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\hypersetup{
  pdftitle={XIX-A: Clock Separation Before Coordinate Identification},
  pdfauthor={Ngo Cheung},
  pdfsubject={Classical relational observables and non-identification},
  pdfkeywords={relational scalar, scalar clock, harmonic coordinates,
    relational lapse, incidence parameter, regulator}
}

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\title{\textbf{XIX-A}\\[3pt]
Clock Separation Before Coordinate Identification:\\
Four Labelled Objects and Five Non-Identifications}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We distinguish four time-related constructions: a finite-time
harmonic relational scalar, an inflaton-clock construction with
original slice-harmonic spatial rods and its relational lapse,
a prescribed conical incidence parameter with a finite register,
and a smooth ramp inserted into a coordinate density. The last
object is a regulator, not a dynamical clock.

The distinctions are formulated as statements about domains,
defining equations, auxiliary data, and functional dependence.
They are supported by explicit calculations. A relational scalar
value does not determine the scalar-clock lapse. Original and
modified cosmological rods have different first variations, and
an explicit compact perturbation makes their quadratic relational
lapses different. A fixed density ramp has vanishing field
variation and fails the harmonic pure-gauge response test in its
transition region. A branch-normal deformation speed depends on
the prescribed parametrization and does not determine a
matter-defined lapse.

These are classical separation results on the stated kinematic
domains. They neither identify the coordinate prescription of
Landmark VI with that required by XVI-D nor establish a quantum
Ward identity, regulator removal, a continuum evolution, or
quantum evolution without an external time. The supplied Family
271 catalogue and its reported Lean scope remain documentary
comparisons, not theorem premises.
\end{abstract}

\noindent\textbf{Keywords:}
relational observables; harmonic coordinates; scalar clocks;
relational lapse; spatial rods; prescribed slicing; density ramp.

\tableofcontents

%=====================================================================
\section{Question, mathematical claim, and source discipline}
\label{sec:scope}
%=====================================================================

The question is whether several constructions called a ``clock'',
``time'', or ``lapse'' can be used as the same functional.
The answer requires more than a shared word. It requires compatible
configuration spaces, prescriptions, normalizations, and dependence
on the declared data.

The comparison is motivated by Cheung's Landmark VI,
VIII-A, VIII-B, X-A, and Em02-E
\cite{CheungVI,CheungVIIIA,CheungVIIIB,CheungXA,CheungEm02E}.
The supplied records separate their classical constructions from
their remaining quantum and dynamical obligations. We do not use
a documentary separation edge as a mathematical premise.
Instead, the definitions needed for the present calculations are
written below, and the separation statements are proved directly.

The general relational-coordinate framework is developed in
\cite{BFHPR,Frob}. Hyperbolic coordinate prescriptions and
slice-elliptic prescriptions are different choices within that
framework. Their equations and boundary data form part of the
observable definition.

\paragraph{The correction to an overstrong reading.}
``Not the same functional'' does \emph{not} mean that the numerical
values must differ at every configuration or point. Nor does it mean
that no future comparison map could relate the constructions.
The claim proved here is that they are not interchangeable under
the declarations in this paper. Where two expressions can be
compared on a common domain, we give a detecting variation or a
counterexample to the proposed identification.

\paragraph{The correction to an underdetermined reading.}
The supplied VI excerpts state that a harmonic-coordinate Cauchy
prescription is fixed, but do not reproduce all of its defining
equations and initial data. Consequently, Section~\ref{sec:harmonic}
declares an explicit finite-time harmonic prescription for the
mathematical comparison. It does not silently authenticate every
detail of the registered VI prescription.
A VI instance can use the results only after its actual prescription
has been matched to the displayed one. This qualification prevents
a separation paper from becoming an unproved identification paper.

\paragraph{Classical domain.}
Unless explicitly stated otherwise, perturbations below are
off shell. They need not solve the coupled Einstein--scalar
equations. A counterexample on this kinematic domain is not
automatically a counterexample on a smaller solution space.

\paragraph{Standing boundary.}
The paper does not establish quantum evolution without an external
time. It constructs no interacting quantum product, state,
field-theoretic instrument, or dynamical continuum limit.

%=====================================================================
\section{Typed objects and the meaning of identification}
\label{sec:typed}
%=====================================================================

\begin{definition}[A declared functional object]
\label{def:object}
A declared functional object consists of
\[
 \OO=(\mathscr D,\mathscr V,F,\mathscr A,\mathscr E),
\]
where $\mathscr D$ is its domain, $\mathscr V$ its target,
$F:\mathscr D\to\mathscr V$ its functional,
$\mathscr A$ its auxiliary data, and $\mathscr E$ its defining
equations and boundary or update conditions.

A literal functional identification requires the same declared
domain and target, with equality of the functionals.
An identification between different domains additionally requires
specified comparison maps and an equality after those maps.
Equality of one evaluation is not either identification.
\end{definition}

The labels used throughout the paper are
\begin{equation}
\label{eq:four-labels}
 \boxed{
 \mathsf C_{\mathrm{VI}},\qquad
 \mathsf C_{\mathrm E},\qquad
 \mathsf C_{\mathrm{VIII}},\qquad
 \mathsf R_{\mathrm X}.}
\end{equation}
These are four labelled entries in the time-related ledger.
They are not four verified physical clocks.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.19\linewidth}YY@{}}
\toprule
Label & Declared object & Data that cannot be omitted\\
\midrule
$\mathsf C_{\mathrm{VI}}$
&
Harmonic coordinate map and relational scalar
$\RR=T\circ\HH_q^{-1}$
&
Metric, scalar, initial surface, coordinate-time Cauchy data
\\[4pt]
$\mathsf C_{\mathrm E}$
&
Scalar clock $T$, original rods $Y_T$, and relational lapse
$\NN$
&
Timelike branch, clock calibration, spatial boundary condition
\\[4pt]
$\mathsf C_{\mathrm{VIII}}$
&
Incidence parameter $s$ and a finite register carrying selected
parameter values
&
Prescribed cone, slicing family, coding, and update
\\[4pt]
$\mathsf R_{\mathrm X}$
&
Prescribed ramp $\sigma_\varepsilon$ in a coordinate density
&
External coordinate time, profile, width, and frozen prescription
\\
\bottomrule
\end{tabularx}
\end{center}

\begin{lemma}[Factorization test]
\label{lem:factor}
Let $\pi:\mathscr D\to\mathscr B$ and $F:\mathscr D\to\mathscr V$.
There exists a map $\overline F:\pi(\mathscr D)\to\mathscr V$
such that $F=\overline F\circ\pi$ if and only if $F$ is constant
on every fiber of $\pi$.
\end{lemma}

\begin{proof}
A composition with $\pi$ is constant on each fiber.
Conversely, if $F$ is constant on fibers, define
$\overline F(\pi(d))=F(d)$. Fiber constancy makes this definition
independent of the representative.
\end{proof}

The lemma will distinguish a scalar value from the lapse depending
on its gradient, and a geometric schedule from a lapse requiring
additional matter data. It is an elementary criterion, not a
transfer of a companion theorem.

%=====================================================================
\section{The finite-time harmonic relational scalar}
\label{sec:harmonic}
%=====================================================================

\subsection{An explicit comparison prescription}

Let
\[
 M=[t_0,t_1]\times\R^3,\qquad
 \eta=-\dd t^2+\delta_{ij}\dd x^i\dd x^j.
\]
All fields extend smoothly to a neighborhood of the closed slab.
We use a formal parameter $e$ and work modulo $e^3$:
\begin{equation}
\label{eq:metric-jet}
 q_e=\eta+e h+\frac{e^2}{2}k.
\end{equation}
The coefficients $h,k$ are real smooth symmetric tensors with
spatial supports contained in one fixed compact set over the slab.
Indices in this subsection are raised with $\eta$.

