A Smearing-Free Constraint-Pressure Mechanism for Dark Energy from Emergent Causal Time

Autor: A. Jaster | Datum: 2026/06/12 | DOI: 10.5281/zenodo.20667781

Abstract

We develop a purely cosmological model in which the cosmological constant is replaced by a causal — that is, retarded-source — constraint pressure built from the retarded matter distribution, resting on two axioms of emergent causal time: time couples to matter, and causality as a construction principle (all local sources are built retarded, on the past lightcone); the homogeneous background closes as a causal system, while full perturbative causal closure remains open. For the late, classical, homogeneous background no quantisation or causal smearing is required, and we work without it. The minimal realisation — a bare, volume-filling retarded bulk measure — introduces no additional equation-of-state parameter: the shape of w(z) is fixed, with no crossing of w=−1, deepening toward −4/3 at high redshift and relaxing to −1 in the de Sitter future; numerically w0 ≃ −1.20 at a fiducial Ωm=0.30 and w0 ≃ −1.19 at the model's own best-fit Ωm=0.265. Throughout we distinguish the fiducial Ωm=0.30 (used to isolate the dark-energy effect) from the fit-preferred Ωm=0.265 (used for all data-facing statements). Two structural problems are addressed: the vacuum-energy fine-tuning is sidestepped because the constraint couples only to real matter, and the coincidence is structurally mitigated: the time dependence ρC/ρm ∝ a4 follows from the constraint structure, while the present ratio is calibrated through μB. Energy conservation requires a non-local companion Tnlμν; its local sources are retarded, while strict causality of the full sector is expected from an in-in construction that has not yet been carried out explicitly. Within the assumed companion limits its effect on the background is bounded: the companion can deepen w0 by at most µ 0.04. A Dirac analysis establishes kinematic ghost-freeness — the constraint sector adds no propagating degree of freedom — while the deeper field-theoretic health is left open. Linear perturbations are lightcone-suppressed, so the constraint sector is smooth on structure scales: to leading order in this suppression Geff=G and the gravitational slip ηgrav=Ψ/Φ=1 are predicted, and dark matter is not eliminated. At fit-consistent parameters the growth signature is a pattern, not a uniform boost: fσ8 enhanced by ∼6% at z ≳ 0.5 but slightly suppressed at z=0, with S8 ≈ 0.79 — comparable to ΛCDM and not a second tension, though this relief itself rests on the low Ωm preferred by the otherwise poor BAO fit. Confronted with DESI DR2 baryon-acoustic-oscillation data plus Planck distance priors, the model is strongly disfavoured relative to ΛCDM, Δχ2 ≃ 28.5 at equal parameter count (ΔAIC=ΔBIC=Δχ2), the conflict residing in the BAO distances — the committed kernel is thus likely already strongly disfavoured by current geometry — while it remains compatible with the CMB within the compressed-prior approximation and drives H0 upward. The single amplitude μB may acquire a physical value, and the background a falsifiable modification, through time-bubble nucleation. The model is sharply falsifiable on independent fronts — expansion, growth pattern, and gravitational slip — which is the point of a fixed-shape prediction.

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\newcommand{\fB}{f_B}
\newcommand{\IH}{I_H}
\newcommand{\chiH}{\chi_H}
\newcommand{\Om}{\Omega_m}
\newcommand{\OC}{\Omega_C}
\newcommand{\rhoC}{\rho_C}
\newcommand{\muB}{\mu_B}
\newcommand{\nf}{n_f}

