and $\rho$ is geodesic distance on $S$ from $X_0$,
pulled back by $\pi$. Translation preserves its geometric estimates.
Changing the reference tangent plane by this translation and the nearby
projection changes squared tilt by at most
\begin{equation}\label{eq:translated-tilt}
                  e\le C(e_i+a_i^2).
\end{equation}
Indeed the two projections differ by $O(a_i)$ and the second fundamental
forms are uniformly bounded; squared tilt is comparable to the squared
distance of the corresponding orthogonal projections.

Define
\[
 d^2=\rho^2+|z|^2,\qquad
 U=\rho\nabla_S\rho,\qquad
 G=\nabla(d^2/2)=J_z^{-1}U+z,\qquad Y=J_zU+z.
\]
These fields are smooth at the central fiber, and
$Y\cdot G=d^2$. Let $W$ be the inverse volume density in geodesic
normal coordinates on $S$ centered at $X_0$, again pulled back by
projection. Thus integration of $W\vol_S$ is ordinary integration in
those coordinates. Enlarging the sequence $\varepsilon_i\to0$ from
\eqref{eq:horizontal-error-bound} if necessary, geodesic coordinate
estimates give
\begin{equation}\label{eq:geodesic-compensation}
\begin{gathered}
 |\nabla_S W|\le C\varepsilon_i\rho,
 \qquad |\nabla^S U-\Id|\le C\varepsilon_i\rho^2,\\
 W(\diver_SU-m)+U\cdot\nabla_SW=0.
\end{gathered}
\end{equation}
The last identity is exact: $U$ is the coordinate radial field and $W$
is the reciprocal volume density.

Apply \eqref{eq:horizontal-error} to $U$ and add the normal trace
\eqref{eq:normal-trace}. Their leading tilt terms cancel. The identity
in \eqref{eq:geodesic-compensation} then cancels the reference-volume
error. The result is
\begin{equation}\label{eq:radial-error}
\begin{aligned}
 \mathcal E&=P_T:D(WY)-mW,\\
 |\mathcal E|&\le C\varepsilon_i\bigl\{
 e(\rho^2+|z|)+(a_i|z|+|z|^2)(1+\rho)
                         +\rho\sqrt e\,|z|\bigr\}.
\end{aligned}
\end{equation}
For completeness, pairing $\nabla^SU-\Id$ with $B$ costs
$C\varepsilon_i\rho^2e$. Differentiating $W$ in the horizontal part
gives $U\cdot\nabla_SW$ and another error of this size. Its vertical
pairing costs $C\varepsilon_i\rho\sqrt e\,|z|$. The remaining terms
are exactly those bounded in \eqref{eq:horizontal-error-bound} and
\eqref{eq:normal-trace}.

Put
\[
 \Phi_t(d)=\left(1-\frac{d^2}{t^2}\right)_+^p,
 \qquad \Psi_t(d)=2p\left(1-\frac{d^2}{t^2}\right)_+^{p-1},
 \qquad
 I_i(X_0,t)=t^{-m}\int W\Phi_t(d)\,\dd\mu_i.
\]
Differentiate in $\log t$ and test stationarity with
$W\Phi_t(d)Y$. Since $Y\cdot G=d^2$, this gives
\begin{equation}\label{eq:radial-derivative}
 \dot I_i(X_0,t)
 =t^{-m}\int\left\{
       \frac{\Psi_t}{t^2}W Y\cdot(I-P_T)G
                         +\Phi_t\mathcal E\right\}\,\dd\mu_i.
\end{equation}
The test is compactly supported within the normal tube; the cutoff power
$p\ge4$ gives more regularity than this calculation requires.

Here the estimate must hold down to arbitrarily small $t$, with no
comparison between $t$ and $a_i$. On $d\le t\le l_*\le1$,
\begin{equation}\label{eq:radial-normal-control}
\begin{aligned}
 Y\cdot(I-P_T)G&\ge\tfrac12|(I-P_T)G|^2
                              -C\varepsilon_i^2|z|^2t^2,\\
 |z|&\le |(I-P_T)G|+Ct\sqrt e.
\end{aligned}
\end{equation}
The first inequality follows from
$|Y-G|\le C\varepsilon_i|z|\rho$. For the second, take the vertical
component of $G$ and use $|P_vP_T|\le C\sqrt e$ and $|G|\le Ct$.
Set $b'=|(I-P_T)G|/t$. Then both $|z|\le t$ and
$|z|\le t(b'+C\sqrt e)$ hold. In particular,
\[
 e|z|\le te,\qquad
 a_i|z|\le Ct\bigl((b')^2+e+a_i^2\bigr),
 \qquad |z|^2\le Ct^2\bigl((b')^2+e\bigr).
\]
Using $\Phi_t\le C\Psi_t$, every adverse term in
\eqref{eq:radial-derivative} is therefore at most
$C\varepsilon_i t\Psi_t((b')^2+e+a_i^2)$.
Absorb the $(b')^2$ term into the positive term from
\eqref{eq:radial-normal-control}. Equation~\eqref{eq:translated-tilt}
and comparison of $d$ with ambient distance yield
\begin{equation}\label{eq:radial-almost-monotonicity}
 \dot I_i(X_0,t)\ge
 -C\varepsilon_i t^{1-m}
       \int_{\mathbf B_{Ct}(X_0)}(e_i+a_i^2)\,\dd\mu_i.
\end{equation}

