\section{The first two integral Gersten positions in degree two}
\label{k2:sec:main}
We study the kernel of generic restriction and the obstruction to
extending an unramified generic class. The groups defined below measure
these two defects. Integral Adams operations give finite-exponent bounds;
divisibility and torsion-lifting arguments then show that both groups vanish.

Throughout this part, $K_i$ denotes Quillen algebraic $K$-theory,
$\KM_i$ denotes the ordinary unit-symbol Milnor group, and
$\Khat_i$ denotes improved Milnor $K$-theory.
For a Noetherian regular local domain $T$ with fraction field $F$, put
\begin{align*}
 K_2^{\mathrm{nr}}(T)&=\ker\left(K_2(F)\xto{\partial}
       \bigoplus_{\mathfrak p\in(\Spec T)^{(1)}}
                  \kappa(\mathfrak p)^\times\right),\\
 J_2(T)&=\ker\bigl(K_2(T)\to K_2(F)\bigr),\qquad
 P_2(T)=\coker\bigl(K_2(T)\to K_2^{\mathrm{nr}}(T)\bigr).
\end{align*}
Thus $J_2(T)$ measures the failure of generic injectivity, while
$P_2(T)$ measures the failure of an unramified generic class to extend
over $T$. The map defining $P_2(T)$ is well-defined by localization
at height-one primes. Each component of $\partial$ is the tame
boundary for the discrete valuation ring (DVR) $T_{\mathfrak p}$.
We use the convention $\partial\{u,\pi\}=\bar u$, with $u$ a unit
and $\pi$ a uniformizer. The opposite uniform sign gives the same
kernels and images.

\begin{theorem}[The first two integral Gersten positions]\label{k2:thm:main}
For every Noetherian regular local domain $T$, the sequence
\[
 0\longrightarrow K_2(T)\longrightarrow K_2(\Frac T)
 \xto{\partial}
 \bigoplus_{\mathfrak p\in(\Spec T)^{(1)}}
              \kappa(\mathfrak p)^\times
\]
is exact at its first two nonzero terms. Equivalently,
$J_2(T)=P_2(T)=0$.
\end{theorem}

The equal-characteristic case is the Quillen--Panin Gersten theorem
\cite[Theorem A]{Panin}. We prove the mixed-characteristic $(0,p)$ case by treating generic
injectivity and image purity separately.

\subsection{Integral Adams operations with support}
\begin{proposition}[Integral Adams operations with support]\label{k2:prop:supportadams}
Let $X$ be a finite-dimensional Noetherian regular scheme. On the
codimension-$c$ field term of the Quillen coniveau spectral sequence,
the integral Adams operation $\psi^k$, for an integer $k\geq1$, is
\[
       k^c\psi^k_{\kappa(x)}.
\]
The operation preserves the exact couple and agrees with the usual
operation on its abutment. In particular, on a field $K_j$ term for
$j=0,1,2$ its scalar is $k^{c+j}$.
\end{proposition}
\begin{proof}
Here $K_m^Z(C)$ denotes the $m$-th $K$-group of the category of perfect complexes over $C$
whose restriction to $\Spec C\setminus Z$ is acyclic.
Soul\'e constructs integral operations with closed supports and their
coniveau action in \cite[Sections 4.1--4.3 and Theorem 4(i),
pp.~521--522]{Soule}. We compute their action by localizing at $x$;
d\'evissage then identifies the local-support group with the
$K$-theory of $\kappa(x)$.

Fix $x\in X^{(c)}$ and an affine neighborhood $\Spec C$ of $x$.
Let $\mathfrak p\subset C$ be the prime corresponding to $x$.
The local-support group is the filtered colimit
\[
 \colim_{s\notin\mathfrak p}K_m^{V(\mathfrak p C_s)}(C_s)
       =K_m^{\{x\}}(C_{\mathfrak p}).
\]
The ideal $\mathfrak p$ is finitely generated. The open complement
of $V(\mathfrak p)$ has a finite principal-open cover. Continuity of perfect-complex
$K$-theory, finite Zariski descent, and the description of supported
$K$-theory as a fibre prove the displayed continuity statement; see
\cite[Proposition 3.20]{TT}. The supported operations are natural for
these restrictions.

The closed immersion
$\Spec\kappa(x)\to\Spec C_{\mathfrak p}$ is regular, with trivial
normal bundle of rank $c$. Apply the integral supported
Riemann--Roch formula of \cite[Theorem 3, p.~517; Lemma 2(i),
p.~515]{Soule}, taking the base scheme to be
$\Spec C_{\mathfrak p}$ itself. The ambient scheme and its closed
point are regular schemes of finite type over that base, satisfying
the hypotheses of the theorem. Under d\'evissage,
\[
 \psi^k(i_*a)=i_*\bigl(\theta^k(N)\psi^k(a)\bigr).
\]
For a trivial rank-$c$ normal bundle, the defining product formula for
$\theta^k$ gives $\theta^k(N)=k^c$ integrally. This computes the
operation on the field term. Its scalar on field $K_0,K_1,K_2$
follows respectively from rank, the power operation on units, and
Matsumoto symbols.
\end{proof}

For $T$ of dimension $d$ we use the finite coniveau spectral sequence
\begin{equation}\label{k2:eq:coniveau}
 E_1^{c,q}=\bigoplus_{x\in(\Spec T)^{(c)}}
 K_{-c-q}(\kappa(x))\ \Longrightarrow\ K_{-c-q}(T),
 \qquad d_s:(c,q)\mapsto(c+s,q-s+1).
\end{equation}
In total degree two, the generic term is $K_2(F)$ and its first
differential is the tame map $\partial$. Thus
$E_2^{0,-2}=K_2^{\mathrm{nr}}(T)$, while the terminal edge
$E_\infty^{0,-2}$ is the image of $K_2(T)\to K_2(F)$.
The first positive filtration stage is $J_2(T)$, and $P_2(T)$ is
the quotient of the unramified edge by the terminal edge image.
The filtration has length at most $d$.

