Part III · Observers of ObserversChapter XIII

The Level-Seven Shadow

Plate XIII.1The finite relativity group does not sit inside the Lorentz group.

XIII.1
XIII.2
XIII.3
XIII.4
XIII.5
XIII.6
XIII.7

If the finite group is not the Lorentz group, what is it the shadow of?

Part III has built a finite relativity: a sky of eight points; seven clocks, each pairing those points as a cube’s diagonals pair its corners; a law that reads the same in seven charts; a comparison between charts that is forced to be curved; and one relation on which observers at different clocks meet without comparing. Its vocabulary, sky, rest frame, aberration and Thomas–Wigner rotation, is borrowed from special relativity, and this chapter asks how literally it can be taken.

The answer comes in three steps. The finite group is not a subgroup of the Lorentz group. It is a reduction of discrete Lorentz groups, taken modulo seven. And one of those reductions is singled out: the reduction, at a prime above seven, of the integral Lorentz group of the program’s own report counts. Divide velocity space, the hyperbolic space of rest frames, by the integral Lorentz transformations that become the identity modulo that prime, and the result is a hyperbolic three-manifold with eight ends whose cells are the program’s objects. The volume calls it the lift.

The central result

Let $\omega$ be a primitive cube root of unity and $\mathfrak p=(3+\omega)$ , a prime of $\Z[\omega]$ of norm seven.

(i) The report counts of one observer form the Eisenstein lattice $\mathrm{Herm}_2(\Z[\omega])$ in Minkowski space; the null vectors of the four reports form a $\Z$ -basis of it, and the determinant is the null report form. So the Bianchi group $\SL(2,\Z[\omega])$ is the integral Lorentz group of report counts.

(ii) Reduction modulo $\mathfrak p$ maps it onto $\SL(2,\F_7)$ . The quotient $M=\Gamma(\mathfrak p)\backslash\mathbb{H}^3$ of velocity space by the kernel is Thurston’s congruence link complement, a hyperbolic three-manifold with eight cusps, whose canonical cells are the program’s objects: the eight cusps are the sky points, the twenty-eight edges are the observers, and the twenty-eight ideal tetrahedra make up the seven clock cubes and the seven line cubes, two to a cube.

(iii) Its isometry group is $\PGL(2,7)$ , and every isometry preserves orientation, so $M$ is chiral; its mirror image is the lift at the conjugate prime.

(iv) The canonical transport between observers is velocity space’s parallel transport, reduced modulo $\mathfrak p$ . Its holonomy is the Thomas–Wigner rotation: a half-turn on every face, a third-turn on every shortest loop that space forgets, and, on spinors, Thomas precession.

(v) Every massless field that $M$ carries is fixed by its values at the eight cusps. The finer lifts of levels $\mathfrak p^2$ and $\mathfrak p^3$ carry nothing else: their first Betti numbers equal their numbers of cusps. Among the levels examined, the first classes that live in the interior appear on covers of $M$ of levels 8 and $4\sqrt{-3}$ , as the irreducible $\mathbf{8}$ and as $\mathbf{3}\oplus\overline{\mathbf{3}}$ of $\PSL(2,7)$ . Both are ramified at $\mathfrak p$ , so they belong to $M$ itself.

Status

Items (i), (ii) and (iv) are exact computations with the lift’s own matrices; item (ii) also identifies the program’s objects with a manifold known from the literature, and item (iii) is exact. In item (v) the group arithmetic is exact and the dimensions at the finer levels were computed numerically, with a gap of more than ten orders of magnitude between the zero and the nonzero singular values. On the finite sky alone, reading the curvature of comparing charts as a Thomas–Wigner rotation was a reading with a test attached; the lift makes the placement canonical, the test has been run, and the reading holds in the $3+1$ lift, where it selects more sharply than it was asked to.

What the lift supplies is kinematics: frames, boosts, holonomy and the massless fields at its ends. It does not supply dynamics, the rate at which records are written and the law by which registers combine, nor the strength of any coupling. One interior mode agrees at every prime computed with an elliptic curve, which is strong evidence and not a proof, and what those numbers mean physically is not yet known.

