# Syncre Canonical Equation Reference

## Canonical objects

Fix a period \(P>0\) and the phase circle \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\).

Let \(M\) be an arena equipped with a slicing by global instants
\[
M=\bigsqcup_{s\in\mathbb{R}} \Sigma_s,
\]
and let \(\Phi:M\to\mathbb{T}_P\) be a phase field.

For each slice, write the restriction
\[
\Phi_s := \Phi|_{\Sigma_s}:\Sigma_s\to\mathbb{T}_P.
\]

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## Canonical Syncre equation (SCL — Syncre Coverage Law)

\[
\boxed{\;\forall s\in\mathbb{R},\qquad \Phi_s(\Sigma_s)=\mathbb{T}_P.\;}
\]

This is the stable “one-line” Syncre identity: every global instant contains a full cycle of phase labels across space.

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## Canonical operational form (SLL — Syncre Lift Law)

A lift form is used to support nonlinear deformation while keeping analysis in \(\mathbb{R}\):

\[
\boxed{\;\Phi = F \bmod P\qquad\text{for some }F:M\to\mathbb{R}.\;}
\]

Then the canonical equation is:

\[
\boxed{\;\forall s,\qquad (F\bmod P)\big|_{\Sigma_s}\ \text{is onto }\mathbb{T}_P.\;}
\]

This is the interface where future constraints will be imposed on \(F\) (monotonicity along a chosen future direction, bounded rates, holonomy consistency, dynamics, and multi-cycle torus coupling).

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## Earth chart (reference example)

On a longitude circle \(S^1\) with coordinate \(\lambda\), a canonical lift is:

\[
F(t,\lambda)=t+\frac{P}{2\pi}\lambda,
\qquad
\Phi(t,\lambda)=F(t,\lambda)\bmod P.
\]

For fixed \(t\), the map \(\lambda\mapsto \Phi(t,\lambda)\) wraps the phase circle once and therefore satisfies SCL.
