The Galois-twisted arithmetic groupoid: its KMS classification, and why the Tannakian obstruction does not compute it

The Galois-twisted groupoid model, in which the Frobenius elements act through a homomorphism of the abelian ideal group (hence with commuting lifts), is classified: its KMS_beta simplex is Prob(E_beta \ G) for the abelian essential-value group E_beta, which agrees with the Tannakian prediction Prob(G/N_beta) only when E_beta is normal. The gap between the two is analyzed and left as the intermediate-structure question, answered in Paper XXVII of the series.