\section{Global Witt, Chow, and Milnor--Witt vanishing}\label{global:section}
We now prove the first vanishing in \eqref{main:arithmetic_endpoints}.
The argument begins with Chow groups and the full Witt row. We then
improve Witt cochains successively by correcting their rank parities
and discriminants. These corrections remove the two-torsion in the
Milnor cohomology group. Finally, we make the Milnor and Witt
preimages compatible in the field pullback, and pass from the field
complex to the naive sheaf by support bounds. Let $R$ and $r$ satisfy the hypotheses of Theorem~\ref{main:quillen}, and put $A=R[1/r]$ and $S=R/(r)$. The ring $S$ is regular local of dimension three; it may have characteristic zero or positive characteristic. The arguments below retain the exceptional lines and the mixed-residue maps supplied by the supported hermitian theory.

\subsection{Field complexes and their normalizations}

For a regular scheme $Y$ write $C_{\mathrm M,n}(Y)$ for the field complex whose codimension-$p$ term is
\[
 C^p_{\mathrm M,n}(Y)=\bigoplus_{x\in Y^{(p)}}K^{\mathrm M}_{n-p}(\kappa(x)),
\]
with negative Milnor degrees zero. In the low degrees used here its maps are the Quillen-supported maps, identified with tame symbols, norms and divisors by the forward field symbol comparison. Write $C_{\mathrm W}(Y)$ for the full Witt row of homotopy-symmetric coniveau, and $C_{I,j}(Y)$ for the filtered row whose codimension-$p$ coefficient is $I^{j-p}(\kappa(x),\omega_x)$, with $I^a=W$ for $a\leq0$. Finally,
\[
 C^p_{\mathrm{MW},4}(Y)=
   \bigoplus_{x\in Y^{(p)}}K_{4-p}^{\mathrm{MW}}(\kappa(x),\omega_x).
\]
Here $\omega_x$ is the exceptional-duality line defined in Section~\ref{gw:supports}. All twists use the same supported duality and the same parameter orientation. The degree-three through degree-zero field comparisons in Theorem~\ref{gw:field_symbols} identify the positive-codimension terms relevant to $R$ and $A$ with the shifted hermitian support row.

By Lemma~\ref{gw:low_residues}, rank parity, the discriminant
$I/I^2\simeq K_1^{\mathrm M}/2$, and
$e_2:I^2/I^3\simeq K_2^{\mathrm M}/2$ commute with the corresponding
supported boundaries. We use these three residue squares below.

The regular divisor has short exact field-complex sequences
\begin{equation}\label{global:divisor_complex}
 0\longrightarrow C_{\mathrm M,n-1}(S)[-1]
  \longrightarrow C_{\mathrm M,n}(R)
  \longrightarrow C_{\mathrm M,n}(A)\longrightarrow0,
\end{equation}
and the analogous full-Witt sequence. The twist on the divisor is the exceptional line trivialized by $r$. The sequences are induced by localization of the support categories.

\subsection{The elementary Chow terms}

For any regular local ring $T$ used below, $\CH^2(T)=0$. Indeed $\CH^2(T)$ occupies $(2,-2)$ in the Quillen coniveau sequence
\begin{equation}\label{global:quillen_ss}
 E_1^{p,q}=\bigoplus_{x\in(\Spec T)^{(p)}}K_{-p-q}(\kappa(x))
       \Longrightarrow K_{-p-q}(T),
 \qquad d_s:(p,q)\longmapsto(p+s,q-s+1).
\end{equation}
Its only possible incoming higher map is $d_2$ from the unramified $K_1$ edge. That edge consists of $T^\times$ and extends to $K_1(T)$, so its classes are permanent. Outgoing terms have negative field $K$-degree. The limiting positive filtration in $K_0(T)=\Z$ is zero, giving the assertion. In particular
\begin{equation}\label{global:ch2}
 \CH^2(R)=\CH^2(S)=\CH^2(A)=0,
\end{equation}
where the assertion for $A$ follows by localization.

A regular parameter DVR curve through the closed point of a regular local ring supplies every terminal divisor coefficient. Consequently $\CH^4(R)=0$ and $\CH^3(S)=0$: the valuation of a uniformizer is one. The calculation takes place on the regular parameter curve.