Not a subgroup

Plate XIII.1The finite relativity group does not sit inside the Lorentz group.

No qubit carries a spin that every clock agrees on, since the double cover has no two-dimensional representation: over all the observers the spin factor is a bundle, the report qubit induced from one clock to all seven, in two forms between which nothing yet chooses. The seven clocks’ own lifts do generate the double cover: on the fiber, the eight-dimensional interior each observer carries, they generate exactly $\SL(2,7)$ , acting as $\mathbf{4}\oplus\overline{\mathbf{4}}$ . They are the lifts of the spinor transport, the fiber transport that the lift forces, each a relabelling of the octonion units followed by a shift, right multiplication by the image of 1, and only the shift moves the unit. Over $\C$ the place of the two-component spinor is taken by the faithful $\mathbf{4}$ . In the Weil representation of $\SL(2,\F_7)$ on functions on $\F_7$ , the even halves are $\mathbf{4}$ and $\overline{\mathbf{4}}$ , since $-1$ is not a square modulo seven, and the odd halves are $\PSL(2,7)$ ‘s $\mathbf{3}$ and $\overline{\mathbf{3}}$ , through which it acts on the plane of Klein’s quartic.

None of this demotes the relativity group to a symmetry of the fiber alone. The program’s sky is $\Proj^1(\F_7)$ with the Möbius action of $\PSL(2,\F_7)$ , the same construction as $\PSL(2,\C)$ on $\Proj^1(\C)$ over a different field. The right relation between the two is not inclusion. It is reduction.

Proposition(no Lorentz inclusion) proved

Every homomorphism from $\PSL(2,7)$ to $O(3,1)$ is trivial, and $\SL(2,7)$ has no nontrivial homomorphism to $\SL(2,\C)$ .

Proof

Since $\PSL(2,7)$ is simple, a nontrivial homomorphism would be injective. A finite subgroup of $O(3,1)$ is compact, so conjugate into $O(3)\times O(1)$ ; a simple nonabelian group maps trivially to $O(1)$ and to the determinant, so it would lie in $\SO(3)$ , whose finite subgroups are cyclic, dihedral, $A_4$ , $S_4$ and $A_5$ , none simple of order 168. The irreducible representations of $\SL(2,7)$ have degrees 1,3,3,4,4,6,6,6,7,8,8, so any two-dimensional representation is a sum of trivial ones.

Reduction

Plate XIII.2The finite sky as a reduction of discrete Lorentz groups, with their quotients of velocity space below.

Over $\F_7$ itself, $\PSL(2,\F_7)$ acts on the trace-free $2\times2$ matrices preserving the determinant, which identifies it with $\Omega_3(7)$ and the eight sky points with the null lines of a three-dimensional quadratic form; counted by dimension the sky looks like the circle of light rays of $2+1$ dimensions, but over $\F_7$ every nondegenerate ternary form is the same up to scale, so the count does not decide. Reduction of entries modulo 7 gives $\PSL(2,\Z)/\overline{\Gamma}(7)\cong\PSL(2,\F_7)$ . The quotient of the hyperbolic plane, the $2+1$ space of velocities, by $\Gamma(7)$ has 24 punctures, and filling them in gives the modular curve $X(7)$ , Klein’s quartic $x^3y+y^3z+z^3x=0$ , whose 24 cusps are its 24 flexes, three over each sky point.

