Abstract

This book develops a constructive method for producing magnetic \(K_0\)-classes of aperiodic and quasicrystalline crossed products. Let \(\Sigma\) be a Cantor dynamical system with a \(\mathbb Z^p\)-action and let

\[ A_{\Sigma,\Theta} =C(\Sigma)\rtimes_{\sigma_\Theta}\mathbb Z^p \]

be its crossed product twisted by a constant magnetic multiplier. Magnetic gap labelling predicts that the trace range of this algebra is assembled from products of pattern frequencies with Pfaffian minors of the magnetic matrix \(\Theta\). Index theory and cohomology can constrain or identify this numerical range, but they need not exhibit finite, pattern-equivariant projective modules carrying the predicted traces. The purpose of the book is to construct such representatives and to identify the obstruction to doing so in a strict finite-propagation form.

Choose a primitive splitting

\[ \mathbb Z^p=G\oplus\mathbb Zh, \qquad G\cong\mathbb Z^{p-1}, \]

and assume that the height homeomorphism \(T_h\) is minimal. The coefficient input is an invariant height-coinvariant class

\[ f\in \left(C(\Sigma,\mathbb Z)_{T_h}\right)^G. \]

A clopen projection already represents the underlying positive class in the untwisted height algebra

\[ B_h=C(\Sigma)\rtimes_{T_h}\mathbb Z. \]

For a block-supported magnetic form, the new step is to enhance this representative to a graded finite projective module \(F_f\) with strict finite-propagation, pattern-equivariant transverse transport. Thus, for every \(s,t\in G\), the construction produces semilinear operators \(V_s\) satisfying

\[ V_s(\xi b)=V_s(\xi)\alpha_s(b), \qquad V_sV_t=V_{s+t}. \]

These identities make \(F_f\) suitable for equivariant descent. If \(E_T\) is a transverse Rieffel--Heisenberg module over the noncommutative torus \(A_{\Theta_G}\), the balanced product

\[ \mathcal E_{T,f} =E_T\widehat\otimes_{A_{\Theta_G}}X_f \]

is a mixed quasicrystalline module: every transverse magnetic time--frequency shift is coupled to the finite-PE height transport determined by the local pattern. Its trace factors as

\[ \tau_\mu([\mathcal E_{T,f}]) =\mu(f)\,\tau_{\Theta_G}([E_T]). \]

Consequently, for a magnetic form with no height--transverse entries, the construction realizes the height-one lower bound

\[ \mathbb Z[\mu] +D_h(\mu)\, \tau_{\Theta_G}\!\left(K_0(A_{\Theta_G})\right) \subseteq \tau_\mu\!\left(K_0(A_{\Sigma,\Theta})\right), \]

where

\[ D_h(\mu) =\mu\!\left( \left(C(\Sigma,\mathbb Z)_{T_h}\right)^G \right). \]

This includes every even Pfaffian minor supported in the transverse block, multiplied by the selected height-one pattern-frequency group. In dimension three it constructs the mixed term in the Benameur--Mathai magnetic frequency group without their finite-sheet Hypothesis (H). When combined with the known upper containment, this gives equality for the corresponding block-supported field. More generally, the result supplies a constructive lower bound in arbitrary dimension; it is not by itself an upper-bound theorem or a calculation of the full \(K\)-theory group.

The proof converts transverse invariance in height coinvariants into bounded clopen transport along height orbits. The resulting partial isometries can have square holonomy. A PV/Barlak comparison identifies its \(K_1\)-class with an integer transfer-function curl, minimality forces that curl to vanish, and a Giordano--Putnam--Skau normalizer retraction upgrades vanishing index to exact commuting transport. Rieffel descent then inserts the magnetic Heisenberg factor and yields the product trace formula.

For a general magnetic matrix, the same argument removes the integral and permutation defects but can leave a locally constant diagonal \(U(1)\)-valued cocycle. Strictification is conditional on this cocycle being a coboundary at an allowed finite stage. The book does not claim that this condition always holds or that it defines a choice-independent stable obstruction. Likewise, the present height-one theorem does not realize every coefficient layer predicted by magnetic gap labelling in higher dimensions. That extension requires a higher-rank absorption theorem coupling several coefficient directions simultaneously to a higher-dimensional Heisenberg module.

The construction therefore contributes to the broader program of understanding quasicrystal \(C^*\)-algebras by connecting three levels that are often studied separately: pattern-equivariant representatives of \(K\)-classes, their magnetic trace pairings, and the cohomological frequency groups appearing in gap labelling. It provides native mixed modules rather than passive torus pullbacks, while making explicit which parts of the expected trace range are proved, which depend on an obstruction-vanishing hypothesis, and which remain higher-rank research problems.

Twisted absorption for pattern-equivariant Heisenberg modules

Readers coming from tiling dynamics can begin with The proof without technical machinery and the tiling-theory dictionary. The complete PV/Barlak--KK comparison and Hilbert-module calculations are retained in Part III as technical certification chapters, so they need not be mastered before the construction itself is understood.

The introductory part also contains a short construction of Rieffel--Heisenberg modules, including their phase-space lattice, Gabor interpretation, projectivity, and trace. It explains the magnetic input used by the theorem without requiring the analytic descent calculations of Part III.

This book explains a constructive theorem about graded finite-projective representatives over twisted crossed products

\[ A_{\Sigma,\Theta} =C(\Sigma)\rtimes_{\sigma_\Theta}\mathbb Z^p, \]

where \(\Sigma\) is a Cantor dynamical system arising from an aperiodic pattern and \(\Theta\) is a constant magnetic field.

Why gap labels are K-theoretic

The observable algebra of an aperiodic solid is assembled from two kinds of information: the local patterns seen in the hull and the translations which move between them. In the Cantor-transversal model these become the coefficient algebra \(C(\Sigma)\) and the action of \(\mathbb Z^p\). A constant magnetic field changes the translation law: magnetic translations commute only up to the multiplier \(\sigma_\Theta\). The twisted crossed product \(A_{\Sigma,\Theta}\) is therefore the natural receptacle for covariant finite-range Hamiltonians and their norm limits.

If a Fermi energy lies in a spectral gap of such a Hamiltonian, continuous functional calculus produces a Fermi projection and hence a class

\[ [P_F]\in K_0(A_{\Sigma,\Theta}). \]

For every invariant probability measure \(\mu\) on the hull, the integrated density of states in that gap is

\[ \operatorname{IDS}(E_F) =\tau_\mu([P_F]). \]

Thus the universal group of possible gap labels is controlled by the ordered, traced \(K_0\)-group, and in particular by

\[ \tau_\mu\left(K_0(A_{\Sigma,\Theta})\right). \]

This statement concerns the allowed label group. It does not assert that every class in that group is realized as a gap of one fixed Hamiltonian. It separates the topological constraint on all covariant Hamiltonians from the additional spectral question of which gaps a particular model actually opens.

In the untwisted theory, the trace range is governed by frequencies of finite patches, encoded by integrals of integer-valued locally constant functions on \(\Sigma\). Magnetic twisting introduces new topological coefficients. The expected magnetic frequency group has the schematic form

\[ \mathcal G_\Theta(\mu)={} \sum_{\substack{I\subseteq\{1,\ldots,p\}\\|I|\text{ even}}} \operatorname{Pf}(\Theta_I)\mathbb Z_I[\mu], \]

where the groups \(\mathbb Z_I[\mu]\) are obtained from pattern frequencies after taking the appropriate directional coinvariants and invariants. Benameur and Mathai's magnetic gap-labelling conjecture is, in its general form, the upper containment

\[ \tau_\mu\left(K_0(A_{\Sigma,\Theta})\right) \subseteq \mathcal G_\Theta(\mu). \]

This upper-bound statement says that no unexpected trace values can occur. Equality, or even the reverse inclusion for a specified summand, is a separate existence problem: one must produce enough \(K_0\) classes with the predicted traces.

Why constructive representatives matter

Index theory, assembly maps, and spectral sequences can detect trace values without producing a finite pattern-equivariant projection or module which realizes them. For gap labelling this abstract information is already powerful, but a constructive representative answers several additional questions:

• how the patch-frequency class is incorporated into the magnetic module;
• whether the representative can be chosen local, finite propagation, and pattern equivariant;
• how the magnetic and aperiodic pieces interact before taking the trace; and
• which cohomological obstruction prevents an abstract class from descending to a strict module of the desired form.

The construction in this book makes the mixed nature of the class visible. If \(f\) is directional coinvariant data and \(x\in K_0(A_{\Theta_G})\) is a transverse noncommutative-torus class, the resulting graded module has

\[ \tau_\mu([P_{f,x}]) =\mu(f)\tau_{\Theta_G}(x). \]

When the torus pairing isolates a Pfaffian term, this becomes \(\operatorname{Pf}(\Theta_I)\mu(f)\). If \(f\) records nonconstant clopen pattern data, the resulting class is genuinely quasicrystalline: it is not merely the pullback of a class from the noncommutative torus. The formula is therefore not only a numerical coincidence in a trace calculation; it is implemented by a pattern-equivariant Rieffel--Heisenberg module.

This is the sense in which the theorem supplies a constructive lower-bound mechanism for magnetic gap labelling. Combined with an independently proved upper containment, such a construction can yield equality for the subgroup or dimension under consideration. The construction alone is not an upper-bound theorem and is not a complete calculation of \(K_0\).

Place in the K-theory program for quasicrystals

Understanding a quasicrystal \(C^*\)-algebra involves more than computing an abstract abelian group. A fuller program has at least four interlocking parts:

1. compute \(K_0\) and \(K_1\), using crossed-product exact sequences, groupoid methods, assembly, or spectral sequences;
2. identify geometric or pattern-equivariant representatives of the classes;
3. compute their pairings with invariant traces and higher cyclic cocycles; and
4. compare the resulting lower bounds with index-theoretic upper bounds to determine the actual gap-labelling group.

Twisted algebras sharpen this program. Even when deformation methods relate the abstract \(K\)-groups of twisted and untwisted crossed products, they do not by themselves identify finite PE representatives, compute the magnetic trace pairing, or explain the appearance of Pfaffian coefficients. Conversely, a trace calculation does not determine all of \(K_0\), since the trace can have a large kernel. Explicit modules provide a bridge between these two levels: they retain the local pattern data while carrying the global magnetic topology.

The present height-one theorem is one step in that larger program. It constructs a substantial family of mixed classes, removes the finite sheet condition from the block-supported height-one representative problem, works in arbitrary ambient dimension for every transverse Rieffel height, and isolates the remaining diagonal phase obstruction when the magnetic field has height--transverse components. It also shows precisely what the method does not yet supply. A full realization of all expected Pfaffian layers would require higher-rank absorption, in which several coefficient directions are absorbed simultaneously into a higher-dimensional Heisenberg module. A full equality theorem would additionally require the corresponding analytic or topological upper containment.

Conceptually, the book connects three constructive traditions: Rieffel's projective modules over noncommutative tori, Gabor modules for quasicrystals, and the cohomological magnetic gap-labelling framework of Benameur and Mathai. The new ingredient is to absorb clopen coinvariant data into finite PE transport and then strictify its dynamical defect before performing Rieffel descent.

The problem addressed here

The problem begins with a clopen coinvariant class. In three dimensions, Benameur and Mathai constructed the desired magnetic \(K_0\)-class when the clopen set admits a finite sheet decomposition, their Hypothesis (H). The construction developed here replaces those sheets by finite pattern-equivariant transport along one minimal “height” direction.

Current proof status. The operator construction, the PV/Barlak comparison identifying the transfer-function curl with \(d_2\), the GPS reduction under the normalizer hypothesis, and the descent/trace calculation are proved. The block-supported height-one theorem and its stated lower bounds are unconditional under the displayed minimal-height and normalizer hypotheses. A full magnetic matrix still requires the separate finite-stage diagonal phase-coboundary hypothesis.

The central result has two forms.

1. Block-supported magnetic field. If the magnetic multiplier has no height–transverse entries, every invariant height-coinvariant class admits a strict finite PE equivariant representative when the height homeomorphism is minimal.
2. General magnetic field. Under the same minimal-height hypothesis, the norm-continuous phase-scaling path (A.20)--(A.22) identifies its primary \(K_1\)-curvature with the block-supported one. The GPS construction then removes the nonabelian permutation defect. What remains in a fixed corner is an explicit locally constant diagonal \(2\)-cocycle. Its vanishing is necessary and sufficient for diagonal strictification in that corner. Across representatives and stabilizations, the book uses only an existential vanishing hypothesis and does not claim a choice-independent stable invariant.

The proof is constructive. It produces matrices whose entries are clopen-weighted powers of the height shift. After Rieffel descent, each generator acts as

\[ \text{ordinary time-frequency shift} \quad\times\quad \text{finite PE height transport}. \]

Technical Engine B proves the balanced correspondence, compactness, and product trace formula. The native time-frequency realization works for every permitted Rieffel height, not only the top-dimensional Pfaffian class.

flowchart LR
    f["Invariant clopen coinvariant f"]
    P["Full clopen projection P"]
    W["Individual PE transports W_i"]
    K["Pairwise K_1 curvature"]
    GPS["GPS normalizer retraction"]
    phase["Diagonal magnetic phase ν"]
    module["Mixed Heisenberg module"]

    f --> P --> W --> K
    K -->|"proved comparison + minimal height"| GPS
    GPS --> phase
    phase -->|"block support: ν = 1"| module
    phase -->|"general Θ: coboundary at an allowed finite stage"| module

Main references

The proof uses:

• Benameur–Mathai’s explicit magnetic gap-label construction;
• Barlak’s formula for the \(d_2\)-differential;
• the normalizer decomposition of Giordano–Putnam–Skau;
• Rieffel’s Heisenberg modules and their Gabor realizations.

Precise links and theorem locations appear in Sources. The consequences for magnetic gap labelling are developed separately in dimension three and in higher dimensions.

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.

The proof without technical machinery

This chapter gives the whole argument in the language of patches, bounded orbit transport, and holonomy. It is intended to be read before the formal proof. Nothing proved later is being replaced: the two technically difficult bridges are stated here as named engines and proved in Part III.

Reader card. This chapter assumes familiarity with tiling hulls or Cantor dynamical systems, clopen patch cylinders, and invariant measures. It does not assume \(KK\)-theory. Its output is a map of the proof and a precise explanation of where the technical machinery enters.

1. The question from gap labelling

Let \(\Sigma\) be the Cantor transversal of an aperiodic pattern and let \(\mathbb Z^p\) act by translations. A finite-range covariant Hamiltonian belongs to, or is represented in, the crossed product

\[ A_{\Sigma,\Theta} =C(\Sigma)\rtimes_{\sigma_\Theta}\mathbb Z^p. \]

If a Fermi energy lies in a spectral gap, its Fermi projection determines a class in \(K_0(A_{\Sigma,\Theta})\). Pairing this class with an invariant measure gives the integrated density of states. The constructive problem is therefore not merely to predict a number, but to exhibit a finite pattern-equivariant projective representative having that trace.

The labels targeted in this book have the form

\[ \text{magnetic Pfaffian}\times\text{pattern frequency}. \]

The two factors come from different pieces of geometry. The pattern frequency is absorbed into a coefficient module. A Rieffel--Heisenberg module supplies the magnetic factor. Its abbreviated phase-space and lattice construction is given in A short introduction to Rieffel--Heisenberg modules.

2. Choose one height direction

Split the translation lattice as

\[ \mathbb Z^p=G\oplus\mathbb Zh, \qquad G\cong\mathbb Z^{p-1}, \]

and assume that the homeomorphism \(T_h\) is minimal. The height algebra is

\[ B_h=C(\Sigma)\rtimes_{T_h}\mathbb Z. \]

At the level of \(K\)-theory, integer-valued locally constant functions are identified when their difference is a height coboundary:

\[ K_0(B_h) \cong C(\Sigma,\mathbb Z)_{T_h} ={} \frac{C(\Sigma,\mathbb Z)}{(T_h-1)C(\Sigma,\mathbb Z)}. \]

Geometrically, a term \((T_h-1)a\) moves a finite, locally determined amount of integer mass along height orbits. Coinvariants therefore remember the amount of pattern mass but forget bounded redistributions in the height direction.

The input to the theorem is a class

\[ [f]\in \left(C(\Sigma,\mathbb Z)_{T_h}\right)^G. \]

The superscript does not say that a particular function \(f\) is fixed by every transverse translation. It says that translating \(f\) changes it by a height coboundary.

3. Turn a signed pattern weight into a projection

The function \(f\) may take negative values. Choose \(N\) so that

\[ g=f+N\geq1. \]

From a nonnegative integer-valued function \(g\), form a diagonal clopen projection \(P(g)\) whose pointwise rank is \(g(x)\). Then

\[ [P(g)]-[1_N]=[f]. \]

This is why the natural output is sometimes a graded module: an arbitrary integer-valued pattern class is a difference of two positive finite projective modules. For positive inputs, the construction can produce an ordinary ungraded module.

4. What is new: the transverse structure, not the projection

The projection in the preceding section is not by itself the new object. A clopen projection already represents a positive pattern-frequency class in the untwisted height algebra, and a graded difference represents an arbitrary integer-valued class. If we write this coefficient module as \(F_f\), then

\[ [F_f]=x_f\in K_0(B_h), \qquad \tau_\mu([F_f])=\mu(f). \]

The real construction is to enhance \(F_f\) by strict transverse transport. For every \(s\in G\), we seek a finite-propagation PE operator \(V_s\) such that

\[ V_s(\xi b)=V_s(\xi)\alpha_s(b), \qquad V_sV_t=V_{s+t}. \tag{O.A} \]

The first identity says that \(V_s\) covers the transverse action on the height algebra. The second says that the transports form an honest group action, rather than a collection of maps agreeing only in \(K\)-theory or up to a phase. Chapters 1--5 are devoted to constructing these operators and removing their holonomy. Thus the first substantive output is the pair

\[ (F_f,V), \]

not merely the underlying projection defining \(F_f\).

This enhancement is designed to accept a transverse magnetic module. Put

\[ A=B_h\rtimes_{\alpha,\sigma}G, \qquad D=C^*(G,\sigma), \qquad X_f=F_f\widehat\otimes_{B_h}A. \]

The coupled operators

\[ L_s(\xi\otimes a)=V_s\xi\otimes U_sa \]

give \(X_f\) a left \(D\)-action. Hence a transverse Rieffel--Heisenberg module \(E_T\) can be balanced against it:

\[ [f] \longmapsto F_f \longmapsto (F_f,V) \longmapsto \mathcal E_{T,f}=E_T\widehat\otimes_DX_f \longmapsto \mu(f)\tau_D([E_T]). \tag{O.B} \]

Keeping the two actions independent would give only an external product in which the pattern factor is passive. Balancing over \(D\) identifies each time-frequency shift on \(E_T\) with the same transverse translation that applies the pattern-dependent transport \(V_s\). This is what makes \(\mathcal E_{T,f}\) a genuinely mixed quasicrystalline module.

This perspective also clarifies the comparison with Benameur--Mathai. Their index theorem detects the relevant trace labels abstractly, while Hypothesis (H) supplies a rigid finite-sheet model for the strict transports needed in their explicit matrix homomorphism. The construction here replaces those sheets by class-dependent PE transports. When Hypothesis (H) holds, its sheet permutations should be regarded as a special source of the \(V_s\). After forgetting transverse equivariance, both constructions represent the same coefficient class in \(K_0(B_h)\). The book does not, however, infer equality of the descended modules merely from equality of their traces. A complete comparison would have to prove, with compatible choices and conventions,

\[ [\Phi_\Lambda(e)]=[\lambda_{F_\Lambda}(e)] \quad\text{in }K_0(A) \]

for the relevant transverse projection \(e\).

Finally, allowing every invariant height-coinvariant \(f\) and every transverse torus class realizes the full subgroup

\[ D_h(\mu),\tau_D(K_0(D)) \]

associated with the chosen height-one splitting, where

\[ D_h(\mu) =\mu!\left( \bigl(C(\Sigma,\mathbb Z)_{T_h}\bigr)^G \right). \]

The product denotes the additive group generated by products of elements from the two displayed groups. In arbitrary dimension this is a constructively realized height-one layer of the Benameur--Mathai frequency group, not automatically the whole predicted group. The remaining coefficient layers require higher-rank absorption.

5. Transverse invariance becomes bounded flow

For every generator \(T_i\) of \(G\), invariance of \([g]\) means that a locally constant integer function \(h_i\) satisfies

\[ (T_i-1)g=(T_h-1)h_i. \tag{O.1} \]

After adding a constant, take \(h_i\geq0\). Equation (O.1) is a conservation law. The discrepancy between \(T_i(g)\) and \(g\) is exactly the boundary of a bounded flow in the height direction.

Because the functions are locally constant and bounded, the matching can be performed on a finite clopen partition. It produces a partial isometry

\[ W_i:\alpha_i(P)B_h^n\longrightarrow PB_h^n \]

whose entries are finite sums of clopen-weighted height shifts \(1_CU_h^m\). In tiling language, \(W_i\) is a finite-range rule: a bounded patch around a point determines how far that component is moved along its height orbit.

The construction proves more than finite Fourier support. Each \(W_i\) is supported on a compact-open bisection, so it has a well-defined geometric symbol and normalizes the diagonal algebra of clopen sets.

6. The transports may have holonomy

Individual transports do not yet define a \(G\)-action. For two transverse generators, compare the two routes around a square. Their ratio is

\[ \Omega_{ij} =W_i\alpha_i(W_j) \bigl(W_j\alpha_j(W_i)\bigr)^*. \tag{O.2} \]

If \(\Omega_{ij}=1\), the two transports commute. In general, \(\Omega_{ij}\) is the holonomy around one transverse square.

