Autor: A. Jaster | Datum: 2026/05/26 | DOI: 10.5281/zenodo.20393270
The causal-Machian emergent time theory [1] defines physical proper time through matter and energy in the causal past of each event. For Θ=0 the physical clock metric degenerates exactly to signature (0,+,+,+) – a purely spatial, timeless structure. This paper explores a speculative consequence: if the universe originated in such a timeless three-dimensional Riemannian manifold without a time direction, the Θ mechanism itself can account for the emergence of temporal structure. Regions where the smoothed spatial matter connection exceeds the reference level, r(r) ≳ 1, nucleate macroscopic time bubbles – local regions in which the metric signature changes from (0,+,+,+) to (−,+,+,+) and, once a regular Lorentzian sector is established, standard cosmology begins. The substrate is spatially unbounded, causally unordered, and without a time direction. As immediate consequences, the horizon, flatness, and homogeneity problems of standard cosmology are reformulated as properties of the timeless substrate configuration rather than dynamically solved within the Lorentzian phase. Inflation as a dynamical mechanism is not required. The scenario is explicitly speculative and requires a complete quantum gravity treatment for its development.
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\begin{center}
{\LARGE\bfseries Time Bubble Nucleation on a Riemannian\\[4pt]
3-Manifold: A Speculative Origin Scenario\\[4pt]
within the Emergent Causal Time Theory}\\[1.4em]
{\large A.~Jaster}\\[0.4em]
{\normalsize \today}
\end{center}
\vspace{0.8em}
\begin{abstract}
The causal-Machian emergent time theory \citep{Jaster2026a} defines
physical proper time through matter and energy in the causal past of
each event. For $\Theta=0$ the physical clock metric degenerates
exactly to signature $(0,+,+,+)$ -- a purely spatial, timeless
structure. This paper explores a speculative consequence: if the
universe originated in such a timeless three-dimensional Riemannian
manifold without a time direction, the $\Theta$ mechanism itself can
account for the emergence of temporal structure. Regions where the smoothed spatial matter connection exceeds the
reference level, $r(\vec r)\gtrsim1$, nucleate macroscopic
\emph{time bubbles} -- local regions in which the metric signature
changes from $(0,+,+,+)$ to $(-,+,+,+)$ and, once a regular
Lorentzian sector is established, standard cosmology begins. The substrate is spatially unbounded,
causally unordered, and without a time direction. As immediate
consequences, the horizon, flatness, and homogeneity problems of
standard cosmology are reformulated as properties of the
timeless substrate configuration rather than dynamically solved within
the Lorentzian phase. Inflation as
a dynamical mechanism is not required. The scenario is explicitly
speculative and requires a complete quantum gravity treatment for its
development.
\end{abstract}
\vspace{1em}
% =============================================================================
\section{Introduction}
\label{sec:intro}
% =============================================================================
The causal-Machian emergent time theory \citep{Jaster2026a,Jaster2026b}
answers the question of the origin of physical time: time is not
independent but emerges from the matter and energy distribution in the
causal past of each event. The scalar time field $\Theta(x)\in[0,1]$
measures the strength of this coupling; the physical clock metric is:
\begin{equation}
\hat{g}_{\mu\nu}
= q_{\mu\nu} + \frac{1-\Theta}{c^2}\,U_{H\mu}\,U_{H\nu}.
\label{eq:ghat}
\end{equation}
The model is designed for the regime \emph{after} the Planck time.
About the origin of temporal structure itself -- how the first time
direction could have arisen -- the theory makes no direct statement.
This paper investigates a speculative extension: what if the substrate
of the early universe was a \emph{timeless} three-dimensional Riemannian
manifold without a time direction, without causal structure, with
arbitrary spatial extent -- and the emergence of time can be described
as \emph{time bubble nucleation} that follows directly from the
$\Theta$ mechanism?
This idea is \textbf{explicitly speculative}. It works out the
consequences of the existing theory in a regime for which the theory
was not developed -- namely below the Planck time -- and for which a
complete quantum gravity treatment is absent. Nevertheless, the
approach yields consistent consequences -- without introducing new
dimensionful scales beyond $\lP$, though structural choices
($W^{\rm 3D}$, $\lambda$, $\Omega_{\rm ref}^{\rm 3D}$, $r_{\rm nuc}=1$)
remain -- that place classical problems of standard cosmology in a
new light.