\title{\textbf{A Smearing-Free Constraint-Pressure Mechanism for\\
Dark Energy from Emergent Causal Time}}
\author{A.\ Jaster}
\date{June 2026}

\begin{document}
\maketitle

\begin{abstract}
\noindent
We develop a purely cosmological model in which the cosmological constant is replaced by a causal ---
that is, retarded-source --- \emph{constraint pressure} built from the retarded matter distribution, resting on two axioms of
emergent causal time: time couples to matter, and causality as a construction principle (all local
sources are built retarded, on the past lightcone); the homogeneous background closes as a causal
system, while full perturbative causal closure remains open. For the late, classical, homogeneous background no
quantisation or causal smearing is required, and we work without it. The minimal realisation --- a
bare, volume-filling retarded bulk measure --- introduces \emph{no additional equation-of-state
parameter}: the \emph{shape} of $w(z)$ is fixed, with no crossing of $w=-1$, deepening toward $-4/3$ at
high redshift and relaxing to $-1$ in the de~Sitter future; numerically $w_0\simeq-1.20$ at a fiducial
$\Om=0.30$ and $w_0\simeq-1.19$ at the model's own best-fit $\Om=0.265$. Throughout we distinguish the
fiducial $\Om=0.30$ (used to isolate the dark-energy effect) from the fit-preferred $\Om=0.265$ (used
for all data-facing statements). Two
structural problems are addressed: the vacuum-energy fine-tuning is sidestepped because the constraint
couples only to real matter, and the coincidence is structurally mitigated: the \emph{time dependence}
$\rhoC/\rho_m\propto a^4$ follows from the constraint structure, while the present ratio is calibrated
through $\muB$. Energy conservation requires a non-local companion $T^{\rm nl}_{\mu\nu}$; its local sources
are retarded, while strict causality of the
full sector is \emph{expected} from an in-in construction that has not yet been carried out explicitly.
Within the assumed companion limits its effect on the background is bounded: the companion can deepen
$w_0$ by at most $\approx0.04$. A Dirac analysis establishes kinematic ghost-freeness --- the constraint sector adds no
propagating degree of freedom --- while the deeper field-theoretic health is left open. Linear
perturbations are lightcone-suppressed, so the constraint sector is smooth on structure scales: to leading
order in this suppression $G_{\rm eff}=G$ and the gravitational slip $\eta_{\rm grav}=\Psi/\Phi=1$ are predicted,
and dark matter is \emph{not} eliminated. At fit-consistent parameters the growth signature is a
\emph{pattern}, not a uniform boost: $f\sigma_8$ enhanced by $\sim6\%$ at $z\gtrsim0.5$ but slightly
suppressed at $z=0$, with $S_8\approx0.79$ --- comparable to $\Lambda$CDM and \emph{not} a second
tension, though this relief itself rests on the low $\Om$ preferred by the otherwise poor BAO fit.
Confronted with DESI DR2 baryon-acoustic-oscillation data plus Planck distance
priors, the model is strongly disfavoured relative to $\Lambda$CDM, $\Delta\chi^2\simeq28.5$ at equal
parameter count ($\Delta{\rm AIC}=\Delta{\rm BIC}=\Delta\chi^2$), the conflict residing in the BAO
distances --- the committed kernel is thus likely already strongly disfavoured by current geometry ---
while it remains compatible with the CMB within the compressed-prior approximation and drives $H_0$
upward. The single amplitude
$\muB$ may acquire a physical value, and the background a falsifiable modification, through time-bubble
nucleation. The model is sharply falsifiable on independent fronts --- expansion, growth pattern, and
gravitational slip --- which is the point of a fixed-shape prediction.
\end{abstract}

\section{Relation to the earlier emergent-time cosmology}\label{sec:relation}
The cosmological implications of the same time field were examined in an earlier study~\cite{jaster2026d}.
There the homogeneous limit was taken to be $\Theta=1$ identically, so that standard cosmology is
reproduced exactly and the cosmological constant remains necessary; the emergent-time field produced
only the small deviations $\Theta<1$ in cosmic voids, whose phenomenological consequences (a possible
contribution to the $\sigma_8$ tension, a modified ISW effect, void clocks, $\Theta$-shell lensing) were
of unmeasurably small amplitude, and the Hubble tension was found unresolved, any background-level
correction having the wrong sign.

