Part III · Observers of ObserversChapter X

The Finite Celestial Sphere

Plate X.1Clock 1 at the centre: its axes are the medians, its four reports the three sides and the incircle, and

x_0=(1,246)

takes the incircle as vantage.

What does an observer see when it looks out at every other observer at once?

An observer in special relativity looks out along light rays, and the rays arriving at one event form a sphere, the celestial sphere. Written as the complex projective line $\Proj^1(\C)$ , that sphere carries the whole Lorentz group: the proper orthochronous Lorentz group is $\PSL(2,\C)$ acting by Möbius transformations, and nothing in this description needs a continuum except the field.

Part II worked at one clock. The program has seven clocks and twenty-eight anchored observers, and together they carry a sky of the same algebraic kind over the field with seven elements: the eight points of $\Omega=\Proj^1(\F_7)=\{0,1,2,3,4,5,6,\infty\}$ , with $\PSL(2,7)$ acting by Möbius maps $z\mapsto(az+b)/(cz+d)$ , $ad-bc=1$ . The twenty-eight anchored observers are exactly the pairs of points of this sky. A clock pairs the eight points as the diagonals of a cube pair its corners, the fiber every observer carries is one space of functions on the eight points, and the same twenty-eight objects are, in the finite data that defines them, the bitangents of Klein’s quartic curve.

The central result

Let $\Omega=\Proj^1(\F_7)$ , with $G=\PSL(2,7)$ acting by Möbius maps. The rotation of an anchored observer’s three rods fixes exactly two points of $\Omega$ , and this pair defines a $G$ -equivariant bijection from the 28 anchored observers onto the 28 unordered pairs of points of $\Omega$ . Under it:

(i) a clock’s four observers have disjoint pairs, the body diagonals of a cube on which the clock’s $S_4$ acts by rotations, and this pairing of the sky is not a Möbius map;

(ii) the fiber is one space $\C[\Omega]=\mathbf{1}\oplus\mathbf{7}$ for all observers, and an observer’s lepton plane, spanned by the shared octonion unit and its clock, is the functions constant on its pair and on the complement; no relabelling of its rods moves it, and every native relabelling carries it to a lepton plane;

(iii) the pairs are the 28 odd theta characteristics of a symplectic $\F_2^6$ whose unique invariant even one is the sky: the mod-two homology data of Klein’s quartic, so the observers correspond to its 28 bitangents.

The eight points are the null lines of the crystal’s metric read modulo seven, and each observer’s pair is the two ends of its report’s axis.

Status

The finite geometry is exact, and it has been checked by two independent constructions, one through the Fano plane and one through the Möbius action. What it does not supply is as definite: it contains no rapidity, no boost and no velocity, and $\PSL(2,7)$ is not a subgroup of the Lorentz group. Counting dimensions does not decide between the continuum skies, since over $\F_7$ every nondegenerate form in three variables is the same up to scale. The program’s form is known, though: it is the crystal’s metric $x^2+y^2+z^2$ read modulo seven, which is definite, so the finite sky is the reduction, at a prime above seven, of the celestial sphere of $3+1$ dimensions, and its clocks follow the $3+1$ pattern.

“A clock is a rest frame” is still a reading here; Chapter XIII makes it exact in the lift. The theta dictionary describes an abstract three-qubit carrier, not the program’s fiber, and no complex curve is constructed from the observers: nothing in their dynamics has yet been read off Klein’s quartic.

Observers on the Fano plane

Plate X.1Clock 1 at the centre: its axes are the medians, its four reports the three sides and the incircle, and

x_0=(1,246)

takes the incircle as vantage.

Write the seven points of the Fano plane as the nonzero vectors of $\F_2^3$ , labelled by the integers $1,\dots,7$ they represent in binary, so that $1+2=3$ and $2+4=6$ . The lines are the triples $\{a,b,a+b\}$ : 123, 145, 167, 246, 257, 347, 356. The collineations form $\GL(3,2)\cong\PSL(2,7)$ , of order 168; throughout Part III this group is $G$ .

