letter

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

A pair of reports; there are six, and every record is a letter. A register’s log is a word in letters, oldest first.

Plate W.7The letter 01 in gold: the pair of reports 0 and 1, ringed, and the edge of the kernel graph that joins them.

Two letters are adjacent when they share a report and antipodal when they are disjoint, and under adjacency the six form the octahedron. Each letter is one answer to one binary question about the reports. On the Fano plane, relative to a clock $p$ , the letters are the six points $a\ne p$ , each on exactly two report lines, the two reports it joins. Read as an unordered pair a letter carries the algebra the forcing theorem selects; read as an oriented edge it is a two-form, and on the observer’s report form an oriented letter generates a Lorentz transformation: the gradients generate boosts and the loops rotations.

In the lift a letter is a rest frame: the sum of its two reports, $T_{ab}=N_a+N_b$ , has determinant one, a unit timelike vector, so consecutive records are related by a boost, with $\gamma=\tfrac32$ between letters sharing a report and $\gamma=2$ between antipodes. The phrase letters as links has a precise content: an antipode is what a shared occurrence leaves, and the letters are links of the kernel graph, not tokens held jointly by the two registers anchored at their reports.

Definition(Letters and axes)

A letter is an unordered pair of reports; there are six. Two letters are adjacent when they share a report and antipodal when they are disjoint. An axis is a pair of antipodal letters, equivalently a perfect matching of the four reports; there are three.

As mathematics

With the clock $c$ fixed, a letter is a point $a\ne c$ of $\mathrm{PG}(2,2)$ , and with its clock an ordered pair $(c,a)$ of distinct points: the forty-two such pairs are the object with stabilizer $V_4^b$ . On $K_4$ it is an edge, on $\mathrm{AG}(2,\F_2)$ a line of two points, and at the prime above three a split line of $\mathfrak{sl}_2(\F_3)$ , whose two eigenlines pick out two points of $\Proj^1(\F_3)$ . In the fiber the six letters act as left multiplications $\gamma_a=L_{e_a}$ , anticommuting and squaring to $-1$ .

In the report lattice $\mathrm{Herm}_2(\Z[\omega])$ the letter $\{a,b\}$ of the base tetrahedron is $T_{ab}=N_a+N_b$ , of determinant one. Every such record $T=gg^\dagger$ splits uniquely into two primitive null vectors, and their cusps modulo $\mathfrak p$ name an observer.

Its name in another field	Bridge
an edge of $K_4$	built
a line of the affine plane $\mathrm{AG}(2,\F_2)$	built
a Lorentz bivector	built
a basis vector $e_i\wedge e_j$ of $\Lambda^2\C^4$	built
a split line of $\mathfrak{sl}_2(\F_3)$	built
a unit rest frame $T_{ab}=N_a+N_b$ of $\mathrm{Herm}_2(\Z[\omega])$	built
a generator of $\mathrm{Cl}(0,6)$	a reading
the value of a record	a reading