The SevenPoints of M23

Every sporadic simple group is now a Galois group over the rationals — and the point that finished the list is still unexplained.

and the permutation image is the largest subgroup of S₇ compatible with fixing one point. Two more results frame that answer: the finite-level Grothendieck–Teichmüller constraints attached to M23 permit every permutation of the seven — so whatever forces the fixed point is invisible there — and at the prime 23, exactly one of the seven survives inertia. The symmetry breaking is real, it is maximal, and it now has a local fingerprint.

Émile Mathieu found the first five sporadic simple groups in 1861 and 1873, a century before anyone understood how strange they were — five permutation groups so symmetric they should not exist, orphans outside every infinite family. The full census took until the classification: twenty-six sporadic groups, ending at the Monster. M23 is one of Mathieu's five, a group of order 10,200,960 acting on twenty-three points.

The inverse Galois problem asks whether every finite group is the Galois group of some polynomial with rational coefficients. Hilbert settled the symmetric and alternating groups in 1892, Shafarevich the solvable groups in 1954, and in the 1980s the rigidity method — Thompson for the Monster, then Matzat's school for most of the rest — worked through the sporadic list group by group. By the mid-1990s the count stood at twenty-five of twenty-six. The rigidity criteria that dispatched the others never found a foothold in M23, and for three more decades it stayed the only sporadic group not known to be a Galois group over Q. The general problem remains open; the sporadic chapter needed one more group.

The 2026 proof went around the obstruction instead of through it. The six authors studied Galois covers of the projective line with monodromy group M23, branched over the three points 0, √−23, and −√−23, with local monodromy in the conjugacy classes 2A, 23A, and 23B. The moduli space of such covers — the Hurwitz space — has exactly seven points in the fiber over that branch configuration, and the absolute Galois group of the rationals shuffles those seven among themselves. A point fixed by the whole group is a cover defined over Q, and such a cover realizes M23 regularly over Q(t), which Hilbert irreducibility then specializes to Q. They exhibited a fixed point. That is the whole game.

A branched cover of the line is a surprisingly combinatorial object. Fix three branch points and a finite group. A Galois cover branched over those points, with that monodromy group, is the same data as a triple of group elements (x, y, z) with xyz = 1, each drawn from a prescribed conjugacy class, together generating the group — counted up to simultaneous conjugation. Riemann supplied the analytic half of the dictionary; the payoff is that a moduli problem over the complex numbers becomes a finite count inside a finite group.

For M23 with classes (2A, 23A, 23B) the count comes out to exactly seven, and the census is small enough to certify outright. Each of the classes 23A and 23B has 443,520 elements. Anchor the involution x and the only surviving symmetry is its centralizer, a group of order 2,688; the set of valid partners y turns out to have exactly 18,816 = 7 × 2,688 elements, falling into seven free orbits. Generation comes for free — every candidate pair already generates all of M23, certified by verified stabilizer chains of order 10,200,960 — and the seven orbit representatives are printed in the paper's appendix.

Seven triples, seven covers, seven points, and one arithmetic symmetry: because the branch points 0, ±√−23 and the covers themselves are algebraic, the absolute Galois group acts on the fiber through some subgroup of S₇. The labels 1 through 7 are fixed once and for all in the appendix, with label 6 the class of the triple the discoverers printed — the fixed point. The question “why is there a rational point” becomes concrete: which permutations does arithmetic actually perform on these seven, and what distinguishes 6?

The absolute Galois group is a profinite monster, but it has one element you can hold in your hand: complex conjugation. Fried's branch cycle argument, from 1977, says exactly how conjugation acts on a Nielsen class when the branch locus is symmetric under it — and this one is: conjugation fixes 0 and exchanges √−23 with −√−23. On triples the recipe reads κ(x, y, z) = (x, z⁻¹, y⁻¹), which lands back in the same class structure because inverting a 23-cycle trades 23A for 23B.

On the anchored slice the map collapses to something a computer can settle: y ↦ x·y. Deciding where each of the seven classes goes is then a finite search for conjugating elements, and the paper exhibits all seven certificates explicitly — short permutations you can multiply out by hand if you distrust every line of software involved. The verdict:

Three of the seven points — labels 3, 4, and 6 — are fixed by the mirror, so exactly three of the seven covers admit real models. That label 6 comes out fixed is a consistency check the certificate passes, not an input fed in. And the involution alone already constrains the answer: every Galois orbit must be a union of the blocks {1, 2}, {5, 7}, {3}, {4}, {6}, which cuts the possible shapes of the remaining six down to a menu of eight. Everything after this point is about choosing from that menu.