Write the inverse metric density as
\begin{equation}
\label{eq:density-jet}
 \mathfrak a_e^{\mu\nu}
 :=\sqrt{-\det q_e}\,q_e^{\mu\nu}
 =\eta^{\mu\nu}+e A_1^{\mu\nu}
      +\frac{e^2}{2}A_2^{\mu\nu}.
\end{equation}
Put $h_{\mathrm{tr}}=\eta^{\mu\nu}h_{\mu\nu}$ and
$k_{\mathrm{tr}}=\eta^{\mu\nu}k_{\mu\nu}$. Direct matrix expansion gives
\begin{align}
\label{eq:A1}
 A_1^{\mu\nu}
 &=\frac12h_{\mathrm{tr}}\eta^{\mu\nu}-h^{\mu\nu},\\
\label{eq:A2}
 A_2^{\mu\nu}
 &=\frac12k_{\mathrm{tr}}\eta^{\mu\nu}-k^{\mu\nu}
   +2h^\mu{}_\rho h^{\rho\nu}
   -h_{\mathrm{tr}}h^{\mu\nu}
   +\left(\frac14h_{\mathrm{tr}}^2
          -\frac12h_{\rho\sigma}h^{\rho\sigma}\right)
             \eta^{\mu\nu}.
\end{align}

\begin{definition}[Finite-time harmonic prescription]
\label{def:harmonic}
The formal coordinate map
\[
 \HH_{q_e}^\mu=x^\mu+e W_1^\mu+\frac{e^2}{2}W_2^\mu
\]
is defined by
\begin{equation}
\label{eq:harmonic}
 \partial_\alpha\!
 \left(\mathfrak a_e^{\alpha\beta}
       \partial_\beta\HH_{q_e}^\mu\right)=0
       \pmod{e^3},
\end{equation}
and the ordinary coordinate-time Cauchy conditions
\begin{equation}
\label{eq:harmonic-data}
 W_j^\mu(t_0,\mathbf x)=0,\qquad
 \partial_tW_j^\mu(t_0,\mathbf x)=0,
 \qquad j=1,2.
\end{equation}
A metric-dependent unit-normal derivative is not substituted for
$\partial_t$ in these conditions.
\end{definition}

Expanding \eqref{eq:harmonic} gives
\begin{align}
\label{eq:W1}
 \Boxeta W_1^\mu
 &=-\partial_\alpha A_1^{\alpha\mu},\\
\label{eq:W2}
 \Boxeta W_2^\mu
 &=-\partial_\alpha A_2^{\alpha\mu}
   -2\partial_\alpha
        \left(A_1^{\alpha\beta}\partial_\beta W_1^\mu\right).
\end{align}

\begin{lemma}[Existence and uniqueness of the displayed prescription]
\label{lem:harmonic-existence}
Equations \eqref{eq:W1}--\eqref{eq:W2} with
\eqref{eq:harmonic-data} have unique smooth coefficient fields.
Uniqueness does not require spatial decay of the compared smooth
solutions.
\end{lemma}

\begin{proof}
For a smooth spatially compact forcing $F$, the solution of
$\Boxeta u=F$ with zero coordinate-time data is
\[
 u(t)
 =-\int_{t_0}^{t}
   \frac{\sin((t-s)|D|)}{|D|}F(s)\,\dd s,
 \qquad |D|=\sqrt{-\Delta}.
\]
The quotient has its continuous value $t-s$ at zero frequency.
Spatial Fourier transformation verifies the equation and data.
Smoothness follows by differentiating the Fourier integral, or by
the standard local wave estimates. Apply this construction first
to \eqref{eq:W1}, and then to \eqref{eq:W2}. The second forcing is
spatially compact because $A_1,A_2$ are.

For uniqueness, let $u$ be a smooth homogeneous solution with zero
data. On a shrinking ball of radius $R-(t-t_0)$, its wave energy
\[
 E(t)=\frac12\int_{\abs{\mathbf x-\mathbf x_*}<R-(t-t_0)}
       \left((\partial_tu)^2+\abs{\nabla u}^2\right)\dd^3x
\]
has nonpositive derivative. The boundary flux is
\[
 -\frac12\int_{\partial B}
   \left((\partial_tu-\partial_ru)^2
                  +\abs{\nabla_{\mathrm{tan}}u}^2\right)\dd S.
\]
Since $E(t_0)=0$, the solution vanishes on the corresponding
interior cone. Taking $R$ large enough treats any chosen point.
The argument is local and imposes no condition at spatial infinity.
Apply it successively to the differences of the first and second
coefficient solutions.
\end{proof}

This calculation is the flat finite-time version of the
Cauchy construction discussed in \cite{BGP}. It does not construct
an interacting gravitational solution.

\subsection{The relational scalar is not the coordinate map}

For a formal scalar
\[
 T_e=T_0+e\theta_1+\frac{e^2}{2}\theta_2,
\]
define
\begin{equation}
\label{eq:rel-scalar}
 \RR[q_e,T_e]:=T_e\circ\HH_{q_e}^{-1}.
\end{equation}
The inverse is formal, so no global numerical convergence of a
perturbation series is asserted.

\begin{proposition}[Scalar expansion]
\label{prop:scalar-expansion}
The coefficients of \eqref{eq:rel-scalar} are
\begin{align}
 \RR_0&=T_0,\nonumber\\
 \RR_1&=\theta_1-W_1^\mu\partial_\mu T_0,\nonumber\\
\label{eq:scalar-second}
 \RR_2
 &=\theta_2-2W_1^\mu\partial_\mu\theta_1
   +\left(2W_1^\nu\partial_\nu W_1^\mu-W_2^\mu\right)
       \partial_\mu T_0
   +W_1^\mu W_1^\nu\partial_\mu\partial_\nu T_0.
\end{align}
\end{proposition}

\begin{proof}
Solving $\HH_{q_e}\circ\HH_{q_e}^{-1}=\id$ coefficientwise gives
\[
 \HH_{q_e}^{-1}(x)
 =x-eW_1(x)
   +\frac{e^2}{2}
       \left(2W_1^\nu\partial_\nu W_1-W_2\right)(x).
\]
Taylor expansion of $T_e$ at this inverse map gives the formula.
\end{proof}

A harmonic coordinate component $\HH_q^0$ and the scalar
$T\circ\HH_q^{-1}$ are different objects. Moreover, a general
relational scalar need not have a timelike gradient and therefore
need not be a scalar clock.

\begin{proposition}[Restricted classical covariance]
\label{prop:harmonic-covariance}
Let $F_e$ be a formal diffeomorphism equal to the identity on a
neighborhood of the initial surface, with compactly supported
positive-order generators. Then
\[
 \HH_{F_e^*q_e}=\HH_{q_e}\circ F_e,\qquad
 \RR[F_e^*q_e,F_e^*T_e]=\RR[q_e,T_e]
 \pmod{e^3}.
\]
\end{proposition}

\begin{proof}
The scalar wave equation is covariant under pullback, so
$\HH_{q_e}\circ F_e$ solves the coordinate equation for $F_e^*q_e$.
The initial data agree because $F_e$ is the identity in an initial
collar. Coefficientwise uniqueness proves the first equality.
Since $F_e^*T_e=T_e\circ F_e$, the inverse-map identity proves the
second.
\end{proof}

For example, if
$F_e=\exp(e\xi+\tfrac12e^2\zeta)$, the factorial convention gives
\[
 h'=h+\Lie_\xi\eta,\qquad
 k'=k+2\Lie_\xi h+\Lie_\zeta\eta+\Lie_\xi^2\eta.
\]
A second-order coordinate expansion is not combined with only a
first-order metric transformation. The covariance statement is
classical and includes the complete retained jet.

%=====================================================================
\section{The scalar-clock lapse and the original spatial rods}
\label{sec:inflaton}
%=====================================================================

\subsection{Clock, lapse, and relational pullback}

Let $q$ have signature $(-,+,+,+)$, and let $T$ be a smooth scalar
with
\begin{equation}
\label{eq:timelike}
 S[q,T]:=-q^{\mu\nu}\partial_\mu T\partial_\nu T>0.
\end{equation}
The clock orientation is chosen so that $T$ increases to the
future. Its future unit normal and raw lapse are
\begin{equation}
\label{eq:clock-normal}
 n_T^\mu=-\frac{q^{\mu\nu}\partial_\nu T}{\sqrt{S[q,T]}},
 \qquad
 N[q,T]=S[q,T]^{-1/2}.
\end{equation}
Thus $n_T(T)=\sqrt{S}$ and $Nn_T(T)=1$.

The lapse is neither the scalar value $T$ nor its normal derivative.
It is a functional of the metric and the first derivatives of $T$.

