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%   pdflatex -interaction=nonstopmode -halt-on-error Em02-E.tex
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%
% Release:
%   Classical identification and renormalization audit.
%   An explicit Einstein--scalar inflationary background is fixed.
%   The inflaton-slice harmonic-coordinate lapse is calculated through
%   second order, without discarding metric components.
%   Its complete classical quadratic kernel is identified.
%   Noncompact functional support, instantaneous-kernel singularities,
%   and transport of diagonal corrections are analyzed.
%
% Quantum boundary:
%   No complete gauge-fixed covariance, state, admissible counterterm
%   classification, quantum remainder, or Ward identity is constructed.
%   The quantum regulator-removal verdict is therefore INCOMPLETE AUDIT.
%   No existence or universal obstruction theorem is asserted.
%
% Verification:
%   No compiler execution, numerical execution, machine proof,
%   registration verification, or referee approval is claimed.

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\hypersetup{
 pdftitle={Em02-E: Relational Cosmological Lapse, Classical Identification and a Renormalization Audit},
 pdfauthor={}
}

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\title{\textbf{Em02-E}\\[0.4em]
Relational Cosmological Lapse:\\
Classical Identification and a Renormalization Audit\\[0.4em]
\large Full-tensor quadratic kernels, nonlocal support,\\
and the boundary of the quantum regulator-removal question}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We identify one relational lapse in an explicitly specified
Einstein--scalar theory on a power-law inflationary background.
The clock is the scalar field. Spatial labels are harmonic
coordinates on its level sets, with the asymptotic Cartesian
boundary condition. We use the original inflaton-slice harmonic
rods, rather than the modified rods employed in some simplified
cosmological formulas.

The complete classical expansion through second order is derived
in tensor components. It retains arbitrary metric perturbations,
second-order field data, the clock displacement, and the spatial
coordinate correction. At this order the lapse depends on the
first spatial rod correction but not on the second. The resulting
quadratic functional has an explicit local part and an
instantaneous spatial-Green part.

We prove that compact external smearing does not generally make
the functional derivatives spatially compact. We also calculate
the off-diagonal wavefront set of the instantaneous Green kernel
and show, by an exact distributional example, that an internal
diagonal correction can have a non-diagonal observable-level
image. Quantitative tail and regulator-comparison criteria
separate spatial integration from ultraviolet removal.

The quantum input gate is not passed: a complete gauge-fixed
covariance, a state or parametrix with the required spatial
control, an admissible counterterm classification, and the
retained-order quantum gauge identity are not constructed.
Accordingly, the release is an incomplete quantum audit, not
a regulator-removal theorem or a universal obstruction.
The classical formulas and auxiliary distributional results
are proved independently of the declared models in Em02-C
and Em02-D.
\end{abstract}

\noindent\textbf{Release boundary.}
This paper completes a classical identification and a mathematical
audit of the proposed quantum calculation. It does not complete
the quantum regulator-removal problem. In particular, the title
does not assert regulator removal.

\tableofcontents

%=====================================================================
\section{Question, chosen observable, and release criteria}
\label{sec:question}
%=====================================================================

The intended quantum question is:
\begin{quote}
For the relational lapse specified here, at a stated quantum
order, does the regulated expression admit a distributional
limit after corrections satisfying its renormalization and
quantum gauge requirements?
\end{quote}
A meaningful answer requires an actual regulated quantum family
and an independently determined correction class. Neither can
be replaced by a model coefficient from another paper.

The relational construction follows the geometric prescription
of Brunetti, Fredenhagen, Hack, Pinamonti, and Rejzner
\cite[Sections 3--4]{BFHPR}. Its general form is
\begin{equation}
 \Nrel_\Gamma=N_{\phi,\Gamma}\circ\alpha_\Gamma,
 \qquad
 \alpha_\Gamma=X_\Gamma^{-1}\circ X_{\Gamma_0},
 \qquad
 \Of(\Gamma)=\int_M f\,\Nrel_\Gamma.
 \label{eq:observable}
\end{equation}
Here \(f\) is a compact test density on the background label
space. It is fixed under active gauge transformations.
It is not the scalar potential, an interaction switch, or
a state.

% External construction checked against the primary source:
% ([arxiv.org](https://arxiv.org/html/1605.02573))

\subsection{A definite coordinate choice}

We choose the scalar clock and the \emph{original}
inflaton-slice harmonic spatial coordinates described in
\cite[equations (28)--(31)]{BFHPR}.
We denote the latter by \(Y_\Gamma^i\), and set
\[
 X_\Gamma=(\widetilde\phi,Y_\Gamma^1,Y_\Gamma^2,Y_\Gamma^3).
\]
The modified spatial rods of equation (35) of that reference
are not used. The two choices generally give different
second-order observables. This distinction is part of the
identification, not an approximation.

All calculations below use cosmic time and the action
normalization in Section~\ref{sec:theory}. We do not import
the reference's on-shell scalar formulas without converting
their conventions.

\subsection{Perturbative and quantum bookkeeping}

The parameter \(\lambda\) counts field perturbations:
\[
 \Gamma_\lambda
 =
 \Gamma_0+\lambda\eta+\frac{\lambda^2}{2}\eta_2
 \pmod{\lambda^3}.
\]
It is not \(\hbar\), a smoothing length, or an interaction
coupling.

The concrete classical target is the complete coefficient
through \(\lambda^2\). The first prospective quantum calculation
is the order-\(\hbar\) contraction associated with quantizing
its quadratic part in a chosen free representation.
Section~\ref{sec:quantum} derives this contraction count.

An interacting order-\(\hbar\) observable can require additional
action vertices and observable coefficients, depending on
the coupling grading. No complete interacting inventory is
claimed from the free quadratic count.

\subsection{Three possible quantum outcomes}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.19\linewidth}Y@{}}
\toprule
Outcome & Required evidence\\
\midrule
Existence &
A defined quantum family, admissible subtraction, a convergent
complete remainder, spatial-tail control, and the quantum gauge
identity through the retained order.\\[0.3em]
Obstruction &
A surviving detecting pairing or anomaly, together with proof
that every correction in the independently derived admissible
class fails the required conditions.\\[0.3em]
Incomplete audit &
One or more of those ingredients is absent or unproved,
with the missing operation identified explicitly.\\
\bottomrule
\end{tabularx}
\end{center}

The present quantum outcome is the third. The positive results
are the classical identification, quadratic kernel bridge,
and distributional support and tail calculations proved below.

%=====================================================================
\section{Einstein--scalar theory and an explicit inflationary background}
\label{sec:theory}
%=====================================================================

\subsection{Conventions and equations}

Use signature \((-+++)\), units \(c=1\), and the action
\begin{equation}
 S[g,\phi]
 =
 \int_M
 \left[
 \frac{\MPl^2}{2}R(g)
 -\frac12g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi
 -V(\phi)
 \right]\dd\operatorname{vol}_g,
 \qquad \MPl>0.
 \label{eq:action}
\end{equation}
The curvature convention is
\[
 R^\rho{}_{\sigma\mu\nu}
 =
 \partial_\mu\Gamma^\rho_{\nu\sigma}
 -\partial_\nu\Gamma^\rho_{\mu\sigma}
 +\Gamma^\rho_{\mu\alpha}\Gamma^\alpha_{\nu\sigma}
 -\Gamma^\rho_{\nu\alpha}\Gamma^\alpha_{\mu\sigma}.
\]
The field equations are
\begin{align}
 \MPl^2G_{\mu\nu}
 &=
 \partial_\mu\phi\,\partial_\nu\phi
 -g_{\mu\nu}
 \left(\frac12g^{\alpha\beta}
                 \partial_\alpha\phi\,\partial_\beta\phi+V\right),
 \label{eq:Einstein}\\
 \Box_g\phi-V'(\phi)&=0.
 \label{eq:KG}
\end{align}
All variations in the classical observable calculation are
off shell. The background alone is required to solve
\eqref{eq:Einstein}--\eqref{eq:KG}.

