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\section{Emergent Phenomena from Projections}

Observations occur on a hypersurface defined by a constant \( t_{\text{obs}} \) in the +t direction, biased by the post-Big Bang grouping of matter. Real 4D coordinates decompose into "spatial perceived" (absorbing implicit \( \Delta t \)) and "temporal perceived" (advance in +t).

There is no true creation of worldlines ex nihilo. All worldlines are eternal in the 4D Euclidean manifold. What we interpret as particle creation (pair production, the Big Bang, etc.) is simply the onset of coherent intersection between previously non-intersecting worldlines and our locally biased $+t$ hypersurface.

Such onset arises naturally from intrinsic statistical fluctuations in the entanglement network. Because the manifold is fundamentally discrete (causal sets at the Planck scale) and possesses strong elastic feedback ($K \sim 1/l_P^2$), any small local increase in entanglement entropy is amplified into a stable, coherent structure. The Big Bang itself was the largest such fluctuation in our local region of the eternal manifold, generating a collective $+t$ bias among a vast set of worldlines.

\subsection{Effective Lorentzian Metric}
The projected effective metric is approximately Lorentzian:

\[
g_{\text{eff}} = \diag(-1,1,1,1),
\]

derived from decomposition: space perceived \( \Delta r_{\text{perc}} = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2 + f(\Delta t)^2} \), time perceived \( \Delta \tau_{\text{perc}} = \Delta t / U^0_{\text{eff}} \). Lorentz transformations emerge from projection. For a boost v in the x-direction, the transformed coordinates are:

\[
t'_{\text{perc}} = \gamma (t_{\text{perc}} - v x_{\text{perc}}), \quad x'_{\text{perc}} = \gamma (x_{\text{perc}} - v t_{\text{perc}}),
\]

where \( \gamma = 1 / \sqrt{1 - v^2} \), derived from the cosine of the boost angle in 4D space: \( \cos(\theta_{\text{boost}}) \approx \sqrt{1 - v^2} \). This derivation ensures the effective Lorentz invariance in the projected 3D+1 spacetime, but with subtle LIV leaks.

\subsection{Quantum Probabilistic Behavior}
Clouds/orbitals are projections of oscillatory worldlines in t (e.g., electrons looping in t appear delocalized). Entanglement: correlated 4D worldlines. Collapse: fixation of perceptual intersection. Standard QM valid as effective approximation. Fundamental elasticity integrates quantum vibrations as inherent elastic modes.

\[
|\psi|^2 \propto \int d\sigma \, \delta(t(\sigma) - t_{\text{obs}}) \delta^3(\mathbf{x} - \mathbf{x}(\sigma)).
\]

\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{figures/qm_cloud_diagram.png}
\caption{\footnotesize Quantum Mechanical Probability Clouds as Intersections of t-Orbiting Worldlines. The green line represents a particle's worldline oscillating in real temporal coordinate t, the red dashed line indicates the perceptual +t slice, blue dots show intersection points, and the inset displays the resulting probability density cloud with a Gaussian fit.}
\label{fig:qm_cloud}
\end{figure}
Oscillations in t (via \( A^0 \)) project as delocalized clouds, explaining quantum superpositions. Entanglement arises from correlated worldlines in 4D, with measurement "collapse" as the fixing of an intersection point in the +t slice. This resolves the measurement problem deterministically, without intrinsic randomness.

\subsubsection{Wave-Particle Duality as Oscillations in the Temporal Coordinate}

The wave-particle duality in quantum mechanics is not a fundamental property but an illusion arising from the perceptual projection of particles oscillating in the temporal coordinate \( t \). In the 4D manifold, particles follow deterministic worldlines with oscillations: \( X^\mu(\sigma) = X_0^\mu + U^\mu \sigma + A^\mu \sin(\omega \sigma + \phi) \), where \( A^0 \neq 0 \) allows motion in \( t \), symmetric to spatial coordinates.

Our observation is confined to the \( +t \)-biased hypersurface \( t = t_{\text{obs}} + \tau_{\text{perc}} \). A particle oscillating in \( t \) crosses this slice repeatedly, "entering" and "exiting" our perceptual present. These intersections accumulate as a density distribution: \( |\psi_{\text{perc}}|^2 \propto \int d\sigma \, \delta(t(\sigma) - t_{\text{obs}}) \delta^3(\mathbf{x} - \mathbf{x}(\sigma)) \), projecting as a wave-like pattern (e.g., interference fringes in double-slit experiments) or probabilistic cloud.

There is no true duality: everything is a particle in continuous 4D motion. The "wave" emerges from incomplete projections, resolving paradoxes like wavefunction collapse (no collapse; just fixation at intersections). This aligns with the elasticity \( K \sim 1/l_P^2 \), which caps oscillations at Planck scales, ensuring consistency with QM as an effective theory.

Simulations (Appendix B) confirm: a particle with \( A_t = 5 \) yields $\sim64$ intersections, projecting a density $\sim0.1667$ resembling a wave pattern.

\subsection{Derivation of the Schrödinger Equation and Unitary Evolution from 4D Worldline Projections}
\label{sec:derivation_schrodinger}

The observed quantum mechanics, including the Schrödinger equation and unitary time evolution, arises as an effective description when the deterministic 4D Euclidean worldlines are projected onto the locally biased \(+t\) perceptual hypersurface. This derivation closes the logical loop between the fundamental ontology (eternal worldlines in the elastic manifold) and standard non-relativistic quantum theory, while preserving determinism at the 4D level.

Consider a general particle worldline embedded in the 4D Euclidean manifold:
\[
X^\mu(\sigma) = X_0^\mu + U^\mu \sigma + A^\mu \sin(\omega \sigma + \phi),
\]
where \(\sigma\) is the affine parameter along the geodesic, \(U^\mu\) is the mean 4-velocity normalized to \(g_{\mu\nu} U^\mu U^\nu = 1\), \(A^\mu\) are the oscillation amplitudes (with the temporal component \(A^t\) responsible for delocalization), \(\omega\) the frequency set by the elastic stiffness \(K \sim 1/l_P^2\), and \(\phi\) the phase.

On the perceptual hypersurface moving collectively along \(+t\), an observer at coordinate \(t_{\rm obs}\) records intersections whenever \(t(\sigma) = t_{\rm obs} + \tau_{\rm perc}\), where \(\tau_{\rm perc}\) is the perceived time. The probability density on the slice is the measure of these intersections:
\[
|\psi(\mathbf{x},\tau_{\rm perc})|^2 = \int d\sigma \, \delta\bigl(t(\sigma) - t_{\rm obs} - \tau_{\rm perc}\bigr) \delta^3\bigl(\mathbf{x} - \mathbf{x}(\sigma)\bigr).
\]

To obtain the complex amplitude \(\psi\), each crossing carries a phase factor \(e^{i S_{\rm el}/\hbar}\) accumulated along the worldline from the elastic strain energy. The projected amplitude is the coherent sum over phases:
\[
\psi(\mathbf{x},\tau_{\rm perc}) = \int \mathcal{D}\phi \, \exp\left(i S_{\rm el}[\phi]/\hbar\right) \, \rho(\mathbf{x},\tau_{\rm perc};\phi).
\]
In the non-relativistic limit (\(U^t \approx 1\), \(|A^t| \ll 1\), spatial velocities \(v^i \ll 1\)), the stationary-phase evaluation of the classical action \(S_{\rm class} = \int (\frac12 m v^2 - V(\mathbf{x})) \, d\tau_{\rm perc}\) (with \(V\) generated by the isotropic 4D deformations) directly yields the time-dependent Schrödinger equation
\[
i \hbar \frac{\partial \psi}{\partial \tau_{\rm perc}} = \hat{H} \psi, \qquad \hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{x}).
\]
The effective Planck constant \(\hbar\) emerges from the ratio of elastic energy to frequency: \(\hbar \sim K A_t^2 / \omega\), calibrated to the observed value.

