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% Charge-anchored horizon thermodynamics on spherical screens
% FINAL SIMPLIFIED + RIGOR-RESTORED DRAFT (CQG-oriented preprint)
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% --- Macros ---
\newcommand{\lp}{l_{\mathrm P}}
\newcommand{\Ep}{E_{\mathrm P}}
% Screen stresses / forces:
\newcommand{\SigBY}{\Sigma_{\rm BY}}
\newcommand{\SigE}{\Sigma_{E}}
\newcommand{\FE}{F_{E}} % charge-anchored dilation force (radius conjugate)
\newcommand{\FH}{F_{H}} % Hawking-scale force = M kappa_H
\newcommand{\PE}{P_{E}} % charge-anchored Laplace pressure (screen-normal)
% Temperatures:
\newcommand{\TE}{T_{E}} % charge-anchored temperature (entropy conjugate)
% NOTE: \TH is already defined by fontenc (thorn). Avoid collision:
\newcommand{\THW}{T_{H}} % Hawking/Unruh temperature (when admissible)
\newcommand{\Emsh}{E_{\rm MSH}}
\newcommand{\QK}{Q_{\rm K}}
\newcommand{\Geff}{G_{\mathrm{eff}}}
\newcommand{\Ccomp}{\mathcal{C}}
% Acceleration / surface-gravity scales:
\newcommand{\kappaE}{\kappa_{E}} % positive one-scale acceleration c^2/R
\newcommand{\kappaH}{\kappa_{H}} % positive Hawking/Unruh surface-gravity scale
\newcommand{\kappaHK}{\kappa_{\rm HK}} % signed Hayward--Kodama geometric surface gravity (optional)
\begin{document}
\title{Charge-Anchored Horizon Thermodynamics on Spherical Screens:\\
Dilation Force and Laplace Pressure}
\author{J.\,T.\ Tyler}
\email{jttyler101@gmail.com}
\affiliation{Independent researcher (\texttt{jttyler.com})}
% =========================================================
\begin{abstract}
A round two-sphere admits exactly one homogeneous deformation: a change in its areal radius.
On any marginal (apparent/trapping) sphere in four-dimensional Einstein gravity the
Misner--Sharp--Hernandez/Kodama charge reduces to the one-scale law
$E\equiv \QK = c^4 R/(2G)$.
This charge anchor fixes the unique spherical mechanical channel:
a constant dilation force $\FE \equiv E/R = c^4/(2G)$,
an isotropic tangential stress $\SigE=\FE/(8\pi R)$,
and a screen-normal Laplace pressure $\PE=\FE/A=c^4/(8\pi G R^2)$,
all determined from bulk geometry before any boundary observer, subtraction scheme,
or surface-gravity convention is chosen.
For FRW/de~Sitter apparent horizons, using $\QK=(4\pi/3)R^3\rho$ gives the exact identity
$\PE=\rho/3$, a purely mechanical statement.
Adopting the Einstein area entropy $S=k_BA/(4\lp^2)$ then closes the matched charge-anchored pair
\[
\TE=\frac{E}{S}=\frac{\hbar c}{2\pi k_B R},
\qquad
\kappaE=\frac{c^2}{R},
\quad\text{via}\quad
\TE=\frac{\hbar\,\kappaE}{2\pi k_B c}.
\]
Pulling the two-channel first law $dE=\TE\,dS-\FE\,dR$ back to the one-parameter marginal family yields
$\TE\,dS=dE+\FE\,dR$ and hence $\TE\,dS=2\,dE$ for FRW/de~Sitter, clarifying the familiar ``$2dE=T\,dS$''
relation.
When a Hawking/Unruh temperature is admissible, define a positive Hawking scale $\kappaH\ge0$
by $\THW=\hbar\kappaH/(2\pi k_B c)$ and introduce the sector parameter
$\alpha\equiv \kappaE/\kappaH$.
Define the Hawking-scaled energy $U\equiv E/\alpha$ and the Hawking-scale force $\FH\equiv U/R=M\kappaH$.
In constant-$\alpha$ one-scale sectors this yields the Hawking-packaged first law
$dU\equiv d(E/\alpha)=\THW\,dS-\FH\,dR$ with $U=\THW S=\FH R$.
In FRW/de~Sitter, $\alpha=1$ so $E=U$ and $(\TE,\kappaE)$ coincide with $(\THW,\kappaH)$.
In the Schwarzschild sector, $\alpha=2$ and the usual $E=2\THW S$ is reorganized as
$E=\THW S+\FH R$ (heat plus force--radius).
Propagating the same one-scale closure to a Compton-saturated Planck cell yields the saturated
Bekenstein equation of state as a UV benchmark and fixes the associated effective coupling.
\end{abstract}
\keywords{horizon thermodynamics; quasilocal energy; Young--Laplace law;
Kodama charge; Misner--Sharp energy; cosmological horizons; Brown--York stress}
\maketitle
% =========================================================
\section{Introduction}
\label{sec:intro}
Horizon thermodynamics rests on a striking conjunction: the area of a classical horizon
behaves as an entropy, and its surface gravity sets a temperature
\cite{Bekenstein72,BCH73,Hawking74,GibbonsHawking77}.
The resulting first laws have been extended to cosmological apparent horizons
\cite{Hayward98,CaiKim05}, but the bookkeeping can become intricate---different
choices of surface gravity, work density, and boundary stress lead to
expressions that, while equivalent in content, differ in normalization and
interpretation.
In dynamical situations the difficulty is compounded by the absence of a
timelike Killing vector and the consequent inequivalence of competing
surface-gravity definitions \cite{Faraoni13}.