\subsection{Fixed background}

Fix
\[
 p>1,\qquad a_*>0,\qquad t_*>0,\qquad \phi_*\in\R,
\]
and define
\begin{align}
 a(t)&=a_*(t/t_*)^p,
 &
 \phi_0(t)&=\phi_*+\sqrt{2p}\,\MPl\log(t/t_*),
 \label{eq:background}\\
 V(\phi)
 &=
 \frac{p(3p-1)\MPl^2}{t_*^2}
 \exp\left[
 -\sqrt{\frac2p}\,\frac{\phi-\phi_*}{\MPl}
 \right].
 \label{eq:potential}
\end{align}
Work on
\[
 M=I\times\R^3,\qquad I=(t_*/2,2t_*),
 \qquad
 g_0=-\dd t^2+a(t)^2\delta_{ij}\dd x^i\dd x^j.
\]
A dot denotes cosmic-time differentiation. Put
\begin{equation}
 H=\frac{\dot a}{a}=\frac pt,\qquad
 v=\dot\phi_0=\frac{\sqrt{2p}\,\MPl}{t}>0.
 \label{eq:vH}
\end{equation}
There is no limit toward \(t=0\) in this paper.

\begin{proposition}[Background and clock]
\label{prop:background}
The pair \((g_0,\phi_0)\) solves
\eqref{eq:Einstein}--\eqref{eq:KG}.
It is expanding and accelerating, and its scalar clock
is nondegenerate on every compact subinterval of \(I\).
\end{proposition}

\begin{proof}
For a spatially flat homogeneous background, the equations
obtained from \eqref{eq:Einstein}--\eqref{eq:KG} are
\[
 3\MPl^2H^2=\frac12v^2+V(\phi_0),
 \qquad
 -2\MPl^2\dot H=v^2,
 \qquad
 \dot v+3Hv+V'(\phi_0)=0.
\]
Substitution gives
\[
 V(\phi_0(t))=\frac{p(3p-1)\MPl^2}{t^2},
 \qquad
 \frac12v^2=\frac{p\MPl^2}{t^2},
\]
which proves the first equation. The second follows from
\(\dot H=-p/t^2\).
Writing \(c=\sqrt{2p}\,\MPl\), the third equation has terms
\[
 -c/t^2,\qquad 3pc/t^2,\qquad -(3p-1)c/t^2,
\]
whose sum is zero.

Finally, \(\dot a>0\), \(\ddot a/a=p(p-1)/t^2>0\),
and \(v>0\) has a positive lower bound on each compact
time interval.
\end{proof}

\subsection{Perturbation domain}

For the calculations through second order, take smooth,
spatially compact perturbations with a common compact support:
\begin{equation}
 \widetilde g_\lambda
 =
 g_0+\lambda h+\frac{\lambda^2}{2}k,
 \qquad
 \widetilde\phi_\lambda
 =
 \phi_0+\lambda\varphi+\frac{\lambda^2}{2}\psi
 \pmod{\lambda^3}.
 \label{eq:field-jet}
\end{equation}
The tensors \(h,k\) are arbitrary symmetric covariant tensors.
No scalar, vector, or tensor metric component is discarded.

The formulas are formal Taylor coefficients. For an actual
nonperturbative coordinate map, one must additionally restrict
to configurations for which the scalar level sets are spacelike,
the harmonic-coordinate problem has the stated boundary
solution, and the resulting coordinate map is invertible
on the region used by \(f\). We do not assert such global
properties for every smooth configuration.

For the formal coefficients considered here, all elliptic
sources are smooth and spatially compact, and the required
decaying solutions are constructed explicitly.

%=====================================================================
\section{Clock slices and harmonic spatial labels}
\label{sec:coordinates}
%=====================================================================

\subsection{Clock-adapted description}

Use \(s\in I\) as a background clock label.
The perturbed slice carrying that label is
\[
 \widetilde\phi_\lambda(t,\mathbf x)=\phi_0(s).
\]
Write it as
\begin{equation}
 t=\tau_\lambda(s,\mathbf x)
 =s+\lambda u_1(s,\mathbf x)
                  +\frac{\lambda^2}{2}u_2(s,\mathbf x)
                  \pmod{\lambda^3}.
 \label{eq:tau}
\end{equation}

\begin{lemma}[Clock displacement]
\label{lem:clock}
The unique formal solution of the clock equation is
\begin{equation}
 u_1=-\frac{\varphi}{v},
 \qquad
 u_2=
 -\frac{\psi+2u_1\dot\varphi+\dot v\,u_1^2}{v}.
 \label{eq:u}
\end{equation}
All quantities on the right are evaluated at \((s,\mathbf x)\).
\end{lemma}

\begin{proof}
Expand
\(\widetilde\phi_\lambda(\tau_\lambda,\mathbf x)\)
through second order:
\[
 \phi_0(s)
 +\lambda(vu_1+\varphi)
 +\frac{\lambda^2}{2}
       (vu_2+\dot v\,u_1^2+2u_1\dot\varphi+\psi).
\]
Setting the coefficients to zero proves the result because
\(v\ne0\).
\end{proof}

Let
\[
 \Psi_\lambda(s,\mathbf x)=(\tau_\lambda(s,\mathbf x),\mathbf x).
\]
The induced positive spatial metric on the clock slice is
\[
 q_{\lambda,ij}
 =
 \widetilde g_{\lambda,\mu\nu}(\Psi_\lambda)
            \partial_i\Psi_\lambda^\mu\partial_j\Psi_\lambda^\nu.
\]
Write
\[
 q_{\lambda,ij}
 =
 a^2\delta_{ij}
 +\lambda q_{1,ij}+\frac{\lambda^2}{2}q_{2,ij}.
\]

\begin{proposition}[Induced metric through second order]
\label{prop:induced}
The coefficients are
\begin{align}
 q_{1,ij}
 &=
 h_{ij}+u_1\,\partial_t(a^2)\delta_{ij},
 \label{eq:q1}\\
 q_{2,ij}
 &=
 k_{ij}+2u_1\dot h_{ij}
 +u_2\,\partial_t(a^2)\delta_{ij}
 +u_1^2\,\partial_t^2(a^2)\delta_{ij}\notag\\
 &\quad
 +2(h_{0i}\partial_ju_1+h_{0j}\partial_iu_1)
 -2\partial_iu_1\partial_ju_1.
 \label{eq:q2}
\end{align}
\end{proposition}

\begin{proof}
The spatial tangent vectors are
\[
 \partial_i\Psi_\lambda
 =
 \partial_i+(\lambda\partial_iu_1
                  +\lambda^2\partial_iu_2/2)\partial_t.
\]
Taylor-expand the spatial-spatial metric component at
\(t=s\). The mixed components first contribute at degree two,
as \(\lambda h_{0i}\lambda\partial_ju_1\).
The background time-time component is \(-1\), so its
degree-two contribution is
\(-\lambda^2\partial_iu_1\partial_ju_1\).
With the factorial convention of \eqref{eq:field-jet},
these are exactly \eqref{eq:q1}--\eqref{eq:q2}.
\end{proof}

\subsection{An explicit Laplace operator expansion}

Set
\[
 Q_1=a^{-2}q_1,\qquad Q_2=a^{-2}q_2,
\]
where spatial indices on \(Q_1,Q_2\) are raised using
\(\delta_{ij}\). Define
\begin{align}
 b_1&=\frac12\tr Q_1,\notag\\
 b_2&=\frac12\tr Q_2
       +\frac14(\tr Q_1)^2-\frac12\tr(Q_1^2),
 \label{eq:b}\\
 A_1&=b_1\mathbf1-Q_1,\notag\\
 A_2&=b_2\mathbf1-2b_1Q_1+2Q_1^2-Q_2.
 \label{eq:A}
\end{align}
These matrices are unrelated to the component maps denoted
by \(\mathsf A\) in Em02-A--D.

For a scalar \(F\), the slice Laplacian is
\[
 \Delta_{q_\lambda}F
 =
 (\det q_\lambda)^{-1/2}
 \partial_i\left((\det q_\lambda)^{1/2}
                              q_\lambda^{ij}\partial_jF\right).
\]

\begin{lemma}[Laplacian coefficients]
\label{lem:laplacian}
One has
\[
 \Delta_{q_\lambda}
 =
 a^{-2}\left(\Delta+\lambda P_1
                         +\frac{\lambda^2}{2}P_2\right),
\]
where
\begin{align}
 P_1F
 &=
 \partial_i(A_1^{ij}\partial_jF)-b_1\Delta F,
 \label{eq:P1}\\
 P_2F
 &=
 \partial_i(A_2^{ij}\partial_jF)
 -2b_1\partial_i(A_1^{ij}\partial_jF)
 +(2b_1^2-b_2)\Delta F.
 \label{eq:P2}
\end{align}
In particular,
\begin{equation}
 P_1x^i
 =
 \frac12\partial_i\tr Q_1-\partial_jQ_1^{ji}.
 \label{eq:B}
\end{equation}
\end{lemma}

\begin{proof}
The determinant and inverse expansions are
\[
 \frac{\sqrt{\det q_\lambda}}{a^3}
 =
 1+\lambda b_1+\frac{\lambda^2}{2}b_2,
\]
\[
 a^2q_\lambda^{-1}
 =
 \mathbf1-\lambda Q_1
       +\frac{\lambda^2}{2}(2Q_1^2-Q_2).
\]
Their product has coefficients \(A_1,A_2\).
The inverse determinant factor is
\(1-\lambda b_1+\lambda^2(2b_1^2-b_2)/2\).
Multiplication proves \eqref{eq:P1}--\eqref{eq:P2}.
Applying \(P_1\) to \(x^i\) gives \eqref{eq:B}.
\end{proof}

\subsection{The spatial inverse and its boundary condition}

Fix
\begin{equation}
 (\Gsp F)(\mathbf x)
 =
 -\frac1{4\pi}\int_{\R^3}
                  \frac{F(\mathbf y)}{|\mathbf x-\mathbf y|}
                  \dd^3\mathbf y,
 \qquad F\in C_c^\infty(\R^3).
 \label{eq:Coulomb}
\end{equation}
Then \(\Delta\Gsp F=F\), and \(\Gsp F\to0\) at spatial infinity.
For compact smooth \(F\), also \(\Gsp\Delta F=F\).
The sign in \eqref{eq:Coulomb} is fixed by
\(\Delta(-1/(4\pi|\mathbf x|))=\delta^{(3)}\).

