Irreducible multiplicity data is not canonical outside type I

A character of $A$ is uniquely determined by $z=\chi (a)\in \{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$, with $\chi (a^n)=z^n$. This identifies $\widehat{A}$ with the torus: compact subsets of the discrete cyclic group are finite, so its compact-open character topology is exactly the topology of this evaluation. Transport normalized torus Haar measure to $\widehat{A}$; do the same for $B$ and $\psi (b)$. These are standard-Borel probability parameter spaces. A transversal $T$ of $A\mathrm{\backslash }F$ is countable. Realize all ${\pi }_z={\mathrm{Ind}}_A^F{\chi }_z$ on the fixed space ${\mathrm{\ell }}^2(T)$ as in [F2]. For fixed $g$ every basis matrix coefficient is zero or a monomial $z^n$, hence continuous in $z$. Thus [F4] defines ${\mathrm{\Pi }}_A={\int }_{\widehat{A}}^{\oplus }{\pi }_{z}d\mu (z)$ on $L^2(\widehat{A};{\mathrm{\ell }}^2(T))$. The pointwise group law is exact for every parameter; since $F$ is discrete, the integrated unitary representation is automatically strongly continuous.

The constant section $\xi (z)={\delta }_e$ is a unit vector. Its diagonal coefficient at $g$ is the integral of the coefficient in [F2]: it is zero if $g\mathrm{\notin }A$, and if $g=a^n$ it is $\int z^{n}d\mu (z)={\mathbf{1}}_{n=0}$. Hence it equals ${\mathbf{1}}_{g=e}$. Crucially, $\xi$ is cyclic for the entire integral. Indeed, for every $t\in T$ and $n\in \mathbb{Z}$, the group law and [F2] give ${\mathrm{\Pi }}_A(t^{-1}{a}^n)\xi (z)=z^n{\delta }_t$. The span of these orbit vectors contains every finite sum of Fourier polynomials times coset basis vectors. Such sums are dense: the squared norm is the sum of the scalar-coordinate squared $L^2$ norms, so truncating the countable coset coordinates makes the tail arbitrarily small, and [F5] approximates each of the finitely many remaining coordinates by Fourier polynomials. This proves global cyclicity, without inferring it from fibrewise cyclicity.

The regular representation and $({\mathrm{\Pi }}_A,\xi )$ now have the same diagonal coefficient and cyclic unit vectors, so [F6] gives ${\lambda }_F\cong {\mathrm{\Pi }}_A$. Repeating steps 1.1 and 2.1 with $B$ gives ${\lambda }_F\cong {\int }_{\widehat{A}}^{\oplus }{\mathrm{Ind}}_A^F\chi d\mu (\chi )\cong {\int }_{\widehat{B}}^{\oplus }{\mathrm{Ind}}_B^F\psi d\nu (\psi )$. Both measures are normalized Haar probabilities.

By self-commensuration in [F1], [F3] makes every fibre in both integrals irreducible. Every intersection $g^{-1}Bg\cap A$ is trivial, hence has infinite index in the infinite group $A$. The cross-family inequivalence hypothesis is therefore vacuous, and [F3] gives ${\mathrm{Ind}}_A^F\chi \cong ̸{\mathrm{Ind}}_B^F\psi$ for every pair of parameters. Within one family, distinct characters also give inequivalent fibres: if the intersection $g^{-1}Ag\cap A$ has finite index in both groups, self-commensuration forces $g\in A$; conjugation then fixes every character of the abelian $A$, and different characters differ on $A$. The same inequivalence criterion applies, and similarly for $B$. Thus each model has multiplicity one on its own irreducible classes, and the two sets of classes are disjoint.

No removal of null parameter sets, change of measure within its class, or parameter identification can match the irreducible fibres of these two probability models: every cross pair is inequivalent by step 4.1. This refutes canonical irreducible measure-class/multiplicity data. No standard-Borel structure on the full dual $\widehat{F}$ is being assumed; the two models already use standard-Borel circles and disjoint images among irreducible classes. Moreover [F7] proves that $F$ is not type I. In this example the central factor datum remains the one-point non-irreducible regular factor of [F7]; the two irreducible disintegrations are not central diagonalizations. Hence the uniqueness appropriate to central factor decompositions cannot be transferred to irreducible multiplicity data outside type I.