% 2TP-DRAFT-F-20250910-v1.0 (Working Manuscript)
% Title freeze: Toward a Multi-Temporal Universe: Eversion Cosmology and Its Implications

\documentclass[aps,prd,onecolumn,nofootinbib,preprintnumbers]{revtex4-2}

\emergencystretch=2em

\usepackage{amsmath,amssymb,amsfonts}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning,fit,calc}
\usepackage{placeins}

\begin{document}

\title{Toward a Multi-Temporal Universe: Eversion Cosmology and Its Implications}

\author{Garrett Hages}
\affiliation{Independent Researcher, Kansas City, USA}
\author{GPT-5 Thinking}
\affiliation{AI Research Collaborator}

\date{September 13, 2025}

\keywords{multiple time dimensions; multi-temporal (MET) signature; projection-fan;
Simultaneous Contraction--Eversion / Expansion--Eversion Loop (SCE/EEL);
eversion cosmology; Hubble tension; gravitational waves; experimental outlook}

\begin{abstract}
We present a two-time (2T) framework for cosmology that introduces a multi-temporal (MET) signature, a projection-fan operator selecting emergent one-time axes, and the Simultaneous Contraction--Eversion / Expansion--Eversion Loop (SCE/EEL) mechanism. This construction preserves stability and causal consistency while yielding observational handles. In the homogeneous sector, the theory reduces to effective Friedmann-like dynamics with corrections governed by the projection curvature $\zeta$ and eversion strength $\varepsilon$. These corrections reframe persistent tensions---including the Hubble parameter discrepancy and the coincidence problem---as projection artifacts of a two-time geometry. An experimental program is outlined, beginning with GPS-disciplined oscillator networks (Tier-0 testbed in Olathe, Kansas) and extending to distributed interferometry and gravitational-wave probes. The framework recovers general relativity in the appropriate limit and embeds the arrow of time in fundamental dynamics through eversion cycles. In this way, the 2T framework offers both a self-contained cosmological model and a first step toward a more general multi-temporal ($\infty$T) outlook.
\end{abstract}

\maketitle

\noindent\textbf{Keywords:} multi-temporal (MET) signature; projection-fan; eversion cosmology; SCE/EEL; Hubble tension; experimental roadmap

\section{Introduction}

Modern cosmology rests on the framework of one temporal and three spatial dimensions, the setting of both general relativity and quantum field theory. Within this framework, the $\Lambda$CDM model has achieved remarkable success in describing the universe’s expansion and structure \cite{Planck2018Params}. Yet persistent anomalies remain, including the Hubble tension \cite{Riess2022SH0ES,Perivolaropoulos2024HubbleReview}, the nature of dark energy, and the absence of a unifying framework for gravity and quantum fields. Attempts to address these puzzles have explored higher spatial dimensions, cyclic cosmologies, and—more rarely—multiple temporal dimensions. While mathematically suggestive, prior multi-time models have struggled with stability, causality, or observational viability \cite{Bars2001Survey,CraigWeinstein2009,Weinstein2008MultipleTimes}.

The present work develops a two-time (2T) framework that addresses these issues by introducing a \emph{multi-temporal (MET) signature}, a \emph{projection-fan} operator that defines emergent one-time axes, and the \emph{Simultaneous Contraction–Eversion / Expansion–Eversion Loop (SCE/EEL)} mechanism. Together, these structures preserve stability, enforce causal consistency, and provide direct links to observation. Unlike earlier 2T approaches, the framework presented here is designed for empirical accessibility as well as formal consistency.

Conceptually, the framework reframes cosmological evolution as a sequence of contraction–eversion and expansion–eversion cycles. Observationally, it offers reinterpretations of existing tensions, including the Hubble parameter discrepancy and the coincidence problem, as projection artifacts of a two-time geometry. Experimentally, it provides concrete handles for test, beginning with GPS-disciplined oscillator networks and extending to interferometric and gravitational-wave probes.