\subsection*{Averaging over the centers}

Let $c_\Phi=\int_{\R^m}(1-|y|^2)_+^p\,\dd y$.
At the center $X_0$, the squared radius $d^2$ agrees with the squared
ambient distance to leading order and $W(X_0)=1$. Since $X_0$ has
density $Q$,
\[
                  \lim_{t\downarrow0}I_i(X_0,t)=Qc_\Phi.
\]
Take any bounded nonnegative continuous center weight $g$. Average
\eqref{eq:radial-almost-monotonicity} against $g\,\dd\sigma_i$ and
integrate in logarithmic radius. Fubini's theorem and the upper ball
bound for $\sigma_i$ give
\begin{align}
 &\int g(X_0)\bigl(I_i(X_0,l)-Qc_\Phi\bigr)\,\dd\sigma_i(X_0)
 \notag\\
 &\hspace{2em}\ge
 -C\|g\|_\infty\varepsilon_i
    \left(\int(e_i+a_i^2)\,\dd\mu_i\right)
                  \int_0^l t^{s-m}\,\dd t
 =-o(a_i^2).\label{eq:averaged-density}
\end{align}
The first integral is over fixed interior room and is $O(a_i^2)$.
The second is finite precisely because $s>m-1$. To justify the lower
endpoint, first integrate between positive radii, then use Fatou's
lemma for the nonnegative functions $gI_i$ and the displayed density
limit. This avoids any uniform rate of density convergence in $X_0$.

The averaged inequality now gives a lower bound with error $o(a_i^2)$.
To obtain the asserted comparison, we need an upper bound for the same
integrand in terms of the original mass defect and squared height.
Translating the reference graph introduces a cross term and an additional
negative square in this upper expansion. The vanishing first moment
removes the cross term, and the negative square can be discarded.

\subsection*{Expansion at a fixed radius}

Keep $l>0$ fixed. Write $y=\pi_iX$ and $z_i=X-y$. After translating
the new base back to $S_i$, its projection and residual are
\[
                   y'=\pi_i(X-w),\qquad z'=X-w-y'.
\]
All relevant support points satisfy $|z_i|+|z'|\le Ca_i$ in common
interior room. Functions such as
$q(y)=W(y+w)\Phi_l(\rho(y+w))$, and the analogous function with
$\Psi_l$, have uniform $C^2$ bounds and compact interior support.
We claim that, uniformly in the centers,
\begin{equation}\label{eq:translated-base-test}
             \int q(y')\,\dd\mu_i
               =\int q(y)\,\dd\mu_i+o(a_i^2).
\end{equation}
Indeed the nearest-projection formula and smooth convergence to a plane
give
\[
 y'-y=-P_h(y)w+o(a_i^2),\qquad
                  \sup_y|P_h(y)w|=o(a_i).
\]
For the first estimate, $D\pi_i(X)=J_{z_i}^{-1}P_h(y)$ differs from
$P_h(y)$ by $O(\varepsilon_i a_i)$, and the second derivative of
$\pi_i$ is $o(1)$. For the second, $w$ is normal to $S_i$ at the
center and all tangent planes tend uniformly to $\Pi$.
Taylor expansion thus leaves only
$-\int\nabla^{S_i}q\cdot P_hw\,\dd\mu_i$, up to $o(a_i^2)$.
Replacing the projected measure by $Q\vol_{S_i}$ costs $o(a_i^2)$
because $|M_i|=O(a_i^2)$. Integration by parts then uses
\[
             \diver_{S_i}(P_hw)=\mathcal H_{S_i}\cdot w=o(a_i^2)
\]
to prove \eqref{eq:translated-base-test}. Compact support eliminates
all boundary terms.

Taylor expansion in the squared normal height gives, uniformly on the
test region,
\[
 \Phi_l\bigl((\rho^2+|z'|^2)^{1/2}\bigr)
 =\Phi_l(\rho)-\frac{|z'|^2}{2l^2}\Psi_l(\rho)+o(a_i^2).
\]
The normalized reference-volume integral of $W\Phi_l(\rho)$ is
exactly $c_\Phi$, by its geodesic-coordinate definition. Hence
\eqref{eq:translated-base-test} expresses the zeroth-order term of
$I_i-Qc_\Phi$ as an integral against $M_i$, with error $o(a_i^2)$.

For the height term, $z'=z_i-w+o(a_i)$ uniformly. Put
$q_\Psi(y)=W(y+w)\Psi_l(\rho(y+w))\ge0$. Moving its argument
from $y'$ to $y$ costs $o(a_i^2)$ in the height-square integral,
and
\[
 \int q_\Psi|z_i-w|^2\,\dd\mu_i
 =\int q_\Psi|z_i|^2\,\dd\mu_i
   +|w|^2\int q_\Psi\,\dd\mu_i
   -2w\cdot\int q_\Psi z_i\,\dd\mu_i.
\]
This integral enters $I_i$ with the negative coefficient
$-l^{-m}/(2l^2)$, so discarding the $|w|^2$ term gives an upper
bound. The mixed term is $o(a_i^2)$ by the centering assumption.
We spell out its uniformity, since the centers move with $i$.
The vector measures
$a_i^{-1}(\pi_i)_\#(z_i\mu_i)$ have uniformly bounded total variation
and converge weakly to zero. At fixed $l$, the above center-dependent
tests and their smooth coefficient factors form a relatively compact
family in the uniform norm on common interior room. A finite
approximation of this family therefore makes the weak convergence
uniform over all its members. Multiplication by the bounded vectors
$w/a_i$ preserves this conclusion; these vectors need not converge.