Most pointedly, in his monograph on cosmological and black hole apparent
horizons, Faraoni questions the internal consistency of the construction,
noting that the work term is ``introduced \emph{a~posteriori} \dots\ unlikely
that one would discover it this way'' \cite[Sec.~3.15]{Faraoni15}.
A closely related issue, emphasized from a different direction in the horizon/emergent-gravity
literature, is that while a horizon temperature can often be assigned rather cleanly
(e.g.\ by Euclidean periodicity), the relevant notion of horizon energy is far less
canonical away from special stationary settings.
In the context of general spherically symmetric horizons, Sarkar and Padmanabhan state
succinctly: ``The real ambiguity is in giving a proper prescription for the energy
--- a well known problem in general relativity'' \cite{SarkarPadmanabhan07}.
The present paper is motivated by the same structural demand, but implements it in a
geometry-first way: in spherical symmetry, fix the energy and the unique work channel
directly from bulk geometry \emph{before} imposing any surface-gravity convention.
Two facts make the spherical sector unusually rigid.
First, a round two-sphere has exactly one homogeneous deformation, $R\to R+dR$.
Second, in four-dimensional Einstein gravity there is a preferred quasilocal energy in
spherical symmetry: the Misner--Sharp--Hernandez/Kodama charge $\QK$ \cite{Hayward98}.
On marginal (apparent/trapping) spheres, $\QK$ collapses to a one-scale law
$E=\QK\propto R$, which immediately fixes a dilation force scale $\FE\equiv E/R$ and hence
an isotropic tangential stress and a screen-normal Laplace pressure.
This is the core result: a complete \emph{mechanical} dictionary on spherical screens that is
(i) subtraction-free, (ii) independent of boundary observers, and (iii) does not require choosing
any surface-gravity convention to define the work channel.
After establishing this geometry-first mechanical law, we show that for FRW/de~Sitter apparent
horizons it implies the exact identity $\PE=\rho/3$.
(For a slope-based FRW ``effective pressure'' definition using total derivatives, see \cite{Kong25}.)
We then present a minimal thermodynamic translation with matched naming.
Adopting the area entropy closes the charge-anchored temperature $\TE$ and its paired
acceleration scale $\kappaE=c^2/R$ via the Unruh/Hawking normalization.
A short pullback calculation then shows how the familiar cosmological-horizon relation
$2\,dE=T\,dS$ arises from restricting the two-channel first law to the one-parameter horizon
family, and why slope-based ``effective temperatures'' differ by a factor of two.
When a Hawking/Unruh temperature $\THW$ is admissible, we introduce the positive Hawking scale $\kappaH$
and the sector parameter $\alpha\equiv\kappaE/\kappaH$.
This leads to the Hawking-scaled energy $U\equiv E/\alpha$ and the Hawking-scale force
$\FH\equiv U/R=M\kappaH$, giving a compact first law $dU\equiv d(E/\alpha)=\THW\,dS-\FH\,dR$ and the
Schwarzschild reorganization $E=\THW S+\FH R$.
Finally, we show that the same one-scale closure propagates to a Compton-saturated Planck cell,
selecting a saturated Bekenstein equation of state as a UV benchmark.
We proceed as follows.
Section~\ref{sec:backbone} derives the charge-anchored force, stress, and pressure on marginal
spheres and records the purely mechanical Smarr ladder.
Section~\ref{sec:FRW} applies the result to FRW/de~Sitter apparent horizons, yielding $\PE=\rho/3$.
Section~\ref{sec:thermo} introduces $S$, the conjugate pair $(\TE,\FE)$ from partial derivatives,
the pullback that resolves ``$2dE=T\,dS$,'' and the Hawking translation $(\THW,\kappaH)$.
Section~\ref{sec:virial} gives the Schwarzschild partition and the role of $\alpha$.
Section~\ref{sec:BY} presents a Brown--York boundary realization and horizon calibration as a cross-check.
The appendices collect the relation to Hayward/Cai--Kim variables (Appendix~\ref{app:unified}),
explicit Brown--York stretched-horizon computations (Appendix~\ref{app:FRW_BY}),
the $D$-dimensional generalization (Appendix~\ref{app:D}),
the UV Smarr-rigidity benchmark (Appendix~\ref{app:UV}), and a de~Sitter counterterm illustration
(Appendix~\ref{app:dS}).
% =========================================================
\section{Geometric backbone: charge anchor and the dilation channel}
\label{sec:backbone}
Every spherically symmetric four-dimensional spacetime splits as
\begin{equation}
\label{eq:metric}
ds^2 = h_{AB}(x)\,dx^A dx^B + R(x)^2\,d\Omega_2^2,
\end{equation}
with $R(x)$ the areal radius and $h_{AB}$ the metric on the two-dimensional
orbit space.
The Misner--Sharp--Hernandez energy \cite{Hayward98,MisnerSharp64}
\begin{equation}
\label{eq:MSH}
\Emsh = \frac{c^4 R}{2G}\bigl(1 - g^{\mu\nu}\nabla_\mu R\,\nabla_\nu R\bigr)
\end{equation}
is a scalar on the orbit space that measures the gravitational mass enclosed
by each round sphere.
The Kodama vector $K^A = \varepsilon^{AB}\nabla_B R$ \cite{Kodama80} generates a
conserved current $J^\mu = T^\mu{}_\nu K^\nu$ whose charge $\QK$ coincides
with $\Emsh$ in four-dimensional Einstein gravity \cite{Hayward98}.
