# Pass C — Syncre Frame Conditions, New Invariants, Obstructions, and a Canonical Nonlinear Class

## Purpose

Elevate Syncre from a topological rephrasing into a theory with:

- explicit conditions on what “global instants” are allowed to mean
- invariants that are not just degree
- a sharp obstruction taxonomy (how coverage can fail)
- a canonical nonlinear deformation class with a quantitative coverage theorem

This pass also introduces new program language terms that will be reused in later passes.

---

## Program language: core terms introduced here

| Term | Tag | Meaning |
|---|---|---|
| Syncre Instant System | **SIS** | A slicing \(\{\Sigma_s\}\) of an arena \(M\) defining “global instants” |
| Admissible Slicing | **ADS** | A slicing meeting regularity and completeness conditions |
| Frame Equivalence | **FEQ** | When two slicings yield the same Syncre invariants and witness structure |
| Syncre Coverage Modulus | **SCM** | A quantitative radius \(\rho\) measuring how strongly coverage holds |
| Syncre Uniformity Defect | **SUD** | A metric distance of the phase distribution from Haar (uniform) |
| Syncre Distortion Functional | **SDF** | A time-indexed measure of how phase distributions evolve |
| Holonomy Class | **HOL** | Loop-integrality data for lifts \(F\) with \(\Phi=F\bmod P\) |
| Holonomy Margin | **HMR** | Quantitative buffer preventing holonomy failure under perturbation |
| Obstruction Taxonomy | **OBT** | Named failure modes: Pinch, Collapse, Tear, Holonomy Defect |
| Syncre Admissible Deformation Class | **SADC** | A structured nonlinear class \(F=\psi+k\lambda+u\) with constraints |

---

## Syncre Instant System and Admissible Slicing

Let \(M\) be an arena and \(\mathbb{T}_P=\mathbb{R}/P\mathbb{Z}\) a phase circle.

### Definition (Syncre Instant System, SIS)

A **Syncre Instant System** on \(M\) is a family of disjoint subsets \(\{\Sigma_s\}_{s\in\mathbb{R}}\) such that
\[
M=\bigsqcup_{s\in\mathbb{R}} \Sigma_s.
\]

This is the bare mathematical encoding of “global instants.”

### Definition (Admissible Slicing, ADS)

A Syncre Instant System \(\{\Sigma_s\}\) is an **Admissible Slicing** if it satisfies:

- **ADS-Regularity:** the map \((s,x)\mapsto x\in \Sigma_s\) can be represented by a continuous (or smooth) parametrization in the chosen model of \(M\)
- **ADS-Ordering:** there is a consistent ordering parameter \(s\) so that “later” slices are labeled by larger \(s\)
- **ADS-Completeness:** every point of \(M\) lies on exactly one slice, and slices are nonempty over the time range of interest

This keeps the slicing from being arbitrary and ensures Syncre statements are comparable across models.

### Program language note

If a later development depends on the slicing being smooth, we will say **ADS-\(C^k\)**.

---

## Frame equivalence and invariant survival

Different slicings can represent different notions of “global moment.” For Syncre to be meaningful, we need a way to compare them.

### Definition (Frame Equivalence, FEQ)

Two admissible slicings \(\{\Sigma_s\}\) and \(\{\widetilde\Sigma_{s}\}\) on the same arena \(M\) are **frame-equivalent** if there exists:

- a reparameterization \(r:\mathbb{R}\to\mathbb{R}\) that is strictly increasing
- a bijection \(\Psi:M\to M\) that maps slices to slices:
  \[
  \Psi(\Sigma_s)=\widetilde\Sigma_{r(s)}
  \]
- and for the Syncre field \(\Phi:M\to\mathbb{T}_P\), the pullback respects phase up to a global gauge shift:
  \[
  \Phi\circ \Psi^{-1} = \Phi \oplus c
  \quad\text{for some constant }c\in\mathbb{T}_P
  \]

Under FEQ, any slice-wise coverage property is preserved, and degree/holonomy invariants computed on embedded loops are stable.

### Invariant survival table under FEQ

| Quantity | Survives FEQ | Reason |
|---|---|---|
| Coverage \(\Phi(\Sigma_s)=\mathbb{T}_P\) | Yes | slice bijection and gauge shift preserve surjectivity |
| Degree on loop witnesses | Yes | loop images transported by \(\Psi\), degree is homotopy invariant |
| Holonomy class (loop integrals mod \(P\)) | Yes | gauge shifts cancel on closed loops |
| Pushforward distribution \(\mu_s\) | Yes, up to reindexing | slices relabeled by \(r(s)\) |

---

## New invariants beyond degree

We introduce invariants that can be measured and used in stability and convergence claims.