The transfer functions have a parallel, purely integer-valued curvature:

\[ \kappa_{ij} =(T_j-1)h_i-(T_i-1)h_j. \tag{O.3} \]

Applying \(T_h-1\) and using (O.1) shows that

\[ (T_h-1)\kappa_{ij}=0. \]

Thus \(\kappa_{ij}\) is constant on each height orbit. Minimality of \(T_h\) makes it constant on all of \(\Sigma\). An invariant measure gives

\[ \int_\Sigma\kappa_{ij}\,d\mu=0, \]

so the constant is zero:

\[ \kappa_{ij}=0. \tag{O.4} \]

This short calculation is the dynamical reason the theorem works.

7. Technical Engine A: certify that the operator obstruction vanishes

Equations (O.2) and (O.3) live in different worlds. The first is a unitary in a crossed-product corner; the second is an integer-valued function. The load-bearing comparison theorem proves

\[ \partial_h[\Omega_{ij}] =\pm[\kappa_{ij}], \tag{O.5} \]

with the target and quotient interpreted precisely. It follows from (O.4) that

\[ [\Omega_{ij}]=0 \quad\text{in the actual corner }K_1. \tag{O.6} \]

This is where the PV sequence, Barlak's \(d_2\), the equivariant Toeplitz triangle, and enhanced \(KK\)-theory enter. They do not create the transports. They certify that the concrete holonomy built above has zero index. The full proof of this bridge is isolated in Technical Engine A.

8. Replace zero-index holonomy by exact commutation

Vanishing in \(K_1\) is still weaker than the equality \(\Omega_{ij}=1\). The next step uses the geometry of a single minimal Cantor homeomorphism.

The full diagonal corner can be written as the transformation groupoid of a coloured minimal successor map. The bisection symbols of the \(W_i\) normalize its topological full group. The Giordano--Putnam--Skau retraction separates such a normalizer into an index-zero orbit rearrangement and a part commuting with the successor.

Because (O.6) says that the orbit rearrangement has zero index, finite-PE gauges can absorb it. One obtains corrected transports whose underlying bisection symbols commute simultaneously. This is the strictification step, developed in the GPS chapter.

9. Separate permutation curvature from magnetic phase

For a block-supported magnetic form,

\[ \Theta_{ih}=0 \qquad(i\in G), \]

transverse automorphisms do not rotate the height unitary. Equality of the corrected bisections then gives equality of the corresponding operators. The coefficient module carries a strict \(G\)-equivariant structure.

For a general magnetic matrix, the corrected operators may differ by a locally constant diagonal phase. The remaining datum is a cocycle

\[ \nu\in Z^2\bigl(G,U(C_{\mathrm{lc}}(Y))\bigr). \]

In a fixed corner, diagonal gauges remove it exactly when \([\nu]=0\). The book does not claim that this phase always vanishes, nor that the phases from all choices form a choice-independent stable invariant.

10. Technical Engine B: attach the magnetic module

Let \(F_f\) be the strict coefficient module just constructed, and let \(E_T\) be a Rieffel--Heisenberg module over the transverse twisted group algebra. Equivariant descent forms a correspondence \(X_f\), and then

\[ \mathcal E_{T,f} =E_T\widehat\otimes X_f. \]

Conceptually, every transverse magnetic time-frequency shift is accompanied by the finite-PE height transport determined by the local pattern. The analytic details prove that the balanced action is well defined, adjointable, compact, and finite projective. They also prove

\[ \tau_\mu([\mathcal E_{T,f}]) =\mu(f)\tau_{\Theta_G}([E_T]). \tag{O.7} \]

The conceptual construction is explained in Attaching the magnetic Heisenberg module; the complete verification is in Technical Engine B.

11. What the theorem contributes

If the torus class isolates a Pfaffian term, (O.7) becomes

\[ \operatorname{Pf}(\Theta_I)\mu(f). \]

For nonconstant clopen data \(f\), this is not merely a noncommutative-torus class pulled back to the quasicrystal algebra. The projective module itself contains local pattern-dependent transport.

The result is a constructive lower bound for magnetic gap labelling. It does not by itself compute all of \(K_0\), prove the upper containment, or show that every allowed label is an open gap of one particular Hamiltonian.

The proof in one diagram

flowchart TB
    patch["locally constant pattern weight f"]
    coinv["G-invariant height coinvariant [f]"]
    flow["bounded height flows h_i"]
    transport["finite PE bisection transports W_i"]
    holonomy["square holonomy Omega_ij"]
    curl["integer curl kappa_ij"]
    kk["PV / Barlak comparison"]
    gps["GPS finite-PE gauge correction"]
    phase["residual diagonal magnetic phase"]
    strict["strict equivariant coefficient module F_f"]
    rieffel["Rieffel--Heisenberg module E_T"]
    mixed["mixed PE projective module"]
    trace["Pfaffian times pattern frequency"]

    patch --> coinv --> flow --> transport --> holonomy
    flow --> curl
    curl -->|"minimality: zero"| kk
    holonomy --> kk -->|"zero K_1 index"| gps
    gps --> phase
    phase -->|"block support, or phase coboundary"| strict
    strict --> mixed
    rieffel --> mixed --> trace

The running Sturmian example makes the flow and curvature steps explicit before the general proof is read.

Dictionary for tiling theorists

This chapter translates the main objects between tiling dynamics, groupoids, operator algebras, and the proof. It is not a replacement for the Glossary; it is a map between mathematical dialects.

Reader card. Read across a row whenever an operator-algebraic term obscures a geometric operation. The last column says why that object is present in the proof.

Four distinctions to keep visible

A function versus its coinvariant class

A function \(f\) is concrete patch data. Its class \([f]\) forgets bounded height redistributions. Transverse invariance is imposed on \([f]\), not necessarily on the chosen representative \(f\).

Equality in \(K\)-theory versus an explicit isomorphism

The equality

\[ [P]=[\alpha_i(P)] \]

says that two projective modules have the same stable class. The theorem needs an explicit, finite-PE transport between chosen representatives. The clopen-flow construction supplies that extra locality.

Vanishing index versus exact commutation

The equation

\[ [\Omega_{ij}]=0\in K_1 \]

does not literally say \(\Omega_{ij}=1\). The GPS step turns zero index into finite-PE gauge corrections whose geometric symbols commute. In a magnetic field one must still check the remaining diagonal phase.

A gap-label group versus open spectral gaps

The trace range of \(K_0\) constrains the values that gaps may carry across covariant Hamiltonians. It does not say that every class is realized by an open gap of one fixed Hamiltonian. That is a further spectral problem.

A minimal amount of \(K\)-theory

Only three ideas are needed on a first reading.

1. A projection represents a stable vector bundle or finite projective module.
2. \(K_0\) permits formal differences of projections, which is why signed pattern data can be used.
3. A unitary represents a \(K_1\)-class. Here that class measures the index of square holonomy.

The PV sequence computes the \(K\)-theory of crossing by one copy of \(\mathbb Z\). The technical curvature chapter uses it to translate the unitary class \([\Omega_{ij}]\) into the integer curl \(\kappa_{ij}\).

A minimal amount of \(KK\)-theory

For this proof, it is enough initially to regard \(KK\)-theory as a category in which:

• homomorphisms and geometric correspondences define generalized maps;
• homotopy-equivalent constructions become equal;
• exact sequences become exact triangles; and
• compatible constructions can be composed by the Kasparov product.

The book uses this language because \(1-\alpha_h\) is an additive \(KK\)-morphism rather than a \(*\)-homomorphism from which one could form an ordinary algebraic mapping cone. The enhanced equivariant framework provides the coherent cofibers needed to compare the PV filtration with Barlak's differential. Readers may use the comparison formula as a named theorem until reaching Part III.

A minimal amount of Morita equivalence

A full corner \(PAP\) and the algebra \(A\) carry the same stable module theory. Passing to a colored full corner lets the proof replace a matrix projection by the orbit groupoid of a single minimal successor map. This is not a claim that the corner and the original algebra are literally equal; it is a controlled equivalence preserving the relevant \(K\)-theory and trace data.

A short introduction to Rieffel--Heisenberg modules

Rieffel--Heisenberg modules provide the magnetic part of the construction in this book. They are explicit finite projective modules over noncommutative tori, built from translations and modulations on an ordinary phase space. Their trace supplies the Pfaffian factor in a magnetic gap label.

Reader card. This chapter is an abbreviated construction, not a replacement for Rieffel's imprimitivity theorem. It assumes elementary Fourier analysis and familiarity with lattices. The exact right-module convention, compactness proof, and trace calculation used later are recorded in Technical Engine B.

1. The noncommutative torus as magnetic translations

Let

\[ G=\mathbb Z^q \]

and let \(\sigma:G\times G\to U(1)\) be a multiplier determined by a skew-symmetric matrix \(\Theta\). The twisted group algebra

\[ D=C^*(G,\sigma)=A_\Theta \]

is generated by unitaries \(u_g\) satisfying

\[ u_gu_h=\sigma(g,h)u_{g+h}. \]

For coordinate generators this is the familiar relation

\[ u_ju_k=e^{2\pi i\Theta_{jk}}u_ku_j. \]

These are magnetic translation relations: translations commute only up to the flux through the parallelogram they span.

2. Translation and modulation on phase space

Choose a locally compact abelian group \(M\), such as

\[ M=\mathbb R^r\times\mathbb Z^{q-2r}, \]

and write \(\widehat M\) for its Pontryagin dual. A point

\[ (x,\omega)\in M\times\widehat M \]

acts on a function \(\xi\) on \(M\) by a time-frequency shift

\[ \bigl(\pi(x,\omega)\xi\bigr)(t) =\omega(t)\xi(t-x), \tag{R.1} \]

up to a harmless Weyl phase used to make the cocycle alternating.

Translations and modulations do not commute:

\[ \pi(x,\omega)\pi(y,\eta) =c\bigl((x,\omega),(y,\eta)\bigr) \pi(x+y,\omega\eta), \tag{R.2} \]

where \(c\) is the Heisenberg cocycle. Thus phase space already carries the same projective behavior as a noncommutative torus.

3. Embed the lattice

Choose an injective homomorphism

\[ T:G\longrightarrow M\times\widehat M \]

whose image is a lattice and whose Heisenberg commutator form agrees with the prescribed magnetic form \(\Theta\). Then

\[ R_g=\pi(Tg) \]

satisfies

\[ R_gR_h=\sigma(g,h)R_{g+h}. \tag{R.3} \]

Consequently the generators \(u_g\) of \(A_\Theta\) can act by the time-frequency operators \(R_g\). The choice of \(M\) and \(T\) is the geometric input. Different values of \(r\) produce Rieffel modules of different heights.

In dimension two, one may take \(M=\mathbb R\). One coordinate generator acts as translation and the other as modulation. Their failure to commute is exactly the phase \(e^{2\pi i\theta}\) defining the rotation algebra \(A_\theta\).

4. Turn test functions into a module

Start with a well-behaved space of functions on \(M\), for example a Schwartz or Feichtinger space. After choosing a consistent left- or right-module convention, the action of a Fourier series is schematically

\[ \xi\cdot\left(\sum_{g\in G}a_gu_g\right) =\sum_{g\in G}a_gR_g\xi. \tag{R.4} \]

The \(A_\Theta\)-valued inner product samples ordinary Hilbert-space coefficients along the lattice:

\[ \langle\xi,\eta\rangle_{A_\Theta} =\sum_{g\in G} \langle \xi,R_g\eta\rangle\,u_g, \tag{R.5} \]

with the order, conjugation, and possible inverse adjusted to the selected right-module convention. Formula (R.5) is the essential idea: its coefficients are samples of the short-time Fourier transform, or ambiguity function, of \(\xi\) and \(\eta\).

Completing in the norm determined by this inner product gives a Hilbert \(A_\Theta\)-module \(E_T\). The exact convention used in this book is written in formulas (B.7)--(B.8b) of Technical Engine B.

5. Why the module is finite projective

The lattice \(T(G)\) has a complementary, or adjoint, lattice in phase space. The two lattice actions commute, and Rieffel's Heisenberg imprimitivity theorem identifies the completed function space as an equivalence bimodule between the corresponding twisted group algebras.

Because the lattice quotient is compact, the module is finitely generated and projective. Concretely, one can choose finitely many vectors \(g_1,\ldots,g_N\) forming a module frame. The matrix

\[ p_{ij}=\langle g_i,g_j\rangle_{A_\Theta} \tag{R.6} \]

is a projection in \(M_N(A_\Theta)\), and

\[ E_T\cong pA_\Theta^N. \]

Under the time-frequency interpretation, a finite module frame corresponds to a multi-window Gabor frame generated by the lattice \(T(G)\). This is a powerful analytic realization of the module, but the absorption proof in this book needs only a finite projection representing \(E_T\); it does not construct new coefficient-valued Gabor windows.

6. Trace, volume, and Pfaffians

The canonical trace on \(A_\Theta\) takes the zero Fourier coefficient. Applied to the projection (R.6), Rieffel's calculation expresses the module trace through the volume or determinant associated with the lattice embedding \(T\).

For the two-dimensional rotation algebra and the basic positive module, normalized with \(0<\theta<1\),

\[ \tau_\theta(E_\theta)=\theta. \]

In higher dimensions, the trace range of \(K_0(A_\Theta)\) is generated by the Pfaffian minors

\[ \operatorname{Pf}(\Theta_I), \qquad |I|\ \text{even}. \]

An elementary Heisenberg module has positive trace determined by its embedding. Depending on the chosen embedding, it may represent a positive combination or a positive multiple of the desired component. An isolated signed Pfaffian term may therefore require a graded difference of positive modules. The book calls the output an actual projective module only when the chosen trace and \(K_0\)-class are positive; otherwise it uses a virtual class.

7. What absorption adds

The Rieffel module by itself depends only on the constant magnetic form. It does not contain nonconstant patch data from \(\Sigma\). Its trace lies in the ordinary noncommutative-torus trace group.

The absorption theorem separately constructs a pattern-dependent coefficient module \(F_f\) carrying strict finite-PE transverse transports. It then forms

\[ \mathcal E_{T,f} =E_T\widehat\otimes_{A_{\Theta_G}}X_f. \]

The lattice shift \(R_g\) is thereby coupled to the PE height transport \(V_g\). Schematically, a generator acts as

\[ \text{time-frequency shift }R_g \quad\times\quad \text{pattern-dependent transport }V_g. \]

The trace factors:

\[ \tau_\mu([\mathcal E_{T,f}]) =\mu(f)\tau_{\Theta_G}([E_T]). \tag{R.7} \]

Thus Rieffel's module supplies the magnetic topology and the Pfaffian, while twisted absorption supplies the quasicrystalline frequency and the finite-PE dependence on local patterns.

For the running Fibonacci cylinder and a basic two-dimensional module,

\[ \mu(f)=\frac1\varphi, \qquad \tau_\mu([\mathcal E_{\theta,f}]) =\frac{\theta}{\varphi}. \]

8. What to remember

1. A phase-space lattice turns translations and modulations into the projective relations of \(A_\Theta\).
2. Sampling time-frequency coefficients along that lattice gives the \(A_\Theta\)-valued inner product.
3. Heisenberg imprimitivity makes the completed space finite projective; finite module frames are multi-window Gabor frames.
4. Rieffel's trace calculation supplies the magnetic Pfaffian factor.
5. The new absorption argument does not reconstruct this magnetic engine. It couples it to a strict, pattern-dependent coefficient module.

The original references and further Gabor-module literature are listed in Sources.

Example: a product of Sturmian systems

Reader card. This is the running example for Parts I and II. It can be read immediately after the proof overview. The example makes the coinvariant equation, transfer functions, zero curl, and product trace explicit; the general chapters explain why the same steps work without these simplifying identities.

This example makes every ingredient visible: a minimal height direction, an irrational clopen frequency, explicit coinvariant transports, and a mixed trace.

The transfer functions in this example have zero curl directly. The passage from that integer calculation to vanishing of the Barlak corner \(K_1\)-defect uses the PV/Barlak comparison proved in Technical Engine A.

The dynamical system

Let \((X_j,S_j)\), for \(j=0,1,2\), be Sturmian subshifts of slopes \(\alpha_j\). Choose the slopes so that the diagonal product map

\[ T_h=S_0\times S_1\times S_2 \]

is minimal. A convenient sufficient choice is that the associated rotation vector is totally irrational; equivalently, choose the factors so that the diagonal product rotation is minimal. Put

\[ \Sigma=X_0\times X_1\times X_2, \qquad T_1=S_0\times 1\times1, \qquad T_2=1\times S_1\times1. \]

The three maps commute. Their exponent vectors

\[ h=(1,1,1),\qquad e_1=(1,0,0),\qquad e_2=(0,1,0) \]

form a unimodular basis of the original product action.

flowchart LR
    X0["Sturmian factor X_0"]
    X1["Sturmian factor X_1"]
    X2["Sturmian factor X_2"]
    S["Sigma = X_0 x X_1 x X_2"]
    H["height: T_h = (S_0,S_1,S_2)"]
    A["transverse: T_1 = (S_0,1,1)"]
    B["transverse: T_2 = (1,S_1,1)"]
    X0 --> S
    X1 --> S
    X2 --> S
    S --> H
    S --> A
    S --> B

The warning here is important: a product of minimal systems need not be minimal. The theorem requires minimality of the chosen single height map, so it must be checked rather than inferred factor by factor.

An explicit coinvariant class

Let \(C_a\subset X_0\) be the cylinder in which the symbol at the origin is \(a\), and define

\[ g=1_{C_a\times X_1\times X_2}. \]

Because \(g\) depends only on the first coordinate,

\[ \alpha_1(g)=\alpha_h(g), \qquad \alpha_2(g)=g. \]

Thus the transverse invariance equations are solved by

\[ h_1=g, \qquad h_2=0, \]

up to the covariance sign convention. Consequently

\[ (\alpha_i-1)g=(\alpha_h-1)h_i, \]

so \([g]\in C(\Sigma,\mathbb Z)_{T_h}\) is fixed by both transverse generators. The pairwise curvature is visibly zero:

\[ \kappa_{12}=(\alpha_2-1)h_1-(\alpha_1-1)h_2=0. \]

The general proof produces the same conclusion without needing these particularly simple transfer functions.

The trace

For the Fibonacci substitution

\[ a\mapsto ab, \qquad b\mapsto a, \]

the frequency of \(a\) is \(\varphi^{-1}\), where \(\varphi=(1+\sqrt5)/2\). Hence the invariant product measure satisfies

\[ \mu(g)=\frac1\varphi. \]

Assume first that the parameter is normalized so that \(0<\theta=\Theta_{12}<1\). Let \(E_\theta\) be the corresponding actual basic Rieffel module for the transverse two-torus. With the usual positive orientation,

\[ \tau([E_\theta])=\theta, \]

and the mixed module has

\[ \tau_\mu([\mathcal E_{\theta,g}]) =\frac{\theta}{\varphi}. \]

For a parameter outside this normalized positive range, the same formula describes the virtual \(K_0\)-class of trace \(\theta\), and the mixed output is a graded difference. Alternatively, choose an actual positive Rieffel class of trace \(m+n\theta>0\); its mixed trace is \((m+n\theta)/\varphi\). Thus no actual projective module with nonpositive trace is being asserted.

This number is the product of a pattern frequency and a magnetic Rieffel trace. It is exactly the kind of mixed gap label that is invisible if one only lists the two factors separately.

Dynamical and operator-algebraic setup

Reader card. This chapter fixes notation and translates the selected height direction into an iterated crossed product. Readers unfamiliar with crossed-product terminology may keep the tiling-theory dictionary open alongside it. The only computation used immediately is the height PV identification of \(K_0\) with coinvariants and \(K_1\) with invariants.

One height direction

Let \(\Sigma\) be a Cantor set. Fix commuting homeomorphisms

\[ T_h,T_1,\ldots,T_q, \qquad q=p-1, \]

and assume that \(T_h\) is minimal. Write

\[ H=\langle T_h\rangle\cong\mathbb Z, \qquad G=\langle T_1,\ldots,T_q\rangle\cong\mathbb Z^q. \]

The coefficient algebra is

\[ B_h=C(\Sigma)\rtimes_{T_h}\mathbb Z. \]

If \(U_h\) denotes its canonical unitary, our covariance convention is

\[ U_h aU_h^{\ast}=\alpha_h(a), \qquad a\in C(\Sigma). \]

Since \(K_1(C(\Sigma))=0\), the Pimsner–Voiculescu sequence gives

\[ K_0(B_h)\cong C(\Sigma,\mathbb Z)_{T_h}, \qquad K_1(B_h)\cong C(\Sigma,\mathbb Z)^{T_h}. \]

Minimality implies

\[ C(\Sigma,\mathbb Z)^{T_h}=\mathbb Z. \]

The magnetic crossed product

Let \(\Theta\in M_p(\mathbb R)\) be skew-symmetric. Splitting \(\mathbb Z^p=G\oplus H\) gives an iterated presentation

\[ A_{\Sigma,\Theta} \cong B_h\rtimes_{\alpha^\Theta,\sigma_{\Theta_G}}G. \]

The automorphism \(\alpha_i^\Theta\) acts on \(C(\Sigma)\) by \(T_i\) and rotates the height unitary by

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

up to the harmless sign determined by the cocycle convention.