Seek harmonic rods on the clock slice in the form
\[
 Y_\lambda^i(s,\mathbf x)
 =
 x^i+\lambda Y_1^i(s,\mathbf x)
                +\frac{\lambda^2}{2}Y_2^i(s,\mathbf x),
 \qquad
 \Delta_{q_\lambda}Y_\lambda^i=0,
\]
with \(Y_\lambda^i-x^i\to0\) at spatial infinity coefficientwise.

\begin{proposition}[Constructed harmonic-coordinate coefficients]
\label{prop:rods}
The unique decaying coefficients are
\begin{equation}
 Y_1^i=-\Gsp P_1x^i,
 \qquad
 Y_2^i=-\Gsp(P_2x^i+2P_1Y_1^i).
 \label{eq:Y}
\end{equation}
They are smooth. On compact time intervals,
\(Y_1=O(|\mathbf x|^{-2})\) and \(Y_2=O(|\mathbf x|^{-1})\)
at spatial infinity.
\end{proposition}

\begin{proof}
The harmonic equation at the two successive orders is
\[
 \Delta Y_1^i+P_1x^i=0,
 \qquad
 \Delta Y_2^i+P_2x^i+2P_1Y_1^i=0.
\]
All coefficients of \(P_1,P_2\) have spatially compact
support. Hence both sources are smooth and compactly
supported, and \eqref{eq:Coulomb} constructs the solutions.

Any difference is a smooth harmonic function tending to zero
at infinity. The maximum principle on successively larger
balls gives zero, proving uniqueness.

The second estimate is the usual direct \(1/|\mathbf x|\)
bound for the displayed compact-source integral.
For the first, \eqref{eq:B} is a spatial divergence, so its
integral vanishes. Subtract
\(|\mathbf x|^{-1}\) from the kernel in
\eqref{eq:Coulomb} outside a ball containing the source.
The remaining kernel is \(O(|\mathbf x|^{-2})\).
The same estimates are uniform on compact time intervals.
\end{proof}

The background slice inverse is \(a(s)^2\Gsp\).
It is an elliptic spatial inverse, not a retarded or advanced
spacetime inverse.

\subsection{Relational coordinate map}

Let \(Z_\lambda(s,\cdot)\) be the formal spatial inverse of
\(Y_\lambda(s,\cdot)\). Then
\begin{align}
 Z_\lambda
 &=
 \mathbf x+\lambda z_1+\frac{\lambda^2}{2}z_2,\notag\\
 z_1&=-Y_1,\qquad
 z_2=-Y_2+2Y_1^j\partial_jY_1.
 \label{eq:Z}
\end{align}
The relational map is
\[
 \alpha_\lambda(s,\mathbf x)
 =
 \Psi_\lambda(s,Z_\lambda(s,\mathbf x)).
\]
Consequently its first and second displacement coefficients are
\begin{equation}
 \alpha_1=(u_1,z_1),\qquad
 \alpha_2=(u_2+2z_1^i\partial_iu_1,z_2).
 \label{eq:alpha}
\end{equation}
These identities follow by composing the Taylor series.
They implement \eqref{eq:observable} with
\(X_{\Gamma_0}(s,\mathbf x)=(\phi_0(s),\mathbf x)\).

%=====================================================================
\section{Complete classical lapse expansion}
\label{sec:lapse}
%=====================================================================

\subsection{The unsmeared scalar lapse}

Define
\begin{equation}
 W_\lambda
 =
 -\widetilde g_\lambda^{\mu\nu}
        \partial_\mu\widetilde\phi_\lambda
        \partial_\nu\widetilde\phi_\lambda,
 \qquad
 N_{\phi,\lambda}=W_\lambda^{-1/2}.
 \label{eq:lapse}
\end{equation}
The timelike-clock branch has \(W_0=v^2>0\).

In this section, raising a spacetime index on \(h,k\) uses
\(g_0\), not the Euclidean spatial metric. Set
\[
 v_\mu=\partial_\mu\phi_0,\qquad
 p_\mu=\partial_\mu\varphi,\qquad
 w_\mu=\partial_\mu\psi.
\]
Write
\(W_\lambda=W_0+\lambda W_1+\lambda^2W_2/2\).

\begin{lemma}[Full-tensor lapse coefficients]
\label{lem:bare-lapse}
The coefficients are
\begin{align}
 W_0&=v^2,\notag\\
 W_1&=h^{\mu\nu}v_\mu v_\nu
                         -2g_0^{\mu\nu}v_\mu p_\nu,
 \label{eq:W1}\\
 W_2&=
 (k^{\mu\nu}-2h^{\mu\alpha}g_{0,\alpha\beta}h^{\beta\nu})
                                      v_\mu v_\nu\notag\\
 &\quad+4h^{\mu\nu}v_\mu p_\nu
       -2g_0^{\mu\nu}p_\mu p_\nu
       -2g_0^{\mu\nu}v_\mu w_\nu.
 \label{eq:W2}
\end{align}
For \(N_{\phi,\lambda}=N_0+\lambda N_1+\lambda^2N_2/2\),
\begin{equation}
 N_0=v^{-1},\qquad
 N_1=-\frac{W_1}{2v^3},\qquad
 N_2=\frac{3W_1^2}{4v^5}-\frac{W_2}{2v^3}.
 \label{eq:N}
\end{equation}
\end{lemma}

\begin{proof}
The inverse metric expansion is
\[
 \widetilde g_\lambda^{\mu\nu}
 =
 g_0^{\mu\nu}-\lambda h^{\mu\nu}
 +\frac{\lambda^2}{2}
 (2h^{\mu\alpha}g_{0,\alpha\beta}h^{\beta\nu}-k^{\mu\nu}).
\]
Multiply it by the two scalar-gradient expansions and retain
the minus sign in \eqref{eq:lapse}.
This gives \eqref{eq:W1}--\eqref{eq:W2}.
The first two derivatives of \(W^{-1/2}\) give
\eqref{eq:N}.
\end{proof}

This tensor formula includes the mixed metric components and
all scalar spatial gradients. No scalar-sector display is used
as a substitute for the full coefficient.

\subsection{Composition with the relational map}

Write
\[
 \Nrel_\lambda=N_0+\lambda n_1+\frac{\lambda^2}{2}n_2
                                      \pmod{\lambda^3}.
\]

\begin{theorem}[Complete relational lapse through second order]
\label{thm:lapse}
The coefficients are
\begin{align}
 n_1&=N_1+\dot N_0u_1,
 \label{eq:n1}\\
 n_2&=
 N_2+2u_1\dot N_1+\dot N_0u_2+\ddot N_0u_1^2
                      -2Y_1^i\partial_in_1.
 \label{eq:n2}
\end{align}
Every quantity on the right is given explicitly by
\eqref{eq:u}, \eqref{eq:q1}, \eqref{eq:B}, \eqref{eq:Y},
and \eqref{eq:W1}--\eqref{eq:N}.

The second spatial rod coefficient \(Y_2\) does not occur
in \(n_2\).
\end{theorem}

\begin{proof}
The composition rule gives
\begin{align*}
 n_2={}&
 N_2+2\alpha_1^\mu\partial_\mu N_1
       +\alpha_2^\mu\partial_\mu N_0
       +\alpha_1^\mu\alpha_1^\nu
                                      \partial_\mu\partial_\nu N_0.
\end{align*}
Since \(N_0\) depends only on time, substitution of
\eqref{eq:alpha} leaves
\[
 N_2+2u_1\dot N_1+2z_1^i\partial_iN_1
 +\dot N_0(u_2+2z_1^i\partial_iu_1)+\ddot N_0u_1^2.
\]
Use \(z_1=-Y_1\) and
\(\partial_i(N_1+\dot N_0u_1)=\partial_in_1\).
The spatial part of \(\alpha_2\) multiplies
\(\partial_iN_0=0\), which proves the last assertion.
\end{proof}

\subsection{Linear sign and convention check}

In cosmic coordinates,
\(h^{00}=h_{00}\) and \(g_0^{00}=-1\). Thus
\begin{equation}
 n_1
 =
 -\frac{h_{00}}{2v}
 -\frac{\dot\varphi}{v^2}
 +\frac{\dot v\,\varphi}{v^3}
 =
 \frac1v\left(-\frac{h_{00}}2
                         -\partial_t\frac{\varphi}{v}\right).
 \label{eq:linear-explicit}
\end{equation}

To compare with conformal time, let \(\dd t=a\,\dd\tau\).
Then \(\phi_0'=av\), \(\mathcal H=aH\), and the conventional
scalar lapse perturbation satisfies \(A=-h_{00}/2\).
Consequently
\[
 \frac a{\phi_0'}
 \left[A-(\partial_\tau+\mathcal H)
                                  \frac{\varphi}{\phi_0'}\right]
 =
 \frac1v\left[A-\partial_t\frac{\varphi}{v}\right].
\]
This matches the off-shell first-order relational lapse
construction in \cite[equation (36)]{BFHPR}.
No on-shell Bardeen-potential identity is needed here.