Let $\Delta_{q,T}$ be the Laplace--Beltrami operator of the
Riemannian metric induced on a $T$-level set.
The original rods satisfy
\begin{equation}
\label{eq:original-rods}
 \Delta_{q,T}Y_T^i=0,\qquad
 Y_T^i-x^i\longrightarrow0
 \quad\hbox{at spatial infinity}.
\end{equation}
These are slice-elliptic equations, not spacetime wave equations.
The original and modified choices in
\cite[arXiv equations (28)--(35)]{BFHPR} must be distinguished.
The supplied Em02-E record explicitly selects the original rods
\cite{CheungEm02E}.

For a fixed background clock $T_0(t)$ with $\dot T_0>0$, let
\begin{equation}
\label{eq:relmap}
 X_{q,T}=(T,Y_T^1,Y_T^2,Y_T^3),\qquad
 \alpha_{q,T}=X_{q,T}^{-1}\circ X_0,
 \qquad X_0=(T_0(t),x^1,x^2,x^3).
\end{equation}
The relational lapse is
\begin{equation}
\label{eq:rel-lapse}
 \NN[q,T]:=N[q,T]\circ\alpha_{q,T}.
\end{equation}
The fixed clock calibration in \eqref{eq:relmap} is part of this
definition.

\subsection{The cosmological comparison background}

Use the displayed background of the supplied Em02-E record:
\begin{align}
\label{eq:background}
 q_0&=-\dd t^2+a(t)^2\delta_{ij}\dd x^i\dd x^j,\qquad
 a(t)=a_*(t/t_*)^p,\\
 T_0(t)&=\phi_*+\sqrt{2p}\,\MPl\log(t/t_*),\qquad p>1,\nonumber\\
 H(t)&=\frac{\dot a}{a}=\frac pt,\qquad
 v(t)=\dot T_0(t)=\frac{\sqrt{2p}\,\MPl}{t}>0.\nonumber
\end{align}
Work on a fixed closed subinterval of $(t_*/2,2t_*)$.
There is no limit toward $t=0$.
The background lapse is
\begin{equation}
\label{eq:N0}
 N_0(t)=\frac1{v(t)}.
\end{equation}

Only the displayed geometry and nondegenerate clock are needed
below. We do not import an Einstein-equation theorem as a premise
of a separation calculation.

\begin{lemma}[The background scalar clock is not a harmonic time coordinate]
\label{lem:clock-not-harmonic}
On \eqref{eq:background},
\[
 \Box_{q_0}T_0
 =-\ddot T_0-3H\dot T_0
 =-\frac{(3p-1)\sqrt{2p}\,\MPl}{t^2}\ne0.
\]
In particular, the inflaton clock is not a spacetime-harmonic
coordinate component on this background.
\end{lemma}

\begin{proof}
For a homogeneous scalar,
$\Box_{q_0}u=-\ddot u-3H\dot u$.
Substitution of \eqref{eq:background} gives the formula.
\end{proof}

This lemma concerns the clock field itself. It does not identify
or compare an independently dressed matter scalar with a
coordinate component.

\subsection{A self-contained second-order composition identity}

All perturbations use the factorial convention
\[
 q_e=q_0+eh+\frac{e^2}{2}k,\qquad
 T_e=T_0+e\theta_1+\frac{e^2}{2}\theta_2.
\]
Write
\[
 q_e^{-1}=R_0+eR_1+\frac{e^2}{2}R_2,\qquad
 S[q_e,T_e]=v^2+eS_1+\frac{e^2}{2}S_2.
\]
Matrix inversion gives
\[
 R_0=q_0^{-1},\qquad
 R_1=-q_0^{-1}hq_0^{-1},\qquad
 R_2=2q_0^{-1}hq_0^{-1}hq_0^{-1}-q_0^{-1}kq_0^{-1}.
\]
Consequently,
\begin{align}
 S_1
 &=-R_1^{\mu\nu}\partial_\mu T_0\partial_\nu T_0
   -2R_0^{\mu\nu}\partial_\mu T_0\partial_\nu\theta_1,
 \nonumber\\
 S_2
 &=-R_2^{\mu\nu}\partial_\mu T_0\partial_\nu T_0
   -4R_1^{\mu\nu}\partial_\mu T_0\partial_\nu\theta_1
 \nonumber\\
 &\hspace{8mm}
   -2R_0^{\mu\nu}\partial_\mu\theta_1\partial_\nu\theta_1
   -2R_0^{\mu\nu}\partial_\mu T_0\partial_\nu\theta_2.
 \label{eq:Scoeff}
\end{align}
No vector or tensor metric components have been discarded.

The raw lapse coefficients are
\begin{equation}
\label{eq:Ncoeff}
 N_1=-\frac{S_1}{2v^3},\qquad
 N_2=\frac{3S_1^2}{4v^5}-\frac{S_2}{2v^3}.
\end{equation}

Write
\[
 Y_e^i=x^i+eY_1^i+\frac{e^2}{2}Y_2^i.
\]
Define
\begin{equation}
\label{eq:ucoeff}
 u_1=-\frac{\theta_1}{v},\qquad
 u_2=-\frac{\theta_2+2u_1\dot\theta_1+\dot v\,u_1^2}{v}.
\end{equation}

\begin{proposition}[Rod dependence of the quadratic relational lapse]
\label{prop:lapse-composition}
The coefficients of $\NN=N\circ\alpha$ are
\begin{align}
\label{eq:n1}
 n_1&=N_1+\dot N_0u_1,\\
\label{eq:n2}
 n_2&=N_2+2u_1\dot N_1+\dot N_0u_2+\ddot N_0u_1^2
                 -2Y_1^i\partial_i n_1.
\end{align}
In particular, $Y_2$ does not occur in $n_2$.
\end{proposition}

\begin{proof}
Let
\[
 \alpha_e=x+e\alpha_1+\frac{e^2}{2}\alpha_2.
\]
The equations $T_e\circ\alpha_e=T_0$ and
$Y_e^i\circ\alpha_e=x^i$ imply
\[
 \alpha_1^0=u_1,\qquad \alpha_1^i=z_1^i=-Y_1^i.
\]
At second order in the clock equation,
\[
 v\alpha_2^0+\dot v\,u_1^2
       +2u_1\dot\theta_1+2z_1^i\partial_i\theta_1+\theta_2=0.
\]
Since $\partial_i u_1=-\partial_i\theta_1/v$, this gives
\[
 \alpha_2^0=u_2+2z_1^i\partial_i u_1.
\]
Taylor expansion yields
\[
 n_2=N_2+2\alpha_1^\mu\partial_\mu N_1
          +\alpha_2^\mu\partial_\mu N_0
          +\alpha_1^\mu\alpha_1^\nu
                    \partial_\mu\partial_\nu N_0.
\]
Because $N_0$ depends only on $t$, this becomes
\[
 N_2+2u_1\dot N_1+\dot N_0u_2+\ddot N_0u_1^2
       +2z_1^i\partial_i(N_1+\dot N_0u_1).
\]
Substitute $z_1=-Y_1$. The second spatial displacement multiplies
$\partial_iN_0=0$, so $Y_2$ is absent.
The first-order formula follows from the same Taylor expansion.
\end{proof}

The result agrees with the composition structure recorded in
Em02-E \cite{CheungEm02E}, but its proof here uses only the displayed
definitions. It is not a quantum gauge identity.

%=====================================================================
\section{First refusal: the harmonic relational scalar is not the lapse}
\label{sec:scalar-lapse}
%=====================================================================

\begin{lemma}[Scalar--lapse separation]
\label{lem:scalar-lapse}
The harmonic relational scalar $\RR$ and the scalar-clock
relational lapse $\NN$ are different functional prescriptions.

More concretely, on a common flat, homogeneous comparison domain:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\RR$ and $\NN$ are not equal as functionals;
\item at a fixed point, $\NN$ does not factor through the scalar
      value $\RR$ alone;
\item original spatial rods are not the relational scalar.
\end{enumerate}
These statements do not exclude identities on specially restricted
families or after additional, explicitly specified constructions.
\end{lemma}

\begin{proof}
Take $q=\eta$. The harmonic map of
Definition~\ref{def:harmonic} is $\HH_q=\id$.
For any homogeneous clock $T(t)$ with $\dot T>0$, the original
spatial rods are $Y_T^i=x^i$. Choose the clock itself as the
background reference, so the relational map is the identity.
Then
\[
 \RR=T(t),\qquad \NN=\frac1{\dot T(t)}.
\]
For $T(t)=ct+b$ with $c>0$, these are $ct+b$ and $1/c$.
They are not the same functional.