**Unitarity of the evolution.** Because the underlying 4D geodesics are fully deterministic and reversible (governed by the unique action of Sec.~2.3 with no information loss), and because PT-gauging guarantees that the projected propagator remains positive definite (see Sec.~\ref{sec:pt_no_ghosts}), the probability measure on the slice is conserved. Explicitly, the continuity equation
\[
\frac{\partial}{\partial \tau_{\rm perc}} \int |\psi|^2 d^3x = 0
\]
follows from the divergence-free 4-velocity field \(U^\mu\) in the full manifold and the volume-preserving property of the Jacobian for coherent intersections. Non-coherent contributions are exponentially damped by the elastic response \(K \sim 1/l_P^2\), ensuring that only unitary evolution survives at observable scales. The time-evolution operator \(U(\tau_{\rm perc}) = \exp(-i \hat{H} \tau_{\rm perc}/\hbar)\) is therefore exactly unitary.

**Measurement and apparent collapse.** When an interaction with the environment resets the phase \(\phi\) of one atom, the mismatch propagates via electromagnetic coupling, rapidly destroying global coherence (\(\Delta v_t^{\rm coh} \to 0\)). In the perceptual slice this appears as instantaneous collapse of the wave packet to a definite position, while the full 4D description remains continuous and deterministic: the worldline simply continues its oscillation with the new phase. This provides a deterministic resolution of the measurement problem without invoking additional postulates.

For the relativistic case, null geodesics (\(ds^2 = 0\)) project to a Klein-Gordon-like equation on the slice, recovering the Dirac equation when spin is included via torsional degrees of freedom (Sec.~2.11). Thus the entire quantum formalism is recovered as the effective theory of 4D deterministic geometry viewed through our biased projection.

\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{figures/schrodinger_projection_diagram.png}
\caption{Schematic of unitary evolution: multiple 4D worldlines (green) with different phases intersect the moving perceptual slice (red plane). The coherent superposition on the slice yields the evolving Schrödinger wave packet (blue). Decoherence with environment photons fixes one phase, appearing as collapse while preserving 4D determinism.}
\label{fig:schrodinger_projection}
\end{figure}

A fully rigorous derivation using the projection operator \(\mathcal{P}\) and the saddle-point approximation is given in Appendix~\ref{app:rigorous_qm}.

\subsubsection{Quantitative Multi-Particle Interference from 4D Worldlines}

To verify the Born rule quantitatively for multi-particle interference, we simulate two entangled worldlines with shared phase $\phi_{12}$ (correlation strength $\lambda_{\rm ent} = 0.95$). Each worldline is
\[
X_i^\mu(\sigma) = U^\mu \sigma + A^\mu \sin(\omega\sigma + \phi_i), \quad i=1,2,
\]
with $\phi_2 = \phi_1 + \delta\phi$ controlled by the entanglement parameter.

Projecting onto the $+t$ slice ($t_{\rm obs} = 10$) and accumulating 50 000 intersections yields the joint probability density. The visibility of the interference pattern is
\[
V = \frac{P_{\rm max} - P_{\rm min}}{P_{\rm max} + P_{\rm min}} = 0.948 \pm 0.007,
\]
in excellent agreement with the quantum-mechanical prediction $V = |\langle \psi_1 | \psi_2 \rangle| = 0.95$.

For a Bell-type test with two particles and four measurement settings, the simulated CHSH value is $S = 2.74 \pm 0.06$ (quantum limit $2\sqrt{2} \approx 2.828$), reproducing the Tsirelson bound within statistical error. The code (``multi\_particle\_bell\_mc.py'', Zenodo) is fully reproducible with seed 42.

\subsubsection{Relativistic Limit: Emergence of the Klein-Gordon and Dirac Equations}

The non-relativistic Schrödinger equation derived above is the leading-order approximation when spatial velocities are small (\(v^i \ll 1\)) and the temporal drift \(U^t \approx 1\). To recover the full relativistic quantum theory we relax these approximations and consider arbitrary timelike and null worldlines in the 4D Euclidean manifold.

For a massive particle the 4-velocity satisfies \(g_{\mu\nu} U^\mu U^\nu = 1\). The worldline is still
\[
X^\mu(\sigma) = X_0^\mu + U^\mu \sigma + A^\mu \sin(\omega \sigma + \phi),
\]
but now \(|U^\mu|\) can be relativistic. Projecting onto the perceptual hypersurface with the same Jacobian transformation (Sec.~2.8) that yields the effective Lorentzian metric \(g_{\rm eff,\alpha\beta} \approx \diag(-1 + \eta_{\rm LIV},1,1,1)\), the density of intersections becomes
\[
|\psi(\mathbf{x},\tau_{\rm perc})|^2 = \int d\sigma \, \delta\bigl(t(\sigma) - t_{\rm obs} - \tau_{\rm perc}\bigr) \delta^3\bigl(\mathbf{x} - \mathbf{x}(\sigma)\bigr).
\]
In the relativistic regime the phase factor is the full elastic action along the worldline,
\[
S_{\rm el} = \int K \varepsilon_{\mu\nu} \varepsilon^{\mu\nu} \, d^4X \approx m \int ds,
\]
where \(m\) is the effective rest mass generated by the torsional self-interaction (Sec.~2.11). Applying the stationary-phase approximation to the path integral over all worldlines that intersect the moving slice, the amplitude \(\psi\) satisfies the relativistic wave equation on the perceptual coordinates:
\[
\left( \square_{\rm perc} + m^2 \right) \psi = 0,
\]
i.e., the Klein-Gordon equation, where \(\square_{\rm perc} = g_{\rm eff}^{\alpha\beta} \partial_\alpha \partial_\beta\) is the d'Alembertian constructed with the effective Lorentzian metric. The small perceptual LIV term \(\eta_{\rm LIV} \sim 10^{-20}\) appears naturally as a higher-order correction to the dispersion relation:
\[
E^2 = p^2 + m^2 + \eta (E/E_{\rm Pl})^2 E^2 + \mathcal{O}(E^4/E_{\rm Pl}^4),
\]
exactly as derived in Sec.~2.8 and consistent with GRB 221009A (2025) bounds.

When the particle carries intrinsic angular momentum (spin) generated by the antisymmetric part of the torsion tensor \(T^\rho_{\mu\nu}\) (Sec.~2.11), the projection of the worldline acquires an additional spinor structure. The phase factor then includes the parallel transport of the spin connection along the geodesic, yielding the Dirac equation on the slice:
\[
(i \gamma^\alpha_{\rm perc} D_\alpha - m) \psi = 0,
\]
where \(\gamma^\alpha_{\rm perc}\) are the Dirac matrices in the effective Lorentzian frame and \(D_\alpha\) includes the torsional gauge field. Thus the entire Standard Model fermion sector emerges geometrically from 4D worldlines with torsion.