We first obtain annihilators for the two defects. The same Adams
operation acts with different weights on the generic group and on the
higher-codimension terms that can obstruct its extension. Subtracting
these scalar actions gives a finite annihilator on each graded piece;
the finite length of the coniveau filtration then gives an annihilator
for the whole defect.

\begin{lemma}[Finite-exponent bounds for the two defects]\label{k2:lem:bounds}
There are nonzero integers
\[
 N_d=\prod_{c=1}^{d}4(2^c-1),\qquad
 M_d=\prod_{s=2}^{d}4(2^{s-1}-1)
\]
such that $N_dJ_2(T)=0$ and $M_dP_2(T)=0$.
An empty product is one.
\end{lemma}
\begin{proof}
Ordinary unit symbols generate $K_2$ of every local ring:
combine the ordinary-to-improved surjection with
$\Khat_2=K_2$ in \cite[Propositions 4.2.8(3), 4.3.3]{KerzThesis}.
Multiplicativity of Adams operations therefore gives
$\psi^2=4$ on $K_2(T)$, including for small finite residue fields.

The kernel $J_2(T)$ is the first positive coniveau filtration stage.
Its codimension-$c$ graded piece is a subquotient of field $K_2$ in
that codimension. Proposition~\ref{k2:prop:supportadams} gives scalar
$2^{c+2}$ there, whereas the same operation on the whole kernel has
scalar four. Thus $4(2^c-1)$ kills this graded piece. Multiplying the
finitely many factors proves the assertion for $J_2$.

The unramified edge $E_2^{0,-2}=K_2^{\mathrm{nr}}(T)$ has scalar four and no incoming
differentials. Its only possible outgoing higher maps have targets
\[
 E_s^{s,-s-1},\qquad 2\leq s\leq d.
\]
Each target is a subquotient of a codimension-$s$ field $K_1$ term,
with scalar $2^{s+1}$. Its differential image is consequently killed
by $4(2^{s-1}-1)$. The terminal edge is precisely the image of $K_2(T)\to K_2(F)$.
The descending filtration of $K_2^{\mathrm{nr}}(T)/\im(K_2(T)\to K_2(F))$ proves the second bound.
In dimension at most one it is simply DVR image exactness.
\end{proof}

\subsection{Prime-to-residue divisibility}
Every regular Noetherian semilocal domain is factorial.
Its height-one prime ideals are locally principal, hence invertible,
because its regular localizations are factorial.
Invertible modules over a semilocal ring are free of rank one, so
these prime ideals are principal; see \cite[Exercise~4.13,
Corollary~10.6, and Theorem~19.19]{Eisenbud1995}.
The definitions of $J_2$, $K_2^{\mathrm{nr}}$, and $P_2$ apply to these
semilocal domains as well. We use this slightly broader setting because
adjoining a root of unity may split the ring into several components.

\begin{lemma}[Prime-to-residue divisibility with roots of unity]\label{k2:lem:tamedivision}
Let $C$ be a regular Noetherian semilocal domain, and let $\ell$ be a
prime invertible in $C$. If $C$ contains a primitive $\ell$-th root of
unity, both $J_2(C)$ and $P_2(C)$ are $\ell$-divisible.
\end{lemma}
\begin{proof}
The norm-residue theorem first provides division modulo classes that
extend from the ring. We then correct the resulting generic torsion or
residues, so that the divided class belongs to the same kernel or
unramified subgroup as the original class.

Write $L=\Frac C$ and fix a primitive root $\zeta$.
Hoobler's natural norm-residue isomorphism
\cite[Theorem 3]{Hoobler} is
\[
 K_2(D)/\ell\cong H^2_{\et}(D,\mu_\ell^{\otimes2})
\]
for each semilocal ring $D$ in which $\ell$ is invertible.
The root untwists $\mu_\ell^{\otimes2}$ to $\mu_\ell$.
Kummer theory, $\Pic(C)=0$, and Brauer generic injectivity
\cite[Theorem 1.2]{Cesnavicius} therefore show that
$K_2(C)/\ell\to K_2(L)/\ell$ is injective.

Take $a\in J_2(C)$ and write $a=\ell b$ using that injection.
Then $b_L\in K_2(L)[\ell]$. The field torsion theorem
\cite[Theorem 14.2, pp.~331--332]{MerkurjevSuslin} expresses this as
$\{\zeta,u\}$. Sums of such symbols combine into one because
$\mu_\ell$ is cyclic. The class is integrally unramified. Its tame
residue at a height-one prime is $\bar\zeta^{v(u)}$, so every
valuation of $u$ is divisible by $\ell$. Factoriality gives
$u=\epsilon h^\ell$, with $\epsilon\in C^\times$.
The symbol $\{\zeta,\epsilon\}$ is defined over $C$ and is killed by
$\ell$ there. Subtracting it from $b$ gives $b_0\in J_2(C)$ with
$a=\ell b_0$. Its generic restriction is zero, while multiplication by
$\ell$ is unchanged. We have therefore divided $a$ within $J_2(C)$.