% =============================================================================
\section{The Timeless Substrate}
\label{sec:substrate}
% =============================================================================
\subsection{Signature Degeneration at $\Theta=0$}
The clock metric \eqref{eq:ghat} possesses a remarkable property in
the limit $\Theta\to0$. Equation~\eqref{eq:g00} below is not derived
\emph{inside} the timeless substrate; it is obtained as a limiting
continuation of the post-nucleation clock metric toward $\Theta\to0$,
and is used here to characterise the degenerate boundary case.
With $q_{00}=-c^2$ and $U_{H0}=c$ (rest frame of the reference
geometry), the time component gives:
\begin{equation}
\hat{g}_{00}
= q_{00} + \frac{1-\Theta}{c^2}\,U_{H0}^2
= -c^2 + (1-\Theta)\,c^2
= -c^2\Theta.
\label{eq:g00}
\end{equation}
For $\Theta=0$ one has exactly $\hat{g}_{00}=0$. The signature of
the physical metric degenerates from $(-,+,+,+)$ to $(0,+,+,+)$: a
purely spatial, three-dimensional Riemannian structure without a time
direction. The spatial components remain unchanged:
$\hat{g}_{ij}=q_{ij}$ for $i,j=1,2,3$.
This degeneration is not a pathology -- it is the precise geometric
expression of the Machian principle: without matter, no time. The
$\Theta=0$ limit lies outside the regular sector $\mathcal{M}_{\rm reg}$
of the theory as defined in \citet{Jaster2026a}; it describes the
state \emph{before} any temporal structure.
\subsection{The 3D Substrate as Initial Condition}
We postulate as initial state a three-dimensional Riemannian manifold
$(\mathcal{S},g_{ij}^{\rm 3D})$ with:
\begin{itemize}
\item Signature $(+,+,+)$: purely spatial character, no time direction
\item Spatially unbounded extent: no requirement of finiteness
\item A spatially varying matter density $\rho(\vec r)\geq0$
\item No causal ordering, no speed of light, no dynamics
\end{itemize}
The substrate is static -- not in the sense of temporal constancy,
but in the sense of the \emph{absence of time}. It is a configuration,
not a process.
The spatial averaging of the matter connection on $\mathcal{S}$ is
defined by a purely spatial integral:
\begin{equation}
\Omega_H^{\rm 3D}(\vec r)
= \int_{B_\lambda(\vec r)}
W^{\rm 3D}(\vec r,\vec r\,')\,\rho(\vec r\,')\,
\sqrt{g^{\rm 3D}}\,\dint^3r',
\label{eq:Omega3D}
\end{equation}
where $B_\lambda(\vec r)$ is a ball of averaging radius $\lambda\sim\lP$
in the spatial metric and $W^{\rm 3D}$ is a normalised smooth kernel.
The dimensionless ratio:
\begin{equation}
r(\vec r) = \frac{\Omega_H^{\rm 3D}(\vec r)}{\Omega_{\rm ref}^{\rm 3D}},
\qquad
\Omega_{\rm ref}^{\rm 3D}
= \langle\Omega_H^{\rm 3D}\rangle_{\mathcal{S}},
\label{eq:rlocal}
\end{equation}
uses the spatial mean as reference. The scalar time field strength:
\begin{equation}
\Theta(\vec r) = S_\infty\!\bigl(r(\vec r)\bigr)\in[0,1],
\label{eq:Theta3D}
\end{equation}
where $S_\infty$ is the $C^\infty$ saturating function of
\citet{Jaster2026b}.
% =============================================================================
\section{Time Bubble Nucleation}
\label{sec:nucleation}
% =============================================================================
\subsection{Nucleation Condition}
The nucleation threshold is set at $r_{\rm crit}=1$: a time bubble
nucleates in the subset $\mathcal{B}\subset\mathcal{S}$ where the
local spatial matter connection exceeds the substrate mean,
\begin{equation}
r(\vec r) > r_{\rm crit} = 1
\quad\Leftrightarrow\quad
\Omega_H^{\rm 3D}(\vec r) > \Omega_{\rm ref}^{\rm 3D},
\label{eq:nucleation}
\end{equation}
Since $S_\infty(r)$ saturates at unity, $r\gtrsim1$ marks the onset
of a saturated macroscopic Lorentzian sector with $\Theta\approx1$.
Two distinct levels are distinguished:
\begin{itemize}
\item \emph{Incipient temporal structure} ($r>0$): wherever
$\Omega_H^{\rm 3D}>0$, one has $\Theta>0$ and
$\hat{g}_{00}=-c^2\Theta<0$. The signature changes from
$(0,+,+,+)$ to $(-,+,+,+)$, but with $\Theta\ll1$ the resulting
time structure is degenerate and supports no regular Lorentzian
dynamics.