% First-order benchmark and coordinate-choice distinction:
% ([arxiv.org](https://arxiv.org/html/1605.02573))

%=====================================================================
\section{Classical covariance and the quadratic kernel bridge}
\label{sec:bridge}
%=====================================================================

\subsection{The relevant gauge group}

Take formal pullbacks generated by smooth vector fields compactly
supported in \(I\times\R^3\). They preserve the prescribed
asymptotic spatial labels. The background label map
\(X_{\Gamma_0}\) remains fixed.

\begin{proposition}[Equivariance and classical invariance]
\label{prop:covariance}
Through the retained order,
\[
 X_{\chi^*\Gamma}=X_\Gamma\circ\chi,
 \qquad
 \alpha_{\chi^*\Gamma}=\chi^{-1}\circ\alpha_\Gamma,
\]
and therefore
\[
 \Nrel_{\chi^*\Gamma}=\Nrel_\Gamma,\qquad
 \Of(\chi^*\Gamma)=\Of(\Gamma)
 \pmod{\lambda^3}.
\]
\end{proposition}

\begin{proof}
The scalar clock transforms by pullback.
A diffeomorphism maps its level hypersurfaces and their induced
metrics into the corresponding level hypersurfaces and metrics
of the transformed configuration.

The pullback of each harmonic spatial coordinate is harmonic
for the pulled-back slice metric. Because \(\chi\) is the identity
at spatial infinity, it has the same asymptotic Cartesian
boundary data. The coefficientwise uniqueness proved in
Proposition~\ref{prop:rods} identifies it with the coordinate
constructed for \(\chi^*\Gamma\).

Invert the resulting scalar-coordinate identity to obtain
the transformation law for \(\alpha\).
The lapse is a scalar built from the metric and scalar gradient,
so \(N_{\phi,\chi^*\Gamma}=N_{\phi,\Gamma}\circ\chi\).
The two compositions cancel. Integration against the fixed
label-space density \(f\) preserves the identity.
\end{proof}

\begin{remark}[Scope of covariance]
This proposition is an off-shell classical statement.
It is neither a quantum BRST identity nor a statement that
a particular gauge-fixed state or regulator is invariant.
It also does not assert invariance under diffeomorphisms that
change the boundary data at spatial infinity.
\end{remark}

\subsection{Linear and quadratic functionals}

Fix a real compact test density
\[
 f=f_0(t,\mathbf x)\dd t\,\dd^3\mathbf x.
\]
For an affine perturbation \(\eta=(h,\varphi)\), set
\(k=\psi=0\) in the coefficient formulas and define
\[
 \Lf(\eta)=\int f\,n_1[\eta],
 \qquad
 \Qf(\eta)=\frac12\int f\,n_2[\eta].
\]
Then
\[
 \Of(\Gamma_0+\lambda\eta)
 =
 \Of(\Gamma_0)+\lambda\Lf(\eta)+\lambda^2\Qf(\eta)
 \pmod{\lambda^3}.
\]

Define the linear differential operations
\begin{equation}
 P\eta=n_1[\eta],
 \qquad
 B^i\eta=
 \frac12\partial_i\tr Q_1[\eta]-\partial_jQ_1^{ji}[\eta].
 \label{eq:PB}
\end{equation}
Set
\begin{equation}
 L_{\mathrm{loc}}[\eta]
 =
 \frac12N_2+u_1\dot N_1
                  +\frac12\dot N_0u_2+\frac12\ddot N_0u_1^2,
 \qquad k=\psi=0.
 \label{eq:Llocal}
\end{equation}
This is a local quadratic expression. Every coefficient in it
is fixed by the preceding formulas.

\begin{theorem}[Complete classical quadratic kernel bridge]
\label{thm:kernel}
The quadratic observable is exactly
\begin{align}
 \Qf(\eta)
 ={}&
 \int f\,L_{\mathrm{loc}}[\eta]\notag\\
 &+\sum_{i=1}^3
 \int_I\int_{\R^3}\int_{\R^3}
 f_0(t,\mathbf x)\,
 G_\Delta(\mathbf x-\mathbf y)\,
 (B^i\eta)(t,\mathbf y)\,
 \partial_i(P\eta)(t,\mathbf x)
 \dd^3\mathbf y\,\dd^3\mathbf x\,\dd t,
 \label{eq:kernel-bridge}\\
 G_\Delta(\mathbf r)&=-\frac1{4\pi|\mathbf r|}.
 \notag
\end{align}
For compact perturbations this is a continuous quadratic
functional. It has a symmetric distributional Hessian
\(\Hess_f\) satisfying
\[
 \Qf(\eta)=\frac12\pair{\Hess_f}{\eta\otimes\eta}.
\]
Its bilocal part is obtained by polarizing the second line of
\eqref{eq:kernel-bridge}. There are no additional classical
quadratic terms for this coordinate prescription.
\end{theorem}

\begin{proof}
The last term of \eqref{eq:n2}, divided by two, is
\(-Y_1^i\partial_i n_1\).
Since \(Y_1^i=-\Gsp B^i\eta\), it is precisely the second
line of \eqref{eq:kernel-bridge}. All other terms are
\eqref{eq:Llocal}.

On any fixed compact input-support class, the kernel
\(|\mathbf x-\mathbf y|^{-1}\) is locally integrable.
The differential operations \(B,P\) have smooth coefficients
on compact time intervals. Direct integral estimates give
continuity in finite smooth seminorms.

Polarization gives the symmetric bilinear form
\[
 \frac12\bigl(\Qf(\eta+\theta)-\Qf(\eta)-\Qf(\theta)\bigr).
\]
The local part is represented by derivatives of the diagonal
delta distribution. The nonlocal part is represented by
\(G_\Delta(\mathbf x-\mathbf y)\delta(t-u)\) with the displayed
differential operations transferred to its two field slots.
These are defined distributions before any quantum
contraction. Their sum is \(\Hess_f\).
\end{proof}

The bridge is a classical Hessian calculation, not a graph
renormalization theorem. Its metric field space has ten
symmetric components plus the scalar component.

\subsection{A quadratic coefficient is not separately the full gauge identity}

For a compact vector field \(\xi\), put
\[
 K\xi=(\mathcal L_\xi g_0,\mathcal L_\xi\phi_0).
\]
Classical invariance implies
\begin{equation}
 \Lf(K\xi)=0,
 \qquad
 \pair{\Hess_f}{\eta\otimes K\xi}
                  +\Lf(\mathcal L_\xi\eta)=0.
 \label{eq:classical-Ward}
\end{equation}

\begin{proof}
Differentiate
\[
 \Of'(\Gamma)[\mathcal L_\xi\Gamma]=0
\]
first at \(\Gamma_0\), and then in direction \(\eta\).
The first derivative gives the first identity.
The direction derivative gives the second, since the
derivative of \(\mathcal L_\xi\Gamma\) is
\(\mathcal L_\xi\eta\).
\end{proof}

Thus linear gauge transformations of the quadratic coefficient
couple to the nonlinear transformation of the linear coefficient.
Dropping the latter is not a valid quadratic gauge audit.

As a direct first-order check,
\[
 h_{00}=-2\partial_t\xi^0,\qquad
 \varphi=v\xi^0
\]
makes \eqref{eq:linear-explicit} zero.
For a pure spatial gauge perturbation,
\(Q_{1,ij}=\partial_i\xi_j+\partial_j\xi_i\), so
\[
 B^i(K\xi)=-\Delta\xi^i,\qquad Y_1^i(K\xi)=\xi^i.
\]
These checks fix the displacement signs.

\subsection{Physical contribution ledger}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.31\linewidth}Y@{}}
\toprule
Contribution & Status\\
\midrule
Bare lapse derivatives &
Derived in \eqref{eq:W1}--\eqref{eq:N}; all components retained.\\
Clock displacement &
Derived in \eqref{eq:u}; second-order scalar data retained.\\
Spatial relational correction &
Derived in \eqref{eq:Y} and \eqref{eq:n2}; one spatial inverse
in the quadratic lapse.\\
Second spatial rod correction &
Constructed, but absent from the quadratic lapse because
\(\partial_iN_0=0\).\\
Complete classical quadratic kernel &
Derived in \eqref{eq:kernel-bridge}.\\
Seven-row retarded architecture of Em02-A--D &
Not identified with this Hessian and not imposed.\\
Quantum contractions and extension terms &
Require the additional inputs in Section~\ref{sec:quantum}.\\
Ghost and quantum gauge contributions &
Not supplied by the bosonic classical Hessian.\\
\bottomrule
\end{tabularx}
\end{center}

%=====================================================================
\section{Noncompact support and instantaneous singularities}
\label{sec:support}
%=====================================================================

\subsection{Compact external smearing does not compactify the Hessian}

\begin{proposition}[A noncompact classical kernel component]
\label{prop:noncompact}
There are compact test densities \(f\) for which
\(\Hess_f\) has noncompact spatial support in a field slot.
This occurs even though the external smearing is compact.
\end{proposition}

\begin{proof}
Use two independent perturbation directions.

