kernel graph

Part I, The Sentence and the Kernel · defined in Chapter I, The Founding Sentence

$K_4$ , with the reports as corners and the letters as edges.

Plate W.6The kernel graph

K_4

of one clock: the four reports 0 to 3 as vertices, report 0 at the centre, and the six letters

01,\dots,23

as edges.

The name records that four reports with $S_4$ , letters as pairs and records as words are the program’s fixed data, the part of every frame that is never varied. Every clock carries its own copy: with the clock deleted the Fano plane is the incidence of $K_4$ , reports as vertices, letters as edges, antipodal letters as opposite edges and the axes as the three perfect matchings.

Chapter V reads the graph two ways. Read unsigned, its letters carry $\mathbf1\oplus\mathbf3\oplus\mathbf2$ under the report group, and the relabellings span exactly $\C\oplus M_3(\C)\oplus M_2(\C)$ , the algebra the forcing theorem selects; read oriented, the letters are the two-forms $\Lambda^2\R^4$ , gradients plus loops, and the loops are the periods of the crystal. It is also the graph of one observer’s frame changes, and its covers organise Part II: the tree keeps the full order of a history, the crystal its signed tally, and $K_4$ itself only the current frame.

Definition(The kernel graph)

The graph of one clock, its reports joined by its letters, is the kernel graph $\Gamma=K_4$ of that clock.

As mathematics

The complete graph $K_4$ , the 1-skeleton of the tetrahedron, with fundamental group free of rank $6-4+1=3$ . For a point $c$ of the Fano plane it is the complete quadrilateral of the four lines missing $c$ , the residue of $c$ . As $\Z_2\times\Z_2$ with its translations it is the affine plane $\mathrm{AG}(2,\F_2)$ , points and lines, with automorphism group $\mathrm{AGL}(2,\F_2)\cong S_4$ , and it is the Cayley graph of $\F_2^2$ on its three nonzero elements.

Its four vertices, the four points of $\Proj^1(\F_3)$ and the four effects of the qubit’s tetrahedral measurement are incarnations of one rigid object of the rotation group $A_4$ , so between any two there is exactly one seam once the groups are marked. Under all of $S_4\cong\PGL(2,3)$ the thirteen lines of $\mathfrak{sl}_2(\F_3)$ are its four vertices, six edges and three perfect matchings.

Its name in another field	Bridge
the 1-skeleton of the tetrahedron	classical
the complete quadrilateral, the residue of a point of the Fano plane	built
the Cayley graph of $\F_2^2$ on its nonzero elements	built
the affine plane $\mathrm{AG}(2,\F_2)$	built
$\Proj^1(\F_3)$ with the lines of $\mathfrak{sl}_2(\F_3)$	built

A second sense

It is not the kernel of Chapter III, what a description forgets; Part II shows how the two meet, since each cover of the graph describes histories and what it forgets is a kernel in the second sense. Seams writes the graph as $K_4$ and keeps the word kernel for what a description forgets.

Built from: reportletter
Builds: branchialcrystaltally
In the dictionary: incarnationthe seven points
In the volume: IThe Founding SentenceIIIThe KernelVFour Reports, Six LettersVIIThe Branchial TreeIXWhat Space ForgetsXThe Finite Celestial SphereXIRulial RelativityXXThe Commuting SquaresXXIIILight, Vacuum and HandednessEp.Forcing, Not Sacred Geometry