The present work departs from that treatment in one decisive respect, which is a change of
\emph{normalisation} of the causal content. There the Machian causal density $\Omega_H$ is normalised by
its own homogeneous-FLRW reference value $\Omega_{\rm ref}$, so that the ratio $\Omega_H/\Omega_{\rm
ref}\equiv1$ in the homogeneous background and the field $\Theta=S_\infty(1)=1$ is inert, leaving
$\Lambda$ necessary and deviations confined to voids. Here, instead, the fundamental cosmological
constant is set to zero and the accumulated causal bulk $\IH$ is normalised by the causal-horizon
four-volume $\chiH^4$; the resulting density $\fB=\IH/\chiH^4\propto a$ does \emph{not} cancel in the
homogeneous limit and therefore gravitates as a constraint pressure $\rhoC=\muB\fB$ (Section~\ref{sec:bg}).
This change of reference --- from the homogeneous-mean density to the horizon four-volume ---
converts an inert causal field into a dynamical dark-energy density, and the homogeneous background is
therefore no longer exactly $\Lambda$CDM. Two clarifications keep the comparison honest. First, the
departure involves \emph{two} assumptions, not one: setting $\Lambda_{\rm fund}=0$, and adopting the
horizon-four-volume normalisation --- the second makes the causal field gravitate, the first makes it the
sole dark-energy source; both are introduced here and modify the underlying emergent-time model
of~\cite{jaster2026a}, which retains the cosmological constant and does not promote the causal bulk to a
homogeneously gravitating component. Second, the earlier cosmology was framed within the quantised,
Planck-scale-smeared theory~\cite{jaster2026b}, whereas the present treatment is deliberately
smearing-free (Section~\ref{sec:nosmear}). We do not conceal that this is
a change of assumption rather than a consequence of the earlier work: the conclusion ``$\Lambda$ remains
necessary'' holds within the earlier assumptions, and we here trade it for a falsifiable dynamical
alternative. The revised predictions, relative to~\cite{jaster2026d}, are:
\begin{itemize}
\item dark energy becomes dynamical (phantom, $w_0\approx-1.20$, deepening to $-4/3$) and $\Lambda$ is no
longer required, in place of a retained cosmological constant ($w=-1$);
\item the vacuum-energy problem is sidestepped (real-matter coupling) and the coincidence structurally
mitigated (time dependence $\rhoC/\rho_m\propto a^4$ structural; present ratio calibrated;
$\rhoC\propto\Om$), rather than left to $\Lambda$;
\item the $\sigma_8/S_8$ effect becomes a measurable, background-driven growth \emph{pattern} (enhanced
at $z\gtrsim0.5$, slightly suppressed at $z=0$ at fit-consistent parameters), in place of
the unmeasurably small void contribution;
\item the model now shifts $H_0$ upward --- the direction that eases the distance-ladder tension --- in
place of a negligible, wrong-sign correction.
\end{itemize}
The void-level $\Theta<1$ phenomenology of~\cite{jaster2026d} is unchanged by, and orthogonal to, the
present mechanism; the two are distinct models within the same framework, differing only in whether the
causal field contributes at the homogeneous level. The trade-off is equally explicit: the earlier model
preserved the exact standard background but explained nothing about dark energy, whereas the present
model explains the dark-energy sector at the cost of a now strong tension with the DESI distance data
(Section~\ref{sec:data}).

\section{Introduction}
Emergent causal time defines physical proper time through a scalar field $\Theta(x)\in[0,1]$ built
from the retarded matter distribution in the past lightcone~\cite{jaster2026a}. The physical clock
metric $\hat g_{\mu\nu}=q_{\mu\nu}+\tfrac{1-\Theta}{c^2}U_{H\mu}U_{H\nu}$ replaces the background
metric in all field equations, and general relativity (GR) is recovered wherever $\Theta=1$.