For the factorization claim, fix a background clock
$T_0(t)=ct+b$, a point $p=(t_c,\mathbf x_c)$, and a smooth
time bump $\chi$ equal to one near $t_c$.
Compare
\[
 T^{(a)}_e(t)=T_0(t)+e\,a(t-t_c)\chi(t).
\]
For sufficiently small $e$, these are timelike clocks.
At $p$ they all have the same scalar value,
\[
 \RR[\eta,T^{(a)}_e](p)=T_0(t_c).
\]
The relational inverse sends $p$ to $p$ in time because the clock
values agree there. Spatial rods remain $x^i$. Thus
\[
 \NN[\eta,T^{(a)}_e](p)=\frac1{c+ea}.
\]
The lapse varies with $a$ on a fiber of the scalar-value map.
Lemma~\ref{lem:factor} excludes factorization through that value.

Finally, on the homogeneous flat family the scalar is independent
of $\mathbf x$, whereas $Y_T^i=x^i$ is not. Even componentwise,
they cannot be equal as fields on an open spatial set.
\end{proof}

\begin{remark}[Quantifier of the witness]
The homogeneous flat comparison is not the power-law
Einstein--scalar solution of Em02-E and is not an assertion about
a restricted VI solution jet. It detects nonidentity of the
displayed kinematic prescriptions. The native manuscript objects
also retain different domains and auxiliary data.
\end{remark}

\begin{lemma}[Classical descent is not regulator removal]
\label{lem:descent-not-removal}
Classical covariance or descent of a functional does not imply
existence of a regulator-removal limit, even when every
positive-regulator member is smooth and covariant.
\end{lemma}

\begin{proof}
Let a gauge group act on any nonempty configuration space and let
$C$ be the constant scalar field $1$ on a spacetime region.
For $\delta>0$, set
\[
 Z_\delta=\delta^{-1}C.
\]
Every $Z_\delta$ is smooth and gauge invariant. For a compact test
$f$ with $\int f\,\dd^4x\ne0$,
\[
 \pair{Z_\delta}{f}
 =\delta^{-1}\int f\,\dd^4x
\]
has no finite limit. Hence covariance does not imply
distributional regulator removal.
\end{proof}

The example is a logical counterexample, not a model of the
Em02-E quantum family. It justifies keeping the following statement
as a lemma-level separation:
\[
 \boxed{\text{VI's classical descent is not Em02-E's
 regulator-removal theorem.}}
\]
The supplied VI record leaves its quantum certificate RG-open;
the supplied Em02-E record leaves its quantum audit incomplete
\cite{CheungVI,CheungEm02E}. Neither status is changed here.

%=====================================================================
\section{Second refusal: original rods are not modified rods}
\label{sec:rods}
%=====================================================================

\subsection{The elliptic prescription and its boundary condition}

Use
\[
 \Delta=\sum_{i=1}^3\partial_i^2,\qquad
 \GDelta f(\mathbf x)
 =-\frac1{4\pi}\int_{\R^3}
       \frac{f(\mathbf y)}{\abs{\mathbf x-\mathbf y}}\,\dd^3y
\]
for $f\in C_c^\infty(\R^3)$.
Then $\Delta\GDelta f=f$, and $\GDelta f$ decays at infinity.

For an induced metric
$\gamma_e=a^2(\delta+e c+\tfrac12e^2d)$ with compactly
supported coefficient perturbations, expand its Laplacian as
\[
 \Delta_{\gamma_e}
 =a^{-2}\Delta+eL_1+\frac{e^2}{2}L_2.
\]
The rod equations recursively read
\[
 \Delta Y_1^i=-a^2L_1x^i,\qquad
 \Delta Y_2^i=-a^2(L_2x^i+2L_1Y_1^i).
\]
Their right-hand sides are spatially compact: the perturbation
operators have compact coefficient support.
Application of $\GDelta$ gives smooth decaying coefficients.
A smooth harmonic difference tending to zero at infinity vanishes
by the maximum principle on expanding balls. This proves
coefficientwise existence and uniqueness for the rod prescriptions
used below.

\subsection{A detecting compact perturbation}

Consider the scalar metric subclass
\begin{equation}
\label{eq:scalar-metric}
 q_e=-(1+2eA)\dd t^2
        +a(t)^2(1+2eD)\delta_{ij}\dd x^i\dd x^j
        \pmod{e^3},
 \qquad T_e=T_0(t),
\end{equation}
where $A,D$ are spatially compact smooth functions.

For the original rods, the clock slices are exactly $t=\text{constant}$.
For the modified rods, it suffices to use the first clock jet
\begin{equation}
\label{eq:modified-clock}
 \Theta_e=T_0(t)+e\frac{v(t)}{H(t)}D(t,\mathbf x)+O(e^2).
\end{equation}
Define $Z_e^i$ to be harmonic on the $\Theta_e$-slices with the
same asymptotic Cartesian boundary condition.

Equation \eqref{eq:modified-clock} has the first jet of the modified
slicing used in \cite{BFHPR}. No choice of its second-order
coefficient is needed for the quadratic distinction proved below.
In both relational observables, the \emph{time coordinate remains
$T_e$}; only the spatial rods are replaced.

\begin{lemma}[First rod variations]
\label{lem:rod-first}
On \eqref{eq:scalar-metric},
\begin{equation}
\label{eq:rod-first}
 Y_1^i=-\partial_i\GDelta D,\qquad Z_1^i=0.
\end{equation}
Thus original and modified rods are not identical functional
prescriptions.
\end{lemma}

\begin{proof}
On an original clock slice,
\[
 \gamma_{ij}=a^2(1+2eD)\delta_{ij}.
\]
Expanding
$\Delta_\gamma f=(\det\gamma)^{-1/2}
 \partial_i((\det\gamma)^{1/2}\gamma^{ij}\partial_jf)$
gives
\[
 \Delta_\gamma x^i=e\,a^{-2}\partial_iD+O(e^2).
\]
The first harmonic equation is
$\Delta Y_1^i=-\partial_iD$, which proves the first formula.

A modified slice through background time $\tau$ has embedding
\[
 t=\tau-e\,\frac{D(\tau,\mathbf x)}{H(\tau)}+O(e^2),
\]
because $v=\dot T_0$. Its induced spatial metric at first order is
\[
 a(\tau)^2
 \left(1+2eD-2eH\,\frac{D}{H}\right)\delta_{ij}
 =a(\tau)^2\delta_{ij}+O(e^2).
\]
The contribution involving $\partial_it\,\partial_jt$ starts
at order $e^2$; the lapse perturbation $A$ therefore does not
alter this first-order calculation.
The modified harmonic equation is $\Delta Z_1^i=0$ with decaying
boundary data, so $Z_1^i=0$.
For any $D$ such that $\partial_i\GDelta D$ is nonzero, the
prescriptions differ.
\end{proof}

\begin{theorem}[Original and modified rods change the quadratic lapse]
\label{thm:rod-detection}
Let $\psi\in C_c^\infty(\R^3;\R)$ be nonconstant and let
$b$ be a smooth time bump supported strictly inside the comparison
interval. In \eqref{eq:scalar-metric}, choose
\begin{equation}
\label{eq:rod-witness}
 D=-b(t)\Delta\psi,\qquad A=v(t)b(t)\psi.
\end{equation}
Let $n_{2,Y}$ and $n_{2,Z}$ be the second coefficients of the
relational lapse using original and modified rods, respectively,
with the same clock $T_e=T_0$. Then
\begin{equation}
\label{eq:rod-lapse-difference}
 n_{2,Y}-n_{2,Z}
 =-2b(t)^2\abs{\nabla\psi}^2.
\end{equation}
The difference is detected by a nonnegative compact spacetime test.
Hence the quadratic relational lapse depends on the rod choice.
\end{theorem}