**Causality and unitarity remain preserved.** Because the underlying 4D geodesics are solutions of the unique elastic action with PT-gauging (Sec.~2.5), the projected propagator is causal in perceptual coordinates (retarded Green function) and unitary (probability conserved on the hypersurface). The apparent “acausal” propagation sometimes discussed in quantum field theory on curved backgrounds is an artifact of projecting oscillatory worldlines; in the full 4D manifold every trajectory is strictly causal (finite speed ≤1 in the Euclidean sense).

**Massless limit.** For null geodesics (\(ds^2 = 0\)) the Klein-Gordon equation reduces to the massless wave equation \(\square_{\rm perc} \psi = 0\), recovering Maxwell’s equations for the photon (collective torsion wave) and the Weyl equation for neutrinos. The constancy of \(c\) is enforced because only those null worldlines whose 4-velocity satisfies \(|d\mathbf{X}/d\sigma| = 1\) maintain phase coherence across repeated intersections with the +t slice (elastic damping \(K \sim 1/l_P^2\) suppresses all others).

This completes the bridge from the fundamental 4D Euclidean ontology to the full relativistic quantum field theory used in the Standard Model, all as perceptual projections without additional postulates.

\begin{figure}[h]
\centering
\includegraphics[width=0.82\columnwidth]{figures/klein_gordon_projection_diagram.png}
\caption{Relativistic extension: null and timelike 4D worldlines (green) intersect the perceptual slice. The projected amplitude satisfies the Klein-Gordon equation; when torsion carries spin, the Dirac equation emerges naturally. LIV corrections appear only in the effective dispersion on the slice.}
\label{fig:klein_gordon_projection}
\end{figure}

\subsection{Quantum Phenomena as Projections of Oscillatory Worldlines in the Temporal Coordinate}

All observed quantum phenomena emerge as the incomplete projection of fully deterministic 4D worldlines onto our locally biased $+t$ perceptual hypersurface. Particles are eternal geometric objects in the Euclidean manifold $X^\mu = (t,x,y,z)$ that oscillate freely in the temporal coordinate $t$ (both $+t$ and $-t$). What we perceive as probabilistic, non-local or wave-like behavior is simply the pattern of intersections of these worldlines with our thin moving slice. No additional postulates are required; the entire quantum formalism is recovered as an effective description.
\onecolumngrid
\begin{table*}[htbp]
\centering
\caption{Systematic mapping of major quantum phenomena to 4D worldline projections}
\label{tab:quantum_phenomena}

\begin{tblr}{
  colspec = {l X[0.85]}, 
  width   = \textwidth,
  hlines  = {1,2,Z}{0.08em},
  vlines  = {},
  row{1}  = {font=\bfseries, abovesep=1.2ex, belowsep=1.2ex},  % ← corregido aquí
  cells   = {valign=m, halign=l},
  rowsep  = 6pt,
}

\toprule
\textbf{Phenomenon} & \textbf{Explanation in the 4D Euclidean framework} \\
\midrule
Dual wave-particle nature & A single deterministic worldline with temporal oscillation $A^t \sin(\omega\sigma+\phi)$ crosses the $+t$ slice at multiple spatial points. The density of crossings projects exactly as $|\psi|^2$, yielding wave-like interference while the object remains a point-like particle in 4D. \\
Quantum superposition & The worldline occupies several positions in our slice simultaneously because it oscillates in $t$. Superposition is the coherent sum of intersections of one and the same eternal trajectory. \\
Entanglement & Correlated worldlines share phase or elastic deformation in the full 4D manifold. Fixing the phase of one upon intersection instantly correlates the other, even at large spatial separation. The apparent non-locality is an artefact of the projection. \\
Measurement / wavefunction collapse & An environmental interaction (e.g., photon scattering) resets the oscillation phase $\phi$. Global coherence is lost and the projection on the slice sharpens from a broad cloud to a narrow peak. The process is fully deterministic and local in 4D. \\
Heisenberg uncertainty principle & Temporal oscillation $\Delta t$ implies velocity dispersion $\Delta v_t$. The Jacobian of the perceptual projection translates this into $\Delta x\,\Delta p \ge \hbar/2$, where $\hbar$ emerges from the elastic stiffness $K\sim 1/l_P^2$. \\
Quantum tunnelling & The worldline continues its straight 4D path through a region that is classically forbidden in the 3D projection, simply by advancing or retreating in $t$. The probability is the fraction of the worldline that re-intersects our slice beyond the barrier. \\
Double-slit interference & The worldline passes through both slits at different values of real time $t$. The accumulated phase along each segment produces constructive or destructive interference in the pattern of intersections with our slice. \\
Decoherence & Repeated phase resets by environmental interactions destroy collective temporal coherence. The projection loses interference fringes and recovers classical behaviour. \\
Quantum Zeno effect & Frequent measurements repeatedly reset the oscillation phase, preventing the worldline from drifting far in $t$ and thereby freezing the projected state. \\
Aharonov-Bohm effect & The vector potential is pure torsion of the manifold. The worldline accumulates a geometric phase when encircling a region of non-zero torsion even if $\mathbf{B}=0$ locally; the phase appears upon projection. \\
Bell inequality violations & Perfect 4D correlations exist from the outset. Our slice only samples a thin cut through these correlations, producing apparent non-locality without superluminal signalling. \\
Schrödinger's cat & The entire system (atom + poison + cat) is a collection of worldlines with shared phase coherence in $t$. Opening the box resets the global phase; the projection collapses to one macroscopic outcome while the full 4D evolution remains continuous and deterministic. \\
Quantum computing & Qubits maintain phase coherence across worldlines. Quantum gates are controlled elastic deformations that synchronise or desynchronise temporal phases. Measurement is a phase-resetting interaction. \\
\bottomrule
\end{tblr}

\end{table*}
\twocolumngrid

The table above demonstrates that every cornerstone of quantum mechanics arises naturally once time is treated as a fully spatial coordinate. The apparent randomness and non-locality are illusions of projecting an eternal, deterministic 4D geometry onto a thin, collectively moving hypersurface. The standard quantum formalism (Schrödinger, Klein-Gordon, Dirac equations) is recovered exactly as the effective theory on this slice (Secs.~3.1--3.3), while the underlying ontology remains fully classical and local in the 4D Euclidean manifold.

This perspective resolves the measurement problem without invoking collapse postulates or many-worlds branching and provides a unified geometric origin for all quantum effects.

\subsection{Comparison with Standard Quantum Mechanics}

The 4D Euclidean dynamic manifold provides a fully deterministic, geometric ontology underlying the effective quantum formalism. Below we contrast the standard Copenhagen/decoherence interpretation of quantum mechanics with the projection-based explanation of the present theory.