It follows that, after division by $a_i^2$, the limiting upper bound
for $I_i(X_0,l)-Qc_\Phi$ at a center tending to $x\in\Pi$ is
\begin{equation}\label{eq:center-expression}
 F_l(x)=\int\phi_{x,l}\,\dd\mathfrak m
                  -\frac1{2l^2}\int\psi_{x,l}\,\dd\nu.
\end{equation}
More explicitly, let $\operatorname{pr}_{\Pi}$ denote orthogonal
projection onto the limiting plane. The same compact-test argument
applied to the bounded measure sequences in \eqref{eq:limit-measures}
gives
\begin{equation}\label{eq:uniform-center-upper}
 a_i^{-2}\bigl(I_i(X_0,l)-Qc_\Phi\bigr)
 \le F_l(\operatorname{pr}_{\Pi}X_0)+o(1)
 \quad\hbox{uniformly for }X_0\in\spt\sigma_i.
\end{equation}
The function $F_l$ is continuous.
Combining this uniform upper bound with
\eqref{eq:averaged-density}, and passing to the limit against any
nonnegative continuous $g$, gives $\int gF_l\,\dd\sigma\ge0$.
Continuity implies $F_l\ge0$ on $\spt\sigma$, which is
\eqref{eq:center-comparison}. Since $l\in(0,l_*)$ was arbitrary, the
comparison holds for every such radius.
\end{proof}
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//06b-density-centers.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//07-dimension-reduction.tex ====
\section{Dimension reduction for the limiting measures}
\label{sec:dimension-reduction}

We now separate the final argument from the varifold construction.
The preceding sections produced a nonzero height measure, its flux
and stress measures, and a density inequality at many centers. The
following proposition shows that these data cannot have more than
$m-1$ dimensions of such centers. Its proof uses a second blow-up:
the frequency becomes constant at several independent centers, and
equality in a positive matrix measure then forces the height to vanish.

Fix an integer $p\ge4$. For $x\in\R^m$ and $l>0$ write
\begin{equation}
 \label{eq:reduction-kernels}
 \phi_{x,l}(y)=l^{-m}\left(1-\frac{|y-x|^2}{l^2}\right)_+^p,
 \qquad
 \psi_{x,l}(y)=2p\,l^{-m}
       \left(1-\frac{|y-x|^2}{l^2}\right)_+^{p-1}.
\end{equation}
We retain the kernels of \eqref{eq:limit-kernels} in order to state
the measure-theoretic argument independently.

\begin{proposition}[Dimension reduction for height and stress]
\label{prop:measure-reduction}
Let $\Omega\subset\R^m$ be connected and open. Let $\nu$ be a
nonzero nonnegative Radon measure, $f$ an $\R^m$-valued Radon measure,
$\beta$ a symmetric matrix-valued Radon measure, and $\mathfrak m$
a signed Radon measure on $\Omega$. Suppose that
\begin{equation}
 \label{eq:abstract-psd}
 \begin{pmatrix}\nu&f^{\rm t}\\ f&\beta\end{pmatrix}
 \quad\hbox{is a positive semidefinite matrix-valued measure},
 \qquad d_0=\tr\beta,
\end{equation}
and that, in distributions,
\begin{equation}
 \label{eq:abstract-identities}
 \nabla\nu=2f,\qquad \diver f=d_0,\qquad
 \int\diver X\,\dd\mathfrak m
       =\int DX:\dd\beta
 \quad(X\in C_c^\infty(\Omega;\R^m)).
\end{equation}
Assume also that for every open $\Omega'\Subset\Omega$ there is a
constant $C_{\Omega'}$ such that, on $\Omega'$,
\begin{equation}
 \label{eq:abstract-sign}
 2\mathfrak m\le d_0,\qquad |\mathfrak m|\le C_{\Omega'}d_0.
\end{equation}
Let $Z\subset\Omega$ be a Borel set such that for each $x\in Z$,
\begin{equation}
 \label{eq:abstract-center}
 2l^2\int\phi_{x,l}\,\dd\mathfrak m
       \ge\int\psi_{x,l}\,\dd\nu
\end{equation}
for every sufficiently small $l>0$. Then
$\dim_{\Hh}Z\le m-1$.
\end{proposition}

All tensor inequalities in this statement are measure inequalities.
For instance, \eqref{eq:abstract-psd} means that contraction with
every constant vector in $\R^{m+1}$ gives a nonnegative measure.
Equivalently, the matrix density with respect to its trace measure is
positive semidefinite almost everywhere. This latter formulation also
allows contraction with bounded, point-dependent vectors.