\begin{proof}
Since $\GDelta\Delta\psi=\psi$ for compact $\psi$,
Lemma~\ref{lem:rod-first} gives
\[
 Y_1^i=b\,\partial_i\psi,\qquad Z_1^i=0.
\]
There is no clock perturbation, so $u_1=u_2=0$.
For \eqref{eq:scalar-metric}, the raw lapse is
\[
 N_e=\frac{\sqrt{1+2eA}}{v},
\]
and therefore
\[
 n_1=N_1=\frac{A}{v}=b\psi.
\]
The local raw second coefficient is the same for the two rod
choices. Proposition~\ref{prop:lapse-composition} consequently gives
\[
 n_{2,Y}-n_{2,Z}
 =-2(Y_1^i-Z_1^i)\partial_i n_1
 =-2b^2\abs{\nabla\psi}^2.
\]
Choose a nonnegative test supported where $b\ne0$ and
$\nabla\psi\ne0$, positive on a nonempty open subset.
Its pairing with the displayed difference is strictly negative.
\end{proof}

\begin{remark}[What this theorem does and does not identify]
The witness is an off-shell compact perturbation of the
cosmological background. It proves that the two prescriptions
are different, not that every configuration distinguishes them.
The proof uses only their first spatial jets, so it applies to
any smooth completion of the modified prescription having
\eqref{eq:modified-clock}. It does not reconstruct or verify every
higher-order formula in the external reference.
\end{remark}

\begin{lemma}[Spatial rods are not harmonic Cauchy coordinates]
\label{lem:elliptic-hyperbolic}
On the cosmological background, original slice-harmonic spatial
rods are not interchangeable with spatial harmonic coordinates
having fixed zero perturbative Cauchy data.
\end{lemma}

\begin{proof}
The background spatial functions $x^i$ are both slice harmonic
and spacetime harmonic. This is a background coincidence.
Choose the perturbation $D=-b(t)\Delta\psi$ with $b(t_0)\ne0$.
The original rods have
\[
 Y_1^i(t_0,\mathbf x)=b(t_0)\partial_i\psi(\mathbf x).
\]
The perturbative spacetime-harmonic coordinates with fixed
zero Cauchy positions have
$W_1^i(t_0,\mathbf x)=0$.
They therefore differ at the initial surface.
Existence of the corresponding hyperbolic coefficient problem
follows from the Cauchy theorem for the smooth background
\cite{BGP}; only the fixed initial position is needed for the
contradiction.
\end{proof}

The lemma concerns the stated Cauchy prescription. Replacing its
initial position by the elliptic rod value changes that prescription
and requires a separate comparison.

\subsection{Localization powers and smoothing coefficients do not transfer}

The supplied Em02-A record distinguishes localization asymptotics
in $\epsilon$ from smoothing asymptotics in $\delta$
\cite{CheungEm02A}. The supplied Em02-D record concerns one
fixed distribution model with leading inverse-cubic smoothing
behavior \cite{CheungEm02D}. Neither result defines the
quantum contraction of the lapse in this paper.

\begin{lemma}[An asymptotic coefficient needs a connecting map]
\label{lem:asymptotic-map}
Let $t_\delta=\delta^{-r}C+R_\delta$ in a topological vector
space, where $\delta^rR_\delta\to0$.
If $B$ is a fixed continuous linear comparison map, then
\[
 \delta^rBt_\delta\longrightarrow BC.
\]
Nonvanishing of $C$ does not imply nonvanishing of $BC$.
Without a specified comparison map and matching regulated families,
this expansion determines no leading coefficient for a different
observable.
\end{lemma}

\begin{proof}
Apply continuity of $B$ to
$\delta^rt_\delta=C+\delta^rR_\delta$.
A nonzero $C$ may belong to $\ker B$, including when $B$ is a
nonzero projection. Thus the source coefficient need not survive
the target operation. An expansion in the source space, by itself,
contains no equality defining a distinct target family.
\end{proof}

Even the parameters cannot be identified by notation alone:
\[
 F(\epsilon,\delta)=\epsilon^2\delta^{-3}
\]
is constant along $\epsilon=\delta^{3/2}$, divergent along
$\epsilon=\delta$, and tends to zero along $\epsilon=\delta^2$.
A joint limit requires its own hypotheses.

Accordingly, neither Em02-A's localization powers nor Em02-D's
inverse-cubic coefficient is assigned to the relational-lapse
kernel. The rod distinction of Theorem~\ref{thm:rod-detection}
is already enough to rule out replacing the original rods by a
modified formula without recalculation.

%=====================================================================
\section{Third refusal: the density ramp is a named non-clock}
\label{sec:ramp}
%=====================================================================

\subsection{The regulated coordinate density}

Choose a fixed $\chi\in C^\infty(\R)$ with
\[
 0\le\chi\le1,\qquad
 \chi(u)=0\quad(u\le0),\qquad
 \chi(u)=1\quad(u\ge1),
\]
and a nonempty transition region. Put
\begin{equation}
\label{eq:ramp}
 \sigma_\varepsilon(t)=
     \chi\!\left(\frac{t-t_0}{\varepsilon}\right),
 \qquad 0<\varepsilon<t_1-t_0.
\end{equation}
The X-A-type regulated density is
\begin{equation}
\label{eq:ramped-density}
 \mathfrak b_{\varepsilon,e}^{\mu\nu}
 =\eta^{\mu\nu}
   +\sigma_\varepsilon(t)
          \bigl(\mathfrak a_e^{\mu\nu}-\eta^{\mu\nu}\bigr).
\end{equation}
It defines a coordinate equation by replacing $\mathfrak a_e$
with $\mathfrak b_{\varepsilon,e}$ in
\eqref{eq:harmonic}, with the same zero coefficient Cauchy data.

This is the density-ramp placement described in the supplied
X-A record \cite{CheungXA}. It is not field-argument convolution,
and it is not a scalar-clock equation.

\begin{definition}[Dynamical scalar-clock criterion]
\label{def:clock-criterion}
In this paper, a dynamical scalar clock is a scalar functional
of the dynamical configurations whose gradient is timelike and
nonzero on its declared clock region, with a fixed calibration.
A prescribed external profile is not a dynamical clock merely
because it is a function of coordinate time.
\end{definition}

\begin{lemma}[The ramp is not the declared dynamical clock]
\label{lem:ramp-nonclock}
With $\sigma_\varepsilon$ held fixed:
\begin{enumerate}[label=\textup{(\roman*)}]
\item its variation with respect to $q$ and $T$ is zero;
\item $\dd\sigma_\varepsilon=0$ on both plateau regions, so it
      fails the nondegeneracy criterion there;
\item the same ramp is compatible with different scalar-clock
      lapses;
\item it is not the harmonic relational scalar or the incidence
      register update.
\end{enumerate}
\end{lemma}

\begin{proof}
The first claim follows from \eqref{eq:ramp}, which contains
no dynamical field. The second follows from the constant values
of $\chi$.

For the third, keep $q=\eta$ and the same ramp, and use
$T^{(a)}_e$ from Lemma~\ref{lem:scalar-lapse}. The ramp is unchanged
while the lapse at the detecting point is $1/(c+ea)$.
Lemma~\ref{lem:factor} excludes determination of that lapse by
the ramp alone.

For the fourth, changing $T$ changes $\RR$ even at $q=\eta$,
whereas it does not change the ramp. The register is a map on a
finite code set with a declared update; a fixed smooth function
on a time slab is not that map or its update.
\end{proof}

The lemma does not deny that a strictly monotone segment of an
external profile can be used to reparametrize an interval.
That operation remains an externally prescribed parametrization;
it does not turn \eqref{eq:ramp} into the matter-defined clock
of Section~\ref{sec:inflaton}.