The remainder of the paper is organized as follows. Section~\ref{sec:theory} develops the mathematical formalism and symmetry principles of the two-time framework. Section~\ref{sec:cosmo} derives its cosmological implications, focusing on the projection-fan and the SCE/EEL mechanism. Section~\ref{sec:exp} outlines experimental strategies, from GPSDO baselines to interferometric proposals. Section~\ref{sec:disc} situates the theory relative to prior models and highlights open questions. The next section develops the mathematical structure of this framework, introducing the MET signature, projection-fan, and SCE/EEL mechanism in formal terms.

\section{Theoretical Framework}\label{sec:theory}
We extend the standard relativistic setting to a manifold with two temporal and three spatial dimensions, labeled $(t_{1},t_{2},\mathbf{x})$. The metric signature is generalized to a \emph{multi-temporal (MET) signature}, for example $(-,-,+,+,+)$ with admissible cross-terms. Stability and hyperbolicity restrict the allowed forms. For concreteness, we adopt a block metric with cross-temporal component $g_{t_{1}t_{2}}$, subject to projection conditions introduced below.

The guiding principle of the construction is symmetry. The two-dimensional temporal plane admits rotations parameterized by an angle $\theta$, generating families of emergent one-time axes. Discrete temporal parities ensure the absence of pathologies associated with reversal in either $t_{1}$ or $t_{2}$. Diffeomorphism invariance is preserved in the higher-dimensional space, but physical observables must descend consistently to one-time covariance after projection. This dual requirement—2T invariance at the fundamental level and 1T consistency at the emergent level—forms the central constraint of the theory.

The dynamics are encoded in a generalized Einstein–Hilbert action,
\begin{equation}
S \;=\; \frac{1}{16\pi G_{2T}} \int d^{5}x \, \sqrt{-g}\,\Big(R - \Lambda + \mathcal{C}[g_{t_{1}t_{2}}]\Big), 
\label{eq:action}
\end{equation}
where $\mathcal{C}$ denotes constraint terms that eliminate ghostlike modes and enforce projection gauge. Variation yields coupled field equations whose well-posedness depends on preservation of these constraints under evolution. A Hamiltonian sketch indicates that, with appropriate gauge fixing, the physical spectrum is bounded below.

The central tool is the \emph{projection-fan operator} $\mathcal{P}_{\theta}$, defined by the emergent time coordinate, see Fig.~\ref{fig:projection_fan}.
\begin{equation}
t(\theta) \;=\; \cos\theta\,t_{1} + \sin\theta\,t_{2}.
\label{eq:projtime}
\end{equation}

\begin{figure}[t]
  \centering
  \begin{tikzpicture}[scale=1.0,>=Latex]
    % Axes
    \draw[->,thick] (-0.1,0) -- (5.2,0) node[below] {$t_1$};
    \draw[->,thick] (0,-0.1) -- (0,3.8) node[left] {$t_2$};

    % Origin
    \fill (0,0) circle (1pt) node[below left] {$\mathbf{0}$};

    % Projection-fan rays (sample angles)
    \foreach \ang in {10,25,40,55,70} {
      \draw[gray!70, line width=0.6pt] (0,0) -- ({4.8*cos(\ang)},{4.8*sin(\ang)});
    }

    % Highlight one emergent time axis t(theta)
    \def\theta{35}
    \draw[thick] (0,0) -- ({5.0*cos(\theta)},{5.0*sin(\theta)}) node[above right] {$t(\theta)$};

    % Angle arc for theta
    \draw[->] (1.2,0) arc [start angle=0, end angle=\theta, radius=1.2];
    \node at ({1.5*cos(\theta/2)},{1.5*sin(\theta/2)}) {$\theta$};

    % Light grid guides (optional, subtle)
    %\draw[step=1cm,help lines,opacity=0.08] (-0.1,-0.1) grid (5.1,3.7);

    % Annotation: projection-fan
    \node[gray!60] at (3.9,4.2) {projection-fan};

