promotion

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

An 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.

Plate W.20Promoting the rod 2 of the base observer

\{0,\infty\}

: its two targets

\{0,5\}=(2,347)

and

\{4,\infty\}=(2,356)

, in blue, keep opposite ends of the pair; the other ten promotion targets, the chords sharing one end with

\{0,\infty\}

, are in hairline.

From $(p,L)$ choose a letter $q$ and let $r=p+q$ complete the axis $\{p,q,r\}$ . A collineation with $g(p)=q$ , $g(q)=r$ and $g(r)=p$ rotates the three roles on that axis: the letter becomes the clock, its antipode becomes a letter, and the old clock becomes that letter’s antipode. There are four such maps, carrying $(p,L)$ to two targets, two maps to each, and none keeps the vantage line, so every observer has twelve promotion targets, two for each new clock.

On the sky the twelve targets are exactly the twelve pairs that share one point with the observer’s pair, and the two targets for one new clock keep opposite ends of it: the finite analogue of aberration. Around a closed loop of promotions every covariant reversible comparison of charts returns the observer’s rods permuted. In the lift a promotion is a pair of edges through a shared cusp, an edge of a cusp torus, carried by a unique null rotation that fixes the shared light direction: the parallel transport of velocity space. Natively no occurrence changes a register’s clock.

Proposition(the chart identities) proved

For every clock $p$ , letter $a$ , line $L$ and collineation $g$ , $g(\bar a^{\,p})=\overline{ga}^{\,gp}$ and $a\in L\iff ga\in gL$ , where $\bar a^{\,p}=a+p$ is the antipode at clock $p$ . Every native rule at clock $p$ is defined from antipodes and rod membership alone, so each is carried by $g$ onto the same rule at clock $gp$ .

Proof

A collineation is a linear map of $\F_2^3$ , so $g(a+p)=ga+gp$ , and it maps lines to lines. The writer uses the forbidden antipode; the exchange rule uses the shared rod and its antipode; re-anchoring and the generators use rod membership and antipodes. All of these are transported.

As mathematics

A promotion joins two antiflags at distance three in the Coxeter graph: their clocks differ, and exactly one of the two clocks lies on the other’s line. On $\Proj^1(\F_7)$ two observers are at distance three exactly when their pairs share exactly one point. The unordered pairs of observers at distance three form a single regular orbit of 168, the object with trivial stabilizer.

On the cusp torus $\C/\mathfrak p$ at $\infty$ the seven observers through $\infty$ are the vertices of a triangulation of the torus as $K_7$ , and its twenty-one edges are the promotions through $\infty$ .

Its name in another field	Bridge
two pairs of $\Proj^1(\F_7)$ sharing one point	built
two antiflags at distance three in the Coxeter graph	built
an edge of a cusp torus of the link complement	built
a change of clock	a reading

Built from: anchored observerclockletter
In the dictionary: the object of size 168
In the volume: XThe Finite Celestial SphereXIRulial RelativityXIIThe Coxeter GraphXIIIThe Level-Seven ShadowXIXRulial InvariantsXXThe Commuting Squares