# Global Syncré Universality Theorem Package

## Purpose

Syncré Form Theory (SFT) is a witness-first theory of phase coverage.

The Syncré Coverage Law (SCL) is the slice-wise statement that, at each global instant, the phase field attains every phase value somewhere on the slice.

This package upgrades SCL from a catalog of examples into a theorem chain:

- cyclic symmetry + equivariance forces coverage
- the witness engines (action, degree, holonomy, coupled target) have a finite obstruction catalog
- forcing is quantitatively stable under admissible perturbations
- in multi-cycle settings, the correct target is a coupled subtorus (CST) and forcing persists on that target
- in nonresonant drives, time averages converge to Haar on the correct target
- under admissible dynamics, witnesses persist in time
- under diffusion/noise, distributions converge exponentially to Haar on the correct target
- the minimal data needed for forcing is explicit
- the boundary is sharp: without cyclic symmetry in the admissible automorphism class, the forcing engine cannot apply

Throughout, \(\mathbb{T}_P := \mathbb{R}/P\mathbb{Z}\) is the phase circle of period \(P>0\).

---

## Core definitions

### Sliced arena

A sliced arena is a decomposition
\[
M = \bigsqcup_{s\in \mathbb{R}} \Sigma_s
\]
with \(\Sigma_s\) representing a global instant.

### Phase field

A phase field is a map
\[
\Phi: M \to \mathbb{T}_P.
\]
Write \(\Phi_s := \Phi|_{\Sigma_s}\).

### Syncré Coverage Law (SCL)

\[
\boxed{\ \forall s,\quad \Phi_s(\Sigma_s) = \mathbb{T}_P.\ }
\]

### Equivariant circle action and phase response

Let \(A: S^1\times \Sigma \to \Sigma\) be a continuous \(S^1\)-action on a slice \(\Sigma\).

A phase observable \(\phi: \Sigma\to \mathbb{T}_P\) is **equivariant** if there exists a continuous group homomorphism
\[
\chi: S^1 \to \mathbb{T}_P
\]
so that
\[
\boxed{\ \phi(A(\theta,x)) = \phi(x) \oplus \chi(\theta)\ \text{for all }\theta\in S^1,\ x\in\Sigma.\ }
\]
The map \(\chi\) is the **phase response**.

For \(x\in\Sigma\), write its stabilizer
\[
\mathrm{Stab}(x):=\{\theta\in S^1: A(\theta,x)=x\}.
\]
The orbit \(O_x:=\{A(\theta,x):\theta\in S^1\}\) is homeomorphic to \(S^1/\mathrm{Stab}(x)\).

We say \(x\) lies on a **\(\chi\)-compatible orbit** if \(\mathrm{Stab}(x)\subseteq \ker\chi\). Equivalently, \(\chi\) descends to a well-defined map on the orbit parameter circle \(S^1/\mathrm{Stab}(x)\).

#### Lemma (stabilizers are forced into the kernel)

If \(\phi\) is \(\chi\)-equivariant, then every \(x\in\Sigma\) lies on a \(\chi\)-compatible orbit.

*Proof.* If \(\theta\in\mathrm{Stab}(x)\), then \(A(\theta,x)=x\). Equivariance gives
\(\phi(x)=\phi(A(\theta,x))=\phi(x)\oplus\chi(\theta)\), hence \(\chi(\theta)=0\). \(\square\)

### Nonzero slope

Any continuous homomorphism \(\chi: S^1\to \mathbb{T}_P\cong S^1\) is either trivial or surjective.
After identifying \(S^1\cong \mathbb{R}/2\pi\mathbb{Z}\) and \(\mathbb{T}_P\cong \mathbb{R}/P\mathbb{Z}\), there exists an integer \(d\in\mathbb{Z}\) such that
\[
\chi(\theta) = \tfrac{P}{2\pi} d\,\theta \ \bmod P.
\]
Nonzero slope means \(d\neq 0\), equivalently \(\chi\) is surjective.

A key consequence of the stabilizer lemma is:

- if \(\chi\) has nonzero slope, then \(\ker\chi\) is a finite subgroup of \(S^1\), so in any \(\chi\)-equivariant domain there are no fixed points and every orbit is a circle (possibly with a finite covering parameterization)

---

## Pass 1: Minimal forcing theorem (coverage forced by cyclic symmetry)

### Theorem A (Circle-action forcing, minimal form)

Let \(\Sigma\) carry a continuous \(S^1\)-action \(A\). Let \(\phi: \Sigma\to \mathbb{T}_P\) be \(\chi\)-equivariant.

