Syncré Symbol and Definition

Syncré is named so the formation stays visible.

• syn- suggests together-in-time.
• cre is from Latin and means to create.
• form names the forced pattern of embeddedness.

Symbol

We denote Syncré by the distinctive symbol

\[
\boxed{\;\mathscr{S}_{\!\mathrm{r}}\;}
\]

read “Syncré,” using a script S with a tight roman subscript \(r\).

Definition (Syncré 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 Syncré 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 Syncré Coverage Law expressed as membership in \(\mathscr{S}_{\!\mathrm{r}}(P)\).

Optional operational form (lift-compatible Syncré)

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 macros

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

Next reading

• Syncré Canonical Equation Reference
• Syncré Form Theory