\subsection{The ramp changes the first harmonic response}

Let $\xi$ be a compact smooth vector field vanishing in an initial
collar, and set
\[
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.
\]
For the first metric-density coefficient,
\begin{equation}
\label{eq:pure-gauge-density}
 \partial_\alpha A_1^{\alpha\mu}[K\xi]
 =-\Boxeta\xi^\mu.
\end{equation}
Let $G_{\mathrm C}$ denote the zero-Cauchy solution operator for
$\Boxeta$ on the slab.

\begin{proposition}[Plateau response and transition-region defect]
\label{prop:ramp-defect}
The unregulated first coordinate correction satisfies
$W_1[K\xi]=\xi$.
The ramped first correction satisfies
\begin{equation}
\label{eq:ramp-defect}
 W_{1,\varepsilon}^\mu[K\xi]-\xi^\mu
 =G_{\mathrm C}\!\left[
  (\sigma_\varepsilon-1)\Boxeta\xi^\mu
       -(\partial_\alpha\sigma_\varepsilon)
                       A_1^{\alpha\mu}[K\xi]\right].
\end{equation}
It returns $\xi$ when its support lies strictly inside the
$\sigma_\varepsilon=1$ plateau.
There is a compact pure-gauge input in the transition region for
which it does not return $\xi$.
\end{proposition}

\begin{proof}
For $h=K\xi$,
\[
 A_1^{\alpha\mu}
  =(\partial_\rho\xi^\rho)\eta^{\alpha\mu}
       -\partial^\alpha\xi^\mu-\partial^\mu\xi^\alpha.
\]
Taking the divergence proves \eqref{eq:pure-gauge-density}.
The unregulated equation is
$\Boxeta W_1=\Boxeta\xi$ with the same zero data as $\xi$;
uniqueness proves $W_1=\xi$.

The ramped equation is
\[
 \Boxeta W_{1,\varepsilon}^\mu
 =\sigma_\varepsilon\Boxeta\xi^\mu
     -(\partial_\alpha\sigma_\varepsilon)
                         A_1^{\alpha\mu}[K\xi].
\]
Subtract the equation for $\xi$ and apply $G_{\mathrm C}$.
On the plateau the forcing in \eqref{eq:ramp-defect} vanishes.

For a detecting input, choose a transition time $t_c$ with
$\sigma_\varepsilon(t_c)<1$, a compact spatial bump $\psi$ equal
to one near $\mathbf x_c$, and a compact time bump $a$ in the
transition region such that
$a'(t_c)=0$ and $a''(t_c)\ne0$.
Set
\[
 \xi^0=a(t)\psi(\mathbf x),\qquad \xi^i=0.
\]
Near the detecting point,
\[
 A_1^{00}=a'\psi,\qquad
 \Boxeta\xi^0=-a''\psi.
\]
The forcing for $W_{1,\varepsilon}^0-\xi^0$ there equals
\[
 (1-\sigma_\varepsilon(t_c))a''(t_c)\ne0.
\]
If the difference field vanished identically, its wave operator
would vanish identically, contradicting this forcing.
\end{proof}

\begin{corollary}[No second-order repair of the fixed first response]
\label{cor:first-order}
A correction beginning at order $e^2$ cannot remove the
nonzero order-$e$ defect of Proposition~\ref{prop:ramp-defect}
while the ramp and first-order prescription remain fixed.
\end{corollary}

\begin{proof}
Equality of formal series modulo $e^3$ requires equality of their
order-$e$ coefficients. Adding an order-$e^2$ term does not change
that coefficient.
\end{proof}

This is a calculation for the displayed density ramp.
It is not an identification with VI's separately reported
compact-switch defect. The two defects retain their own
prescriptions and domains.

If the external ramp is also pulled back under a diffeomorphism,
one instead studies a construction depending covariantly on the
enlarged data $(q,T,\sigma_\varepsilon)$.
Treating a profile as an additional transforming background does
not make it a dynamical matter clock.

%=====================================================================
\section{Fourth refusal: incidence parameter and branch-normal speed}
\label{sec:incidence}
%=====================================================================

\subsection{The declared geometric and register interfaces}

Let $C_\mu$ be a spinless spatial cone with a supplied deficit
$\mu$, and use the regular-part ultrastatic metric
\[
 q_{\mathrm{cone}}=-\dd U^2+\gamma_\mu.
\]
The cone and its metric are declared inputs.

A slicing family is a graph
\begin{equation}
\label{eq:graph}
 \iota_s(x)=(F(s,x),x),
 \qquad \abs{\nabla_{\gamma_\mu}F}<1
\end{equation}
on each regular branch. The incidence parameter $s$ is supplied
by the controller, not solved from a scalar-field equation.

A finite implementation consists of a finite code set
$\mathscr J$, an installed parameter table
\[
 \tau:\mathscr J\longrightarrow I_s,
\]
and a declared update $U_{\mathscr J}:\mathscr J\to\mathscr J$.
When a reversible realization is intended, the corresponding
complete finite update is a permutation of its declared code space.
These definitions do not acquire a matter expectation or define
a Hamiltonian.

This separates the geometric parameter from its finite code,
as required by the supplied VIII-A/B records
\cite{CheungVIIIA,CheungVIIIB}. The clock code is denoted by $j$;
it is not the finite Hermitian source matrix also written $j[f]$
in that track.

\subsection{The normal-speed calculation}

\begin{lemma}[Kinematical normal component]
\label{lem:normal-speed}
On a regular branch of \eqref{eq:graph}, the future unit normal is
\[
 n_s=\frac{\partial_U+\nabla_{\gamma_\mu}F}
             {\sqrt{1-\abs{\nabla_{\gamma_\mu}F}^2}}.
\]
For the deformation $V_s=(\partial_sF)\partial_U$,
its positive normal component is
\begin{equation}
\label{eq:normal-speed}
 L_s:=-q_{\mathrm{cone}}(n_s,V_s)
 =\frac{\partial_sF}
          {\sqrt{1-\abs{\nabla_{\gamma_\mu}F}^2}}.
\end{equation}
\end{lemma}

\begin{proof}
A tangent vector to the graph is
$w+(\dd_xF)(w)\partial_U$.
Its inner product with $\partial_U+\nabla F$ is zero.
The squared norm of the latter is $-1+\abs{\nabla F}^2$.
Normalization and contraction with $V_s$ give the formulas.
\end{proof}

For the local branch data reported in VIII-B, the calculation is
\[
\begin{array}{c|c|c|c}
\text{branch}&\abs{\nabla F}^2&\partial_sF&L_s\\ \hline
\text{core}&8/9&1&3\\
\text{inner}&9/25&2&5/2\\
\text{resting}&0&1&1
\end{array}
\]
The arithmetic follows directly from \eqref{eq:normal-speed}.
The geometric construction establishing those branch data belongs
to VIII-B \cite{CheungVIIIB}; it is not reconstructed by the table.

In oblique cone coordinates, the gradient norm must be computed
with the inverse physical Gram matrix, not with the Euclidean norm
of the coordinate covector. Formula \eqref{eq:normal-speed} is
coordinate independent.

\begin{proposition}[Parametrization dependence]
\label{prop:retiming}
Let $r=f(s)$ with $f'>0$. Keeping the same unparametrized graph
family,
\begin{equation}
\label{eq:retiming}
 L_r=\frac{L_s}{f'(s)}.
\end{equation}
Consequently, the unparametrized slices and the cone packet do
not determine the branch-normal speed without the parameter
calibration.
\end{proposition}

\begin{proof}
The spatial gradient is unchanged at corresponding slices, while
$\partial_rF=(\partial_sF)/f'(s)$.
Apply \eqref{eq:normal-speed}.
For example, $r=2s$ preserves the slices and their ordering but
halves the speed. Lemma~\ref{lem:factor} gives the final assertion.
\end{proof}

\begin{lemma}[The incidence parameter is not the matter lapse]
\label{lem:incidence-lapse}
The declared parameter $s$, the finite reading $j$, the
branch-normal speed $L_s$, and the matter-defined relational
lapse $\NN$ are not one functional.
In particular:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a parameter value is not its deformation speed;
\item graph ordering and cone matching alone do not determine
      a scalar-clock lapse;
\item the reported source parameter $s_c=-1/2$ is not any of
      the branch speeds $3,5/2,1$;
\item the finite update supplies no missing scalar or rod field.
\end{enumerate}
\end{lemma}

\begin{proof}
The first statement follows from the distinct formulas:
$s$ is the independent family argument, whereas $L_s$ depends on
$\partial_sF$ and $\nabla F$. Proposition~\ref{prop:retiming}
also shows their different dependence on a parameter change.