For $x\in K_2^{\mathrm{nr}}(C)$, DVR localization lifts $x$ over every height-one
local ring. Hoobler's theorem and the root untwisting identify its
mod-$\ell$ class with a Brauer class extending at each height-one
point. Brauer purity extends it over $C$; generic injection makes its
extension $\ell$-torsion. Kummer theory and $\Pic(C)=0$, followed by
Hoobler, supply $y\in K_2(C)$ with
$x-y_L=\ell z$ for some $z\in K_2(L)$.
The residues of $z$ are $\ell$-th roots of unity, with finite support.
Choose for each nontrivial component the corresponding global root
$\zeta_{\mathfrak p}$ and a generator $\pi_{\mathfrak p}$ of its prime.
The $\ell$-torsion symbol
$\{\zeta_{\mathfrak p},\pi_{\mathfrak p}\}$ has that residue and no
others. Subtract their sum from $z$. The corrected $z_0$ is
integrally unramified and $\ell z_0=\ell z$.
Thus $[x]=\ell[z_0]$ in $P_2(C)$.
\end{proof}

\begin{corollary}[Removal of prime-to-residue torsion]\label{k2:cor:primeaway}
If $T$ has mixed characteristic $(0,p)$ and $\ell$ is a prime
different from $p$, then
$J_2(T)=\ell J_2(T)$ and $P_2(T)=\ell P_2(T)$.
Consequently each of these groups is killed by a power of $p$.
\end{corollary}
\begin{proof}
The algebra $T'=T[z]/\Phi_\ell(z)$ is finite \'etale and free of rank
$\ell-1$. For $\ell=2$ it is $T$ itself. Its regular semilocal domain
components contain the required primitive root, which remains
primitive in every residue field. Apply
Lemma~\ref{k2:lem:tamedivision} componentwise.

Restriction and Quillen transfer preserve both generic kernels and
integral unramified subgroups, the latter by their compatibility with
DVR residues and norms. They also preserve the images of the ring
$K_2$ groups, so they act on the purity defects. Their composite is
multiplication by $\ell-1$ because $T'$ is free of that rank.
For $x$ in either group write $\res(x)=\ell y$ and put $z=\cor(y)$.
Then $(\ell-1)x=\ell z$ and $x=\ell(x-z)$, proving divisibility.

Factor either finite annihilator of Lemma~\ref{k2:lem:bounds} as
$p^a m$ with $(p,m)=1$. Repeated divisibility gives $G=mG$ for the
relevant group $G$, and $p^a mG=0$ then implies $p^aG=0$.
\end{proof}

\section{The ordinary local motivic edge in weight two}
\label{k2:sec:motivic}
We use Bouis's mixed-characteristic motivic complexes. Their weight-two
cohomology in degree two is denoted $H^2_{\mot}(C,\Z(2))$.

\subsection{The presentation coequalizer}
We use ordinary Milnor groups defined by unit symbols, multilinearity, and Steinberg relations. We denote the ordinary and improved Milnor groups by $\KM_n$ and $\Khat_n$, respectively.

The local resolution used to construct motivic cohomology gives a
coequalizer in the degree of interest. To compare it with unit symbols,
we must know that a symbol lifted to the resolution is independent of
its chosen unit lifts. The following lemma proves this directly from
the simplicial identities.

\begin{lemma}[The ordinary symbol coequalizer]\label{k2:lem:symbolcoequalizer}
Let $P_\bullet\to T$ be a simplicial resolution of a local ring by local rings. Suppose its augmentation and its face maps in degrees zero and one are local surjections. Then for every $n\geq1$,
\[
 \operatorname{coeq}\bigl(\KM_n(P_1)\rightrightarrows\KM_n(P_0)\bigr)
 \cong\KM_n(T).
\]
\end{lemma}
\begin{proof}
Let $C$ denote the coequalizer. Unit symbols lift along $P_0\to T$, so $C\to\KM_n(T)$ is onto. We construct its inverse by lifting the units of a symbol. If $u,v\in P_0^\times$ have the same image, then $u-v$ is in the image of $d_0-d_1$ on the additive group of $P_1$: this is the formula for $\pi_0$ of the underlying simplicial abelian group. Choose $z$ with $d_0z-d_1z=u-v$ and put
\[
 w=z+s_0(v-d_1z).
\]
Then $d_0w=u$ and $d_1w=v$. Since $d_0$ is local, $w$ is a unit. A symbol with $w$ in the chosen coordinate and degeneracies of the other lifted units in every other coordinate identifies the two lifts in $C$. This proves independence in every coordinate.

For a product of units choose the product of their lifts; this gives multilinearity. If $a$ and $1-a$ are units in $T$, any lift $\widetilde a$ has both $\widetilde a$ and $1-\widetilde a$ invertible, since the augmentation is local. The Steinberg relation therefore already holds upstairs. This construction is well-defined and gives the inverse of the natural map.
\end{proof}

\begin{theorem}[The integral weight-two edge for local rings]\label{k2:thm:alllocalq2}
For every local ring $T$, the natural multiplicative weight-two edge induces an isomorphism
\[
 q_{2,T}:K_2(T)\xrightarrow{\sim}H^2_{\mot}(T,\Z(2)).
\]
It agrees with the unit-symbol map and is natural in ring homomorphisms between local rings.
\end{theorem}
\begin{proof}
We first identify the edge when the residue field is large enough for
ordinary and improved Milnor theory to agree. The coequalizer then gives
surjectivity for every local ring. For a small residue field, injectivity
will descend from two finite extensions of coprime degrees.