Let \((\Sigma,\nu)\) be a measured slice and \(\Phi:\Sigma\to\mathbb{T}_P\) a phase field. Define the pushforward phase distribution:
\[
\mu := \Phi_\#\nu.
\]

### Definition (Syncre Coverage Modulus, SCM)

Define the **Syncre Coverage Modulus** \(\rho(\Phi;\Sigma,\nu)\) as the supremum of radii \(\rho\ge 0\) such that for every \(y\in\mathbb{T}_P\),
\[
\mu(B_\rho(y))>0,
\]
where \(B_\rho(y)\) is the arc-neighborhood of radius \(\rho\).

- \(\rho>0\) is a strong quantitative coverage statement.
- \(\rho=0\) is consistent with coverage but allows extreme concentration.

### Definition (Syncre Uniformity Defect, SUD)

Let \(\mu_{\mathrm{H}}\) be Haar (uniform) measure on \(\mathbb{T}_P\).  
Choose a metric \(d\) on probability measures on \(\mathbb{T}_P\) (Wasserstein-1 or total variation for absolutely continuous cases). Define
\[
\mathrm{UD}(\Phi;\Sigma,\nu):= d(\mu,\mu_{\mathrm{H}}).
\]

A small uniformity defect means “all phases are present and represented evenly.”

### Definition (Syncre Distortion Functional, SDF)

For a time-indexed family \(\Phi_s:\Sigma_s\to\mathbb{T}_P\) with measures \(\nu_s\), define
\[
\mathrm{DF}(s):=\mathrm{UD}(\Phi_s;\Sigma_s,\nu_s).
\]

This is a single scalar observable capturing how distributional phase structure evolves over the slicing parameter.

---

## Holonomy language for lifts

To deform time nonlinearly, we often pass to a lift \(F\) with \(\Phi = F \bmod P\). Holonomy measures loop-consistency.

### Definition (Holonomy Class, HOL)

Let \(\Sigma\) be a manifold and \(F:\Sigma\to\mathbb{R}\) a lift. For a closed loop \(\gamma:S^1\to\Sigma\), define the holonomy increment
\[
\mathrm{Hol}_F(\gamma) := F(\gamma(1)) - F(\gamma(0)).
\]
When \(\gamma\) is closed, this is defined by choosing a lift along the loop, and it is meaningful modulo \(P\) for \(\Phi=F\bmod P\).

The **holonomy class** condition for Syncre consistency is:
\[
\mathrm{Hol}_F(\gamma)\in P\mathbb{Z}
\quad\text{for every closed loop }\gamma.
\]

### Definition (Holonomy Margin, HMR)

In applications, one often has a specific set of loops \(\Gamma\) that matter (fundamental cycles, longitude loops, or waveguide loops). Define the holonomy margin as:
\[
\mathrm{HM}_F(\Gamma):=\inf_{\gamma\in\Gamma}\ \mathrm{dist}\big(\mathrm{Hol}_F(\gamma),\,P\mathbb{Z}\big).
\]
A positive margin gives a quantitative buffer against small lift perturbations causing a holonomy defect.

---

## Obstruction taxonomy (how coverage fails)

We name failure modes to support classification, diagnosis, and robustness.

### Obstruction (Pinch)

**Pinch** occurs when \(\Phi(\Sigma)\) misses an open arc in \(\mathbb{T}_P\). Equivalently, coverage fails and the image is contained in a proper closed arc.

### Obstruction (Collapse)

**Collapse** occurs when \(\Phi\) is surjective but the pushforward \(\mu=\Phi_\#\nu\) concentrates most mass in a small arc, forcing a tiny coverage modulus \(\rho\) and large uniformity defect.

This is the typical obstruction to “even presence” even when bare coverage holds.

### Obstruction (Tear)

**Tear** occurs when regularity fails in a way that breaks continuity of the phase field or its lift consistency. A tear can destroy degree and invalidate loop-witness arguments.

### Obstruction (Holonomy Defect)

**Holonomy Defect** occurs when a lift \(F\) exists locally but fails global consistency:
\[
\exists\gamma\text{ closed with }\mathrm{Hol}_F(\gamma)\notin P\mathbb{Z}.
\]

Holonomy defects are the natural “global obstructions” in a nonlinear time theory.

---

## Canonical nonlinear deformation class on \(S^1\)

We now pick a canonical class that is broad enough to express nonlinear time but structured enough for theorems.

Let \(\lambda\in S^1\cong \mathbb{R}/2\pi\mathbb{Z}\). Let \(\psi:\mathbb{R}\to\mathbb{R}\) be a global time warp, \(k\in\mathbb{Z}\) a winding parameter, and \(u:\mathbb{R}\times S^1\to\mathbb{R}\) a bounded correction.