The block-supported case is

\[ \Theta_{ih}=0 \qquad (i=1,\ldots,q). \]

Then every transverse automorphism fixes \(U_h\). This is exactly the form \(i_*\sigma\) used in Benameur–Mathai’s explicit construction.

Equivalently, the selected height vector lies in the radical of the magnetic form. Thus a nondegenerate standard symplectic cocycle on an entire even-dimensional ambient lattice is not block-supported for any height-one splitting. What is included is a standard symplectic cocycle on an even-dimensional transverse block, for example \(J_{2m}\oplus0\) on \(\mathbb Z^{2m}\oplus\mathbb Zh\).

flowchart LR
    full["Z^p"]
    G["G = Z^(p−1)<br/>transverse"]
    H["H = Z<br/>height"]
    Bh["B_h = C(Σ) ⋊ H"]
    A["A_(Σ,Θ) = B_h ⋊ G"]

    full --> G
    full --> H
    H --> Bh
    G --> A
    Bh --> A

The input class

Let

\[ f\in\bigl(C(\Sigma,\mathbb Z)_{T_h}\bigr)^G. \]

Its image in \(K_0(B_h)\) will be denoted \(x_f\). The invariance is only an equality of \(K_0\)-classes. It does not yet supply commuting module isomorphisms.

The construction problem is therefore:

Find a graded finite projective \(B_h\)-module representing \(x_f\), together with finite-propagation semilinear maps \(V_i\) satisfying \(V_iV_j=V_jV_i\).

Once this is done, equivariant descent pairs the coefficient module with any Rieffel–Heisenberg module over the transverse noncommutative torus.

Pattern equivariance

For this book, a finite PE height transport is a matrix whose entries are finite sums

\[ \sum_n 1_{C_n}U_h^n, \]

where the \(C_n\) are clopen and only finitely many integers \(n\) occur. In the regular representation on \(\ell^2(\mathbb Z)\), it is a finite-range shift whose displacement is determined by a finite clopen partition—equivalently, by a finite patch.

This broad finite-Fourier condition does not imply that a unitary or partial isometry normalizes the diagonal. Chapter 3 and the exact two-stage lemma use the stronger notion of a finite PE bisection transport: its source and range corners are diagonal corners, it normalizes those diagonals, and its support is the graph of a coefficient-free compact-open height bisection. Such a transport has a well-defined full-group symbol and GPS index. The transports constructed from the clopen orbit partitions in Chapter 1 have this stronger form; an arbitrary matrix of finite Fourier sums need not.

The height-one twisted absorption theorem

Reader card. The first section is the precise strictification lemma. The second is the main theorem for classes constructed from patch data. The third explains how a transverse magnetic module supplies the Pfaffian trace factor. For the geometric narrative behind the hypotheses, read The proof without technical machinery first.

Plain-language statement

Choose one translation as a minimal height direction. Suppose a locally constant integer pattern weight is unchanged by every transverse translation after allowing bounded redistribution along height orbits. The construction turns that weight into a graded finite projective module and builds finite-range PE transports for all transverse generators.

When the magnetic form has no height--transverse entries, those transports can be corrected to a strict commuting family. Pairing the resulting coefficient module with a transverse Rieffel--Heisenberg module produces a mixed quasicrystalline class whose trace is

\[ \text{pattern frequency}\times\text{transverse magnetic trace}. \]

For a general magnetic form, the same construction removes the integer and permutation defects but may leave a locally constant diagonal phase. Its finite-stage coboundary condition is an additional hypothesis, not an automatic consequence of minimality.

We first state the exact absorption lemma under its required normalizer hypothesis, then the minimal-height consequence of the PV/Barlak comparison proved in Technical Engine A.

Exact two-stage lemma

Let a \(G=\mathbb Z^q\)-invariant clopen finite projective class over \(B_h\) be represented, after a common finite stabilization, by a full diagonal projection \(P\) and individual finite PE bisection transports

\[ W_i^{\ast}W_i=\alpha_i(P), \qquad W_iW_i^{\ast}=P. \]

This means more than being a finite Fourier partial isometry. Put \(D_P=PM_n(C(\Sigma))P\) and \(D_i=\alpha_i(P)M_n(C(\Sigma))\alpha_i(P)\). We require

\[ W_iD_iW_i^\ast=D_P, \]

and require the coefficient-free support of \(W_i\) to be a compact-open height bisection. Equivalently, \(W_i\) is a diagonal-normalizing, finite-propagation PE partial isometry with a well-defined bisection symbol. The transports constructed in Chapter 1 have this property. A general unitary matrix of finite Fourier sums need not have it and is not an input to this lemma.

For each pair define the square defect

\[ \Omega_{ij} =W_i\alpha_i(W_j) \bigl(W_j\alpha_j(W_i)\bigr)^{\ast} \in U(PM_n(B_h)P). \]

Theorem (height-one twisted absorption).

1. If \([\Omega_{ij}]=0\) in \(K_1(PM_n(B_h)P)\) for every pair \(i,j\), finite zero-index PE gauges make all underlying symbol transports commute simultaneously.
2. After this correction, the remaining Busby–Smith cocycle is a locally constant diagonal cocycle \[ \nu\in Z^2\!\left(G,U(C_{\mathrm{lc}}(Y))\right), \] where \(Y\) is the clopen unit space of the corner.
3. In a fixed corner, strictification by diagonal PE gauges is equivalent to \([\nu]=0\). For the flow representative constructed in Chapter 1, strictification also follows if this fixed-corner condition holds after an allowed finite stabilization and an integer index change. This is an existential hypothesis; no choice-independent stable cohomology class is asserted.
4. If \(\Theta_{ih}=0\) for every \(i\), the cocycle is already \(1\) in the constructed corner.

The first obstruction is integral and geometric. The second is circle-valued and magnetic. They should not be conflated.

flowchart TD
    start["Individual finite PE transports"]
    k1{"All pairwise K_1 curvatures zero?"}
    gps["GPS gauges"]
    symbols["Commuting symbols"]
    block{"Height–transverse magnetic entries?"}
    strict["Strict commuting PE transports"]
    phase["Diagonal cocycle ν"]
    stable{"ν a coboundary at an allowed finite stage?"}
    obstruction["Unresolved strictification defect"]

    start --> k1
    k1 -->|"no"| obstruction
    k1 -->|"yes"| gps --> symbols --> block
    block -->|"all zero"| strict
    block -->|"some nonzero"| phase --> stable
    stable -->|"yes"| strict
    stable -->|"no"| obstruction

Minimal-height theorem

Theorem (arbitrary-dimensional block-supported absorption). Let \(T_h\) be minimal and let \(G=\mathbb Z^{p-1}\) commute with \(T_h\). For the block-supported multiplier \(i_*\sigma_{\Theta_G}\), every \[ f\in\bigl(C(\Sigma,\mathbb Z)\_{T\_h}\bigr)^G \] has a graded finitely generated projective \(B_h\)-representative carrying a strict finite-propagation PE \(G\)-equivariant structure. Its graded \(K_0\)-class is exactly \(x_f\), not a multiple of it.

Technical Engine A proves both parts of the comparison: the square defect is Barlak's compressed unitary, and its height PV image is the integer curvature. Minimality kills that curvature in the actual \(K_1\)-group. Chapter 3 then supplies the corner, normalizer, orientation, and \(K_1\)-index bridge lemmas needed to apply the GPS retraction. Block support makes the remaining coefficient-one bisections exactly equal.

For a full magnetic matrix, keep the same minimal-height hypothesis. The norm-continuous path (A.20)--(A.22) identifies every primary \(K_1\)-curvature with the block-supported one, so items 1--2 of the exact two-stage lemma apply. A strict representative then follows under the fixed-corner, or allowed finite-stage, diagonal coboundary hypothesis in item 3; the path does not make either hypothesis automatic.

Mixed Heisenberg consequence

Assume the hypotheses of the minimal-height theorem.

Let \(E_T\) be an elementary Rieffel–Heisenberg module over the transverse torus \(A_{\Theta_G}\), of any permitted height. Technical Engine B constructs the balanced correspondence and produces the graded class

\[ [\mathcal E_{T,f}] =[E_T]\widehat\otimes_{A_{\Theta_G}}[X_f] \]

over

\[ C(\Sigma)\rtimes_{i_*\sigma_{\Theta_G}}\mathbb Z^p, \]

with trace

\[ \tau_\mu([\mathcal E_{T,f}]) =\mu(f)\,\tau_{\Theta_G}([E_T]). \]

For a two-dimensional transverse torus and the basic Rieffel class, this becomes the following formula whenever the corresponding positive module exists (otherwise interpret the isolated signed term virtually):

\[ \tau_\mu([\mathcal E_{f}]) =\Theta_{12}\mu(f) \]

up to the standard choice of orientation and normalization.

If the coefficient input and torus input are actual positive modules, the product is an actual finitely generated projective module. For a signed \(f\), an arbitrary torus \(K_0\)-class, or an isolated Pfaffian term obtained by subtraction, the output is a graded finite-projective representative of a virtual \(K_0\)-class.

What the theorem does—and does not—say

Reader card. Use this chapter as the hypothesis checklist. It separates the unconditional block-supported height-one result from the full-field phase condition, higher-rank research program, upper-bound problem, and spectral gap-opening problem.

Main conclusion

The PV/Barlak comparison (A.4) is proved in Technical Engine A. The construction therefore produces strict graded finite-projective representatives for arbitrary invariant classes in

\[ \bigl(C(\Sigma,\mathbb Z)_{T_h}\bigr)^{\mathbb Z^{p-1}} \]

when:

• the coefficient group has rank one;
• the height generator \(T_h\) is minimal;
• the magnetic multiplier is supported on the transverse block.

For a general magnetic matrix, the phase-scaling homotopy is proved, and the exact two-stage lemma gives a fixed-corner phase criterion once the corner \(K_1\)-defects vanish; the proved comparison derives that vanishing from the integer curl under minimality. A choice-independent stable phase invariant is not constructed.

The theorem removes three restrictions that appeared in earlier approaches:

• no finite sheet decomposition such as Hypothesis (H);
• no \(2\)-divisibility assumption on \(K_0(B_h)\);
• no restriction to the top Rieffel height.

What is not automatic

A merely minimal \(\mathbb Z^p\)-action

Minimality of the whole \(\mathbb Z^p\)-action does not imply that a chosen generator \(T_h\) is minimal. The proof uses the stronger height-minimal condition at one exact point:

\[ C(\Sigma,\mathbb Z)^{T_h}=\mathbb Z. \]

Without it, the pairwise curvature may be a nonconstant \(T_h\)-invariant integer function.

A full magnetic matrix

Nonzero entries \(\Theta_{ih}\) rotate the height unitary. Under the same minimal-height assumption, the phase-scaling path (A.20)--(A.22) identifies the primary \(K_1\)-curvature with the block-supported class. The GPS retraction therefore still makes the bisection symbols commute, but the coefficient-one lifts may differ by a diagonal phase. The theorem identifies this phase in a selected corner; it does not assert that it always vanishes or that fixed-corner classes from different representatives have been canonically compared.

More than one coefficient direction

If

\[ B_J=C(\Sigma)\rtimes\mathbb Z^J, \qquad |J|>1, \]

the coefficient groupoid is no longer the orbit groupoid of one Cantor minimal homeomorphism. The specific GPS retraction used in the proof is then unavailable, and the LHS spectral sequence may have higher differentials.

Relationship to gap labelling

The theorem is a constructive lower-bound mechanism: it exhibits classes and computes their traces. It is not a complete calculation of \(K_0(A_{\Sigma,\Theta})\), nor does it claim that every gap label is obtained by a height-one presentation.

The construction is especially useful when \(\mu(f)\) is a distinctive pattern frequency. Pairing it with a Rieffel class yields a visibly magnetic trace such as

\[ \operatorname{Pf}(\Theta_I)\,\mu(f). \]

The precise dimension-three comparison with Benameur--Mathai is given in Dimension three: comparison and equality. The arbitrary-dimensional subgroup constructed by the theorem is stated in Higher dimensions: an absorption lower bound.

1. From a coinvariant to finite PE transports

Reader card. Input: a transverse-invariant height-coinvariant class. Output: one explicit finite-PE bisection-normalizer transport for each transverse generator, all acting on a common stabilized projection. This chapter is constructive and uses no \(KK\)-theory.

The running Sturmian example has the particularly simple transfers \(h_1=g\) and \(h_2=0\). It may be helpful to compare each general block construction below with that example.

Let

\[ M=C(\Sigma,\mathbb Z), \qquad [f]\in(M_{\alpha_h})^G. \]

The purpose of this chapter is to construct, for every transverse generator, an explicit module isomorphism between a projection and its translate. No stable-rank theorem is needed for existence.

1.1 Replace a signed class by a full positive projection

Choose \(N\) so large that

\[ g=f+N\geq1. \]

Put \(d=\lVert g\rVert_\infty\). All copies of \(P(g)\) below are regarded as projections in \(M_d(C(\Sigma))\).

For a nonnegative locally constant integer function \(a\), define

\[ P(a)=\operatorname{diag} \left( 1_{\{a\geq1\}}, \ldots, 1_{\{a\geq\lVert a\rVert_\infty\}} \right). \]

At a point \(x\), the rank of \(P(a)(x)\) is exactly \(a(x)\). Hence

\[ [P(g)]-[1_N]=[f]\in K_0(B_h). \]

Because \(g\geq1\), \(P(g)\) is nonzero everywhere and is full.

1.2 Convert coinvariant invariance into a bounded flow

The equality

\[ \alpha_i([g])=[g]\quad\text{in }M_{\alpha_h} \]

means that there is a locally constant integer function \(h_i\) with

\[ (\alpha_i-1)g=(\alpha_h-1)h_i. \]

Adding a constant does not change the right side. Choose a constant so that

\[ r_i=h_i+C_i\geq0. \]

Rearranging the preceding equation gives the pointwise equality

\[ \alpha_i(g)+r_i=g+\alpha_h(r_i). \tag{1.1} \]

Think of \(r_i(x)\) as a finite amount of auxiliary mass. Equation (1.1) says that the translated mass \(\alpha_i(g)\) can be matched with \(g\) once the unmatched part is moved by one height step.

flowchart LR
    source["α_i(g) + r_i"]
    perm["Clopen permutation Π_i"]
    target["g + α_h(r_i)"]
    shift["Unmatched free block<br/>moved by U_h"]

    source --> perm --> target
    source -. complement .-> shift -.-> target

1.3 The clopen permutation

Choose the constants in the preceding section for all generators, and put

\[ L=\max_i\lVert r_i\rVert_\infty. \]

Write

\[ Q_i =\operatorname{diag} \left( 1_{\{r_i\geq1\}},\ldots,1_{\{r_i\geq L\}} \right) \in M_L(C(\Sigma)). \]

Thus \(Q_i\) is \(P(r_i)\) padded by zero coordinates to the same size \(L\) for every \(i\). Refine \(\Sigma\) by a finite clopen partition on which

\[ g,\ \alpha_i(g),\ r_i,\ \alpha_h(r_i) \]

are all constant. On each atom of the partition, equation (1.1) is an equality between two finite cardinalities. Choose a permutation matching the corresponding coordinates. Since the partition is clopen, these permutations assemble to a unitary \(\Pi_i\in M_{d+L}(C(\Sigma))\) such that

\[ \Pi_i \bigl(\alpha_i(P(g))\oplus Q_i\bigr) \Pi_i^{\ast} =P(g)\oplus\alpha_h(Q_i). \tag{1.2} \]

Every matrix entry of \(\Pi_i\) is a characteristic function of a clopen set.

1.4 Complete the partial isometry with the height shift

On the complement of \(Q_i\) in the common \(L\)-dimensional free block,

\[ U_h(1_L-Q_i) \]

has initial projection \(1_L-Q_i\) and final projection

\[ \alpha_h(1_L-Q_i)=1_L-\alpha_h(Q_i). \]

Combining this complement map with (1.2) gives a partial isometry

\[ W_i ={} \Pi_i\bigl(\alpha_i(P(g))\oplus Q_i\bigr) + \bigl(0_d\oplus U_h(1_L-Q_i)\bigr). \tag{1.3} \]

The two summands have orthogonal initial projections \(\alpha_i(P(g))\oplus Q_i\) and \(0_d\oplus(1_L-Q_i)\), and orthogonal final projections \(P(g)\oplus\alpha_h(Q_i)\) and \(0_d\oplus(1_L-\alpha_h(Q_i))\). Consequently, with

\[ P=P(g)\oplus1_L. \]

Then

\[ W_i^{\ast}W_i=\alpha_i(P), \qquad W_iW_i^{\ast}=P. \tag{1.4} \]

The same matrix size and the same \(P\) work for all \(i=1,\ldots,q\). This explicit block calculation is the projectivity check; no stable-rank or cancellation theorem is being used here.

1.5 Why the transport is a finite PE bisection normalizer

There is a little more structure in (1.3) than finite Fourier support. Work in the stabilized height transformation groupoid

\[ \mathcal G_h=(\Sigma\rtimes_{T_h}\mathbb Z)\times \mathcal R_{d+L}, \]

whose matrix colour set is \(\{1,\ldots,d+L\}\). On every atom of the clopen partition used above, the first summand in (1.3) is the characteristic function of a finite union of height-zero arrows. The chosen coordinate permutation is injective on both source and range, so this union is a compact-open bisection. The second summand is the characteristic function of the union, over the free-block colours, of the height-one arrows restricted to the clopen sets on which the corresponding diagonal entry of \(1_L-Q_i\) is one (with the orientation fixed by our covariance convention). Distinct colours give disjoint sources and ranges, so this too is a compact-open bisection.

The orthogonality calculation preceding (1.4) says more precisely that the source sets of these two bisections are disjoint and that their range sets are disjoint. Their union is therefore a compact-open bisection \(B_i\) from the diagonal support of \(\alpha_i(P)\) to that of \(P\), and

\[ W_i=1_{B_i}. \]

Consequently \(W_i\) is a diagonal normalizer. If

\[ D_i=\alpha_i(P)M_{d+L}(C(\Sigma))\alpha_i(P), \qquad D_P=PM_{d+L}(C(\Sigma))P, \]

then the usual bisection-convolution formula gives

\[ W_iD_iW_i^*=D_P, \qquad W_i^*D_PW_i=D_i. \tag{1.5} \]

Thus the transports constructed here satisfy the normalizer/bisection hypothesis used in Chapter 3; this is not a property of an arbitrary finite Fourier partial isometry.

They are also finite pattern-equivariant transports. Indeed, each entry of \(W_i\) is a finite sum of terms of the form

\[ 1_C \quad\text{or}\quad 1_CU_h, \]

after block permutation and padding. More general choices of transfer function lead to finitely many powers \(U_h^n\), but never infinitely many. Thus the transport is:

• finite propagation in the height direction;
• locally constant in the Cantor variable;
• a native pattern-equivariant time shift in the regular representation.

At this point the transports exist individually. Their commutation is a different problem:

\[ W_i\alpha_i(W_j) \stackrel{?}{=} W_j\alpha_j(W_i). \]

The next chapter identifies the \(K_1\)-class of this failure.

1.6 Preserve the signed class

The stabilization does not multiply the desired class. The positive module is represented by \(P(g)\oplus1_L\); place the trivial strict action on a negative free module of rank \(N+L\). The graded class is

\[ [P(g)\oplus1_L]-[1_{N+L}] =[P(g)]-[1_N] =x_f. \]

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:

1. each transverse generator has an explicit finite-PE bisection transport;
2. 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.

3. Correcting the transports: the GPS normalizer retraction

Reader card. Input: finite-PE bisection transports whose pairwise corner \(K_1\)-defects vanish. Output: finite-PE gauge corrections making all underlying bisection symbols commute simultaneously. This chapter turns a stable index statement into exact geometric compatibility; it does not remove the separate magnetic diagonal phase.

In plain language, the argument puts the matrix colors into one ordered height orbit, separates each transverse transport into an index-zero local rearrangement and a successor-commuting part, and absorbs the local rearrangement by a PE gauge.

This chapter is the geometric heart of the proof. It upgrades \(K_1\)-triviality to exact commutation of the underlying finite PE bisections.

3.1 The corner as a one-dimensional orbit groupoid

Write \(P=\operatorname{diag}(p_1,\ldots,p_n)\). The full diagonal projection determines the clopen unit space

\[ Y=\{(x,a):p_a(x)=1\} \subset\Sigma\times\{1,\ldots,n\}. \tag{3.1} \]

Fullness here is concrete: Chapter 1 arranged that at least one \(p_a(x)\) is \(1\) for every \(x\). For fixed \(x\), order

\[ \Lambda_x =\{(m,a)\in\mathbb Z\times\{1,\ldots,n\}: p_a(T_h^m x)=1\} \]

lexicographically, first by \(m\) and then by the colour \(a\). Define \(\psi(x,a)\) to be the next coloured point in this ordered set. Since every height level has at least one colour, the height displacement of one successor step is either \(0\) or \(1\). It is locally constant because the \(p_a\) are clopen. Hence

\[ \psi:Y\to Y \tag{3.2} \]

is a homeomorphism. Its orbit through \((x,a)\) is exactly the set of all coloured points above the \(T_h\)-orbit of \(x\), so minimality of \(T_h\) implies minimality of \(\psi\).

