# Syncré Form Theory (SFT): Core Principles and Stable Notation

## Project name

**Syncré Form Theory (SFT)**

SFT starts with Syncré as a Principle-Identical Proxy (PIP) for a higher Relational Form Principle (RFP), then strengthens the proxy using invariants, obstructions, diagnostics, coupling laws, and dynamics.

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## Core principles

- **RFP — Relational Form Principle**
  Form is an invariant of embeddedness: points are studied as \((M,p)\), not as isolated \(p\).

- **PIP — Principle-Identical Proxy**
  A minimal observable structure that preserves the invariant/gauge logic of a higher principle.

- **PIP statement (SFT foundation)**
  Syncré is the PIP of RFP: it is the minimal observable phase-field framework retaining the embeddedness-first, gauge-stable, invariant-first logic of the higher relational claim.

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## Canonical Syncré laws (SFT spine)

Fix \(P>0\), \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\).

- **SCL — Syncré Coverage Law**

  For a sliced arena \(M=\bigsqcup_{s\in\mathbb{R}}\Sigma_s\) and phase field \(\Phi:M\to\mathbb{T}_P\),
  \[
  \forall s,\quad \Phi(\Sigma_s)=\mathbb{T}_P.
  \]

- **SLL — Syncré Lift Law**

  \[
  \Phi = F \bmod P
  \]
  for some real lift \(F\), enabling nonlinear deformation while keeping analysis in \(\mathbb{R}\).

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## Syncré symbol

- \(\mathscr{S}_{\!\mathrm{r}}(P)\): the class of frames satisfying SCL
- \(\mathscr{S}^{\uparrow}_{\!\mathrm{r}}(P)\): the lift-compatible class satisfying SLL + SCL

See `SYNCRE_SYMBOL_DEFINITION.md`.

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## Cyclic Observability Class (COC)

SFT uses a named maximal applicability class for universality statements.

- **Phase observable (PO):** a declared cycle-valued observable on the intended slice/subsystem
- **Witness mechanism (WM):** a declared admissible forcing mechanism (symmetry, loop/holonomy, defect loop, embedded subsystem witness, etc.)
- **Target rule (TR):** the correct target type and baseline convention (Haar on the target; coupled Haar on CST when coupling is present)

Global universality means: universal on COC.
