Choose a reading route

The book is organized in layers. A reader can understand the construction, its hypotheses, and its gap-labelling significance before learning the technical \(KK\)-theoretic comparison used to certify one obstruction.

flowchart TB
    story["Part I: geometric story"]
    construction["Part II: explicit construction"]
    certification["Part III: technical certification"]
    consequences["Part IV: gap-labelling consequences"]

    story --> construction --> certification
    construction --> consequences
    certification --> consequences

Numbering convention. The constructive chapters in Part II are Chapters 1--6. The certification chapters in Part III have their own lettered namespaces: Technical Engine A uses sections and equations \(A.x\), while Technical Engine B uses \(B.x\). Thus “Chapter 2” refers only to the geometric square-holonomy chapter, whereas “Technical Engine A” refers to the complete PV/Barlak proof.

Route A: tiling theorists and newcomers

This route explains what is constructed and why the argument works, while treating the PV/Barlak comparison and analytic Hilbert-module checks as named theorems.

1. Preface, for the gap-labelling motivation.
2. The proof without technical machinery.
3. Dictionary for tiling theorists.
4. A short introduction to Rieffel--Heisenberg modules.
5. The running Sturmian example.
6. Setup and the theorem.
7. Finite PE transports.
8. Square holonomy and integer curvature.
9. The GPS correction, reading first the reader card and the end-of-chapter summary.
10. Attaching the Heisenberg module.
11. Dimension three, for the clearest gap-labelling application.

On a first pass, it is reasonable to accept the following bridge as a black-box theorem:

\[ \partial_h[\Omega_{ij}]=\pm[\kappa_{ij}]. \]

Part III proves it.

Route B: the constructive operator-algebra proof

Readers familiar with crossed products and elementary \(K\)-theory can follow the complete construction in this order:

1. Setup and the theorem.
2. Chapters 1--5 of Part II.
3. Technical Engine A, especially the statement of the comparison and the admissible-lift calculation.
4. The conceptual descent chapter.
5. Technical Engine B.
6. Proof audit.

The construction has three mathematical layers:

• algebraic: coinvariants, transfer functions, and integer curl;
• geometric: compact-open bisections, colored corners, and full groups;
• analytic: equivariant descent, Heisenberg modules, and trace pairing.

Keeping those layers separate prevents equality in \(K\)-theory from being confused with an explicit PE transport, or zero index from being confused with literal commutation.

Route C: the technical or referee route

For checking every load-bearing implication:

1. Read Conventions.
2. Read Technical Engine A together with the source locations listed in Sources.
3. Read the corner, orientation, and index comparison in the GPS chapter.
4. Read the magnetic phase chapter, paying attention to the difference between a fixed-corner criterion and the finite-stage existential hypothesis.
5. Read Technical Engine B.
6. Check every dependency against the proof audit.

The proof audit, not an informal overview, is the authoritative ledger for the status of individual implications.

Route D: gap labelling and applications

Readers chiefly interested in trace ranges can read:

1. the Preface sections on gap labels and constructive representatives;
2. the statement and scope;
3. Dimension three;
4. Higher-dimensional lower bounds;
5. the odometer example and model-set application template.

This route should retain three distinctions:

• upper containment versus constructive lower bound;
• abstract existence of a \(K_0\)-class versus a native finite-PE module;
• an allowed label versus an open gap of a fixed Hamiltonian.

Route E: time-frequency and Gabor modules

Begin with the Rieffel--Heisenberg primer, then read Attaching the magnetic Heisenberg module before Technical Engine B. The primer explains the lattice and Gabor construction, the second chapter explains why the module is genuinely mixed, and the technical engine fixes the cocycle convention, balanced tensor product, adjoints, compactness, and trace formula.

The book does not claim an explicit finite family of coefficient-valued Gabor windows. Constructing such windows is a stronger research problem.

The genuinely twisted route

For the block-supported multiplier, the constructive route through Chapters 1--3 and 5 gives the coefficient absorption; the magnetic phase is already one. If \(\Theta_{ih}\neq0\) for some transverse direction, also read The residual magnetic phase and the cross-phase example.

The GPS theorem removes the permutation defect, but a locally constant diagonal phase remains. Its vanishing is a separate condition.

Status labels used throughout

Theorem means all displayed hypotheses and proof steps are supplied in the book.

Conditional theorem means a proof-critical assertion is retained as an explicit hypothesis.

Criterion is a necessary-and-sufficient reformulation, not an automatic vanishing theorem.

Existential hypothesis asks for vanishing at some allowed finite stage and is not being promoted to a choice-independent stable invariant.

Research direction is not used as an input to an unconditional result.

Each technical chapter now begins with a reader card describing its input, output, and role. The proof audit records the exact status of every nontrivial bridge.