In the first direction take
\[
 h_{ij}=2a^2D\,\delta_{ij},\qquad
 h_{00}=h_{0i}=0,\qquad\varphi=0.
\]
Then \(P\eta=0\), \(Q_1=2D\mathbf1\), and
\(B^i\eta=\partial_iD\).

In the second direction take
\[
 \theta_{00}=-2v\,u,\qquad
 \theta_{0i}=\theta_{ij}=0,\qquad\theta_\phi=0.
\]
Then \(P\theta=u\), \(B^i\theta=0\).
The local mixed term is zero: with both scalar perturbations
zero, the lapse's local quadratic metric term has no mixed
spatial-trace/time-time contribution.

Consequently the polarized quadratic form is
\[
 \frac12\int f_0(t,\mathbf x)
        \partial_i(\Gsp D)(t,\mathbf x)
        \partial_i u(t,\mathbf x)
        \dd t\,\dd^3\mathbf x.
\]
Integrating the compact \(D\) source by parts gives its
kernel coefficient
\[
 K_u(t,\mathbf z)
 =
 \frac12\int_{\R^3}
 f_0(t,\mathbf x)\,
 \partial_{x^i}G_\Delta(\mathbf x-\mathbf z)
 \partial_i u(t,\mathbf x)\dd^3\mathbf x.
\]

Choose \(f_0(t,\mathbf x)=\eta_0(t)F(\mathbf x)\), with
\(F\ge0\), \(\int F>0\), and let
\(u(t,\mathbf x)=\beta(t)x^1\) on a neighborhood of its support,
using a compact spatial cutoff outside that neighborhood.
Then for a unit vector \(\mathbf n\),
\[
 K_u(t,R\mathbf n)
 =
 -\frac{\eta_0(t)\beta(t)n_1}{8\pi R^2}
                 \int_{\R^3}F(\mathbf x)\dd^3\mathbf x
 +O(R^{-3}).
\]
At times where \(\eta_0\beta\ne0\), this is nonzero in cones
with \(n_1\ne0\) for arbitrarily large \(R\).
A compact \(D\) supported near any such point detects the
coefficient. Hence the kernel support is not spatially compact.
\end{proof}

This does not prevent testing the Hessian on compact field
directions. It does prevent inferring the compact functional
support required by some local-functional constructions.

\subsection{The instantaneous kernel has off-diagonal wavefront}

The background spatial inverse, considered as a spacetime
kernel with coordinate-volume pairing, is
\begin{equation}
 \mathcal G_0(t,\mathbf x;u,\mathbf y)
 =
 -\frac{a(t)^2}{4\pi}
       \frac{\delta(t-u)}{|\mathbf x-\mathbf y|}.
 \label{eq:instantaneous}
\end{equation}

\begin{proposition}[Off-diagonal singularity and scaling]
\label{prop:instantaneous}
On the region \(\mathbf x\ne\mathbf y\),
\begin{equation}
 \WF(\mathcal G_0)
 =
 \left\{
 \begin{array}{l}
 (t,\mathbf x;u,\mathbf y;
       \omega,0;-\omega,0):\\
 t=u,\ \mathbf x\ne\mathbf y,\ \omega\ne0
 \end{array}
 \right\}.
 \label{eq:WF}
\end{equation}
In relative variables
\((\sigma,\mathbf r)=(t-u,\mathbf x-\mathbf y)\),
the principal kernel
\[
 \delta(\sigma)|\mathbf r|^{-1}
\]
is homogeneous of degree \(-2\) and has scaling degree two
at the origin.
\end{proposition}

\begin{proof}
For \(\mathbf x\ne\mathbf y\), the spatial factor in
\eqref{eq:instantaneous} is smooth and nowhere zero.
Multiplication by it does not change the wavefront set
of \(\delta(t-u)\). In local coordinates
\((t-u,u,\mathbf x,\mathbf y)\), this is the tensor product
of a one-dimensional delta with a smooth nonvanishing factor.
Its nonzero covectors are exactly the conormal covectors to
\(t=u\), proving \eqref{eq:WF}.

Under positive dilation of all four relative coordinates,
\[
 \delta(\epsilon\sigma)|\epsilon\mathbf r|^{-1}
 =
 \epsilon^{-2}\delta(\sigma)|\mathbf r|^{-1}.
\]
The distribution is nonzero, so its scaling degree is exactly
two. The smooth nonzero background coefficient preserves
the leading scaling degree.
\end{proof}

\begin{remark}[A necessary microlocal correction]
A spacetime instantaneous Green kernel is not singular only on
the full spacetime diagonal. Equation \eqref{eq:WF} exhibits
equal-time singularities at spatially separated points.
Accordingly, a shorthand comparison with a full diagonal delta
cannot be used as a literal wavefront-set identity.

The explicit instantaneous kernel in
\cite[Section 5.2]{BFHPR} must be analyzed together with the
other graph factors. The present proposition concerns that
kernel type alone; it does not assert that a particular
complete graph has an off-diagonal divergence.
\end{remark}

% Instantaneous kernel and the separate tail qualification checked in:
% ([arxiv.org](https://arxiv.org/html/1605.02573))

\subsection{What scaling degree does and does not decide}

For a distribution \(t\) in \(\R^n\), use
\[
 \sd_0(t)
 =
 \inf\{\omega:\epsilon^\omega t(\epsilon\cdot)\to0
                           \text{ in }\Dp\text{ as }\epsilon\downarrow0\}.
\]
The scaling-degree extension theorems apply after an actual
distribution is defined away from the specified coincidence
set \cite[Theorems 5.2--5.3]{BF}.

\begin{lemma}[Difference of scaling-bounded point extensions]
\label{lem:ambiguity}
Suppose two extensions \(t,\widetilde t\) agree away from
the origin and have scaling degrees at most \(s<\infty\).
Then
\[
 t-\widetilde t
 =
 \sum_{|\alpha|\le\lfloor s-n\rfloor}
                 c_\alpha\partial^\alpha\delta^{(n)}
\]
when \(s\ge n\). If \(s<n\), they agree.
\end{lemma}

\begin{proof}
Their difference is supported at the origin.
A finite-order distribution supported at one point is a
finite linear combination of delta derivatives.
A nonzero highest derivative of order \(m\) has scaling
degree \(n+m\). The bound \(n+m\le s\) gives the result.
For \(s<n\), no nonzero point-supported term satisfies it.
\end{proof}

For a diagonal with tangential variables, the corresponding
coefficients can depend on those variables. Finite normal order
alone does not make the complete ambiguity space
finite-dimensional.

No scaling degree of a complete quantum lapse graph has been
calculated here. The degree two of the single kernel
\eqref{eq:instantaneous} is not the degree of its product with
a differentiated Feynman distribution.

% Extension theorem and its exact hypotheses:
% ([arxiv.org](https://arxiv.org/pdf/math-ph/9903028))

%=====================================================================
\section{Transport of internal counterterms}
\label{sec:counterterms}
%=====================================================================

The admissible correction space cannot be obtained by simply
naming the final attachment diagonal. It must be transported
through the actual observable construction.

The following calculation is a distributional support test.
It does not declare any particular counterterm physically
admissible.

\begin{proposition}[A diagonal correction with a non-diagonal image]
\label{prop:image}
Let \(c\in C_c^\infty(\R^3)\) be nonzero and define
\[
 C_c(\mathbf y,\mathbf z)
 =
 c(\mathbf y)\delta^{(3)}(\mathbf y-\mathbf z).
\]
Applying a spatial Green kernel in the first slot gives the
well-defined distribution
\begin{equation}
 \Pi C_c(\mathbf x,\mathbf z)
 =
 G_\Delta(\mathbf x-\mathbf z)c(\mathbf z).
 \label{eq:image}
\end{equation}
For every \(\mathbf z\) where \(c(\mathbf z)\ne0\), it is
nonzero on open sets with \(\mathbf x\ne\mathbf z\).
Its support is therefore not contained in the output diagonal.
\end{proposition}

\begin{proof}
The right side is locally integrable because
\(|\mathbf x-\mathbf z|^{-1}\) is locally integrable in
three relative spatial dimensions. For a compact smooth
test \(\Phi(\mathbf x,\mathbf z)\), its action is
\[
 \int c(\mathbf z)
       \left[\int G_\Delta(\mathbf x-\mathbf z)
                              \Phi(\mathbf x,\mathbf z)
                              \dd^3\mathbf x\right]\dd^3\mathbf z.
\]
This is also the distributional evaluation of
\[
 \int G_\Delta(\mathbf x-\mathbf y)
                   c(\mathbf y)\delta^{(3)}(\mathbf y-\mathbf z)
                   \dd^3\mathbf y.
\]
At separated points the kernel is smooth and nonzero, proving
the support assertion.
\end{proof}

Time can be retained as a common parameter in
\eqref{eq:image}. No product of two time deltas is required.