This paper asks a deliberately narrow question: starting from the two foundational axioms alone, can
one build a cosmological model with few parameters and sharp, testable predictions, in which the
cosmological constant is replaced by a pressure derived from matter? We adopt:
\begin{enumerate}
\item \textbf{Time couples to matter:} $\Theta$ and the constraint pressure follow from the matter
distribution.
\item \textbf{Causality:} all local source terms are retarded, built on the past lightcone $J^-(x)$.
\end{enumerate}
Mach's principle and global Lorentz-violation are taken over from the underlying theory~\cite{jaster2026a}
as intended consequences; we do not re-derive them here. We make no quantum-field-theoretic claims; in
particular, causal (Planck-scale) smearing is not invoked (Section~\ref{sec:nosmear}).

\section{The smearing-free choice}\label{sec:nosmear}
Causal smearing regularises the sharp null-cone boundary on a Planck width and is needed in the
underlying theory for two purposes~\cite{jaster2026b}: ultraviolet regularisation of the quantised field, and local
null-cone localisation. For the object of interest here --- the late, classical, homogeneous background
--- neither applies, and we work without smearing. This is a controlled reduction: the cosmological
background is governed by the classical lightcone quadrature alone, as verified by explicit
self-consistent integration below. We note one caveat for completeness: in the absence of smearing the
lightcone boundary is sharp, so the metric variation and the perturbative analysis can in principle
generate distributional boundary terms at the null cone. These belong to the open construction of the
non-local companion (Sections~\ref{sec:tnl},~\ref{sec:pert}) and do not affect the homogeneous
background.

\section{Model}\label{sec:model}
\subsection{Constraint pressure}
We set the fundamental cosmological constant to zero and replace it by a causal density. Define the
accumulated four-dimensional causal content
\begin{equation}
\IH(x)=\int_{J^-(x)} \rho_{\rm real}(x')\,\sqrt{-q}\;\mathrm d^4x',
\label{eq:IH}
\end{equation}
with a volume-filling (uniformly weighted) measure over the interior of the past lightcone, and the
local four-dimensional causal density
\begin{equation}
\fB(t)=\frac{\IH(t)}{\chiH^4(t)},\qquad
\chiH(a)=\int_0^a\frac{\mathrm da'}{a'^2 H(a')}\quad(\text{comoving particle horizon}).
\label{eq:fB}
\end{equation}
Dimensionally, $\IH$ carries (energy density)$\times$(four-volume); dividing by the comoving four-volume
$\chiH^4$ (with $c=1$, all lengths comoving and the scale factors carried explicitly by $\sqrt{-q}=a^4$
in \eqref{eq:IH}) returns an energy density, as the matter-era check $\fB\propto\rho_{m,0}a$ of
Section~\ref{sec:bg} confirms. All residual normalisation and convention freedom (comoving versus
physical lengths, the choice of reference four-volume) is concentrated in the single constant $\muB$
and fixed by the flatness calibration; $\muB$ thus carries convention content as well as physical
amplitude. The
dark-energy (constraint) density is
\begin{equation}
\rhoC(a)=\muB\,\fB(a),
\label{eq:rhoC}
\end{equation}
where $\muB$ is the single free amplitude parameter, taking over the role of $\Lambda$.

\paragraph{Model choices.} Three ingredients are adopted as motivated choices, not derivations.
(a)~The power $n=4$ in $\IH/\chiH^4$ is the one consistent with the four-dimensional definition
\eqref{eq:IH}, so that $\fB$ has the dimensions of an energy density. (b)~The normalisation length is
the comoving particle horizon $\chiH=\eta$, selected by causality (built from the past) and by
saturation (it approaches a constant in a de~Sitter future). (c)~The measure is the \emph{bare} uniform
bulk weight --- the minimal choice with no additional depth profile, denoted $m=0$ in the one-parameter
family of Section~\ref{sec:alt}. Choice~(c) is made on grounds of parsimony alone --- the bare measure carries no additional depth
profile --- and is not derivable from the axioms; we do not invoke the (non-robust) CMB preference of
Section~\ref{sec:data} in its support. We refer
to the resulting fully specified model as the \emph{committed model}; the broader family is discussed
only in Section~\ref{sec:alt}.

