The Galois action on the seven M23 Hurwitz points

Huang, Jackson, Lee, Poonen, Pries, and Zhang recently realized M23 as a Galois group over Q by finding a G_Q-fixed point among the seven inner Nielsen classes of type (2A, 23A, 23B), and noted that they had no conceptual explanation for this fixed point. We determine the Galois action on the full seven-point Hurwitz fiber. First, we compute the action of complex conjugation exactly. With branch points (0, √−23, −√−23), it acts on the seven classes as (1 2)(5 7), fixing 3, 4, 6. We then compute the associated genus-4 quotient curves by numerical uniformization and construct marked Petri invariants which we prove algebraic and G_K-equivariant for K = Q(√−23). Trace and norm along the marking descend these to algebraic G_Q-equivariant invariants of the unmarked fiber. Integer-relation recognition, independently replicated by two descended coordinates with ~70 digits of confirmation margin, gives a high-precision numerical determination of the arithmetic decomposition {1,…,7} = {6} ⊔ {1,2,3,4,5,7}. The two coordinates generate the same sextic residue field F; it has signature (2,2), and its Galois closure has group S_6. Thus, subject only to the stated integer-relation recognitions, ρ(G_Q) = StabS_7(6) ≅ S_6 and ρ(G_K) = S_6. Arithmetic fixes exactly one point and acts maximally on the remaining six. This symmetry breaking is not forced by the finite-level Grothendieck–Teichmüller constraints: using Guillot's description of GT_1(G) for finite simple groups, we prove that GT_1(M23) induces the full S_7 on these seven Nielsen classes. Finally, under the same recognition hypothesis, at the prime 23 the seven-point Hurwitz algebra decomposes locally as Q_23 × F_2 × F_4, where F_2 and F_4 are totally ramified tame extensions of degrees 2 and 4. Hence inertia has cycle type (1)(2)(4), making label 6 the unique inertia-fixed point at 23 and giving a concrete local target for a conceptual explanation of the exceptional rational point.