For a source derivative acting on the internal incidence,
the precise formula is
\[
 \Pi\bigl(c(\mathbf y)\partial_{\mathbf y}^\alpha
                 \delta^{(3)}(\mathbf y-\mathbf z)\bigr)
 =
 (-1)^{|\alpha|}
 \left[
 \partial_{\mathbf y}^\alpha
       \{G_\Delta(\mathbf x-\mathbf y)c(\mathbf y)\}
 \right]_{\mathbf y=\mathbf z},
\]
interpreted distributionally. This retains derivatives of
both the coefficient and the Green kernel.

\subsection{The prospective physical support image}

For the quantum calculation, one would first classify
graph-level corrections on the actual singular subgraphs.
If the derived observable construction defines a map
\(\Pi\) on that class, its image is
\begin{equation}
 \Cobs=\Pi(\Cgraph).
 \label{eq:physical-image}
\end{equation}
Equation \eqref{eq:physical-image} is an interface to be
instantiated. It is not an evaluated physical correction space
in this paper.

Proposition~\ref{prop:image} rules out the inference
\[
 \text{internal diagonal support}
 \quad\Longrightarrow\quad
 \text{final diagonal support}.
\]
It does not prove that every extended support is allowed.
Covariance, scaling, field content, gauge compatibility,
and normalization still constrain \(\Cgraph\).

\subsection{Independent admissibility requirements}

Before a subtraction is used to decide the quantum limit,
the chosen theory and order must determine:
\begin{enumerate}
 \item the graph distributions and their coincidence sets;
 \item the tensor structures and background/clock dependence
 permitted in their extensions;
 \item the derivative and scaling bounds;
 \item causal and local covariance conditions in the applicable
 renormalization framework;
 \item quantum BRST/BV conditions and the relevant equation ideal;
 \item allowed finite normalizations;
 \item existence and tail control for their observable-level images.
\end{enumerate}
This list is an admissibility contract, not a completed
classification of \(\Cadm\).

Neither successful cancellation nor failure of
attachment-diagonal support alone settles admissibility.

%=====================================================================
\section{Spatial tails and ultraviolet removal are separate limits}
\label{sec:tails}
%=====================================================================

\subsection{Elementary quantitative tail bounds}

\begin{lemma}[Uniform Coulomb tails]
\label{lem:tail}
Let \(K\subset\R^3\) be compact, and suppose
\[
 |F_\delta(\mathbf y)|
 \le C(1+|\mathbf y|)^{-q}
\]
uniformly in \(\delta\).
For sufficiently large \(R\),
\[
 \sup_{\mathbf x\in K}
 \int_{|\mathbf y|>R}
 |G_\Delta(\mathbf x-\mathbf y)F_\delta(\mathbf y)|
                                       \dd^3\mathbf y
 \le C_KR^{2-q}
 \qquad(q>2).
 \label{eq:tail-G}
\]
For a first spatial derivative of \(G_\Delta\),
\[
 \sup_{\mathbf x\in K}
 \int_{|\mathbf y|>R}
 |\partial_iG_\Delta(\mathbf x-\mathbf y)F_\delta(\mathbf y)|
                                       \dd^3\mathbf y
 \le C_KR^{1-q}
 \qquad(q>1).
 \label{eq:tail-dG}
\]
The estimates remain valid for time or spectator derivatives
when their sources satisfy the same uniform bounds.
\end{lemma}

\begin{proof}
For \(R\) larger than twice a ball containing \(K\),
\[
 |\mathbf x-\mathbf y|\ge|\mathbf y|/2.
\]
The Coulomb kernel is bounded by \(C|\mathbf y|^{-1}\),
and its first derivative by \(C|\mathbf y|^{-2}\).
Spherical integration therefore reduces the tails to
\[
 C\int_R^\infty r^{1-q}\dd r,
 \qquad
 C\int_R^\infty r^{-q}\dd r,
\]
respectively.
\end{proof}

Merely requiring \(F_\delta(\mathbf y)\to0\) at infinity
does not imply either estimate. For example,
\[
 F(\mathbf y)=(1+|\mathbf y|^2)^{-q/2}
\]
has a divergent absolute Coulomb integral when \(0<q\le2\).
At \(q=2\), the divergence is logarithmic.
Cancellations can improve particular integrals, but require
their own proof.

The classical sources in Proposition~\ref{prop:rods} are compact,
so their classical integrals exist. Quantum correlation
functions are not compact sources. Their required tail bounds
have not been inferred from the classical argument.

\subsection{A joint regulator-removal criterion}

Let \(R\) be a spatial cutoff radius, independent of the
ultraviolet scale \(\delta\).

\begin{theorem}[Uniform-tail joint-limit criterion]
\label{thm:joint}
Let \(v_{\delta,R}\in\Dp(X)\).
Suppose that for each fixed sufficiently large \(R\),
\[
 v_{\delta,R}\longrightarrow T_R
 \quad\text{strongly as }\delta\downarrow0,
\]
and, for every compact test support \(H\), there are
\(C_H,j,\beta>0\) such that
\begin{equation}
 |(v_{\delta,R'}-v_{\delta,R})(\Phi)|
 \le C_HR^{-\beta}\|\Phi\|_{j,H},
 \qquad R'\ge R,
 \label{eq:uniform-tail}
\end{equation}
uniformly for sufficiently small \(\delta\).
Then \(T_R\) converges strongly to a distribution \(T\), and
\[
 v_{\delta,R}\longrightarrow T
\]
jointly as \(\delta\downarrow0\) and \(R\to\infty\).
\end{theorem}

\begin{proof}
Pass to the fixed-\(R,R'\) limits in
\eqref{eq:uniform-tail}. The same bound holds for \(T_{R'}-T_R\).
It makes the family Cauchy on each compact test-support class.

An anchor \(T_{R_0}\), together with the tail bound, gives
a finite-order compact-local bound for the limit, so the limit
is a distribution. The estimate is uniform on bounded test
sets and gives strong convergence.

For joint convergence, fix a bounded test set and choose
\(R_0\) so its tail bound is small. For all \(R\ge R_0\),
compare \(v_{\delta,R}\) to \(v_{\delta,R_0}\).
The latter converges at fixed \(R_0\), and \(T_{R_0}\) is
close to \(T\). The three differences prove the joint limit.
\end{proof}

Without a uniform tail condition, fixed-\(R\) ultraviolet
convergence is insufficient. For a nonzero distribution \(D\),
\[
 v_{\delta,R}=\delta\log(1+R)D
\]
converges to zero for each fixed \(R\), but not along
\(R=\exp(1/\delta)-1\).

\subsection{A complete scale certificate remains conditional}

\begin{proposition}[Dini certificate with an anchor]
\label{prop:Dini}
Let \(u_\delta,S_\delta\in\Dp(X)\) have fixed-test pairings
locally absolutely continuous in \(\log\delta\).
Suppose
\[
 \scaleder(u_\delta-S_\delta)=R_\delta,
 \qquad
 |R_\delta(\Phi)|
 \le\omega_H(\delta)\|\Phi\|_{j,H},
\]
and
\[
 \int_0^{\delta_*}\omega_H(t)\frac{\dd t}{t}<\infty.
\]
Then \(u_\delta-S_\delta\) has a strong distributional limit
\(T\), with
\[
 |(u_\delta-S_\delta-T)(\Phi)|
 \le\|\Phi\|_{j,H}
          \int_0^\delta\omega_H(t)\frac{\dd t}{t}.
\]
\end{proposition}

\begin{proof}
Absolute continuity gives the scalar integral identity between
two positive scales. The bound makes every pairing Cauchy.
A fixed positive-scale anchor and its finite-order estimate,
combined with the integrable derivative bound, show that the
limit is a distribution. Uniformity on bounded test sets
proves strong convergence and the stated rate.
\end{proof}

For the quantum lapse, no complete \(S_\delta\) or
\(R_\delta\) satisfying this proposition has been calculated.
A leading-order bound from Em02-D is not such a certificate.

%=====================================================================
\section{The quantum input gate and the first contraction target}
\label{sec:quantum}
%=====================================================================

\subsection{What is identified and what is not}

The classical theory, background, clock, boundary condition,
observable, and quadratic Hessian are fixed by explicit formulas.
The quantum package is still incomplete.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.22\linewidth}Yp{.18\linewidth}@{}}
\toprule
Input & Required specification & Present status\\
\midrule
Action and background &
Equations \eqref{eq:action}--\eqref{eq:vH}. &
Specified\\[0.3em]
Relational observable &
Clock-slice harmonic rods and \eqref{eq:lapse},
with \eqref{eq:n1}--\eqref{eq:n2}. &
Specified classically\\[0.3em]
Gauge fixing &
Gauge fermion, auxiliary fields, ghost action, complete coupled
quadratic operator, and its domain. &
Not constructed\\[0.3em]
Green distributions &
Retarded and advanced inverses of that operator, causal
propagator, Feynman convention, component normalization. &
Not constructed\\[0.3em]
State or parametrix &
Identified two-point matrix with the required singular and
spatial-decay properties. &
Not selected\\[0.3em]
Quantum insertion &
Choice of Wick/time-ordered representation and retained coupling
and \(\hbar\) grading. &
Not completed\\[0.3em]
Graph corrections &
Complete scaling, covariance, causal, and gauge-compatible
classification. &
Not classified\\[0.3em]
Spatial images &
Well-defined image of those graph corrections through
the observable construction. &
Not calculated\\[0.3em]
Quantum gauge identity &
Breaking term and its admissible repair at the retained order. &
Not verified\\
\bottomrule
\end{tabularx}
\end{center}

Linearized Einstein--scalar quantization is a separate construction
with its own gauge and field-space hypotheses \cite{Hack}.
Naming that construction does not select a covariance for
\eqref{eq:action} and does not prove the tail bounds needed here.