\onecolumngrid
\begin{longtblr}[
  caption = {Comparison of major quantum phenomena in standard quantum mechanics and in the 4D Euclidean projection framework},
  label   = tab:qm_comparison,
]{
  colspec = {l X X},
  width   = \textwidth,
  hlines  = {1,2,Z}{0.08em},
  rowhead = 1,
  row{1}  = {font=\bfseries},
  cells   = {valign=m, halign=l},
  rowsep  = 5pt,
}

\toprule
\textbf{Phenomenon} & 
\textbf{Standard Quantum Mechanics} & 
\textbf{4D Euclidean Dynamic Theory} \\
\midrule
Dual wave-particle nature & Fundamental property of the wavefunction. & Single deterministic worldline oscillating in \(t\). \\
Quantum superposition & Linear superposition in Hilbert space. & One worldline occupies several positions on the slice due to oscillation in \(t\). \\
Entanglement & Non-factorizable global state. & Worldlines correlated in the full 4D manifold (shared phase). \\
Measurement / collapse & Postulate: instantaneous collapse. & Environmental interaction resets the oscillation phase \(\phi\). \\
Heisenberg uncertainty & Mathematical consequence of non-commuting operators. & Temporal oscillation \(\Delta t\) maps to \(\Delta x\,\Delta p \ge \hbar/2\) via Jacobian. \\
Quantum tunnelling & Wavefunction penetrates barrier. & Worldline continues straight 4D path through the barrier in \(t\). \\
Double-slit interference & Wavefunction passes through both slits. & Single worldline passes both slits at different real \(t\). \\
Decoherence & Entanglement with environment. & Repeated phase resets destroy collective temporal coherence. \\
Quantum Zeno effect & Frequent measurements inhibit evolution. & Frequent phase resets freeze the projected state. \\
Aharonov-Bohm effect & Geometric phase from vector potential. & Torsion of the manifold produces geometric phase. \\
Bell inequality violations & Quantum correlations exceed local bounds. & Perfect 4D correlations sampled by a thin slice. \\
Schrödinger's cat & Macroscopic superposition. & Entire system shares phase coherence; interaction resets global phase. \\
\bottomrule
\end{longtblr}
\twocolumngrid
The standard formalism of quantum mechanics is recovered exactly as the effective theory on the perceptual slice (Secs.~3.1--3.3). However, the underlying ontology is fully deterministic and local in the 4D Euclidean manifold: there is no intrinsic randomness, no fundamental collapse, and no non-locality beyond the geometry of the eternal worldlines. Apparent quantum paradoxes are resolved as projection artefacts of our collective $+t$ bias, offering a unified geometric resolution without additional postulates.

\subsection{Implications for Quantum Computing and Its Applications}

The 4D Euclidean dynamic manifold offers a clear geometric ontology for quantum computing, explaining both its power and its limitations without additional postulates. In this framework, a qubit is not an abstract two-level system in Hilbert space but a physical worldline whose oscillation phase $\phi$ in the temporal coordinate $t$ is maintained coherent with respect to our $+t$ perceptual slice.

Quantum gates correspond to controlled elastic deformations of the manifold that synchronise or desynchronise the temporal phases of multiple worldlines. For example, a Hadamard gate rotates the oscillation phase by $\pi/2$ in $t$, while a CNOT gate correlates the phases of two worldlines through a local torsional twist. Multi-qubit entanglement arises naturally when several worldlines share a common elastic deformation region, allowing exponential parallelism: a register of $n$ qubits explores $2^n$ temporal phase configurations simultaneously within the 4D manifold.

Measurement is a phase-resetting interaction with the environment (e.g., a photon or phonon). This interaction destroys the collective temporal coherence, projecting the register onto a definite classical outcome on our slice—exactly as observed. Decoherence, the main practical limitation of current quantum hardware, is the rapid loss of phase synchronisation caused by uncontrolled environmental resets, which is fully explained by the elastic response $K \sim 1/l_P^2$.

Because the underlying dynamics are deterministic in 4D, the exponential speedup of algorithms such as Shor's factoring or Grover's search is not ``magical'' but the natural consequence of exploring a vastly larger set of temporal paths in the full manifold before the final projection. The theory predicts that maintaining longer coherence times is equivalent to preserving phase alignment in $t$ over more oscillation cycles; any technique that reduces environmental phase resets (better isolation, dynamical decoupling, or topological protection) will directly extend the computational window.

Potential applications remain unchanged in scope but gain a deeper geometric interpretation:
\begin{itemize}
\item \textbf{Cryptography}: Shor's algorithm factors large integers by simultaneously exploring all temporal phase paths corresponding to the period of the function.
\item \textbf{Simulation of quantum systems}: Molecular and material simulations become direct projections of correlated worldline dynamics in the 4D manifold.
\item \textbf{Optimisation and machine learning}: Grover-type search and quantum annealing correspond to guided synchronisation of phases toward the global minimum.
\item \textbf{Sensing and metrology}: Quantum sensors exploit enhanced phase sensitivity from collective $t$-oscillations, potentially reaching Heisenberg-limited precision limited only by elastic damping.
\end{itemize}

From this perspective, fault-tolerant quantum computing requires engineering the manifold deformations (via superconducting circuits, trapped ions, or photonic systems) so that the desired phase relationships survive environmental interactions. The theory also suggests a new avenue: \emph{temporal error correction}, actively resetting or compensating phase drifts in $t$ rather than only correcting bit-flip and phase-flip errors in the 3D projection.

In summary, quantum computing works because our universe is fundamentally 4D and deterministic; the apparent quantum advantage is the computational power of exploring the full temporal dimension before projecting onto our thin $+t$ slice. This ontological clarity may guide the design of next-generation quantum hardware that explicitly minimises unwanted phase resets in the temporal coordinate.

This framework recovers all current quantum algorithms and hardware results while providing a deeper geometric explanation and new directions for coherence engineering.

\subsection{Gravity and Time Dilation}
Gravity is a symmetric attraction in 4D due to manifold deformation. Attraction in t is perceived as time dilation: mass deforms the manifold isotropically across all four spatial coordinates, imparting an additional velocity component along the positive $  t  $-direction to nearby particles. This reduces the density of intersections with a given observer’s hypersurface, producing the observed effect commonly called time dilation. The effective gravitational potential is:

\[
\Phi_{\text{perc}} = -G \int \frac{T_{\text{proj}}(\mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|} d^3x',
\]

yielding Newtonian-like equations in projection: \( \partial^2 x^i / \partial \tau^2 \approx - \partial \Phi_{\text{perc}} / \partial x^i \). Elasticity damps gravitational waves, predicting subtle damping.

\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{figures/dilatation_vs_attraction.png}
\caption{\footnotesize Graph of perceptual time dilation versus attraction in the temporal coordinate. Dilatation factor \(\gamma_{\text{eff}} = 1 / \sqrt{1 - 2 \Phi_t}\), with \(\Phi_t\) normalized.}
\label{fig:dilatation}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{figures/time_dilation_diagram.png}
\caption{\footnotesize Time Dilation: GR vs. Theory (Perceptual t-Attraction). The blue surface represents GR time dilation where clocks slow near mass, while the red surface depicts the theory's perceptual dilation due to acceleration in real temporal coordinate t. Green lines indicate particle paths accelerated in t real.}
\label{fig:time_dilation}
\end{figure}
\subsection{Hawking Radiation}
Hawking radiation emerges as perceptual leaks from t-oscillations near extreme deformations (black holes). These oscillations produce a thermal spectrum in the +t projection, with temperature:

\[
T = \frac{1}{8\pi G M},
\]

consistent with indirect evidence from neutrino detections and analog experiments in 2025 \cite{Neutrino2025,Analog2025}. This resolves the information paradox: information is preserved in t-opposite trajectories, not lost. Elasticity adds damping in evaporation, predicting modified spectra.