\subsection{Kernel identities and positive height}

For the moment fix an arbitrary center $x\in\Omega$, not necessarily
in $Z$, and restrict to radii with
$\overline B_l(x)\subset\Omega$. Define
\begin{equation}
 \label{eq:abstract-frequencies}
 \begin{aligned}
 H(l)&=\int\psi_{x,l}\,\dd\nu,&
 A(l)&=l^2\int\phi_{x,l}\,\dd\mathfrak m,\\
 D(l)&=l^2\int\phi_{x,l}\,\dd d_0,&
 R(l)&=\int\psi_{x,l}(y)\,\dd\beta(y)[y-x,y-x].
 \end{aligned}
\end{equation}
Dots denote derivatives with respect to $\log l$. The distributional
identities imply
\begin{equation}
 \label{eq:exact-frequency-identities}
 \dot A=2A-D+R,\qquad
 D=\int\psi_{x,l}(y)(y-x)\cdot\dd f(y),\qquad
 \dot H=2D,\qquad RH\ge D^2,\qquad 2A\le D.
\end{equation}
Here is the calculation. The identities
\[
 \nabla\phi_{x,l}=-l^{-2}(y-x)\psi_{x,l},\qquad
 \dot\psi_{x,l}=-\diver\big((y-x)\psi_{x,l}\big)
\]
give the second and third formulas. In the stress identity use
$X=(y-x)\phi_{x,l}$; differentiating $A$ then gives the first
formula. Cauchy--Schwarz for the positive matrix measure gives
$D^2\le RH$, and the signed inequality gives $2A\le D$.
Although the kernels are compactly supported powers rather than
$C^\infty$ functions, their use is justified by approximation in
$C^2$ on a compact interior set. The choice $p\ge4$ provides that
regularity. All quantities and their displayed derivatives are
continuous in the admissible center and radius.

We need positive height before dividing by $H$. The next lemma is
the finite-radius continuation argument used in the failed-fit
analysis of AE; we give it in the measure form required here
\cite[Proof of Lemma~7.2]{AE}.

\begin{lemma}[No open zero-height region]
\label{lem:positive-limit-height}
For the nonzero measure $\nu$ on the connected set $\Omega$ in
Proposition~\ref{prop:measure-reduction}, equations
\eqref{eq:abstract-psd}--\eqref{eq:abstract-sign} imply $H(l)>0$
whenever $\overline B_l(x)\subset\Omega$ and $l>0$.
\end{lemma}

\begin{proof}
Suppose that $\nu$ vanishes on a nonempty open subset. Because
$\nu\ne0$ and $\Omega$ is connected, choose a point $x$ outside
$\spt\nu$ so close to its boundary that
\[
 l_0=\operatorname{dist}(x,\spt\nu)>0,
 \qquad \overline B_{2l_0}(x)\subset\Omega.
\]
Then $H(l)=0$ for $l\le l_0$ and $H(l)>0$ for
$l_0<l\le l_1$, where $l_1>l_0$ is fixed and interior. Put
$t=\log l$, $t_j=\log l_j$, and on $(t_0,t_1]$ define
\[
 P=A/H,\qquad n=D/H,\qquad q=R/H.
\]
By \eqref{eq:exact-frequency-identities},
\begin{equation}
 \label{eq:finite-radius-frequency}
 n\ge0,\quad n\ge2P,\quad q\ge n^2,\qquad
 \dot P=2P-n+q-2Pn
       \ge(n-1)(n-2P).
\end{equation}
For $P\ge1$ the right-hand side is nonnegative. Therefore, going
inward from $t_1$,
$P(t)\le\max\{1,P(t_1)\}$.

On $B_{l_0}(x)$, the distributional identities give $f=0$ and
$d_0=0$, and \eqref{eq:abstract-sign} gives $\mathfrak m=0$.
The compact kernels converge uniformly and vanish on the boundary
sphere, so $A(t)\to0$ as $t\downarrow t_0$, even if
$\mathfrak m$ charges that sphere. Moreover,
\[
 (e^{-2t}A)^{\displaystyle\cdot}
   =e^{-2t}(R-D)\ge-\tfrac14e^{-2t}H,
\]
because $q-n\ge n^2-n\ge-1/4$. Since $\dot H=2D\ge0$,
integration from $t_0$ yields
\[
 P(t)\ge-\frac{e^{2(t-t_0)}-1}{8}.
\]
Thus $P$ is bounded on this finite interval. Using $q\ge n^2$ in
the exact equation for $\dot P$ now gives
\[
 \dot P\ge\tfrac12 n^2-C.
\]
Integrating from $t_0+\varepsilon$ to $t_1$ bounds
$\int_{t_0+\varepsilon}^{t_1}n^2\,\dd t$ independently of
$\varepsilon$. In particular $n$ is integrable down to $t_0$.
But $(\log H)^{\displaystyle\cdot}=2n$, which prevents $H$ from
tending to zero there. This contradiction excludes every nonempty
open zero-height region. Since the kernel is positive in its open
ball, the stated conclusion follows.
\end{proof}

\subsection{A second blow-up at the centers}

\begin{proof}[Proof of Proposition~\ref{prop:measure-reduction}]
The preceding lemma makes $P(x,l)=A(x,l)/H(x,l)$ well defined at
every interior ball. In the harmonic-map model from
Section~\ref{sec:linearized}, this quotient is half the weighted
Dirichlet frequency
\[
 \frac{l^2\int\phi_{x,l}|Du|^2\,\dd y}
      {\int\psi_{x,l}|u|^2\,\dd y}.
\]
For a nonzero harmonic map homogeneous of degree $d$ about $x$, the
flux identity in \eqref{eq:exact-frequency-identities} makes this
frequency equal to $d$, so $P=d/2$. For the measure data, constant
frequency at the retained centers, together with the lower bound
$2P\ge1$ supplied by their density comparison, will force the radial
measure identity used below.