There is also an integral bound
\begin{equation}\label{global:ch3_bound}
 2\CH^3(R)=0.
\end{equation}
To see this, $E_2^{3,-3}=\CH^3(R)$ has zero limiting piece in $K_0(R)$. The possible $d_3$ source is the extending unit edge; hence $d_2:E_2^{1,-2}\to\CH^3(R)$ is onto. On its source the integral supported Adams weight is two, whereas the target has weight three. The supported Adams operations of Part~II are defined for every positive integer $k$. Thus the target is killed by $k^3-k^2$ for every positive $k$. The values $4$ and $18$ at $k=2,3$ have greatest common divisor two, proving \eqref{global:ch3_bound}.

\subsection{Witt cochains and the third Chow group}

The codimension-three lifting theorem, Theorem~\ref{gw:codim3_lift}, gives
\begin{equation}\label{global:witt3_R}
 H^3(C_{\mathrm W}(R))=0.
\end{equation}
The divisor sequence gives $H^3(C_{\mathrm W}(R))\to H^3(C_{\mathrm W}(A))\to H^3(C_{\mathrm W}(S))$. Its rightmost group is a terminal residue cokernel on the regular local three-dimensional ring $S$. A regular parameter DVR curve lifts each diagonal residue form with coefficient one, so this cokernel vanishes in every characteristic. Therefore
\begin{equation}\label{global:witt3_A}
 H^3(C_{\mathrm W}(A))=0.
\end{equation}

For $Y=\Spec R$ or $\Spec A$, we first improve a full-Witt preimage
so that its codimension-two coefficients lie in $I^1$. Let $\alpha$
be a degree-three $C_{I,3}(Y)$ cocycle. By
\eqref{global:witt3_R}--\eqref{global:witt3_A}, choose a full-Witt
cochain $\beta$ whose boundary is $\alpha$. Its codimension-two rank parities form a cycle modulo two in the terminal term of $C_{\mathrm M,2}(Y)$. By \eqref{global:ch2}, choose a codimension-one unit cochain whose divisor has these parities. Lift each unit to a one-fold Pfister form. The rank/discriminant boundary square shows that its Witt boundary has the required parities. Subtracting this boundary from $\beta$ makes its coefficients
$I^1$-valued. Its boundary remains $d\beta$, since the differential
squares to zero. Hence
\begin{equation}\label{global:i3}
 H^3(C_{I,3}(Y))=0.
\end{equation}

We next show $\CH^3(R)/2=0$. The preceding filtered vanishing will
provide a preimage, but first we must lift the given cycle to a Witt
cocycle. Lift a degree-three cycle modulo two of $C_{\mathrm M,3}(R)$
to a Witt degree-three cochain $\alpha$ with the same rank parities. Its Witt boundary at the closed point need not initially vanish. The terminal map from the codimension-three $I^1$ terms to the closed-point Witt group is onto: on a regular parameter DVR curve, the form $\langle -u\rangle\langle1,-\pi\rangle=\langle -u,\pi u\rangle$ lies in $I^1$ and has residue $\langle\bar u\rangle$ with the fixed line orientation. Choose an $I^1$ cochain $\gamma$ with $d\gamma=d\alpha$. Then $\alpha-\gamma$ is a full-Witt cocycle with unchanged rank parities.

By \eqref{global:i3}, it is the boundary of a codimension-two
$I^1$ cochain. The discriminants of this cochain give a
codimension-two cochain of unit square classes. The low-degree boundary square identifies its divisor modulo two with the original degree-three cycle. Thus every such cycle is a boundary. Together with \eqref{global:ch3_bound} this proves
\begin{equation}\label{global:ch3}
 \CH^3(R)=\CH^3(A)=0.
\end{equation}
The second equality follows by Chow localization.