\proofstep{1. The local weight-two edge.}
The local resolution in \cite[proof of Lemma 2.19]{BouisWeibel} is a simplicial resolution $P_\bullet\to T$ by local ind-smooth $\Z$-algebras, with henselian-surjective face maps. Its augmentation to $\pi_0(P_\bullet)=T$ is surjective and hence local. Every term has the same residue field as $T$. The local left-Kan-extension theorem and the displayed coequalizer there give
\begin{equation}\label{k2:eq:motcoequalizer}
 H^2_{\mot}(T,\Z(2))=
 \operatorname{coeq}\bigl(H^2_{\mot}(P_1,\Z(2))
 \rightrightarrows H^2_{\mot}(P_0,\Z(2))\bigr).
\end{equation}

On a local essentially smooth $\Z$-algebra the weight-two comparison is $H^2_{\mot}=K_2$ by \cite[Lemma 2.16]{BouisWeibel}; continuity gives the same on the ind-smooth terms. Its identification on products of weight-one unit classes is the ordinary $K_2$ symbol, by multiplicativity of the filtration. Kerz identifies improved Milnor $K_2$ with Quillen $K_2$ on every local ring and identifies ordinary with improved Milnor $K_2$ once the residue field is sufficiently large \cite[Proposition 4.2.8(3),(5)]{KerzThesis}. Hence for sufficiently large residue fields, \eqref{k2:eq:motcoequalizer} is the symbol coequalizer of Lemma~\ref{k2:lem:symbolcoequalizer}, proving the assertion.

\proofstep{2. Surjectivity of the local edge.}
For an arbitrary residue field, surjectivity follows from the coequalizer: the map from the $P_0$ term in \eqref{k2:eq:motcoequalizer} is onto, and its natural identification with $K_2(P_0)$ factors through $q_{2,T}$. It remains to prove injectivity for a small finite residue field $\F_{p^f}$.

\proofstep{3. Injectivity by coprime extensions.}
Choose two sufficiently large coprime positive integers $a,b$, each coprime to $f$. A monic lift to $\Z_{(p)}$ of an irreducible polynomial of degree $a$ over $\F_p$ gives a finite free local \'etale algebra $V_a$ with residue field $\F_{p^a}$. Its discriminant is a unit, so it is \'etale. Put
\[
 T_a=T\otimes_{\Z_{(p)}}V_a.
\]
Its closed residue algebra is $\F_{p^f}\otimes_{\F_p}\F_{p^a}=\F_{p^{fa}}$, a field. Every maximal ideal of this finite algebra lies over the maximal ideal of $T$, so $T_a$ is local. It is free of rank $a$, with large residue field. Construct $T_b$ similarly.

If $x\in\ker q_{2,T}$, its restriction to either extension is zero by naturality and the proved large-residue injectivity. Quillen transfer composed with restriction is multiplication by $a$, respectively $b$: the extension module is free of that rank, and the projection formula applies. Thus $ax=bx=0$, whence $x=0$. This proves the theorem for all local rings.
\end{proof}

\subsection{Finite coefficients and arithmetic constants}
For the remainder of this part, let $T$ be a Noetherian regular local
domain of mixed characteristic $(0,p)$, with fraction field $F$,
and put $Y=\Spec T[1/p]$.

\begin{lemma}[The finite-coefficient comparison]\label{k2:lem:finitecomparison}
The comparison gives natural isomorphisms
\[
 H^a_{\mot}(T,\Z/p(2))\cong H^a_{\et}(Y,\mu_p^{\otimes2}),\qquad a=0,1,
\]
and an injection in degree two. Generic restriction on the degree-one
group is injective.
\end{lemma}
\begin{proof}
A characteristic-zero domain is $\Z$-flat, and a regular Noetherian
ring is $F$-smooth by \cite[Theorem 4.15]{BhattMathew}.
The fibre in \cite[Corollary 5.12]{BouisMotivic}, applied in weight
two, has cohomology only in degrees at least three. Its long exact
sequence gives exactly these comparisons. A torsor under the finite \'etale group sheaf $\mu_p^{\otimes2}$ on the normal
integral scheme $Y$ which is generically trivial has a generic
section. The component selected by that section is finite birational;
normality makes it the base itself. The torsor is therefore trivial,
which proves generic injectivity on $H^1$.
\end{proof}

We will also use the natural weight-two edge
\begin{equation}\label{k2:eq:k3edge}
 q_{3,T}:K_3(T)\longrightarrow H^1_{\mot}(T,\Z(2)).
\end{equation}
The first two filtration triangles identify
$\pi_3\Fil^2_{\mot}K(T)$ with $K_3(T)$. Indeed, the weight-zero and
weight-one graded layers have vanishing $\pi_3$ and $\pi_4$.
For weight zero, this follows from \cite[Example~3.8]{BouisMotivic} and
\cite[Corollary~2.12]{BouisWeibel}; for weight one, it follows from
\cite[Example~2.15]{BouisWeibel}. The long exact sequences therefore
make the maps from the second filtration stage isomorphisms in the
degree under consideration.
Projection to the weight-two graded layer gives
\eqref{k2:eq:k3edge}. Multiplicativity and naturality of the filtration
identify its field version with the quotient
$K_3(F)\to K_3^{\ind}(F)=H^1_{\mot}(F,\Z(2))$.

\begin{lemma}[Lifting the arithmetic constants]\label{k2:lem:constants}
Restriction is surjective in
\[
 H^1_{\mot}(T,\Z(2))/p\longrightarrow
 H^1_{\mot}(F,\Z(2))/p.
\]
\end{lemma}
\begin{proof}
Represent a given field class over a finitely generated
characteristic-zero subfield $F_0\subset F$. Its algebraic constant
field $K$, the algebraic closure of $\Q$ in $F_0$, is a number field.
The finite-level clause of \cite[Theorem 4.13, p.~336]{Levine} gives
\[
 K_3^{\ind}(K)/p\xrightarrow{\sim}K_3^{\ind}(F_0)/p,
\]
including for $p=2$. Thus the class modulo $p$ comes from $K_3(K)$.

