2. Square holonomy and integer curvature
Chapter 1 constructed one finite-PE transport for each transverse generator. This chapter explains, without the filtered \(KK\)-theoretic proof, why their failure to commute is measured by an integer curl and why minimality forces that curl to vanish.
Reader card. Input: the projection \(P\), transports \(W_i\), and transfer functions \(h_i\) from Chapter 1. Output: the square defect \(\Omega_{ij}\), the integer curvature \(\kappa_{ij}\), and the precise comparison theorem used in Chapter 3. The proof of that comparison is deferred to Technical Engine A.
2.1 Why individual transports are not enough
For a generator \(T_i\), the semilinear map represented by \(W_i\) carries the translated module back to the original one. For two generators there are two composites from \(\alpha_i\alpha_j(P)\) to \(P\):
\[ W_i\alpha_i(W_j) \qquad\text{and}\qquad W_j\alpha_j(W_i). \]
Their ratio is the unitary
\[ \Omega_{ij} =W_i\alpha_i(W_j) \bigl(W_j\alpha_j(W_i)\bigr)^* \in U(PM_n(B_h)P). \tag{2.1a} \]
This is square holonomy. The transports form a strict action exactly when every such square closes, together with the corresponding group-word coherence.
flowchart TD
Pij["alpha_i alpha_j(P)"]
Pi["alpha_i(P)"]
Pj["alpha_j(P)"]
P["P"]
Pij -->|"alpha_i(W_j)"| Pi -->|"W_i"| P
Pij -->|"alpha_j(W_i)"| Pj -->|"W_j"| P
2.2 The same square at the level of bounded flows
The transfer equations are
\[ (\alpha_i-1)g=(\alpha_h-1)h_i. \]
Their curl is
\[ \kappa_{ij} =(\alpha_j-1)h_i-(\alpha_i-1)h_j. \tag{2.1b} \]
This measures the net height flow left after traversing the same transverse square. Commutativity of the actions gives
\[ \begin{aligned} (\alpha_h-1)\kappa_{ij} &=(\alpha_j-1)(\alpha_h-1)h_i -(\alpha_i-1)(\alpha_h-1)h_j\\ &=(\alpha_j-1)(\alpha_i-1)g -(\alpha_i-1)(\alpha_j-1)g\\ &=0. \end{aligned} \]
Hence \(\kappa_{ij}\) is constant along height orbits.
2.3 Where minimality is used
Minimality is not a background convenience. It is used at this exact point. An integer-valued continuous function invariant under a minimal Cantor homeomorphism must be constant, so
\[ \kappa_{ij}=m_{ij}1_\Sigma. \]
Choose a probability measure invariant under the height and transverse actions. Integrating the two coboundaries in (2.1b) yields
\[ m_{ij}=0. \]
Therefore
\[ \kappa_{ij}=0. \tag{2.1c} \]
This argument works for every pair of transverse generators in any ambient dimension. What is special about the theorem is that the coefficient group has rank one: there are no higher height directions producing further layers of coherence.
2.4 The bridge that cannot be skipped in the proof
It is tempting to infer directly from \(\kappa_{ij}=0\) that \([\Omega_{ij}]=0\). That inference needs proof. The transports were built from clopen matchings and height shifts, while \(\Omega_{ij}\) is a unitary in a crossed-product corner.
The technical comparison theorem identifies the height PV boundary of the operator holonomy with the integer curl:
\[ \partial_h[\Omega_{ij}] =\pm[\kappa_{ij}]. \tag{2.1d} \]
It also checks that the compressed generalized Bott representative is the actual corner holonomy and that minimality removes the relevant quotient. Consequently (2.1c) gives
\[ [\Omega_{ij}]=0 \quad\text{in }K_1(PM_n(B_h)P). \tag{2.1e} \]
For the constructive narrative, (2.1d) is the named PV/Barlak curvature comparison. Its full proof occupies Technical Engine A.
2.5 The running example
In the Sturmian product example, the chosen cylinder function satisfies
\[ h_1=g, \qquad h_2=0. \]
Because \(g\) is fixed by the second transverse generator,
\[ \kappa_{12} =(\alpha_2-1)g-(\alpha_1-1)0 =0 \]
before minimality is invoked. The comparison theorem still performs a nontrivial job: it certifies that this visible integer cancellation kills the actual operator \(K_1\)-defect.
2.6 What passes to the next chapter
At this stage we know:
- each transverse generator has an explicit finite-PE bisection transport;
- every pairwise square defect has zero corner \(K_1\)-class.
We do not yet know that the transports commute as operators. Chapter 3 uses the one-dimensional orbit structure and the GPS normalizer retraction to replace zero index by exact commutation of the bisection symbols.