Part III · Observers of ObserversChapter XI

Rulial Relativity

Plate XI.1Promoting the rod 2 of

x_0=(1,246)

: the roles on the axis 123 rotate, and

(2,347)

is one of the two targets.

XI.1
XI.2
XI.3
XI.4
XI.5
XI.6

Does the law read the same to every observer, and what does comparing them cost?

Einstein’s principle of relativity makes two demands: the laws must take the same form in every frame, and passing from one frame to another must be a definite transformation, so that two observers can compare what they see. In the program the frames are the twenty-eight anchored observers and the transformations are the 168 collineations of the Fano plane, the group $G\cong\PSL(2,7)$ that moves the eight points of the finite sky.

The first demand holds exactly: at every clock the native rules are written in the incidence of the Fano plane and in nothing else. The second has a price. Carried around a closed loop of clock changes, the comparison of charts comes back with the observer’s three rods permuted, and the price cannot be avoided across clocks, because $G$ is simple, though it can be avoided within one clock. The word rulial is Wolfram’s: the program’s rulial level is a finite section of his space of rules, the rules its own observers can use.

The central result

(i) One law, seven charts. At clock $p$ the antipode of a letter $a$ is $a+p$ , and every collineation satisfies $g(a+p)=ga+gp$ , so $g$ carries every rule the program writes at clock $p$ to the same rule at $gp$ , and every finite native history to a native history. Around a closed loop of clock changes the re-description returns up to a permutation of the observer’s three rods.

(ii) Rulial curvature. Along any $G$ -invariant, symmetric, connected set of moves between anchored observers, every covariant, reversible transport has holonomy equal to the whole stabilizer $S_3$ of an observer, so none is flat. Among the four observers of one clock a flat covariant transport exists.

(iii) The lift selects the transport. In the lift of Chapter XIII each promotion, a change of clock that keeps one sky point, is carried by a unique null rotation $T$ fixing the shared point. $T$ is the parallel transport of velocity space: flat on the torus around every end of the lift, a half-turn around every triangular face, and the only transport covariant under the lift’s whole isometry group $\PGL(2,7)$ . The two connections $F_0$ and $F_1$ are rotations of cells.

(iv) Three relativities, one triangle. Every step of $T$ is a product $T=\sigma\tau$ of the half-turn $\sigma$ of the face behind it and a third-turn $\tau$ about that face’s axis, with $\sigma^2=\tau^3=1$ . On the lift the two generate $\PSL(2,\Z)=\Z_2*\Z_3$ freely; on the sky they satisfy Hurwitz’s relation of type $(2,3,7)$ . Around a loop, the abelian part of the holonomy is fixed by the loop’s spatial tally, and the third-turns are carried only by loops of zero tally, the loops that space forgets.

Status

All four parts are exact. Part (i) is a statement about descriptions: it shows that one family of rules is read in seven charts, and it does not construct an occurrence that changes a register’s clock. Natively none exists; the law as it stands never changes clock, so the native world falls into seven fixed-clock worlds, one for each copy of the kernel graph, and the physical content of the other parts is conditional on a clock-changing dynamics.

Parts (iii) and (iv) are exact computations in the lift. They say which comparison the geometry itself makes, and how the program’s three relativities, of space, of history and of rule, are one group. Whether the order-three twist that separates $F_1$ from $T$ on the cusp tori is physical is a real question, and nothing yet decides it.

One law, seven charts

Plate XI.1Promoting the rod 2 of

x_0=(1,246)

: the roles on the axis 123 rotate, and

(2,347)

is one of the two targets.

At a clock $p$ a register’s record is a word in the six letters of $p$ , its newest letter the top. The writer appends any of the five letters other than the antipode $t+p$ of the top $t$ . Two registers at one clock with different vantages share exactly one rod $e$ , and the exchange rule is enabled when their tops are $e$ and $e+p$ ; then each appends the other’s top in one shared occurrence. A promotion changes the clock: from $(p,L)$ choose a letter $q$ , put $r=p+q$ , and apply a collineation with $g(p)=q$ , $g(q)=r$ , $g(r)=p$ . There are four such maps, two to each of two targets, so each observer has twelve promotion targets.

A negative control shows the statement is not empty: changing the clock but keeping the old names of the points fails in $210{,}768$ of $312{,}768$ transition tests, while the translation maps every two-register history of length three, under all 24 promotion maps at the base observer, without a failure. Promoting the rod 2 of $x_0=(1,246)$ by $g=(1\,2\,3)(5\,6\,7)$ , the forbidden transition $2\to3$ at clock 1 becomes $3\to1$ , forbidden again at clock 2 because $3+2=1$ .

Proposition(the chart identities) proved

For every clock $p$ , letter $a$ , line $L$ and collineation $g$ ,

g\bigl(\bar a^{\,p}\bigr)=\overline{ga}^{\,gp},\qquad 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 the shared rod and its antipode, and re-anchoring and the generators use rod membership and antipodes. All of these are transported.