The map which sends a power of the successor to the corresponding height arrow and colour change is an isomorphism

\[ Y\rtimes_\psi\mathbb Z \cong \bigl((\Sigma\rtimes_{T_h}\mathbb Z) \times\mathcal R_n\bigr)|_Y, \tag{3.3} \]

where \(\mathcal R_n\) is the full equivalence relation on the colour set. Thus the coloured full corner is exactly a Cantor minimal transformation groupoid, not merely Morita equivalent to one.

flowchart LR
    y0["(x, colour 1)"]
    y1["(T_h^a x, colour 3)"]
    y2["(T_h^b x, colour 2)"]
    y3["(T_h^c x, colour 1)"]

    y0 -->|"ψ"| y1 -->|"ψ"| y2 -->|"ψ"| y3

3.2 The topological full group and its index

Let

\[ \Gamma=\tau[\psi] \]

be the topological full group. Every \(\gamma\in\Gamma\) has a locally constant orbit cocycle \(n_\gamma:Y\to\mathbb Z\) such that

\[ \gamma(y)=\psi^{n_\gamma(y)}(y). \]

There is a canonical index homomorphism

\[ I:\Gamma\to\mathbb Z, \qquad \Gamma_0=\ker I. \]

For the corresponding coefficient-one normalizer unitary \(u_\gamma\), the \(K_1\)-class is exactly this index under

\[ K_1(C(Y)\rtimes_\psi\mathbb Z)\cong\mathbb Z. \]

3.3 Transverse transports normalize the full group

Let \(Y_i\) be the unit space of \(\alpha_i(P)\). The automorphism \(\alpha_i\) gives an orientation-preserving groupoid isomorphism from the reduction over \(Y\) to the reduction over \(Y_i\). The coefficient-free symbol of \(W_i\) is a compact-open height bisection from \(Y_i\) to \(Y\). Their composite is therefore a homeomorphism

\[ S_i=\operatorname{symb}(W_i\alpha_i)|_Y. \]

It induces an automorphism of the groupoid in (3.3). Conjugating a compact-open full bisection by this automorphism again gives a compact-open full bisection. Equivalently, if \(\gamma(y)=\psi^{n_\gamma(y)}y\) with locally constant \(n_\gamma\), then the orbit cocycle of \(S_i\gamma S_i^{-1}\) is a finite composition of the locally constant cocycles of \(S_i,S_i^{-1}\), and \(\gamma\). Consequently

\[ S_i\in N(\Gamma), \]

the normalizer of \(\Gamma\) in \(\operatorname{Homeo}(Y)\).

We determine the orientation without identifying two different integer cocycles. For \(x\in\Sigma\), let \(\Lambda_x\) be the coloured set of units over the original height orbit of \(x\). It is ordered first by the original height coordinate \(m\in\mathbb Z\) and then by the finite colour. A level contributes at most the fixed finite number of colours. Because the corner is full and clopen and the height system is minimal, the contributing levels are syndetic: finitely many height translates cover the ambient unit space, so gaps between consecutive contributing levels are uniformly bounded. It follows that the two ends of a \(\psi\)-orbit are exactly the ends

\[ m\longrightarrow+\infty, \qquad m\longrightarrow-\infty \tag{3.4} \]

of the original height coordinate.

The map induced by \(\alpha_i\) preserves these two ends because \(\alpha_iT_h=T_h\alpha_i\). The compact-open bisection supplied by \(W_i\) has only finitely many branches. Hence there is an \(R_i<\infty\) such that each branch changes the original height coordinate by an integer of absolute value at most \(R_i\). It also preserves both ends. Their composite \(S_i\) therefore satisfies

\[ S_i(+\infty)=+\infty, \qquad S_i(-\infty)=-\infty. \tag{3.5} \]

This argument concerns the original height coordinate only. It does not assert that the successor exponent \(n_\gamma\), or the cocycle of the coloured system, equals that height coordinate.

3.4 The Giordano–Putnam–Skau decomposition

Define

\[ C^\varepsilon(\psi) ={} \{c\in\operatorname{Homeo}(Y): c\psi c^{-1}=\psi^{\pm1}\}. \]

Giordano–Putnam–Skau prove that multiplication gives

\[ N(\Gamma) \cong \Gamma_0\rtimes C^\varepsilon(\psi). \]

Consequently there is a homomorphic retraction

\[ r:N(\Gamma)\to C^\varepsilon(\psi), \qquad \ker r=\Gamma_0. \]

Write the GPS factorization first as

\[ S_i=\eta_i c_i, \qquad \eta_i\in\Gamma_0, \qquad c_i\psi c_i^{-1}=\psi^{\varepsilon_i}, \quad \varepsilon_i\in\{1,-1\}. \]

Every element of \(\Gamma\) has a bounded, locally constant successor exponent on compact \(Y\), so \(\eta_i\) preserves the two ends. If \(\varepsilon_i=-1\), then \(c_i\psi^ky=\psi^{-k}c_i y\), and \(c_i\) exchanges them. This contradicts (3.5), because both \(S_i\) and \(\eta_i\) preserve the ends. Thus \(\varepsilon_i=1\), and the factorization is

\[ S_i=\eta_i c_i, \qquad \eta_i\in\Gamma_0, \qquad c_i\psi=\psi c_i. \tag{3.6} \]

3.5 Apply the retraction to the square defect

The symbol of the square defect is the commutator

\[ \operatorname{symb}(\Omega_{ij})=[S_i,S_j] \]

up to the selected commutator convention. Applying \(r\) gives

\[ [c_i,c_j] =r\!\left(\operatorname{symb}(\Omega_{ij})\right). \tag{3.7} \]

Here is the required \(K_1\)-to-index comparison, with all maps typed. Let

\[ \partial_\psi: K_1(C(Y)\rtimes_\psi\mathbb Z) \longrightarrow K_0(C(Y)) \tag{3.8} \]

be the height PV boundary. Minimality gives \(K_0(C(Y))^\psi=\mathbb Z[1_Y]\), so exactness and the usual crossed-product convention give

\[ \partial_\psi[u_\gamma] =\varepsilon_\psi I(\gamma)[1_Y], \qquad \varepsilon_\psi\in\{1,-1\}. \tag{3.9} \]

Indeed, GPS Section 5 constructs the Fredholm homomorphism \(K_1(C(Y)\rtimes_\psi\mathbb Z)\to\mathbb Z\), proves that it sends \([u_\gamma]\) to \(I(\gamma)\), and sends the implementing unitary \([u_\psi]\) to \(1\). Since the height PV boundary sends \([u_\psi]\) to \(\varepsilon_\psi[1_Y]\), the two homomorphisms agree up to this single sign. Equation (3.9), rather than a boundary map into \(K_1\), is the precise comparison.

Let

\[ \Phi:C(Y)\rtimes_\psi\mathbb Z \xrightarrow{\cong}C=PM_n(B_h)P \tag{3.10} \]

be the algebra isomorphism induced by the groupoid isomorphism (3.3). A finite PE normalizer \(w\in C\) with symbol \(\gamma\) satisfies

\[ w=\Phi(d\,u_\gamma) \]

for a locally constant diagonal unitary \(d\in C(Y)\). On a finite clopen partition choose logarithms of the finitely many values of \(d\); hence \([d]=0\) and

\[ [w]=\Phi_{\ast}[u_\gamma]. \tag{3.11} \]

This avoids any assertion that a preselected numerical PV boundary is literally preserved by corner Morita equivalence. What is needed is only vanishing: \(\Phi_*\) is an isomorphism, and the inclusion of the full corner \(j:C\hookrightarrow M_n(B_h)\) is a \(K_1\)-isomorphism. The complete comparison actually used is therefore

\[ \begin{array}{ccccc} K_1(C(Y)\rtimes_\psi\mathbb Z) &\xrightarrow[\cong]{\ \Phi_{\ast}\ }& K_1(C) &\xrightarrow[\cong]{\ j_{\ast}\ }& K_1(M_n(B_h))\\ [u_\gamma]&\longmapsto&[w]&\longmapsto&j_{\ast}[w]. \end{array} \tag{3.11a} \]

Therefore (3.9)--(3.11) give

\[ [w]=0\text{ in }K_1(C) \quad\Longleftrightarrow\quad I(\gamma)=0. \tag{3.12} \]

Applied to the square defect, this says

\[ I\!\left(\operatorname{symb}(\Omega_{ij})\right)=0 \quad\Longleftrightarrow\quad [\Omega_{ij}]=0\text{ in }K_1(C). \tag{3.13} \]

Assume \([\Omega_{ij}]=0\) in the actual corner \(K_1\)-group. This is an input to the exact two-stage lemma. In the minimal-height theorem it follows from the integer-curl calculation and the PV/Barlak comparison (A.4); minimality removes the coinvariant quotient. Under this actual \(K_1\)-vanishing hypothesis,

\[ \operatorname{symb}(\Omega_{ij})\in\Gamma_0 =\ker r. \]

Equation (3.7) therefore yields

\[ [c_i,c_j]=1 \qquad\text{for every }i,j. \tag{3.14} \]

3.6 Lift the correction back to PE operators

Lift \(\eta_i^{-1}\) by its coefficient-one compact-open bisection unitary \(z_i\), and set

\[ W_i'=z_iW_i. \]

Since \(\eta_i\) has a finite clopen orbit partition, \(z_i\) is a finite PE height transport. The corrected semilinear operator \(W_i'\alpha_i\) has symbol \(c_i\).

The same retraction \(r\) is applied to every generator. Thus (3.14) holds simultaneously for all \(q\) directions; there is no iterative pairwise correction that could spoil an earlier pair.

flowchart LR
    S["S_i in N(Γ)"]
    factor["S_i = η_i c_i"]
    eta["η_i in Γ_0<br/>finite PE gauge"]
    c["c_i commutes with ψ"]

    S --> factor
    factor --> eta
    factor --> c
    eta -->|"multiply by η_i^−1"| c

3.7 What has—and has not—been proved

We have proved exact commutation of the symbols \(c_i\). If all operators are coefficient-one bisections and the transverse action preserves coefficient-one bisections, this already gives exact operator commutation. That is the block-supported case.

If \(\alpha_i(U_h)\) carries a magnetic phase, two operators with the same bisection can differ by a diagonal unitary. The next chapter identifies that residual defect.

4. The residual magnetic phase

Reader card. Input: transports whose bisection symbols already commute. Output: an exact fixed-corner criterion for lifting that commutation to the twisted operators. In the block-supported theorem the phase is automatically one. For a full magnetic matrix, finite-stage phase triviality remains an explicit hypothesis.

The GPS retraction removes the permutation part of the defect. This chapter shows that the remaining problem is abelian. It proves an exact criterion in one fixed corner. Across different representatives and stabilizations, only an existential strictifiability hypothesis is used; no choice-independent cohomology class is claimed.

4.1 Kernel of the symbol map

Let

\[ D_{\mathrm{lc}}=C_{\mathrm{lc}}(Y). \]

A finite PE normalizer with trivial bisection symbol is multiplication by a locally constant circle-valued function. Therefore, after (3.9),

\[ \nu_{ij} =W_i'\alpha_i(W_j') \bigl(W_j'\alpha_j(W_i')\bigr)^{\ast} \in U(D_{\mathrm{lc}}). \tag{4.1} \]

A diagonal unitary contributes nothing to \(K_1(C)\): a locally constant circle-valued function has a logarithm on each member of a finite clopen partition. This explains why \(K_1\)-curvature cannot see (4.1).

4.2 The full Busby–Smith cocycle

Because the symbols \(c_i\) commute, define

\[ c_g=c_1^{g_1}\cdots c_q^{g_q} \qquad(g\in G). \]

Choose finite PE lifts \(W_g'\) with \(W_0'=P\) and set

\[ \nu(g,h) =W_g'\alpha_g(W_h')W_{g+h}'{}^{\ast} \in U(D_{\mathrm{lc}}). \tag{4.2} \]

On the diagonal define

\[ \bar\beta_g(d) =W_g'\alpha_g(d)W_g'{}^{\ast}. \]

Although the lifts are only projective, \(\bar\beta\) is a genuine action on \(D_{\mathrm{lc}}\): the inner defect is diagonal, and conjugation by a diagonal element acts trivially on the diagonal.

Associativity gives

\[ \nu(g,h)\nu(g+h,k) =\bar\beta_g(\nu(h,k))\nu(g,h+k), \tag{4.3} \]

so

\[ \nu\in Z^2\bigl(G,U(D_{\mathrm{lc}})_{\bar\beta}\bigr). \]

4.3 Diagonal gauges are exactly coboundaries

Let \(d_g\in U(D_{\mathrm{lc}})\) and put

\[ W_g''=d_gW_g'. \]

Then

\[ \nu''(g,h) =d_g\bar\beta_g(d_h)\nu(g,h)d_{g+h}^{\ast}. \tag{4.4} \]

Therefore:

Fixed-corner criterion. For fixed symbol indices, diagonal finite PE gauges make the transports into a genuine \(G\)-action if and only if \[ [\nu]=0 \quad\text{in}\quad H^2\bigl(G,U(D_{\mathrm{lc}})_{\bar\beta}\bigr). \]

Trivializing the full cocycle—not only its generator commutators—shows that every group word is coherent. There is no hidden \(3\)-cocycle.

4.4 Index freedom

If a transport is multiplied by a corner full-group element of index \(k_i\), its retracted symbol changes from \(c_i\) to

\[ \psi^{k_i}c_i. \]

Indeed, for \(\gamma\in\Gamma\),

\[ r(\gamma)=\psi^{I(\gamma)}. \]

For the selected corner, one may therefore ask whether some index vector \(k\in\mathbb Z^q\) makes the corresponding diagonal cocycle a coboundary. This is still a statement about the selected corner. It does not identify the cohomology groups belonging to different corners.

4.5 The finite-stable existential hypothesis

An allowed finite stabilization replaces \(P\) by

\[ P\oplus1_L \]

and subtracts the same free block in the graded class. The associated unit space acquires finitely many colours, and one can repeat the GPS and diagonal reductions there.

Finite-stable phase hypothesis for a constructed representative. Starting with the flow representative of Chapter 1, there is an allowed finite stabilization, a GPS reduction, and an integer index choice for which the resulting fixed-corner class \[ [\nu]\in H^2\bigl(G,U(D_{\mathrm{lc}})_{\bar\beta}\bigr) \] is zero.

If this hypothesis holds, (4.4) supplies the final diagonal gauges, so it is a sufficient condition for strictification. Conversely, any strictification reached through this same staged construction yields such a zero class. This is the strongest statement established here.

We deliberately do not denote this condition by a class \(\mathfrak m_{\Theta,\mathrm{st}}(x_f)\). To construct such an invariant one would need:

1. comparison maps between the cohomology groups for different corners and finite stabilizations;
2. invariance under changing the transfer functions and the projection representative; and
3. a proof that every finite PE equivalence is generated by the allowed colour additions, index shifts, and diagonal gauges.

Those comparison results are not proved. The present hypothesis is therefore existential, not a choice-independent obstruction attached to the \(K_0\)-class.

4.6 Why block support makes the phase one

Assume

\[ \Theta_{ih}=0 \qquad(i=1,\ldots,q). \]

Then

\[ \alpha_i(U_h)=U_h, \]

so \(\alpha_i\) sends every coefficient-one height bisection to another coefficient-one bisection.

Use the coefficient-one transports constructed in Chapter 1 and the coefficient-one GPS gauges from Chapter 3. If \(B_i\) is the compact-open bisection representing \(W_i'\), then

\[ W_i'\alpha_i(W_j') =1_{B_i\alpha_i(B_j)}, \qquad W_j'\alpha_j(W_i') =1_{B_j\alpha_j(B_i)}. \]

The two symbols commute. Their transverse group labels agree, and their remaining difference is a height arrow. Since a minimal Cantor homeomorphism is free, the reduced height groupoid is principal; equal source-to-range maps give equal bisections. Hence

\[ B_i\alpha_i(B_j)=B_j\alpha_j(B_i) \]

and therefore

\[ \nu_{ij}=1 \]

as an exact operator equality.

4.7 Where a cross phase comes from

If

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

then a branch \(1_CU_h^n\) of \(W_j'\) acquires the factor

\[ e^{2\pi i\Theta_{ih}n} \]

under \(\alpha_i\). Around a transverse square, the bisection exponents cancel but these factors need not. The result is precisely the diagonal cocycle (4.1): the magnetic transgression of the integer return-time data.

This also explains why parity and \(2\)-divisibility are not the right general obstruction. They concern the permutation kernel or a particular matrix factorization; the surviving class is circle-valued and depends on the height–transverse magnetic block.

5. Minimality and arbitrary dimension

Reader card. This chapter performs the short dynamical calculation that kills every pairwise integer curvature and then assembles the block-supported theorem for any number of transverse generators. The coefficient group still has rank one; this is not a higher-rank absorption theorem.

We now prove that the integer transfer-function curvatures vanish for any finite number of transverse generators. The PV/Barlak comparison in Technical Engine A turns that calculation into vanishing of the operator \(K_1\)-defects.

5.1 Pairwise curvature is height-invariant

For each \(i\), choose \(h_i\in C(\Sigma,\mathbb Z)\) satisfying

\[ (\alpha_i-1)g=(\alpha_h-1)h_i. \]

For every pair define

\[ \kappa_{ij} =(\alpha_j-1)h_i-(\alpha_i-1)h_j. \]

Exactly as in Technical Engine A,

\[ \begin{aligned} (\alpha_h-1)\kappa_{ij} &=(\alpha_j-1)(\alpha_h-1)h_i -(\alpha_i-1)(\alpha_h-1)h_j\\ &=(\alpha_j-1)(\alpha_i-1)g -(\alpha_i-1)(\alpha_j-1)g\\ &=0. \end{aligned} \]

Thus

\[ \kappa_{ij}\in C(\Sigma,\mathbb Z)^{T_h}. \]

5.2 Minimality makes curvature constant

If an integer-valued continuous function is \(T_h\)-invariant and \(T_h\) is minimal, it is constant. Therefore

\[ \kappa_{ij}=m_{ij}\,1_\Sigma \]

for some integer \(m_{ij}\).

5.3 The constant has integral zero

The compact convex set of \(T_h\)-invariant probability measures is nonempty and is preserved by every \(T_i\). Since \(G=\mathbb Z^q\) is amenable, there is a probability measure \(\mu\) invariant under \(T_h\) and all of \(G\).

Integrating gives

\[ \begin{aligned} m_{ij} &=\int_\Sigma\kappa_{ij}\,d\mu\\ &=\int_\Sigma(\alpha_j-1)h_i\,d\mu -\int_\Sigma(\alpha_i-1)h_j\,d\mu\\ &=0. \end{aligned} \]

Hence

\[ \kappa_{ij}=0 \qquad\text{for all }i,j. \tag{5.1} \]

Minimality does identify the Barlak target with the actual \(K_1(B_h)\cong\mathbb Z\). The comparison theorem (A.4) and (5.1) therefore imply

\[ [\Omega_{ij}]=0\in K_1(C) \qquad\text{for all }i,j. \tag{5.2} \]

5.4 There are no higher LHS differentials

The class lies in

\[ E_2^{0,1} =H^0\!\left(G,H^1(H,M)\right). \]

The only possible outgoing differential is

\[ d_2:E_2^{0,1}\to E_2^{2,0}. \]

For \(r\geq3\),

\[ d_r:E_r^{0,1}\to E_r^{r,2-r} \]

has negative second degree, so its target is zero. Thus (5.1) removes the complete cohomological obstruction in the height-one row.

5.5 Completion of the block-supported proof

The remaining argument has four finite steps.

flowchart LR
    flow["Bounded clopen flows h_i"]
    curvature["κ_ij = 0"]
    gps["GPS: commuting symbols c_i"]
    block["Block support: ν = 1"]
    equivariant["Strict PE G-module"]

    flow --> curvature -->|"PV/Barlak comparison"| gps --> block --> equivariant

1. Chapter 1 constructs all \(W_i\) in one finite corner.
2. The PV/Barlak comparison theorem and (5.1) kill all actual \(K_1\)-curvatures.
3. The corner-groupoid, normalizer, orientation, and index lemmas in Chapter 3 allow one GPS retraction to make all symbols commute simultaneously.
4. Block support turns symbol equality into operator equality.

This proves the arbitrary-dimensional block-supported height-one theorem.

5.6 Why the result covers non-top heights

Nothing in Chapters 1–5 depends on a Rieffel height. They construct the coefficient equivariant module once. It may then be paired with any class in the transverse noncommutative torus that has a Rieffel–Heisenberg model.

If the transverse rank is \(q\), an elementary Rieffel module of height \(r\) uses

\[ M_T=\mathbb R^r\times\mathbb Z^{q-2r}, \qquad 0\leq2r\leq q. \]

The discrete factor is nontrivial below top height. The absorption proof does not alter it and imposes no top-dimensional condition.

6. Attaching the magnetic Heisenberg module

The previous chapters constructed a finite projective module \(F_f\) over the height algebra \(B_h\), together with strict finite-PE transverse transport. This chapter explains conceptually how that coefficient module is combined with a Rieffel--Heisenberg module. The analytic verification is deferred to Technical Engine B.

Reader card. Input: a strict \(G\)-equivariant coefficient module \(F_f\) and a transverse Rieffel module \(E_T\). Output: the mixed projective module and its product trace. No knowledge of unbounded operators or Kasparov cycles is required for this overview.

6.1 Why an ordinary tensor product is not enough

Put

\[ B=B_h, \qquad A=B\rtimes_{\alpha,\sigma}G, \qquad D=C^*(G,\sigma). \]

The algebra \(D\) is the transverse noncommutative torus. The final quasicrystal algebra \(A\) contains both the pattern-dependent coefficient algebra \(B\) and the magnetic transverse translations.