For the second, augment the same regular cone patch and graph
family by either scalar $T_c=cU$ with $c>0$.
Every such scalar is a timelike clock, and its raw lapse is
$1/c$. With its corresponding homogeneous reference and Cartesian
rods on the regular flat patch, its relational lapse is also $1/c$.
The prescribed graph ordering and matching data are unchanged as
$c$ varies. They therefore cannot determine this additional
matter-defined lapse. This is a counterexample to an inference
from geometry alone, not an identification with the cosmological
background of Em02-E.

The third assertion is a distinction between the named parameter
value and the displayed branch values. For the fourth,
$U_{\mathscr J}$ is defined on finite codes and has no scalar
field or slice-elliptic rod equation in its definition.
\end{proof}

\subsection{An important equality that does not erase the separation}

\begin{proposition}[A prescribed foliation can be encoded by an added scalar]
\label{prop:foliation-clock}
Suppose $\partial_sF>0$ on a regular patch.
Define the local scalar $S_F(U,x)$ implicitly by
$U=F(S_F(U,x),x)$. Then its raw scalar-clock lapse equals
the branch-normal speed:
\[
 N[q_{\mathrm{cone}},S_F]=L_s
\]
at $U=F(s,x)$.
\end{proposition}

\begin{proof}
Implicit differentiation gives
\[
 \dd S_F=\frac{\dd U-\dd_xF}{\partial_sF}.
\]
Therefore
\[
 -q_{\mathrm{cone}}^{-1}(\dd S_F,\dd S_F)
 =\frac{1-\abs{\nabla F}^2}{(\partial_sF)^2}.
\]
Taking the positive inverse square root proves the equality.
\end{proof}

This proposition prevents an incorrect universal claim that a
kinematical lapse can never equal a scalar-clock lapse.
Here equality is obtained by \emph{adding the schedule-defined
scalar $S_F$}. It does not identify $S_F$ with the original
inflaton clock, supply inflaton-slice rods, or derive the prescribed
family from matter dynamics.

The two uses of ``lapse'' accordingly remain separate:
\[
 \boxed{
 L_s\text{ is a normal deformation component of a prescribed family};
 \quad
 \NN\text{ is a relational observable of }(q,T,Y_T).}
\]

%=====================================================================
\section{Fifth refusal: Family 271 and reported Lean scope}
\label{sec:catalogue}
%=====================================================================

The supplied Family 271 catalogue concerns the quantum Heisenberg
ferromagnet and records Bloch-type low-temperature statements.
The companion marks those records as unverified and non-premissory
\cite{Catalogue271}. The reported Lean scope names 271-04 only.
That is a documentary statement about the supplied scope record,
not a claim that a Lean build has been inspected or executed here.

For orientation, rigorous spin-wave free-energy work such as
\cite{CGS} belongs to statistical mechanics. It is not a
relational-coordinate construction, and no result from that
article is used in a proof in this paper.

\begin{proposition}[Scope records do not transfer between identifiers]
\label{prop:catalogue}
Let a documentary scope record list the singleton identifier set
$\{271\text{-}04\}$. That record supplies no formalization claim
for VI, Em02-E, X-A, VIII-A/B, or the other Family 271 identifiers.
It supplies no premise for a clock-separation theorem.
\end{proposition}

\begin{proof}
The listed identifiers are part of the record's declared scope.
None of the other identifiers belongs to the singleton set.
A mathematical identification or proof transfer would require
an additional statement and its hypotheses; membership of one
identifier supplies no such statement. The proposition concerns
the scope of a record, not correctness of the recorded proof.
\end{proof}

In particular, VI is not 271-04. The Family 271 landmark comparison
is retained as a separation, not as a bridge.

%=====================================================================
\section{Main separation theorem and exact release}
\label{sec:main}
%=====================================================================

\begin{theorem}[Clock separation before coordinate identification]
\label{thm:main}
Retain the declarations of
Sections~\ref{sec:typed}--\ref{sec:incidence}, including the
Cauchy data, positive timelike branch, clock calibration,
spatial boundary condition, prescribed parameter table,
and fixed ramp.

The four labelled objects
$\mathsf C_{\mathrm{VI}}$, $\mathsf C_{\mathrm E}$,
$\mathsf C_{\mathrm{VIII}}$, and $\mathsf R_{\mathrm X}$
are not interchangeable functional prescriptions.
The following five separations hold:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The harmonic relational scalar and scalar-clock relational
      lapse are distinct. A scalar value alone does not determine
      the lapse. Classical descent does not imply regulator removal.
\item Original and modified rods are distinct, and the compact
      perturbation \eqref{eq:rod-witness} distinguishes their
      quadratic relational lapses. Their formulas and asymptotic
      coefficients cannot be transferred without a connecting
      construction.
\item The density ramp is a prescribed non-clock. Its first
      harmonic response has a transition-region pure-gauge defect
      that a second-order addition cannot remove while the first
      prescription remains fixed.
\item The incidence parameter, its finite code, and the branch-normal
      speed do not supply the matter-defined clock, original rods,
      or relational lapse. A foliation-defined scalar can reproduce
      the normal speed only after that additional scalar is declared.
\item The supplied Family 271 catalogue and reported Lean scope
      supply no premise or identification for these constructions.
\end{enumerate}
The substantive detecting calculations are on the stated kinematic
domains. The theorem makes no universal claim that numerical
coincidences or future explicitly constructed comparison maps
are impossible.
\end{theorem}

\begin{proof}
Item (i) is Lemma~\ref{lem:scalar-lapse} together with
Lemma~\ref{lem:descent-not-removal}.
Item (ii) is Lemma~\ref{lem:rod-first},
Theorem~\ref{thm:rod-detection},
Lemma~\ref{lem:elliptic-hyperbolic}, and
Lemma~\ref{lem:asymptotic-map}.
Item (iii) is Lemma~\ref{lem:ramp-nonclock},
Proposition~\ref{prop:ramp-defect}, and
Corollary~\ref{cor:first-order}.
Item (iv) is Lemma~\ref{lem:incidence-lapse} and
Propositions~\ref{prop:retiming} and~\ref{prop:foliation-clock}.
Item (v) is Proposition~\ref{prop:catalogue}.

The objects also have the different domains, targets, equations,
and auxiliary data recorded in Definition~\ref{def:object}.
No cross-model identification is supplied by the labels.
The detecting witnesses prove the stated nonidentities without
assuming that every evaluation must differ.
\end{proof}

\subsection{What is released}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.35\linewidth}p{.18\linewidth}Y@{}}
\toprule
Object or statement & Ending here & Exact extent\\
\midrule
Four-object ledger & Constructed & Domains and prescriptions named\\
Scalar--lapse separation & Proved & Flat common-domain witness\\
Original--modified rod separation & Proved & Compact cosmological perturbation\\
Quadratic lapse difference & Evaluated & Equation \eqref{eq:rod-lapse-difference}\\
Fixed-ramp non-clock and defect & Proved & Displayed density prescription\\
Incidence--matter-clock separation & Proved & Additional-data and calibration tests\\
Family 271 comparison & Documentary & No proof or Lean verification\\
VI--XVI-D coordinate identification & Not decided & Actual Cauchy comparison still required\\
Quantum regulator removal & Not constructed & No quantum family defined here\\
Continuum evolution of the register & Not constructed & Not the target of this paper\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{What the next identification paper must supply}

The supplied XVI-D record keeps inherited coordinate compatibility
conditional on a specified larger-domain realization
\cite{CheungXVID}. This paper does not discharge that condition.

A successor comparing VI with XVI-D must provide:
\begin{enumerate}[label=\textup{(\arabic*)}]
\item VI's actual coordinate equation and Cauchy data, including
      the derivative convention and admissible source space;
\item XVI-D's actual coordinate forcing and required Cauchy data;
\item a common domain on which both prescriptions are defined;
\item either equality of the two problems followed by uniqueness,
      or a detecting mismatch in forcing, initial data, or domain;
\item a separate analysis of any switched assignment whose first
      coefficient is used in the comparison.
\end{enumerate}
A reported first-order switch obstruction is not automatically
an obstruction to an independently defined coordinate Cauchy
problem. Conversely, a second-order correction cannot retroactively
change a fixed defective first coefficient.