\subsection{Coupling to real matter only}
The source $\rho_{\rm real}$ in \eqref{eq:IH} is the real matter and radiation density; vacuum energy is
excluded by construction. This is a statement about what the time field responds to --- real excitations,
not the vacuum --- and it sidesteps (defers, rather than dynamically solves) the cosmological-constant fine-tuning problem as a property of the
coupling rather than through any dynamical cancellation (Section~\ref{sec:solved}).

\section{Background dynamics}\label{sec:bg}
\subsection{Matter-era scaling}
In flat Friedmann--Lema\^itre--Robertson--Walker (FLRW) geometry (conformal time $\eta$, $c=1$),
$\sqrt{-q}\,\mathrm d^4x'=a^4\,\mathrm d\eta'\,\mathrm d^3x_{\rm com}$ and
$\rho_{\rm real}=\rho_{m,0}a^{-3}$. The spatial integral over the past-lightcone interior at conformal
time $\eta'$ is the comoving ball of radius $(\eta-\eta')$, so
\begin{equation}
\IH(\eta)\simeq\frac{4\pi}{3}\,\rho_{m,0}\int_0^\eta a(\eta')\,(\eta-\eta')^3\,\mathrm d\eta'.
\label{eq:quad}
\end{equation}
With $a\propto\eta^2$, $\int_0^\eta\eta'^2(\eta-\eta')^3\mathrm d\eta'=\eta^6B(3,4)\propto a^3$, hence
\begin{equation}
\IH\propto a^{3},\qquad \chiH=\eta\propto a^{1/2}\ \Rightarrow\ \chiH^4\propto a^2,\qquad
\fB=\IH/\chiH^4\propto a^{+1}.
\label{eq:scal}
\end{equation}

\paragraph{Radiation era.} In the radiation era $a\propto\eta$ and $\rho_{\rm real}a^4\to$ const, so
\eqref{eq:quad} gives $\IH\propto\int_0^\eta(\eta-\eta')^3\mathrm d\eta'\propto\eta^4\propto a^4$, while
$\chiH^4=\eta^4\propto a^4$; hence $\fB\to$ const, $\nf\to0$ and $w_C\to-1$. The constraint density is
then a small constant with $\rhoC/\rho_r\propto a^4\to0$, so the constraint sector is explicitly
negligible at big-bang nucleosynthesis and recombination, independently of the matter-era result.

\subsection{Effective equation of state}
The gravitating constraint density $\muB\fB(a)$ is local and time-dependent; continuity gives, at
leading order in the companion term (Section~\ref{sec:tnl}),
\begin{equation}
w_C(a)=-1-\tfrac13\,\nf(a),\qquad \nf\equiv\frac{\mathrm d\ln\fB}{\mathrm d\ln a}.
\label{eq:wC}
\end{equation}
Deep in matter domination $\nf\to1$, so $w_C\to-4/3$ (the matter-era, high-redshift limit); in the
de~Sitter future $\IH$ and $\chiH$ saturate, $\nf\to0$, and $w_C\to-1$. Thus, read forward in time, the
equation of state \emph{relaxes to $-1$}; read backward (toward high redshift) it \emph{deepens toward
$-4/3$}. In neither direction does it cross $w=-1$.

\subsection{Self-consistent fixed point}
Because the geometry enters $\fB$ through $\chiH$ and the measure, \eqref{eq:rhoC} is a fixed-point
equation. The Friedmann equation
\begin{equation}
H^2(a)=H_0^2\!\left[\Om a^{-3}+\Omega_r a^{-4}+\OC\,\hat g(a)\right],\qquad \hat g(a)=\fB(a)/\fB(1),
\end{equation}
(written in the negligible-companion-density limit, the upper --- least negative --- edge of the band of
Section~\ref{sec:tnl}) is solved jointly with \eqref{eq:quad} by iteration; convergence in a few tens of steps provides the
numerical demonstration that the fixed point exists. That it is unique is made plausible analytically: in the
matter era the shape $\fB\propto a$ follows from the matter scalings alone and is \emph{independent} of
the dark-energy amplitude, so the fixed-point condition reduces to setting the present amplitude through
flatness; the only amplitude dependence enters through the weak geometric feedback in $\chiH$, which
renders the iteration contractive in practice --- a plausibility argument rather than a proof; the
numerical convergence, independent of the starting profile, provides the supporting evidence. We fix the amplitude by flatness, $\OC=1-\Om-\Omega_r$, which determines $\muB$; thus
$\muB$ is calibrated exactly as $\Lambda$ is in $\Lambda$CDM --- with the caveat of
Section~\ref{sec:model} that $\muB$, unlike $\Lambda$, is a combined amplitude, normalisation, and (in the
scenario of Section~\ref{sec:mub}) substrate parameter. The result for $\Om=0.30$ is shown in
Fig.~\ref{fig:wz}.