Here the paper leaves finite group theory and starts doing analysis, because the seven points are not abstract tokens — each one is a curve. Quotient the Galois cover attached to a triple by a point stabilizer and you get a degree-23 map to the line, branched over the same three points; its local permutations have 15, 1, and 1 cycles, and Riemann–Hurwitz prices the genus at exactly 4. The seven labels are seven genus-4 curves, concrete objects with equations — if you can find them.

Finding them is numerical uniformization. Each curve is the quotient of the hyperbolic disc by an index-23 subgroup of the triangle group Δ(23, 2, 23), and the power-series method of Klug, Musty, Schiavone, and Voight computes its holomorphic differentials directly: expand in the disc coordinate around the order-23 elliptic point, impose automorphy on a circle of radius ρ = 0.98128…, and extract fixed vectors of a large structured operator. At working precision 100 digits the series run to order 12,187, matched at 12,207 collocation points, and the four differentials of each curve satisfy the defining equation with residuals near 10⁻⁹¹ — against an acceptance gate of 10⁻⁸⁸. Underneath, the geometry stack passes exact receipts at every collocation point, and an independently rebuilt low-precision solve pins the same four-dimensional space down to its roundoff floor.

Four differentials embed each curve canonically in P³, and a classical theorem of Petri says a non-hyperelliptic genus-4 canonical curve is cut out by exactly one quadric and one cubic. The Petri quadric is the part that matters: after the expansion forces four linear relations, six of its coefficients survive, and that is where the moduli of the curve live. Non-hyperellipticity is checked rather than assumed — it is precisely the statement that the quadric is unique.

Now the obstruction. Numbers read off a power-series model depend on choices — rescale the disc coordinate and every coefficient rescales with it, and that scale is a transcendental with no arithmetic meaning. The standard remedy is to take ratios in which the scale cancels; the paper calls these marked invariants, weight-matched ratios of Petri coefficients. The delicate question is why such a ratio should be an algebraic number that moves correctly under Galois, when it was computed from a floating-point expansion in a normalization no algebraic geometer would recognize.

The answer is one inequality. Near the marked point of the curve over √−23, the cover is exactly ϕ = √−23 + t²³ in a suitable formal coordinate; such adapted coordinates exist after a finite extension and any two differ by a 23rd root of unity — they form a µ₂₃-torsor. The analytic disc coordinate agrees with an adapted one up to corrections of order w²³. Meanwhile, the Petri quadric of a genus-4 curve is determined by 12-jets: a section of twice the canonical bundle has degree 4g − 4 = 12, so it cannot vanish to order thirteen without vanishing outright.

Because 23 beats 12, the entire quadric computation is blind to the discrepancy between the analytic coordinate and the algebraic one: their 12-jets agree up to precisely the scalings that weight-matched ratios kill. This is the hinge of the paper, and it is a theorem, not a heuristic — the invariants are algebraic numbers attached to the marked cover, and every field embedding moves them the way it moves the cover. No truncation hypothesis, no numerical input.

The ground-truth test is the point we already understand. For label 6 — the rational point — integer-relation search recognizes all six invariants as elements of Q(√−23) with small height:

Recognition at 100 working digits leaves roughly 85 digits of confirmation margin — the relation keeps holding long after the digits it was found from run out. And as a negative control, a ratio with deliberately mismatched weights, which still contains the transcendental scale, fails to be recognized as anything at all.

The marked invariants live over K = Q(√−23), because marking a point above √−23 is itself a choice of square root. Arithmetic over Q needs quantities that forget the marking, and the classical move works verbatim: trace and norm. S = J + τJ and N = J·τJ, with τ the conjugation of K, are honest functions of the unmarked seven points — and the computed mirror action is exactly what makes them evaluable from the marked data alone. Among small combinations, U₁ = S₁ + N₁ separates all seven points, and so, independently, does U₂ = S₂ + N₂.