\subsection{What remains outside}

The quantum audit would require, at minimum, a gauge-fixed
covariance, a state or suitable parametrix, an admissible
counterterm class, controlled quantum remainders, and the
retained-order quantum gauge identity. The supplied Em02-E audit
lists these as missing constructions \cite{CheungEm02E}.
Classical smoothness and covariance do not supply the
distributional product and extension data required for such a
calculation; see the general distinction in \cite{BF}.

The finite measurement characterization of Cheung's Enter
calculus \cite{CheungEnter} is a separate operational result.
A prescribed register update is not, by itself, a field-theoretic
measurement instrument. No measurement theorem is used to
construct a clock or a lapse here.

The following documentary boundaries are unchanged:
\[
 \begin{gathered}
 \text{VI quantum certificate: RG-open},\\
 \text{Em02-E quantum gauge identity: not verified here},\\
 \sid{O.1 = NOT_CLOSED},\\
 \text{XV-B downstream quantum-admission exit: not opened},\\
 \text{XVI-N and Route B: not discharged}.
 \end{gathered}
\]
These are source-status boundaries, not new universal
nonexistence theorems.

No inherited characteristic quantum coefficient is admitted.
No Einstein dynamics, de Sitter state, physical invariant speed,
spacetime emergence, or evolution without an external time is
derived.

%=====================================================================
\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}
%=====================================================================

\paragraph{Mathematical scope.}
The results are classical kinematic separation statements and
explicit perturbative calculations. Formal equalities mean
coefficientwise equalities through second order, not convergence
of a numerical series.

\paragraph{Source scope.}
The Cheung manuscript citations retain the supplied versions and
filenames. They identify documentary comparisons, not independently
verified predecessor proofs. No unpublished theorem number has
been invented for an unavailable passage. In particular, the
displayed harmonic prescription is an explicit comparison
definition, not an unproved authentication of VI's complete source.

\paragraph{Verification.}
The proofs are the arguments printed in this paper.
No successful pdfLaTeX execution, proof-assistant verification,
independent symbolic or numerical execution, referee approval,
or acceptance of this manuscript is asserted.

\paragraph{Authorship.}
The source is formatted for anonymous submission.
Citation of N.~Cheung does not attribute authorship of this
manuscript to that author. A signed submission must supply the
responsible author's authorized metadata.

\paragraph{Availability of cited manuscript records.}
The unpublished comparison records should be supplied in their
cited versions, or deposited at stable archival locations, with
a signed submission. None of their unprinted proofs is required
to verify the detecting calculations in this paper.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}
%=====================================================================

\bibitem{CheungVI}
N.~Cheung,
\emph{Landmark VI},
version 1, supplied unpublished manuscript,
\sid{VI.tex}, n.d.
Supplied records on finite-time harmonic relational descent,
the compact-switch obstruction, and the RG-open quantum audit.
Enter Calculus Companion:
\url{https://enter.grok.me}.

\bibitem{CheungVIIIA}
N.~Cheung,
\emph{Landmark VIII-A},
version 3, supplied unpublished manuscript,
\sid{VIII-A-v3.tex}, n.d.
Especially ``The exact clock is not a relational lapse''
and the finite matching and record--residual interfaces.
Enter Calculus Companion:
\url{https://enter.grok.me}.

\bibitem{CheungVIIIB}
N.~Cheung,
\emph{Landmark VIII-B},
version 3, supplied unpublished manuscript,
\sid{VIII-B-v3.tex}, n.d.
Especially \sid{rem:lapse}, \sid{lem:gradients},
\sid{thm:dictionary}, and the parameter--relational-lapse
non-identification.
Enter Calculus Companion:
\url{https://enter.grok.me}.

\bibitem{CheungXA}
N.~Cheung,
\emph{X-A},
version 1, supplied unpublished manuscript,
\sid{X-A.tex}, n.d.
Smooth time ramp in a divergence-form coordinate density,
its plateau covariance test, and the excluded transition-region
response.
Enter Calculus Companion:
\url{https://enter.grok.me}.

\bibitem{CheungEm02E}
N.~Cheung,
\emph{Em02-E},
version 1, supplied unpublished manuscript,
\sid{Em02-E-v1.tex}, n.d.
Original inflaton-slice harmonic rods, classical relational lapse,
\sid{prop:background}, \sid{thm:lapse}, and the incomplete
quantum audit.
Enter Calculus Companion:
\url{https://enter.grok.me}.

\bibitem{CheungEm02A}
N.~Cheung,
\emph{Em02-A},
version 6, supplied unpublished manuscript,
\sid{Em02-A-v6.tex}, n.d.
Identification-first boundary and
\sid{thm:transition}; localization in $\epsilon$ is kept
distinct from smoothing in $\delta$.
Enter Calculus Companion:
\url{https://enter.grok.me}.

\bibitem{CheungEm02D}
N.~Cheung,
\emph{Em02-D},
version 4, supplied unpublished manuscript,
\sid{Em02-D-v4.tex}, n.d.
\sid{thm:main}; the leading inverse-cubic expansion belongs
to its fixed distribution model.
Enter Calculus Companion:
\url{https://enter.grok.me}.

\bibitem{CheungXVID}
N.~Cheung,
\emph{Landmark XVI-D},
version 1, supplied unpublished manuscript,
\sid{XVI-D.tex}, n.d.
\sid{thm:response-package} and the conditional
inherited-coordinate compatibility interface.
Enter Calculus Companion:
\url{https://enter.grok.me}.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus with an
Operational Characterization of Measurement,''
\emph{Frontiers in Physics} \textbf{14} (2026),
article 1954943.
Publisher accepted-article record; accepted 28 September 2026.
\url{https://doi.org/10.3389/fphy.2026.1954943}.
Operational comparison only; no clock or gravitational theorem
is imported.

\bibitem{BFHPR}
R.~Brunetti, K.~Fredenhagen, T.-P.~Hack, N.~Pinamonti,
and K.~Rejzner,
``Cosmological perturbation theory and quantum gravity,''
\emph{Journal of High Energy Physics} \textbf{08} (2016), 032.
\url{https://doi.org/10.1007/JHEP08(2016)032}.
\url{https://arxiv.org/abs/1605.02573}.
Original and modified spatial-coordinate prescriptions are
distinguished in the arXiv equations (28)--(35).

\bibitem{Frob}
M.~B.~Fr\"ob,
``Gauge-invariant quantum gravitational corrections to
correlation functions,''
\emph{Classical and Quantum Gravity} \textbf{35} (2018), 055006.
\url{https://doi.org/10.1088/1361-6382/aaa74c}.
\url{https://arxiv.org/abs/1710.00839}.
Relational-coordinate comparison; its propagator choice and
quantum results are not adopted as the finite-time prescription
of this paper.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and Quantization},
ESI Lectures in Mathematics and Physics,
European Mathematical Society, Z\"urich, 2007.
\url{https://doi.org/10.4171/037}.
\url{https://arxiv.org/abs/0806.1036}.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
``Microlocal analysis and interacting quantum field theories:
Renormalization on physical backgrounds,''
\emph{Communications in Mathematical Physics}
\textbf{208} (2000), 623--661.
\url{https://doi.org/10.1007/s002200050004}.
\url{https://arxiv.org/abs/math-ph/9903028}.
General distributional construction and extension framework;
no application to an unconstructed quantum lapse family is
asserted.

\bibitem{CGS}
M.~Correggi, A.~Giuliani, and R.~Seiringer,
``Validity of the spin-wave approximation for the free energy
of the Heisenberg ferromagnet,''
\emph{Communications in Mathematical Physics}
\textbf{339} (2015), 279--307.
\url{https://doi.org/10.1007/s00220-015-2402-0}.
\url{https://arxiv.org/abs/1312.7873}.
Statistical-mechanical context only, not a clock-paper premise.

\bibitem{Catalogue271}
\emph{OpenAI mathematics catalogue: Family 271 and reported
Lean scope},
attributed to OpenAI in the supplied companion record,
record dated 7 October 2026.
Enter Calculus Companion:
\url{https://enter.grok.me/openai}.
The supplied scope names 271-04 only.
Catalogue attribution, mathematical proofs, and Lean execution
are not independently authenticated here.
Documentary separation only.

\end{thebibliography}

\end{document}