One dimension up, the Bianchi groups $\PSL(2,\mathcal{O}_d)$ act on $\mathbb{H}^3$ , the velocity space of $3+1$ Minkowski space, whose boundary is the celestial sphere; for $d=3$ the prime 7 splits, $7=(3+\omega)(3+\bar\omega)$ , and for $d=7$ it ramifies, and in both the finite sky is the reduction of the rational points of the true celestial sphere. The lift must be a Bianchi group: the Thomas–Wigner rotations of loops at one observer of a $2+1$ lift generate a cyclic group of order one, two or three, which cannot reduce onto $S_3$ ; and a clock, whose pairing of the sky has the rotation group of a cube as its centralizer, has the pattern of the $3+1$ sphere, whose antipodal map is anti-Möbius, not of a $2+1$ rest frame, whose centralizer has order eight. The crystal’s metric $x^2+y^2+z^2$ is definite. It is anisotropic over $\Q(\sqrt{-7})$ and isotropic over $\Q(\sqrt{-3})$ , where the ends of the four report axes are Eisenstein null lines and complex conjugation swaps the two ends of each axis. At the split primes above seven conjugation becomes the clock’s pairing; at the ramified prime of $\Q(\sqrt{-7})$ the antipodal maps reduce to the twenty-one fixed-point-free half-turns. Both shadows are of the $3+1$ type, but only $d=3$ carries the crystal’s light cone.

The report lattice

Plate XIII.3The Eisenstein lattice in the plane at infinity, with the base tetrahedron’s finite cusps 0, 1 and

1+\omega

On an observer’s report qubit the Hermitian $2\times2$ matrices carry the form $\langle A,B\rangle=\tfrac12(\operatorname{Tr}A\operatorname{Tr}B-\operatorname{Tr}AB)$ , with $\langle A,A\rangle=\det A$ , of signature $(1,3)$ , and the four reports are null in it.

The formula in the proposition is the null report form of Chapter I, there derived from the reports alone and here from the unimodularity of four Eisenstein spinors. The letters sit in the same lattice as frames: the sum $T_{ab}=N_a+N_b$ of a letter’s two reports has determinant 1, a unit timelike vector, a rest frame. The letter joining $\infty$ and 0 is the identity matrix, and every letter of the lift is its image, $T_{ab}=gg^\dagger$ for some $g$ in the Bianchi group. An own move, to an adjacent letter, is a boost with $\gamma=\tfrac32$ (speed $\sqrt5/3$ ), and a received exchange, to the antipode, a boost with $\gamma=2$ (speed $\sqrt3/2$ ). These are rapidities per record; how many records are written per unit time is not fixed by them. The lattice also blocks a reflection. Let $X$ be a spacelike lattice vector with $3\mid\det X$ , not divisible by the prime $\pi=1-\omega$ above three. Reduced modulo $\pi$ it is $\lambda\,\ell\ell^{\mathsf T}$ over $\F_3$ , and whether $\lambda$ is a square modulo three is unchanged by every Lorentz transformation of the lattice, $X\mapsto gXg^\dagger$ with $g\in\GL(2,\Z[\omega])$ , and by the mirror; since $-1$ is not a square modulo three, none of them carries $X$ to $-X$ . At these separations the reflection that the continuum uses to show that a local field needs antiparticles is missing, and a field’s locality would need the joint reversal of charge, parity and time, CPT, as an input.

Proposition(report counts are Eisenstein) proved

Take the base tetrahedron’s four cusps $\infty$ , 0, 1 and $1+\omega$ , with primitive spinors $\xi_\infty=(1,0)$ , $\xi_0=(0,1)$ , $\xi_1=(1,1)$ , $\xi_{1+\omega}=(1+\omega,1)$ , and the null matrices $N_a=\xi_a\xi_a^\dagger$ . Then $N_\infty$ , $N_0$ , $N_1$ , $N_{1+\omega}$ form a $\Z$ -basis of $\mathrm{Herm}_2(\Z[\omega])$ , and

\det\Bigl(\sum_a x_aN_a\Bigr)=\tfrac12\Bigl[\Bigl(\sum_a x_a\Bigr)^2-\sum_a x_a^2\Bigr].

So $\SL(2,\Z[\omega])$ , acting by $X\mapsto gXg^\dagger$ , is the integral Lorentz group of report counts.