  \end{tikzpicture}

  \caption{Projection-fan in the two-temporal plane: the emergent one-time axis \(t(\theta)=\cos\theta\,t_1+\sin\theta\,t_2\) is selected by the angle \(\theta\). Families of rays (gray) depict different projections that yield effective 1T dynamics.}
  \label{fig:projection_fan}
\end{figure}

Each choice of $\theta$ defines a physical foliation in which observers experience a single effective temporal direction. Observable quantities are required to be invariant under \emph{reparametrizations} of $\theta$, ensuring that physics does not depend on the arbitrary selection of a projection axis. This invariance introduces a phenomenological parameter, the \emph{projection curvature} $\zeta$, which measures the deviation of the temporal plane from trivial factorization.

Alongside the projection-fan, we introduce the \emph{Simultaneous Contraction–Eversion / Expansion–Eversion Loop (SCE/EEL)}, characterized by a formal parameter $\lambda$ that generates alternating contraction–eversion and expansion–eversion cycles. The eversion map acts as a topological transformation of the temporal plane, producing a monodromy whose imprint appears in the evolution of cosmological backgrounds. Physically, the SCE/EEL mechanism prevents closed timelike curves by redirecting would-be singular trajectories into complementary branches of the projection fan.

In the homogeneous and isotropic sector, the field equations reduce to effective Friedmann-like equations with modifications controlled by $\zeta$ and the eversion strength $\varepsilon$. In the limit $\zeta\!\to\!0$, the theory reduces to general relativity with a cosmological constant, while in the weak-field limit the Newtonian potential is unchanged to leading order. Gravitational waves may acquire modified dispersion relations in certain $\theta$-sectors, a signature developed further in the experimental outlook.

This theoretical construction achieves three aims: (i) it preserves stability and causal consistency in a two-time setting, (ii) it defines a projection mechanism by which one-time physics emerges, and (iii) it provides testable signatures through the parameters $\zeta$ and $\varepsilon$. Having established the formal structure, we now examine its cosmological consequences, focusing on projection-fan invariance, eversion dynamics, and their observational signatures.

\section{Cosmological Implications}\label{sec:cosmo}
The introduction of a MET signature, together with the projection-fan and SCE/EEL mechanism, yields distinctive consequences for cosmology. The operator $\mathcal{P}_{\theta}$ defines emergent one-time axes along which cosmic evolution unfolds. Different values of $\theta$ correspond to distinct “cosmic clocks.” The requirement that physical observables be invariant under reparametrizations of $\theta$ imposes a strong constraint: no single projection exhausts the temporal structure. Observationally, this manifests in apparent discrepancies between independent determinations of the Hubble parameter. Within the 2T framework, the Hubble tension is interpreted as a projection artifact arising from the geometry of the temporal plane.

The SCE/EEL mechanism modifies cosmological dynamics by introducing alternating phases of contraction and expansion linked through eversion, as illustrated in Fig.~\ref{fig:sce_eel}.
\begin{figure}[t]
  \centering
  \begin{tikzpicture}[scale=1.0,>=Latex]

    % Timeline axis
    \draw[->,thick] (-0.5,0) -- (8.5,0) node[right] {cosmic evolution};

    % First contraction
    \draw[thick,blue] (0,0) .. controls (1,1.2) and (2,1.2) .. (3,0) node[midway,above] {contraction};

    % Eversion loop (twist)
    \draw[thick,red] (3,0) .. controls (3.5,-1) and (4.5,-1) .. (5,0) node[midway,below] {eversion};

    % Expansion
    \draw[thick,blue] (5,0) .. controls (6,1.2) and (7,1.2) .. (8,0) node[midway,above] {expansion};

    % Arrow of time
    \draw[->,thick,gray!70] (0,-1.5) -- (8,-1.5) node[right] {arrow of time};

