# Pass H — Syncre Dynamics (Fluid Form) and Preservation Results

## Purpose

Introduce a minimal “fluid” dynamics for Syncre lifts and phase fields, then prove preservation results that are specifically Syncre-relevant:

- preservation of winding/degree on \(S^1\)
- preservation of holonomy class on general \(\Sigma\) under natural dynamics
- sufficient conditions under which coverage and coverage modulus persist

This is the first pass where Syncre behaves like a field theory rather than a static labeling.

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## Program language introduced here

| Term | Tag | Meaning |
|---|---|---|
| Syncre Transport Dynamics | **STD** | Evolution law \(\partial_s F + v\cdot\nabla F = \omega\) |
| Admissible Dynamics Class | **ADC** | Conditions on \((v,\omega)\) ensuring well-posedness and admissibility preservation |
| Degree Preservation | **DPR** | Winding \(k\) is invariant under dynamics |
| Holonomy Preservation | **HPR** | Loop integrals remain in \(P\mathbb{Z}\) under dynamics |
| Coverage Preservation | **CPR** | SCL persists under dynamics |
| Modulus Preservation | **MPR** | SCM lower bounds persist or degrade controllably |

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## Syncre transport dynamics on lifts

Let the arena be \(M=\mathbb{R}\times \Sigma\) with slicing \(\Sigma_s=\{s\}\times\Sigma\). Fix period \(P>0\).

Let \(F(s,x)\) be a real lift and define \(\Phi(s,x)=F(s,x)\bmod P\).

### Definition (Syncre Transport Dynamics, STD)

We say \(F\) evolves by **STD** if
\[
\boxed{\;\partial_s F(s,x) + v(s,x)\cdot\nabla_x F(s,x) = \omega(s,x)\;}
\]
where

- \(v(s,x)\) is a vector field on \(\Sigma\) (a “flow” on space)
- \(\omega(s,x)\) is a source term (“phase production rate”)

This is a standard transport equation (advection + forcing) applied to the Syncre lift.

### Admissible dynamics class (ADC)

We call \((v,\omega)\) **admissible** if:

- \(v\) is regular enough to generate a flow map \(\varphi_{s_0\to s}\) on \(\Sigma\) (Lipschitz in space suffices for classical well-posedness)
- \(\omega\) is bounded and measurable (or smooth) as needed
- boundary/periodic conditions are compatible with \(\Sigma\)

These conditions are chosen to preserve the lift structure in later proofs.

---

## Evolution along characteristics

Let \(\varphi\) be the flow generated by \(v\):
\[
\frac{d}{ds}\varphi(s;x_0)=v(s,\varphi(s;x_0)),\qquad \varphi(s_0;x_0)=x_0.
\]

Then along a characteristic curve \(x(s)=\varphi(s;x_0)\),
\[
\frac{d}{ds}F(s,x(s))=\omega(s,x(s)).
\]

So \(F\) is transported by the flow and accumulated by \(\omega\).

This formula is the basis for Syncre preservation results.

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## Degree preservation on \(S^1\)

Let \(\Sigma=S^1\) with coordinate \(\lambda\) and periodic boundary conditions. Suppose \(F\) satisfies the winding relation:
\[
F(s,\lambda+2\pi)=F(s,\lambda)+kP
\]
for some integer \(k\).

### Theorem (DPR — degree is preserved under STD)

Assume:

- \(\Sigma=S^1\) with periodic \(v\) and \(\omega\) in \(\lambda\)
- \(F\) solves STD
- \(F(s,\lambda)\) is continuous in \(\lambda\) and the winding relation holds at \(s=s_0\)

Then the same winding relation holds for all \(s\), with the same integer \(k\). In particular, degree/winding is preserved.

#### Proof sketch

Define \(G(s,\lambda)=F(s,\lambda+2\pi)-F(s,\lambda)\). Using periodicity of \(v,\omega\) in \(\lambda\), the equation for \(G\) is homogeneous transport with zero forcing, so \(G\) is constant along characteristics. Since \(G(s_0,\lambda)\equiv kP\), it remains \(kP\) for all \(s\). ∎

### Corollary (Coverage preservation on \(S^1\))

If \(k\neq 0\), then for each \(s\) the slice map \(\Phi_s(\lambda)=F(s,\lambda)\bmod P\) is surjective, hence SCL holds on every slice.

This is **CPR** in the canonical 1D setting.

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## Holonomy preservation on general \(\Sigma\)

Now let \(\Sigma\) be a manifold and assume the lift atlas framework of Pass D. The connection form is \(\alpha=dF\) (locally) and holonomy is \(\int_\gamma \alpha\).

### Theorem (HPR — holonomy class preserved under transport)

Assume:

- \(\Sigma\) has no boundary (or boundary conditions compatible with the flow)
- \(v\) generates a flow of diffeomorphisms \(\varphi\)
- \(F\) evolves by STD with sufficiently smooth data so that \(\alpha=dF\) exists
- \(\omega\) is a globally single-valued function on \(\Sigma\) (not multi-valued modulo \(P\))

Then for any closed loop \(\gamma\) transported by the flow, the holonomy increment changes by the loop integral of \(d\omega\) along the transported loop. In particular, if \(d\omega=0\) (for example \(\omega=\omega(s)\) depends only on slice parameter), then holonomy is preserved.

#### Proof idea

Differentiate \(\int_{\gamma_s}\alpha\) with respect to \(s\) using the transport equation and Cartan’s formula for Lie derivatives. Under \(d\omega=0\), the derivative vanishes, giving invariance. ∎

### Syncre consequence

Holonomy preservation means that the SOC class (Pass D) is stable under admissible dynamics, which is essential if you want nonlinear time deformations that do not destroy global consistency.

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## Coverage modulus preservation (a first usable sufficient condition)

On \(S^1\), Pass C gave a quantitative lower bound on mass of arcs assuming a stretch bound on \(\partial_\lambda F\).

Under STD, such a stretch bound is preserved if the spatial derivative evolves without degeneration.

### Proposition (MPR under uniform stretch control)

Assume \(\Sigma=S^1\) and:

- \(F\) is \(C^1\) in \(\lambda\)
- there exist constants \(0<a\le b\) such that \(a\le \partial_\lambda F(s_0,\lambda)\le b\)
- along the evolution, \(\partial_\lambda F\) remains bounded above by \(b'\) (no gradient blow-up)

Then the lower mass bound from Pass C holds at later times with \(b\) replaced by \(b'\), giving an explicit lower bound on SCM.

This converts dynamics control (prevent gradient blow-up) into a quantitative “no collapse” guarantee.

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## Acceptance gates

- STD is stated with clear roles for \(v\) and \(\omega\)
- A DPR theorem is stated and proved at the level needed to preserve winding \(k\)
- HPR is formulated linking holonomy to \(d\omega\) and flow transport
- CPR is stated as a corollary of DPR when \(k\neq 0\)
- A first MPR proposition links stretch bounds to SCM lower bounds under dynamics control
