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\title{Solar--Lunar Local Time and Convergence on a Rotating Globe}
\author{}
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\begin{document}
\maketitle

\section*{Purpose}
We formalize:
\begin{itemize}
\item Local time-of-day as a positional phase field on a globe.
\item ``At one global instant, every local time exists somewhere'' as a surjectivity claim.
\item A solar-cycle--aware refinement (annual modulation) and a lunar-cycle companion.
\item A matching identity: the solar--lunar local-time difference equals a global lunar-cycle phase.
\item A convergence-ready formulation using torus dynamics and a daily stroboscopic map.
\end{itemize}

\section*{Core setting}
Let longitudes be the circle $S^1=\mathbb{R}/2\pi\mathbb{Z}$ with longitude $\lambda\in S^1$ (east-positive).
Let $t\in\mathbb{R}$ be global reference time.

\section*{Earth rotation and direction angles}
Let $\Theta(t)\in S^1$ be the Earth rotation angle in an inertial coordinate. Let $\alpha_\odot(t)\in S^1$ be the Sun direction angle and $\alpha_{\leftmoon}(t)\in S^1$ the Moon direction angle in the same coordinate.

\begin{definition}[Local solar phase]
Define the local solar phase at longitude $\lambda$ by
\[
\theta_\odot(t,\lambda)=\big(\Theta(t)+\lambda-\alpha_\odot(t)\big)\bmod 2\pi\in S^1.
\]
A clock-seconds version is $\tau_\odot(t,\lambda)=\frac{86400}{2\pi}\theta_\odot(t,\lambda)\in \mathbb{R}/86400\mathbb{Z}$.
\end{definition}

\begin{definition}[Local lunar phase]
Define the local lunar phase at longitude $\lambda$ by
\[
\theta_{\leftmoon}(t,\lambda)=\big(\Theta(t)+\lambda-\alpha_{\leftmoon}(t)\big)\bmod 2\pi\in S^1.
\]
A clock-seconds version is $\tau_{\leftmoon}(t,\lambda)=\frac{86400}{2\pi}\theta_{\leftmoon}(t,\lambda)\in \mathbb{R}/86400\mathbb{Z}$.
\end{definition}

\begin{lemma}[All local times exist at once]
Fix $t=t_0$. Then
\[
\{\theta_\odot(t_0,\lambda):\lambda\in S^1\}=S^1,
\qquad
\{\theta_{\leftmoon}(t_0,\lambda):\lambda\in S^1\}=S^1.
\]
\end{lemma}

\begin{proof}
For fixed $t_0$, $\Theta(t_0)-\alpha_\odot(t_0)$ is constant, so $\theta_\odot(t_0,\lambda)=(\mathrm{constant}+\lambda)\bmod 2\pi$, which is onto $S^1$. Identical for $\theta_{\leftmoon}$.
\end{proof}

\begin{definition}[Lunar-cycle (synodic) phase]
Define
\[
\phi(t)=\big(\alpha_{\leftmoon}(t)-\alpha_\odot(t)\big)\bmod 2\pi\in S^1.
\]
\end{definition}

\begin{lemma}[Solar--lunar matching identity]
For all $t$ and $\lambda$,
\[
\theta_\odot(t,\lambda)-\theta_{\leftmoon}(t,\lambda)\equiv \phi(t)\pmod{2\pi},
\]
and the right-hand side is independent of $\lambda$.
\end{lemma}

\begin{proof}
Compute:
\[
\theta_\odot-\theta_{\leftmoon}\equiv (\Theta+\lambda-\alpha_\odot)-(\Theta+\lambda-\alpha_{\leftmoon})
\equiv \alpha_{\leftmoon}-\alpha_\odot\equiv \phi(t)\pmod{2\pi}.
\]
\end{proof}

\section*{Solar-cycle and lunar-cycle modulation}
A convenient decomposition is
\[
\alpha_\odot(t)=\omega_\odot t + A_\odot(\Omega t),\quad \Omega=\frac{2\pi}{T_{\text{yr}}},\quad A_\odot\ \text{$2\pi$-periodic},
\]
\[
\alpha_{\leftmoon}(t)=\omega_{\leftmoon} t + A_{\leftmoon}(\Lambda t),\quad \Lambda=\frac{2\pi}{T_{\leftmoon}},\quad A_{\leftmoon}\ \text{$2\pi$-periodic}.
\]
Then
\[
\phi(t)=\big((\omega_{\leftmoon}-\omega_\odot)t + A_{\leftmoon}(\Lambda t)-A_\odot(\Omega t)\big)\bmod 2\pi.
\]