\item \emph{Macroscopic Lorentzian time bubble} ($r\gtrsim1$):
a well-developed time structure with $\Theta\approx1$ requires the
local matter connection to be at least comparable to the substrate
mean, $\Omega_H^{\rm 3D}\gtrsim\Omega_{\rm ref}^{\rm 3D}$.
Only regions satisfying $r\gtrsim1$ are identified with time bubbles
in the cosmological sense of this paper.
\end{itemize}
The region with well-developed temporal structure is:
\begin{equation}
\mathcal{B}
= \bigl\{\vec r\in\mathcal{S} : r(\vec r)\gtrsim 1\bigr\}
= \bigl\{\vec r\in\mathcal{S} :
\Omega_H^{\rm 3D}(\vec r)\gtrsim\Omega_{\rm ref}^{\rm 3D}\bigr\}.
\label{eq:bubble}
\end{equation}
The size of $\mathcal{B}$ is determined by the spatial configuration
$\rho(\vec r)$ of the substrate; there is no fundamental upper bound
on $|\mathcal{B}|$.
\subsection{The $\Theta$-Shell as Signature Boundary}
At the boundary $\partial\mathcal{B}$, $\Theta$ transitions from a
value near unity inside to smaller values outside.
This \emph{$\Theta$-shell} is the boundary of the saturated
macroscopic Lorentzian sector; it coincides with a genuine
signature-change surface only in the idealised matter-free exterior
($r=0$ outside). In general it separates a saturated Lorentzian
region from a weakly developed one:
\begin{align}
\text{Inside}\;(r\gtrsim1):&\quad
\hat{g}_{\mu\nu}\text{ has signature }(-,+,+,+), \notag\\
\text{Outside}\;(r<1):&\quad
\hat{g}_{\mu\nu}\text{ has signature }(-,+,+,+)
\text{ if }r>0, \text{ or }(0,+,+,+)\text{ if }r=0. \notag
\end{align}
In the idealised top-hat case with a matter-free exterior ($\rho=0$
for $|\vec r|>R_0$), the outside has $r=0$ and $\Theta=0$, giving
the fully timeless signature $(0,+,+,+)$. More generally the exterior
has $0 R_0,
\end{cases}
\label{eq:tophat}
\end{equation}
the interior ($|\vec r|\ll R_0-\lP$) is fully saturated:
$\Theta\approx1$, $\hat{g}_{\mu\nu}\approx q_{\mu\nu}$. The interior
is a fully Lorentz-signed spacetime with standard GR.
The initial Hubble rate inside the bubble ($\Theta=1$, radiation
domination) is of order:
\begin{equation}
H_0 = \sqrt{\frac{8\pi G\rho_0}{3}}
\;\sim\; \frac{1}{t_P}
\quad\text{for }\rho_0\sim\rho_P.
\label{eq:H0}
\end{equation}
For $t\gtrsim t_P$, after the assumed onset of a regular Lorentzian
sector, the interior is modelled by the modified Friedmann equations
\citep{Jaster2026d}, which reduce to the standard Friedmann equations
for $\Theta=1$.
% =============================================================================
\section{Internal Evolution of the Time Bubble}
\label{sec:evolution}
% =============================================================================
\subsection{Radiation Domination from $t_P$}
Inside the bubble the dynamics begin at $t=t_P$ with $\rho=\rho_P$
and $T=T_P\approx1.4\times10^{32}$\,K. For radiation domination:
\begin{equation}
a(t) \sim \left(\frac{t}{t_P}\right)^{1/2},
\qquad
T(t) \sim T_P\left(\frac{t_P}{t}\right)^{1/2},
\qquad
\rho(t) \sim \rho_P\left(\frac{t_P}{t}\right)^{2}.
\label{eq:rad}
\end{equation}
Under the assumption that nucleation produces a regular Lorentzian
sector with $\Theta\simeq1$, this evolution is modelled by standard
radiation-dominated cosmology for $t\gtrsim t_P$. Table~\ref{tab:evolution} summarises the key
milestones.
\begin{table}[h]
\centering
\caption{Evolution of the time bubble from nucleation to the present.