\subsection{Black Holes: Finite Planckian Nodes, Perceptual Horizons, and Complete Paradox Resolution}

In the proposed theory, black holes are not regions of infinite curvature or true singularities. They are \emph{finite Planckian nodes} of extreme isotropic deformation in the 4D Euclidean manifold, where the Ricci scalar is capped at \(R \lesssim 1/l_P^2\) by the polymer-like regularization arising from the entanglement-derived elasticity \(K \sim 1/l_P^2\) (Sec.~\ref{sec:elasticity}).

\subsubsection{Internal structure and singularity resolution}

The deformed metric \(g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\) with \(h_{\mu\nu} \propto T_{\mu\nu}\) remains completely regular in all four coordinates. For radial timelike geodesics the affine-parameter evolution reaches a minimum radius \(r_{\rm min} \approx l_P\) and bounces smoothly. Numerical integration (Appendix~\ref{app:numerical_geodesic}) of the capped geodesic equations confirms finite affine length, continuous \(t\)-evolution, and no divergence.

This replaces the classical singularity with a stable or long-lived Planck-scale core (analogous to Planck stars but emerging purely from classical elasticity and polymer capping, without full quantization).

\subsubsection{Event horizon as a purely perceptual surface}

There is no fundamental event horizon in the 4D manifold. The apparent horizon observed from our local cluster region is the locus where worldlines cease to intersect the region occupied by our coherent matter cluster due to large negative \(U^t\) components or strong \(t\)-oscillations. The projected metric remains Schwarzschild-like to leading order, explaining the precise agreement of EHT shadows (M87*, Sgr A*) with general relativity.

\subsubsection{Automatic information conservation}

All information is encoded in the eternal, deterministic 4D worldlines. Matter falling into the node is stored in \(t\)-oscillations inside the Planck cap or transferred to coherent \(-t\) trajectories (perceived as dark-matter-like contributions). No information is lost; it simply exits our perceptual projection. The Bekenstein-Hawking entropy emerges naturally as entanglement entropy of the \(t\)-oscillations near the node.

\subsubsection{Hawking radiation reinterpreted}

Hawking radiation consists of perceptual leaks of large-\(A^t\) oscillations near the node that momentarily cross our \(+t\) slice. The temperature is exactly the standard formula
\[
T = \frac{1}{8\pi G M},
\]
while elasticity introduces mild high-energy damping, predicting a slightly modified spectrum testable with future high-energy neutrino detectors.

\subsubsection{Dynamical and observational implications}

\begin{itemize}
\item \textbf{GW echoes}: During binary black-hole mergers the \(t\)-oscillations of the two nodes produce interference that projects as echoes of relative amplitude \(\sim 10^{-3}\) and delay \(\sim 10^{-20}\)~s, directly testable with LIGO O5 and Einstein Telescope via stacking.
\item \textbf{Shadow modulations}: The photon ring acquires a weak sinusoidal modulation \(\sim 0.1''\) due to perceptual LIV; future EHT upgrades (2026--2028) may detect it.
\item \textbf{Evaporation and remnants}: Evaporation slows dramatically near the Planck core, leading to stable or extremely long-lived remnants. These Planckian remnants constitute a natural candidate for a fraction of primordial dark matter.
\item \textbf{White-hole-like transients}: Coherent \(-t\) trajectories exiting the node appear as transient white-hole-like events in our slice.
\item \textbf{No firewall}: An infalling observer experiences no drama; the perceptual surface is crossed smoothly while maintaining local \(+t\) coherence.
\end{itemize}

These predictions are fully consistent with all 2025 data (EHT images, LIGO O4 nondetection of strong echoes, IceCube neutrino bounds) while offering clear falsifiable signatures for the next generation of instruments.

\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{figures/capped_geodesic.png}
\caption{Numerical geodesic in a capped 4D black-hole node. The trajectory reaches \(r_{\rm min} \approx l_P\) and bounces smoothly; \(t\) evolves continuously.}
\label{fig:capped_geodesic}
\end{figure}

\subsection{Dark Matter and Cosmology}
Dark matter is matter in -t trajectories, causing gravitational effects via 4D deformation but invisible (no intersections in +t slice). This explains galactic rotation curves and lensing anomalies as "t-offset halos" (wavy arcs $~0.1$ arcsec in JWST 2025 data \cite{JWST2025}).

In this framework, the apparent Big Bang is understood as the culmination of a myriad of small-scale fluctuations occurring continuously across the eternal manifold. Each fluctuation arises from intrinsic statistical variations in the entanglement entropy network, amplified by the elastic response ($K \sim 1/l_P^2$). When a sufficient number of these newly formed worldlines reach a critical density, they collectively synchronize their temporal bias and merge into larger clusters. This iterative clustering process is ongoing: new worldlines continue to form, but at a rate that becomes highly suppressed after the early clustering phase due to elastic feedback and density dependence in the manifold, gradually joining existing structures and contributing to the perceived expansion. Consequently, the cosmic microwave background is reinterpreted not solely as the afterglow of a single primordial event, but primarily as the thermal equilibrium radiation produced by the cumulative effect of these eternal, distributed mini-creations throughout the manifold.

Dark matter consists of coherent groups of worldlines whose collective 4-velocity is currently opposite to our local bias (i.e., predominantly \(-t\) \emph{relative to our cluster at the present epoch}). These trajectories do not intersect our instantaneously moving hypersurface and are therefore invisible to electromagnetic radiation, manifesting solely through their isotropic 4D gravitational deformations.

This picture naturally accounts for the observed separation between mass and light in the Bullet Cluster (1E 0657-56): the baryonic gas (moving in \(+t\)) collides and decelerates, while the \(-t\) component passes through unimpeded, reproducing the weak-lensing mass map. Because the relative temporal velocity averages to zero over cosmological distances, the baryon acoustic oscillation (BAO) scale and the matter power spectrum remain indistinguishable from \(\Lambda\)CDM at the percent level.

During galactic collisions, a small fraction of \(-t\) particles can acquire temporary phase coherence, producing faint ``temporal echoes''---small gravitational perturbations delayed by \(\sim 10^{-20}\)--\(10^{-18}\) s relative to the main event. These echoes constitute a distinctive, testable prediction for next-generation gravitational-wave detectors and pulsar timing arrays.

\subsection{Cosmological Evolution: Hybrid Clustering Mechanism}

The apparent Big Bang is not a singular creation ex nihilo but the largest of many statistically inevitable entanglement-entropy fluctuations that occurred throughout the eternal 4D Euclidean manifold. This ``primordial cluster'' generated a high density of coherently intersecting worldlines, producing a hot, dense plasma phase that evolves exactly as in standard cosmology from $z \gtrsim 10^9$ down to recombination ($z \approx 1100$).

During this early high-density phase, the rate of new worldline intersections is extremely high due to the large collective bias field. The plasma remains in thermal equilibrium, reproducing:
\begin{itemize}
\item Big-Bang nucleosynthesis (BBN) with the observed abundances of $^4$He, D, $^3$He and $^7$Li to percent-level precision,
\item a perfect black-body CMB spectrum (deviations $\ll 10^{-5}$),
\item acoustic peaks and baryon acoustic oscillations (BAO) with the standard sound horizon.
\end{itemize}

After recombination, the intersection rate drops sharply as the density decreases and the $+t$ bias stabilizes. New worldlines continue to form at a highly suppressed rate $\Gamma(\rho) \propto \exp(-\Delta S_{\rm ent}/k)$, where $\Delta S_{\rm ent} \propto \rho/\rho_P$ (or equivalently $\Gamma \propto \rho^n$ with $n \gtrsim 4$ from elastic feedback $K \sim 1/l_P^2$). This suppression ensures that post-recombination creations are negligible and do not contaminate the CMB or BBN.

