# Syncre Symbol and Definition

## Symbol

We denote Syncre by the distinctive symbol
\[
\boxed{\;\mathscr{S}_{\!\mathrm{r}}\;}
\]
read “Syncre,” using a script S with a tight roman subscript \(r\).

---

## Definition (Syncre class)

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.

Define Syncre as the class/property:
\[
\boxed{\;
(M,\{\Sigma_s\},\Phi)\in \mathscr{S}_{\!\mathrm{r}}(P)
\quad\Longleftrightarrow\quad
\forall s\in\mathbb{R},\ \Phi(\Sigma_s)=\mathbb{T}_P.
\;}
\]

Equivalently, writing \(\Phi_s:=\Phi|_{\Sigma_s}\),
\[
\boxed{\;
(M,\{\Sigma_s\},\Phi)\in \mathscr{S}_{\!\mathrm{r}}(P)
\quad\Longleftrightarrow\quad
\forall s,\ \Phi_s \text{ is onto } \mathbb{T}_P.
\;}
\]

This is the Syncre Coverage Law expressed as membership in \(\mathscr{S}_{\!\mathrm{r}}(P)\).

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## Optional operational form (lift-compatible Syncre)

If \(\Phi = F \bmod P\) for a real lift \(F:M\to\mathbb{R}\), define the lift-compatible class:
\[
\boxed{\;
(M,\{\Sigma_s\},F)\in \mathscr{S}^{\uparrow}_{\!\mathrm{r}}(P)
\quad\Longleftrightarrow\quad
\forall s,\ (F\bmod P)\big|_{\Sigma_s}\ \text{is onto } \mathbb{T}_P.
\;}
\]

This is the stable interface where nonlinear admissibility constraints and dynamics will later be imposed on \(F\).

---

## LaTeX macro

```tex
% Syncre symbol
\newcommand{\Sr}{\mathscr{S}_{\!\mathrm{r}}}
\newcommand{\SrLift}{\mathscr{S}^{\uparrow}_{\!\mathrm{r}}}
```