If \(\chi\) has nonzero slope, then
\[
\boxed{\ \phi(\Sigma)=\mathbb{T}_P.\ }
\]

#### Proof

Pick any \(x\in\Sigma\). By equivariance and the stabilizer lemma, \(\mathrm{Stab}(x)\subseteq\ker\chi\), so the orbit \(O_x\cong S^1/\mathrm{Stab}(x)\) is a circle and the formula
\(
\phi(A(\theta,x))=\phi(x)\oplus\chi(\theta)
\)
is consistent on the orbit.

Since \(\chi\) is surjective, \(\phi(O_x)=\phi(x)\oplus\chi(S^1)=\mathbb{T}_P\). Because \(O_x\subseteq\Sigma\), we have \(\phi(\Sigma)=\mathbb{T}_P\). \(\square\)

This is the core inevitability move: in the nonzero-slope equivariant regime, coverage does not require a separate "free orbit" hypothesis.

---

## Pass 1 upgrade: Necessary and sufficient conditions under equivariant circle actions

### Theorem B (Slice-wise coverage under an equivariant circle action)

Let \((\Sigma,A,\phi,\chi)\) be as above with \(\phi\) \(\chi\)-equivariant.

Then the following are equivalent:

- \(\phi(\Sigma)=\mathbb{T}_P\)
- either

  - **nonzero-slope regime**: \(\chi\) has nonzero slope, or
  - **quotient regime**: \(\chi\) is trivial and the induced map on orbit space \(\bar\phi:\Sigma/S^1\to\mathbb{T}_P\) is surjective

In particular, if \(\chi\) has nonzero slope and equivariance holds on \(\Sigma\), slice-wise coverage is automatic.

#### Proof

If \(\chi\) has nonzero slope, Theorem A gives \(\phi(\Sigma)=\mathbb{T}_P\).

If \(\chi\) is trivial, equivariance says \(\phi\) is constant on each orbit, hence factors through \(\Sigma/S^1\). Coverage is then equivalent to surjectivity of the factor \(\bar\phi\). \(\square\)

**Clarification (slope-zero is not itself an obstruction).** When \(\chi\) is trivial, equivariance only says \(\phi\) is constant along orbits. Coverage may still hold if the induced quotient factor \(\bar\phi\) is surjective. Therefore obstruction is the *pair* “slope-zero + quotient nonsurjectivity,” not slope-zero alone.


---

## Pass 2: Obstruction classification theorem (finite failure catalog)

SFT is certificate-driven: a strong universality sentence must be traceable to either a **witness certificate** (WSC) or a **witness obstruction** (WOB).

### Definition (SCL inside the witness-first model)

Fix an admissible model class (regularity, domain, coupling conventions). On a slice \(\Sigma\), SFT treats "SCL holds" as:

- **witnessed coverage** on the correct target (full circle/torus or CST), certified by one of the witness engines (action witness, degree witness, holonomy witness, coupled-target witness)

Write this as \(\mathrm{SCL}^{\mathrm{SFT}}(\Sigma)\).

### Theorem C (Witness-or-obstruction dichotomy)

For a fixed admissible model class, exactly one of the following occurs on a slice \(\Sigma\):

- **Witness:** a finite certificate exists (WSC) proving the correct target coverage (full \(\mathbb{T}_P\) / full \(\mathbb{T}_{\mathbf{P}}\) / or the appropriate coupled subtorus \(\mathrm{CST}_s\))
- **Obstruction:** a finite obstruction object exists (WOB) from the following catalog:

    - **WOB-12-slope-zero-quotient-nonsurjective:** \(\chi\) is trivial (slope zero) **and** the induced quotient factor map \(\bar\phi:\Sigma/S^1\to\mathbb{T}_P\) is not surjective (slope-zero alone is not an obstruction)
  - **WOB-08-orbit-collapse:** the claimed forcing domain contains no \(\chi\)-active circle-type orbit on which an equivariant response can act (for example the symmetry is not realized on the domain)
  - **WOB-09-equivariance-defect:** the observable is not genuinely equivariant in the adopted model class (or defect exceeds admissible tolerance)
  - **WOB-05-holonomy-defect:** a lift/connection class needed for holonomy certification is inconsistent
  - **WOB-10-resonant-locking:** rank deficiency or rational dependence collapses the intended torus sweep or target
  - **WOB-06-coupling-mismatch:** the wrong target (full torus) is used when coupling forces a CST
  - **WOB-11-disconnected-slice:** the measurement domain misses the orbit-bearing or loop-bearing component needed for the claimed target
  - **WOB-07-mixing-not-justified:** a mixing claim is made without an admissible diffusion/noise model (NDB/MXC not provided)


Equivalently,
\[
\boxed{\ \neg\mathrm{SCL}^{\mathrm{SFT}}(\Sigma)\ \Longleftrightarrow\ \text{some WOB obstruction code occurs.}\ }
\]

This is the universality leverage: in the certified regime, there is no "mysterious failure." One either has a witness, or a named structural obstruction.