Comparison within one clock

Plate XI.2The four observers of clock 1 as the kernel graph: each edge is the letter two vantages share, and the transport across it is the element of

V

fixing that letter’s axis.

A transport 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 $T_{gy,gx}=gT_{yx}g^{-1}$ . Around a closed path the product of the $T$ ‘s fixes the starting observer, so it lies in $H_x\cong S_3$ and permutes the rods. These products form the holonomy group, and the transport is flat when it is trivial.

Within one clock a flat comparison exists. At clock 1 the maps $z\mapsto z+\varphi(z)\cdot1$ , for the linear functionals $\varphi$ with $\varphi(1)=0$ , form the Klein four-group $V=\{\id,(4\,5)(6\,7),(2\,3)(6\,7),(2\,3)(4\,5)\}$ . Each fixes the clock and every axis as a set, fixes one axis pointwise, and permutes the four reports regularly, so between two observers of clock 1 exactly one element of $V$ carries the first vantage to the second. Around any triangle the three elements used are the three nonidentity elements of $V$ , whose product is the identity.

The rulial curvature theorem

Plate XI.3The promotion moves from

x_0

, the twelve chords sharing an end with

\{0,\infty\}

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

S_3

The argument says more than the theorem: a flat covariant comparison exists exactly when the relativity group has a normal subgroup acting regularly on the frames. The clock’s $S_4$ has one, the Klein four-group, and the simple group $\PSL(2,7)$ has none. The cocycle $w(g,x)=s_{gx}^{-1}gs_x$ used in the proof is the finite form of the little-group cocycle of Wigner’s classification, in which the frames are boosts and $w$ is the Wigner rotation.

The theorem was also checked on the natural move sets by exhaustive enumeration: same clock or same vantage, the promotion shell, Coxeter distance two. On Coxeter adjacency no covariant reversible transport exists at all, which Chapter XII turns to advantage. Corollary: no connected invariant move set carries a consistent orientation of the rods.

Theorem(rulial curvature) proved

Let $E$ be a $G$ -invariant, symmetric, connected set of moves between the 28 anchored observers, and $T$ a covariant reversible transport along $E$ . Then $\operatorname{Hol}_x=H_x\cong S_3$ for every observer $x$ . On the four observers of one clock, with the clock’s $S_4$ in place of $G$ , the transport by the Klein four-group is covariant, reversible and flat.

Proof

Write $H=H_{x_0}$ and $N=\operatorname{Hol}_{x_0}$ . By covariance, carrying a loop by $h\in H$ conjugates its product by $h$ , so $N$ is normal in $H$ .

Suppose $N\ne H$ and let $\pi\colon H\to H/N$ . In a frame $s_x$ with $s_xx_0=x$ the transport becomes $A_{yx}=s_y^{-1}T_{yx}s_x\in H$ , and $\pi\circ A$ has trivial holonomy, so, $E$ being connected, $\pi(A_{yx})=c_yc_x^{-1}$ . Covariance then makes $\varphi(g)=c_{gx}^{-1}\pi(w(g,x))c_x$ independent of $x$ , and the cocycle identity makes it a homomorphism $G\to H/N$ that equals $\pi$ on $H$ . Its kernel would be a normal subgroup of index 2 or 6, which is impossible because $\PSL(2,7)$ is simple. Hence $N=H$ .

Within one clock, $V$ is normal in $S_4$ and permutes the four observers regularly. Taking $T_{yx}$ to be the unique element of $V$ carrying $x$ to $y$ gives a covariant, reversible transport whose loop products lie in $V$ and fix a point, hence are trivial.

A loop to follow by hand

Plate XI.4The loop on the sky: three observers through the point 0, whose other end swings

\infty\to5\to6\to\infty

A promotion splits into a re-anchoring and a change of clock along a fixed vantage line, and each has two covariant forms: the Klein four-group element, $d$ , or a transposition, $r$ . Only two of the four combinations realize the promotion’s role cycle on every edge, $F_0=(d,d)$ and $F_1=(d,r)$ , the program’s two promotion connections, and nothing in the law selects between them.

The whole census agrees with the example. Over all $2{,}016$ rooted, oriented three-clock promotion triangles, $F_0$ gives 336 identities and $1{,}680$ transpositions, while $F_1$ gives 672 identities, 336 transpositions and $1{,}008$ three-cycles; on every triangle the two give different classes.

Example(a three-clock loop)

Follow $x_0=(1,246)\to(2,347)\to(4,356)\to(1,246)$ . The two connections give the collineations

\begin{array}{llll} & (1,246)\to(2,347) & (2,347)\to(4,356) & (4,356)\to(1,246)\\ F_0: & (1\,2\,3)(5\,6\,7) & (1\,7\,5)(2\,4\,6) & (1\,5\,4)(2\,7\,3)\\ F_1: & (1\,2\,3)(4\,7\,6) & (2\,4\,6)(3\,5\,7) & (1\,5\,4)(2\,3\,6)\end{array}

and the holonomies $(4\,6)(5\,7)$ for $F_0$ and $(2\,4\,6)(3\,5\,7)$ for $F_1$ , both fixing the clock 1 and the vantage 246. On the sky the three observers are $\{0,\infty\}$ , $\{0,5\}$ , $\{0,6\}$ , and the holonomies are $z\mapsto-1/z$ , exchanging the two ends of the pair, and $z\mapsto4z$ , a third-turn about it. The lift’s transport fixes 0 at every step; in $w=1/z$ it is a translation, the swinging end runs through $w=0,3,6,0$ by steps 3, 3 and 1, and $3+3+1=7\equiv0$ , so its holonomy on this loop is the identity.