Every algebraic integer of $K$ is integral over $T$, because its monic
polynomial over $\Z$ is also over $T$. Normality therefore puts
$\mathcal O_K$ inside $T$. Contracting the maximal ideal gives a prime
$\mathfrak p_K$ over $p$, and locality extends the map to
$\mathcal O_{K,\mathfrak p_K}\subset T$.
DVR localization gives a surjection
\[
 K_3(\mathcal O_{K,\mathfrak p_K})\longrightarrow K_3(K),
\]
since its next term is $K_2$ of a finite field, zero by
\cite{QuillenFinite}. The resulting class of $K_3(T)$ maps under
\eqref{k2:eq:k3edge} to the required finite-level motivic class.
\end{proof}

The preceding two lemmas complement one another. A class killed by
$p$ in the integral generic kernel comes from
$H^1_{\mot}(T,\Z/p(2))$ by the Bockstein. The generic restriction
of that degree-one class can be removed using a lifted constant
class, after which the degree-one generic injectivity in
Lemma~\ref{k2:lem:finitecomparison} applies.

\begin{proposition}[Absence of residue-prime torsion in the kernel]\label{k2:prop:noptorsion}
The kernel of $H^2_{\mot}(T,\Z(2))\to H^2_{\mot}(F,\Z(2))$ has no $p$-torsion.
\end{proposition}
\begin{proof}
If $z$ belongs to that kernel and $pz=0$, the integral coefficient
sequence supplies $a\in H^1_{\mot}(T,\Z/p(2))$ with $\beta(a)=z$.
The generic class $a_F$ has zero Bockstein, so it is the reduction of
a class in $H^1_{\mot}(F,\Z(2))/p$. Lift the latter by
Lemma~\ref{k2:lem:constants} and subtract its reduction from $a$.
The Bockstein remains $z$, while the new class is generically zero.
Lemma~\ref{k2:lem:finitecomparison} makes that new class zero. Thus $z=0$, proving that the integral generic kernel has trivial
$p$-torsion.
\end{proof}

\begin{theorem}[Integral generic injectivity]\label{k2:thm:genericinjection}
For every Noetherian regular local domain of mixed characteristic,
$K_2(T)\to K_2(F)$ is injective.
\end{theorem}
\begin{proof}
By Corollary~\ref{k2:cor:primeaway}, $J_2(T)$ is killed by a power of
the residue prime $p$. The natural isomorphism $q_{2,T}$ of
Theorem~\ref{k2:thm:alllocalq2}, together with the field isomorphism
$q_{2,F}$, identifies this kernel with the corresponding motivic kernel.
Proposition~\ref{k2:prop:noptorsion} rules out $p$-torsion in it.
Successive multiplication by $p$ therefore kills no nonzero element,
and a group of bounded $p$-power exponent with this property is zero.
\end{proof}

\section{Image purity at the residue prime}
\label{k2:sec:image}
Generic injectivity settles $J_2(T)$; extension of unramified generic
classes requires the separate assertion $P_2(T)=0$.
Corollary~\ref{k2:cor:primeaway} gives $P_2(T)$ a bounded $p$-power
exponent. Any nonzero group with this property has a nonzero element
killed by $p$, so it suffices to prove $P_2(T)[p]=0$.
For odd $p$, we combine the divided-Chern calculation with unramified
torsion lifting. For $p=2$, the quotient comparison and the
Dennis--Stein lifts provide the corresponding steps.

\subsection{Root-free lifting of unramified torsion for odd primes}
\begin{lemma}[Unramified torsion at an odd residue prime]\label{k2:lem:oddtorsion}
Assume $p$ is odd. Every element of $K_2(F)[p]\cap K_2^{\mathrm{nr}}(T)$
extends to $K_2(T)$.
\end{lemma}
\begin{proof}
\proofstep{1. The torsion class.}
For $\tau$ in the indicated group, choose
$\theta\in H^1_{\et}(F,\mu_p^{\otimes2})=H^1_{\mot}(F,\Z/p(2))$ whose integral
Bockstein is $q_{2,F}(\tau)$. Such a choice exists by the coefficient
sequence and the field comparison.
Use the finite \'etale primitive-root torsor
\[
 Y'=\Spec\bigl(T[1/p][z]/\Phi_p(z)\bigr)\longrightarrow Y.
\]
Its group is $G=(\Z/p)^\times$, its degree is $p-1$, and it may be
disconnected. Work on each regular integral component with function
field $F'$ and tautological primitive root $\zeta$. Untwisting by
$\zeta$ writes $\theta_{F'}=\zeta\cup[u]$ for a Kummer class $[u]$.
Multiplicativity of the integral coefficient triangle gives
\begin{equation}\label{k2:eq:bocksteinsymbol}
       \beta(\zeta\cup[u])=\{\zeta,u\}.
\end{equation}
Here the weight-one Bockstein sends the finite-coefficient root to the
integral unit $\zeta$, and $[u]$ lifts integrally to a unit class.
This verifies the symbol normalization.

\proofstep{2. Extension of the Kummer class.}
At each height-one point of the component, the tame boundary of the
right side is $\bar\zeta^{v(u)}$. It is trivial since $\tau$ is
integrally unramified and finite \'etale restriction preserves this
property. The root remains primitive, so $p$ divides every valuation.
Write $\Div(u)=pD$. Regularity makes $D$ a Cartier divisor.
The Kummer pair consisting of $\mathcal O(D)$ and its $p$-th-power
trivialization defined by $u$ extends $[u]$ to the component of $Y'$;
see \cite[tag 03PK]{Stacks}. A change to the inverse divisor and
trivialization gives the same assertion under the opposite Kummer
convention.