At a center in $Z$, the additional hypothesis
\eqref{eq:abstract-center} gives $2P\ge1$ for small radii. Hence
\eqref{eq:finite-radius-frequency} improves to
\begin{equation}
 \label{eq:center-frequency-monotonicity}
 2P\ge1,\qquad n\ge2P,\qquad
 \dot P\ge(n-1)(n-2P)\ge0.
\end{equation}
In particular the finite limit $P(x,0+)$ exists and is at least
$1/2$.

Suppose that $\dim_{\Hh}Z>m-1$, and choose
$m-1<s'<\dim_{\Hh}Z$. We may first restrict to centers in a compact
interior region for which \eqref{eq:abstract-center} holds on one
common interval $0<l<l_*$. To justify this restriction, exhaust
$\Omega$ by compact sets with positive boundary distance and impose
the comparison for all rational radii below $1/k$. The resulting
sets are Borel by continuity of the kernels, their union is $Z$, and
continuity in radius supplies the comparison at every radius. One
of them still has dimension greater than $s'$.

The compact-subset theorem and Egoroff's theorem now give a compact
set $K$ of positive finite $\Hh^{s'}$ measure on which
\begin{equation}
 \label{eq:uniform-center-frequency}
 P(y,l)\longrightarrow P(y,0+)\quad\hbox{uniformly as }l\downarrow0.
\end{equation}
Indeed first apply Egoroff at dyadic radii; monotonicity supplies the
same conclusion between them. Since each function $P(\cdot,l)$ is
continuous, its uniform limit is continuous on $K$. A further
compact restriction gives
\begin{equation}
 \label{eq:second-center-frostman}
 \Hh^{s'}(K\cap B_u(y))\le C u^{s'}.
\end{equation}
Choose $x\in K$ with positive upper $s'$-density, and radii
$l_j\downarrow0$ such that
$\Hh^{s'}(K\cap B_{l_j}(x))\ge c l_j^{s'}$.

We next normalize the measure tuple at these scales. Fix an open
neighborhood $\Omega'\Subset\Omega$ of $x$, so that the constant
in \eqref{eq:abstract-sign} is fixed on $\Omega'$. Put
$T_j(y)=(y-x)/l_j$ and $N_j=l_j^m H(x,l_j)$. On the expanding
domains $T_j(\Omega)$ define
\begin{equation}
 \label{eq:second-measure-normalization}
 \nu_j=N_j^{-1}(T_j)_\#\nu,\qquad
 f_j=l_jN_j^{-1}(T_j)_\# f,\qquad
 (\beta_j,d_j,\mathfrak m_j)
   =l_j^2N_j^{-1}(T_j)_\#(\beta,d_0,\mathfrak m).
\end{equation}
The identities and inequalities
\eqref{eq:abstract-psd}--\eqref{eq:abstract-sign} are invariant under
this normalization, and the new height at center zero and radius one
equals one.

Here are the local mass bounds needed to pass to a limit. At the
fixed center $x$, $P$ is bounded on $0<l<l_*/2$. The exact quotient
equation and $n\le\sqrt q$ therefore give
\[
 \dot P=q-(1+2P)n+2P\ge\tfrac12q-C.
\]
On every interval of radii with bounded ratio, this bounds
$\int q\,\dd\log l$. Since
$\dot H/H=2n\le2\sqrt q$, Cauchy--Schwarz gives two-sided
comparison of $H$ on the same interval.

To see the resulting mass estimates explicitly, fix $R>0$ and discard
finitely many $j$ so that all balls used below lie in $\Omega'$.
Comparison on the interval from $\min\{1,R\}l_j$ to
$\max\{1,4R\}l_j$ gives
$H(x,\tau l_j)\le C_RH(x,l_j)$ for $R\le\tau\le4R$.
The kernel $\psi_{x,2Rl_j}$ is at least $c_Rl_j^{-m}$ on
$B_{Rl_j}(x)$; hence
\[
 \nu(B_{Rl_j}(x))\le C_Rl_j^mH(x,l_j).
\]
The integrated bound for $q$ on $[2Rl_j,4Rl_j]$ supplies
$\tau_j\in[2R,4R]$ with $n(x,\tau_jl_j)\le C_R$.
The kernel $\phi_{x,\tau_jl_j}$ is also at least
$c_Rl_j^{-m}$ on $B_{Rl_j}(x)$, so the definition of $D=nH$ gives
\[
 d_0(B_{Rl_j}(x))\le C_Rl_j^{m-2}H(x,l_j),\qquad
 \nu_j(B_R)+d_j(B_R)\le C_R.
\]
Positivity bounds $\beta_j$ by its trace $d_j$, and
Cauchy--Schwarz for \eqref{eq:abstract-psd} bounds $f_j$.
Finally, $|\mathfrak m_j|\le C_{\Omega'}d_j$ on every fixed ball
for all sufficiently large $j$. In particular the same constant
is available on all fixed balls of the eventual limit.