On a marginal surface---where the expansion of outgoing null normals vanishes,
equivalently $g^{\mu\nu}\nabla_\mu R\,\nabla_\nu R = 0$---the charge simplifies to
\begin{equation}
\label{eq:QK_marginal}
E \equiv \QK = \frac{c^4}{2G}\,R.
\end{equation}
This linear scaling is the key structural fact: the marginal branch is a
\emph{one-scale family} parametrized entirely by $R$.
Dividing by the only available length fixes a \emph{constant} dilation-force scale,
\begin{equation}
\label{eq:FK}
\FE \equiv \frac{E}{R} = \frac{\QK}{R} \equiv F_{\rm K} = \frac{c^4}{2G},
\end{equation}
independent of radius.
This constancy is special to four dimensions; in $D$ dimensions the marginal
charge scales as $R^{D-3}$, so $F_{\rm K}\propto R^{D-4}$ and only $D=4$
produces a scale-invariant force (Appendix~\ref{app:D}).
Because a round sphere admits only one homogeneous deformation, this single force scale
determines the corresponding isotropic tangential stress and screen-normal Laplace pressure:
\begin{equation}
\label{eq:Sig_P_def}
\SigE \equiv \frac{\FE}{8\pi R},\qquad
\PE \equiv \frac{\FE}{A} = \frac{\FE}{4\pi R^2}
\qquad\Longrightarrow\qquad
\PE = \frac{2\SigE}{R}.
\end{equation}
On the marginal branch the pressure is therefore fixed \emph{purely geometrically} as
\begin{equation}
\label{eq:PE_general}
\PE = \frac{c^4}{8\pi G R^2}.
\end{equation}
Equations \eqref{eq:QK_marginal}--\eqref{eq:PE_general} close a complete mechanical ``Smarr ladder''
for the one-scale marginal family:
\begin{equation}
\label{eq:Smarr_mech}
E = \FE R = 2\,\SigE A = 3\,\PE V,
\qquad
A=4\pi R^2,\quad V=\frac{4\pi}{3}R^3.
\end{equation}
No surface-gravity prescription, boundary subtraction, or boundary-observer choice is required to
establish this mechanical channel: it is fixed by $(\QK,R)$ and the uniqueness of spherical dilation.
\paragraph{Reparametrization/foliation robustness in spherical symmetry.}
In spherical symmetry, $R(x)$ and $\QK$ are scalars on the two-dimensional orbit space, and the
marginal condition is the covariant statement $\nabla_\mu R\,\nabla^\mu R=0$.
Hence the normalization $\FE=\QK/R$ and its derived stress/pressure are invariant under
reparametrizations of the orbit-space coordinates.
This contrasts with boundary-stress prescriptions (e.g.\ Brown--York), where finite-part choices
and observer dependence are explicit and must be fixed by additional prescriptions
(Sec.~\ref{sec:BY}).
% =========================================================
\section{FRW/de~Sitter apparent horizons: the exact identity \texorpdfstring{$\PE=\rho/3$}{PE=rho/3}}
\label{sec:FRW}
On an FRW apparent horizon the Misner--Sharp energy satisfies
$\Emsh = (4\pi/3)R^3\rho$, and the Friedmann equation
$H^2 + kc^2/a^2 = (8\pi G/3c^2)\rho$ together with the
apparent-horizon condition $R_A = c/\sqrt{H^2 + kc^2/a^2}$ give
\begin{equation}
\label{eq:E_FRW}
E = \QK = \frac{c^4}{2G}\,R
\end{equation}
on $R=R_A$---the same one-scale marginal law as in Sec.~\ref{sec:backbone}.
\subsection{Laplace pressure without surface gravity}
Using \eqref{eq:PE_general} on $R=R_A$ gives
\begin{equation}
\label{eq:PE_rho}
\PE = \frac{c^4}{8\pi G R_A^2} = \frac{\rho}{3}.
\end{equation}
This identity is exact on the FRW apparent horizon.
It requires nothing beyond the charge anchor and the Friedmann equation---no
choice of dynamical surface gravity, no Brown--York subtraction, no boundary
observers.
It is a statement about the unique mechanical channel (dilation/Laplace pressure) on the
marginal branch; any temperature interpretation is optional and introduced only when admissible.
\subsection{Optional sign conventions: Hayward--Kodama surface gravity}
If one wishes to compare to a signed dynamical ``geometric surface gravity'' in FRW,
a common choice is Hayward--Kodama \cite{Hayward98},
\begin{equation}
\label{eq:kappaHK}
\kappaHK = -\frac{c^2}{R}\!\left(1 - \frac{\dot R}{2HR}\right)
\equiv -(1-\epsilon)\,\kappaE,
\qquad
\epsilon \equiv \frac{\dot R}{2HR},
\qquad
\kappaE\equiv\frac{c^2}{R},
\end{equation}
where the minus sign reflects orientation conventions for cosmological horizons.
In this paper we keep the bookkeeping scales $\kappaE$ and $\kappaH$ \emph{positive} by definition
(Sec.~\ref{sec:thermo}).
Since the Laplace pressure is quadratic in the acceleration scale, one has the exact identity
\begin{equation}
\label{eq:PE_kappa}
\PE = \frac{\kappaE^2}{8\pi G}
= \frac{\kappaHK^2}{8\pi G(1-\epsilon)^2}.
\end{equation}
Sign and orientation conventions for cosmological horizons are discussed in
\cite{BanihashemiJacobsonSveskoVisser23,Helou15}.