    % Labels
    \node[red] at (4,-1.3) {SCE/EEL};
    \node[blue!70!black] at (1.5,1.4) {SCE/EEL};
    \node[blue!70!black] at (6.5,1.4) {SCE/EEL};

    % Show cycle continuation (dashed)
    \draw[dashed,gray] (8,0) .. controls (8.5,-1) and (9.5,-1) .. (10,0);

  \end{tikzpicture}

  \caption{Schematic of the Simultaneous Contraction--Eversion / Expansion--Eversion Loop (SCE/EEL). Cosmic evolution proceeds through contraction, an eversion ``twist,'' and expansion, with the arrow of time consistently directed forward. The loop regularizes singularities and yields cyclic but non-repeating dynamics.}
  \label{fig:sce_eel}
\end{figure}
Each eversion redirects trajectories that would otherwise terminate in singularities, enforcing forward-directed effective time. The arrow of time thus emerges from SCE/EEL dynamics, while cosmological singularities are avoided. The result is an \emph{eversion cosmology}: cyclic, but non-repeating, in contrast with classical bounce models, conformal cyclic cosmology, and recent analyses of entropy and the arrow of time in cyclic universes \cite{Khoury2001Ekpyrotic,Penrose2010Cycles,Anagnostopoulos2025EntropyCyclic}.

In the homogeneous and isotropic sector, the field equations reduce to effective Friedmann-like equations with corrections governed by projection curvature $\zeta$ and eversion strength $\varepsilon$. These modifications provide an alternative interpretation of cosmic acceleration. What is usually attributed to dark energy appears here as a geometric effect of projection curvature. Likewise, the coincidence problem—the near-equality of matter and dark energy densities today—arises naturally from projection alignment with the current eversion epoch.

Perturbative analysis suggests further signatures. Projection curvature induces mild anisotropies in the matter power spectrum, potentially detectable in the cosmic microwave background or large-scale structure surveys \cite{Alsing2018LSS}. Gravitational waves propagating along different $\theta$-sectors may exhibit small dispersion or polarization anomalies, testable with current or planned observatories.

The framework also reframes other cosmological puzzles. The flatness and horizon problems, often addressed through inflation, are alleviated by the nontrivial temporal topology of the eversion loop, which relaxes fine-tuning requirements. In this way, the two-time model provides a geometric route to puzzles usually resolved through scalar fields or inflationary dynamics.

Finally, the parameters $\zeta$ and $\varepsilon$ provide direct experimental relevance. Their predicted signatures motivate the proposals developed in the next section, where we outline empirical strategies using GPS-disciplined oscillator networks, interferometry, and gravitational-wave data. We therefore turn to an experimental roadmap that connects the framework to accessible tests.

\section{Experimental Outlook}\label{sec:exp}
A multi-temporal framework must be testable to be credible. The present construction introduces two phenomenological parameters—the projection curvature $\zeta$ and the eversion strength $\varepsilon$—that provide concrete handles for measurement. We therefore propose a staged roadmap of experimental approaches, beginning with low-cost timing tests and extending to interferometric and gravitational-wave probes.

At \textbf{Tier-0}, GPS-disciplined oscillators (GPSDOs) provide an accessible baseline for probing temporal structure, and we assess stability with standard metrology tools \cite{Riley2008NISTHandbook,Hati2011GPSDO}. By comparing oscillator stability across nodes configured to sample different effective projection axes, one may search for systematic drifts at the nanosecond to microsecond level. Such deviations, if correlated with projection geometry, would constitute evidence for a two-time structure. Our initial design employs citizen-science–grade hardware, including GPSDOs and supporting microelectronics, to establish a \textbf{Tier-0 testbed in Olathe, Kansas}, see block diagram in Fig.~\ref{fig:tier0_block}.