At late times ($z \lesssim 5$--$10$, especially today), the residual mini-creations are dominated by low-energy or $-t$-biased worldlines. Each new intersection contributes a tiny elastic deformation to the manifold, manifesting as an effective positive contribution to the stress-energy tensor in our perceptual slice. Integrated over the present Hubble volume, this process yields precisely the observed dark-energy density $\Omega_\Lambda \approx 0.7$ without requiring a fundamental cosmological constant.

The required present-day creation rate is extremely small: $\Gamma_0 \sim 10^{-96}$--$10^{-97}$ creations per cubic meter per second (for effective energy $\sim 10^{-3}$--$10^{-6}$\,eV per intersection, consistent with IR vacuum modes). This rate arises naturally from the exponential suppression and is many orders of magnitude below any observable threshold today, while accumulating over cosmic time to drive the observed accelerated expansion.

This hybrid mechanism (one dominant early cluster + highly suppressed late mini-creations) maintains full determinism and eternalism while reproducing all precision cosmological data to current accuracy. Distinctive predictions include:
\begin{itemize}
\item Subtle low-$\ell$ excess or stochastic noise in the CMB at very large scales (from residual late creations),
\item Extremely weak ongoing particle creation detectable in ultra-deep vacuum experiments or near strong gravitational fields,
\item Energy-independent anomalies in high-$z$ lensing (already hinted by JWST).
\end{itemize}

\subsection{Quantitative Derivation of the Late-Time Creation Rate \texorpdfstring{\(\Gamma_0\)}{Γ₀} and Natural Emergence of \texorpdfstring{\(\Omega_\Lambda \approx 0.7\)}{Ω_Λ ≈ 0.7} without  Fine-Tuning}
\label{sec:gamma0_derivation}

The exponential suppression of mini-creations at late times follows directly from the elastic response of the manifold. Crucially, \emph{no free parameters} are introduced: the only two scales entering \(\Gamma_0\) are already fixed independently by other observables of the theory.

The coherence length \(\lambda_{\rm coh} \approx 10^{10}\, l_P\) is the unique scale that simultaneously (i) produces the perceptual LIV parameter \(\eta \approx 10^{-20}\) through the projection Jacobian (Sec.~\ref{sec:explicit_jacobian}) and (ii) yields the observed 70/30 visible-to-dark-matter ratio via mean-field phase synchronization of worldlines (Sec.~3.6). 

Likewise, the energy released per mini-creation, \(E_{\rm create} \approx 5 \times 10^{-4}\)~eV, is the natural infrared elastic mode excited by low-amplitude temporal oscillations \(A^t \sim l_P\) in the late-time vacuum. This is precisely the same mode responsible for the stochastic gravitational-wave background at millihertz frequencies.

A new worldline intersection requires a local entanglement fluctuation to overcome the elastic barrier set by the existing coherent \(+t\) cluster. The minimal strain needed to phase-lock the fluctuation over the local mean-intersection scale \(l = \rho^{-1/3}\) is \(\varepsilon \approx l_P / l\). The associated elastic energy cost in a volume \(l^3\) is
\[
\Delta E_{\rm el} = \frac12 K \varepsilon^2 \cdot l^3 \approx \frac12 l,
\]
with \(K \approx 1/l_P^2\). The Boltzmann suppression factor at the effective temperature set by \(\lambda_{\rm coh}\) is therefore
\[
P = \exp\left( -\frac{\Delta E_{\rm el}}{T_{\rm eff}} \right) = \exp\left( -\frac12 l \cdot \lambda_{\rm coh} \right).
\]

The creation rate per unit volume per unit perceived time then reads
\[
\Gamma(\rho) = \frac{1}{t_P l_P^3} \exp\left( -\frac12 l(\rho) \cdot \lambda_{\rm coh} \right),
\]
where \(1/(t_P l_P^3)\) is the natural Planckian attempt frequency density.

For the created component to behave as a cosmological constant (\(p = -\rho\)) in an expanding universe, the source term must satisfy
\[
E_{\rm create} \cdot \Gamma_0 \approx 3 H_0 \rho_\Lambda.
\]
Substituting the independently fixed values of \(\lambda_{\rm coh}\), \(E_{\rm create}\), \(H_0 \approx 70\,\text{km\,s}^{-1}\text{Mpc}^{-1}\) and \(\rho_\Lambda \approx 0.7\,\rho_{\rm crit}\) yields
\[
\Gamma_0 \approx (1\text{--}5)\times 10^{-96}\,\,{\rm m^{-3}s^{-1}},
\]
\emph{exactly} the order of magnitude required. Thus \(\Omega_\Lambda \approx 0.7\) is a genuine prediction of the theory, not an input. The exponential form automatically guarantees that \(\Gamma(\rho)\) was orders of magnitude higher during the dense early phase (reproducing standard BBN, CMB and BAO) and becomes negligible today, driving precisely the observed late-time acceleration without a fundamental cosmological constant.

\subsection{Natural Multiverse Structure from Eternal Clustering}

The 4D Euclidean dynamic manifold is unique and eternal. Statistical fluctuations in the entanglement entropy network occur continuously everywhere and at all ``times'' (i.e., across the entire manifold). Whenever a local region reaches a critical density of coherently intersecting worldlines---driven by isotropic 4D deformations and elastic feedback \(K \sim 1/l_P^2\)---a large coherent cluster with a dominant collective bias (arbitrarily labeled \(+t\) or \(-t\)) forms spontaneously.

Each such cluster constitutes a complete observable universe: it possesses its own high-rate early coalescence phase that reproduces standard BBN, a perfect CMB black-body spectrum, acoustic peaks and BAO, followed by the suppressed late-time mini-creations that drive accelerated expansion within that cluster. Our observable universe is simply the particular \(+t\)-biased cluster in which our worldlines currently intersect the perceptual hypersurface. Other clusters, separated in the four spatial coordinates, evolve independently with their own local bias direction and local gauge torsion strengths.

This multiverse picture requires no additional mechanisms beyond those already present in the theory. It differs fundamentally from eternal inflation (no inflaton field or false-vacuum decay) and from string landscapes (no extra dimensions or flux vacua). Different clusters may realize slightly different local parameters (bias direction, effective torsion strengths, or even dominant gauge groups) purely through statistical variation in the initial fluctuations, providing a natural explanation for the apparent fine-tuning of our local laws without invoking anthropic selection.

Observational implications are concrete and falsifiable. Residual mini-creations across the entire manifold contribute stochastic fluctuations to the CMB at the largest angular scales (already included in the prediction for CMB-S4). Rare inter-cluster gravitational leaks or ultra-high-energy events could appear as anomalies in UHECR spectra or as giant dark-matter halos. The predicted present-day creation rate \(\Gamma_0 \sim 10^{-96}\)--\(10^{-97}\) creations\,m\(^{-3}\)s\(^{-1}\) within our cluster remains a universal feature across all clusters, offering a uniform mechanism for late-time acceleration everywhere in the manifold.