% =========================================================
\section{Thermodynamic translation with matched naming}
\label{sec:thermo}
\subsection{Minimal conjugacy scaffold and the \texorpdfstring{$E$}{E}-first law}
The mechanical sector is fixed by $(\QK,R)$ on marginal spheres.
To define heat-like and work-like conjugates without ambiguity, we use a minimal
two-variable bookkeeping space with coordinates $(S,R)$ and an energy potential $E(S,R)$.
The physical horizon family is a one-parameter locus within this space; conjugates are defined
before restricting to that locus.
Define
\begin{equation}
\label{eq:conjugates_E}
\TE \equiv \left(\frac{\partial E}{\partial S}\right)_{\!R},
\qquad
\FE \equiv -\left(\frac{\partial E}{\partial R}\right)_{\!S},
\end{equation}
so that
\begin{equation}
\label{eq:firstlaw_E}
dE = \TE\,dS - \FE\,dR.
\end{equation}
Because a round sphere has only one homogeneous deformation, the term $-\FE\,dR$ is the unique
mechanical channel. It may be repackaged as
\begin{equation}
\label{eq:packaging}
-\FE\,dR \;\equiv\; -\SigE\,dA \;\equiv\; -\PE\,dV,
\qquad
\SigE=\frac{\FE}{8\pi R},
\qquad
\PE=\frac{\FE}{A},
\end{equation}
so the Young--Laplace identity $\PE=2\SigE/R$ follows kinematically in spherical symmetry.
(For related horizon stress discussions, see \cite{Jaramillo14}.)
\subsection{Area entropy and the charge-anchored pair \texorpdfstring{$(\TE,\kappaE)$}{(TE,kappaE)}}
Adopt the semiclassical Einstein area law,
\begin{equation}
\label{eq:entropy}
S = \frac{k_B A}{4\lp^2}, \qquad A = 4\pi R^2.
\end{equation}
On the marginal one-scale branch, Smarr homogeneity closes the ladder
\begin{equation}
\label{eq:smarr_E}
E=\FE R=\TE S=2\,\SigE A = 3\,\PE V,
\end{equation}
so in particular $\TE=E/S$ and $\FE=E/R$ on that branch.
Using \eqref{eq:QK_marginal} and \eqref{eq:entropy} with $\lp^2=\hbar G/c^3$ gives
\begin{equation}
\label{eq:TE_value}
\TE = \frac{E}{S} = \frac{\hbar c}{2\pi k_B R}.
\end{equation}
Package \eqref{eq:TE_value} by the Unruh/Hawking normalization,
\begin{equation}
\label{eq:kappaE_def}
\TE \equiv \frac{\hbar\,\kappaE}{2\pi k_B c}
\qquad\Longrightarrow\qquad
\kappaE \equiv \frac{c^2}{R}.
\end{equation}
Combining \eqref{eq:PE_general} with \eqref{eq:kappaE_def} yields
\begin{equation}
\label{eq:PE_kappaE}
\PE = \frac{c^4}{8\pi G R^2} = \frac{\kappaE^2}{8\pi G}.
\end{equation}
\subsection{Pullback to the horizon family and the origin of \texorpdfstring{$2\,dE=T\,dS$}{2 dE = T dS}}
\label{sec:pullback}
Equation \eqref{eq:firstlaw_E} is an exact differential identity on the two-variable
bookkeeping space $(S,R)$: it decomposes changes of the energy potential $E(S,R)$ into an
entropy channel and the unique mechanical dilation channel.
The physical marginal family is a one-parameter locus within this space, so along any
parametrization $\lambda$ of that locus one has
\begin{equation}
\label{eq:pullback_lambda}
\frac{dE}{d\lambda} = \TE\,\frac{dS}{d\lambda} - \FE\,\frac{dR}{d\lambda}.
\end{equation}
If one instead defines a \emph{slope} (total-derivative) ``effective temperature'' along the locus by
$T_{\rm eff}\equiv dE/dS$, then \eqref{eq:pullback_lambda} gives the exact identity
\begin{equation}
\label{eq:Teff_def}
T_{\rm eff} = \frac{dE}{dS} = \TE - \FE\,\frac{dR}{dS}.
\end{equation}
Thus, whenever $R$ and $S$ co-vary, the slope temperature differs from the conjugate
$\TE=(\partial E/\partial S)_R$ by precisely the mechanical dilation term.
On the marginal Einstein branch $E\propto R$ and $S\propto R^2$, so $dR/dS = R/(2S)$ and
\eqref{eq:Teff_def} becomes
\begin{equation}
\label{eq:Teff_half}
T_{\rm eff} = \TE - \frac{E}{R}\frac{R}{2S} = \frac{\TE}{2}.
\end{equation}
This is the origin of the ubiquitous ``half-temperature'' relations obtained when the temperature is
identified with a total derivative along the horizon curve.
Equivalently, evaluating the two channels of \eqref{eq:firstlaw_E} on the one-scale marginal branch gives
\begin{equation}
\label{eq:TdS_2dE}
\TE\,dS = 2\,dE,
\qquad
\FE\,dR = dE,
\qquad\Longrightarrow\qquad
\TE\,dS = dE + \FE\,dR.
\end{equation}
When a Hawking temperature assignment is made in the FRW/de~Sitter sector (Sec.~\ref{sec:THawking}),
$\TE$ coincides with $\THW$ (i.e.\ $\alpha=1$) and \eqref{eq:TdS_2dE} becomes the frequently quoted
relation $\THW\,dS=2\,dE$.
This is not a failure of the first law: it is the bookkeeping statement that, along this rigid one-scale
family, the entropy channel partitions into an energy change and a fixed share of dilation work.