---

## Pass 3: Robustness forcing theorem (quantitative stability)

The forcing theorem is only useful if its hypotheses persist under perturbations.

### Perturbation model and defect

Use the geodesic metric on \(\mathbb{T}_P\):
\[
d_{\mathbb{T}_P}([u],[v]) := \min_{n\in\mathbb{Z}} |u-v+nP|.
\]

Define the equivariance defect on a domain \(D\subseteq\Sigma\):
\[
\mathrm{Def}_D(A,\phi,\chi) := \sup_{\theta\in S^1,\ x\in D} d_{\mathbb{T}_P}(\phi(A(\theta,x)),\phi(x)\oplus\chi(\theta)).
\]

### Theorem D (Robust forcing on an orbit)

Assume \(\chi\) has nonzero slope and \(\mathrm{Def}_D(A,\phi,\chi)\le \varepsilon\). Then for any \(x\in D\), the image \(\phi(O_x\cap D)\) is \(\varepsilon\)-dense in \(\mathbb{T}_P\).

Moreover, if on some orbit \(O\cong S^1\) the restriction \(\phi|_O\) admits a lift
\(f(t)=\tfrac{dP}{2\pi}t+\psi(t)\)
with \(\psi\in C^1\) and
\[
\|\psi'\|_{\infty} < \tfrac{|d|P}{2\pi},
\]
then \(\phi(O)=\mathbb{T}_P\) (exact coverage persists).

A Fourier-defect sufficient condition is:
if \(\psi(t)=\sum_{n\ne 0} a_n e^{int}\) with \(\sum_{n\ne 0} |n a_n| < \tfrac{|d|P}{2\pi}\), then \(\phi(O)=\mathbb{T}_P\).

---

## Pass 4: Multi-cycle forcing (torus universality) and correct coupled targets

Let \(\mathbb{T}_{\mathbf{P}}:=\prod_{j=1}^k \mathbb{T}_{P_j}\).

A \(\mathbb{T}^k\)-action \(A: \mathbb{T}^k\times\Sigma\to\Sigma\) with an equivariant observable \(\Phi: \Sigma\to\mathbb{T}_{\mathbf{P}}\) is governed by a homomorphism
\(\chi: \mathbb{T}^k\to\mathbb{T}_{\mathbf{P}}\).

After identifying each torus with \(\mathbb{R}^k\) modulo an integer lattice (with the appropriate period scalings), \(\chi\) is represented by an integer matrix \(D\in\mathbb{Z}^{k\times k}\). Let \(r:=\mathrm{rank}(D)\).

- if \(r=k\), then \(\chi\) is surjective onto \(\mathbb{T}_{\mathbf{P}}\)
- if \(r<k\), then \(\mathrm{Im}(\chi)\) is a subtorus of dimension \(r\), and that subtorus is the correct symmetry-forced target

### Theorem E (Torus forcing, correct target form)

If \(\Phi\) is \(\chi\)-equivariant and \(r=\mathrm{rank}(D)\), then
\[
\boxed{\ \Phi(\Sigma)=\mathrm{Im}(\chi).\ }
\]
In particular, if \(r=k\) then \(\Phi(\Sigma)=\mathbb{T}_{\mathbf{P}}\), and if \(r<k\) then \(\Phi(\Sigma)\) is the \(r\)-dimensional subtorus \(\mathrm{Im}(\chi)\).

### Coupled torus theorem (CST targeting)

If additional coupling constraints restrict the image to a coupled subtorus \(\mathrm{CST}_s\subseteq\mathbb{T}_{\mathbf{P}}\), then the correct coverage target is \(\Phi(\Sigma_s)=\mathrm{CST}_s\) and the correct statistical baseline is Haar on \(\mathrm{CST}_s\).

---

## Pass 5: Ergodic universality upgrade (statistical inevitability)

### Theorem F (Unique ergodicity of irrational torus translations)

Let \(R_{\omega}: \mathbb{T}^k\to \mathbb{T}^k\) be translation by \(\omega\in\mathbb{R}^k\). If \(1,\omega_1,\dots,\omega_k\) are linearly independent over \(\mathbb{Q}\), then \(R_{\omega}\) is uniquely ergodic.

Equivalently, for every continuous \(f\),
\[
\lim_{N\to\infty} \frac{1}{N}\sum_{n=0}^{N-1} f(x+n\omega) = \int_{\mathbb{T}^k} f\,d\mu_{\mathrm{H}}
\]
uniformly in \(x\), where \(\mu_{\mathrm{H}}\) is Haar.