Pass to a diagonal subsequence with local weak limits on $\R^m$,
and denote them by
\[
 \nu_\infty,\quad f_\infty,\quad\beta_\infty,\quad
 d_\infty,\quad\mathfrak m_\infty.
\]
They satisfy the same measure identities and
inequalities. The normalization gives
$H_\infty(0,1)=1$, so $\nu_\infty\ne0$ and
Lemma~\ref{lem:positive-limit-height} applies on all of $\R^m$.

The rescaled center measures
\[
 \lambda_j=l_j^{-s'}(T_j)_\#
       \big(\Hh^{s'}\!\llcorner(K\cap\overline B_{l_j}(x))\big)
\]
have a nonzero weak limit $\lambda$ on $\overline B_1$, with the
upper ball bound \eqref{eq:second-center-frostman}. Consequently
$\spt\lambda$ spans $\R^m$. Otherwise its bounded support lies in
an $(m-1)$-dimensional linear space and can be covered by
$O(\varepsilon^{-(m-1)})$ balls of radius $\varepsilon$. The ball
bound would give
$\lambda(\R^m)\le C\varepsilon^{s'-m+1}\to0$, contradicting
its nonzero mass.

We have now retained enough centers in the second blow-up. Its
remaining feature is that they all have the same constant frequency.
If $z\in\spt\lambda$, choose $y_j\in K$ with
$T_j(y_j)\to z$. At every fixed radius $r>0$,
\[
 P_j(T_j(y_j),r)=P(y_j,l_jr)
       \longrightarrow P(x,0+)=:P_0.
\]
This follows from \eqref{eq:uniform-center-frequency} and continuity
of the limiting frequency on $K$. Local mass bounds and continuous
compact kernels permit passage to the weak limits even with these
moving centers. Since the limiting height is positive,
\begin{equation}
 \label{eq:constant-frequency-centers}
 P_\infty(z,r)=P_0
 \quad(z\in\spt\lambda\cup\{0\},\ r>0),
 \qquad P_0\ge\tfrac12.
\end{equation}
The comparison inequality also passes to these centers.

\subsubsection*{Equality at independent centers}

Fix $z\in\spt\lambda\cup\{0\}$. In the limit,
\eqref{eq:center-frequency-monotonicity} and
\eqref{eq:constant-frequency-centers} give
\[
 0=\dot P_\infty\ge(n-1)(n-2P_0)\ge0.
\]
As $n\ge2P_0\ge1$, this forces $n=2P_0=:n_0$, including the
case $P_0=1/2$. The exact quotient equation then gives
$q=n_0^2$, or equivalently $D=n_0H$ and $R=n_0^2H$.

Let $\mathcal M_\infty$ be the positive matrix measure in
\eqref{eq:abstract-psd} for the limiting data. Contract it with
$(-n_0,y-z)$ and integrate against $\psi_{z,r}$. The result is
\[
 n_0^2H-2n_0D+R=0.
\]
With respect to the trace measure, a positive semidefinite matrix
whose quadratic form vanishes on a vector annihilates that vector.
Since $\psi_{z,r}>0$ on $B_r(z)$, the first row therefore gives
\[
 (y-z)\cdot\dd f_\infty(y)=n_0\,\dd\nu_\infty(y)
 \quad\hbox{on }B_r(z).
\]
The radius is arbitrary, so this is an identity on $\R^m$.
Subtracting the identity for center zero yields
$z\cdot f_\infty=0$. The support of $\lambda$ spans $\R^m$,
and hence $f_\infty=0$. The identity at zero then gives
$n_0\nu_\infty=0$, contradicting $n_0\ge1$ and
$H_\infty(0,1)=1$. This proves the proposition.
\end{proof}

\subsection{Completion of the varifold argument}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Suppose that $\dim_{\Hh}\Sing(V)>m-1$, and choose
$m-1<s<\dim_{\Hh}\Sing(V)$. Proposition~\ref{prop:flat-scales}
gives the fixed singular point, flat target scales, and center
measures satisfying \eqref{eq:center-measure-bounds}.
Proposition~\ref{prop:centered-blowup} supplies the centered,
nonzero height normalization at these scales.
Proposition~\ref{prop:linearized} then gives on the connected base
ball $B_4\subset\Pi$ the nonzero measure tuple satisfying
\eqref{eq:limit-psd}, \eqref{eq:limit-identities}, and
\eqref{eq:limit-sign}. These are precisely the hypotheses
\eqref{eq:abstract-psd}--\eqref{eq:abstract-sign}.

Pass to a weak limit $\sigma$ of the center measures. It is nonzero,
is supported in $\Pi\cap\overline{\mathbf B}_1$, and retains the
upper ball bound with exponent $s$. The mass distribution principle
therefore gives
$\dim_{\Hh}\spt\sigma\ge s$.
Proposition~\ref{prop:density-comparison} supplies
\eqref{eq:center-comparison} at every point of $\spt\sigma$,
which is \eqref{eq:abstract-center}. Applying
Proposition~\ref{prop:measure-reduction} with
$Z=\spt\sigma$ gives
\[
 s\le\dim_{\Hh}\spt\sigma\le m-1,
\]
a contradiction.

