# Pass G — Diagnostics and Convergence Program (Slice Measures, Stroboscopic Maps, Torus Equidistribution)

## Purpose

Convert Syncre from “coverage exists” into an analyzable program with diagnostics and convergence targets.

This pass:

- formalizes phase distributions on slices via pushforward measures
- defines metrics that quantify coverage strength and uniformity
- introduces stroboscopic sampling as a canonical discrete-time reduction
- states baseline convergence theorems in the simplest (linear/rotation) setting
- explains how coupling (Pass F) changes the convergence target from full torus to a coupled subtorus

---

## Program language introduced here

| Term | Tag | Meaning |
|---|---|---|
| Syncre Slice Measure | **SSM** | A chosen measure \(\nu_s\) on each slice \(\Sigma_s\) |
| Phase Pushforward | **PPF** | \(\mu_s := (\Phi_s)_\# \nu_s\) |
| Coverage Modulus | **SCM** | \(\rho(\Phi_s;\Sigma_s,\nu_s)\) (from Pass C) |
| Uniformity Defect | **SUD** | distance \(d(\mu_s,\mu_{\mathrm{H}})\) (from Pass C) |
| Distortion Functional | **SDF** | \(s\mapsto d(\mu_s,\mu_{\mathrm{H}})\) or related statistic |
| Stroboscopic Section | **STS** | A chosen sampling schedule \(s_n\) (often arithmetic progression) |
| Stroboscopic Process | **STP** | The discrete sequence \(\Phi_{s_n}\) or \(\mu_{s_n}\) |
| Equidistribution Target | **EQT** | Haar measure on torus or the natural measure on a coupled subtorus |
| Resonance Flag | **RNF** | whether an increment is rational/irrational (rotation number test) |

---

## Slice measures and phase distributions

Let \(M=\bigsqcup_s \Sigma_s\) be an admissible slicing and \(\Phi:M\to\mathbb{T}_{\mathbf{P}}\) a Syncre torus field (Pass F).

### Definition (Syncre Slice Measure, SSM)

Choose a measure \(\nu_s\) on each slice \(\Sigma_s\). Typical choices:

- uniform measure on a geometric slice
- surface area measure on \(S^2\)
- a physically weighted density (mass density, emissivity, sampling density)

The pair \((\Sigma_s,\nu_s)\) is the measured slice.

### Definition (Phase pushforward, PPF)

Define the phase distribution on the torus at slice \(s\) by
\[
\boxed{\;\mu_s := (\Phi_s)_\# \nu_s,\qquad \Phi_s=\Phi|_{\Sigma_s}.\;}
\]

This is the central object for convergence analysis.

---

## Diagnostics: coverage strength and uniformity

### Coverage Modulus (SCM)

For each slice \(s\), SCM is the largest radius \(\rho\) such that every torus-ball of radius \(\rho\) has positive \(\mu_s\)-mass. For a circle, SCM is the largest arc-radius such that every arc-neighborhood has positive mass.

- SCM \(>0\) is a quantitative “no-hole” statement
- SCM small indicates near-collapse even if coverage holds

### Uniformity defect (SUD)

Choose a metric \(d\) on probability measures on the codomain:

- Wasserstein-1 is natural for circle/torus
- total variation is natural when densities exist

Then
\[
\mathrm{UD}(s):=d(\mu_s,\mu_{\mathrm{H}}),
\]
where \(\mu_{\mathrm{H}}\) is Haar measure on the full torus.

When coupling constraints hold (Pass F), replace \(\mu_{\mathrm{H}}\) by the natural invariant measure on the coupled subtorus.

### Distortion functional (SDF)

Define a time-indexed diagnostic such as
\[
\mathrm{DF}(s):=\mathrm{UD}(s),
\]
or a vector of diagnostics:
\[
\mathrm{DF}(s):=\big(\mathrm{UD}(s),\ \mathrm{SCM}(s)\big).
\]

The “Syncre convergence” program aims to show \(\mathrm{UD}(s)\to 0\) or that \(\mathrm{UD}(s)\) remains bounded/small under dynamics.