\proofstep{3. Descent of the extended class.}
Generic restriction on this Kummer $H^1$ is injective, either by the
normal torsor argument or by the divisor description. Thus the
extension $\theta'$ of $\theta_{F'}$ is unique on all of $Y'$, and it is
$G$-invariant because its generic class comes from $F$. Group
cohomology in positive degree with $\F_p$-coefficients vanishes for
the group of order $p-1$. Hochschild--Serre therefore identifies
\[
 H^1_{\et}(Y,\mu_p^{\otimes2})=H^1_{\et}(Y',\mu_p^{\otimes2})^G.
\]
The descended class restricts to $\theta$: restriction to the generic
primitive-root torsor is itself injective by the same argument.
Lemma~\ref{k2:lem:finitecomparison} lifts it to
$\theta_T\in H^1_{\mot}(T,\Z/p(2))$.
Its integral Bockstein extends $q_{2,F}(\tau)$ in $H^2_{\mot}(T,\Z(2))$.
Surjectivity of $q_{2,T}$ lifts that Bockstein to $K_2(T)$, and the field
isomorphism $q_{2,F}$ shows that its image is exactly $\tau$.
The possible $\Pic(Y')[p]$ contribution was retained throughout.
\end{proof}

\subsection{A divided Chern class with the Picard term retained}
\begin{lemma}[A divided Chern class retaining the Picard term]\label{k2:lem:dividedchern}
Suppose $p$ is odd, $\alpha\in K_2^{\mathrm{nr}}(T)$, and $b\in K_2(T)$ satisfy
$b_F=p\alpha$. Then the coefficient reduction of $q_{2,T}(b)$ is zero in
$H^2_{\mot}(T,\Z/p(2))$.
\end{lemma}
\begin{proof}
We shall test the coefficient class after adjoining a primitive root
of unity. On that cover, the Chern class of the difference between the
ring class and the divided generic class is controlled by Adams
operations. Removing supports of codimension at least two makes this
calculation accessible; Picard invariance and Brauer injectivity then
bring the vanishing back to the whole cover.

\proofstep{1. The primitive-root cover.}
Consider the same degree-$(p-1)$ torsor $Y'\to Y$, and let $S$ be an
integral component, with function field $L$. Since all rational primes
other than $p$ are already units of $T$, the scheme $S$ is a regular
affine Noetherian $\Q$-scheme. The restricted class $\alpha_L$ is
integrally unramified at all its height-one points.

\proofstep{2. The normalized Chern map.}
For a quasi-compact open $V\subset S$ use the ordinary integral
higher Chern homomorphism, normalized on unit products,
\[
 c_V:K_2(V)\longrightarrow H^2_{\et}(V,\mu_p^{\otimes2}),
 \qquad c_V(\{u,v\})=[u]\cup[v].
\]
It is additive and obeys the integral identity
\begin{equation}\label{k2:eq:chernadams}
                  c_V(\psi^2x)=4c_V(x).
\end{equation}
For the product normalization see \cite[Section 2, equation (2.2),
Lemma 2.3, and Section 3]{WeibelChern}; for the integral Adams
identity and its finite \'etale coefficient examples see
\cite[Section 4, pp.~25 and 29]{Schneider}.
Schneider constructs these homomorphisms for smooth quasi-projective
schemes over a field (p.~14). The integral identity
$c_{i,j}\psi^m=m^j c_{i,j}$ on p.~25 applies to the root-of-unity
coefficient complexes listed on p.~29. A common sign change of
$c_{2,2}$ gives the displayed unit-product normalization and preserves
the Adams identity. Thus $c_V$ is a homomorphism from integral
$K$-theory to torsion cohomology.

The coordinate ring of $S$ is geometrically regular over the perfect
field $\Q$, so these maps extend to $S$ by continuity.
Popescu's theorem \cite[tag 07GC]{Stacks} expresses it as a filtered
colimit of smooth $\Q$-algebras. A quasi-compact open is a finite union
of principal opens and descends to a finite stage. Continuity of
$K$-theory \cite[Proposition 3.20]{TT} and of finite \'etale cohomology
\cite[Expos\'e VI, Theorem 8.7.3]{SGA4} carries the Chern maps,
naturality, and \eqref{k2:eq:chernadams} to these opens.

\proofstep{3. The codimension-one calculation.}
Put
\[
 C(S)=\colim_{\operatorname{codim}Z\geq2}K_2(S-Z),\qquad
 H(S)=\colim_{\operatorname{codim}Z\geq2}H^2_{\et}(S-Z,\mu_p^{\otimes2}).
\]
The colimits run over closed subsets of codimension at least two. Quillen localization gives
\begin{equation}\label{k2:eq:skeleton}
 \bigoplus_{P\in S^{(1)}}K_2(\kappa(P))\longrightarrow C(S)
 \longrightarrow K_2(L)\longrightarrow
 \bigoplus_{P\in S^{(1)}}\kappa(P)^\times .
\end{equation}
It is the exact localization couple after deeper supports are removed.
Let $J(S)=\ker(C(S)\to K_2(L))$. Exactness identifies this group with
the image of the first arrow. Proposition~\ref{k2:prop:supportadams}
therefore gives the scalar $\psi^2=2\cdot2^2=8$ on the whole kernel:
one factor comes from codimension one, and the other from field degree
two. The colimit removes all supports of codimension at least two. Passing the Chern maps to
it yields $8c(x)=c(\psi^2x)=4c(x)$ for $x\in J(S)$.
Since the target is killed by the odd prime $p$, this forces $c(x)=0$.