### Corollary (Empirical universality)

If a physical system factors to a torus translation on phase coordinates and the factor is uniquely ergodic, then observed phase statistics converge to the appropriate Haar target (full torus Haar or coupled Haar on \(\mathrm{CST}_s\)).

---

## Pass 6: Dynamical preservation theorem

Let \(T_t: M\to M\) be dynamics that preserve the slicing: \(T_t(\Sigma_s)\subseteq \Sigma_{s+t}\) (or a relabeling).

### Theorem G (Witness preservation under admissible dynamics)

Assume:

- **slice compatibility:** \(T_t\) maps each slice to a slice
- **symmetry compatibility:** \(T_t(A(\theta,x)) = A(\theta, T_t(x))\) for the relevant symmetry action (or preserves its conjugacy class)
- **invariant compatibility:** the witness invariants used by the certificate (degree / holonomy / coupled-target relation) are preserved under \(T_t\)

Then any witness certificate at time \(s\) transports to a witness certificate at time \(s+t\). In particular, \(\mathrm{SCL}^{\mathrm{SFT}}\) is invariant under the admissible dynamics class.

---

## Pass 7: Noise and diffusion universality

### Theorem H (Diffusion enforces Haar on a torus)

On \(\mathbb{T}^k\), the heat semigroup \(e^{t\Delta}\) drives any initial probability density to Haar exponentially fast in \(L^2\), with explicit decay rates given by the Laplacian eigenvalues.

For advection–diffusion \(\partial_t\rho + \nabla\cdot(\rho u)=\kappa\Delta\rho\) with \(\kappa>0\), if the resulting Markov semigroup is irreducible and strong Feller (standard under nondegenerate noise/ellipticity assumptions), the stationary measure on the correct target torus is unique and convergence is exponential in total variation.

This upgrades stability into strengthening uniformity over time.

---

## Pass 8: Minimal structure theorem

### Theorem I (Minimal sufficiency for forcing)

For the forcing implication
\(
\text{cyclic symmetry + equivariance + nonzero slope}\ \Rightarrow\ \text{coverage}
\)
the only required ingredients are:

- a sliced arena \(M=\bigsqcup_s \Sigma_s\)
- a nontrivial cyclic symmetry action on slices
- an equivariant phase observable with nonzero slope

No additional geometric, measure-theoretic, or dynamical structure is needed for the forcing implication.

---

## Pass 9: Universality boundary theorem

### Definition (cyclic symmetry in the admissible automorphism class)

Let \(\mathrm{Aut}_{\mathrm{adm}}(\Sigma)\) denote the admissible automorphisms of a slice (homeomorphisms or diffeomorphisms, depending on the declared model class). We say \(\Sigma\) admits admissible cyclic symmetry if \(\mathrm{Aut}_{\mathrm{adm}}(\Sigma)\) contains a nontrivial continuous subgroup isomorphic to \(S^1\), equivalently if \(\Sigma\) admits a nontrivial continuous \(S^1\)-action in the admissible class.

### Theorem J (Sharp boundary for the forcing engine)

If \(\Sigma\) admits no admissible cyclic symmetry, then the action-forcing witness engine cannot apply on \(\Sigma\). Any Syncré-type coverage claim must therefore be witnessed by an embedded mechanism (degree/holonomy on a loop subsystem) or by adding new structure not present in the admissible class.

In particular, in strictly ordered one-dimensional systems admitting only monotone scalar observables and no admissible cyclic symmetry, circle-valued forcing coverage is structurally excluded.

---

## Pass 10: Global Syncré Universality Theorem (capstone)

### Theorem K (Global universality)

Let \(M=\bigsqcup_{s\in\mathbb{R}}\Sigma_s\) be a sliced arena. Suppose:

- each slice admits admissible cyclic symmetry (an \(S^1\) or \(\mathbb{T}^k\)-action in the declared class)
- there exists a phase field \(\Phi:M\to\mathbb{T}_P\) (or \(\mathbb{T}_{\mathbf{P}}\)) that is equivariant with nonzero slope on each slice, with no degree/holonomy obstruction in the adopted lift/connection model
- in coupled multi-cycle settings, the correct target \(\mathrm{CST}_s\) is selected and the witness claims are made on that target

Then:

- slice-wise coverage holds on the correct target (SCL in the witness-first sense)
- coverage is structurally stable under admissible perturbations preserving the obstruction-free regime
- witnesses are preserved under admissible dynamics
- in the presence of rationally independent drive frequencies, time averages on phase coordinates converge to the appropriate Haar target
- under weak diffusion/noise, phase distributions converge exponentially to Haar on the correct target

Moreover:

- failures are classified by the finite WOB obstruction catalog

This is the theorem form of global universality: Syncré is the structurally forced phase-field layer of cyclic systems, exactly on the cyclic-symmetry side of the boundary.