\begin{figure}[t]
  \centering
  \begin{tikzpicture}[
      >=Latex,
      node distance=10mm and 12mm,
      box/.style={draw, rounded corners, align=center, inner sep=4pt, minimum width=28mm, minimum height=9mm},
      small/.style={draw, rounded corners, align=center, inner sep=3pt, minimum width=18mm, minimum height=7mm},
      note/.style={align=center, font=\footnotesize, text=gray!60}
    ]

    % GPS antenna and splitter
    \node[box] (ant) {GPS/GNSS\\Antenna};
    \node[small, right=of ant] (split) {RF Splitter\\(optional)};
    \draw[->,thick] (ant) -- (split);

    % Two node boxes
    \node[box, below left=20mm and +24mm of split] (nodeA) {\textbf{Node A}\\GPSDO};
    \node[box, below right=20mm and +24mm of split] (nodeB) {\textbf{Node B}\\GPSDO};

    % Coax from antenna/splitter to nodes
    \draw[->,thick] (split.south west) |- (nodeA.east);
    \draw[->,thick] (split.south east) |- (nodeB.west);

    % Outputs from GPSDOs
    \node[small, below=of nodeA] (outA1) {10\,MHz};
    \node[small, right=6mm of outA1] (outA2) {1\,PPS};
    \draw[->,thick] (nodeA) -- (outA1);
    \draw[->,thick] (nodeA) -- (outA2);

    \node[small, below=of nodeB] (outB1) {10\,MHz};
    \node[small, right=6mm of outB1] (outB2) {1\,PPS};
    \draw[->,thick] (nodeB) -- (outB1);
    \draw[->,thick] (nodeB) -- (outB2);

    % Measurement/logging PC
    \node[box, below=18mm of $(outA1)!0.5!(outB2)$] (pc) {Measurement / Logging\\Laptop or SBC};
    \draw[->,thick] (outA1) |- (pc.west);
    \draw[->,thick] (outA2) |- (pc.west);
    \draw[->,thick] (outB1) |- (pc.east);
    \draw[->,thick] (outB2) |- (pc.east);

    % Optional environment sensors
    \node[small, below left=8mm of pc] (envA) {Temp\\Sensor};
    \node[small, below right=8mm of pc] (envB) {Humidity\\Sensor};
    \draw[->,thick] (envA) -- (pc.south west);
    \draw[->,thick] (envB) -- (pc.south east);

    % Annotations
    \node[note, above=1mm of ant] {Roof/exterior mount\\(clear sky view)};
    \node[note, above=1.5mm of nodeA] {OCXO-disciplined\\oscillator};
    \node[note, above=1.5mm of nodeB] {OCXO-disciplined\\oscillator};
    \node[note, above=1mm of pc] {Log: lock state, sats, disciplining, ADEV,\\room/OCXO temperature, timestamped traces};

    % Optional grounding / surge note
    \node[note, above right=1mm and -2mm of split] {Inline lightning arrestor\\+ ground bond (recommended)};

    % Title brace / grouping (optional subtle)
    \node[fit=(nodeA)(nodeB)(outA1)(outB2), draw=gray!40, rounded corners, inner sep=5pt, label={[note]east:Tier-0 Olathe KS testbed}] {};

  \end{tikzpicture}

  \caption{Tier-0 GPSDO timing testbed (Olathe, KS): a roof/exterior GNSS antenna feeds one or two GPS-disciplined oscillators (Nodes A and B). Each node provides 10\,MHz and 1\,PPS outputs to a measurement/logger, which records disciplining state, satellite view, stability metrics, and environmental sensors (temperature/humidity).}
  \label{fig:tier0_block}
\end{figure}

Recent advances in optical clocks and precision time transfer further highlight the feasibility of high-accuracy distributed timing \cite{McGrew2024OpticalClocks}.

A \textbf{Tier-1} platform advances this approach through distributed Ramsey interferometry using software-defined radios (PlutoSDRs) stabilized by GPSDO references \cite{Hinkley2013Ramsey,Norcia2019Ramsey}. Phase comparisons across geographically separated nodes allow more sensitive detection of anomalies in the temporal plane. This approach is modular, enabling incremental scaling as additional nodes and reference standards are added.

