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.