Application template: regular cut-and-project sets

Application status. This is a checklist for turning a regular model set into an example, not a completed verification of the theorem's dynamical hypotheses.

Regular model sets provide a large supply of computable clopen frequencies. This chapter records how the construction could become a practical recipe. It is not a completed example: no concrete global \(\mathbb Z^p\)-action, minimal height homeomorphism, and directionally invariant cylinder are verified here.

Acceptance domains

Start from a cut-and-project scheme

\[ \begin{array}{ccccc} \mathbb R^d & \xleftarrow{\ \pi\ } & \mathbb R^d\times H & \xrightarrow{\ \pi_{\mathrm{int}}\ } & H,\\ && \mathcal L && \end{array} \]

and a compact window \(W\subset H\) whose boundary has Haar measure zero. The associated regular model set is

\[ \Lambda(W)=\{\pi(\ell):\ell\in\mathcal L, \ \pi_{\mathrm{int}}(\ell)\in W\}. \]

A finite patch \(P\) determines a clopen cylinder \(C_P\) in the canonical transversal. Its acceptance domain \(W_P\subset H\) is built by intersecting translates of \(W\) for required points and complements for forbidden points. Under the standard transversal normalization,

\[ \mu(C_P)=\operatorname{dens}(\mathcal L)\,m_H(W_P). \]

flowchart LR
    P["finite patch P"]
    A["acceptance domain W_P in internal space"]
    C["clopen cylinder C_P"]
    F["frequency mu(C_P)"]
    K["coinvariant class [1_C_P]"]
    M["mixed module E_(T,P)"]
    P --> A --> F
    P --> C --> K --> M
    F --> M

Checks required before applying the theorem

For a chosen lattice direction \(h\), an actual application must perform the following checks and finite computations.

1. Produce a global commuting \(\mathbb Z^p\)-action, or a precisely identified clopen return system, in which the chosen height map \(T_h\) is defined and minimal.
2. Choose a cylinder \(C_P\) and verify that its height-coinvariant class is fixed by every transverse generator. Equivalently, solve \[ (\alpha_i-1)1_{C_P}=(\alpha_h-1)h_i. \] In a finite-local-complexity system this is a bounded pattern-equivariant flow problem on collared patches.
3. Build the clopen bisection partial permutations encoded by the \(h_i\).
4. Use the PV/Barlak comparison theorem to identify each height PV square defect with the corresponding transfer-function curl.
5. Apply the GPS retraction to make the zero-index symbols commute.
6. If the magnetic form is block-supported, tensor with an appropriate transverse Rieffel class, respecting positivity for an actual module.

If all of these checks hold, the predicted trace is

\[ \tau_\mu([\mathcal E_{T,P}]) =\operatorname{dens}(\mathcal L)m_H(W_P) \,\tau_{\Theta_G}([E_T]). \]

In particular, regularity of the model set alone does not establish any of the dynamical or comparison hypotheses above. Not every model-set presentation supplies a globally defined minimal height map on the first transversal one writes down. Passing to a clopen corner or a return system may be necessary. This is a geometric input to check, not a consequence of regularity alone.