The transport the lift selects

Plate XI.5The cusp torus at

\infty

: the seven observers through

\infty

, one at each clock, triangulated as

K_7

by the corners of clock cubes and of line cubes.

In the lift the observers are the edges of a hyperbolic three-manifold whose eight ends, its cusps, are the sky points. A promotion triangle, three observers each two of which share a sky point, is either three edges from one cusp, cutting a triangle on the cusp torus there, or the three edges of a face. Its four orbits are the manifold’s cells: the 56 faces, the 56 corners of clock cubes, the 56 corners of line cubes, and the 168 empty triangles of the cusp tori, which are the corner of no tetrahedron.

The connections are sorted by these cells. $F_0$ is the identity exactly on the faces; $F_1$ is the identity on the corners, a half-turn on each face and a third-turn on each empty triangle, a flat connection on each torus with a twist of order three. The null rotation $T$ is the identity on all $1{,}680$ rooted triangles of the cusp tori and a half-turn on each of the 336 rooted faces; it is the only unipotent choice and the only one covariant under all of $\PGL(2,7)$ , and $F_1$ is $T$ followed by a third-turn about the source’s line of sight, $F_0$ is $T$ followed by a reversal of that line. The question the finite law leaves open has an answer in the lift, and the answer is neither.

Three relativities, one triangle

Plate XI.6The three relativities as one triangle group, read at the lift’s two places.

A history is an element of the free group $F_3=\pi_1(K_4)$ , its tally is its image in $H_1(K_4)\cong\Z^3$ , and what the tally forgets is the commutator subgroup. On one clock cube $T$ ‘s holonomy group is $S_3$ , and the image of a loop’s holonomy in its largest abelian quotient is $(z_1+z_2+z_3)\bmod2$ , a function of the tally alone, the spatial flux; every decagon has tally zero and a nontrivial third-turn, the branchial residue. On spinors the holonomy group is the binary dihedral group of order 12, with abelian label $(z_1-z_2+z_3)\bmod4$ . Space sees the flux; only history sees the residue.

Read at the lift’s two places the triangle closes differently: at $\sqrt{-3}$ , where $\omega\equiv1$ , the generators reduce to type $(2,3,3)$ and generate $A_4$ , the report group on the four reports; at $\mathfrak p$ they reduce to $(2,3,7)$ and the sky’s group. In the lift space and history generate freely, and the rule’s prime seven appears only on the sky, as the order of their product.

Theorem(the modular triangle)

Let $x=\{c,a\}\to y=\{c,b\}$ be a promotion with null rotation $T$ , and $\Delta$ the face spanned by $x$ and its predecessor along $c$ . The holonomy $\sigma$ of $\Delta$ reverses $x$ ‘s line of sight, and $\tau=\sigma T$ has order three and fixes $\Delta$ ‘s axis, so $T=\sigma\tau$ with $\sigma^2=\tau^3=1$ . On the sky $T^7=1$ as well, and the 336 steps are exactly the $(2,3,7)$ -generating pairs of $\PSL(2,7)$ , the Hurwitz pairs of Klein’s quartic. In $\SL(2,\Z[\omega])$ , at the step $\{\infty,0\}\to\{\infty,1\}$ ,

\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}0&1\\-1&-1\end{pmatrix}=\begin{pmatrix}1&1\\0&1\end{pmatrix},

the first factor of order two and the second of order three in $\PSL(2,\Z)$ , which they generate freely as $\Z_2*\Z_3$ .

No comparison across clocks is flat, and the finite law does not choose among the covariant ones; the lift chooses, by its parallel transport, and lifted through the double cover that transport is the transport of spin (Chapter XV). Matter’s internal labels are carried instead by relabellings that keep the octonion product and its unit. Democracy and comparison cannot both be kept at seven: under the full symmetry $G_2(2)$ of the twenty-eight time directions there is no covariant transport between distinct observers, and only once arrows of time are chosen, dropping the group to $\PSL(2,7)$ , does comparison exist and become curved.

The holonomy is the program’s version of a monodromy of identifications: what a system of chart comparisons fails to bring back around a loop. The next chapter finds the one relation between clocks on which two observers can meet without comparing at all.

Words defined here: promotionrulial
In the Esquisse: 4La monodromie des sutures9Immeubles et réseauxÉp.L’horizon : dessins d’enfants
Concepts: objectstabilizer classseam monodromycoherencecontinuum
Objects: the twenty-eightthe skythe seven pointsthe seven linesthe object of size 42, class athe object of size 56the object of size 14, class athe object of size 14, class b