\subsection{Low-degree Milnor cohomology}

We now turn to the integral Milnor complex. By
Theorem~\ref{k2:thm:main}, every class in the unramified $K_2$ edge
of \eqref{global:quillen_ss} extends from the local ring. Such
classes are permanent cycles. We use this observation to prove
\begin{equation}\label{global:milnor_h2}
 H^2(C_{\mathrm M,3}(R))=
 H^2(C_{\mathrm M,3}(S))=
 H^2(C_{\mathrm M,3}(A))=0.
\end{equation}
For $R$, the group in question is $E_2^{2,-3}$. Its only higher
incoming differential is from the extending $K_2$ edge, and is
therefore zero. Its only possible outgoing target is a subquotient
of $\CH^4(R)=0$. Its limiting positive filtration in $K_1(R)=R^\times$ is zero. For $S$ there is no outgoing target. The assertion for $A$ follows from \eqref{global:divisor_complex}, because the following group is $H^2(C_{\mathrm M,2}(S))=\CH^2(S)=0$. Only field $K_2,K_1,K_0$ occur here, so Matsumoto's degree-two comparison suffices.

Set $B_Y=H^3(C_{\mathrm M,4}(Y))$. Then
\begin{equation}\label{global:milnor_bound}
 B_R\simeq B_A,\qquad 2B_R=2B_A=0.
\end{equation}
Indeed $B_R=E_2^{3,-4}$. The $d_3$ source is the extending unramified $K_2$ edge; the limit is zero in positive-coniveau $K_1(R)$. Hence $d_2:E_2^{1,-3}\to B_R$ is onto. Its source is a subquotient of codimension-one field $K_2$ cochains, of supported weight three; its target is a subquotient of codimension-three field $K_1$ cochains, of supported weight four. Consequently $k^3(k-1)B_R=0$. At $k=2,3$ the annihilators are $8$ and $54$, again with greatest common divisor two. The divisor sequence gives
\[
 H^2(C_{\mathrm M,3}(S))\longrightarrow B_R\longrightarrow B_A
                    \longrightarrow H^3(C_{\mathrm M,3}(S)),
\]
whose outer groups vanish by \eqref{global:milnor_h2} and $\CH^3(S)=0$. This proves the isomorphism.

To strengthen this exponent-two bound to vanishing, it remains to prove
that every class of $B_A$ is divisible by two. The filtered Witt
corrections below provide that divisibility.

We will also use
\begin{equation}\label{global:modtwo_h2}
 H^2(C_{\mathrm M,3}(Y)/2)=0\qquad(Y=\Spec R,\Spec A).
\end{equation}
Lift a degree-two cocycle modulo two to $c$ and write $dc=2a$ in the free degree-three cycle group. By \eqref{global:ch3}, choose $b$ with $db=a$. Then $c-2b$ is an integral cocycle, filled by \eqref{global:milnor_h2}. Reduction of its filler proves \eqref{global:modtwo_h2}. The point of the correction $c\mapsto c-2b$ is that its boundary
vanishes integrally. We can therefore apply the integral exactness
before reducing the resulting preimage modulo two.

\subsection{Filtered Witt cochains and Milnor two-torsion}

The next correction moves the degree-two preimage one step further
into the fundamental-ideal filtration. Let $\alpha$ be a degree-three
$C_{I,4}(Y)$ cocycle. Its coefficients lie in $I^1$, and
\eqref{global:i3} gives a degree-two $I^1$ cochain $\beta$ with
boundary $\alpha$. Its discriminants form a degree-two cocycle of $C_{\mathrm M,3}(Y)/2$, because the next boundary is the rank parity of $\alpha$ and is zero. By \eqref{global:modtwo_h2}, these discriminants are the boundary
of a degree-one $K_2/2$ cochain. Lift that cochain fieldwise to
$I^2$ forms. The $e_2$/discriminant residue square makes the boundary of this lift have the same discriminant as $\beta$. Subtract it. The resulting degree-two cochain has zero discriminant and hence
lies in $I^2$. Its boundary is still $\alpha$. Thus
\begin{equation}\label{global:i4}
 H^3(C_{I,4}(R))=H^3(C_{I,4}(A))=0.
\end{equation}
The residue squares for $e_1$ and $e_2$ give the two successive corrections.

On $A$ a degree-three cochain of $C_{\mathrm M,4}/2$ is a tuple of unit square classes and has no further boundary. Lift it to a tuple of $I^1$ forms. By \eqref{global:i4}, this tuple is the boundary of a degree-two
$I^2$ cochain. Its $e_2$ invariant has the prescribed unit-square-class boundary. Therefore
\[
 H^3(C_{\mathrm M,4}(A)/2)=0.
\]
As this is the final term of the complex, its cokernel is $B_A/2B_A$. Hence $B_A=2B_A$, and the exponent-two bound \eqref{global:milnor_bound} gives
\begin{equation}\label{global:milnor_h3}
 H^3(C_{\mathrm M,4}(A))=H^3(C_{\mathrm M,4}(R))=0.
\end{equation}
The equality for $R$ follows from \eqref{global:milnor_bound}.