### Definition (Syncre Admissible Deformation Class, SADC)

A lift \(F:\mathbb{R}\times S^1\to \mathbb{R}\) lies in **SADC** if it has the form
\[
\boxed{\;F(t,\lambda)=\psi(t) + \frac{kP}{2\pi}\lambda + u(t,\lambda)\;}
\]
with constraints:

- **SADC-Periodicity:** \(u(t,\lambda+2\pi)=u(t,\lambda)\)
- **SADC-Time-Monotone:** \(\partial_tF(t,\lambda)>0\) for all \((t,\lambda)\) where derivatives exist, or an equivalent monotone condition in the continuous setting
- **SADC-Bounded-Rate:** there exist \(0<m\le M\) with \(m\le \partial_tF\le M\) (when differentiable)
- **SADC-Spatial-Regularity:** \(u(t,\cdot)\) is continuous in \(\lambda\) and optionally Lipschitz with constant \(L\)

Define the phase field \(\Phi(t,\lambda)=F(t,\lambda)\bmod P\in\mathbb{T}_P\).

---

## Theorem: coverage is guaranteed by the winding parameter

### Theorem (SADC coverage by degree)

Fix \(t\). Assume \(F(t,\cdot)\) is continuous and satisfies
\[
F(t,\lambda+2\pi)=F(t,\lambda)+kP
\]
for some integer \(k\neq 0\). Then the circle map
\[
\Phi_t(\lambda):=\Phi(t,\lambda)=F(t,\lambda)\bmod P
\]
is surjective onto \(\mathbb{T}_P\). Hence \(\Phi(t,S^1)=\mathbb{T}_P\).

### Proof

Define a lift map \(g(\lambda)=\frac{2\pi}{P}F(t,\lambda)\). Then
\[
g(\lambda+2\pi)=g(\lambda)+2\pi k.
\]
The induced circle map \(e^{i\lambda}\mapsto e^{ig(\lambda)}\) has degree \(k\neq 0\), hence is surjective. Translating back under the identification \(\mathbb{T}_P\cong S^1\) gives \(\Phi_t(S^1)=\mathbb{T}_P\). ∎

This theorem is the canonical “coverage certificate” for the nonlinear class: the complicated term \(u\) cannot destroy coverage as long as the winding data \(k\neq 0\) remains intact.

---

## Quantitative theorem: lower bounds on phase distribution from spatial stretch

Coverage is qualitative. To enable convergence and “evenness” analysis, we introduce a quantitative condition.

Assume \(F(t,\cdot)\) is \(C^1\) in \(\lambda\) and satisfies a spatial stretch bound:
\[
0<a\le \partial_\lambda F(t,\lambda)\le b
\quad\text{for all }\lambda.
\]
This makes \(\Phi_t\) locally monotone along each wrap.

Let \(\nu\) be normalized Lebesgue measure on \(S^1\), so \(\nu(S^1)=1\), and let \(\mu_t=(\Phi_t)_\#\nu\).

### Theorem (Stretch bound yields a uniform lower mass bound)

Assume \(\Phi_t\) has winding number \(k\neq 0\) and the stretch bound \(0<a\le \partial_\lambda F\le b\) holds. Then for any arc \(I\subseteq\mathbb{T}_P\) of length \(|I|\) (measured in \(\mathbb{R}/P\mathbb{Z}\)),
\[
\boxed{\;\mu_t(I)\ \ge\ \frac{|k|}{2\pi b}\,|I|.\;}
\]

Consequently, the Syncre coverage modulus satisfies a positive lower bound depending on \(b\) and \(k\), and collapse is quantitatively prevented.

### Proof

On each wrap branch, preimages of an arc of length \(|I|\) have \(\lambda\)-length at least \(|I|/b\) by the upper derivative bound. Since the map wraps \(|k|\) times, the total preimage length is at least \(|k||I|/b\). Normalizing by the total circle length \(2\pi\) yields
\[
\mu_t(I)=\nu(\Phi_t^{-1}(I))\ge \frac{|k||I|/b}{2\pi}=\frac{|k|}{2\pi b}|I|.
\]
∎

This is a first quantitative bridge from topology to analysis, phrased in Syncre language.

---

## Acceptance gates

- SIS, ADS, and FEQ are defined and linked to invariant survival
- SCM, SUD, and SDF are defined in terms of pushforward measures
- HOL and HMR are defined and identified as nonlinear-time consistency data
- OBT includes Pinch, Collapse, Tear, and Holonomy Defect with precise criteria
- SADC is specified and a degree-based coverage theorem is proved
- A quantitative stretch-based bound is proved to prevent collapse


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## Syncre symbol

See `SYNCRE_SYMBOL_DEFINITION.md` for the canonical Syncre symbol \(\mathscr{S}_{\!\mathrm{r}}\) and its formal definition.