Proof

For spinors $\xi,\eta$ , $\det(\xi\xi^\dagger+\eta\eta^\dagger)=\lvert\det(\xi,\eta)\rvert^2$ , so $\langle N_a,N_b\rangle=\tfrac12\lvert\det(\xi_a,\xi_b)\rvert^2$ and $\det N_a=0$ . The six determinants are 1 for each pair containing $\infty$ , then $-1$ for $\{0,1\}$ , $-(1+\omega)$ for $\{0,1+\omega\}$ and $-\omega$ for $\{1,1+\omega\}$ , all units of $\Z[\omega]$ since $1+\omega=-\omega^2$ . So every $\langle N_a,N_b\rangle$ is $\tfrac12$ and $\det\sum x_aN_a=\sum_{a<b}x_ax_b$ . The diagonal entries of $N_\infty$ and $N_0$ generate the integer diagonals, and the off-diagonal entries 1 and $1+\omega$ of $N_1$ and $N_{1+\omega}$ generate $\Z[\omega]$ .

The lift and its cells

Plate XIII.4The cusp torus at

\infty

, the plane modulo the lattice

\mathfrak p

: seven observers, one at each clock, triangulated as

K_7

by corners of clock cubes and of line cubes.

On the celestial sphere a cusp is a light direction, so the eight cusps are the eight directions from which light can arrive, and each edge, joining two of them, is the observer whose pair they are. The manifold is not new: it is Thurston’s eight-component congruence link complement, one of the principal congruence link complements classified by Baker, Goerner and Reid, and its 2-skeleton is an immersed punctured copy of Klein’s quartic, so the $2+1$ quotient is a slice of the $3+1$ one. The identification holds cell by cell, with the stabilizers matching; it is not an analogy between two lists of the same lengths. Seams reads the cells as one more column of its table, with built seams to the other incarnations, and refutes the coincidence of twenty-eight tetrahedra with twenty-eight observers.

The permutations of the sky that preserve the twenty-eight tetrahedra form exactly $\PGL(2,7)$ , and all 336 preserve orientation: $\PSL(2,7)$ keeps the two colours of tetrahedra, and the other half exchanges the clock cubes with the line cubes, as the Fano plane’s duality exchanges points and lines. So $M$ has no orientation-reversing isometry. Complex conjugation, the anti-Möbius map that nothing on the finite sky distinguishes, does not preserve $M$ : it carries it to the lift at the conjugate prime $\bar{\mathfrak p}$ , its mirror image. The two lifts differ only in which cube root of unity modulo seven, 2 or 4, plays the role of $\omega$ , and over $\F_7$ no automorphism tells them apart. Every $3+1$ lift of the sky makes that choice, and the choice is a handedness: this is where the volume’s chirality begins.

Theorem(the lift and its cells) computed

Reduction modulo $\mathfrak p=(3+\omega)$ maps $\SL(2,\Z[\omega])$ onto $\SL(2,\F_7)$ , with $\omega\mapsto4$ . Its kernel $\Gamma(\mathfrak p)$ is torsion-free, and the stabilizer of a cusp maps onto a Borel subgroup, of order 21 in $\PSL(2,7)$ . The quotient $M=\Gamma(\mathfrak p)\backslash\mathbb{H}^3$ has eight cusps, one over each point of $\Proj^1(\F_7)$ , and its canonical decomposition consists of twenty-eight regular ideal tetrahedra. The 8 cusps are the sky points; the 28 edges are the pairs of sky points, each with stabilizer $S_3$ , the anchored observers; the 56 triangles are the triples, each with stabilizer $C_3$ ; and the 28 tetrahedra form two disjoint Steiner systems $S(3,4,8)$ , the orbits of $(\infty,0,1,1+\omega)$ and $(\infty,0,1,-\omega)$ , which reduce to $\{\infty,0,1,5\}$ and $\{\infty,0,1,3\}$ : the halves of the seven clock cubes and of the seven line cubes. Each triple lies in one block of each system, and each pair in three blocks of each. Every cusp torus is $K_7$ with two Fano classes of faces, its seven vertices the observers through that sky point, one at each clock.