A naive external tensor product would keep these pieces independent. The desired class is different: when a transverse magnetic translation acts, it must also apply the pattern-dependent transport carried by \(F_f\). That coupling is the meaning of absorption.

6.2 Extend the coefficient module to the full algebra

Form

\[ X_f=F_f\widehat\otimes_B A. \]

This allows the coefficients of \(F_f\) to range in the full crossed product. If \(F_f\cong pB^n\), then

\[ X_f\cong pA^n, \]

so it remains finite projective.

Let \(V_g\) denote the strict semilinear transport on \(F_f\), and let \(U_g\) be the magnetic translation in \(A\). Define

\[ L_g(\xi\otimes a) =V_g\xi\otimes U_ga. \tag{6.G1} \]

The first factor moves the pattern-dependent coefficient module; the second performs the magnetic translation. Strict equivariance of the \(V_g\) ensures that the only projective phase in the product is the prescribed magnetic multiplier:

\[ L_gL_h=\sigma(g,h)L_{g+h}. \]

Thus \(D\) acts on \(X_f\). In correspondence language, \(X_f\) is a \(D\)-to-\(A\) bridge.

6.3 Insert a Rieffel--Heisenberg module

Let \(E_T\) be a finite projective module over \(D\). Its vectors may be realized as time-frequency windows, and the generators of \(D\) act by ordinary magnetic time-frequency shifts.

The mixed module is

\[ \mathcal E_{T,f} =E_T\widehat\otimes_DX_f. \tag{6.G2} \]

The balancing over \(D\) identifies the torus translation acting on \(E_T\) with the coupled operator (6.G1) acting on \(X_f\). Consequently a generator acts schematically as

\[ \boxed{ \text{magnetic time-frequency shift} \times \text{finite-PE height transport}. } \]

This is why the result is a genuinely mixed quasicrystalline module rather than a torus module carrying a passive scalar coefficient.

6.4 Why the result is finite projective

Represent \(E_T\) by a finite projection \(e\in M_m(D)\). The action of \(D\) on \(X_f\) sends \(e\) to a projection

\[ \lambda_f^{(m)}(e) \]

on the finite projective module \(X_f^m\). The mixed module is its range. This gives a concrete projectivity proof without asking the newcomer to manipulate an abstract Kasparov product.

For signed pattern data or a virtual torus class, apply the construction to the positive and negative summands separately. The result is then a graded difference, not an assertion that a negative trace is carried by an actual positive module.

6.5 Why the trace factors

The canonical crossed-product trace sees only the zero Fourier coefficient. On the finite projection representing (6.G2), the coefficient contribution and the torus contribution therefore separate:

\[ \tau_\mu([\mathcal E_{T,f}]) =\tau_\mu([F_f])\,\tau_{\Theta_G}([E_T]) =\mu(f)\,\tau_{\Theta_G}([E_T]). \tag{6.G3} \]

When \(E_T\) represents a Pfaffian component,

\[ \tau_\mu([\mathcal E_{T,f}]) =\operatorname{Pf}(\Theta_I)\mu(f), \]

up to the selected orientation and normalization.

For the running Fibonacci cylinder, \(\mu(f)=\varphi^{-1}\), and a basic positive two-torus module gives

\[ \tau_\mu([\mathcal E_{\theta,f}]) =\frac{\theta}{\varphi}. \]

6.6 What Technical Engine B proves

The full analytic chapter verifies that:

• (6.G1) is well defined on the balanced tensor product;
• the operators are adjointable and satisfy the exact cocycle convention;
• the left action is by compact operators;
• the projection defining (6.G2) has finite projective range; and
• the zero-Fourier-coefficient calculation gives (6.G3).

Those checks are essential for the proof, but they are logically downstream of the geometric absorption argument. A reader interested first in the construction and gap-label formula can treat them as a certified analytic engine and return to them later.

Technical Engine A: the PV/Barlak curvature comparison

Reader card. This chapter proves the load-bearing bridge between the concrete square holonomy and the integer transfer-function curl. Its input is the construction of Chapter 1; its output is the typed identity between the height PV boundary of the holonomy and the curl. Readers seeking the construction before the certification should first read Square holonomy and integer curvature.

The equation numbers retain the prefix \(2\) because this is the technical proof of Step 2 in the constructive argument.

The individual transports in (1.4) need not commute. For two transverse generators, define

\[ \Omega =W_1\alpha_1(W_2) \bigl(W_2\alpha_2(W_1)\bigr)^{\ast} \in U(C), \qquad C=PM_n(B_h)P. \]

This is the holonomy around one transverse square.

flowchart TD
    P00["P"]
    P10["α_1(P)"]
    P01["α_2(P)"]
    P11["α_1α_2(P)"]

    P11 -->|"α_1(W_2)"| P10
    P10 -->|"W_1"| P00
    P11 -->|"α_2(W_1)"| P01
    P01 -->|"W_2"| P00

The two paths have the same source and range. Their ratio is \(\Omega\).

A.1 The discrete curvature

Recall the transfer equations

\[ (\alpha_i-1)g=(\alpha_h-1)h_i. \]

Set

\[ \kappa_{12} =(\alpha_2-1)h_1-(\alpha_1-1)h_2. \tag{A.1} \]

Commutativity of the three automorphisms gives

\[ \begin{aligned} (\alpha_h-1)\kappa_{12} &=(\alpha_2-1)(\alpha_h-1)h_1 -(\alpha_1-1)(\alpha_h-1)h_2\\ &=(\alpha_2-1)(\alpha_1-1)g -(\alpha_1-1)(\alpha_2-1)g\\ &=0. \end{aligned} \]

Thus

\[ \kappa_{12}\in M^{\alpha_h}. \]

Changing \(h_i\) changes \(\kappa_{12}\) by the appropriate transverse coboundary, so its cohomology class depends only on \([f]\).

A.2 The Pimsner–Voiculescu target

Write \(M=C(\Sigma,\mathbb Z)\). Because \(K_1(C(\Sigma))=0\), the height Pimsner–Voiculescu sequence gives canonical, transverse-equivariant identifications

\[ K_0(B_h)\cong M_{\alpha_h}, \qquad \partial_h:K_1(B_h)\xrightarrow{\cong}M^{\alpha_h}. \tag{A.2} \]

For the \(\mathbb Z^2\)-action generated by \(\alpha_i,\alpha_j\), Barlak's target is

\[ E_2^{2,-1} =H^2\!\left(\mathbb Z^2,K_1(B_h)\right) \cong K_1(B_h)_{\langle\alpha_i,\alpha_j\rangle}. \tag{A.3} \]

The following comparison for the block-supported action is the main result of this chapter:

\[ \partial_h[\Omega_{ij}] =\varepsilon\, [\kappa_{ij}] \quad\text{in}\quad \left(M^{\alpha_h}\right)_{\langle\alpha_i,\alpha_j\rangle}, \qquad \varepsilon\in\{1,-1\}. \tag{A.4} \]

The sign \(\varepsilon\) is fixed once the covariance convention, the orientation \(i\wedge j\), and Barlak's Bott convention are fixed. Only vanishing is used below.

When \(T_h\) is minimal, \(M^{\alpha_h}=\mathbb Z1_\Sigma\). Every transverse automorphism acts trivially on this copy of \(\mathbb Z\); equivalently, \(\alpha_i(U_h)=\lambda_iU_h\) is homotopic to \(U_h\). Hence the quotient in (A.3) is canonically \(K_1(B_h)\cong\mathbb Z\), and (A.4) is an equality in the actual \(K_1\)-group, not merely in a quotient.

A.3 The PV/Barlak comparison

There are two comparisons to make: the corner defect must be Barlak's compressed unitary, and Barlak's abstract \(d_2\) must become the integer curl (A.1) under (A.2). Section A.3.1 proves the operator comparison. Section A.3.2 constructs the filtered equivariant PV comparison and computes its exact-couple staircase.

A.3.1 The compressed unitary is the corner holonomy

For a fixed pair \(i,j\), put

\[ q=\begin{pmatrix}P&0\\0&0\end{pmatrix}. \]

The partial isometry \(W_i\) has initial projection \(\alpha_i(P)\) and final projection \(P\). Therefore

\[ v_i= \begin{pmatrix} W_i^\ast&1-\alpha_i(P)\\ 1-P&W_i \end{pmatrix} \in M_{2n}(B_h) \tag{A.5} \]

is a unitary and satisfies

\[ \alpha_i(q)=v_iqv_i^\ast. \]

Apply Barlak's Theorem 4.4 with \(v=v_i\) and \(w=v_j\). Its generalized Bott unitary is

\[ \kappa\!\left( q,\, v_j^\ast\alpha_j(v_i)^\ast\alpha_i(v_j)v_i \right). \tag{A.6} \]

Compressing the second entry of (A.6) by \(q\) gives

\[ W_j\alpha_j(W_i)\alpha_i(W_j^\ast)W_i^\ast =\Omega_{ij}^\ast. \tag{A.7} \]

The generalized Bott notation in (A.6) means the class of this corner unitary, extended by \(1-q\). Since \(P\) is full, corner Morita equivalence sends it to \([\Omega_{ij}^\ast]\). This proves the operator part needed for (A.4), including the only possible global sign. It does not prove the transfer-curl comparison.

A.3.2 The equivariant PV comparison with the integer curl

Theorem 4.4 of Barlak identifies his \(d_2\) with the generalized Bott unitary in (A.6), but a separate naturality argument is needed to identify that differential with the transfer-function curl. We give that argument using Barlak's Baum--Connes skeletal model before passing to his mapping-torus model. This makes the height PV triangle a cofiber sequence of filtered objects by applying one exact functor; no cone of \(1-\alpha_h\) is formed in the category of \(C^*\)-algebras.

Put \(A_0=C(\Sigma)\) and \(G=G_{ij}=\langle\alpha_i,\alpha_j\rangle\cong\mathbb Z^2\). In the block-supported case, \(G\) acts on \(B_h=A_0\rtimes_{\alpha_h}\mathbb Z\) by its action on the coefficients and fixes the height unitary. The height Toeplitz extension

\[ 0\longrightarrow A_0\otimes\mathcal K \longrightarrow\mathcal T_{\alpha_h} \longrightarrow B_h\longrightarrow0 \]

is therefore \(G\)-equivariant: \(G\) acts trivially on the Toeplitz shift and by \(\alpha_i,\alpha_j\) on \(A_0\). Its standard completely positive Toeplitz splitting is \(G\)-equivariant as well.

We record why the two equivalences used here are equivariant. For the correspondence \({}_{\alpha_h}A_0\), its Fock module \(\mathcal F_{\alpha_h}=\bigoplus_{n\geq0}({}_{\alpha_h}A_0)^{\otimes n}\) carries \[ g(\delta_n\otimes a)=\delta_n\otimes\alpha_g(a), \] because every \(\alpha_g\) commutes with \(\alpha_h\). The coefficient representation is covariant for this action, and both the Fock shift and the degree projections commute with it. The standard Toeplitz Kasparov inverse to the coefficient inclusion is built from this representation, the Fock shift, and the degree projections; the usual rotation/shift homotopy proving that the two Kasparov products are identities uses the same operators. It therefore commutes with the displayed \(G\)-action at every parameter value. This is the ordinary Toeplitz \(KK\)-equivalence proof, now regarded as a proof in \(KK^G\), and shows \[ A_0\longrightarrow\mathcal T_{\alpha_h} \quad\text{is a }KK^G\text{-equivalence}. \] Similarly, \(G\) acts trivially on \(\mathcal K\), so the rank-one corner and the standard \(G\)-equivariant stabilization bimodule show that \(A_0\to A_0\otimes\mathcal K\) is a \(KK^G\)-equivalence. Under these two equivalences, the ideal inclusion is the difference of the identity and the correspondence class \([\alpha_h]\), as in the standard equivariant Toeplitz proof of the PV sequence. Hence the extension gives the equivariant PV triangle

\[ \mathrm{kk}^G(A_0) \xrightarrow{\ u\ } \mathrm{kk}^G(A_0) \longrightarrow \mathrm{kk}^G(B_h) \longrightarrow \Sigma\mathrm{kk}^G(A_0), \tag{A.8} \]

where \(u=1-[\alpha_h]\), up to the standard rotation/sign convention. Below we use \(u_*=\alpha_h-1\) on \(K_0(A_0)\); replacing \(u\) by its negative changes only the global sign \(\varepsilon\) in (A.4).

We use the stable equivariant enhancement of Bunke--Engel--Land. Their Theorems 1.3--1.4 make \(\mathrm{KK}^G_{\mathrm{sep}}\) a stable \(\infty\)-category and send equivariantly semisplit extensions to cofiber sequences. Their Proposition 1.7 makes tensor product bi-exact, and Theorem 1.22 makes crossed product an exact functor on the enhanced categories.

Let

\[ \varnothing=Y_{-1}\subset Y_0\subset Y_1\subset Y_2=\mathbb R^2 \]

be the \(G\)-invariant cubical skeletal filtration used by Barlak. For a \(G\)-object \(X\) of equivariant \(KK\), define

\[ \mathscr C_p(X) =\Bigl(C_0(Y_p)\otimes X\Bigr)\rtimes G \quad\text{in }\mathrm{KK}_{\mathrm{sep}}. \]

Here tensor product carries the diagonal action. Proposition 1.7 and Theorem 1.22 cited above show that every \(\mathscr C_p\) is exact. The reduced and maximal choices agree because \(G\cong\mathbb Z^2\) is amenable. The inclusions \(Y_{p-1}\subset Y_p\) induce restriction \(\ast\)-homomorphisms \(C_0(Y_p)\to C_0(Y_{p-1})\), natural in \(X\). The \(G\)-action is proper on \(Y_p\), and \(Y_{p-1}\) is second countable, so the restriction extensions are equivariantly semisplit by Bunke--Engel--Land, Proposition 1.12(1). Thus the collection

\[ \mathscr C_\bullet: \mathrm{KK}^G_{\mathrm{sep}}\longrightarrow \operatorname{Fun}(\{2\to1\to0\to-1\}, \mathrm{KK}_{\mathrm{sep}}) \]

is one exact filtered functor. Applying it to (A.8) gives, for \(p=0,1,2\), the commutative diagram

\[ \begin{array}{ccccccc} \mathscr C_p(A_0)&\xrightarrow{u_p}&\mathscr C_p(A_0) &\longrightarrow&\mathscr C_p(B_h)&\longrightarrow& \Sigma\mathscr C_p(A_0)\ \big\downarrow&&\big\downarrow&&\big\downarrow&&\big\downarrow\ \mathscr C_{p-1}(A_0)&\xrightarrow{u_{p-1}}& \mathscr C_{p-1}(A_0)&\longrightarrow& \mathscr C_{p-1}(B_h)&\longrightarrow& \Sigma\mathscr C_{p-1}(A_0), \end{array} \tag{A.9} \]

whose rows are cofiber sequences. Equivalently, in the stable functor category there is a canonical filtered equivalence

\[ \Phi_\bullet: \operatorname{cofib}\!\left( u_\bullet:\mathscr C_\bullet(A_0)\to \mathscr C_\bullet(A_0) \right) \xrightarrow{\ \simeq\ } \mathscr C_\bullet(B_h). \tag{A.10} \]

This is the required coherent comparison. It is not a choice of unrelated triangulated-category cones: (A.10) is obtained by applying a single exact \(\infty\)-functor to the actual equivariant Toeplitz triangle.

For completeness, let

\[ \mathscr J_p(X)=\operatorname{fib}\!\left( \mathscr C_p(X)\to\mathscr C_{p-1}(X)\right). \]

The stable \(3\times3\) lemma applied to (A.9) gives the following diagram in which every displayed row and column is a cofiber sequence:

\[ \begin{array}{ccccc} \mathscr J_p(A_0)&\longrightarrow&\mathscr C_p(A_0)& \longrightarrow&\mathscr C_{p-1}(A_0)\ {\scriptstyle u}\big\downarrow&& {\scriptstyle u_p}\big\downarrow&& {\scriptstyle u_{p-1}}\big\downarrow\ \mathscr J_p(A_0)&\longrightarrow&\mathscr C_p(A_0)& \longrightarrow&\mathscr C_{p-1}(A_0)\ \big\downarrow&&\big\downarrow&&\big\downarrow\ \mathscr J_p(B_h)&\longrightarrow&\mathscr C_p(B_h)& \longrightarrow&\mathscr C_{p-1}(B_h). \end{array} \tag{A.11} \]

To justify the cell identification and its signs, note that \(Y_p\setminus Y_{p-1}\) is the disjoint union of free \(G\)-orbits of open \(p\)-cells, one orbit for each \(\mu\in T(p,2)\). The restriction extension identifies the fibre with the crossed product of the corresponding open-cell ideal. On each orbit, Green imprimitivity gives

\[ \left(C_0(G\times(0,1)^p)\otimes X\right)\rtimes G \ \simeq\ C_0((0,1)^p)\otimes X \ \simeq\ \Sigma^pX . \]

Taking the finite direct sum over the cell orbits gives

\[ \mathscr J_p(X)\simeq \bigoplus_{\mu\in T(p,2)}\Sigma^p X. \tag{A.12} \]

This equivalence is natural in \(X\): restriction, crossed-product descent, and the cell imprimitivity bimodule above all tensor with \(\operatorname{id}_X\). Fix the coordinate orientations \(e_i,e_j\) and choose the two-cell incidence convention for which

\[ \partial_1(e_k)=(\alpha_k-1)v,\qquad \partial_2(e_i\wedge e_j) =(\alpha_j-1)e_i-(\alpha_i-1)e_j . \]

The endpoint identifications of the \(e_k\)-cell give the first formula; traversing the four oriented sides of the square gives the second. Commutativity of \(\alpha_i\) and \(\alpha_j\) makes \(\partial_1\partial_2=0\). Applying the coefficient action therefore gives Barlak's Koszul maps

\[ \begin{array}{ccccc} M&\xrightarrow{\ d_G^0\ }&M\oplus M& \xrightarrow{\ d_G^1\ }&M,\\[2mm] g&\longmapsto& \bigl((\alpha_i-1)g,(\alpha_j-1)g\bigr),&&\\[-1mm] && (h_i,h_j)&\longmapsto& (\alpha_j-1)h_i-(\alpha_i-1)h_j . \end{array} \tag{A.13} \]

Equivalently, these are the degree-one and degree-two attaching maps in Barlak's Corollary 2.5. Reversing the chosen two-cell orientation changes the final formula by one global sign.

Combining the cell identifications (A.12) with the \(3\times3\) diagram (A.11) gives, through transverse degree two, the following diagram on the relevant homotopy groups:

\[ \begin{array}{ccccc} C^0(G;M)_1&\xrightarrow{\ d_G\ }& C^1(G;M)_1&\xrightarrow{\ d_G\ }& C^2(G;M)_1\\ {\scriptstyle\alpha_h-1}\big\downarrow&& {\scriptstyle\alpha_h-1}\big\downarrow&& {\scriptstyle\alpha_h-1}\big\downarrow\\ C^0(G;M)_0&\xrightarrow{\ d_G\ }& C^1(G;M)_0&\xrightarrow{\ d_G\ }& C^2(G;M)_0 . \end{array} \tag{A.13a} \]

The subscripts are height homological degrees. Diagram (A.13a) is not an extra chain model imposed on the filtered object: it is the homotopy-group diagram of the cell-fibre cofiber sequences in (A.11). This observation is what makes its staircase the exact-couple staircase.

Barlak's \(C_p\) in his Baum--Connes cofiltration is the concrete \(C^*\)-algebra representing \(\mathscr C_p\). Proposition 1.4 of his paper constructs compatible imprimitivity bimodules and, after Brown stabilization, a commutative isomorphism of the entire cofiltrations \(C_\bullet\otimes\mathcal K\cong F_\bullet\otimes\mathcal K\). Consequently (A.10)--(A.11) induce an isomorphism of exact couples, including the maps \(i,j,k\), with the mapping-torus exact couple used in his Proposition 4.2 and Theorem 4.4. Thus it remains only to calculate the \(d_2\) of the filtered cofiber.

The height PV sequence and \(K_1(A_0)=0\) give the exact sequence of \(G\)-modules

\[ 0\longrightarrow K_1(B_h) \xrightarrow{\ \partial_h\ }M \xrightarrow{\ \alpha_h-1\ }M \xrightarrow{\ \iota_h\ }K_0(B_h) \longrightarrow0. \tag{A.14} \]

The maps are \(G\)-equivariant because they come from the equivariant triangle (A.8). In particular, (A.14), not merely its kernel and cokernel, is the two-extension carried by the filtered comparison.

Filtered-cofiber staircase lemma. Under the edge identifications in (A.14), the differential \[ d_2^{0,0}:H^0(G,K_0(B_h))\longrightarrow H^2(G,K_1(B_h)) \] of the exact couple of \(\mathscr C_\bullet(B_h)\) is, up to the fixed global orientation sign, the two-extension connecting map of (A.14). Explicitly, if \(x\) is represented by \(g\in M\) and \[ (\alpha_i-1)g=(\alpha_h-1)h_i,\qquad (\alpha_j-1)g=(\alpha_h-1)h_j, \tag{A.15} \] then \[ d_2^{0,0}(x)= \left[(\alpha_j-1)h_i-(\alpha_i-1)h_j\right]. \tag{A.16} \]

Proof. Apply the spectrum-valued functor \(\operatorname{map}_{\mathrm{KK}}(\mathbb C,-)\) to (A.11). Through transverse degree two, its homotopy-group diagram consists of two copies of the Koszul complex (A.13), with vertical map \(\alpha_h-1\), and the homotopy groups of their cofiber. This assertion uses only the cell fibres (A.12); no vanishing is asserted for the odd \(K\)-groups of the intermediate stages \(\mathscr C_1(A_0)\).