Any point $p$ can serve as a clock. The three lines through $p$ are its axes, the axis $\{p,a,a+p\}$ carrying the letter $a$ and its antipode $a+p$ , and the four lines missing $p$ are its reports. Every letter lies on two reports and any two reports meet in one letter, so with $p$ deleted the plane is the kernel graph $K_4$ . For the base observer $x_0=(1,246)$ the rods are 2,4,6 and the antirods 3,5,7.

Definition(anchored observer)

An anchored observer is a pair $x=(p,L)$ of a clock $p$ and a report $L$ of $p$ , a line not through $p$ , called its vantage. The three points of $L$ are its rods, one on each axis; their antipodes are its antirods. There are $7\times4=28$ anchored observers, $G$ permutes them transitively, and the stabilizer $H_x$ acts faithfully on the three rods, so $H_x\cong S_3$ .

Observers are pairs

Plate X.2The finite celestial sphere with every observer as a chord; the base observer is the gold chord

\{0,\infty\}

$\PSL(2,7)$ acts two-transitively on the eight points of $\Omega$ , so it acts transitively on the 28 unordered pairs, each with a stabilizer of order six. Every element of order three is conjugate to $z\mapsto2z$ , which fixes 0 and $\infty$ and cycles the other six points in two three-cycles.

The base observer has the pair $\{0,\infty\}$ . The Möbius maps fixing it are $z\mapsto2z$ , $4z$ , $-1/z$ , $-2/z$ , $-4/z$ and the identity, and in the volume’s chart the rod rotation $(2\,4\,6)(3\,5\,7)$ is $z\mapsto4z$ and the rod transposition $(4\,6)(5\,7)$ is $z\mapsto-1/z$ . In space’s terms the pair is the two ends of the observer’s report axis: a rotation of order three fixes exactly the two null lines at the ends of its axis.

Theorem(observers are pairs) proved

For each anchored observer $x$ , the subgroup of order three in $H_x$ fixes exactly two points of $\Omega$ ; write $\operatorname{pair}(x)$ for them. The map $x\mapsto\operatorname{pair}(x)$ is a $G$ -equivariant bijection from the anchored observers onto the unordered pairs of points of $\Omega$ .

Proof

$H_x\cong S_3$ has a unique subgroup $C_x$ of order three, and it fixes exactly two points. Since $H_{gx}=gH_xg^{-1}$ , also $C_{gx}=gC_xg^{-1}$ , whose fixed points are $g\cdot\operatorname{pair}(x)$ , so the map is equivariant. Its image is a nonempty $G$ -invariant set of pairs, hence all 28, and a surjection between two sets of 28 elements is a bijection.

A point of the sky is a tour

Plate X.3The seven chords through the sky point 0: one observer at each clock, a single tour.

The seven pairs containing 0 belong to seven observers, one at each clock: $(1,246)$ , $(2,347)$ , $(3,167)$ , $(4,356)$ , $(5,123)$ , $(6,257)$ , $(7,145)$ . The Möbius map $z\mapsto z/(1-z)$ fixes 0 and cycles the other seven points as $\infty\to6\to3\to2\to5\to4\to1\to\infty$ ; the corresponding collineation is the Singer cycle $(1\,4\,6\,5\,2\,3\,7)$ , which carries $(1,246)$ to $(4,356)$ , then to $(6,257)$ , and so on through all seven clocks.

Choosing a point of the sky is therefore choosing one observer at every clock, and the seven so chosen form a single tour. A subgroup of order seven is generated by a Singer cycle; $G$ has eight of them and permutes them as it permutes $\Omega$ . This was the form in which the correspondence was first found.

A clock is a cube

Plate X.4Clock 1 as a cube: its four observers are the chords

\{0,\infty\}

\{4,5\}

\{2,6\}

and

\{1,3\}

, the body diagonals.

Each clock’s four observers have pairs that partition $\Omega$ , and so do the four observers sharing a vantage line. These fourteen perfect matchings have as stabilizers the fourteen subgroups $S_4$ of $G$ , in two classes of seven: the clock groups and the line groups.