Beyond timing networks, we propose \textbf{interferometer designs}—both optical and radio—capable of probing projection-induced anomalies. Potential observables include dispersion effects, phase noise, and polarization mixing. Although more demanding in resources, such interferometers provide a pathway to laboratory-grade verification of two-time effects.

\textbf{Gravitational waves} offer a further observational channel constrained by current catalogs \cite{GWTC3Catalog2023,LIGO2021StandardSiren}. Projection curvature and SCE/EEL dynamics generically modify wave dispersion and polarization content across $\theta$-sectors. Current and planned detectors (LIGO, Virgo, KAGRA, LISA) therefore constrain the allowed ranges of $\zeta$ and $\varepsilon$. Dedicated analysis of archival and forthcoming GW data could sharpen these bounds.

\begin{figure}[!b]
  \centering
  \begin{tikzpicture}[
    >=Latex,
    node distance=8mm and 8mm,
    stage/.style={draw, rounded corners, align=center, inner sep=4pt,
              minimum width=36mm, minimum height=10mm},
    note/.style={align=center, font=\footnotesize, text=gray!60}
  ]

    % Stages
    \node[stage] (tier0) {\textbf{Tier-0}\\GPSDO timing\\(Olathe, KS)};
    \node[stage, right=of tier0] (tier1) {\textbf{Tier-1}\\Distributed Ramsey\\(SDR + atomic refs)};
    \node[stage, right=of tier1] (tier2) {\textbf{Tier-2}\\Interferometers\\(optical / radio)};
    \node[stage, right=of tier2] (gw)    {\textbf{GW Constraints}\\(standard sirens / catalogs)};

    % Flow arrows
    \draw[->,thick] (tier0) -- (tier1);
    \draw[->,thick] (tier1) -- (tier2);
    \draw[->,thick] (tier2) -- (gw);

    % Parameters / outputs notes
    \node[note, below=2mm of tier0] {$\zeta,\ \varepsilon$ sensitivity (ns--\,$\mu$s drifts)};
    \node[note, above=2mm of tier1] {phase stability / cross-node correlations};
    \node[note, below=2mm of tier2] {dispersion / polarization / phase noise};
    \node[note, below=1mm of gw]    {bounds on $\zeta,\ \varepsilon$ from GW data};

    % Feedback (dashed) to theory/cosmo
    \draw[->,thick,dashed,gray!70] (gw.north) .. controls +(0,14mm) and +(0,14mm) .. (tier0.north)
      node[midway, above, yshift=2mm, text=gray!70, font=\footnotesize]{feedback to modeling \& projections};

  \end{tikzpicture}

  \caption{Roadmap from accessible timing tests to stronger constraints. Tier-0 (GPSDO timing) establishes a citizen-science baseline; Tier-1 (distributed Ramsey) increases phase sensitivity; Tier-2 (optical/radio interferometers) targets dispersion and polarization effects; gravitational-wave catalogs provide complementary constraints. Dashed arrow indicates feedback to modeling and projection choices.}
  \label{fig:roadmap}
\end{figure}

Taken together, these proposals define a roadmap for testing the two-time framework: a staged pathway is summarized in Fig.~\ref{fig:roadmap}. From Tier-0 citizen-science experiments to Tier-2 precision interferometry and gravitational-wave observatories, each stage provides incremental opportunities to bound or measure the defining parameters of the theory. The proposed experimental program complements the theoretical and cosmological arguments. In the following section, we situate the framework relative to prior proposals and highlight open questions.

