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.