4. The residual magnetic phase
Reader card. Input: transports whose bisection symbols already commute. Output: an exact fixed-corner criterion for lifting that commutation to the twisted operators. In the block-supported theorem the phase is automatically one. For a full magnetic matrix, finite-stage phase triviality remains an explicit hypothesis.
The GPS retraction removes the permutation part of the defect. This chapter shows that the remaining problem is abelian. It proves an exact criterion in one fixed corner. Across different representatives and stabilizations, only an existential strictifiability hypothesis is used; no choice-independent cohomology class is claimed.
4.1 Kernel of the symbol map
Let
\[ D_{\mathrm{lc}}=C_{\mathrm{lc}}(Y). \]
A finite PE normalizer with trivial bisection symbol is multiplication by a locally constant circle-valued function. Therefore, after (3.9),
\[ \nu_{ij} =W_i'\alpha_i(W_j') \bigl(W_j'\alpha_j(W_i')\bigr)^{\ast} \in U(D_{\mathrm{lc}}). \tag{4.1} \]
A diagonal unitary contributes nothing to \(K_1(C)\): a locally constant circle-valued function has a logarithm on each member of a finite clopen partition. This explains why \(K_1\)-curvature cannot see (4.1).
4.2 The full Busby–Smith cocycle
Because the symbols \(c_i\) commute, define
\[ c_g=c_1^{g_1}\cdots c_q^{g_q} \qquad(g\in G). \]
Choose finite PE lifts \(W_g'\) with \(W_0'=P\) and set
\[ \nu(g,h) =W_g'\alpha_g(W_h')W_{g+h}'{}^{\ast} \in U(D_{\mathrm{lc}}). \tag{4.2} \]
On the diagonal define
\[ \bar\beta_g(d) =W_g'\alpha_g(d)W_g'{}^{\ast}. \]
Although the lifts are only projective, \(\bar\beta\) is a genuine action on \(D_{\mathrm{lc}}\): the inner defect is diagonal, and conjugation by a diagonal element acts trivially on the diagonal.
Associativity gives
\[ \nu(g,h)\nu(g+h,k) =\bar\beta_g(\nu(h,k))\nu(g,h+k), \tag{4.3} \]
so
\[ \nu\in Z^2\bigl(G,U(D_{\mathrm{lc}})_{\bar\beta}\bigr). \]
4.3 Diagonal gauges are exactly coboundaries
Let \(d_g\in U(D_{\mathrm{lc}})\) and put
\[ W_g''=d_gW_g'. \]
Then
\[ \nu''(g,h) =d_g\bar\beta_g(d_h)\nu(g,h)d_{g+h}^{\ast}. \tag{4.4} \]
Therefore:
Fixed-corner criterion. For fixed symbol indices, diagonal finite PE gauges make the transports into a genuine \(G\)-action if and only if \[ [\nu]=0 \quad\text{in}\quad H^2\bigl(G,U(D_{\mathrm{lc}})_{\bar\beta}\bigr). \]
Trivializing the full cocycle—not only its generator commutators—shows that every group word is coherent. There is no hidden \(3\)-cocycle.
4.4 Index freedom
If a transport is multiplied by a corner full-group element of index \(k_i\), its retracted symbol changes from \(c_i\) to
\[ \psi^{k_i}c_i. \]
Indeed, for \(\gamma\in\Gamma\),
\[ r(\gamma)=\psi^{I(\gamma)}. \]
For the selected corner, one may therefore ask whether some index vector \(k\in\mathbb Z^q\) makes the corresponding diagonal cocycle a coboundary. This is still a statement about the selected corner. It does not identify the cohomology groups belonging to different corners.
4.5 The finite-stable existential hypothesis
An allowed finite stabilization replaces \(P\) by
\[ P\oplus1_L \]
and subtracts the same free block in the graded class. The associated unit space acquires finitely many colours, and one can repeat the GPS and diagonal reductions there.
Finite-stable phase hypothesis for a constructed representative. Starting with the flow representative of Chapter 1, there is an allowed finite stabilization, a GPS reduction, and an integer index choice for which the resulting fixed-corner class \[ [\nu]\in H^2\bigl(G,U(D_{\mathrm{lc}})_{\bar\beta}\bigr) \] is zero.
If this hypothesis holds, (4.4) supplies the final diagonal gauges, so it is a sufficient condition for strictification. Conversely, any strictification reached through this same staged construction yields such a zero class. This is the strongest statement established here.
We deliberately do not denote this condition by a class \(\mathfrak m_{\Theta,\mathrm{st}}(x_f)\). To construct such an invariant one would need:
- comparison maps between the cohomology groups for different corners and finite stabilizations;
- invariance under changing the transfer functions and the projection representative; and
- a proof that every finite PE equivalence is generated by the allowed colour additions, index shifts, and diagonal gauges.
Those comparison results are not proved. The present hypothesis is therefore existential, not a choice-independent obstruction attached to the \(K_0\)-class.
4.6 Why block support makes the phase one
Assume
\[ \Theta_{ih}=0 \qquad(i=1,\ldots,q). \]
Then
\[ \alpha_i(U_h)=U_h, \]
so \(\alpha_i\) sends every coefficient-one height bisection to another coefficient-one bisection.
Use the coefficient-one transports constructed in Chapter 1 and the coefficient-one GPS gauges from Chapter 3. If \(B_i\) is the compact-open bisection representing \(W_i'\), then
\[ W_i'\alpha_i(W_j') =1_{B_i\alpha_i(B_j)}, \qquad W_j'\alpha_j(W_i') =1_{B_j\alpha_j(B_i)}. \]
The two symbols commute. Their transverse group labels agree, and their remaining difference is a height arrow. Since a minimal Cantor homeomorphism is free, the reduced height groupoid is principal; equal source-to-range maps give equal bisections. Hence
\[ B_i\alpha_i(B_j)=B_j\alpha_j(B_i) \]
and therefore
\[ \nu_{ij}=1 \]
as an exact operator equality.
4.7 Where a cross phase comes from
If
\[ \alpha_i(U_h)=e^{2\pi i\Theta_{ih}}U_h, \]
then a branch \(1_CU_h^n\) of \(W_j'\) acquires the factor
\[ e^{2\pi i\Theta_{ih}n} \]
under \(\alpha_i\). Around a transverse square, the bisection exponents cancel but these factors need not. The result is precisely the diagonal cocycle (4.1): the magnetic transgression of the integer return-time data.
This also explains why parity and \(2\)-divisibility are not the right general obstruction. They concern the permutation kernel or a particular matrix factorization; the surviving class is circle-valued and depends on the height–transverse magnetic block.