\proofstep{4. Vanishing of the divided Chern class.}
Lift $\alpha_L$ to $a\in C(S)$ by \eqref{k2:eq:skeleton}.
Then $b_S-pa\in J(S)$, so the image of $c_S(b_S)$ in $H(S)$ is zero.
We verify that $H^2_{\et}(S,\mu_p^{\otimes2})\to H(S)$ is injective.
The root identifies $\mu_p^{\otimes2}$ with $\mu_p$. Removing a closed subset of
codimension at least two from a regular scheme preserves Weil
divisors and principal divisors, hence its Picard group. In
\[
 0\longrightarrow\Pic(S)/p\longrightarrow H^2_{\et}(S,\mu_p)
       \longrightarrow\Br(S)[p]\longrightarrow0,
\]
a class killed on such an open has generically zero Brauer component.
Brauer generic injection kills that component already on $S$.
The remaining $\Pic(S)/p$ component is zero by Picard invariance.
Thus restriction to every such open, and therefore to the filtered
colimit, is injective. We conclude $c_S(b_S)=0$ on every component of
$Y'$.

\proofstep{5. The local motivic edge.}
Finally ordinary unit products generate $K_2(T)$, including for small
finite residue fields. Both the normalized Chern map and the
motivic-to-\'etale class of $b$ on $Y$ send these generators to the
same Kummer cup products. These computations identify the two global classes. Pullback to $Y'$ kills this class.
Trace composed with restriction is multiplication by $p-1$, invertible
on the $p$-torsion target. Thus the class is zero on $Y$.
The degree-two injection of Lemma~\ref{k2:lem:finitecomparison}
proves the asserted motivic vanishing.
\end{proof}

\begin{proposition}[Image purity at an odd residue prime]\label{k2:prop:oddimage}
For an odd residue prime $p$, one has $P_2(T)[p]=0$.
\end{proposition}
\begin{proof}
Represent a class killed by $p$ by $\alpha\in K_2^{\mathrm{nr}}(T)$ and choose
$b\in K_2(T)$ with $b_F=p\alpha$.
Lemma~\ref{k2:lem:dividedchern} and the integral coefficient sequence
give $q_{2,T}(b)=pm$ for some $m\in H^2_{\mot}(T,\Z(2))$.
Lift $m$ to $c\in K_2(T)$ using the surjectivity of $q_{2,T}$.
The field isomorphism gives $p(\alpha-c_F)=0$.
The difference remains integrally unramified, so
Lemma~\ref{k2:lem:oddtorsion} lifts it to $K_2(T)$.
Thus $\alpha\in \im(K_2(T)\to K_2(F))$.
\end{proof}

\subsection{The dyadic divided class and explicit torsion lifts}
\begin{lemma}[The dyadic quotient comparison]\label{k2:lem:dyadicquotient}
If $p=2$, the map $H^2_{\mot}(T,\Z(2))/2\to H^2_{\mot}(F,\Z(2))/2$ is injective.
\end{lemma}
\begin{proof}
Bouis's local vanishing \cite[Corollary 2.10]{BouisWeibel} gives
$H^3_{\mot}(T,\Z(2))=0$; the coefficient triangle identifies
$H^2_{\mot}(T,\Z(2))/2$ with $H^2_{\mot}(T,\Z/2(2))$.
Lemma~\ref{k2:lem:finitecomparison} injects this into
$H^2_{\et}(Y,\mu_2^{\otimes2})$.
The regular local ring $T$ is a unique factorization domain (UFD), and its localization is again a UFD, so
$\Pic(Y)=0$. The global root $-1$ untwists
$\mu_2^{\otimes2}$ to $\mu_2$ on $Y$, where two is invertible.
Kummer theory and Brauer generic injection give
\[
 H^2_{\et}(Y,\mu_2^{\otimes2})\cong\Br(Y)[2]
          \hookrightarrow\Br(F)[2]
          \cong H^2_{\et}(F,\mu_2^{\otimes2}).
\]
Naturality with the motivic coefficient sequence proves the claim.
\end{proof}

For the dyadic argument we also need to lift the torsion left after
dividing a ring class. At primes above two, the usual tame residue
condition does not force an even valuation. The Dennis--Stein symbols
in the next proof supply the required integral lifts at exactly those
primes.

\begin{lemma}[Dyadic two-torsion lifts]\label{k2:lem:dyadictorsion}
In mixed characteristic $(0,2)$ every class in
$K_2(F)[2]\cap K_2^{\mathrm{nr}}(T)$ has a lift in $K_2(T)[2]$.
\end{lemma}
\begin{proof}
The field torsion theorem writes the class as $\tau=\{-1,u\}$.
Factor in the UFD $T$:
\[
       u=\epsilon\prod_{\mathfrak p}\pi_{\mathfrak p}^{n_{\mathfrak p}},
       \qquad \epsilon\in T^\times.
\]
If $2\notin\mathfrak p$, the unramified tame value
$(-1)^{n_{\mathfrak p}}$ forces $n_{\mathfrak p}$ to be even, so that
prime contributes zero to the two-torsion symbol. At a prime dividing
two, put $a=-2/\pi_{\mathfrak p}\in T$.
The Dennis--Stein symbol
\[
        d_{\mathfrak p}=\langle-2/\pi_{\mathfrak p},
                               \pi_{\mathfrak p}\rangle
\]
is defined because $1+a\pi_{\mathfrak p}=-1$ is a unit. The
Dennis--Stein identity with invertible second entry gives, over $F$,
\[
       (d_{\mathfrak p})_F=\{-1,\pi_{\mathfrak p}\}.
\]
We use the convention and relations (D1)--(D4) in
\cite[Section 2.1]{LudersMorrowK2}; they are stated for arbitrary rings.
The Dennis--Stein relation with a fixed second entry yields
\[
 2\langle a,\pi\rangle
   =\langle 2a+a^2\pi,\pi\rangle=\langle0,\pi\rangle=0,
\]
since $a\pi=-2$. Thus
$\{-1,\epsilon\}+\sum_{\mathfrak p\ni2}n_{\mathfrak p}d_{\mathfrak p}$
is the required two-torsion lift. Negative exponents are allowed in
this abelian-group sum.
\end{proof}

