rulial

Part III, Observers of Observers · defined in Chapter XI, Rulial Relativity

Wolfram’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.

Plate W.21The promotion moves from the base observer, the twelve chords sharing an end with

\{0,\infty\}

: along such moves every covariant reversible transport has holonomy all of

S_3

In Wolfram’s usage rulial space is the space of rules, its limit the ruliad, and rulial relativity the claim that observers using different rules perceive equivalent laws. The program’s rulial level is a finite section of that space. An observer’s clock and vantage are the rule by which it describes its records; they are themselves records and coordinates, and the laws are to be covariant under them, which is the rulial clause of Chapter II. The first demand of relativity holds exactly: every native rule at a clock is written in antipodes and rod membership alone, so a collineation carries it onto the same rule at the image clock. The second has a price: along any invariant, symmetric, connected set of moves between observers, every covariant reversible transport has holonomy the whole stabilizer $S_3$ , so none is flat across clocks, while within one clock a flat one exists.

In the lift that curvature is velocity space’s: the canonical transport is its parallel transport reduced modulo $\mathfrak p$ , and the holonomy is the Thomas–Wigner rotation. Chapter XIX extends the relativity from charts to the program’s unselected rule parameters: an observable is a rulial invariant of a family of rules if it does not depend on the unselected item, and the item is a rulial coordinate relative to a class of observables if every observable in that class is invariant.

Definition(covariant transport)

A transport along a set of moves $E$ between the anchored observers assigns to each move $x\to y$ an element $T_{yx}\in G$ with $T_{yx}x=y$ . It is reversible if $T_{xy}=T_{yx}^{-1}$ and covariant if $T_{gy,gx}=gT_{yx}g^{-1}$ for all $g\in G$ . For a closed path $x=x_0\to x_1\to\dots\to x_n=x$ the product $T_{x_nx_{n-1}}\cdots T_{x_1x_0}$ fixes $x$ and so lies in $H_x$ ; these products form the holonomy group $\mathrm{Hol}_x\subseteq H_x$ . The transport is flat if its holonomy is trivial.

As mathematics

The groupoid of frames: its objects are the anchored observers, a point with an antiflag of the Fano plane, and its arrows the maps between charts. A transport along a set of moves assigns to each move $x\to y$ an element $T_{yx}$ of the group with $T_{yx}x=y$ ; it is covariant if $T_{gy,gx}=gT_{yx}g^{-1}$ , and its holonomy around a loop lies in the stabilizer $H_x\cong S_3$ . Because $\PSL(2,7)$ is simple, no covariant reversible transport across clocks is flat.

On the lift every step of the canonical transport is a product $\sigma\tau$ of the half-turn of the face behind it and a third-turn about that face’s axis. On the lift the two generate $\PSL(2,\Z)=\Z_2*\Z_3$ freely, and on the sky they satisfy Hurwitz’s relation of type $(2,3,7)$ .

Its name in another field	Bridge
Wolfram’s rulial space	name only
the parallel transport of velocity space reduced modulo $\mathfrak p$ , a Thomas–Wigner rotation	built
the level of rules	a reading

Built from: clockanchoranchored observer
In the volume: IThe Founding SentenceIIBlind ObserversVFour Reports, Six LettersVIIThe Branchial TreeXThe Finite Celestial SphereXIRulial RelativityXIIThe Coxeter GraphXIIIThe Level-Seven ShadowXIXRulial InvariantsXXThe Commuting SquaresXXIVA Number Nature Could RefuteEp.Forcing, Not Sacred Geometry