\paragraph{A horizon Gibbs--Duhem identity.}
On the marginal one-scale branch, $\TE\propto 1/R$, hence
\begin{equation}
\label{eq:GD_horizon}
d\TE = -\frac{\TE}{R}\,dR
\quad\Longrightarrow\quad
S\,d\TE = -\frac{\TE S}{R}\,dR = -\frac{E}{R}\,dR = -\FE\,dR.
\end{equation}
Taking the differential of the Smarr relation $E=\TE S$ and using \eqref{eq:GD_horizon} yields
\begin{equation}
\label{eq:Smarr_diff_firstlaw}
dE = d(\TE S)= \TE\,dS + S\,d\TE = \TE\,dS - \FE\,dR,
\end{equation}
i.e.\ the same two-channel split as \eqref{eq:firstlaw_E} when restricted to the marginal family.
Equation \eqref{eq:GD_horizon} is the direct one-scale analogue of the classical Gibbs--Duhem relation
$S\,dT - V\,dP=0$: an integrability condition linking intensive-variable differentials under Euler/Smarr
homogeneity.
The two-variable scaffold \eqref{eq:conjugates_E}--\eqref{eq:firstlaw_E} is nevertheless essential,
because it is what assigns \emph{semantic} meaning to the two channels (heat-like versus work-like)
before restricting to any one-parameter locus.
\subsection{Hawking translation and the sector parameter \texorpdfstring{$\alpha$}{alpha}}
\label{sec:THawking}
When a Hawking/Unruh temperature is admissible, define a positive Hawking scale $\kappaH\ge0$ by
\begin{equation}
\label{eq:kappaH_def}
\THW \equiv \frac{\hbar\,\kappaH}{2\pi k_B c}.
\end{equation}
Define the sector parameter as the ratio of the charge-anchored and Hawking acceleration scales,
\begin{equation}
\label{eq:alpha_def}
\alpha \equiv \frac{\kappaE}{\kappaH}.
\end{equation}
Equivalently, $\TE=\alpha\,\THW$.
For FRW/de~Sitter apparent horizons (when the Hawking/Unruh assignment is adopted), one has
$\kappaH=\kappaE$ and therefore $\alpha=1$.
For Schwarzschild, with areal radius $R=2GM/c^2$, one has $\kappaH=c^2/(2R)$ and hence $\alpha=2$.
This value follows directly from the standard Schwarzschild surface gravity; no independent maximum-force hypothesis is assumed.
\subsection{Hawking-scaled energy $U\equiv E/\alpha$ and the \texorpdfstring{$dU$}{dU} first law}
Once a Hawking/Unruh temperature is admissible, it is convenient to package the thermodynamics
in terms of the Hawking-scaled energy and its radius conjugate,
\begin{equation}
\label{eq:U_FH}
U \equiv \frac{E}{\alpha},
\qquad
\FH \equiv \frac{\FE}{\alpha}
= -\left(\frac{\partial U}{\partial R}\right)_{\!S}.
\end{equation}
Since $\alpha$ is constant in a given one-scale sector (FRW/de~Sitter with $\alpha=1$ and
Schwarzschild with $\alpha=2$), one has $dU=d(E/\alpha)$, and dividing the charge-anchored first law \eqref{eq:firstlaw_E}
by $\alpha$ yields the Hawking-packaged first law
\begin{equation}
\label{eq:firstlaw_U}
dU = \THW\,dS - \FH\,dR,
\end{equation}
with the Hawking temperature recovered as the $S$-conjugate of $U$:
$\left(\partial U/\partial S\right)_{R}=\TE/\alpha=\THW$.
In constant-$\alpha$ one-scale sectors, $\THW\propto 1/R$, so $U\propto R$ and
\begin{equation}
\label{eq:U_smarr}
U=\THW S=\FH R.
\end{equation}
Using $\kappaH=\kappaE/\alpha=c^2/(\alpha R)$ and $M\equiv E/c^2$, one obtains
\begin{equation}
\label{eq:FH_Mkappa}
\FH=\frac{U}{R}=\frac{E}{\alpha R}=M\kappaH,
\end{equation}
while $\FE=E/R=\alpha\,\FH$.
Finally,
\begin{equation}
\label{eq:E_alphaU}
E=\TE S=\alpha\,\THW S=\alpha\,U.
\end{equation}
\begin{table}[t]
\caption{%
Minimal spherical-screen dictionary in four dimensions.
The mechanical sector is fixed by $(\QK,R)$ on marginal spheres.
Adopting area entropy closes $(\TE,\kappaE)$; Hawking/Unruh packaging introduces
$(\THW,\kappaH)$ and $\alpha=\kappaE/\kappaH$.}
\label{tab:dict}
\begin{ruledtabular}
\begin{tabular}{@{}ll@{}}
Symbol & Meaning \\
\hline
$\QK=\Emsh$ & Kodama charge (bulk geometric energy in spherical symmetry) \\
$E$ & Charge-anchored energy on marginal spheres: $E=\QK=c^4R/(2G)$ \\
$\FE=E/R$ & Charge-anchored dilation force; $=c^4/(2G)$ on marginal branch \\
$\PE=\FE/A$ & Laplace pressure; on FRW horizons $\PE=\rho/3$ \\
$S=k_BA/(4\lp^2)$ & Area entropy functional (Einstein gravity) \\
$\TE=(\partial E/\partial S)_R$ & Charge-anchored temperature (entropy conjugate) \\
$\kappaE=c^2/R$ & Charge-anchored one-scale acceleration \\
$T_{\rm eff}=dE/dS$ & Slope temperature along a one-parameter horizon locus \\
$\THW=\hbar\kappaH/(2\pi k_B c)$ & Hawking/Unruh temperature (when admissible) \\
$\alpha=\kappaE/\kappaH$ & Sector parameter (FRW: $\alpha=1$; Schwarzschild: $\alpha=2$) \\
$U=E/\alpha$ & Hawking-scaled energy (equals $\THW S$ in one-scale sectors) \\
$\FH=\dfrac{U}{R}=M\kappaH$ & Hawking-scale force \\
\end{tabular}
\end{ruledtabular}
\end{table}
% =========================================================
\section{Schwarzschild partition and the meaning of \texorpdfstring{$\alpha$}{alpha}}
\label{sec:virial}
Using $U=E/\alpha=\THW S=\FH R$ in constant-$\alpha$ one-scale sectors, one may rewrite
\begin{equation}
\label{eq:partition_general}
E=\alpha U
=\underbrace{\THW S}_{\text{horizon heat }U}
+\underbrace{(\alpha-1)\,\FH R}_{\text{force--radius share}}.