For sharpness, take two distinct $m$-planes intersecting in an
$(m-1)$-plane, each with multiplicity one. Their sum is stationary
by additivity of first variation. Away from the intersection it is
locally a single smooth plane, whereas at the intersection its
support is not a smooth embedded $m$-submanifold. Its singular set
therefore has dimension exactly $m-1$.
\end{proof}
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//07-dimension-reduction.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//macros.tex ====
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==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//macros.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//main.tex ====
\documentclass[11pt]{article}
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\input{macros}
\title{A Codimension-One Bound for the Singular Set\\of a Stationary Integral Varifold}
\author{OpenAI}
\date{October 5, 2026}
\hypersetup{pdftitle={A Codimension-One Bound for the Singular Set of a Stationary Integral Varifold},pdfauthor={OpenAI}}
\begin{document}
\maketitle
\begin{abstract}
The singular set of every stationary integral $m$-varifold in an open
Euclidean set has Hausdorff dimension at most $m-1$, in every positive
dimension and codimension. This sharp bound proves the Euclidean
singular-set conjecture recorded by Brena, Decio, and De Lellis.
\end{abstract}
\input{sections/01-introduction}
\input{sections/02-flat-scales}
\input{sections/03-normal-inputs}
\input{sections/04-fitting-frequency}
\input{sections/05-centered-blowup}
\input{sections/06-linearized-measures}
\input{sections/06b-density-centers}
\input{sections/07-dimension-reduction}
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//main.tex ====

==== BEGIN A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//references.bib ====
@misc{AE,
  author = {{OpenAI}},
  title = {{Almost-everywhere regularity of stationary integral varifolds}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026/paper.pdf}{OAI:Almost-everywhere-regularity-of-stationary-integral-varifolds-September-23-2026}},
  year = {2026}
}

@article{Allard1972,
  author = {Allard, William K.},
  title = {On the first variation of a varifold},
  journal = {Annals of Mathematics. Second Series},
  volume = {95},
  number = {3},
  year = {1972},
  pages = {417--491},
  doi = {10.2307/1970868},
  url = {https://www.jstor.org/stable/1970868}
}

@incollection{Allard1987,
  author = {Allard, William K.},
  title = {Notes on the theory of varifolds},
  booktitle = {Th\'eorie des vari\'et\'es minimales et applications},
  series = {Ast\'erisque},
  number = {154--155},
  publisher = {Soci\'et\'e math\'ematique de France},
  year = {1987},
  pages = {73--93},
  url = {https://www.numdam.org/item/AST_1987__154-155__73_0/}
}

@misc{Simon2018,
  author = {Simon, Leon},
  title = {Introduction to Geometric Measure Theory},
  year = {2018},
  note = {Author's revised lecture notes, last updated March 7, 2018},
  url = {https://math.stanford.edu/~lms/ntu-gmt-text.pdf}
}

@book{Simon1983,
  author = {Simon, Leon},
  title = {Lectures on Geometric Measure Theory},
  series = {Proceedings of the Centre for Mathematical Analysis, Australian National University},
  volume = {3},
  publisher = {Australian National University, Centre for Mathematical Analysis},
  address = {Canberra},
  year = {1983},
  url = {https://openresearch-repository.anu.edu.au/items/7151bcb4-6ce1-4958-8576-049e09dc79b1}
}

@book{Federer1969,
  author = {Federer, Herbert},
  title = {Geometric Measure Theory},
  series = {Die Grundlehren der mathematischen Wissenschaften},
  volume = {153},
  publisher = {Springer-Verlag},
  address = {Berlin--New York},
  year = {1969},
  doi = {10.1007/978-3-642-62010-2}
}

@article{Federer1970,
  author = {Federer, Herbert},
  title = {The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension},
  journal = {Bulletin of the American Mathematical Society},
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  number = {4},
  year = {1970},
  pages = {767--771},
  doi = {10.1090/S0002-9904-1970-12542-3}
}

@book{Mattila1995,
  author = {Mattila, Pertti},
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  series = {Cambridge Studies in Advanced Mathematics},
  volume = {44},
  publisher = {Cambridge University Press},
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  doi = {10.1017/CBO9780511623813}
}

@book{Almgren2000,
  author = {Almgren, Jr., Frederick J.},
  title = {Almgren's Big Regularity Paper: {$Q$}-Valued Functions Minimizing {Dirichlet}'s Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension {$2$}},
  series = {World Scientific Monograph Series in Mathematics},
  volume = {1},
  publisher = {World Scientific},
  address = {River Edge, NJ},
  year = {2000}
}

@book{Brakke1978,
  author = {Brakke, Kenneth A.},
  title = {The Motion of a Surface by Its Mean Curvature},
  series = {Mathematical Notes},
  volume = {20},
  publisher = {Princeton University Press},
  address = {Princeton, NJ},
  year = {1978},
  url = {https://kenbrakke.com/papers/downloads/motionbook.pdf}
}

@article{NV2020,
  author = {Naber, Aaron and Valtorta, Daniele},
  title = {The singular structure and regularity of stationary varifolds},
  journal = {Journal of the European Mathematical Society},
  volume = {22},
  number = {10},
  year = {2020},
  pages = {3305--3382},
  doi = {10.4171/JEMS/987},
  url = {https://arxiv.org/abs/1505.03428}
}

