UniversalKernel

An illustrated journal of the program’s mathematics

Wes Johnson

October 2026

One object, many names

Many finite objects of classical mathematics carry several names, one in each theory that meets them. The Fano plane has 28 antiflags; the projective line over $\F_7$ has 28 two-element subsets; the Klein quartic has 28 bitangents; the simple group of order 168 has 28 Sylow 3-subgroups; the Coxeter graph has 28 vertices. In each case a group of order 168 acts, and the five sets are, in a precise sense, one set: once the groups are identified, any two of them can be matched compatibly with the group in exactly one way, and the matchings agree with one another.

A transitive set of the group is determined by the conjugacy class of its stabilizer, so the group of order 168 has exactly fifteen such objects, one for each class of its subgroups. Many of the program’s finite objects are among them.

The twenty-eight

Size 28, stabilizer $S_3$ · rigid, one seam between any two incarnations · in the program, anchored observers

The fifteen objects of

G=\PSL(2,7)

on the line of their sizes

|G:H|

. The outer automorphism exchanges each class marked

a

, above the line, with its class

b

below. Each dot is an automorphism of the object, and the dots number the seams between any two of its incarnations; the six rigid objects carry none. Dashed brackets join the three pairs with one permutation character. Choose an object to see its maps.

Esquisse des Sutures

A sketch of seams: one group in many theories

Nineteen chapters in four parts and an epilogue; a dictionary of thirty-nine concepts

Seam theory is the study of seams: whether incarnations of one object in different theories can be joined, in how many ways, and whether the joins agree around a cycle of theories. The Esquisse tells it as a story, each step beside its plate; its dictionary holds the same concepts as entries, on the floors of a tower.

en chantierChapters 16 and 19 and the epilogue are still being built, and each says how far it stands.

Something Lawfully Occurs

Observers, Kernel and World

Twenty-four chapters in five parts and an epilogue; thirty words of its own

The program assumes one sentence:

Something lawfully occurs; occurrence leaves records; records shape occurrence.

It reads the same objects physically: the object of size 28 is its anchored observers, each a clock and a vantage line not through it, and the object of size 8 is its finite sky. Every result carries its status in words: proved, conditional, a reading, or open.