\begin{proposition}[Image purity at residue characteristic two]\label{k2:prop:dyadicimage}
When the residue characteristic is two, $P_2(T)[2]=0$.
\end{proposition}
\begin{proof}
If $\alpha\in K_2^{\mathrm{nr}}(T)$ represents a class killed by two, choose
$b\in K_2(T)$ with $b_F=2\alpha$.
Lemma~\ref{k2:lem:dyadicquotient} shows that $q_{2,T}(b)=2m$ in $H^2_{\mot}(T,\Z(2))$.
Choose $c\in K_2(T)$ with $q_{2,T}(c)=m$.
The field isomorphism implies $2(\alpha-c_F)=0$.
This difference is integrally unramified; its lift supplied by
Lemma~\ref{k2:lem:dyadictorsion}, added to $c$, lifts $\alpha$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{k2:thm:main}]
For mixed characteristic, generic injectivity is
Theorem~\ref{k2:thm:genericinjection}. The purity defect has bounded
$p$-power exponent by Corollary~\ref{k2:cor:primeaway} and has no
$p$-torsion by Proposition~\ref{k2:prop:oddimage} or
\ref{k2:prop:dyadicimage}. It is therefore zero.
The equal-characteristic case follows from \cite[Theorem A]{Panin},
so all cases in the theorem are covered.
\end{proof}

\section{Low-dimensional consequences and coefficient boundaries}
\label{k2:sec:consequences}
The following elementary consequences will be used in the subsequent
Milnor--Witt calculations. They concern Quillen differentials.

\begin{lemma}[The low-dimensional degree-two row]\label{k2:lem:lowrow}
If $T$ is regular local of dimension at most three, the augmented row
\[
 0\to K_2(T)\to K_2(F)\to
 \bigoplus_{T^{(1)}}\kappa(x)^\times\to
 \bigoplus_{T^{(2)}}\Z\to0
\]
is exact. Also $\CH^2(T)=0$ for a regular local $T$ in any dimension.
\end{lemma}
\begin{proof}
The first two nonzero positions are Theorem~\ref{k2:thm:main}.
In \eqref{k2:eq:coniveau}, $\CH^2(T)=E_2^{2,-2}$ has zero positive
$K_0(T)=\Z$ limiting piece. Its sole possible higher incoming
map is $d_2$ from unramified $K_1$, namely $T^\times$; those units
extend and are permanent cycles. All higher outgoing targets have
negative field $K$-degree. Thus $\CH^2(T)=0$ without a dimension bound.

The row's degree-one cohomology is $E_2^{1,-2}$, a positive filtration
piece of $K_1(T)=T^\times$ in the limit, hence zero there. There are no
incoming higher differentials. In dimension at most three its only
possible outgoing higher target is $E_2^{3,-3}=\CH^3(T)$.
In dimension three this is the terminal Chow group, which is zero:
a DVR curve cut out by two members of a regular system of parameters
has a uniformizer whose divisor is the closed point with coefficient one. In smaller dimensions the target
is absent. This proves degree-one exactness as well.
\end{proof}

We will use a finite-quotient consequence of the integral row. Its
proof lifts a cocycle modulo the coefficient integer and corrects its
integral boundary. Torsion-freeness of the terminal divisor group is
what makes the correction possible.

\begin{corollary}[Finite-quotient image exactness]\label{k2:cor:quotientimage}
For $T$ as in Lemma~\ref{k2:lem:lowrow} and any integer $m\geq1$,
\[
 \im\bigl(K_2(T)/m\to K_2(F)/m\bigr)=
 \ker\left(K_2(F)/m\to
       \bigoplus_{T^{(1)}}\kappa(x)^\times/\kappa(x)^{\times m}\right).
\]
Only image exactness is asserted.
\end{corollary}
\begin{proof}
Lift a class of the kernel to $x\in K_2(F)$ and choose a finitely
supported unit cochain $b$ with $\partial x=mb$, in additive notation.
The next group is a direct sum of copies of $\Z$, so
$m\partial b=0$ implies $\partial b=0$.
Lemma~\ref{k2:lem:lowrow} supplies $y\in K_2(F)$ with $\partial y=b$.
The class $x-my$ is integrally unramified and extends by
Theorem~\ref{k2:thm:main}. The reverse inclusion follows from
localization.
\end{proof}

\begin{remark}\label{k2:rem:scope}
At an odd residue prime, the divided-Chern argument starts with
$b_F=p\alpha$ and an integrally unramified class $\alpha$.
The Kummer calculation uses $\Pic(Y')/p$, while the torsion-lifting
calculation uses $\Pic(Y')[p]$. At the dyadic prime,
Lemma~\ref{k2:lem:dyadicquotient} gives the motivic quotient injection
used to divide the class. These two constructions supply the
residue-prime steps in the integral theorem.
\end{remark}