\section*{Torus dynamics and convergence}
Define annual and lunar phases $u(t)=\Omega t\bmod 2\pi$ and $v(t)=\Lambda t\bmod 2\pi$.
At fixed longitude $\lambda$, the pair $X(t)=(\theta_\odot(t,\lambda),\phi(t))\in S^1\times S^1$ is a torus trajectory (or higher-dimensional torus if additional phases are retained).

A natural convergence notion is time-average convergence: for continuous $F:S^1\times S^1\to\mathbb{R}$,
\[
\frac{1}{T}\int_0^T F(X(t))\,dt.
\]
In the ideal mean-motion model (no periodic corrections), this is a linear flow on a torus; if the frequency vector is rationally independent, time averages converge to the Haar torus average.

\section*{Daily stroboscopic map}
Sampling once per mean solar day $D=86400$ seconds gives $\phi_n=\phi(t_0+nD)$.
In the ideal model,
\[
\phi_{n+1}=\phi_n+\rho \pmod{2\pi},\qquad \rho=(\omega_{\leftmoon}-\omega_\odot)D.
\]
If $\rho/2\pi$ is irrational, $\{\phi_n\}$ is equidistributed on $S^1$ and discrete time-averages converge to the uniform circle average.



\section*{Syncre at massive scale: a global phase field, Earth as a pullback}

The Earth-local picture can be understood as a restriction of a more general structure.

\subsection*{Syncre field on a large arena}
Let $M$ be an ambient arena (e.g.\ a spacetime manifold or a configuration/state manifold). Fix a period $P>0$ and phase circle $\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}$.
A \emph{Syncre field} is a phase-valued map
\[
\Phi:M\to\mathbb{T}_P.
\]

\subsection*{Global instants as slices}
To represent a ``global moment'' at massive scale, specify a slicing (foliation)
\[
M=\bigsqcup_{s\in\mathbb{R}}\Sigma_s,
\]
where each $\Sigma_s$ represents ``everything simultaneous'' in the chosen Syncre frame.

\subsection*{Syncre coverage axiom (big-picture form)}
The statement ``all times are present at once'' becomes the slice-level axiom
\[
\forall s\in\mathbb{R},\quad \Phi(\Sigma_s)=\mathbb{T}_P,
\]
i.e.\ each restriction $\Phi|_{\Sigma_s}:\Sigma_s\to\mathbb{T}_P$ is surjective.

\subsection*{Earth as a pullback (restriction)}
At each slice $\Sigma_s$, represent the Earth (or any local subsystem) as an embedding
\[
i_s:\mathcal{E}\hookrightarrow \Sigma_s,
\qquad \mathcal{E}\approx S^2\ \text{(Earth surface), or a longitude subcircle }S^1.
\]
The induced Earth-local time field is the pullback
\[
\tau_s=\Phi\circ i_s:\mathcal{E}\to\mathbb{T}_P.
\]
Thus the Earth model is a special case obtained by restricting the global phase field.

\subsection*{Multi-cycle and nonlinear extensions}
For multiple coupled cycles, replace $\mathbb{T}_P$ by a torus $\prod_j \mathbb{T}_{P_j}$ and take $\Phi:M\to \prod_j \mathbb{T}_{P_j}$.
To study nonlinear time, use a lift $F:M\to\mathbb{R}$ with $\Phi=F\bmod P$ and impose structural constraints (monotone progression along chosen future directions, bounded rates, regularity, and holonomy consistency).

\subsection*{Syncre frame (minimal data)}
A compact package for the massive-scale structure is the triple
\[
\big(M,\{\Sigma_s\}_{s\in\mathbb{R}},\Phi\big),
\]
where $\{\Sigma_s\}$ defines what ``global instant'' means and $\Phi$ is the phase field whose slice restrictions cover the full cycle.


\section*{Proof program and universe-scale witnesses}
For a proof-program framing (Syncre frames, witness theorems, and universe-scale witness constructions), see \texttt{Syncre\_ProofProgram.md} in the accompanying bundle.

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