The reading, tested

Plate XIII.5The ideal triangle

(0,1,\infty)

: three parabolic steps at its cusps compose to

z\mapsto-1/z

, a half-turn about

i

In special relativity a sequence of boosts that returns an object to its original velocity leaves a rotation, the Thomas–Wigner rotation, the holonomy of velocity space around the loop; for a geodesic triangle it is a rotation by the triangle’s area. The test of the reading had three steps: place the observers on velocity space, lift loops of promotions and compute their rotations, reduce modulo seven and compare. The lift supplies the transport. For a promotion $(c;a\to b)$ , $T$ is the unique null rotation of order seven that fixes the light direction $c$ and carries $a$ to $b$ ; around the ideal triangle $(0,1,\infty)$ the three parabolic steps compose to $z\mapsto-1/z$ , a rotation by $\pi$ , the Gauss–Bonnet holonomy of a triangle of area $\pi$ . Each face’s half-turn is the involution of a meeting of Chapter XII: for a face with base observer $\{a,b\}$ and third light direction $c$ , the harmonic conjugate $c'$ of $c$ with respect to $\{a,b\}$ makes a second observer $\{c,c'\}$ that meets the first, and the half-turn is the one involution of the sky exchanging $a$ with $b$ and $c$ with $c'$ . Each meeting serves four corners of faces and each of the twenty-one involutions eight, so the curvature of velocity space, read on the lift, is carried by the meetings of observers.

So the reading holds in the $3+1$ lift, and the comparison selects something the test did not anticipate: neither $F_0$ nor $F_1$ . Each is the rotation of a cell, and each differs from $T$ by an element of the observer’s stabilizer; the cell rotations need the octonion table’s distinction between clock cubes and line cubes, which $\PGL(2,7)$ does not respect. $T$ needs nothing, and it stays the comparison of frames and, through the double cover, the transport of spin; matter’s internal labels are carried instead by relabellings that keep the octonion product, which the program takes as physical. Spin–orbit coupling, which the flat crystal lacks, is native to the lift: it couples spin to changes of frame, and changes of frame are records. Over $\F_7$ the words rotation and boost are not intrinsic and there is no rapidity; the rapidity lives in the lift, per record.

Theorem(rulial curvature is velocity space’s curvature) computed

On $M$ : (a) $T$ is the identity around all 1680 triangles of the cusp tori, and around each of the 336 faces it is the half-turn that reverses the base observer’s line of sight. (b) On one clock cube, based at a letter, the holonomy group is $S_3$ , each triangle of $K_4$ carries a half-turn, and all thirty decagons have holonomy of order three, the commutator of two Gauss–Bonnet half-turns. (c) The spinor lift of $T$ is the identity around corners, has trace 0 and square $-I$ around faces, the Thomas rotation by $\pi$ , and has order exactly three on every decagon. (d) For each promotion exactly three elements of $\PSL(2,7)$ keep the shared ray: $T$ , and the rotations of order three of the clock cube and of the line cube on the face. $F_1$ is the line cube’s rotation and $F_0$ the face’s own; $T$ is the only one of the three that is unipotent and the only one covariant under all of $\mathrm{Isom}(M)$ .

The lift at its primes

Plate XIII.6The lift read at the three primes of 168: the reports at

\sqrt{-3}

, the sky at

\mathfrak p

, and at 2 nothing of the clocks.

The order of $\PSL(2,7)$ is $168=2^3\cdot3\cdot7$ , and its three primes behave differently. At $\sqrt{-3}$ , ramified with residue field $\F_3$ , $\PSL(2,\F_3)\cong A_4$ acts on four cusps, extended to $S_4$ , the crystal’s $\SO(3,\F_3)$ : the four reports are the cusps of the level- $\sqrt{-3}$ quotient. At $\mathfrak p$ the lift is $M$ , with $\PGL(2,7)=\SO(3,\F_7)$ on eight cusps. At 2, inert with residue field $\F_4$ , $\SL(2,\F_4)\cong A_5$ acts on five cusps, which has nothing to do with the clocks’ $\GL(3,2)$ since $60\nmid168$ ; the clocks enter instead as the crystal modulo 2. So the program’s finite structures come from the lift at $\sqrt{-3}$ and at $\mathfrak p$ , and from the crystal at 2.