% Linearized Einstein--scalar framework checked in the primary record:
% ([arxiv.org](https://arxiv.org/abs/1403.3957))

\subsection{A precise classical smoothing convention}

A smoothing convention can be fixed before a quantum state
is available.

Choose a real even nonnegative
\(\rho\in C_c^\infty(\R^4)\) with \(\int\rho=1\).
Let
\[
 \rho_\delta(x)=\delta^{-4}\rho(x/\delta),
 \qquad J_\delta\eta=\rho_\delta*\eta
\]
in the displayed background chart.
Choose a spatial cutoff \(\chi_R\) equal to one on
\(|\mathbf x|\le R\), supported in
\(|\mathbf x|<2R\), and a fixed time cutoff equal to one
on a neighborhood of the time projection of \(\supp f\).

Apply these cutoffs to field slots before smoothing.
The regulated Hessian is
\begin{equation}
 \Hess_{f,\delta,R}
 =
 \bigl(\chi_RJ_\delta^*\otimes\chi_RJ_\delta^*\bigr)\Hess_f,
 \label{eq:regulated-Hessian}
\end{equation}
with the fixed time cutoffs understood.
For finite \(R,\delta>0\), it is a smooth compact test kernel
on the two field slots.

This chart smoothing is an explicitly chosen regulator.
It is not claimed to be diffeomorphism covariant.
Any gauge breaking it causes must be calculated in the quantum
construction rather than presumed absent.

\subsection{The contraction count}

\begin{proposition}[First free contraction of the quadratic insertion]
\label{prop:count}
In a Gaussian free representation, the quadratic insertion
\[
 \Qf=\frac12\pair{\Hess_f}{\eta\otimes\eta}
\]
has at most one complete pair contraction.
If a two-point matrix \(H^{AB}\) is specified, the prospective
regulated scalar contraction is
\begin{equation}
 \frac{\hbar}{2}
 \sum_{A,B}
 \pair{H^{AB}}{\Hess_{f,\delta,R}^{AB}},
 \label{eq:prospective-contraction}
\end{equation}
with the slot convention fixed by the chosen representation.
The linear insertion has no complete self-contraction.
\end{proposition}

\begin{proof}
A complete pair contraction consumes two field factors.
There are two in the quadratic insertion and only one
possible pair. The factorial coefficient is the \(1/2\)
in the Hessian representation. A linear insertion contains
no pair.
For the smoothed compact kernel at finite regulators,
the pairing in \eqref{eq:prospective-contraction} is a
defined distributional pairing whenever \(H\) is supplied.
\end{proof}

For a bosonic symmetric Hessian, only the corresponding
symmetric part of a two-point matrix contributes to the
complete scalar contraction. Whether one is using a
Wick contraction, a state expectation, or a Feynman
time-ordering must nevertheless be specified.
These are not interchangeable normalization conventions.

Equation \eqref{eq:prospective-contraction} identifies the
first free scalar contraction to be studied. It is not
an evaluated quantum family, because \(H\) has not been chosen.
It does not enumerate interacting diagrams, and a field-degree
count alone does not justify calling it the complete one-loop
gravity observable.

\subsection{Why the stop is substantive}

The quantum discussion of the reference distinguishes Feynman
distributions from internal spatial Green kernels and separates
spatial-decay and Ward-identity requirements
\cite[Section 5]{BFHPR}. Those are additional operations beyond
the classical Hessian derived here.

No complete quantum coefficient is inferred from:
\begin{itemize}
 \item the existence of the classical Green integral;
 \item a local Hadamard singularity statement without tail control;
 \item the scaling degree of one kernel rather than its graph products;
 \item a subtraction chosen because it cancels a scalar divergence;
 \item a classical covariance identity without the quantum differential.
\end{itemize}

The stop is a statement about the incomplete construction
in this manuscript. It is not a proof that suitable states,
extensions, or quantum observables cannot exist.

%=====================================================================
\section{Quantum gauge compatibility as a separate obligation}
\label{sec:gauge}
%=====================================================================

The relevant quantum identity requires the deformed differential
on the actual renormalized domain.
Renormalized BV methods provide a framework for such a
construction \cite{FR}; they do not determine its anomaly
or counterterms for this lapse.

% Renormalized BV framework checked in the primary record:
% ([arxiv.org](https://arxiv.org/abs/1110.5232))

The following elementary interface isolates the required
calculation.

\begin{proposition}[Retained-order gauge repair criterion]
\label{prop:gauge}
Let \(V^0,V^1,V^2\) be the specified coefficient spaces of a
graded complex. Suppose
\[
 s_{\mathrm{cl}}^2=0,\qquad
 s_{\mathrm{cl}}s_1+s_1s_{\mathrm{cl}}=0,
 \qquad
 s_\hbar=s_{\mathrm{cl}}+\hbar s_1\pmod{\hbar^2}.
\]
Let \(O_{\mathrm{cl}}\in V^0\) satisfy
\(s_{\mathrm{cl}}O_{\mathrm{cl}}=0\), and let \(T\in V^0\)
be a constructed order-\(\hbar\) coefficient.
For a specified linear space of allowed finite corrections
\(A\subset V^0\), define
\[
 \mathcal A=s_{\mathrm{cl}}T+s_1O_{\mathrm{cl}}.
\]
Then
\[
 s_{\mathrm{cl}}\mathcal A=0,
\]
and the corrected expression
\[
 O_{\mathrm{cl}}+\hbar(T+a),\qquad a\in A,
\]
is closed modulo \(\hbar^2\) if and only if
\begin{equation}
 -\mathcal A\in s_{\mathrm{cl}}(A).
 \label{eq:gauge-image}
\end{equation}
\end{proposition}

\begin{proof}
Nilpotence and the anticommutation relation give
\[
 s_{\mathrm{cl}}\mathcal A
 =
 s_{\mathrm{cl}}^2T+s_{\mathrm{cl}}s_1O_{\mathrm{cl}}
 =
 -s_1s_{\mathrm{cl}}O_{\mathrm{cl}}=0.
\]
The coefficient of \(\hbar\) in the corrected closure
condition is
\[
 \mathcal A+s_{\mathrm{cl}}a=0.
\]
Its solvability in the stipulated correction space is
exactly \eqref{eq:gauge-image}.
\end{proof}

The actual coefficient spaces may be quotients by equations
of motion or exact terms. Those quotients and all maps must
be defined before using this proposition.
At a simultaneous field/coupling truncation, the differential
must preserve the retained coefficient equations, including
the coupling in \eqref{eq:classical-Ward}.

No \(s_1\), anomaly \(\mathcal A\), or physical space \(A\)
for the present lapse has been constructed here.
Gauge-fixing independence likewise remains a separate
unproved condition.

A convergent scalar pairing is not, by itself, a class in
quantum BRST cohomology.

%=====================================================================
\section{Release, reproducibility, and remaining proof obligations}
\label{sec:release}
%=====================================================================

\begin{theorem}[Classical and distributional release]
\label{thm:release}
For the theory, background, clock, and harmonic boundary
prescription fixed in this paper:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item the displayed background solves the Einstein--scalar
 equations and has a nondegenerate scalar clock;

 \item the clock displacement, induced spatial metric,
 harmonic-coordinate coefficients, and relational map
 are constructed through second order;

 \item the full-tensor relational lapse coefficients are
 \eqref{eq:n1}--\eqref{eq:n2};

 \item the complete classical quadratic functional has the
 local-plus-spatial-Green representation
 \eqref{eq:kernel-bridge};

 \item its classical covariance and coefficient identities
 are those of Proposition~\ref{prop:covariance} and
 \eqref{eq:classical-Ward};

 \item compact external smearing does not generally give
 spatially compact Hessian support;

 \item the instantaneous kernel has the off-diagonal wavefront
 set \eqref{eq:WF}, and an internal diagonal correction can
 have the non-diagonal image \eqref{eq:image};

 \item the tail bounds and conditional joint-limit, Dini,
 and gauge-repair criteria hold on their stated domains.
\end{enumerate}
These statements do not establish quantum regulator removal,
quantum gauge closure, gauge-fixing independence, or a universal
obstruction to the relational lapse.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:background} proves (i).
Lemma~\ref{lem:clock}, Proposition~\ref{prop:induced},
Lemma~\ref{lem:laplacian}, Proposition~\ref{prop:rods},
and the series inversions prove (ii).
Theorem~\ref{thm:lapse} proves (iii).
Theorem~\ref{thm:kernel} proves (iv).
Proposition~\ref{prop:covariance} and differentiation of its
identity prove (v).
Propositions~\ref{prop:noncompact},
\ref{prop:instantaneous}, and \ref{prop:image} prove (vi)--(vii).
Lemma~\ref{lem:tail}, Theorem~\ref{thm:joint}, and
Propositions~\ref{prop:Dini} and \ref{prop:gauge} prove (viii).
\end{proof}