We spell out the exact-couple chase. Lift \(x\in K_0(B_h)^G\) first to \(g\in M\). Its first cellular boundary is \(d_G^0g\). Equations (A.15) say precisely that this boundary is the vertical boundary of \(h=(h_i,h_j)\). In the first two columns of the \(3\times3\) diagram (A.11), exactness therefore produces a class

\[ \bar x_{\mathrm{cof}} \in K_0\!\left( \operatorname{cofib}(u_1:\mathscr C_1(A_0)\to \mathscr C_1(A_0))\right) \]

which maps to \(x\) at stage zero. Concretely, it is the cofiber lift represented by \((g;h_i,h_j)\): its first total boundary is

\[ \bigl((\alpha_i-1)g-(\alpha_h-1)h_i, (\alpha_j-1)g-(\alpha_h-1)h_j\bigr)=0. \]

This equality is a vanishing statement in the homotopy group of the relevant \(K\)-theory mapping spectrum. Exactness of the \(3\times3\) diagram supplies the displayed cofiber class; no literal homotopy between the original projections is being asserted or chosen. The filtered equivalence (A.10) sends \(\bar x_{\mathrm{cof}}\) to a class in \(K_0(\mathscr C_1(B_h))\) mapping to \(x\). It is therefore exactly an admissible first-stage lift in Barlak's definition (Appendix A, Remark A.2), rather than merely an element with the same eventual image.

Apply the next oriented attaching map. The \(g\)-part has already been cancelled by (A.15), so the upper boundary is

\[ d_G^1h=(\alpha_j-1)h_i-(\alpha_i-1)h_j. \]

Moreover,

\[ (\alpha_h-1)d_G^1h=d_G^1(\alpha_h-1)h =d_G^1d_G^0g=0, \]

so exactness of the left column of (A.11) identifies this element with a class in \(K_1(B_h)=\ker(\alpha_h-1)\). By the definition \(d_2=ji^{-1}k\), this is the second-page differential.

For clarity, the choice indeterminacy is entirely the algebraic indeterminacy of the two-extension. Put \(u=\alpha_h-1\) and \(N=\ker u\). If another lift of \(x\) is \(g'=g+ua\), then

\[ d_Gg'=u(h+d_Ga). \]

Every compatible choice \(h'\) with \(uh'=d_Gg'\) consequently has

\[ h'=h+d_Ga+n,\qquad n\in C^1(G;N). \]

It follows that

\[ d_Gh'=d_Gh+d_Gn, \]

so the resulting element of \(H^2(G;N)\) changes only by a degree-two Koszul coboundary. This simultaneously handles the choices of \(g\), of \(h\), and of the exact-couple lift, without invoking projection homotopies. This proves (A.16). \(\square\)

The argument is coefficient-functorial and proves the promised universal form:

Universal PV/Barlak comparison lemma. Let \(A\) be a separable \(\mathbb Z^2\)-\(C^*\)-algebra with generators \(\alpha_i,\alpha_j\), let \(\theta\in\operatorname{Aut}(A)\) commute with that action, and put \(B=A\rtimes_\theta\mathbb Z\). Assume that the induced \(\mathbb Z^2\)-action on \(B\) fixes the implementing unitary and that \(K_1(A)=0\). Set \(M=K_0(A)\). For \(x\in K_0(B)^{\mathbb Z^2}\), choose \(g,h_i,h_j\in M\) with \[ \iota_\theta(g)=x,\qquad (\alpha_i-1)g=(\theta-1)h_i,\qquad (\alpha_j-1)g=(\theta-1)h_j. \] Then Barlak's differential satisfies \[ \partial_\theta\!\left(d_2^{0,0}(x)\right) =\varepsilon\left[ (\alpha_j-1)h_i-(\alpha_i-1)h_j \right] \in H^2\!\left(\mathbb Z^2,\ker(\theta-1)\right). \] The sign \(\varepsilon\) depends only on the PV and cellular orientation conventions.

Indeed, replace \(A_0,\alpha_h,B_h\) by \(A,\theta,B\) in (A.8)--(A.16). The exact skeletal functor, the cell-fibre \(3\times3\) diagrams, and the staircase proof are unchanged; the only coefficient hypothesis used in the two-extension is \(K_1(A)=0\).

At the group level the height direction is homological. Put the two copies of \(M\) in homological degrees one and zero:

\[ C_1=M\xrightarrow{\alpha_h-1}C_0=M, \qquad H_0(C_\bullet)=M_{\alpha_h}, \qquad H_1(C_\bullet)=M^{\alpha_h}. \tag{A.17} \]

Thus the mixed page is written

\[ H^a\!\left(G_{ij},H_b(\langle\alpha_h\rangle,M)\right), \]

not with height group cohomology in the same degree labels. Equivalently, Poincare duality for \(\mathbb Z\) rewrites the input as height cohomological degree one and the target as degree zero: \(H^1(\mathbb Z,M)=M_{\alpha_h}\) and \(H^0(\mathbb Z,M)=M^{\alpha_h}\). We use homological labels because they match the two-term PV complex \(C_1\to C_0\) directly. The edge maps in degrees zero and one are respectively the height PV identifications \(\iota_h^{-1}\) and \(\partial_h\) from (A.2). The filtered exact-couple isomorphism (A.10)--(A.11) gives the commutative comparison

\[ \begin{array}{ccc} H^0\!\left(G_{ij},K_0(B_h)\right) &\xrightarrow{\ d_{2,\mathrm{Barlak}}^{0,0}\ }& H^2\!\left(G_{ij},K_1(B_h)\right)\\ {\scriptstyle\iota_h^{-1}}\quad\big\downarrow&& {\scriptstyle\partial_h}\quad\big\downarrow\\ H^0\!\left(G_{ij},M_{\alpha_h}\right) &\xrightarrow{\ \varepsilon d_{2,\mathrm{mix}}\ }& H^2\!\left(G_{ij},M^{\alpha_h}\right). \end{array} \tag{A.18} \]

The lower arrow is the transgression from \(H^0(G_{ij},H_0)\) to \(H^2(G_{ij},H_1)\). The sign \(\varepsilon\) is fixed by the PV boundary and square orientations. Under \(\iota_h^{-1}\), the projection \(q\) in Section A.3.1 is represented by \(g+L\), where \(L\) is the trivial stabilization from Chapter 1. Constants have zero transverse boundary and do not change \(\kappa_{ij}\). Thus (A.16) sends \([q]\) to \([\kappa_{ij}]\). Combining (A.18) with the operator calculation (A.6)--(A.7) proves (A.4). The comparison is therefore a theorem, not an additional hypothesis.

A.4 More than two transverse generators

For \(G=\mathbb Z^q\), define for every pair

\[ \kappa_{ij} =(\alpha_j-1)h_i-(\alpha_i-1)h_j. \tag{A.19} \]

Barlak's operator-theoretic \(d_2\)-class has one component for each oriented coordinate two-plane. Its primary corner curvatures vanish exactly when

\[ [\Omega_{ij}]=0\in K_1(C) \qquad\text{for all }i,j. \]

The integer curls (A.19) have the same coordinate indexing. Applying the proved comparison (A.4) to every coordinate two-plane identifies the two families.

This is necessary for exact commuting transports. It is not yet sufficient: a \(K_1\)-trivial normalizer defect can still be nontrivial as a homeomorphism or as a diagonal phase. The following two chapters remove those layers in order.

A.5 The full-field primary curvature

The exact-couple comparison above uses block support because the transverse automorphisms had to fix the height unitary. The conclusion that the full-field and block-supported defects have the same primary \(K_1\)-class, however, follows by a direct homotopy.

For \(0\leq t\leq1\), define automorphisms of \(B_h\) by

\[ \beta_i^t(a)=\alpha_i(a)\quad(a\in C(\Sigma)), \qquad \beta_i^t(U_h)=e^{2\pi it\Theta_{ih}}U_h. \tag{A.20} \]

Thus \(\beta_i^0\) is the block-supported action and \(\beta_i^1\) is the action induced by the full field. The clopen projection \(P\) and the finite PE transports \(W_i\) constructed in Chapter 1 do not depend on \(t\), and \(\beta_i^t(P)=\alpha_i(P)\) for every \(t\). Hence the source and range identities for \(W_i\) hold for every \(t\), and

\[ \Omega_{ij}(t) =W_i\beta_i^t(W_j) \bigl(W_j\beta_j^t(W_i)\bigr)^{\ast} \in U(PM_n(B_h)P). \tag{A.21} \]

Each \(W_i\) is a finite sum of clopen-weighted powers of \(U_h\). On a term \(aU_h^m\), the only new factor in \(\beta_i^t\) is \(e^{2\pi it m\Theta_{ih}}\). Therefore (A.21) is a norm-continuous path in the fixed corner, and

\[ [\Omega_{ij}(1)]=[\Omega_{ij}(0)] \quad\text{in }K_1(PM_n(B_h)P). \tag{A.22} \]

This proves the full-field homotopy lemma. By the block-supported PV/Barlak comparison, minimality of the selected height then forces the primary \(K_1\)-curvatures for the full field to vanish as well. What the path does not remove is the locally constant diagonal phase left after the GPS symbol correction; Chapter 4 treats that phase and retains its finite-stage coboundary hypothesis.

Technical Engine B: analytic descent and the product trace

Reader card. This chapter certifies the analytic part of Step 6: the balanced action, adjoints, compactness, finite projectivity, and trace factorization. For the construction and its meaning without the detailed Hilbert-module checks, first read Attaching the magnetic Heisenberg module.

The equation numbers retain the prefix \(6\) because this is the technical proof of Step 6 in the constructive argument.

This chapter gives the analytic construction in full. In particular, it fixes the cocycle convention, writes the balanced correspondence, verifies adjointability and compactness, and proves the trace formula on a finite projection.

B.1 Conventions and the equivariant coefficient module

Put

\[ B=B_h, \qquad A=B\rtimes_{\alpha,\sigma}G, \qquad D=C^\ast(G,\sigma)=A_{\Theta_G}. \]

The canonical unitaries satisfy

\[ U_gbU_g^\ast=\alpha_g(b), \qquad U_gU_h=\sigma(g,h)U_{g+h}, \]

and the generators \(u_g\) of \(D\) satisfy the same scalar cocycle relation.

First let \(F\) be an actual finitely generated projective right \(B\)-module. Its strict equivariant structure consists of semilinear unitaries \(V_g:F\to F\) such that

\[ \begin{aligned} V_g(\xi b)&=V_g(\xi)\alpha_g(b),\\ \langle V_g\xi,V_g\eta\rangle &=\alpha_g(\langle\xi,\eta\rangle),\\ V_gV_h&=V_{g+h}. \end{aligned} \tag{B.1} \]

The signed coefficient class is represented by a graded pair \(F^+\ominus F^-\), and the construction below is applied separately to its two summands.

B.2 The balanced correspondence

Define the right Hilbert \(A\)-module

\[ X_F=F\widehat\otimes_B A. \]

On the algebraic balanced tensor product its inner product is

\[ \langle\xi\otimes a,\eta\otimes c\rangle_A =a^\ast\langle\xi,\eta\rangle_Bc. \tag{B.2} \]

Positivity and completion are therefore the standard positivity and completion of an interior tensor product.

For \(g\in G\), define

\[ L_g(\xi\otimes a)=V_g\xi\otimes U_ga. \tag{B.3} \]

This is balanced. Indeed,

\[ \begin{aligned} L_g(\xi b\otimes a) &=V_g\xi\,\alpha_g(b)\otimes U_ga\\ &=V_g\xi\otimes\alpha_g(b)U_ga\\ &=V_g\xi\otimes U_gba =L_g(\xi\otimes ba). \end{aligned} \]

It is an isometry because

\[ \begin{aligned} \langle L_g(\xi\otimes a),L_g(\eta\otimes c)\rangle_A &=a^\ast U_g^\ast\alpha_g(\langle\xi,\eta\rangle)U_gc\\ &=a^\ast\langle\xi,\eta\rangle c. \end{aligned} \]

The adjoint normalization is explicit:

\[ L_g^{\ast}=L_g^{-1} =\overline{\sigma(g,-g)}\,L_{-g}, \]

because \(L_gL_{-g}=\sigma(g,-g)L_0\). Thus every \(L_g\) is adjointable. Finally,

\[ L_gL_h=\sigma(g,h)L_{g+h}. \tag{B.4} \]

Consequently \(u_g\mapsto L_g\) extends to a nondegenerate \(\ast\)-representation

\[ \lambda_F:D\longrightarrow\mathcal L_A(X_F), \]

and \(X_F\) is a \(D\)-\(A\) correspondence. This is the concrete form of twisted equivariant descent.

B.3 Compactness

Choose \(p\in M_n(B)\) with \(F\cong pB^n\). Extension of scalars gives

\[ X_F\cong pA^n. \tag{B.5} \]

Thus \(X_F\) is finitely generated projective as a right \(A\)-module, and

\[ \mathcal L_A(X_F)=\mathcal K_A(X_F) \cong pM_n(A)p. \]

In particular, the identity of \(X_F\) is compact and the entire left action \(\lambda_F(D)\) is by compact operators. This is stronger than the bare adjointability required for a correspondence.

B.4 Product with a Rieffel module

Let \(E_T\) be an elementary Rieffel--Heisenberg module over \(D\). The mixed module is the interior tensor product

\[ \mathcal E_{T,F} =E_T\widehat\otimes_DX_F. \tag{B.6} \]

To see projectivity without an abstract compactness shortcut, represent \(E_T\) by a projection \(e\in M_m(D)\). Then (B.6) is the range of the projection

\[ \lambda_F^{(m)}(e)\in M_m(\mathcal K_A(X_F)) \cong\mathcal K_A(X_F^m). \]

Since \(X_F^m\) is finitely generated projective, so is this range. For graded inputs, the output is the graded difference

\[ \mathcal E_{T,F^+}\ominus\mathcal E_{T,F^-}. \]

Likewise, an arbitrary virtual torus class is handled by a graded difference of Rieffel or standard finite-projective representatives. An ungraded projective output is asserted only when the chosen inputs are actual positive modules.

B.5 The coefficient-valued time-frequency formula

At height \(r\), the coefficients are ordinary time-frequency coefficients on

\[ M_T=\mathbb R^r\times\mathbb Z^{q-2r}. \]

With an embedding \(T=(T',T'')\), one matching normalization is

\[ (R_g\psi)(x) ={} \exp\!\left( 2\pi i \left\langle x-\tfrac12T'(g),T''(g) \right\rangle \right) \psi(x-T'(g)), \tag{B.8} \]

where Hilbert-space inner products are conjugate-linear in the first variable. Put \(a=T'(g)\), \(b=T''(g)\), \(c=T'(h)\), and \(d=T''(h)\). Direct multiplication gives

\[ R_gR_h=\sigma_T(g,h)R_{g+h}, \qquad \sigma_T(g,h) =\exp\!\left( \pi i(\langle c,b\rangle-\langle a,d\rangle) \right). \tag{B.8a} \]

Indeed, the exponent in \(R_gR_h\) minus that in \(R_{g+h}\) is \(\tfrac12(\langle c,b\rangle-\langle a,d\rangle)\). Thus \(\sigma_T\) is the alternating Weyl cocycle: \(\sigma_T(h,g)=\overline{\sigma_T(g,h)}\) and \(\sigma_T(g,-g)=1\). The embedding \(T\) is chosen to realize the fixed multiplier, so from this point \(\sigma_T=\sigma\). A cohomologous nonsymmetric normalization is obtained by the corresponding generator gauge and uses the general adjoint formula from Section B.2.

Here is the right-module convention used below. On the conjugate Hilbert space \(\overline{\mathcal H_T}\), set

\[ Q_g\overline\psi=\overline{R_g\psi}, \qquad \overline\psi\cdot u_g=Q_g\overline\psi. \tag{B.8b} \]

Since

\[ Q_hQ_g =\overline{\sigma_T(h,g)}Q_{g+h} =\sigma_T(g,h)Q_{g+h}, \]

(B.8b) is a right action of \(D=C^{\ast}(G,\sigma_T)\). Its \(D\)-valued inner product is

\[ \left\langle\overline\phi,\overline\psi\right\rangle_D =\sum_{g\in G}c_g(\overline\phi,\overline\psi)u_g, \qquad c_g(\overline\phi,\overline\psi) =\langle Q_g\overline\phi,\overline\psi\rangle_{\overline{\mathcal H_T}} =\langle\psi,R_g\phi\rangle_{\mathcal H_T}. \tag{B.7} \]

This convention has no hidden sign or inverse. For example, \(Q_h^{\ast}=Q_{-h}\) and the coefficient of \(u_k\) in \(\langle\overline\phi, \overline\psi\cdot u_h\rangle_D\) is

\[ \sigma_T(-h,k)c_{k-h} =\sigma_T(k-h,h)c_{k-h}, \]

which is exactly the coefficient of \(u_k\) in \(\langle\overline\phi,\overline\psi\rangle_Du_h\). Also \(u_g^{\ast}=u_{-g}\) for this Weyl normalization, and the coefficient formula gives \(\langle\overline\phi,\overline\psi\rangle_D^{\ast} =\langle\overline\psi,\overline\phi\rangle_D\).

For the dense vectors \(\overline\phi\otimes(\xi\otimes1)\) and \(\overline\psi\otimes(\eta\otimes1)\) in (B.6), the interior-product rule gives

\[ \left\langle \overline\phi\otimes(\xi\otimes1), \overline\psi\otimes(\eta\otimes1) \right\rangle_A {}= \sum_{g\in G} c_g(\overline\phi,\overline\psi) \langle\xi,V_g\eta\rangle_B U_g. \tag{B.9} \]

For Schwartz or Feichtinger vectors the usual Rieffel/Gabor estimates give convergence in the smooth crossed-product norm. Positivity of (B.9) is not a new assertion: it follows from (B.2) and the positive \(D\)-valued inner product (B.7).

Formula (B.9) is the native mixed Gabor picture. Each Fourier coefficient contains both the ordinary time-frequency coefficient and the finite PE height transport.

B.6 Trace factorization

Let \(\mu\) be jointly invariant. Write \(\tau_B\) for the trace on \(B\), and define the canonical crossed-product trace by

\[ \tau_A\!\left(\sum_gb_gU_g\right)=\tau_B(b_0). \]

Use the model \(F=pB^n\). There are partial unitaries \(w_g\) such that

\[ V_g\xi=w_g\alpha_g(\xi), \]

and under (B.5) the operator \(\lambda_F(u_g)\) is left multiplication by \(w_gU_g\). Therefore the induced trace on \(\mathcal K_A(X_F)\cong pM_n(A)p\) satisfies

\[ \operatorname{Tr}_{\tau_A}^{X_F}(\lambda_F(u_g))

\begin{cases} \operatorname{Tr}_{\tau_B}(p),&g=0,\\ 0,&g\ne0. \end{cases} \tag{B.10} \]

By continuity,

\[ \operatorname{Tr}\{\tau_A}^{X_F}\circ\lambda_F =\operatorname{Tr}\{\tau_B}(p)\,\tau_D \quad\text{on }D. \tag{B.11} \]

If \(e\in M_m(D)\) represents \(E_T\), apply (B.11) entrywise to the projection \(\lambda_F^{(m)}(e)\). This gives

\[ \tau_A([\mathcal E_{T,F}]) =\tau_D([E_T])\,\tau_B([F]). \tag{B.12} \]

The same equation is bilinear for graded differences. Since the graded coefficient representative constructed in Chapter 1 satisfies

\[ \tau_B([F^+]-[F^-])=\mu(f), \]

we obtain

\[ \boxed{ \tau_A([\mathcal E_{T,f}]) =\mu(f)\,\tau_{\Theta_G}([E_T]). } \tag{B.13} \]

Rieffel's trace calculation supplies the second factor. The argument did not use the height of the embedding \(T\), so every permitted Rieffel height is included.

B.7 A nontrivial diagonal phase

If the fixed-corner cocycle of Chapter 4 does not vanish, the maps \(V_g\) satisfy a coefficient-valued projective relation rather than (B.1). Then (B.4) acquires an operator-valued multiplier and the present left action of the scalar torus \(D\) is unavailable.

One could instead seek a finite-frame Heisenberg module over the corresponding Fell bundle. The regular Packer--Raeburn stabilization only gives a countably generated construction and does not prove the finite-projective or trace statement required here. That extension remains open.

Proof audit and repair ledger

Reader card. This is the authoritative dependency and status ledger. Conceptual chapters may summarize a technical engine, but claims of proof, conditionality, or remaining obstruction should be checked here and at the cited theorem location.

This chapter records the status of every nontrivial implication after the external proof critique. A checked item means that the relevant construction and bridge lemma are now written in this book. An open item is not used as an unconditional theorem.

Dependency table

Corrections forced by the audit

Fixed corner versus stable invariant

The class \([\nu]\) is well defined after fixing the corner, commuting symbols, coefficient action, and lift convention. The book does not construct comparison maps between the cohomology groups attached to different corners. It therefore does not use the former notation \(\mathfrak m_{\Theta,\mathrm{st}}(x_f)\) as though it were an invariant of \(x_f\).

