# Start here

Syncré Form Theory (SFT) is a target-correct way to state and verify **phase coverage** on a declared slice or subsystem.

A phase-coverage claim is meaningful in SFT only after the claim specifies:

- a **phase observable** (circle/torus-valued),
- a **slice or subsystem** where the phase is observed,
- the **target** on which coverage is asserted (circle, torus, or a coupled subtorus coset),
- a **witness mechanism** that can force coverage in the declared model class,
- a **perturbation model** that fixes what “stable under perturbation” means.

Within the declared applicability class (COC), the engine resolves any declared claim into exactly one audited object:

- **WSC**: a Witness Stability Certificate proving coverage on the correct target, with explicit robustness margins
- **WOB**: a Witness Obstruction naming the first failed validator predicate in a finite codebook, with evidence pointers

## Core vocabulary

| Term | Meaning in SFT |
|---|---|
| Phase | A value in a circle Τₚ = ℝ/Pℝ (or a torus of periods) |
| Slice | The set Σₛ representing a single global instant in a slicing M = ⋃ₛ Σₛ |
| Coverage | The image of the phase on the slice equals the full target set |
| Witness | A finite mechanism that forces coverage (action/degree/holonomy/coupled target) |
| Coupling | Constraints among multiple phases that reduce the correct target |
| Target | The set where coverage and baselines are asserted (circle, torus, or CST coset) |

## Witness mechanisms in this bundle

- **Action witness (AWT):** an equivariant circle action forces surjectivity of the observable on the target.
- **Loop degree witness (LDW):** a nonzero degree of a loop map S¹→S¹ forces coverage on the circle target along an embedded loop.
- **Holonomy witness (HW):** a nontrivial holonomy multiple forces coverage when the phase is defined via a lift atlas or connection.
- **Coupled-target witness (CMT):** coupling identifies a coupled subtorus coset (CST), and forcing is asserted on that CST rather than the ambient product torus.

## What to read next

- Contract and predicates: `engine/contracts/engine_contract.md`, `engine/contracts/VALIDATOR_PREDICATE_SPEC.md`
- Exhaustiveness of WSC/WOB outcomes inside COC: `ENGINE_COMPLETENESS.md`
- Sharp boundary examples (wrong target, missing witness, missing slice): `BOUNDARY_GALLERY.md`
- The main paper: `finish_ready/sft_form_theory.pdf`
- Target inference and certificate construction from data: `finish_ready/sft3_target_inference_and_certificates.pdf`
- Witness atlas tied to shipped artifacts: `finish_ready/witness_atlas.pdf`
