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.