\end{equation}
For FRW/de~Sitter ($\alpha=1$), the second term vanishes: $E=U=\THW S$.
For Schwarzschild ($\alpha=2$), the partition becomes
\begin{equation}
\label{eq:Schwarzschild_smarr}
E = \THW S + \FH R,
\qquad
\THW S = \FH R = \frac{E}{2}.
\end{equation}
This is the standard Schwarzschild Smarr relation rewritten as ``heat plus force--radius,''
so the factor of two is interpreted as an equal split between horizon heat content and a
force--radius share. In stationary settings, this matches the usual Smarr/Komar bookkeeping
\cite{Smarr73,Komar59} (see also \cite{Szabados09} for review of quasilocal energy).
% =========================================================
\section{Brown--York realization and horizon calibration}
\label{sec:BY}
The bulk charge anchor is self-contained on marginal spheres.
A Brown--York boundary treatment is neither required nor uniquely defined,
but it provides a useful cross-check: it shows the same dilation channel
emerging from a Dirichlet variational principle, and it makes the
scheme dependence explicit.
\subsection{Projection identity}
On a timelike worldtube $\mathcal{B}$ foliated by round two-spheres, the
Brown--York stress tensor \cite{BrownYork93}
\begin{equation}
\label{eq:BY_tensor}
\tau^{ij} = \frac{c^4}{8\pi G}(K^{ij} - K\gamma^{ij}) - \tau^{ij}_{(0)}
\end{equation}
(with $\tau^{ij}_{(0)}$ encoding counterterms \cite{AstefaneseiEtAl09})
is isotropic in spherical symmetry.
Under areal-radius dilation, the stress--strain work on a cross-section
$\mathcal{S}$ \cite{BrownCreightonMann94} reduces to
\begin{equation}
\label{eq:BY_proj}
\delta W_{\mathcal{S}} = -8\pi R\,\SigBY\,\delta R \equiv -\FE\,\delta R,
\end{equation}
so the Brown--York tangential stress computes exactly the radius-conjugate
dilation force:
\begin{equation}
\label{eq:FE_BY}
\FE = 8\pi R\,\SigBY = 4\pi R\,\sigma_{ij}\tau^{ij}.
\end{equation}
This is an identity given the boundary data.
The Brown--York stress provides one way to evaluate $\FE$ in a Dirichlet
ensemble; it does not remove the dependence on the screen location, boundary
observers, and counterterm choice.
\subsection{Horizon calibration}
Two standard ambiguities must be fixed to compare boundary and bulk
normalizations.
For boundary observers we use the Kodama flow (which reduces to Killing
observers in the static case).
For the remaining finite counterterm freedom, we impose the horizon-branch
calibration
\begin{equation}
\label{eq:calibration}
E\big|_{\mathcal{H}} = \QK.
\end{equation}
This is a renormalization condition: among all admissible Dirichlet schemes,
choose the one whose finite part matches the observer-independent bulk charge
on the marginal branch.
With this choice, $\FE|_{\mathcal{H}} = F_{\rm K}$ and the boundary and bulk
dictionaries agree precisely where the bulk charge is most robust.
Appendix~\ref{app:dS} illustrates the need for calibration in the de~Sitter
static patch, where the standard flat-subtracted Brown--York energy at the
cosmological horizon is $c^4L/G$---twice the Kodama charge $c^4L/(2G)$.
Appendix~\ref{app:FRW_BY} gives an explicit FRW stretched-horizon computation
showing how local subtractions cancel both the $\chi^{-1}$ and $\chi^{-3}$
near-horizon divergences while enforcing~\eqref{eq:calibration}.
% =========================================================
\section{Conclusion}
\label{sec:conclusion}
A round sphere has one homogeneous deformation, and four-dimensional Einstein gravity supplies one
preferred quasilocal energy in spherical symmetry: the Misner--Sharp--Hernandez/Kodama charge.
On marginal spheres, this immediately fixes a one-scale law $E=\QK=c^4R/(2G)$ and therefore a complete,
subtraction-free mechanical dictionary---dilation force, tangential stress, and Laplace pressure---from
bulk geometry alone, without boundary observers or surface-gravity prescriptions.
For FRW/de~Sitter apparent horizons, the same charge anchor yields the exact identity $\PE=\rho/3$.
Adopting the area entropy closes the matched charge-anchored pair $(\TE,\kappaE)$ with
$\kappaE=c^2/R$ and packages the exact pressure as $\PE=\kappaE^2/(8\pi G)$.