For clock 1 the diagonals are $\{0,\infty\}$ , $\{4,5\}$ , $\{2,6\}$ , $\{1,3\}$ , so $\sigma_1=(0\ \infty)(1\ 3)(2\ 6)(4\ 5)$ . It is not Möbius: a Möbius map exchanging 0 and $\infty$ is $z\mapsto a/z$ , $\sigma_1(1)=3$ forces $a=3$ , and then $2\mapsto5$ , whereas $\sigma_1(2)=6$ . The two inscribed tetrahedra are $\{0,1,2,4\}$ and $\{3,5,6,\infty\}$ , zero with the nonzero squares and infinity with the non-squares. Reading a clock as a finite rest frame, whose rotations are the maps respecting its pairing of the sky, is a reading at this point; it is tested in Chapters XI and XIII, and the second confirms it in a $3+1$ lift.

Theorem(a clock is a cube) computed

For each clock $p$ , its group $G_p\cong S_4$ acts transitively on the eight points of $\Omega$ with point stabilizer of order three, as the rotation group of a cube acts on its vertices. The four observers of $p$ are the four body diagonals. The pairing $\sigma_p$ is not a Möbius map, not even in $\PGL(2,7)$ , and its centralizer in $G$ is exactly $G_p$ . The same holds for the seven line groups.

Promotion keeps one point

Plate X.5The twelve promotion targets of

x_0

: the chords that share one end with

\{0,\infty\}

Clock promotion, defined in Chapter XI, takes an observer $(p,L)$ to a new clock $q$ , one of its letters, by rotating the roles on the axis $\{p,q,p+q\}$ . Each observer has twelve promotion targets, two for each new clock, and in celestial terms they are exactly the twelve pairs that share one point with $\operatorname{pair}(x)$ , the two targets for one new clock keeping opposite ends. From $x_0$ , promoting 2 gives $(2,347)$ with pair $\{0,5\}$ and $(2,356)$ with pair $\{4,\infty\}$ . Promotion keeps one celestial direction and moves its opposite, the finite analogue of aberration.

Inside the functions on $\Omega$ , an observer’s pair defines the unit vector $q_x=\sqrt{2/3}\,(\mathbf{1}_{\operatorname{pair}(x)}-\tfrac14)$ among the functions with zero sum. Since $\langle\mathbf{1}_A-\tfrac14,\mathbf{1}_B-\tfrac14\rangle=\lvert A\cap B\rvert-\tfrac12$ , two such vectors meet at $+\tfrac13$ when the pairs share a point and at $-\tfrac13$ when they are disjoint: a clock’s four vectors form a regular tetrahedron, and the twenty-eight lines are equiangular.

One fiber for every observer

Plate X.6A square of elementary moves on the sky; carried around it, the base observer’s two ends are exchanged.

Each observer carries an eight-dimensional fiber, the complexified octonions, in which its clock is the imaginary unit $e_p$ . The ways of letting $H_x\cong S_3$ act on it compatibly with the octonion structure, the covariant lifts, have the character $(8,0,2)$ on the identity, the transpositions and the three-cycles, and so does $\C[\Omega]$ : that coincidence is the whole content of the theorem.

The identifications are not unique; the intertwiners form a space of dimension twelve. In a frame obtained by reversing the signs of six basis columns, the octonion product becomes a single global one, invariant under all 168 elements of $G$ , and the program takes this product, with its unit, as physical. A trivial bundle is not a flat comparison, however: the square of moves $(1,246)\to(1,257)\to(3,257)\to(3,246)\to(1,246)$ , made by the Klein four-group transport of Chapter XI, returns with the celestial holonomy $z\mapsto-1/z$ , which exchanges the two ends of the pair.