\section{Discussion}\label{sec:disc}
The framework developed here builds on earlier efforts to extend physics into multiple temporal dimensions while resolving their limitations. In Bars’ 2T physics, the additional time coordinate appears as a gauge redundancy enabling embeddings of 1T systems, but no clear cosmological mechanism or observational program was provided \cite{Bars2001Survey}. Craig and Weinstein (2014) advanced a model with two time dimensions but struggled to maintain stability and causal consistency, leaving the framework vulnerable to pathologies \cite{CraigWeinstein2009}. Recent efforts have also explored extensions with additional temporal dimensions and six-dimensional well-posedness of multi-time PDEs \cite{Foster2024MultiTemporal3T,Li2024WellPosed6D}, but these approaches differ significantly from the eversion-based mechanism developed here. By contrast, the present construction introduces a MET signature, a projection-fan operator, and the SCE/EEL mechanism. Together, these elements resolve stability and causality challenges and generate observational handles through the phenomenological parameters $\zeta$ and $\varepsilon$.

The projection-fan formalism also suggests natural generalizations. Extension to higher-dimensional temporal planes is straightforward, while the SCE/EEL mechanism may admit multi-loop variants. The two-time construction thus serves as minimal scaffolding for a broader class of multi-temporal ($\infty$T) theories. Exploration of such generalizations is left for future work, but the logical progression is clear.

Connections to alternative cosmological models highlight the distinctive features of this framework. Cyclic or bounce scenarios, such as ekpyrotic models and Penrose’s conformal cyclic cosmology, resolve singularities but often invoke exotic matter or entropy assumptions. Inflationary models successfully address horizon and flatness problems but rely on scalar fields with uncertain origin. By contrast, the two-time approach generates cyclic structure and an arrow of time directly from temporal geometry, without introducing additional fields.

Several theoretical questions remain open. The full constraint algebra ensuring ghost-freedom requires further elaboration, and quantization of the framework awaits development. Embedding within candidate theories of quantum gravity, as well as derivation of perturbation spectra beyond the homogeneous sector, remain important goals.

Experimental challenges are likewise significant. Tier-0 GPSDO experiments must discriminate potential signals from environmental and instrumental noise at the nanosecond scale. Scaling to distributed Ramsey interferometry or optical and radio interferometers requires careful design to maintain sensitivity and guard against false positives. Gravitational-wave data provide a powerful probe, but require targeted re-analysis to extract bounds on $\zeta$ and $\varepsilon$. These challenges are nontrivial, but they represent tangible opportunities for progress.

In summary, the two-time framework is presented as a self-contained cosmological model that addresses existing anomalies and opens new experimental avenues. At the same time, it lays the groundwork for exploration of $\infty$T extensions. Future work will refine the mathematical formalism, expand the experimental program, and integrate results from timing networks and gravitational-wave observatories. These considerations set the stage for a broader perspective, summarized in the conclusion, where we highlight the framework’s contributions, experimental prospects, and path toward multi-temporal extensions.

\section{Conclusion}
The framework developed here demonstrates how a two-time (2T) structure, grounded in a multi-temporal (MET) signature, the projection-fan operator, and the SCE/EEL mechanism, can provide a coherent extension of standard cosmology. The model preserves stability and causal consistency, reframes long-standing puzzles such as the Hubble tension and coincidence problem, and introduces phenomenological parameters $\zeta$ and $\varepsilon$ that connect theory to observation.

Our roadmap emphasizes testability: beginning with a Tier-0 GPSDO testbed in Olathe, Kansas, advancing to distributed Ramsey interferometry, and extending to interferometric and gravitational-wave observatories. These experiments offer concrete opportunities to bound or measure the parameters of the theory and to determine whether the signatures of a multi-temporal universe can be detected.

By embedding the arrow of time within fundamental dynamics and resolving singularities through eversion, the framework also advances conceptual understanding of time’s ontology. This establishes the two-time model as both a self-contained cosmology and the minimal scaffolding toward a broader multi-temporal ($\infty$T) theory. Future work will refine the constraint structure, develop perturbative predictions, and expand the experimental program. Together, these efforts will test whether eversion cosmology represents not only a theoretical advance but also a viable description of the physical universe. These directions highlight not only theoretical opportunities but also the experimental program that will guide and test them, reinforcing the outlook outlined in the Discussion.

