Example: why the cross magnetic block matters

Reader card. This toy calculation isolates the diagonal magnetic phase left after the permutation defect has been corrected. It is diagnostic fixed-corner data, not a claimed stable obstruction for every choice of representative.

This is a local toy model for the residual obstruction. It is not a claim that every such scalar occurs for every quasicrystal; it isolates the term that a full example must compute.

Suppose the GPS step has produced commuting bisection symbols. On one clopen component, assume the two corresponding transports have constant height exponents \(n_1,n_2\). If

\[ \alpha_i(U_h)=e^{2\pi i\Theta_{ih}}U_h, \]

then the two paths around the transverse square acquire different magnetic phases:

\[ \nu_{12} =\exp\!\left(2\pi i (\Theta_{1h}n_2-\Theta_{2h}n_1)\right). \]

flowchart LR
    O["start in a clopen fibre"]
    P1["W_1, then alpha_1(W_2)"]
    P2["W_2, then alpha_2(W_1)"]
    E["same bisection endpoint"]
    N["phase ratio nu_12"]
    O --> P1 --> E
    O --> P2 --> E
    P1 -. "exp(2 pi i Theta_1h n_2)" .-> N
    P2 -. "exp(2 pi i Theta_2h n_1)" .-> N

When \(\Theta_{ih}=0\), both factors are one and the block-supported theorem is recovered. When a cross entry is nonzero, equality of symbols does not imply equality of operators.

If the induced diagonal action is trivial on this component, a scalar commutator cannot be changed by scalar generator gauges. At a fixed finite corner the obstruction is the class of the full locally constant cocycle in that corner's \(H^2\). The theorem assumes only that, for the constructed representative, this class becomes a coboundary at some allowed finite stage. No comparison maps between stages, and therefore no representative-independent stable class, have been constructed. The displayed scalar is diagnostic data in one corner, not a stable invariant.

This example also explains why a blanket assumption such as ``K_0 is 2-divisible'' is neither necessary nor sufficient. The remaining datum lives in circle-valued group cohomology and depends on the height--transverse block of the multiplier and on return-time functions.