\section{Structural results}\label{sec:solved}
\paragraph{Vacuum-energy problem.} Since only real matter sources \eqref{eq:IH}, the quantum vacuum
energy does not contribute to $\IH$, and the $10^{123}$ discrepancy does not arise. This is a property
of the coupling (a choice of reference, analogous to a renormalisation condition), \emph{not} a dynamical
cancellation; the radiative stability of this choice is listed among the open problems.

\paragraph{Coincidence.} Because $\IH$ is linear in the matter density, the fixed point ties the
amplitude of $\rhoC$ to the matter content. In the pure matter-era reduction one finds the linear
relation
\begin{equation}
\rhoC\propto\Om,
\label{eq:coin}
\end{equation}
exact in that reduction and weakly modified by the geometric feedback through $\chiH(\Om)$
(Section~\ref{sec:bg}). The ratio $\OC/\Om$ is thereby fixed by structure rather than being accidental ---
an improvement over a free $\Lambda$. This \emph{structurally mitigates} rather than solves the
coincidence: the value of the ratio is still set by $\muB$ (an $O(1)$ coincidence rather than a
$10^{123}$ tuning, see Section~\ref{sec:mub}), and the ``why now'' --- that $\rhoC/\rho_m\propto a^4$
reaches order unity at the present epoch --- is not itself explained.

\section{Background predictions}\label{sec:pred}
\paragraph{What is and is not predicted.} The committed model has no \emph{additional} parameter beyond
those of $\Lambda$CDM: it shares $(\Om,H_0,r_d)$ and replaces the assumption $w=-1$ by a fixed functional
\emph{shape} $w(z)$. The shape --- phantom, monotonic from matter domination onward (at very high
redshift the radiation-era limit is again $w\to-1$, Section~\ref{sec:bg}), no crossing at any epoch,
matter-era asymptote $-4/3$ --- is the genuine
prediction. The present value $w_0\approx-1.20$ depends on $\Om$ (as any quantity does) and the
overall amplitude is set by $\muB$ through the flatness calibration above; we therefore claim a fixed
equation-of-state \emph{shape}, not a calibration-free number.

\begin{figure}[t]
\centering
\includegraphics[width=0.72\textwidth]{fig_wz.pdf}
\caption{Self-consistent background prediction of the bare bulk kernel ($m=0$, $\Om=0.30$):
$w_0=-1.20$, deepening to $-4/3$ at high redshift, with no crossing of $w=-1$ (the relaxation to $-1$
occurs in the future, $a>1$, outside the plotted range). The shaded band spans the two representative
companion assumptions of Section~\ref{sec:tnl}; it is a bounded estimate within those assumptions, not a
rigorous envelope over an arbitrary companion equation of state. Shown for comparison are $\Lambda$CDM, a
representative DESI CPL trend with $(w_0,w_a)=(-0.75,-0.9)$, approximately the DESI~DR2
BAO+CMB+SNe~(DESY5) combination, which crosses $-1$; and the tightly constrained DESI pivot at
$z\approx0.4$.}
\label{fig:wz}
\end{figure}