---

## Stroboscopic sampling: the canonical discrete reduction

Continuous flows on circles and tori are often best analyzed by discrete sampling.

### Definition (Stroboscopic Section, STS)

Choose a sampling schedule \(s_n\). The canonical choice is an arithmetic progression:
\[
s_n = s_0 + n\Delta.
\]

### Definition (Stroboscopic Process, STP)

Define the sampled sequences:

- phase fields: \(\Phi^{(n)} := \Phi_{s_n}\)
- measures: \(\mu^{(n)} := \mu_{s_n}\)

In coupled settings, one often tracks the synodic offset \(\Psi(s_n)\) (Pass F).

---

## Baseline convergence theorem: circle rotation equidistribution

The simplest stroboscopic model is a rigid rotation on \(S^1\).

### Theorem (Equidistribution of an irrational rotation)

Let \(\varphi_{n+1}=\varphi_n\oplus \rho\) on \(\mathbb{T}_P\), where \(\rho/P\) is irrational. Then for every continuous \(g:\mathbb{T}_P\to\mathbb{R}\),
\[
\boxed{\;\frac{1}{N}\sum_{n=0}^{N-1} g(\varphi_n)\ \longrightarrow\ \int_{\mathbb{T}_P} g\,d\mu_{\mathrm{H}}\quad\text{as }N\to\infty.\;}
\]

This is the canonical “convergence in time-average” notion for cycle variables.

### Program interpretation

If the synodic offset samples \(\Psi(s_n)\) evolve like an irrational rotation, then synodic phase is equidistributed, and diagnostic statistics averaged over samples converge to Haar averages.

---

## Torus equidistribution: frequency vector and RNF

On a torus \(\mathbb{T}_{\mathbf{P}}\), a rigid translation is
\[
x_{n+1} = x_n \oplus \rho
\]
with increment vector \(\rho\in\mathbb{T}_{\mathbf{P}}\).

### Theorem (Torus equidistribution under nonresonance)

If the increment vector \(\rho\) has rationally independent components relative to the periods (nonresonant), then discrete time averages converge to Haar measure on the torus:
\[
\frac{1}{N}\sum_{n=0}^{N-1} G(x_n)\ \to\ \int_{\mathbb{T}_{\mathbf{P}}} G\,d\mu_{\mathrm{H}}
\]
for all continuous \(G\).

We record this as the **Resonance / Nonresonance Flag (RNF)** condition.

---

## Coupling changes the convergence target

When coupling constraints force \(\Phi(\Sigma_s)\subseteq \mathrm{CST}_s\) (Pass F), full-torus Haar is not the correct target.

### Definition (Equidistribution target, EQT)

- If TCL holds (full torus coverage), EQT is Haar measure on \(\mathbb{T}_{\mathbf{P}}\).
- If coupling defines \(\mathrm{CST}_s\), EQT is the natural invariant measure on that subtorus (Haar measure on \(\mathrm{CST}_s\), or the pushforward of Haar from a parameter circle).

### Example (SOL coupling)

If \(\Phi_{\leftmoon}\ominus \Phi_\odot=\Psi(s)\), then at a fixed \(s\) the coupled subtorus is parameterized by the free coordinate \(a\in\mathbb{T}_P\):
\[
a\mapsto (a,\ a\oplus \Psi(s)).
\]
The natural measure is the pushforward of Haar on \(a\).

---

## How this links to Pass C and Pass D

- Pass C provides SCM/SUD/SDF as quantitative diagnostics
- Pass D provides lift/holonomy language that controls perturbations of phase fields and sampling increments
- Pass F provides coupling laws that define the correct EQT target

This pass completes the “analysis layer” needed before dynamics (Pass H): it defines what we measure and what convergence means.

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## Acceptance gates

- SSM and PPF are defined and used to define slice distributions \(\mu_s\)
- SCM/SUD/SDF are positioned as primary diagnostics for “Syncre convergence”
- STS/STP are defined and a baseline equidistribution theorem is stated for circle rotations
- Torus equidistribution is stated with a clear RNF condition
- EQT is defined and adjusted correctly under coupling constraints