The values tell the story. U₁ of label 6 is −5541/10237, exactly, matching the label-6 ground truth. The other six values are the six roots of a single monic sextic over Q with common denominator 50,019,668,646,075,143; each independently computed root satisfies it to 6 × 10⁻⁸⁸. The sextic is irreducible, its Galois closure has group S₆, and its signature is (2, 2) — forced in advance by the mirror, which fixes exactly two of the six non-rational points. The two independent coordinates generate the same sextic field: purely symbolically, with no numerics, the second sextic acquires a linear factor over the field defined by the first. A Frobenius inventory over 199 good primes sees nine cycle types with frequencies matching the Chebotarev densities of S₆.

So the partition is 1 + 6, and the permutation image is as large as a fixed point permits: the full stabilizer of label 6. Sharper still, the image is all of S₆ already over K — an S₆-closure has a unique quadratic subfield, and since conjugation acts on the six roots by an even permutation, that subfield is real, so the imaginary field K never enters and restriction loses nothing. The action on the six-point orbit is primitive: no invariant partition, no intermediate structure, nothing for an explanation to grab except the fixed point itself. One rational point, and otherwise maximal Galois mixing.

With the answer in hand, the question sharpens: is there a structural reason for the fixed point? The natural suspect is the Grothendieck–Teichmüller machinery, the combinatorial shadow through which the absolute Galois group acts on covers. Guillot identified the level-one group GT₁(G) of a finite simple group with an explicit centralizer construction — a product of wreath products — so for M23 the suspect can actually be interrogated.

The interrogation comes back empty, and provably so. M23 has no outer automorphisms; the six ordered class-vectors in the orbit of (2A, 23A) are pairwise distinct; the tautological S₃ of symmetries acts freely on the 42 pair-classes above them; and the relevant wreath factor collapses to the full symmetric group. GT₁(M23) induces every permutation of the seven Nielsen classes. Any invariant attached functorially to generating pairs of M23 alone is constant across the seven — whatever singles out label 6 is invisible at this level.

The paper adds an instructive footnote. There is a perfectly natural invariant that does separate the seven: the order of the commutator [x, y], which is 4 on labels 1 through 4 and 6 on labels 5 through 7. By the theorem, that partition cannot be respected by GT₁ — and the computed action confirms arithmetic tramples it too, since the six-point orbit crosses both blocks. Natural invariants of generating pairs are not automatically Galois-functorial; here is one failing in public.

There is, finally, a place where label 6 does look different — and you find it by reducing at the prime that has been in the room the whole time. Everything here is soaked in 23: the group acts on 23 points, two of the three local monodromy orders are 23, the branch points are ±√−23, and Q(√−23) is the quadratic field inside the 23rd cyclotomic field. Reduce the sextic modulo 23 and it collapses into two clusters:

Newton polygons and a discriminant valuation turn that factorization into local arithmetic: over the 23-adic numbers the sextic field splits as F₂ × F₄, two totally ramified extensions of degrees 2 and 4, both tame. Together with the rational point, the seven-point Hurwitz algebra at 23 reads Q₂₃ × F₂ × F₄ — so tame inertia at 23 acts on the seven points with cycle type (1)(2)(4), and label 6 is the unique point fixed by inertia. A local distinction, weaker than global rationality, but the first structural property found that singles out the special point at all. A Magma computation over Q₂₃ confirms the factorization independently.

This lands in a known neighborhood of theory — stable reduction of three-point covers with cyclic p-Sylow subgroup, the setting of Raynaud, Bouw–Wewers, and Obus. The proven theorems stop short of M23 (they require p-solvability or a normalizer ratio of 2; here the ratio is 11), yet the behavior they predict inside their range — tameness at p — is exactly what the computation finds outside it. Which makes the open question sharp enough to fit in one sentence: derive the inertia cycle type (1)(2)(4), and the identity of the fixed point, from the deformation data of the stable model at 23, without using the sextic. Answer that, and what remains of the discoverers' question is the passage from the decomposition group at one prime to the whole Galois group.

Seven points; a mirror that swaps two pairs; six points fused into a single orbit carrying the largest symmetry the seventh allows; a tower of would-be explanations that turns out to permit everything; and one prime where the special point sits alone, untouched by inertia. Arithmetic breaks S₇ down to the stabilizer of a point — and nothing weaker. Why 6?