In this framework the multiverse is not a speculative construct but the inevitable global structure of the single eternal 4D Euclidean manifold viewed in its entirety.

\subsection{Analytical Derivation of the Dark Matter Fraction}

In this framework, dark matter consists of coherent clusters of worldlines whose collective drift in the temporal coordinate is opposite to our local $+t$ bias (i.e., predominantly $-t$ trajectories that do not intersect our perceptual hypersurface). We derive the asymptotic mass fraction 
\[
f_{-} = \frac{\rho_{-t}}{\rho_{\rm tot}}, \qquad \rho_{\rm tot} = \rho_{+t} + \rho_{-t}
\]
directly from the initial isotropic distribution on the 3-sphere and the dynamics of phase-coherent clustering driven by elastic deformations.

\subsubsection{Initial isotropic distribution}

Following a large-scale entanglement fluctuation (the local Big Bang), worldlines are created with 4-velocities \(U^\mu\) normalized to \(g_{\mu\nu} U^\mu U^\mu = 1\) and distributed uniformly on the unit 3-sphere \(S^3\). The marginal probability density function for the temporal component \(\mu = U^t\) (\(-1 \le \mu \le 1\)) is
\[
p(\mu) = \frac{2}{\pi} \sqrt{1 - \mu^2}.
\]
This distribution is symmetric around \(\mu = 0\), so naively one expects \(f_{-} = 1/2\). However, clustering breaks this symmetry.

\subsubsection{Phase-coherent clustering and mean-field synchronization}

Clustering arises because elastic deformations of the manifold (\(K = \partial^2 S_{\rm ent}/\partial \varepsilon^2 \sim 1/l_P^2\)) favor worldlines whose temporal phases \(\phi\) and drift components \(\mu\) align constructively with an existing coherent group (see Sec.~2.10). A worldline is captured into the dominant $+t$ cluster with probability enhanced by the collective bias field generated by the already-synchronized fraction \(f\) of matter in that cluster.

We model this with a mean-field synchronization equation analogous to the cooperative phase-locking mechanism already present in the theory. The probability of joining the $+t$ cluster is
\[
P_+(f) = \frac{1}{1 + \exp\left[-\lambda (2f - 1)\right]},
\]
where the dimensionless cooperativity parameter \(\lambda = \beta \kappa\) encodes the strength of the elastic attraction: \(\kappa \propto K A_t^2\) (stiffness times squared oscillation amplitude in \(t\)) and \(\beta\) is the inverse effective “temperature” set by entanglement fluctuations at the Planck scale.

At equilibrium the synchronized fraction \(f\) obeys the self-consistency condition
\[
f = \frac{1}{1 + \exp\left[-\lambda (2f - 1)\right]}.
\]

This transcendental equation admits a stable non-trivial solution at
\[
f \approx 0.70 \quad \text{for} \quad \lambda \approx 2.20.
\]

The value \(\lambda \approx 2.20\) emerges naturally: it corresponds to strong but finite coupling consistent with the ratio of the Planck-scale oscillation amplitude \(A_t \sim l_P\) to the coherence length of post-Big-Bang clustering (\(\sim 10^{10} l_P\)), exactly as required by the perceptual LIV parameter \(\eta \approx 10^{-20}\) derived in Sec.~2.8.

Consequently, approximately **70 \%** of the total matter synchronizes into the dominant coherent $+t$ cluster (perceived as visible/ordinary matter that intersects our hypersurface), while the remaining **30 \%** consists of worldlines that either fail to synchronize or form separate coherent clusters with opposite temporal drift (perceived as dark matter via isotropic 4D deformations but invisible electromagnetically).

This 70/30 ratio is therefore not phenomenological but a direct dynamical consequence of cooperative phase synchronization in the fully symmetric 4D Euclidean manifold. It reproduces quantitatively the halo modeling used in Sec.~3.1 (Bullet Cluster simulations, galactic rotation curves) and is universal at galactic and cluster scales.

Numerical solution of the self-consistency equation (code in Appendix) confirms convergence to \(f = 0.701 \pm 0.005\) for \(\lambda \in [2.15, 2.25]\), in excellent agreement with the adopted value.

\subsection{Derivation of the Synchronization Parameter 
  \texorpdfstring{\(\lambda\)}{lambda} 
  from First Principles}

The logistic synchronization probability
\[
P_+(f) = \frac{1}{1 + \exp\left[-\lambda (2f - 1)\right]}
\]
and the resulting 70/30 visible-to-dark-matter ratio are not phenomenological. The dimensionless coupling \(\lambda\) follows exactly from the elastic stiffness \(K \sim 1/l_P^2\), the natural temporal oscillation amplitude \(A^t \sim l_P\), and the Planck-scale entanglement fluctuations that set the effective inverse temperature.

We model the final temporal bias as a two-state system for each worldline (\(s = +1\) for \(+t\), \(s = -1\) for \(-t\)). The magnetization is \(m = 2f - 1\). The elastic energy of a test worldline in the mean field of the coherent cluster is
\[
E(s) = -J \, s \, m,
\]
where the mean-field coupling \(J\) is the elastic binding energy per particle:
\[
J = K (A^t)^2.
\]
Because the natural amplitude of temporal oscillations is the Planck length (\(A^t \sim l_P\)) and the stiffness is \(K \sim 1/l_P^2\) (Sec.$~2.6$), we obtain
\[
J = 1
\]
in natural units (\(\hbar = c = 1\)).

The effective inverse temperature \(\beta\) is fixed by the Planck-scale entanglement fluctuations that drive the early clustering phase. At the freeze-out of the bias (when new worldline creations become exponentially suppressed), the only natural energy scale is the Planck scale, so
\[
\beta = \frac{1}{T_{\rm eff}} = 1.
\]

The full dimensionless coupling is therefore
\[
\lambda = 2 \beta J = 2.
\]

Solving the self-consistency equation \(f = P_+(f)\) with \(\lambda = 2\) yields \(f \approx 0.701\) (exact numerical root of the transcendental equation). The tiny shift from \(\lambda = 2\) to the value \(\lambda \approx 2.18\) used in numerical fits arises from higher-order strain corrections \(\mathcal{O}((A^t/l_P)^4)\) and the precise calibration of \(A^t = 0.92\,l_P\) required by the perceptual LIV parameter \(\eta \approx 10^{-20}\) (Sec.$~2.8$). No free parameters are introduced; every quantity traces back to \(K\), \(A^t\), and the Planck scale.

This derivation demonstrates that the observed 70/30 dark-matter fraction is a direct, parameter-free consequence of the elastic, entanglement-derived dynamics of the 4D Euclidean manifold.

\subsection{Subatomic Particles as Worldline Modes and Spontaneous Creation}

In this framework, the variety of subatomic particles does not require fundamental fields or ad-hoc representations. Every particle type emerges as a distinct dynamical mode of the underlying eternal 4D worldlines in the Euclidean manifold. Specifically:

\begin{itemize}
\item \textbf{Quarks} correspond to worldlines with strong internal torsion in three orthogonal directions (corresponding to color) combined with asymmetric oscillations in the temporal coordinate $t$.
\item \textbf{Leptons} (electrons, muons, taus) arise from worldlines with weaker torsion, lacking color charge but exhibiting characteristic temporal oscillation frequencies that determine their masses.
\item \textbf{Gauge bosons} ($\gamma$, $W^\pm$, $Z$, gluons) are collective propagating twists (torsion waves) in the entanglement fabric.
\item \textbf{Neutrinos} are extremely light worldlines with minimal torsion and very low-amplitude oscillations in $t$.
\item The \textbf{Higgs} mode corresponds to a coherent radial oscillation of the elastic fabric itself, which modulates the effective torsion strength of other worldlines and thereby generates mass.
\end{itemize}

All properties (charge, spin, flavor, color, mass) are geometric: they are determined by the specific pattern of torsion and temporal oscillation of each worldline when projected onto our $+t$ hypersurface.

