# Embedded Witness Formalism and Obstruction Classification

## Purpose

Many realizations of Syncré are not global on an entire slice; they are observable on a loop subsystem or a controlled submanifold.

This file:

- defines embedded Syncré witnesses precisely
- isolates the two core embedded mechanisms (degree and holonomy)
- states a finite obstruction catalog in the witness-first sense: every failed certificate attempt has a named structural reason

---

## Witness tiers

On a slice \(\Sigma_s\), SFT uses:

- Global witness: \(\Phi(\Sigma_s)=\mathbb{T}_P\)
- Embedded witness: there exists \(E\hookrightarrow\Sigma_s\) with \(\Phi(E)=\mathbb{T}_P\)
- Coupled-target witness: \(\Phi(\Sigma_s)=\mathrm{CST}_s\) in multi-cycle settings

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## Embedded Syncré witness

Let \(\Phi:M\to\mathbb{T}_P\) and fix a slice \(\Sigma=\Sigma_s\).

### Definition (embedded witness)

An embedded Syncré witness is a pair \((E,i)\) where:

- \(E\) is connected
- \(i:E\hookrightarrow\Sigma\) is an embedding
- \(\Phi\circ i(E)=\mathbb{T}_P\)

When \(E\cong S^1\), this is a loop witness.

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## Two core embedded mechanisms

### Loop degree witness

Let \(\gamma:S^1\to\Sigma\) be a loop and set \(f:=\Phi\circ\gamma:S^1\to\mathbb{T}_P\cong S^1\). The integer degree \(\deg(f)\) is defined when \(f\) is continuous.

If
\[
\boxed{\ \deg(\Phi\circ\gamma)\neq 0\ }
\]
then \(f\) is surjective and \(\gamma\) certifies an embedded witness.

### Holonomy witness

If \(\Phi=F\bmod P\) is described by a lift atlas, the 1-form \(\alpha=dF\) is globally defined on atlas overlaps even when \(F\) is not globally single-valued.

For a loop \(\gamma\), define
\[
\mathrm{Hol}(\gamma):=\int_\gamma \alpha.
\]

If
\[
\boxed{\ \mathrm{Hol}(\gamma)=mP\ \text{for some integer }m\neq 0\ }
\]
then \(\gamma\) is an embedded witness.

On \(S^1\), nonzero holonomy matches nonzero degree under the standard identifications.

---

## Obstruction catalog (finite list)

SFT is certificate-driven. A claim is either witnessed by a finite certificate (WSC) or blocked by a finite obstruction object (WOB) with evidence and next moves.

### Slice-level forcing obstructions (circle-action regime)

In the regime where one attempts to force coverage from a cyclic symmetry action and an equivariant observable:

- \(\mathcal{O}_0\) slope zero: the available phase response is trivial
- \(\mathcal{O}_2\) orbit collapse: the claimed forcing domain contains no circle-type orbit on which an equivariant response can act (for example, the symmetry is not realized on the domain)
- \(\mathcal{O}_3\) equivariance defect: the observable is not genuinely equivariant in the adopted model class (or defect exceeds admissible tolerance)
- \(\mathcal{O}_6\) disconnected slice topology: the orbit-bearing or loop-bearing component does not intersect the measurement domain

### Lift and holonomy obstructions

When the model uses lift-atlas or connection language:

- \(\mathcal{O}_4\) holonomy obstruction: no consistent lift/connection class exists in the intended sense, so holonomy cannot certify or is not stable

### Multi-cycle obstructions

In \(\mathbb{T}^k\) regimes:

- \(\mathcal{O}_5\) resonant locking: rank deficiency in frequency/slope collapses the intended torus target
- \(\mathcal{O}_{5'}\) coupling mismatch: the wrong target torus is used (full torus Haar when a CST is the correct target)

---

## Obstruction classification theorem (witness-first)

### Theorem (certificate-or-obstruction dichotomy)

Fix an admissible SFT model class (regularity, domain conventions, coupling conventions).

On a slice \(\Sigma_s\), exactly one of the following holds:

- a witness certificate exists proving the correct target coverage (global, embedded, or coupled-target)
- a WOB obstruction exists from the finite catalog above

Equivalently,
\[
\boxed{\ \neg\mathrm{SCL}^{\mathrm{SFT}}(\Sigma_s)\ \Longleftrightarrow\ \text{some obstruction }\mathcal{O}_i\ \text{occurs.}\ }
\]

This is the precise sense in which failures are classified: a failed coverage claim in the declared model class always reduces to a named structural obstruction.

---

## Certificate format

A witness is most useful when it is auditable. SFT packages witnesses as finite artifacts.

- Witness Stability Certificate (WSC)
  - tier (global, embedded, coupled)
  - mechanism (action, degree, holonomy, coupled target)
  - integer witness value (degree or holonomy multiple)
  - margins ensuring stability under perturbations

- Witness Obstruction (WOB)
  - obstruction code
  - evidence
  - recommended next moves

The obstruction classes above are the mathematical meaning behind WOB codes.
