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.