seam theory

Existence: two sets are joined exactly when their stabilizer classes agree. Number: the seams between two incarnations form a torsor under $N_G(H)/H$. Consistency: seams are unique and coherent exactly when the object is rigid, and otherwise natural seams can carry monodromy. Dependence on markings: an inner change of marking changes nothing up to isomorphism, while an outer automorphism moves the stabilizer class, as it exchanges the points and the lines of the Fano plane, and a bridge refuted for one marking is built over the outer automorphism. Forgetting: a map between theories that is not a seam is a description, and what it forgets at a point is its kernel, a stabilizer. Impossibility: the negative space of absences, with their windows and imprints.

The last question gives the subject its shape: a table of objects against theories in which some cells are empty by necessity, a negative space bounded by the seams that do exist.

Each floor is joined to the one below it by a theorem. The stabilizer principle joins incarnation to the classical floor. Seams join incarnations, unique when the stabilizer is self-normalizing and otherwise carrying monodromy. A bridge has a status, which the stabilizer principle decides for two marked sets of one group, and beside the statuses stand the absences. Artin’s absence leaves exactly four groups with a double life. Residue fields make each life the link of a vertex of a building over a local field: at 2 the octonion table glues such links into a building over $\F_2(\!(t)\!)$ and Kato’s hermitian form glues them, without symmetry, into the building over $\Q_2$; a gluing with the Frobenius symmetry of order 21 is the octonion one; and over $\Z[1/14]$ the group of order 168 is the stabilizer of the vertex of Klein’s lattice, carrying both lives, where the finite geometry is the geometry of short vectors and neighbours; one step beyond the link the two trees at 7 carry a doublet and its symmetric square, the object of the points is carried along one tree and not the other, and the two parents carry independent flips, of which only the first is seen by the oriented cells of the link complement. The two parents are joined only by fiber products, which across scales keep no attachment of the finite line and at one scale force it; around the loops of the scale tree a single relabelling carries the signed table without reversals exactly on the Iwahori subgroup, and on seven loops in eight must reverse two units, while carried observer by observer every loop returns each fiber changed only by colour. The archimedean place gives, over $\C$, Thurston’s link complement, whose cells are objects of the group, with the Fano incidence among them the absence of a shared face; the group has two arithmetic parents at 7, which share the sky but not its completion; absences decide which continua exist; and in the Cayley plane one point and one imaginary unit carry the intersection of Todorov and Dubois-Violette.

The group of order 168 is worked through every floor. It has fifteen objects, six of them rigid; three pairs of them share a permutation character and are refuted bridges for one marking; the object of size 24 carries seam monodromy, the flex-tangent map $\tau$ of power 4, which is the Frobenius at 2 made equivariant; the group has a double life, as the line and the plane; it sits at three places, the primes 2 and 7 and infinity; its double cover adds four new objects, over the rows of odd order; and seven of its objects, of sizes 7, 8, 14, 21, 24, 28 and 56, are followed up the tower floor by floor.