# Coupled Atlas

## Purpose

Extend SFT beyond the single SOL example so “coupled universality” is a reusable framework.

We define two additional coupling classes that commonly arise:

- affine linear coupling on \(\mathbb{T}_P^2\)
- integer-matrix coupling on \(\mathbb{T}_{\mathbf{P}}\)

For each, we specify CST targets and the correct coupled Haar baseline.

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## Coupling class D1 — Affine diagonal coupling (generalized SOL)

Codomain: \(\mathbb{T}_P^2\), coordinates \((a,b)\).

Constraint:
\[
b \ominus (\alpha\odot a) = \Psi(s),
\]
where \(\alpha\in\mathbb{Z}\) acts by \(\alpha\odot a := \alpha a \ (\bmod P)\).

CST target:
\[
\mathrm{CST}_s=\{(a,\ \alpha\odot a\oplus \Psi(s)):\ a\in\mathbb{T}_P\}.
\]

Parameterization:
\[
\eta_s(a)=(a,\alpha\odot a\oplus\Psi(s)).
\]

Coupled Haar:
\[
\mu_{\mathrm{H},s}^{\mathrm{CST}}=(\eta_s)_\#\mu_{\mathrm{H}}.
\]

Coverage inheritance:
If \(a(\Sigma_s)=\mathbb{T}_P\), then \(\Phi(\Sigma_s)=\mathrm{CST}_s\) by the same CMT argument.

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## Coupling class D2 — Integer-matrix subtorus constraints

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

Let \(A\in\mathbb{Z}^{r\times d}\). Define a linear constraint:
\[
A y = \Psi(s)\quad \text{in }\prod_{i=1}^r \mathbb{T}_{Q_i},
\]
with appropriate target periods \(Q_i\) determined by the \(P_j\) and the integer combinations.

Then \(\mathrm{CST}_s\) is a (possibly lower-dimensional) subtorus (or a coset of one).

Parameterization:
Choose a basis of the kernel lattice to define a map
\[
\eta_s:\mathbb{T}_{P_*}^m\to \mathrm{CST}_s
\]
for the free coordinates.

Coupled Haar:
\[
\mu_{\mathrm{H},s}^{\mathrm{CST}}=(\eta_s)_\#\mu_{\mathrm{H}}^{(m)}.
\]

Universality mechanism:
Coverage of the free coordinates implies coverage of the CST coset.

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## Acceptance gates

- at least two coupling classes beyond SOL are defined
- each includes CST target description and coupled Haar baseline
- a clear inheritance statement is given (free coordinate coverage ⇒ CST coverage)
