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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

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

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.