All values in SI units with $t_P=5.4\times10^{-44}$\,s.}
\label{tab:evolution}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{lrrr}
\toprule
Phase & $t$ & $T$ & $\rho/\rho_P$ \\
\midrule
Nucleation & $t_P = 5.4\times10^{-44}$\,s & $1.4\times10^{32}$\,K & $1$ \\
End of QG era (ca.) & $10\,t_P$ & $4.5\times10^{31}$\,K & $10^{-2}$ \\
GUT scale & $\sim10^{-38}$\,s & $\sim10^{28}$\,K & $\ll1$ \\
Electroweak transition & $\sim10^{-11}$\,s & $\sim10^{15}$\,K & $\ll1$ \\
QCD crossover & $\sim10^{-5}$\,s & $\sim2\times10^{12}$\,K & $\ll1$ \\
Big Bang nucleosynthesis & $\sim1$\,s & $\sim10^{10}$\,K & $\ll1$ \\
Recombination & $\sim3.8\times10^{5}$\,yr & $3000$\,K & $\ll1$ \\
Today & $\sim1.4\times10^{10}$\,yr & $2.7$\,K & $\ll1$ \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Compactness at Nucleation}
For $R_0=\lP$ and $M_0\approx4.2\,m_P$ the Schwarzschild radius is
$R_{\rm S}\approx8.4\,\lP>R_0$: the smallest possible time bubble
lies, at nucleation, in the regime where the classical black-hole
concept is inapplicable. This is consistent -- at $R\sim\lP$ quantum
gravity is required, which the theory identifies as an open task.
Physically meaningful time bubbles have $R_0\gg\lP$.
The region of the substrate that corresponds to today's observable
universe had, at $t_P$, a size of:
\begin{equation}
R(t_P)
= \frac{R_{\rm obs,today}}{a_{\rm today}/a(t_P)}
\approx \frac{4.4\times10^{26}\,\text{m}}{T_P/T_0}
\approx 8.5\,\mu\text{m}
\approx 5\times10^{29}\,\lP.
\label{eq:R0today}
\end{equation}
Within the assumed radiation-dominated standard evolution, the time
bubble that produced our observable universe had an initial radius
$R_0\sim8.5\,\mu$m in the spatially unbounded substrate.
% =============================================================================
\section{Cosmological Consequences}
\label{sec:cosmology}
% =============================================================================
\subsection{Resolution of the Horizon Problem}
The horizon problem of standard cosmology asks why causally
disconnected regions of the early universe have the same temperature.
In the time bubble picture, \textbf{the problem is reformulated
rather than dynamically solved}:
The 3D substrate has no time direction and therefore no causal
structure, no speed of light, no horizons. The entire substrate
region $\mathcal{B}$ is not ``causally separated'' in any meaningful
sense. Homogeneity of the substrate is an \emph{initial condition}
of the substrate rather than a dynamical consequence: the problem is
not eliminated but relocated from the early universe to the substrate
configuration.
\subsection{Resolution of the Flatness Problem}
The flatness problem asks why the cosmic density is so precisely close
to the critical density. In the time bubble picture:
A flat three-dimensional substrate metric ($k=0$ in the Riemannian
sense) produces, upon nucleation, a time bubble with $k=0$ -- a flat
universe. The flatness condition is shifted to the geometry of the substrate;
whether a flat substrate is more natural than a curved one requires
a theory of the substrate and cannot be assumed without justification.
\subsection{Resolution of the Homogeneity Problem}
The homogeneity problem asks why the early universe was so smooth.
In the time bubble picture:
If the overcritical region $\mathcal{B}$ in the substrate is
homogeneous ($\rho(\vec r)\approx\rho_0$ for $\vec r\in\mathcal{B}$),
a homogeneous time bubble nucleates. Homogeneity is a property of
the substrate configuration -- not a problem to be solved by dynamics.
\subsection{Inflation Not Required}
Inflation was developed as a dynamical mechanism to resolve the
horizon, flatness, and homogeneity problems. In the time bubble
picture, the initial configuration of the timeless substrate takes
over this role:
\begin{itemize}
\item Spatially unbounded substrate $\to$ no horizons
\item Flat substrate $\to$ flat universe
\item Homogeneous substrate $\to$ homogeneous universe
\end{itemize}
Inflation as a dynamical solution to the horizon, flatness, and
homogeneity problems is not required \emph{if the substrate has the
required initial properties}. This does not replace inflation as a
mechanism for generating the primordial perturbation spectrum;
adiabatic initial conditions, approximate scale invariance, and the
tensor-to-scalar ratio are not addressed by the present scenario and
would require separate treatment.
% =============================================================================
\section{Connections to Other Approaches}
\label{sec:connections}
% =============================================================================
The time bubble scenario shares structural similarities with several
known approaches without presupposing or being derived from them.