@misc{HSgraphs,
  author = {Hirsch, Jonas and Spolaor, Luca},
  title = {Dimension of the singular set for {$2$}-valued stationary {Lipschitz} graphs},
  year = {2023},
  eprint = {2401.00279},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  note = {arXiv:2401.00279v1, December 30, 2023},
  url = {https://arxiv.org/abs/2401.00279v1}
}

@article{DLS2011,
  author = {De Lellis, Camillo and Spadaro, Emanuele Nunzio},
  title = {{$Q$}-valued functions revisited},
  journal = {Memoirs of the American Mathematical Society},
  volume = {211},
  number = {991},
  year = {2011},
  doi = {10.1090/S0065-9266-10-00607-1},
  url = {https://www.math.ias.edu/delellis/sites/math.ias.edu.delellis/files/memoDeSp.pdf}
}

@article{DLS2016II,
  author = {De Lellis, Camillo and Spadaro, Emanuele},
  title = {Regularity of area minimizing currents {II}: center manifold},
  journal = {Annals of Mathematics. Second Series},
  volume = {183},
  number = {2},
  year = {2016},
  pages = {499--575},
  doi = {10.4007/annals.2016.183.2.2},
  url = {https://arxiv.org/abs/1306.1191v5}
}

@article{DLS2016III,
  author = {De Lellis, Camillo and Spadaro, Emanuele},
  title = {Regularity of area minimizing currents {III}: blow-up},
  journal = {Annals of Mathematics. Second Series},
  volume = {183},
  number = {2},
  year = {2016},
  pages = {577--617},
  doi = {10.4007/annals.2016.183.2.3},
  url = {https://arxiv.org/abs/1306.1194v4}
}

@article{Menne2013,
  author = {Menne, Ulrich},
  title = {Second order rectifiability of integral varifolds of locally bounded first variation},
  journal = {Journal of Geometric Analysis},
  volume = {23},
  number = {2},
  year = {2013},
  pages = {709--763},
  doi = {10.1007/s12220-011-9261-5},
  url = {https://arxiv.org/abs/0808.3665v3}
}

@article{BDD2025,
  author = {Brena, Camillo and Decio, Stefano and De Lellis, Camillo},
  title = {Remarks and conjectures on stationary varifolds dedicated to {Enrico Bombieri}},
  journal = {Rendiconti Lincei. Matematica e Applicazioni},
  volume = {36},
  number = {3},
  year = {2025},
  pages = {409--443},
  doi = {10.4171/RLM/1077},
  url = {https://ems.press/journals/rlm/articles/14299484}
}

@misc{BDLF2025,
  author = {Brena, Camillo and De Lellis, Camillo and Franceschini, Federico},
  title = {{$C^\infty$} rectifiability of stationary varifolds},
  year = {2025},
  eprint = {2503.00649},
  archivePrefix = {arXiv},
  primaryClass = {math.AP},
  note = {arXiv:2503.00649v1},
  url = {https://arxiv.org/abs/2503.00649v1}
}

@misc{Kolasinski2026,
  author = {Kolasi{\'n}ski, S{\l}awomir},
  title = {Rectifiability of every finite order for stationary integral varifolds},
  year = {2026},
  eprint = {2609.19979},
  archivePrefix = {arXiv},
  primaryClass = {math.AP},
  note = {arXiv:2609.19979v1, September 17, 2026},
  url = {https://arxiv.org/abs/2609.19979v1}
}

@misc{DLHLS2026,
  author = {De Lellis, Camillo and Hirsch, Jonas and Lihn, Zachary and Spolaor, Luca},
  title = {Stationary varifolds with singularities {II}},
  year = {2026},
  eprint = {2609.20647},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  note = {arXiv:2609.20647v1, September 17, 2026},
  url = {https://arxiv.org/abs/2609.20647v1}
}

@misc{DLHS2026,
  author = {De Lellis, Camillo and Hirsch, Jonas and Spolaor, Luca},
  title = {Stationary and stable varifolds with singularities},
  year = {2026},
  eprint = {2509.21508},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  note = {arXiv:2509.21508v2, revised April 12, 2026},
  url = {https://arxiv.org/abs/2509.21508v2}
}

@misc{HS2025,
  author = {Hirsch, Jonas and Spolaor, Luca},
  title = {Measure {$0$} of the singular set for {$2$}-valued stationary hypercurrents},
  year = {2025},
  eprint = {2509.05544},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  note = {arXiv:2509.05544v1, September 6, 2025},
  url = {https://arxiv.org/abs/2509.05544v1}
}

@misc{KMW2026,
  author = {Krummel, Brian and Minter, Paul and Wickramasekera, Neshan},
  title = {A Branch Set Stratification for Stationary Varifolds with Epsilon-Regularity},
  year = {2026},
  eprint = {2606.01511},
  archivePrefix = {arXiv},
  primaryClass = {math.DG},
  note = {arXiv:2606.01511v1, June 1, 2026},
  url = {https://arxiv.org/abs/2606.01511v1}
}
==== END A-Codimension-One-Bound-for-the-Singular-Set-of-a-Stationary-Integral-Varifold//references.bib ====