The program’s words

A dictionary of Something Lawfully Occurs

The volume names its objects in words of its own, each explained where it first appears. Here they stand under the parts that define them, each with the volume’s definition, what it is as mathematics in the Esquisse des Sutures, its names in other fields with the status of each bridge, and the chapters where it does its work.

IThe Sentence and the Kernel

IIOne Graph, Four Covers

IIIObservers of Observers

IVThe Exceptional Interior

VDynamics and Limits

anchored observer

Part I, The Sentence and the Kernel

A clock together with an anchor; there are twenty-eight. On the sky it is a pair of points, and in the lift an edge.

Built from: clock, anchor
As mathematics: object, incarnation, seam, rigid object
Defined in: Chapter I, The Founding Sentence

The entry for anchored observer

Plate WThe program’s words in the parts of the volume that define them. Choose a word: what it is built from is set in blue and traced up the parts, and the words built on it are dotted.

IThe Sentence and the Kernel

IanchorThe report an observer stands on. Moving it along a letter is a re-anchoring.
Ianchored observerA clock together with an anchor; there are twenty-eight. On the sky it is a pair of points, and in the lift an edge.
IantipodeOf a letter: the letter made of the two other reports.
IaxisA letter together with its antipode; there are three.
IclockOne of the seven points of the Fano plane, chosen as the direction of time. Relative to it the other six points are letters and the four lines missing it are reports.
Ikernel graph $K_4$ , with the reports as corners and the letters as edges.
IletterA pair of reports; there are six, and every record is a letter. A register’s log is a word in letters, oldest first.
IoccurrenceOne lawful event. It makes one distinction and leaves records.
IrecordA mark left by an occurrence that later occurrences can read.
IregisterA record-keeping unit. Its state is its past, up to what no future can reveal.
IreportOne of the four classes of past that the smallest register whose records matter can tell apart. On the observer’s Minkowski space it is a light-like direction.
IrodOf an anchored observer, one of the three letters through its anchor; the other three are its odd letters.
IIIkernelWhat a description forgets. Unqualified, what the world’s permanent record does not keep, where interference lives; in Part II, what the tally forgets.
IVarenaThe positions a register reaches when each record adds its letter’s rest frame to a running sum of report counts. It is the crystal’s lattice read as positions.

IIOne Graph, Four Covers

VIIbranchialWolfram’s word for the level of alternative histories. Here it is the tree of ordered histories.
VIItowerThe program’s model of a register’s memory at a fixed age, words of fixed length with moves among them, extended to composites of registers. Composites of odd radius are read as matter and those of even radius as radiation.
VIIIcrystalThe three-dimensional net of tallies, Sunada’s $K_4$ crystal, which the program reads as space.
VIIItallyThe net signed number of crossings of each letter in a history of re-anchorings, with the order forgotten.

IIIObservers of Observers

XskyThe finite celestial sphere: the eight points of the projective line over the field with seven elements, on which the observers’ group $\PSL(2,7)$ acts.
XIpromotionAn observer’s change of clock to one of its letters. On the sky it keeps one of the observer’s two points and moves the other.
XIrulialWolfram’s word for the space of rules. Here it is the level of frames: which clock and anchor a description uses, and how charts compare.
XIImeetingTwo anchored observers $(p,L)$ and $(q,M)$ at different clocks meet when they are joined in the Coxeter graph: when the triples of points outside $L\cup\{p\}$ and outside $M\cup\{q\}$ are disjoint, equivalently when their pairs of sky points are disjoint and harmonic, or when their report axes are orthogonal in the crystal read modulo seven. The relation is cubic, so each observer has three meetings, and there are forty-two in all, two over each of the twenty-one pairs of clocks.
XIIIcuspOne of the lift’s eight ends. They are the sky’s eight points, the directions from which light can arrive.
XIIIliftThe quotient of velocity space by the integral Lorentz transformations of the report counts that become the identity modulo a prime above seven: Thurston’s congruence link complement. The finite sky is its shadow.
XIIIreport countsInteger combinations of the four reports.
XIIIvelocity spaceThe three-dimensional hyperbolic space of rest frames, on which the Lorentz group acts.

IVThe Exceptional Interior

XIVfiberThe eight-dimensional interior each observer carries, built on the octonions; it holds the quark–lepton quartet.
XVrelabellingAn automorphism of the octonions that permutes the units with signs. There are 1344 of them, every one fixes 1, and each covers a collineation of the Fano plane: the relativity group reaches the fiber through them, up to colour signs.
XVIlepton lineThe line spanned by the octonion unit 1 and an observer’s clock $e_p$ , $\mathrm{span}\{1,e_p\}$ , on which the clock’s complex structure sends 1 to $e_p$ and $e_p$ to $-1$ . Chapter X calls it the lepton plane, the plane spanned by the shared octonion unit and the clock; on the sky it is the functions constant on the observer’s pair and on the complement, and no relabelling of the observer’s rods moves it.

VDynamics and Limits

XXIVclass constantThe gap of the tower’s matter walk, its smallest eigenvalue in modulus, which the program reads as a mass.