\paragraph{Equation of state.} At the fiducial $\Om=0.30$, $w_0\approx-1.20$; at the fit-preferred
$\Om=0.265$ (Section~\ref{sec:data}), $w_0\approx-1.19$ and $w(0.4)\approx-1.26$, the values relevant to
all data comparisons. A CPL fit over $00$), which fixes the sign of the band. We note a tension to be kept
in view: the estimate $\rho_{\rm nl}/(\muB\fB)\approx0.20$ below shows the companion is \emph{not}
negligible in density, so that limit is a bookkeeping anchor rather than a physical regime; that the
two limits bracket the true background effect is an \emph{assumption} of the analysis, plausible at
leading order in $\nf$ but not demonstrated for an arbitrary companion. The magnitude of the companion is bounded by the source: from
\eqref{eq:cons}, $\rho_{\rm nl}/(\muB\fB)\sim\nf/3\approx0.20$ today (with $\nf(0)\approx0.6$), and
vanishes toward the de~Sitter limit. The shaded band in Fig.~\ref{fig:wz} spans these two limits; it is a
bounded \emph{estimate} of the companion uncertainty within these two representative assumptions, not a
rigorous envelope over an arbitrary companion equation of state. Quantitatively the band width tracks
$O(\nf^2)$ with coefficient $\approx0.1$: since $\nf(0)\approx0.6$ (the value unity is the matter-era,
not the present-day, figure), the bare kernel gives $w_0=-1.20$ and the companion can deepen it to at
most $\approx-1.24$, i.e.\ the $z=0$ band is $w_0\in[-1.24,-1.20]$ (width $\approx0.04$); it widens to
$\approx0.11$ by $z=2$, where $\nf\to1$ and dark energy is in any case negligible. The bound is thus a
bound \emph{within the two representative companion assumptions}, not a general one. The DESI pivot value at the fit-preferred $\Om=0.265$ is $w(0.4)\in[-1.33,-1.26]$ (bare kernel
$-1.26$; at the fiducial $\Om=0.30$ the interval is $[-1.34,-1.27]$) and is therefore not relieved by
$T^{\rm nl}$, and the qualitative picture (phantom,
no crossing, $\to-4/3$) is robust \emph{within these two companion assumptions}; a fully general
companion equation of state is not excluded from altering it. Equation~\eqref{eq:wC} should be read as a
background, leading-order result, contingent on this companion treatment.

\section{Kinematic ghost-freeness}\label{sec:ghost}
Written as a local partial differential equation, the bare bulk measure would correspond to a
fourth-order operator ($\Box^2\varphi=\rho$), whose free homogeneous solutions would be Ostrogradsky
ghosts. Two interlocking observations resolve this at the kinematic level.

First, $\fB$ is not a fundamental field with a higher-derivative kinetic term: it is, by definition, the
retarded integral over matter --- the source-driven (particular) solution with \emph{no} free homogeneous
part. The would-be ghost is precisely that homogeneous part, which the retarded prescription excludes.

Second, a Dirac--Bergmann analysis shows the constraint sector is built entirely from non-propagating
quantities (the action and constraint structure --- the multiplier term enforcing the $\Theta$--matter
relation --- are those of the underlying theory~\cite{jaster2026a}, to which we refer for definitions):
the Lagrange multiplier has no kinetic term and gives a primary constraint $p_\lambda\approx0$
(whose consistency enforces the constraint relation); the amplitude $\rhoC$ and the global scalar are
algebraic/global; $\Theta$ and $\fB$ are functionals chained to the matter. The sector adds \emph{zero}
new propagating degrees of freedom, and the physical content remains GR (two graviton polarisations) plus
matter; Ostrogradsky's theorem, which requires a non-degenerate higher-derivative \emph{propagating}
field, does not apply.

This is a statement about degree-of-freedom counting, not a full stability proof. The deeper questions ---
the non-local metric self-coupling $\fB[\hat g]$ (a retarded, lightcone-windowed, sub-horizon-suppressed
correction to the graviton sector; Section~\ref{sec:pert}), strong coupling, the radiative stability of
the real-matter condition, and a complete BRST treatment --- remain open.