\subsection{Compatibility of the Milnor and Witt preimages}

\begin{proposition}[The top Milnor--Witt field-complex vanishing]\label{global:mw4_field}
For $Y=\Spec R$ and $Y=\Spec A$, the twisted field complex satisfies $H^3(C_{\mathrm{MW},4}(Y))=0$.
\end{proposition}
\begin{proof}
The two projections of a Milnor--Witt cocycle can already be filled
separately. To obtain a Milnor--Witt preimage, their chosen preimages
must agree modulo two in the field pullback. We shall correct the
Milnor preimage by an integral cocycle, so that its boundary stays
fixed while its reduction acquires the required value.

Let $\alpha$ be a degree-three Milnor--Witt cocycle. Its Witt and Milnor projections have degree-two preimages under the
respective boundary maps,
\[
 \beta_I\in\bigoplus_{Y^{(2)}}I^2(\kappa(x),\omega_x),
 \qquad
 \beta_M\in\bigoplus_{Y^{(2)}}K_2(\kappa(x))
\]
by \eqref{global:i4} and \eqref{global:milnor_h3}. Their discrepancy modulo two is a degree-two cocycle $\gamma$ of $C_{\mathrm M,4}(Y)/2$, since their boundaries are the two coordinates of the same $\alpha$.

Choose an integral $K_2$ cochain $c$ lifting $\gamma$ and write $dc=2a$. On $R$, the equality $2da=0$ takes place in the free final group of codimension-four integers, so $da=0$. On $A$ the next group is absent. Equation~\eqref{global:milnor_h3} gives $b$ with $db=a$. Thus $c-2b$ is an integral degree-two cocycle with reduction $\gamma$. Subtract the resulting discrepancy cocycle from $\beta_M$.
Because the correction is an integral cocycle, the boundary of
$\beta_M$ does not change; its reduction now agrees with $e_2\beta_I$.

The field pullback, including characteristic two and the purity line,
\[
 K_2^{\mathrm{MW}}(E,\omega)
       =K_2^{\mathrm M}(E)\mathbin{\times}_{K_2^{\mathrm M}(E)/2}I^2(E,\omega),
\]
now supplies a degree-two Milnor--Witt cochain $\beta$. Its two boundary projections agree with those of $\alpha$. The corresponding degree-one field pullback detects equality, so $d\beta=\alpha$. Every boundary used is the compatible supported map of Lemma~\ref{gw:low_residues}.
\end{proof}

\subsection{From the field complex to the naive Zariski sheaf}

\begin{theorem}[The top cohomology of the naive sheaf]\label{global:naive_mw4}
For $A=R[1/r]$ as in Theorem~\ref{main:quillen},
\[
 H^3_{\Zar}(\Spec A,\mathcal K_4^{\mathrm{MW}})=0.
\]
\end{theorem}
\begin{proof}
We compare the naive sheaf with the point-sheaf complex whose global
sections were studied in Proposition~\ref{global:mw4_field}. The
augmentation need not be an isomorphism in every cohomological degree.
What matters here is that its error sheaves are supported in
codimensions large enough to make the comparison an isomorphism in
third cohomology.

\proofstep{1. Residue fields on the parameter open.}
First, every point of $X=\Spec A$ has infinite residue field. Its prime $\mathfrak p$ in the local ring $R$ is properly below $\fm$, which contains $r$. The local domain $R/\mathfrak p$ is therefore not a field. If its fraction field were finite, the domain itself would be a finite domain and hence a field, a contradiction. Also $\dim X\leq3$.