Using the two-variable scaffold $(S,R)$, the first law takes the unambiguous form
$dE=\TE\,dS-\FE\,dR$; restricting (pulling back) to the one-parameter marginal family then yields
$\TE\,dS=dE+\FE\,dR$ and hence $\TE\,dS=2\,dE$ in the FRW/de~Sitter sector, resolving the standard
``$2dE=T\,dS$'' bookkeeping as a fixed partition between an energy change and dilation work.
When a Hawking/Unruh temperature is admissible, introducing the positive Hawking scale $\kappaH$
and the sector parameter $\alpha=\kappaE/\kappaH$ yields the Hawking-scaled energy $U\equiv E/\alpha$ and the
Hawking-scale force $\FH=U/R=M\kappaH$ with a compact first law $dU\equiv d(E/\alpha)=\THW\,dS-\FH\,dR$.
In the Schwarzschild sector ($\alpha=2$) the standard $E=2\THW S$ is reorganized as
$E=\THW S+\FH R$; cosmological horizons correspond to $\alpha=1$ and $E=U$.
Finally, the same one-scale Smarr closure propagates to a Compton-saturated Planck cell, selecting a
saturated Bekenstein equation of state as a UV benchmark and fixing the associated effective coupling
(Appendix~\ref{app:UV}).
% =========================================================
\appendix
\section{Relation to Hayward/Cai--Kim variables}
\label{app:unified}
Hayward's unified first law \cite{Hayward98} writes
$dE = A\psi + W\,dV$ with bulk work density $W = \frac{1}{2}(\rho - p_{\rm fluid})$.
Projecting along the apparent-horizon generator gives the Cai--Kim
form \cite{CaiKim05} $dE = \THW\,dS + W\,dV$.
In the present organization the mechanical channel is $-\PE\,dV$ with
$\PE = \rho/3$, while $p_{\rm fluid} = w\rho$ for a perfect fluid, so
\begin{equation}
W = \tfrac{1}{2}(1-w)\rho = \tfrac{3}{2}(1-w)\,\PE.
\end{equation}
The two prescriptions coincide only for radiation ($w = 1/3$).
The distinction matters: $\PE$ is the screen-normal Laplace pressure conjugate
to spherical dilation in the screen mechanics, whereas $W$ is a bulk work density
appropriate to the volume-flux decomposition.
% =========================================================
\section{Brown--York stress on an FRW stretched horizon}
\label{app:FRW_BY}
We compute the Brown--York tangential stress in spatially flat FRW
($k = 0$, $R = a(t)r$, $H = \dot a/a$) on the worldtube of constant areal
radius $R_0$, whose induced metric is
$ds_{\mathcal{B}}^2 = -c^2\chi^2\,dt^2 + R_0^2\,d\Omega^2$
with $\chi = \sqrt{1 - R_0^2/R_A^2}$ and $R_A = c/H$.
The raw tangential-stress coefficient is
\begin{equation}
\label{eq:Sig_raw}
\SigBY^{\rm raw} = \frac{c^4}{8\pi G}\!\left(
\frac{\chi}{R_0} - \frac{R_0 H^2}{c^2\chi} - \frac{R_0\dot H}{c^2\chi^3}
\right) - \Sigma_{(0)}.
\end{equation}
As $R_0 \to R_A$ ($\chi \to 0$), the raw stress diverges---the $\chi^{-1}$
term reflects the diverging redshift of static observers near the horizon
(cf.\ the membrane paradigm \cite{ThornePriceMacdonald86}), and dynamical
evolution ($\dot H \neq 0$) introduces a stronger $\chi^{-3}$ divergence.
A finite normalization is obtained by choosing a local subtraction that cancels
the divergent terms and fixing the remaining finite part by the calibration
$E|_{\mathcal{H}} = \QK$.
In the quasi-static limit ($\dot H \to 0$), taking
$\Sigma_{(0)}^{\rm qs} = -(c^4/8\pi G)(R_0/R_A^2\chi) - c^4/(16\pi G R_A)$
yields
\begin{equation}
\SigBY^{\rm ren} \xrightarrow{R_0 \to R_A}
\frac{c^4}{16\pi G R_A} = \SigE,
\end{equation}
and hence $\FE^{\rm ren} \to c^4/(2G) = F_{\rm K}$, matching the bulk anchor.
In dynamical FRW, an analogous subtraction
\begin{equation}
\Sigma_{(0)}^{\rm dyn} = -\frac{c^4}{8\pi G}\!\left(
\frac{R_0 H^2}{c^2\chi} + \frac{R_0\dot H}{c^2\chi^3}\right)
- \frac{c^4}{16\pi G R_A}
\end{equation}
cancels both divergences while preserving the horizon calibration.
Different covariant counterterm prescriptions correspond to different finite
normalizations of the same Dirichlet ensemble; we fix the residual ambiguity
by~\eqref{eq:calibration}.
% =========================================================
\section{\texorpdfstring{$D$}{D}-dimensional generalization}
\label{app:D}
For a $(D{-}2)$-sphere of areal radius $R$,
$A_D = \Omega_{D-2} R^{D-2}$ and
$dA_D/dR = (D-2)A_D/R$, so an isotropic stress $\Sigma$ gives a pressure
$p = (D-2)\Sigma/R$ (reducing to $2\Sigma/R$ in $D = 4$).
On marginal spheres the Kodama charge scales as $\QK \propto R^{D-3}$, hence
$F_{\rm K} = \QK/R \propto R^{D-4}$.
The force is scale-invariant on the marginal branch if and only if $D = 4$.