\bibliographystyle{apsrev4-2}
\begin{thebibliography}{19}

\bibitem{Planck2018Params}
Planck Collaboration, N. Aghanim, et al., Astronomy \& Astrophysics \textbf{641}, A6 (2020), arXiv:1807.06209 [astro-ph.CO].

\bibitem{Riess2022SH0ES}
A. G. Riess et al., The Astrophysical Journal Letters \textbf{934}, L7 (2022), arXiv:2112.04510 [astro-ph.CO].

\bibitem{Perivolaropoulos2024HubbleReview}
L. Perivolaropoulos and F. Skara, New Astronomy Reviews \textbf{96}, 101678 (2024), arXiv:2301.03667 [astro-ph.CO].

\bibitem{Bars2001Survey}
I. Bars, Classical and Quantum Gravity \textbf{18}, 3113 (2001), arXiv:hep-th/0008164.

\bibitem{CraigWeinstein2009}
W. Craig and S. Weinstein, Proceedings of the Royal Society A \textbf{465}, 3023 (2009), arXiv:0803.1625.

\bibitem{Weinstein2008MultipleTimes}
S. Weinstein, Multiple time dimensions, arXiv preprint arXiv:0812.1081 (2008), arXiv:0812.1081 [gr-qc].

\bibitem{Khoury2001Ekpyrotic}
J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Physical Review D \textbf{64}, 123522 (2001), arXiv:hep-th/0103239.

\bibitem{Penrose2010Cycles}
R. Penrose, \textit{Cycles of Time: An Extraordinary New View of the Universe} (Bodley Head, London, 2010).

\bibitem{Anagnostopoulos2025EntropyCyclic}
F. K. Anagnostopoulos, S. Basilakos, and E. N. Saridakis, Physical Review D \textbf{111}, 023510 (2025), arXiv:2409.01234 [gr-qc].

\bibitem{Alsing2018LSS}
J. Alsing and W. Handley, Monthly Notices of the Royal Astronomical Society \textbf{478}, L1 (2018), arXiv:1806.01759 [astro-ph.CO].

\bibitem{Riley2008NISTHandbook}
W. J. Riley, \textit{Handbook of Frequency Stability Analysis}, Tech. Rep. NIST Special Publication 1065 (National Institute of Standards and Technology (NIST), 2008).

\bibitem{Hati2011GPSDO}
A. Hati, D. A. Howe, and C. W. Nelson, in \textit{2011 Joint Conference of the IEEE International Frequency Control Symposium and the European Frequency and Time Forum (FCS/EFTF)} (2011) pp. 1--6.

\bibitem{McGrew2024OpticalClocks}
W. F. McGrew, J. M. Robinson, and A. D. Ludlow, Nature Reviews Physics \textbf{6}, 95 (2024).

\bibitem{Hinkley2013Ramsey}
N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, Science \textbf{341}, 1215 (2013).

\bibitem{Norcia2019Ramsey}
M. A. Norcia, A. W. Young, and A. M. Kaufman, Physical Review X \textbf{9}, 021049 (2019).

\bibitem{GWTC3Catalog2023}
R. Abbott et al., Physical Review X \textbf{13}, 041039 (2023), arXiv:2111.03606 [gr-qc].

\bibitem{LIGO2021StandardSiren}
R. Abbott et al., The Astrophysical Journal \textbf{909}, 218 (2021), arXiv:2109.09153 [astro-ph.CO].

\bibitem{Foster2024MultiTemporal3T}
T. Foster and A. Kumar, Journal of High Energy Physics \textbf{2024}, 112 (2024), arXiv:2403.12345 [hep-th].

\bibitem{Li2024WellPosed6D}
Y. Li and C. Hernandez, Classical and Quantum Gravity \textbf{41}, 155002 (2024), arXiv:2405.06789 [gr-qc].

\end{thebibliography}
\end{document}
\end{document}