The prime above three also carries the alphabet of memory. The thirteen lines of $\mathfrak{sl}_2(\F_3)$ are of three kinds, a nilpotent one picking one point of $\Proj^1(\F_3)$ , a report, one with two eigenlines picking a letter, and one with none pairing the points off, an axis: $4+6+3$ . What a record is, which letter it writes, is one digit read at $\sqrt{-3}$ . How records pile up is not: appending generates the free group on the three axes, and on two axes the memory’s rewrites act as a binary counter, structure at the prime 2, while the lift’s own structure at 2 is built on $\F_4$ . Where the counter lives is open.

Fields on the lift

Plate XIII.7First Betti number against number of cusps up the tower of levels: equal through

\mathfrak p^3

, then eight and six interior dimensions on the first covers with depth at 2.

A field on the lift is a cohomology class of $M$ with coefficients in a representation of the Lorentz group, and near each cusp the rotation about that light direction acts on it with a definite helicity. The massless fields are classes of degree one, in three sectors at the same eight light directions: light with helicity $\pm1$ , the fiber with $\pm\tfrac32$ , and the deformations of the lift’s own shape with $\pm2$ . The fiber’s match is with its Clifford module $\mathbf{4}\oplus\overline{\mathbf{4}}$ , on which the spinor transport acts; how the field relates to matter’s labelled fiber $\mathbf{1}\oplus\mathbf{7}$ is open. Each class is fixed by its boundary data, so every question of how massless fields scatter or combine is decided at the eight light directions: light entering at one leaves through each other with the share $\tfrac17$ , by a table built from a cubic character modulo seven and a constant built from the prime itself. The helicity-two sector scatters by the same table up to a constant, the fiber by a different one, the Paley matrix.

Through level $\mathfrak p^3$ the tower of finer lifts is purely kinematic: at level 49 the first Betti number is 392 and at level 343 it is $19{,}208$ , exactly the numbers of cusps, and the modes sit on observers and on sky points, none on the twenty-one half-turn axes. The first interior classes, attached to no light direction, appear with depth at the prime 2: at level 8 the cover has 64 cusps and first Betti number 72, eight interior dimensions forming the irreducible $\mathbf{8}$ , and at level $4\sqrt{-3}$ it has 80 cusps and Betti number 86, six interior dimensions forming $\mathbf{3}\oplus\overline{\mathbf{3}}$ . Neither contains the trivial representation, as a form inherited from the Bianchi group would, so both are ramified at $\mathfrak p$ . Their first Hecke eigenvalues are whole numbers at every prime tested, new arithmetic, and one mode is, on all the evidence, an elliptic curve over the lift’s own field with $j$ -invariant $1024(3+\omega)$ . The 3 in $\mathbf{3}\oplus\overline{\mathbf{3}}$ is the dimension of a representation and says nothing about generations. Both classes need depth at 2, the prime of memory’s counter, but the counter is binary and the lift’s structure at 2 is built on $\F_4$ : that the two meet at one prime is recorded, not explained.

The finite group does not sit inside the Lorentz group; it is the reduction of the integral Lorentz group of the report counts, and the quotient of velocity space by the kernel is a manifold whose cells are the program’s objects. What the reduction forgets is rapidity, and the lift restores it: frames are points of velocity space, records are boosts of fixed rapidity, and comparing charts is parallel transport. If continuous special relativity comes from the finite sky at all, it comes from a lift, and from this one, as kinematics.

The story continues in three directions: the massless fields at the eight light directions (Chapters XV and XXIII); the seven clocks as the seven $E_8$ orders of the sky’s octonions (Chapter XIV); and the dynamics, which is not in the lift. Two later chapters return to the lift itself: the part of its group that reduces to the identity on the sky already keeps exactly one light cone, where Chapter XXII finds the common speed, and Chapter XVIII finds the sky’s second parent, compact at its real place, on whose building the interior lives. In Seams this is the arithmetic continuum of the finite sky, the only kind it has: an absence decides the kind, since by Klein’s list of the finite groups of Möbius maps the sky has no embedded continuum in $\Proj^1(\C)$ . Part IV turns inward, to the interior each observer carries.