Theorem(one global fiber)

There are unitary identifications $F_x$ of the fiber at each observer with $\C[\Omega]$ such that every change of observer becomes the permutation of $\Omega$ : $F_{gx}\,U(g,x)=\rho(g)\,F_x$ , with $U$ the covariant lift and $\rho$ the permutation representation. In $\C[\Omega]=\mathbf{1}\oplus\mathbf{7}$ the $\mathbf{7}$ is irreducible. Every $F_x$ sends the unit 1 to the normalized constant function and $e_p$ to $q_x$ , so the lepton plane $\operatorname{span}\{1,e_p\}$ becomes the functions constant on the pair and on its complement, exactly the $H_x$ -invariant functions.

Odd thetas and Klein’s quartic

{0, ∞}(1, 246)d₀⟨z ↦ 2z⟩

Projective line

a 2-subset of P¹(F₇)

Fano plane

an antiflag (p, L), p ∉ L

Graphs

a vertex of the Coxeter graph

The group

a Sylow 3-subgroup of PSL(2,7)

generator: z ↦ 2z
on the eight points: (1 2 4)(3 6 5)
fixes: {0, ∞}, and nothing else

Klein quartic

a bitangent of x³y + y³z + z³x = 0

the line: x + y + z = 0
touching: at (1 : ω : ω²) and (1 : ω² : ω), the two points fixed by ρ of the subgroup

Choose a vertex of the Coxeter graph, or step through all twenty-eight.

Plate X.7The base observer named five ways: an antiflag, a pair of sky points, a subgroup of order three, a bitangent of Klein’s quartic and a vertex of the Coxeter graph.

Let $V$ be the even subsets of $\Omega$ modulo complementation, a six-dimensional space over $\F_2$ , with $b(A,B)=\lvert A\cap B\rvert\bmod2$ and $q_0(A)=\tfrac12\lvert A\rvert\bmod2$ . The 28 pairs are the vectors with $q_0=1$ and the 35 four-and-four splits are the nonzero vectors with $q_0=0$ . The 64 quadratic refinements of $b$ divide into 28 odd ones, indexed by the pairs, and 36 even ones; for a surface of genus three these are its theta characteristics.

In this language the tour through 0 is an Aronhold set of bitangents. The tempting next step fails. On an abstract three-qubit carrier an even theta is a real structure and an odd one a Kramers structure, but the fiber carries $G_2(2)$ through $\mathbf{1}\oplus\mathbf{7}$ , in which an element of order six has trace 3, while for a three-qubit Clifford unitary $\lvert\operatorname{Tr}U\rvert^2$ is 0 or a power of two. The theta dictionary describes an abstract carrier, not the fiber.

Theorem(observers as odd thetas)

The observers are the 28 odd theta characteristics $q_{\operatorname{pair}(x)}$ , and the sky is the even characteristic $q_0$ , fixed by $\PGL(2,7)$ . The orbits of $\PSL(2,7)$ on the 36 even characteristics have sizes $1+7+7+21$ , the two orbits of seven being the cube splits of the clocks and of the vantage lines. The identification extends equivariantly to the group $G_2(2)$ , of order $12{,}096$ , of the 28 time lines with their octonion product. The action of $\PSL(2,7)$ on $V$ is symplectically conjugate to the action of the automorphisms of Klein’s quartic on its first homology modulo two, so the observers correspond to the quartic’s 28 bitangents and the sky to its unique invariant even characteristic.

The rulial layer now has an exact geometry: the projective line over $\F_7$ with its Möbius group, each clock a cube inscribed in it, and the fiber one space of functions on it. In the language of Seams, the antiflags, the pairs of sky points, the bitangents, the subgroups of order three and the vertices of the Coxeter graph are incarnations of one rigid object, with stabilizer class $S_3$ , so the seams between them are unique.

The reading that goes with the geometry, clock as rest frame, promotion as aberration and comparison as a Thomas–Wigner rotation, is not established by the finite geometry alone. Chapter XI proves that the law reads the same at every clock and that comparison across clocks must be curved; Chapter XII finds the one relation on which a meeting needs no comparison; Chapter XIII identifies what the finite group is the reduction of.