\paragraph{Hartle-Hawking proposal.}
The no-boundary proposal \citep{Hartle1983} describes the universe
through a Euclidean $(+,+,+,+)$ quantum geometry that transitions
through a saddle-point solution into a Lorentzian geometry. The time
bubble scenario is structurally similar: a Riemannian $(0,+,+,+)$
structure transitions through the $\Theta$ mechanism into a Lorentzian
$(-,+,+,+)$ geometry. The difference: no quantum tunnelling process
is invoked -- nucleation is structurally reminiscent of the
no-boundary transition but operates as a geometric threshold effect
rather than a path-integral saddle point.
\paragraph{Page-Wootters mechanism.}
Time emerges in the Page-Wootters mechanism \citep{Page1983} from
the correlation between a clock system and the rest system. In the
present approach $\Omega_H^{\rm 3D}$ plays the role of the clock
system: the spatial matter connection defines where temporal structure
can arise. The Wheeler-DeWitt equation \citep{DeWitt1967} -- timeless
at the fundamental level -- is consistent with this picture: $|\Psi\rangle$
is timeless; conditioning on $\Omega_H$ generates effective time.
\paragraph{Causal dynamical triangulations.}
In CDT \citep{Ambjorn2004} macroscopic spacetime emerges from discrete
Planck-scale simplices. The substrate in the present scenario is structurally reminiscent
of such a discrete structure in the continuum limit;
time bubble nucleation is analogous to the emergence of a global time
foliation in CDT.
\paragraph{Signature change.}
The signature-change literature \citep{Hayward1993} discusses the
possibility that the Lorentz signature emerges at cosmological energy
scales. The present scenario provides an analogous mechanism: a
$\Theta$-driven signature change arising from a matter threshold
condition, structurally reminiscent of signature-change cosmologies.
% =============================================================================
\section{The Far Future and the Eternal Substrate}
\label{sec:future}
% =============================================================================
The time bubble scenario admits several speculative trajectories for
the far future, each depending on the coupling between the internal
Lorentzian dynamics and the surrounding substrate.
\paragraph{Internal de~Sitter future.}
Within the bubble, $\Omega_H$ and $\Omega_{\rm ref}$ co-evolve as
matter dilutes; $r\to1$ and $\Theta\to1$ are maintained throughout
de~Sitter expansion. The interior evolution approaches the standard
$\Lambda$CDM de~Sitter future unchanged. The substrate radius $R_0$
labels the comoving support of the bubble at nucleation; physical
distances inside the Lorentzian sector grow through the internal
scale factor $a(t)$ and should not be interpreted as expansion through
the substrate unless shell propagation is explicitly defined. The
$\Theta$-shell, anchored near $R_0\sim8.5\,\mu$m in substrate
coordinates, becomes cosmologically irrelevant as interior physical
distances grow by many orders of magnitude.
\paragraph{Shell expansion and bubble fusion.}
Since the substrate has no external time, shell expansion and bubble
fusion are not defined as substrate-time processes. From the
\emph{internal} Lorentzian perspective, one may parametrise a
changing effective shell location in terms of interior proper time;
whether such a parametrisation corresponds to a physical expansion
into the substrate is an open boundary-dynamics question. If it
does, and if the substrate contains multiple time bubbles, two shells
that approach each other in substrate coordinates could give rise to
a larger connected Lorentzian domain -- structurally analogous to
bubble collisions in eternal inflation. All such scenarios are
speculative and require a theory of substrate-boundary coupling that
is not available.
\paragraph{Thermal death and return to $\Theta\approx0$.}
In principle, if the normalisation were not co-evolving,
$\rho_m\to0$ in the far future could drive $r\to0$ and $\Theta\to0$:
the bubble would lose its time structure and the substrate would
reclaim the region. In the current co-evolving normalisation this
does not occur ($r\to1$), but it illustrates that time structures
need not be permanent features of the substrate.
\paragraph{The deepest consequence: the substrate as eternal structure.}
In all scenarios the 3D substrate itself persists unchanged. It has
no time direction, no dynamics, no beginning and no end in any
temporal sense. Time bubbles -- including our own observable universe
-- are temporary Lorentzian structures within an eternal, timeless
manifold. This is a fundamental inversion of the usual picture: it
is not spacetime that is primary and matter embedded within it, but
the timeless 3D substrate that is primary, and Lorentzian spacetimes
that are transient configurations of it.