\section{Linear perturbations and structure growth}\label{sec:pert}
The constraint density is a retarded bulk integral over the past lightcone. For a Fourier mode
$\delta\rho=\bar\rho_m\,\delta_k\,e^{i\mathbf k\cdot\mathbf x}$, the spatial integration over the lightcone
slice of comoving radius $(\eta-\eta')$ is a top-hat window $W(x)=3[\sin x-x\cos x]/x^3$. The matter-era
quadrature then gives
\begin{equation}
\frac{\delta\fB}{\fB}\sim
\begin{cases}
\;c_0\,\delta_k,\ \ c_0=B(5,4)/B(3,4)\approx0.21 & k\eta\lesssim1\ \ (\text{super-horizon, }W\to1),\\[4pt]
\;\dfrac{360}{(k\eta)^{4}}\,\delta_k & k\eta\gg1\ \ (\text{sub-horizon}),
\end{cases}
\label{eq:window}
\end{equation}
The super-horizon coefficient is the ratio of perturbed to background lightcone integrals, both
Beta-function moments of the matter-era profile. For the sub-horizon power, substitute
$s=k(\eta-\eta')$ in the windowed quadrature
$\int_0^\eta a(\eta')^2(\eta-\eta')^3 W(k(\eta-\eta'))\,\mathrm d\eta'$: the physical matter--geometry
weight contributes a factor $(1-s/k\eta)^4$ that tapers the integrand to zero at the causal vertex, so
the relevant moment is the \emph{regularised}
$\int_0^{k\eta}(1-s/k\eta)^4 s^3 W(s)\,\mathrm ds\to6$ as $k\eta\to\infty$, yielding the coefficient
$60\times6=360$. (The bare moment $\int_0^\infty s^3 W(s)\,\mathrm ds$ is only conditionally defined ---
$s^3W=3(\sin s-s\cos s)$ has an oscillatory, non-decaying tail --- and it is the finite causal range
together with this physical taper that regularises it; we verified $360/(k\eta)^4$ numerically against
the full quadrature.) For
structure scales ($k\eta\sim10^3$ today) the suppression $360/(k\eta)^4$ is $\sim10^{-10}$ (the
prefactor $360$ included). The constraint sector is
therefore \emph{smooth} on all structure scales --- and is expected to remain so under the inclusion of
$T^{\rm nl}$, since any lightcone-sourced companion perturbation is windowed, and thus suppressed, in the
same way (an expectation, pending the explicit construction); sub-leading near-horizon and boundary
corrections do not affect this conclusion on structure scales.

Two consequences follow. \emph{(i)} On structure scales $G_{\rm eff}=G$ and the gravitational slip
$\eta_{\rm grav}=\Psi/\Phi=1$ (we write $\eta_{\rm grav}$ for the slip to avoid collision with conformal
time $\eta$): to leading order in the lightcone suppression --- i.e.\ within the
background-plus-suppression approximation, pending the explicit $T^{\rm nl}$ and lightcone boundary
terms --- the model behaves as $\Lambda$CDM
perturbations with a different $H(a)$. This is a sharp \emph{discriminator}: generic modified-gravity
dark energy predicts $\eta_{\rm grav}\neq1$, and clustering or coupled dark energy predicts scale-dependent growth,
whereas this model predicts $\eta_{\rm grav}=1$ together with a perfectly smooth dark-energy sector. \emph{(ii)} A
smooth sector cannot supply the small-scale potential wells that dark matter provides, so \textbf{dark
matter is not eliminated}; $\Om\approx0.3$ of clustering matter remains required.

Growth is therefore modified only through the background $H(a)$. Since the constraint-pressure density
$\rhoC\propto a$ was smaller in the past, the matter-dominated era is effectively prolonged and structure
growth is enhanced. Solving the standard sub-horizon growth equation with the bulk $H(a)$ (and
$G_{\rm eff}=G$), \emph{at a common $\Om=0.30$ so as to isolate the dark-energy effect}, gives
Table~\ref{tab:growth}: $f\sigma_8$ is raised by $6$--$11\%$ over $0    
    

Index