Esquisse des Sutures

This is about one finite group and the many theories that meet it. The simple group of order 168 acts on the Fano plane, on the projective line over the field with seven elements, on the Klein quartic and its twenty-eight bitangents, on the Coxeter and Heawood graphs, on the integral octonions, and on Thurston’s manifold of twenty-eight ideal tetrahedra. Each theory gives the group’s objects names of its own. The question is when two such names name one object, and the answer is a single principle. A transitive set is determined by the conjugacy class of a stabilizer, and the matchings between two incarnations of one object, which we call seams, form a torsor under the object’s automorphism group.

Three things follow from taking the matchings themselves as the subject. First, a coincidence of numbers is not an identification. Every bridge between two theories carries a status (built, type, name or refuted), and only built bridges are theorems. Second, where an object has automorphisms, seams carried around a cycle need not close up, and the failure to close, the monodromy, is exactly the holonomy of a lattice gauge connection. Third, the theorems that bound the subject by saying that something does not exist (Galois’s theorem on actions of prime degree, the Hurwitz bound, the end of composition algebras at dimension eight) leave imprints: structures that exist because something else cannot.

The concepts stand in a tower of floors, each built on the one below and joined to it by a theorem: the classical theory, incarnations, seams, what is known with the negative space beside it, double lives, completions at the primes, and continua, under a roof that is the subject itself. Floor three meets floor four through Artin’s absence, since isomorphisms between members of different families of simple groups are rare and the few that survive are the double lives. Floor four meets floor five through residue fields, each life being a geometry over $\F_p$ that is the link of a vertex of the Bruhat–Tits building over $\Q_p$. Floor five meets floor six at the archimedean place, where the congruence that gives a residue field at a prime gives a hyperbolic manifold whose cusps are the points of the finite projective line.

The Esquisse tells this as a story in four parts. The first sets up the language and the finite theory, the second the double lives, and the third climbs the tower to the continua. The fourth gathers the results that only make sense in seam theory, statements about the whole network of incarnations at once. They are being proved as this is read: each is a chantier, a building site, and says how far it stands. Beside the story, the dictionary holds every concept and object as an entry of its own. The book it is drawn from is a draft begun on 2 October 2026 and changing daily, so each chapter says which draft it was read from.

A companion volume, Something Lawfully Occurs, reads these objects physically: the twenty-eight as observers, the projective line over $\F_7$ as a finite celestial sphere, the octonions as the interior each observer carries. Those readings play no part in the proofs here, and the mathematics stands without them. Each chapter ends with the chapters of the volume that read it.