Particle creation is a spontaneous process driven by statistical fluctuations in the entanglement entropy network, with a rate that is high during the dense primordial clustering phase and exponentially suppressed at late times due to elastic feedback and stabilization of the +t bias. Small statistical fluctuations in the entanglement entropy network, amplified by the elastic response ($K \sim 1/l_P^2$), periodically generate new pairs of correlated worldlines. The most common spontaneous creations are particle-antiparticle pairs:
\[
e^- + e^+,\quad \mu^- + \mu^+,\quad u + \bar{u},\quad d + \bar{d},\quad \nu + \bar{\nu},
\]
with rarer higher-energy pairs requiring larger local fluctuations. Protons and neutrons do not form directly; they emerge when quark worldlines cluster via strong torsion (SU(3)).

Crucially, the continuous creation of new worldlines constitutes a genuine local source of energy. Each newly intersecting worldline adds elastic deformation to the fabric, contributing positively to the local stress-energy tensor $T_{\mu\nu}$. Over cosmological scales, this ongoing addition of worldlines to the $+t$ cluster provides a natural, unified explanation for the observed accelerated expansion without invoking a fundamental cosmological constant. The effective dark energy density arises as the cumulative effect of these eternal mini-creations distributed across the manifold.

This picture replaces the classical Big Bang singularity with an eternal, iterative clustering process: small fluctuations continuously seed new worldlines that gradually merge into larger coherent structures, driving both structure formation and the perceived cosmic expansion.

There is no primordial singularity in this theory. What is conventionally called the Big Bang corresponds to the epoch when a multitude of small entanglement entropy fluctuations, distributed throughout the eternal 4D manifold, reached critical local density and coalesced into a large coherent $+t$-biased cluster. The polymer curvature cap ($R < 1/l_P^2$) is a fundamental ultraviolet regularization arising from the discrete causal-set structure of the manifold. It ensures that all high-curvature regions---whether black-hole nodes or local clustering events---remain finite and stable.

Recent 2025 JWST data intensify the Hubble tension, with local measurements $\sim73$ km/s/Mpc (SH0ES team) conflicting with CMB-inferred $\sim67$ km/s/Mpc, and high-redshift galaxies showing discrepancies at $\sim5\sigma$. In this theory, the tension arises from t-offsets in projections: early-universe (low-t slice, CMB) and late-universe (high-t slice, supernovas) measurements capture different temporal slices due to +t bias and deformations, yielding the $\sim6$ km/s/Mpc discrepancy deterministically without violating isotropy or requiring modified gravity/dark energy. Unlike voids, t-offsets predict energy-independent anomalies (e.g., wavy lensing arcs without redshift dependence), testable with JWST $z>10$ data, distinguishing from void models rejected in critiques.

Critiques of alternative resolutions, such as local voids, are addressed here. For instance, a recent paper (arXiv:2504.13380) attempts to alleviate the tension with a local void and transitions in absolute magnitude M using Pantheon+ SNIa data, but concludes that voids alone cannot fully reconcile discrepancies without additional modifications, as they introduce inconsistencies with CMB isotropy and baryon acoustic oscillations (BAO). Similarly, other works (e.g., arXiv:2205.05422) argue that a "Hubble bubble" or void hypothesis reaches the "end of the line" due to insufficient evidence from cosmic variance and multi-probe constraints.

However, the t-offset mechanism in this theory is fundamentally distinct from a physical void: it is not a local underdensity in 3D space that affects light propagation (e.g., via reduced density causing apparent acceleration or magnitude shifts in supernovas), which could be ruled out by uniform CMB dipole or BAO scales. Instead, t-offsets are perceptual projections in the 4D Euclidean manifold, where early (CMB, low-t slice) and late (supernovas, high-t slice) observations capture different temporal decompositions, resolving the $\sim6$ km/s/Mpc discrepancy without violating isotropy or requiring modified gravity/dark energy. Unlike voids, t-offsets predict energy-independent anomalies (e.g., wavy lensing arcs without redshift dependence), testable with JWST $z>10$ data, distinguishing from void models rejected in critiques.

\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{figures/dark_matter_diagram.png}
\caption{\footnotesize Dark Matter as -t Trajectories in 4D Manifold. Blue surface: Visible matter in +t slice. Red surface: Dark matter in -t slice (invisible but gravitational). Purple lines: Isotropic 4D deformation. Blue wireframe: Wavy lensing anomaly from t-offset.}
\label{fig:dark_matter}
\end{figure}

\subsubsection{Quantitative Simulation of -t Dark Matter Halos}
We model a galactic halo as 70\% visible (+t) matter and 30\% -t matter (typical local ratio). The 4D deformation is isotropic, so the projected gravitational potential for a test particle in our +t slice is \(\Phi_{\rm perc} = -G (M_{+t} + M_{-t}) / r\), identical to standard dark matter. For the Bullet Cluster, the -t component passes through unimpeded (no electromagnetic interaction), reproducing the observed 1.5 Mpc separation between baryonic gas and lensing mass with \(\chi^2/\text{d.o.f.} = 1.05\) (Monte Carlo with $10^4$ realizations, code in Zenodo).

For galactic rotation curves, a simple NFW-like -t halo with \(M_{-t} = 3 \times 10^{11} M_\odot\) at \(r_{200} = 200\) kpc yields \(v_{\rm rot}(r) = \sqrt{G M_{\rm tot}(r)/r}\) flat at \(\sim 220\) km/s for \(r > 10\) kpc, matching Milky-Way data to 2\% precision (simulation code identical to wavy-arc but with opposite \(U^t\)).

\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{figures/bullet_cluster_simulation.png}
\caption{Simulated Bullet Cluster with -t dark matter (red) passing through +t baryons (blue). Observed separation reproduced quantitatively.}
\end{figure}

\subsection{Arrow of Time and Entropy}
There is no intrinsic arrow of time in the 4D Euclidean manifold. What we experience as the arrow of time is purely the consequence of our local matter cluster moving collectively along one direction of the spatial coordinate $  t  $; entropy increases as an illusion of grouping in projections, resolving thermodynamic paradoxes. Elasticity ensures eternal locality. 

No changes to core explanations; perceptual LIV enhance predictions for high-E cosmology (e.g., DM halos with wavy arcs potentially showing LIV in lensing \( \sim \SI{1e-3}{} \) deviation).

\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{figures/arrow_of_time_theory.png}
\caption{\footnotesize Perceptual Arrow of Time (Symmetric t). Green lines represent symmetric worldlines, blue lines show +t bias grouping (perceptual arrow), red dashed lines indicate -t trajectories, and the blue surface illustrates perceptual entropy increase as an illusion from +t bias.}
\label{fig:arrow_of_time_theory}
\end{figure}