# Pass 07 — Coupled Atlas Tightening (Beyond SOL)

## Purpose

Push coupled universality to “score-10” by providing a compact, consistent atlas that:

- lists coupling classes
- states CST targets and coupled Haar baselines
- provides a free-coordinate inheritance lemma for each class
- makes diagnostics target selection unambiguous

This consolidates earlier DI Pass D into a paper-facing atlas.

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## Coupled Target Principle (CTP)

If coupling constrains \(\Phi(\Sigma_s)\subseteq \mathrm{CST}_s\), then the correct coverage target is \(\mathrm{CST}_s\) and the correct uniformity baseline is Haar on \(\mathrm{CST}_s\), not Haar on the full torus.

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## Atlas entries

### Class A — SOL (synodic offset coupling)

Codomain: \(\mathbb{T}_P^2\), coordinates \((a,b)\). Constraint:
\[
b\ominus a=\Psi(s).
\]
CST:
\[
\mathrm{CST}_s=\{(a,a\oplus\Psi(s)):a\in\mathbb{T}_P\}.
\]
Parameterization:
\[
\eta_s(a)=(a,a\oplus\Psi(s)).
\]
Coupled Haar:
\[
\mu_{\mathrm{H},s}^{\mathrm{CST}}=(\eta_s)_\#\mu_{\mathrm{H}}.
\]
Inheritance lemma: if \(a(\Sigma_s)=\mathbb{T}_P\), then \(\Phi(\Sigma_s)=\mathrm{CST}_s\).

---

### Class B — Affine diagonal coupling (generalized SOL)

Constraint:
\[
b\ominus (\alpha\odot a)=\Psi(s),\qquad \alpha\in\mathbb{Z}.
\]
CST:
\[
\mathrm{CST}_s=\{(a,\alpha\odot a\oplus\Psi(s)):a\in\mathbb{T}_P\}.
\]
Coupled Haar: pushforward via \(\eta_s(a)=(a,\alpha\odot a\oplus\Psi(s))\).
Inheritance lemma: coverage of \(a\) implies CST coverage.

---

### Class C — Integer-matrix subtorus constraints

Codomain: \(\mathbb{T}_{\mathbf{P}}=\prod_{j=1}^d\mathbb{T}_{P_j}\). Constraint:
\[
A y = \Psi(s)
\]
in an appropriate target torus, where \(A\in\mathbb{Z}^{r\times d}\). Then \(\mathrm{CST}_s\) is a coset of a subtorus.

Parameterization:
\[
\eta_s:\mathbb{T}_{P_*}^m\to \mathrm{CST}_s
\]
built from a basis of the kernel lattice.

Coupled Haar:
\[
\mu_{\mathrm{H},s}^{\mathrm{CST}}=(\eta_s)_\#\mu_{\mathrm{H}}^{(m)}.
\]
Inheritance lemma: coverage of free coordinates implies CST coverage.

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## Diagnostics alignment (must be stated in any draft)

- SUD/FUD must measure deviation from \(\mu_{\mathrm{H},s}^{\mathrm{CST}}\) under coupling.
- Coverage profile CP on CST is computed in free-coordinate parameter neighborhoods.

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## Acceptance gates

- at least three coupling classes are listed with CST + coupled Haar definitions
- each class includes an inheritance lemma statement
- diagnostics alignment rules are explicit and copy-ready