Actual modules versus virtual classes

For an actual coefficient module \(F\) and an actual Rieffel module \(E_T\), Technical Engine B produces an actual finite projective mixed module. For signed \(f\), arbitrary torus \(K_0\)-classes, or isolated Pfaffian terms obtained by subtraction, the output is a graded difference of finite projective modules. No positivity statement is inferred from a possibly negative trace.

Principal solenoidal tori

Benameur--Mathai's principal-solenoidal theorem proves their magnetic gap-labelling conjecture, which is the upper containment

\[ \tau_\mu(K_0(A_{\Sigma,\Theta})) \subseteq\mathcal G_\Theta(\mu). \]

It does not prove equality of these groups in nonzero magnetic field. The higher-dimensional chapter now states only this containment.

Completion checklist

• The stabilization preserves the prescribed graded class.
• The clopen-flow block sizes and orthogonal complements are explicit.
• The Barlak unitary is compressed to \(\Omega_{ij}^\ast\).
• The enhanced equivariant PV triangle is carried through Barlak's exact skeletal functor; the resulting filtered cofiber and \(3\times3\) diagrams construct the exact-couple morphism and the admissible transfer-function lift without forming an ordinary \(C^*\)-algebra cone of \(1-\alpha_h\).
• The Baum--Connes/PV target and minimality removal of coinvariants are proved.
• The coloured corner, normalizer membership, end-preservation orientation argument, and typed \(K_1\)-index comparison are proved.
• For finite-PE bisection normalizers with vanishing corner \(K_1\)-defects, one GPS retraction corrects all symbol generators simultaneously.
• The phase-scaling path preserves the corner and identifies full-field primary \(K_1\)-curvature with the block-supported class; it is not used to trivialize the diagonal phase.
• Block support kills the diagonal phase by exact bisection equality.
• The three-dimensional all-plane statement assumes a minimal selected height in every cyclic splitting.
• The descent correspondence, compact identity, and product trace formula are proved.
• Actual and virtual outputs are distinguished throughout the theorem.
• The principal-solenoid attribution is corrected.
• Construct comparison maps making the finite-stable phase a genuine invariant of \(x_f\).
• Find dynamical conditions forcing the cross-block phase to vanish.
• Extend the coefficient absorption argument beyond height rank one.

The remaining unchecked items concern extensions beyond the proved block-supported height-one theorem.

Dimension three: comparison and equality

Reader card. This chapter compares three logically different claims: an upper trace bound, abstract existence of classes, and explicit finite-PE representatives. It also identifies exactly what is new relative to Benameur--Mathai and when equality follows.

This chapter separates three statements that are easy to conflate:

1. an upper bound on the range of the canonical trace;
2. existence of abstract \(K_0\)-classes producing the expected mixed traces; and
3. construction of native graded finite PE projective representatives of those classes.

The first two are addressed cohomologically by Benameur--Mathai in dimension three. The absorption theorem strengthens the third. Its block-supported constructive conclusions use the proved PV/Barlak comparison (A.4). The full-field conclusion remains conditional only on the separate finite-stage diagonal phase hypothesis.

1. The magnetic frequency group

Let

\[ A_{\Sigma,\Theta} =C(\Sigma)\rtimes_{\sigma_\Theta}\mathbb Z^3, \]

and let \(\mu\) be a \(\mathbb Z^3\)-invariant probability measure. Put

\[ M=C(\Sigma,\mathbb Z). \]

For a cyclic permutation \((i,j,k)\) of \((1,2,3)\), define

\[ \mathbb Z_{ij}[\mu] ={} \mu\!\left( \left(M_{\langle T_k\rangle}\right)^{ \langle T_i,T_j\rangle} \right). \]

The dimension-three magnetic frequency group is

\[ \mathcal G_\Theta(\mu) ={} \mathbb Z[\mu] +\Theta_{12}\mathbb Z_{12}[\mu] +\Theta_{13}\mathbb Z_{13}[\mu] +\Theta_{23}\mathbb Z_{23}[\mu]. \tag{7.1} \]

Benameur--Mathai prove for a minimal \(\mathbb Z^3\)-action that

\[ \tau_\mu\!\left(K_0(A_{\Sigma,\Theta})\right) \subseteq \mathcal G_\Theta(\mu). \tag{7.2} \]

This is the difficult or upper-bound half: no unexpected trace values occur. Their proof uses the measured twisted foliated index theorem, the Baum--Connes assembly map, and a direct group-cohomology calculation.

2. What their cohomological calculation already detects

Choose \(k=3\). If

\[ x_f\in \left(M_{\langle T_3\rangle}\right)^{ \langle T_1,T_2\rangle}, \]

there are integer-valued transfer functions \(h_1,h_2\) satisfying

\[ (T_1-1)f=(T_3-1)h_1, \qquad (T_2-1)f=(T_3-1)h_2. \tag{7.3} \]

The boundary computed in their Lemma 7.5 is represented, up to the action and sign convention, by

\[ \kappa_{12} =(T_2-1)h_1-(T_1-1)h_2. \tag{7.4} \]

This is the same integer curvature that appears as the \(K_1\)-index of the square defect in the absorption proof. If \(T_3\) is minimal, then \(\kappa_{12}\) is constant. Its integral is zero, so

\[ \kappa_{12}=0. \]

Benameur--Mathai consequently obtain

\[ \Theta_{12}\mathbb Z_{12}[\mu] \subseteq \tau_\mu\!\left(K_0(A_{\Sigma,\Theta})\right) \tag{7.5} \]

whenever \(T_3\) is minimal. Permuting the generators gives the analogous statements for the other two mixed groups. If all three generators are minimal -- their strong minimality condition in odd dimension -- they combine these inclusions with (7.2) to obtain equality.

Thus the absorption theorem should not be advertised as discovering the dimension-three trace labels that their cohomological proof missed. It constructs representatives for labels that their index theorem already detects abstractly.

3. Why Hypothesis (H) appears in their explicit construction

Section 8 of their paper asks for a particularly rigid realization. For a clopen set \(\Lambda\), they construct a homomorphism

\[ \Phi_\Lambda: C^{\ast}(\langle T_1,T_2\rangle,\sigma) \longrightarrow M_\infty\!\left(C(\Sigma)\rtimes_{i_*\sigma}\mathbb Z^3\right) \tag{7.6} \]

whose induced map multiplies the canonical trace by \(\mu(\Lambda)\). Their Hypothesis (H) supplies finitely many disjoint height sheets which the transverse generators permute with exact integer height corrections. That hypothesis is natural for a literal finite-matrix homomorphism of the form (7.6).

The absorption theorem changes the representative category. It permits:

• stabilization and graded differences;
• locally constant transfer functions rather than one finite tower;
• partial isometries between projective summands; and
• finite PE gauge corrections inside the height crossed product.

The output is a class-dependent graded finite-projective module or correspondence. For positive inputs it may be ungraded; arbitrary signed classes require a difference. It need not arise from a single global homomorphism \(\Phi_\Lambda\). This is the flexibility that removes Hypothesis (H).

4. Equality for a block-supported field

Suppose

\[ \Theta_{13}=\Theta_{23}=0, \qquad \theta=\Theta_{12}, \]

and assume \(T_3\) is minimal. The absorption theorem applies with height \(T_3\) and transverse group

\[ G=\langle T_1,T_2\rangle. \]

For every

\[ x_f\in \left(M_{\langle T_3\rangle}\right)^G \]

it constructs a graded finite PE module \(P_{f,12}\) satisfying

\[ \tau_\mu([P_{f,12}]) =\theta\,\mu(f). \tag{7.7} \]

If the selected Rieffel representative has trace \(m+\theta\), absorption first gives

\[ m\mu(f)+\theta\mu(f). \]

Subtracting \(m\) copies of the coefficient module isolates (7.7) in graded \(K_0\). Therefore

\[ \theta\mathbb Z_{12}[\mu] \subseteq \tau_\mu(K_0(A_{\Sigma,\Theta})). \tag{7.8} \]

Clopen projections already give

\[ \mathbb Z[\mu] \subseteq \tau_\mu(K_0(A_{\Sigma,\Theta})). \]

Combining these lower bounds with (7.2) yields

\[ \boxed{ \tau_\mu(K_0(A_{\Sigma,\Theta})) ={} \mathbb Z[\mu]+\theta\mathbb Z_{12}[\mu]. } \tag{7.9} \]

As a numerical trace theorem, (7.9) also follows from Benameur--Mathai's Corollary 7.6. The new content is that every term on the right has a native finite-propagation graded PE representative, with no Hypothesis (H).

5. Torus pullbacks and genuinely quasicrystalline classes

The constant coefficient class \(n[1]\) gives \(n\) copies of the pulled-back noncommutative-torus Rieffel class. In contrast, if

\[ x_f\notin\mathbb Z[1] \quad\text{in}\quad \left(M_{\langle T_3\rangle}\right)^G, \]

then \(P_{f,12}\) carries nonconstant pattern data. In particular, let \(\Lambda\) be a clopen pattern cylinder whose indicator satisfies the required invariance condition

\[ [1_\Lambda]\in \left(M_{\langle T_3\rangle}\right)^G. \]

Only for such a cylinder does the theorem directly construct \(P_{\Lambda,12}\), and its trace is

\[ \tau_\mu([P_{\Lambda,12}]) =\Theta_{12}\operatorname{freq}(\Lambda). \tag{7.10} \]

If this number is not in the torus trace group

\[ \mathbb Z +\Theta_{12}\mathbb Z +\Theta_{13}\mathbb Z +\Theta_{23}\mathbb Z, \]

then the class cannot lie in the torus-pullback image. If the numerical trace happens to coincide with a torus trace, one must instead use the associated-graded coefficient component or a higher cyclic pairing. The canonical trace is not generally injective on \(K_0\).

6. Minimal but not strongly minimal

There are useful examples outside strong minimality. Let \((X_r,S_r)\), \(r=0,1,2\), be Sturmian systems chosen so that the diagonal product is minimal, and set

\[ \begin{aligned} \Sigma&=X_0\times X_1\times X_2,\\ T_1&=S_0\times1\times1,\\ T_2&=1\times S_1\times1,\\ T_3&=S_0\times S_1\times S_2. \end{aligned} \]

The full action is minimal because \(T_3\) is minimal, but \(T_1\) and \(T_2\) are not minimal on the product. For a block-supported field the absorption theorem gives (7.9) and explicit labels such as

\[ \frac{\Theta_{12}}{\varphi} \]

from a Fibonacci cylinder depending only on the \(X_0\)-coordinate. Its class is \(T_2\)-fixed, and \((T_1-1)1_\Lambda=(T_3-1)1_\Lambda\), so it satisfies the required invariance in the \(T_3\)-coinvariants. This demonstrates that strong minimality is not needed for the one magnetic plane. It does not produce a trace equality beyond Benameur--Mathai's full dimension-three paper, since their one-height Corollary 7.6 already covers this numerical inclusion.

7. The conditional full-field statement

For a general \(3\times3\) magnetic matrix, repeat the construction for the three cyclic splittings. Assume first that the selected height \(T_k\) is minimal in each splitting; for all three cyclic splittings this requires \(T_1,T_2,T_3\) to be minimal, namely the strong-minimality hypothesis used here. This hypothesis is needed both in the curvature argument and in the GPS reduction. The comparison theorem and the full-field homotopy then remove the integral \(K_1\)-defects. If, in addition, for every pair \((i,j)\) and every class

\[ x\in \left(M_{\langle T_k\rangle}\right)^{ \langle T_i,T_j\rangle}, \]

the corresponding fixed-corner diagonal cocycle becomes a coboundary at an allowed finite stage, then absorption supplies all three reverse inclusions. This is an existential hypothesis for the constructed representatives, not the vanishing of a proved class invariant. Equation (7.2) then gives

\[ \boxed{ \tau_\mu(K_0(A_{\Sigma,\Theta})) =\mathcal G_\Theta(\mu). } \tag{7.11} \]

The final equality argument is formal under all of these hypotheses. A single minimal height proves only the magnetic plane complementary to that height; it does not justify the other two cyclic applications. The unresolved work is replacing strong minimality by suitable class-dependent directional hypotheses and eliminating the height--transverse magnetic phases. Equality of trace ranges also does not classify \(K_0\): torsion, extension data, and trace-zero classes remain invisible.

8. Comparison at a glance

The primary source for the first two rows is Benameur--Mathai, Gap-labelling conjecture with nonzero magnetic field.

Higher dimensions: an absorption lower bound

Reader card. The conclusions here are constructive lower bounds from the proved height-one theorem. The conjectural full magnetic frequency group, higher-rank absorption program, and upper containment are displayed for comparison but are not silently promoted to consequences of the construction.

The proved height-one theorem is arbitrary-dimensional in the transverse group. Together with the PV/Barlak comparison (A.4), proved in Technical Engine A, it gives the lower bound stated in this chapter.

1. The conjectural magnetic frequency group

Let

\[ M=C(\Sigma,\mathbb Z), \qquad A_{\Sigma,\Theta} =C(\Sigma)\rtimes_{\sigma_\Theta}\mathbb Z^p. \]

For every even subset \(I\subseteq\{1,\ldots,p\}\), Benameur--Mathai define

\[ \mathbb Z_I[\mu] ={} \mu\!\left( \left(M_{\mathbb Z^{I^c}}\right)^{\mathbb Z^I} \right). \tag{8.1} \]

Here one first takes coinvariants in the complementary directions \(I^c\), and then invariants under the induced \(\mathbb Z^I\)-action. Their magnetic frequency group is

\[ \mathcal G_\Theta(\mu) ={} \sum_{\substack{I\subseteq\{1,\ldots,p\}\\|I|\text{ even}}} \operatorname{Pf}(\Theta_I)\mathbb Z_I[\mu]. \tag{8.2} \]

The conventions include

\[ \operatorname{Pf}(\Theta_\varnothing)=1, \qquad \mathbb Z_\varnothing[\mu]=\mathbb Z[\mu]. \]

Benameur--Mathai conjecture that the trace range is contained in (8.2) for minimal Cantor actions and discuss reverse inclusions under stronger dynamical hypotheses. They prove the containment in low dimensions and later prove the all-dimensional upper containment for principal solenoidal tori. Their magnetic conjecture is this containment, not equality in nonzero field.

2. The height-one frequency group

Choose a primitive splitting

\[ \mathbb Z^p=\mathbb Zh\oplus G, \qquad G\cong\mathbb Z^{p-1}, \]

and assume that \(T_h\) is minimal. Put

\[ H_h ={} \left(M_{\langle T_h\rangle}\right)^G, \qquad D_h(\mu)=\mu(H_h). \tag{8.3} \]

Assume first that the magnetic form is block-supported:

\[ \Theta(h,G)=0. \]

For \(x_f\in H_h\) and any transverse class \(x\in K_0(A_{\Theta_G})\), the projection argument of Technical Engine B constructs a graded finite-projective class \([P_{f,x}]\) with

\[ \tau_\mu([P_{f,x}]) =\mu(f)\tau_{\Theta_G}(x). \tag{8.4} \]

For actual positive inputs this can be represented by an ungraded projective module; arbitrary signed inputs require a graded difference. After a common stabilization the construction is additive in both inputs. Indeed, if \(x=[e^+]-[e^-]\), with \(e^\pm\in M_{m_\pm}(A_{\Theta_G})\) projections, apply the homomorphism \(\lambda_F\) of Technical Engine B entrywise to \(e^+\) and \(e^-\). Formula (B.12) gives (8.4) for each projection, and subtraction gives it for \(x\). Thus the whole torus trace range occurs here without requiring a separate claim that a selected list of elementary Rieffel modules generates \(K_0\). When the relevant Pfaffian class has positive pairing, Rieffel modules give explicit Heisenberg/Gabor representatives (possibly first for the positive multiple supplied by Rieffel's elementary-module theorem). A general isolated Pfaffian generator is represented here virtually, by a graded difference of positive classes.

It follows immediately that

\[ \boxed{ \mathbb Z[\mu] +D_h(\mu)\, \tau_{\Theta_G}\!\left(K_0(A_{\Theta_G})\right) \subseteq \tau_\mu\!\left(K_0(A_{\Sigma,\Theta})\right). } \tag{8.5} \]

The product in (8.5) means the additive group generated by finite products \(d\,t\), with \(d\in D_h(\mu)\) and \(t\in\tau_{\Theta_G}(K_0(A_{\Theta_G}))\).

3. Pfaffian form of the lower bound

The trace range of the transverse noncommutative torus is generated by its even Pfaffian minors:

\[ \tau_{\Theta_G}(K_0(A_{\Theta_G})) ={} \sum_{\substack{I\subseteq G\\|I|\text{ even}}} \operatorname{Pf}(\Theta_I)\mathbb Z. \tag{8.6} \]

Consequently (8.5) becomes

\[ \boxed{ \mathbb Z[\mu] + \sum_{\substack{I\subseteq G\\0<|I|\text{ even}}} \operatorname{Pf}(\Theta_I)D_h(\mu) \subseteq \tau_\mu(K_0(A_{\Sigma,\Theta})). } \tag{8.7} \]

This is the absorption lower bound associated with the chosen height splitting.

The non-top Rieffel heights are essential here. A proof restricted to the top transverse Heisenberg module would produce only the largest Pfaffian. The theorem instead produces every even transverse minor allowed by \(K_0(A_{\Theta_G})\).

4. Why the lower bound lies in the conjectural group

For every even \(I\subseteq G\), there is a natural map

\[ H_h \longrightarrow \left( M_{\langle h,G\setminus I\rangle} \right)^I. \tag{8.8} \]

It takes the additional coinvariants in the directions \(G\setminus I\). Because the original class was invariant under all of \(G\), its image is invariant under \(I\). Integration descends through coinvariants, so

\[ D_h(\mu)\subseteq\mathbb Z_I[\mu]. \tag{8.9} \]

Term by term,

\[ \operatorname{Pf}(\Theta_I)D_h(\mu) \subseteq \operatorname{Pf}(\Theta_I)\mathbb Z_I[\mu]. \tag{8.10} \]

Thus (8.7) is not a rival to the Benameur--Mathai formula. It is a constructively realized subgroup of their predicted magnetic frequency group.

5. Odd dimensions: the complete codimension-one term

Let

\[ p=2m+1. \]

Then \(G\) has even rank \(2m\). For the top transverse subset \(I=G\), the complement is exactly the height direction. Therefore

\[ H_h ={} \left(M_{\langle h\rangle}\right)^G, \qquad D_h(\mu)=\mathbb Z_G[\mu]. \tag{8.11} \]

Taking a top transverse Rieffel module in (8.4) realizes the entire codimension-one summand:

\[ \boxed{ \operatorname{Pf}(\Theta_G)\mathbb Z_G[\mu] \subseteq \tau_\mu(K_0(A_{\Sigma,\Theta})). } \tag{8.12} \]

This is the direct generalization of

\[ \Theta_{12}\mathbb Z_{12}[\mu] \subseteq\tau_\mu(K_0) \]

from dimension three to every odd dimension. It realizes a full predicted Benameur--Mathai summand, rather than merely a subgroup of one.

6. A dimension-five formula

Take \(p=5\), height \(h=5\), and

\[ G=\langle T_1,T_2,T_3,T_4\rangle. \]

Assume \(T_5\) is minimal and \(\Theta_{i5}=0\). Then

\[ D_5(\mu)=\mathbb Z_{1234}[\mu]. \]

The absorption lower bound contains

\[ \begin{aligned} \mathbb Z[\mu] &+ \sum_{1\le i<j\le4}\Theta_{ij}D_5(\mu)\\ &+ \left( \Theta_{12}\Theta_{34} -\Theta_{13}\Theta_{24} +\Theta_{14}\Theta_{23} \right) \mathbb Z_{1234}[\mu]. \end{aligned} \tag{8.13} \]

The last coefficient is

\[ \operatorname{Pf}(\Theta_{1234}). \]

For a clopen pattern class \(f\), the corresponding top mixed module has trace

\[ \mu(f) \left( \Theta_{12}\Theta_{34} -\Theta_{13}\Theta_{24} +\Theta_{14}\Theta_{23} \right). \tag{8.14} \]

This is a genuinely higher-order pattern--magnetic label. It mixes a pattern frequency with a quadratic expression in the magnetic field.

7. Even dimensions and the rank gap

If \(p=2m\), the transverse group \(G\) has odd rank. Its largest even subsets have size \(p-2\). If \(I\) is such a subset, then

\[ I^c=\{h,j\} \]

has rank two. The Benameur--Mathai coefficient group first takes coinvariants in both directions \(h,j\), whereas \(H_h\) takes coinvariants only in \(h\) and requires invariance under \(j\). Therefore

\[ D_h(\mu)\subseteq\mathbb Z_I[\mu] \]

may be a proper containment.

For example, in dimension four with height \(4\), block support gives

\[ \mathbb Z[\mu] +D_4(\mu) \left( \Theta_{12}\mathbb Z +\Theta_{13}\mathbb Z +\Theta_{23}\mathbb Z \right) \subseteq \tau_\mu(K_0). \tag{8.15} \]

The conjectural frequency group contains the potentially larger terms

\[ \Theta_{12}\mathbb Z_{12}[\mu] +\Theta_{13}\mathbb Z_{13}[\mu] +\Theta_{23}\mathbb Z_{23}[\mu]. \]

This explains exactly why a one-height theorem is not expected to prove the full lower inclusion in every degree.

8. A constructive strengthening of magnetic gap labelling

The numerical conjecture (8.2) suggests the following stronger, representative-level statement.