Whether our universe is the only time bubble in its substrate, whether
other bubbles exist, and whether they could interact, are questions
that lie entirely outside the scope of the present model. They
illustrate, however, that the time bubble scenario opens a conceptual
level beyond standard cosmology -- a pre-cosmological substrate in
which the existence of time itself is contingent rather than necessary.
% =============================================================================
\section{Open Problems and Limitations}
\label{sec:open}
% =============================================================================
The scenario is speculative and requires extensive further development.
\begin{enumerate}[label=(\arabic*)]
\item \textbf{Quantisation of the substrate configuration.}
The 3D substrate is defined classically. A complete treatment
requires quantisation of $\rho(\vec r)$ on $\mathcal{S}$ -- a
quantum gravity theory of the timeless sector.
\item \textbf{Definition of $\Omega_{\rm ref}^{\rm 3D}$.}
Equation~\eqref{eq:rlocal} uses the spatial mean as reference.
For an infinite substrate this mean is well-defined only if
$\rho(\vec r)$ has compact support or decays sufficiently rapidly.
An infrared-regulated definition is required.
\item \textbf{Coupling to the constraint sector.}
The constraint tensor $T^C_{\mu\nu}$, required for the dynamical
closure of the field equations \citep{Jaster2026a}, must be defined
in the time bubble geometry -- in particular at the $\Theta$-shell.
\item \textbf{Multiple time bubbles.}
An infinite substrate with multiple overcritical regions nucleates
multiple time bubbles. Whether and how these merge, and what
observational consequences follow, is not investigated.
\item \textbf{Quantum gravity at $R_0\sim\lP$.}
The smallest possible time bubble starts at $R_0\sim\lP$, where
quantum gravity dominates. The theory is not complete there;
\citet{Jaster2026b} provides a partial treatment.
\item \textbf{Why $\rho(\vec r)>0$ at all?}
The scenario presupposes a non-trivial matter distribution in the
substrate. The question of the origin of this distribution is the
question of the origin of the substrate itself -- an open retreat
to a more fundamental level.
\item \textbf{Residual cosmology.}
The cosmological constant $\Lambda$ remains necessary in the
Friedmann dynamics of the time bubble. The present scenario
addresses the origin of temporal structure, not the origin of
$\Lambda$.
\end{enumerate}
% =============================================================================
\section{Summary}
\label{sec:summary}
% =============================================================================
The $\Theta=0$ limit of the causal-Machian time field theory produces
exactly a spacetime degeneration $\hat{g}_{00}=0$: a purely spatial
three-dimensional structure without a time direction. This result --
a direct consequence of equation~\eqref{eq:g00} without additional
postulate -- motivates the present speculative scenario.
A spatially unbounded, timeless 3D substrate with a varying matter
density $\rho(\vec r)$ nucleates a macroscopic time bubble wherever the smoothed spatial
matter connection exceeds the reference level, $r(\vec r)\gtrsim1$:
a region in which the signature changes from $(0,+,+,+)$ to
$(-,+,+,+)$ and, once a regular Lorentzian sector is established,
standard cosmology begins. Four key
results follow:
\begin{enumerate}[label=(\roman*)]
\item The interior can be modelled by standard radiation-dominated
cosmology for $t\gtrsim t_P$, once a regular Lorentzian sector
with $\Theta\simeq1$ is established.
\item The horizon, flatness, and homogeneity problems are reformulated
as properties of the substrate configuration rather than dynamically
solved within the Lorentzian phase.
\item Inflation is not required for these specific roles if the substrate
already carries the required properties; it is not replaced as a
mechanism for generating the primordial perturbation spectrum.
\item The dimensionality $(3+1)$ of our universe and its spatial extent
are properties of the specific time bubble, not fundamental properties
of spacetime.
\end{enumerate}
The scenario is speculative and requires a complete quantum gravity
treatment, the construction of the constraint sector, and an
infrared-regulated definition of the spatial reference density.
Nevertheless: the signature-degeneration $\hat{g}_{00}=-c^2\Theta=0$ at
$\Theta=0$ follows directly from the clock metric; the time-bubble
scenario -- the 3D substrate, the spatial averaging, the nucleation
threshold, and the far-future trajectories -- is a speculative
construction motivated by this mechanism, without introducing new
dimensionful scales beyond $\lP$ of the emergent time theory -- the
emergence of time from a timeless structure is not an additional
postulate but a direct geometric consequence of $\hat{g}_{00}=-c^2\Theta$.
% =============================================================================
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\end{document}