\subsection{Notation and conventions}\label{intro:notation}

In the main theorem, $R$ is the four-dimensional regular local ring,
$r$ is the chosen parameter, and $A=R[1/r]$. We write $S=R/(r)$ when
working on the divisor and $X=\Spec A$ when working on the parameter
open. Auxiliary local rings are denoted $T$, with fraction field
$F=\Frac T$ and residue field $k$. For a regular scheme $Y$, the set of
codimension-$c$ points is $Y^{(c)}$; for a ring $T$, the abbreviation
$T^{(c)}$ means $(\Spec T)^{(c)}$. The generic point is denoted $\xi$.

We follow the unit-symbol notation for Milnor and Milnor--Witt theory
in \cite{SchlichtingEuler,DegliseMW}. The genuine and homotopy-symmetric
structures are distinguished as in \cite{GWCDHIII}; the square-bracket
shift in $\GW^{[j]}$ or $\KO^{[j]}$ records the shifted duality,
including its biduality sign. The conventions used throughout are as follows.

\begingroup
\renewcommand{\arraystretch}{1.08}
\begin{longtable}{@{}>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.65\textwidth}@{}}
\toprule
Notation & Meaning\\
\midrule
\endfirsthead
\toprule
Notation & Meaning (continued)\\
\midrule
\endhead
\bottomrule
\endfoot
$K_i(T)$ & Quillen $K$-theory, with its nonconnective extension in localization sequences.\\
$\KM_i(T)$, $\Khat_i(T)$ & Ordinary unit-symbol Milnor $K$-theory and improved Milnor $K$-theory.\\
$\KMW_i(T)$ & The degree-$i$ naive Milnor--Witt unit-symbol group of Definition~\ref{notation:mw_definition}.\\
$\mathcal K_i^{\mathrm{MW}}$ & The Zariski sheafification of $T\mapsto\KMW_i(T)$.\\
$H^i_{\mot}(T,\Z(j))$ & Motivic cohomology in cohomological degree $i$ and weight $j$. In particular the weight-two groups are written explicitly as $H^0_{\mot}(T,\Z(2))$, $H^1_{\mot}(T,\Z(2))$, and $H^2_{\mot}(T,\Z(2))$.\\
$q_{i,T}$ & The weight-two edge $K_i(T)\to H^{4-i}_{\mot}(T,\Z(2))$, for $i=2,3,4$.\\
$\GW_i^{[j]}$, $\KO_i^{[j]}$ & Genuine symmetric and homotopy-symmetric Grothendieck--Witt groups, respectively.\\
$Q^{\mathrm{gs}},Q^{\mathrm s},Q^{\mathrm q}$;
$L^{\mathrm{gs}},L^{\mathrm s},L^{\mathrm q}$ & The genuine symmetric, homotopy-symmetric, and pure quadratic structures and their $L$-theories.\\
$\Hyp(P)$, $\Hyp$ & The alternating hyperbolic space on $P\oplus P^*$ and the free alternating plane $\Hyp=\Hyp(T)$. A power $\Hyp^m$ is an $m$-fold orthogonal sum.\\
$\Hypq$, $\hsym$ & The positive-quadratic hyperbolic space used in the formation argument, and the symmetric Grothendieck--Witt class $\hsym=\langle1\rangle+\langle-1\rangle$.\\
$f$, $h^{[j]}$ & Forgetful and hyperbolic maps. In $h^{[j]}$ the superscript specifies the target duality shift.\\
$\eta$, $\eta_{\mathrm g}$ & The Milnor--Witt generator of degree $-1$ and the genuine Bott class. The symbol comparison sends $\eta$ to $-\eta_{\mathrm g}$.\\
$\mu_n$, $\mult$ & The degree-$n$ hermitian symbol comparison and the Milnor--Witt multiplication map used in the completion criterion. In \'etale cohomology, $\mu_n$ instead denotes the sheaf of $n$th roots of unity.\\
$W(k)$, $I(k)$ & The bilinear Witt ring and its fundamental ideal; $I^a(k)=W(k)$ for $a\le0$.\\
$W_{\mathrm{alt}}(A)$ & The stable Witt group of perfect alternating forms over $A$.\\
$K_2^{\mathrm{nr}}(T)$, $J_2(T)$, $P_2(T)$ & The unramified generic subgroup, generic kernel, and image-purity cokernel in degree two; see Section~\ref{k2:sec:main}.\\
$\omega_x$ & The exceptional line $\operatorname{Ext}^{c}_{\mathcal O_{Y,x}}(\kappa(x),\mathcal O_{Y,x})$ for $x\in Y^{(c)}$, with its induced duality.\\
$C_{\mathrm{MW},4}(Y)$ & The compatible weight-four field complex of Definition~\ref{loc:mw_row}; its cochain degree is codimension.\\
$C_{\mathrm W}(Y)$, $C_{I,j}(Y)$ & The full bilinear Witt row and its filtered row with degree-$c$ coefficient $I^{j-c}(\kappa(x),\omega_x)$.\\
$\Fil^c_{\mathrm{con}}$ & The codimension-of-support filtration in coniveau theory.\\
$\Fil^i_{\mot}K(T)$,
$\Fil^i_{\mathrm{lis}}K(T)$ & The motivic filtration and the filtration obtained by realizing the classical filtration on a simplicial local resolution; see Lemma~\ref{loc:lisse_window}.\\
\end{longtable}
\endgroup

For a complex of sheaves $\mathcal C^\bullet$, we write
$\mathcal H^i(\mathcal C^\bullet)$ for its cohomology sheaf and
$\mathbb H^i(X,\mathcal C^\bullet)$ for hypercohomology.
Unadorned $H^i(X,\mathcal A)$ denotes Zariski sheaf cohomology;
$H^i(C^\bullet)$ denotes the cohomology of a complex of abelian groups.
Finite-coefficient groups and integral quotients are distinguished:
$H^i_{\mot}(T,\Z/m(j))$ denotes cohomology with finite coefficients,
whereas $H^i_{\mot}(T,\Z(j))/m$ is a quotient of an integral group.
For an abelian group $M$, $M[m]=\ker(m:M\to M)$.

We use homological shifts for perfect complexes:
$H_i(E[a])=H_{i-a}(E)$. The derived category is $\mathrm D(T)$,
its derived duality is $\Ddual_T E=\operatorname{RHom}_T(E,T)$,
and $\mathrm H M$ denotes an Eilenberg--Mac Lane spectrum.
The category of spectra is $\mathsf{Sp}$, whereas $\Sp_{2n}$ denotes
the symplectic group. The classifying-space functor is written
$\Bclass$; the elementary symplectic subgroup is $\ESp_{2n}$.

For a rank-two projective $B$-module $P$, an orientation is an
isomorphism $\theta:\det P=\bigwedge^2P\xrightarrow{\sim}B$.
It defines the perfect alternating pairing
$b_\theta(v,w)=\theta(v\wedge w)$.
A frame in the completion argument is a specified isometry
$P\perp\Hyp\simeq\Hyp^2$.