Constructive magnetic gap-labelling conjecture. For every even \(I\) and every \[ x\in \left(M_{\mathbb Z^{I^c}}\right)^{\mathbb Z^I}, \] there is a graded finite PE projective representative \(P_{I,x}\) satisfying \[ \tau_\mu([P_{I,x}]) =\operatorname{Pf}(\Theta_I)\mu(x), \] whenever the associated integral coherence classes vanish and the fixed-corner magnetic phase becomes a coboundary at an allowed finite stage.

If these modules exist for every even \(I\), then

\[ \mathcal G_\Theta(\mu) \subseteq \tau_\mu(K_0(A_{\Sigma,\Theta})). \tag{8.16} \]

Combining (8.16) with the conjectural upper containment gives equality. This statement strengthens numerical magnetic gap labelling by prescribing native representatives for all its labels.

9. What a full proof would require

For \(J=I^c\), the required coefficient algebra is

\[ B_J =C(\Sigma)\rtimes_{\Theta_J}\mathbb Z^J. \]

One must represent every class in

\[ \left(M_{\mathbb Z^J}\right)^{\mathbb Z^I} \]

by a graded finite-projective \(B_J\)-representative with a strict finite PE \(\mathbb Z^I\)-action. Pairing it with an \(I\)-direction Rieffel class would then produce the term

\[ \operatorname{Pf}(\Theta_I)\mathbb Z_I[\mu]. \]

When \(|J|>1\), two new problems appear:

1. the LHS spectral sequence can have higher differentials, not only the pairwise \(d_2\)-curvature; and
2. there is no known direct replacement for the cyclic GPS normalizer retraction used for one height direction.

These issues are developed further in Beyond a rank-one height direction.

For a full magnetic matrix one must additionally kill the fixed-corner diagonal phase caused by height--transverse entries, possibly after an allowed finite stabilization. This is presently an existential representative-level hypothesis, not a choice-independent stable invariant. Separately, proving that no other trace values occur requires the Chern-integrality input behind the Benameur--Mathai upper bound.

10. Logical status

The conclusions can be summarized as follows.

The existing numerical target is the Benameur--Mathai magnetic frequency group. Their later principal-solenoidal-torus theorem proves the predicted upper containment in all dimensions for that class of systems by Chern-integrality and inverse-limit methods. It does not prove equality for nonzero magnetic field. The contribution here is different: it constructs explicit graded pattern-equivariant representatives for the stated lower subgroup and isolates the additional module-level obstructions.

Example: an odometer and dyadic labels

Reader card. This is the cleanest finite-stage model of the clopen-flow construction. It is useful after Chapter 1 and before the technical curvature comparison.

The odometer is the cleanest laboratory for the finite clopen-flow part of the proof.

The finite-quotient transport calculation below is explicit. Technical Engine A's PV/Barlak comparison makes it an input to the proved absorption theorem.

Height dynamics

Let

\[ \Sigma=\mathbb Z_2 \]

be the two-adic integers and let \(T_h(x)=x+1\). This is a minimal Cantor homeomorphism. For \(m\geq0\), the cylinder

\[ C_m=2^m\mathbb Z_2 \]

has measure

\[ \mu(C_m)=2^{-m}. \]

Choose commuting transverse translations \(T_i(x)=x+a_i\), with \(a_i\in\mathbb Z_2\). Every locally constant function factors through a finite quotient \(\mathbb Z/2^N\mathbb Z\). On that finite cyclic orbit, the difference

\[ (\alpha_i-1)f \]

has sum zero, and therefore is a discrete derivative \((\alpha_h-1)h_i\). The transfer function \(h_i\) is obtained by taking partial sums around the cycle.

flowchart LR
    C0["one clopen tower of height 2^N"]
    D["difference (alpha_i - 1)f"]
    Z["sum around tower = 0"]
    H["partial sums give h_i"]
    W["finite PE transport W_i"]
    C0 --> D --> Z --> H --> W

This is the finite-quotient version of the clopen-flow construction in Chapter 1.

Mixed dyadic-magnetic traces

Take \(f=1_{C_m}\). For any elementary transverse Rieffel module \(E_T\), the absorption theorem gives

\[ \tau_\mu([\mathcal E_{T,f}]) =2^{-m}\tau_{\Theta_G}([E_T]). \]

For a basic transverse two-torus class this reads

\[ \tau_\mu([\mathcal E_{f}])=2^{-m}\Theta_{12} \]

up to orientation. Thus a single system produces an entire dyadic family of mixed labels.

The abstract theorem does not require the full \(\mathbb Z^p\)-action to be faithful. If one wants a free geometric model, the odometer can instead be used as the transversal of a limit-periodic Delone suspension or placed in a suitable skew product. The height-one proof itself is unchanged.

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.

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.

Beyond a rank-one height direction

Research direction. This chapter explains the obstruction tower and the missing higher-rank normalizer theorem. It is a roadmap, not an input to the height-one theorem. In particular, pairwise square curvature need not exhaust the coherence data when the coefficient rank exceeds one.

The theorem is arbitrary-dimensional in the number of transverse directions, but it uses one distinguished height generator. This distinction is structural.

What changes for \(H=\mathbb Z^s\)

Let \(H\cong\mathbb Z^s\) be the coefficient subgroup and \(G\cong\mathbb Z^q\) the transverse subgroup. For \(M=C(\Sigma,\mathbb Z)\), the Lyndon--Hochschild--Serre page becomes

\[ E_2^{a,b}=H^a\!\left(G,H^b(H,M)\right), \qquad 0\leq b\leq s. \]

This chapter uses cohomological height degree. Technical Engine A instead labels the height PV complex homologically. For the duality group \(H\cong\mathbb Z^s\), with its standard orientation,

\[ H^b(H,M)\cong H_{s-b}(H,M). \]

Thus in rank one \(H^1(\mathbb Z,M)=H_0(\mathbb Z,M)=M_{\mathbb Z}\), the coinvariants. The two chapters use Poincare-dual labels for the same height term; no grading reversal is intended.

When \(s=1\), the class used in this book lies in \(E_2^{0,1}\). Its only possible outgoing differential is \(d_2\). That is why a pairwise square curvature is the complete integral obstruction.

For \(s>1\), coefficient classes can have height degree \(b>1\), and then

\[ d_r:E_r^{0,b}\longrightarrow E_r^{r,b-r+1} \]

can be nonzero for several values of \(r\). Rectangular commutators no longer exhaust the coherence data.

flowchart TD
    R1["rank-one height: b = 1"]
    D2["only d_2 can leave E_r^(0,1)"]
    SQ["pairwise square curvature"]
    GPS["GPS retraction for one minimal homeomorphism"]
    RS["higher-rank height: b > 1 possible"]
    DR["d_2, d_3, ..., d_(b+1)"]
    HC["higher coherence data"]
    NG["no direct GPS replacement known"]
    R1 --> D2 --> SQ --> GPS
    RS --> DR --> HC --> NG

The second missing ingredient

The Giordano--Putnam--Skau normalizer splitting is tied to the ordered orbit structure of a single minimal Cantor homeomorphism. It converts an index-zero full-group element into the AF kernel and retracts the normalizer onto a cyclic centralizer.

A higher-rank coefficient groupoid has no canonical analogue of that cyclic retraction. A complete extension therefore needs both:

1. a cohomological analysis of all higher differentials; and
2. a finite PE normalizer theorem that turns their vanishing into coherent compact-open bisections.

The rank-one theorem should be viewed as the base case in which these two problems collapse to \(d_2\) plus GPS.

Research problems opened by the proof

The proof-critical PV/Barlak comparison for block-supported height-one absorption is closed in Technical Engine A. It uses the equivariant PV triangle, the exact Baum--Connes skeletal functor, and a filtered-cofiber exact-couple chase; it does not form an ordinary \(C^*\)-algebra cone of \(1-\alpha_h\). The remaining problems concern the full magnetic phase, higher coefficient rank, and stronger realizations.

1. Construct comparison maps for the magnetic phase

For a full magnetic matrix, the book produces fixed-corner classes

\[ [\nu]\in H^2\bigl(G,U(D_{\mathrm{lc}})_{\bar\beta}\bigr). \]

Construct comparison maps under corner conjugacy, finite colour stabilization, changes of transfer function, and changes of projection representative. Only after proving invariance under these maps is it legitimate to define a stable phase attached to \(x_f\). In parallel, compute the fixed-corner classes in explicit cut-and-project and skew-product systems. The raw data are the locally constant return-time functions and the entries \(\Theta_{ih}\).

2. Build the twisted module directly as a Gabor module

The descent construction proves finite projectivity through the explicit projection \(\lambda_F^{(m)}(e)\). A stronger result would write a finite family of coefficient-valued windows whose time-frequency shifts give the projection or frame operator directly for the full Fell-bundle cocycle. This should make the magnetic phase visible as a projective representation identity on the windows.

3. Replace GPS in higher height rank

For \(H\cong\mathbb Z^s\), identify a compact-open normalizer category in which vanishing spectral-sequence obstructions imply strict finite PE transport. One possible route is a cubical system of Kakutani--Rokhlin partitions with coherent face maps.

4. Compare the three appearances of cohomology

There are at least three related but distinct groups:

• pattern-equivariant or groupoid cohomology controlling shape data;
• \(H^2\) of the transformation groupoid controlling twists;
• even cohomology entering the Chern character of \(K_0\).

The useful unification is not an identification by slogan. It is a diagram of concrete maps showing which deformation class changes the multiplier, which changes the groupoid, which yields a Kasparov element, and how its Chern character pairs with the invariant trace.

flowchart LR
    PE["PE deformation class"]
    GR["deformed groupoid or action"]
    TW["groupoid H^2 twist"]
    KK["KK correspondence"]
    K0["new K_0 class"]
    CH["Chern character"]
    TR["trace / gap label"]
    PE --> GR --> KK --> K0
    TW --> KK
    K0 --> CH --> TR

5. Turn constructions into lower bounds

For a family of clopen classes \(f_a\) and transverse Rieffel classes \(E_b\), determine the rank of the subgroup generated by

\[ [\mathcal E_{b,a}] \quad\text{and by their trace values}\quad \mu(f_a)\tau(E_b). \]

Trace independence gives a practical lower bound on \(K_0\), but equal traces do not imply equal classes. Cyclic cocycles or higher Chern pairings are needed when the canonical trace has a kernel.

6. Compare with torus pullback classes

When the hull fibers over a torus and pullback is injective on \(K_0\), compare the pulled-back Rieffel lattice with the mixed classes constructed here. The trace formula predicts a tensor-product pattern: torus Pfaffian minors multiplied by clopen frequencies. Proving independence requires tracking the relevant Chern components, not only the scalar trace.

Conventions and identities

Actions

All group actions are written additively. For \(g\in G\), \(\alpha_g\) denotes the induced automorphism. We use

\[ U_h aU_h^{\ast}=\alpha_h(a). \]

Changing to the inverse covariance convention changes some displayed curvatures by an overall sign but does not change any vanishing result.

Coinvariants and invariants

For a \(\mathbb Z\)-module \(M\) with automorphism \(\alpha\),

\[ M_\alpha=M/(\alpha-1)M, \qquad M^\alpha=\ker(\alpha-1). \]

For \(M=C(\Sigma,\mathbb Z)\),

\[ K_0(C(\Sigma)\rtimes_\alpha\mathbb Z)\cong M_\alpha, \qquad K_1(C(\Sigma)\rtimes_\alpha\mathbb Z)\cong M^\alpha. \]

Partial-isometry direction

Our transports satisfy

\[ W_i^{\ast}W_i=\alpha_i(P), \qquad W_iW_i^{\ast}=P. \]

Thus \(W_i\alpha_i\) acts from the module \(PB_h^n\) back to itself.

Square defect

\[ \Omega_{ij} =W_i\alpha_i(W_j) \bigl(W_j\alpha_j(W_i)\bigr)^{\ast}. \]

It is the identity precisely when the two semilinear generator transports commute.

Gauge change

For \(z_i\in U(PM_n(B_h)P)\), set \(W_i'=z_iW_i\) and

\[ \beta_i(a)=W_i\alpha_i(a)W_i^{\ast}. \]

Then

\[ \Omega_{ij}' =z_i\beta_i(z_j)\Omega_{ij} \beta_j(z_i)^{\ast}z_j^{\ast}. \]

For a full group cochain \(d_g\), the Busby–Smith cocycle changes by

\[ \nu'(g,h) =d_g\bar\beta_g(d_h)\nu(g,h)d_{g+h}^{\ast}. \]

These formulas fix the cohomology convention used throughout the book.

Glossary

For a side-by-side translation from tiling language to operator-algebraic language, see the Dictionary for tiling theorists.

Bisection normalizer. A partial isometry supported on a compact-open groupoid bisection. Conjugation by it carries the diagonal algebra on its source to the diagonal algebra on its range. This is stronger than merely having finite Fourier support.

Busby--Smith action. A twisted action consisting of automorphisms \(\alpha_g\) and multiplier unitaries \(\nu(g,h)\). The automorphisms compose up to inner conjugation by \(\nu\), and \(\nu\) satisfies a cocycle identity. After the GPS step, the residual magnetic phase has this form on the diagonal corner.

Crossed product. The operator algebra generated by a coefficient algebra and unitaries implementing a group action. Here the coefficient algebra records local patterns and the unitaries record translations. A magnetic multiplier changes the multiplication law of those unitaries.

Exact couple. A diagram of exact maps that generates a spectral sequence. In the curvature comparison, filtered mapping-torus and PV data give exact couples whose compatibility identifies the concrete transfer staircase with Barlak's \(d_2\).

Full corner. A corner \(PAP\) cut out by a projection \(P\) whose ideal is all of \(A\). A full corner is Morita equivalent to \(A\), so it retains the stable module and \(K\)-theory information while permitting a more concrete colored-orbit model.

Holonomy. The mismatch obtained by transporting around a closed loop. The square holonomy in this book is the unitary \(\Omega_{ij}\).

Kasparov product. The composition operation in \(KK\)-theory. The mixed class can be viewed abstractly as a product. Chapter 6 gives the conceptual balanced Hilbert-module construction, and Technical Engine B supplies its complete analytic verification.

\(KK\)-equivalence. An invertible generalized morphism in Kasparov's bivariant \(K\)-theory. It implies isomorphisms on \(K\)-theory and permits homotopy-coherent replacements that need not be literal algebra isomorphisms.

Mapping torus. An algebra of paths whose endpoints are related by an automorphism. Its cellular filtration encodes group-cohomological data used in the PV/Barlak comparison.

Morita equivalence. An equivalence of module categories implemented by an imprimitivity bimodule. Morita-equivalent algebras need not be literally isomorphic, but have canonically related \(K\)-theory and trace pairings.

Pimsner--Voiculescu sequence. The six-term exact sequence computing the \(K\)-theory of a crossed product by \(\mathbb Z\). For the height algebra it identifies \(K_0\) with integer coinvariants and \(K_1\) with integer invariants.

Strict equivariant module. A module equipped with semilinear maps \(V_g\) satisfying the group law exactly. Equality of the underlying \(K_0\)-class with all its translates does not by itself provide such maps.

AF kernel. The index-zero subgroup \(\Gamma_0\) of the topological full group of a minimal Cantor system. Its elements are generated inside finite Kakutani--Rokhlin models.

Block-supported multiplier. A multiplier whose skew form has entries only among the transverse generators. Equivalently, \(\Theta_{ih}=0\) for every transverse \(i\).

Clopen coinvariant class. An element of \(C(\Sigma,\mathbb Z)_{T_h}\), usually represented by an integer-valued locally constant function or a difference of clopen characteristic functions.

Finite PE. Finite pattern-equivariant: determined by a finite clopen partition, or geometrically by a bounded patch radius.

Graded module. A formal difference of two finite projective modules, implemented concretely by a \(\mathbb Z/2\)-graded projection. This permits an arbitrary integer-valued class, not only a positive one.

Height direction. The distinguished generator \(h\) for which \(T_h\) is a minimal Cantor homeomorphism and \(B_h=C(\Sigma)\rtimes_{T_h}\mathbb Z\).

Heisenberg module. Rieffel's projective module obtained from an embedding of a lattice into a phase space over a locally compact abelian group. Its smooth vectors form a Gabor-module model. See the introductory construction.

Mixed class. A \(K_0\)-class combining a coefficient coinvariant class with a transverse noncommutative-torus class.

Pattern frequency. The invariant-measure value of a clopen patch cylinder. It is the coefficient factor \(\mu(f)\) in the mixed trace.

Square defect. The unitary \[ \Omega_{ij}=W_i\alpha_i(W_j) \bigl(W_j\alpha_j(W_i)\bigr)^{\ast} \] measuring failure of two semilinear transports to commute.

Fixed-corner magnetic phase. The cohomology class of the diagonal Busby--Smith cocycle after fixing the corner, commuting symbols, coefficient action, and lift convention. Its vanishing is exactly the diagonal strictification criterion in that corner.

Finite-stable phase hypothesis. The existential assertion that the fixed-corner phase vanishes after some allowed finite stabilization and index change. The book does not promote this hypothesis to an invariant of the underlying \(K_0\)-class.

Transfer function. A locally constant integer-valued function \(h_i\) solving \((\alpha_i-1)g=(\alpha_h-1)h_i\). It encodes a bounded clopen flow.

Transverse directions. The quotient group \(G\cong\mathbb Z^{p-1}\) complementary to the chosen height direction.

Twisted absorption. The construction of a strict finite PE \(G\)-equivariant representative of a coefficient class so that it can be descended together with a transverse twisted module.

Sources and where they enter

This book separates established ingredients from the new synthesis. The following are the principal primary sources.

Twisted gap labelling

• M. T. Benameur and V. Mathai, Gap-labelling conjecture with nonzero magnetic field. Their explicit morphism supplies the three-dimensional mixed-class construction and Hypothesis (H). Proposition 8.3 uses the multiplier extended from the transverse two-plane, which is precisely the block-supported case isolated here.

• M. T. Benameur and V. Mathai, Proof of the magnetic gap-labelling conjecture for principal solenoidal tori. This proves the magnetic gap-labelling conjecture in every dimension for principal solenoidal tori by an inverse-limit and Chern-integrality argument. In that paper the conjecture is the upper containment of the trace range in the magnetic frequency group; it is not equality in nonzero field. It is not an explicit PE-module construction.

• M. Kreisel, Gabor frames for quasicrystals and K-theory. This is the source for the quasicrystal Gabor module and its density trace.

Heisenberg and Gabor modules

• M. A. Rieffel, Projective Modules Over Higher Dimensional Noncommutative Tori. The elementary modules of arbitrary height and their traces come from this paper.

• F. Luef, Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces. This gives the Gabor-frame realization of smooth projective modules.

• M. Gerhold, A. Lamando, and F. Luef, Linear Deformations of Heisenberg Modules and Gabor Frames. This is a modern deformation-theoretic treatment of the Rieffel--Gabor side.

Spectral sequence and curvature

• S. Barlak, On the spectral sequence associated with the Baum--Connes conjecture for \(\mathbb Z^n\). Proposition 4.2 identifies \(d_2\) with the Snake-Lemma map for the natural PV diagram, and Theorem 4.4 identifies it with a generalized Bott unitary. Technical Engine A proves that its compression is the corner holonomy and compares its Snake-Lemma boundary with the height-transfer integer curl.

• U. Bunke, A. Engel, and M. Land, A stable \(\infty\)-category for equivariant \(KK\)-theory. Theorems 1.3--1.4 supply the stable \(\infty\)-categorical enhancement and semiexactness used in Technical Engine A; Proposition 1.7 supplies bi-exact tensor product, Proposition 1.12 supplies the semisplit proper-skeleton extensions, and Theorem 1.22 supplies the enhanced crossed product functor. The manuscript applies these functors to the equivariant Toeplitz triangle and then constructs the manuscript-specific exact-couple lift. It does not form an ordinary \(C^{\ast}\)-algebra cone of \(1-\alpha_h\).

• H. Matui, \(\mathbb Z\)-actions on AH algebras and \(\mathbb Z^2\)-actions on AF algebras. This is useful background for cohomological obstructions to commuting lifts.

Cantor full groups and finite corners

• T. Giordano, I. F. Putnam, and C. F. Skau, Full groups of Cantor minimal systems. Section 5 identifies the full-group index with the \(K_1\) Fredholm pairing. Proposition 5.11 and Corollary 5.12 give the normalizer splitting used in Chapter 3.

• N. C. Phillips, Crossed products of the Cantor set by free minimal actions of \(\mathbb Z^d\). This provides structural background for the Cantor crossed products and their finite-projective behavior.

Logical status of the present argument

The Pimsner--Voiculescu computation, Barlak's curvature formula, the GPS splitting, and Rieffel's modules are established results. The contribution assembled here is the following constructive combination:

1. clopen coinvariant invariance gives bounded finite PE transports;
2. an enhanced equivariant Toeplitz cofiber and explicit exact-couple morphism carry the transfer-function staircase to Barlak's \(d_2\);
3. one GPS retraction corrects all symbols simultaneously;
4. block support kills the remaining magnetic phase exactly; and
5. once strict coefficient transport is available, the coefficient module pairs with every permitted Rieffel height in every ambient dimension.

Item 2 is proved in Technical Engine A. Consequently the block-supported absorption theorem and its constructive lower bounds no longer require a separate PV/Barlak comparison hypothesis.

For a full magnetic matrix, the book proves a fixed-corner criterion rather than an automatic vanishing theorem. Across finite stabilizations it assumes existential vanishing at one allowed stage; it does not claim a choice-independent stable phase invariant.