\proofstep{2. The point-sheaf complex.}
On the small Zariski site form the point-sheaf complex
\[
 \mathcal C^p=\bigoplus_{x\in X^{(p)}}i_{x*}
                   K_{4-p}^{\mathrm{MW}}(\kappa(x),\omega_x),\qquad0\leq p\leq3.
\]
Its global sections are the complex of Proposition~\ref{global:mw4_field}. Unit symbols have zero first residue, giving the natural augmentation $\mathcal K_4^{\mathrm{MW}}[0]\to\mathcal C^\bullet$. Its first differential is the DVR residue: the DVR image theorem, Lemma~\ref{loc:dvr_image}, expresses classes using unit symbols and uniformizer-times-unit symbols; the latter have the expected coefficient-one boundary by the unit-product formula and the degree-three field comparison.

\proofstep{3. Local cohomology of the complex.}
At the ordinary local stalk $T=\mathcal O_{X,x}$, the complex is
$C_{\mathrm{MW},4}(T)$. To see this, represent a cochain and any
relations involving it by finitely many field coefficients. They
descend to one common neighborhood of $x$. Passing to the stalk
removes the primes whose closures do not contain $x$, while the
purity lines and their boundary maps localize to those of $T$.

\proofstep{4. Supports of the comparison sheaves.}
Let $\varepsilon:\mathcal K_4^{\mathrm{MW}}\to\mathcal H^0(\mathcal C^\bullet)$ be the induced augmentation. The local results of Part~III give the following support bounds:
\begin{itemize}
\item $\ker\varepsilon$ is zero at the generic point;
\item $\coker\varepsilon$ is zero through codimension one, by the ordinary DVR \emph{image} theorem;
\item $\mathcal H^1(\mathcal C^\bullet)$ is zero through codimension two, by Corollary~\ref{loc:original_open}, which combines the surface and DVR results with the empty positive-degree complex at the generic point;
\item $\mathcal H^2(\mathcal C^\bullet)=0$, by Theorem~\ref{loc:mw4_h2} on every local ring of dimension at most three;
\item $\mathcal H^3(\mathcal C^\bullet)=0$: in a dimension-three
regular local ring $T$, a regular system of parameters $t_1,t_2,t_3$ gives the
DVR $T/(t_1,t_2)$, with the image of $t_3$ as uniformizer
\cite[Corollary~10.15 and Proposition~11.1]{Eisenbud1995}.
Its divisor is the closed point with coefficient one.
The supported unit-product residue formula of
Lemma~\ref{gw:low_residues}, with the compatible purity-line orientation,
therefore lifts every terminal unit-symbol coefficient.
Lower dimensions have no such term.
\end{itemize}
These stalkwise bounds concern the augmentation kernel and cokernel
and the cohomology sheaves of the point-sheaf complex.
They control the augmentation cone, while
Proposition~\ref{global:mw4_field} supplies the vanishing of the third
cohomology of the global field cochain complex. Both enter the final
comparison in Step~6.

\proofstep{5. The augmentation cone.}
By Lemma~\ref{bridge:support}, the cohomological dimension of $\ker\varepsilon$ is at most $2$, that of $\coker\varepsilon$ at most $1$, and that of $\mathcal H^1(\mathcal C^\bullet)$ at most $0$. Let $\mathcal E$ be the cone of the augmentation. Its only possibly nonzero cohomology sheaves are these three, in degrees $-1,0,1$.
For this bounded complex, the hypercohomology spectral sequence has
$E_2^{a,b}=H^a(X,\mathcal H^b(\mathcal E))$ and total degree $a+b$;
see \cite[Application~5.7.10]{WeibelHomological1994}.
The sheaf in degree
$-1$ has cohomology only through degree two, the sheaf in degree zero
only through degree one, and the sheaf in degree one only in degree
zero. Thus every possibly nonzero term has total degree at most one.
In particular,
\[
 \mathbb H^2(X,\mathcal E)=\mathbb H^3(X,\mathcal E)=0.
\]

\proofstep{6. Hypercohomology and global cochains.}
Each point pushforward in $\mathcal C^\bullet$ is flasque. The direct sums remain flasque here: a section is locally supported in finitely many summands, and a finite subcover of a Noetherian open makes that support finite. Restriction merely omits components. Hence $\mathbb H^3(X,\mathcal C^\bullet)$ is computed by its global field complex and is zero by Proposition~\ref{global:mw4_field}.
By the augmentation triangle and the two cone vanishings above,
$H^3(X,\mathcal K_4^{\mathrm{MW}})
\simeq\mathbb H^3(X,\mathcal C^\bullet)$.
This proves the asserted vanishing.
\end{proof}