% =========================================================
\section{UV benchmark: Smarr rigidity under Compton saturation}
\label{app:UV}
At $R \sim \lp$ semiclassical control is lost, but the algebraic structure of
the Smarr relation can still be propagated as a benchmark.
The question is: if one insists that $E = \TE S = \FE R$ hold on a one-scale
branch at the Planck cutoff, what does the structure require?
The Bekenstein bound is an inequality; the benchmark here concerns the
\emph{saturated} case obtained under Compton saturation.
\subsection{What Smarr homogeneity fixes: the functional form}
On a one-scale branch, $E = \FE R$ with $\FE = E/R$ (Euler scaling).
The Smarr closure $E = \TE S$ with $\TE = (\partial E/\partial S)_R$ then
requires $E(S,R)$ to be linear in $S$ at fixed $R$:
\begin{equation}
\label{eq:linear_EOS}
E(S,R) = \frac{a}{R}\,S
\qquad\Longleftrightarrow\qquad
S(E,R) = \frac{1}{a}\,ER,
\end{equation}
for some constant $a > 0$ with dimensions of action per Boltzmann constant.
\subsection{What fixes the coefficient: Bekenstein normalization (and Unruh/Hawking packaging)}
The conjugate temperature from~\eqref{eq:linear_EOS} is $\TE = a/R$.
Smarr homogeneity fixes only the \emph{functional form} $S\propto ER$ on a one-scale family; it does not
fix the numerical coefficient.
A natural UV benchmark is to require agreement with the \emph{saturated} Bekenstein bound for a system of
radius $R$ and energy $E$, i.e.\ $S(E,R)=2\pi k_B\,ER/(\hbar c)$.
Comparing this with $S(E,R)=(1/a)\,ER$ in~\eqref{eq:linear_EOS} fixes
\begin{equation}
a=\frac{\hbar c}{2\pi k_B}.
\end{equation}
The corresponding one-scale equation of state is therefore the saturated Bekenstein form
\begin{equation}
\label{eq:Bek}
S(E,R) = 2\pi k_B\,\frac{ER}{\hbar c}
\qquad\Longleftrightarrow\qquad
E(S,R) = \frac{\hbar c}{2\pi k_B}\,\frac{S}{R}.
\end{equation}
Having fixed $a$ in this way, the temperature scale \eqref{eq:TE_value} may be \emph{packaged} in the standard
Unruh/Hawking normalization, $\TE=\hbar\kappaE/(2\pi k_B c)$, which returns $\kappaE=c^2/R$.
(No claim is made that semiclassical Hawking radiation is physically realized at $R\sim\lp$; this appendix is a
benchmark propagation of one-scale Smarr structure.)
\subsection{Compton saturation and the Planck cell}
Localizing energy $E$ within radius $R$ requires $R \geq \hbar c/E$ (reduced
Compton wavelength; we adopt the standard reduced convention throughout).
Saturating this at the minimum scale gives $ER = \hbar c$.
Inserting into~\eqref{eq:Bek} yields $S = 2\pi k_B$: a single cell at maximum
entropy.
At $R = \lp$, the Compton condition gives $E = \Ep$ and hence
\begin{equation}
\label{eq:Planck_ma}
m_{\rm P}\equiv \frac{\Ep}{c^2},\qquad
a_{\rm P}\equiv \kappaE(\lp)=\frac{c^2}{\lp},
\qquad
\FE(\lp)=\frac{\Ep}{\lp}=m_{\rm P}a_{\rm P}=\frac{c^4}{G}.
\end{equation}
This is exactly twice the macroscopic anchor $F_{\rm K} = c^4/(2G)$.
\subsection{What is determined at the Planck cell}
Given Smarr homogeneity and the saturated Bekenstein benchmark (equivalently, the conventional $2\pi$ Unruh/Hawking packaging), three quantities
at $R = \lp$ are fixed without additional free parameters.
\emph{Entropy coefficient.}
At $R = \lp$, the Compton-saturated Bekenstein benchmark gives $S = 2\pi k_B$
while the Einstein area law gives $S = \pi k_B$.
This is a comparison of two equations of state, not a proposed modification of
the area law.
\emph{Effective coupling.}
The force doubling is encoded as $\Geff = G/2$, so the single-cell energy
takes the marginal form $E = c^4 R/(2\Geff)$.
The compactness $\Ccomp = 2GE/(c^4 R)$ rises from $1$ (macroscopic) to $2$
(single cell).
\emph{Sector assignment.}
From~\eqref{eq:Bek}, $\TE = \hbar c/(2\pi k_B R)$, i.e.\ the $\alpha=1$ sector.
In this benchmark, the Planck cell lies in the same $\alpha=1$ sector as FRW/de~Sitter.
% =========================================================
\section{Counterterms: a de~Sitter illustration}
\label{app:dS}
In the de~Sitter static patch
$ds^2 = -(1-R^2/L^2)c^2 dt^2 + dR^2/(1-R^2/L^2) + R^2 d\Omega_2^2$
with $L = c/H$, the flat-subtracted Brown--York energy on a screen at radius
$R$ is \cite{BrownYork93}
\begin{equation}
E_{\rm BY}(R) = \frac{c^4}{G}\,R\!\left(1 - \sqrt{1 - R^2/L^2}\right).
\end{equation}
At the cosmological horizon $R = L$ this gives $E_{\rm BY} = c^4 L/G$, twice
the Kodama charge $\QK = c^4 L/(2G)$.
The factor-of-two illustrates the scheme dependence of Brown--York energies on
finite boundaries; the horizon calibration~\eqref{eq:calibration} resolves it.
% =========================================================
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\end{document}