\subsection{Exact quantum outcome}

\begin{center}
\textbf{Quantum outcome: incomplete audit.}
\end{center}

The first missing quantum operation is the construction and
selection of the gauge-fixed two-point package needed in
\eqref{eq:prospective-contraction}, with a domain appropriate
to the noncompact Hessian. Even after that operation,
the correction classification, complete ultraviolet remainder,
uniform spatial tails, and quantum gauge identity remain
necessary.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}Yp{.31\linewidth}@{}}
\toprule
Milestone & Outcome\\
\midrule
Explicit Einstein--scalar background and classical observable &
Constructed\\[0.3em]
Complete classical quadratic expansion and kernel bridge &
Constructed\\[0.3em]
Physical quantum covariance and state/parametrix package &
Not constructed\\[0.3em]
Complete graph-level admissible correction class &
Not classified\\[0.3em]
Observable-level image of that class &
Not evaluated\\[0.3em]
Complete quantum singular expansion or Dini remainder &
Not calculated\\[0.3em]
Uniform quantum spatial-tail estimates &
Not proved\\[0.3em]
Quantum gauge identity and gauge-fixing independence &
Not verified\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Relationship to the supplied Em02 papers}

Em02-A-v6 distinguishes executable physical identification
from generic seed geometry. Its localization powers do not
determine the powers of the present quantum contraction.

Em02-C completely decides a separately declared crossed-delta
model. Its correction spaces are not the physical correction
spaces of this lapse.

Em02-D-v4 states a leading expansion for a fixed retarded
two-component model. Neither its coefficient nor its
incidence-supported correction obstruction is transferred to
\eqref{eq:kernel-bridge}. In particular, the spatial support
transport in Proposition~\ref{prop:image} must be accounted
for before a physical support obstruction can be formulated.

The supplied documents are not used as premises for the
classical calculations in this paper. No registration or
external programme status is verified or changed here.

\subsection{Reproducibility statement}

All exact inputs and classical coefficient operations used in
the release theorem are printed in this source.
The checks are algebraic substitutions, elliptic integral
identities, and distributional support estimates.

No symbolic script, numerical output, immutable artifact,
or successful compilation is claimed.
Such artifacts could support a later audit, but would not
replace the quantum remainder or gauge proof.

%=====================================================================
\appendix
\section{Factorial conventions and an explicit implementation dictionary}
\label{app:dictionary}
%=====================================================================

The convention
\[
 F_\lambda=F_0+\lambda F_1+\frac{\lambda^2}{2}F_2
\]
is used throughout.
For a general second-order field jet,
\[
 [\lambda^2]\Of(\Gamma_\lambda)
 =
 \frac12\Lf(\eta_2)+\Qf(\eta).
\]
Thus the second-order field data \(k,\psi\) in the full lapse
are not additional quadratic free-field factors.

The classical implementation dictionary is:
\begin{enumerate}[label=\textup{C\arabic*.}]
 \item Evaluate \(a,H,v,\dot v,N_0,\dot N_0,\ddot N_0\)
 from \eqref{eq:background}--\eqref{eq:vH}.

 \item Compute \(u_1,u_2\) from \eqref{eq:u}.

 \item Compute \(q_1,q_2\) from \eqref{eq:q1}--\eqref{eq:q2}.
 For the quadratic lapse, only \(q_1\) is needed in its
 spatial rod term.

 \item Form \(Q_1,Q_2,b_1,b_2,A_1,A_2,P_1,P_2\)
 from \eqref{eq:b}--\eqref{eq:P2}.

 \item Apply the fixed decaying inverse \eqref{eq:Coulomb}
 to obtain \(Y_1,Y_2\) using \eqref{eq:Y}.

 \item Raise spacetime indices with
 \(g_0^{00}=-1\), \(g_0^{0i}=0\),
 \(g_0^{ij}=a^{-2}\delta^{ij}\).
 Evaluate \eqref{eq:W1}--\eqref{eq:N}.

 \item Assemble \(n_1,n_2\) using
 \eqref{eq:n1}--\eqref{eq:n2}.

 \item Multiply by the fixed background test density and
 integrate. No configuration-dependent volume factor is
 added to \eqref{eq:observable}.
\end{enumerate}

This dictionary determines the classical coefficients without
a scalar-vector-tensor decomposition or unspecified component
maps.

%=====================================================================
\section{Checks that prevent false quantum conclusions}
\label{app:debug}
%=====================================================================

\begin{enumerate}[label=\textup{D\arabic*.}]
 \item The release title is an identification and renormalization
 audit, not a completed regulator-removal claim.

 \item The theory and background are explicit.
 They are not a flat declared model renamed cosmological.

 \item The original inflaton-slice rods are distinguished from
 the modified rods of the reference. Their quadratic observables
 need not agree.

 \item Cosmic-time normalization is fixed independently.
 The conformal-time comparison is performed explicitly at
 first order.

 \item The lapse uses the positive timelike-clock branch.
 The scalar gradient is not normalized with a Euclidean
 contraction.

 \item Metric, clock, and coordinate-map perturbations retain
 the same factorial convention.

 \item Vector and tensor metric components are not discarded.
 The second-order lapse is expressed in full tensor contractions.

 \item The spatial Green operator keeps its decaying boundary
 condition. It is not replaced by a compact retarded cone.

 \item Second-order spatial rods are constructed, but their
 absence from the quadratic lapse is proved rather than assumed.

 \item Classical covariance is proved with the asymptotic
 boundary condition and the stated gauge group.

 \item The Hessian and linear observable enter one coupled
 classical gauge identity. The quadratic coefficient is not
 asserted to be separately invariant under a linearized
 transformation.

 \item A complete classical kernel bridge is not a quantum
 contraction or a renormalized graph.

 \item Equal-time singularities at separated spatial points
 are retained in the instantaneous wavefront audit.

 \item The scaling degree of a single spatial Green kernel is
 not assigned to a product of quantum distributions.

 \item Compact external smearing is not confused with compact
 functional support.

 \item Internal diagonal counterterms are transported before
 final support restrictions are inferred.

 \item The counterterm-image example proves a support mechanism,
 not physical admissibility of its coefficient.

 \item Ultraviolet and spatial cutoffs are independent.
 Fixed-radius convergence does not prove their joint limit.

 \item A quantum covariance, state, and gauge-fixed operator
 are not supplied by names or by a classical background.

 \item One free pair contraction is an \(\hbar\)-count.
 It is not a complete interacting-loop inventory.

 \item No complete singular subtraction, integrable quantum
 remainder, or anomaly calculation is hidden in a conditional
 decision theorem.

 \item No absence of a construction is promoted to a universal
 nonexistence theorem.
\end{enumerate}

%=====================================================================
\begin{thebibliography}{9}

\bibitem{BFHPR}
R.~Brunetti, K.~Fredenhagen, T.-P.~Hack, N.~Pinamonti,
and K.~Rejzner,
\emph{Cosmological perturbation theory and quantum gravity},
Journal of High Energy Physics \textbf{2016}, no.~8, 032.
\href{https://arxiv.org/abs/1605.02573}{arXiv:1605.02573}.
\newblock
Used for the relational prescription and the distinction
between the original and modified spatial rods.
The present tensor expansion and support calculations are
derived in the stated cosmic-time conventions.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
\emph{Microlocal Analysis and Interacting Quantum Field Theories:
Renormalization on Physical Backgrounds},
Communications in Mathematical Physics \textbf{208}
(2000), 623--661.
\href{https://arxiv.org/abs/math-ph/9903028}
     {arXiv:math-ph/9903028}.
\newblock
Theorems 5.2--5.3 supply the scaling-degree extension framework.
They are not an evaluated extension or counterterm
classification for the nonlocal lapse.

\bibitem{Hack}
T.-P.~Hack,
\emph{Quantization of the linearised Einstein--Klein--Gordon
system on arbitrary backgrounds and the special case of
perturbations in inflation},
Classical and Quantum Gravity \textbf{31}
(2014), 215004.
\href{https://arxiv.org/abs/1403.3957}{arXiv:1403.3957}.
\newblock
A linearized gauge-theoretic quantization framework.
No covariance or spatial-tail certificate for the present
quadratic insertion is selected by this citation.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Batalin--Vilkovisky formalism in perturbative algebraic
quantum field theory},
Communications in Mathematical Physics \textbf{317}
(2013), 697--725.
\href{https://arxiv.org/abs/1110.5232}{arXiv:1110.5232}.
\newblock
Framework background for the separate renormalized gauge
obligation. No quantum closure theorem for the lapse is
inferred from it.

\end{thebibliography}

\end{document}