# Univ Pass 5 — Coupled Universality (Multi-Cycle Targets, SOL Meta-Theorem, Coupled Haar)

## Purpose

Perfect the universality argument in the coupled multi-cycle setting:

- define universality targets correctly under coupling (CST rather than full torus)
- state a meta-theorem that turns a single “free coordinate” witness into coupled coverage
- define the correct coupled Haar target and diagnostics baseline for SUD/FUD/CP
- clarify what is universal in coupled systems: mechanism + target selection

This prevents a common referee objection: “your target is wrong once coupling exists.”

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## Program language (coupled universality)

| Term | Tag | Meaning |
|---|---|---|
| Coupled Target Principle | **CTP** | in coupled settings the target is CST, not the full torus |
| Coupled Haar Target | **CHT** | Haar measure on CST (pushforward from free coordinate) |
| Coupling Meta-Theorem | **CMT** | free-coordinate coverage ⇒ CST coverage |
| SOL Class | **SOL** | synodic offset coupling: \(b\ominus a=\Psi(s)\) |
| Coupled Witness Tier | **WTR-C** | witness level for coupled targets |

---

## 1) Coupled Target Principle (CTP)

Let \(\Phi:M\to\mathbb{T}_{\mathbf{P}}\) be a multi-cycle field and \(M=\bigsqcup_s \Sigma_s\).

A coupling constraint defines a coupled subtorus (or submanifold) \(\mathrm{CST}_s\subseteq \mathbb{T}_{\mathbf{P}}\) such that
\[
\Phi(\Sigma_s)\subseteq \mathrm{CST}_s.
\]

### CTP

\[
\boxed{\;\text{Universality target is } \Phi(\Sigma_s)=\mathrm{CST}_s,\ \text{not } \mathbb{T}_{\mathbf{P}},\ \text{whenever coupling is present.}\;}
\]

---

## 2) Coupled Haar Target (CHT)

Assume \(\mathrm{CST}_s\) is parameterized by a free coordinate \(u\in\mathbb{T}_{P_*}\) through a continuous map
\[
\eta_s:\mathbb{T}_{P_*}\to \mathrm{CST}_s\subseteq \mathbb{T}_{\mathbf{P}}.
\]

Define coupled Haar:
\[
\boxed{\;\mu_{\mathrm{H},s}^{\mathrm{CST}} := (\eta_s)_\# \mu_{\mathrm{H}}^{(P_*)}\;}
\]
where \(\mu_{\mathrm{H}}^{(P_*)}\) is Haar on \(\mathbb{T}_{P_*}\).

This is the correct baseline for SUD/FUD in coupled settings.

---

## 3) SOL coupling: the canonical coupled universality case

Take \(\mathbb{T}_P^2\) with coordinates \((a,b)\) (solar phase, lunar phase) and SOL:
\[
b\ominus a=\Psi(s).
\]

Then
\[
\mathrm{CST}_s=\{(a,a\oplus\Psi(s)):\ a\in\mathbb{T}_P\}.
\]
A natural parameterization is
\[
\eta_s(a)=(a,a\oplus\Psi(s)).
\]
Thus
\[
\mu_{\mathrm{H},s}^{\mathrm{CST}} = (\eta_s)_\# \mu_{\mathrm{H}}.
\]

---

## 4) Coupling Meta-Theorem (CMT): free coordinate coverage implies coupled coverage

### Theorem 1 (CMT-SOL)

Assume SOL holds on \(\Sigma_s\) and assume the solar component covers:
\[
\Phi_\odot(\Sigma_s)=\mathbb{T}_P.
\]
Define \(\Phi=(\Phi_\odot,\Phi_{\leftmoon})\).

Then
\[
\boxed{\;\Phi(\Sigma_s)=\mathrm{CST}_s.\;}
\]

#### Proof

For any \(a\in\mathbb{T}_P\), pick \(p\in\Sigma_s\) with \(\Phi_\odot(p)=a\). SOL gives \(\Phi_{\leftmoon}(p)=a\oplus\Psi(s)\). Hence \((a,a\oplus\Psi(s))\in\Phi(\Sigma_s)\), and these points exhaust \(\mathrm{CST}_s\). ∎

### Interpretation

The universal mechanism in the coupled case is:

- coupling reduces dimension
- coverage is inherited from the free coordinate

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## 5) Coupled diagnostics alignment

In coupled settings, define:

- pushforward measure \(\mu_s=(\Phi_s)_\#\nu_s\) on \(\mathbb{T}_{\mathbf{P}}\)
- SUD with baseline \(\mu_{\mathrm{H},s}^{\mathrm{CST}}\)
- CP/SCP computed relative to neighborhoods inside \(\mathrm{CST}_s\) (using the parameter coordinate \(u\))

This ensures diagnostics respect the correct target.

---

## Acceptance gates

- CTP and CHT are defined and framed as correctness requirements
- SOL case is written with explicit CST and parameterization \(\eta_s\)
- CMT-SOL is stated and proved
- Diagnostics baselines are aligned to CST (not full torus)
