Two Parents of the Sky

Why the plane has order two

Fix a point $c$, the clock, in a projective plane of order $q$. The lines missing $c$ number $q^2$, the points other than $c$ number $q^2+q$, and a point other than $c$ lies on $q+1$ lines, one of which passes through $c$, so it lies on exactly $q$ lines that miss $c$. In the program a report is a line missing the clock and a letter is a point other than it, so a letter is a pair of reports exactly when $q=2$. Then the four lines missing $c$ and the six points other than it are the reports and letters of $K_4$, and the plane is the Fano plane.

Two further facts hold only at the same order. The four lines missing $c$ form a complete quadrilateral, and its three diagonals meet in one point exactly when $q$ is even; at $q=2$ the point is the clock, so the clock’s three axes meet at the clock. And $\PSL(3,q)$ has the order of a group $\PSL(2,p)$ only when $q=2$ and $p=7$, which is the double life of the group of order 168 (Artin). All three follow from the first. A projective plane, finally, contains the complete graph $K_n$ as the residue of a point only for $n=4$, so the plane exists exactly when the observers have four reports, the lock of Chapter V: the lock and the Fano plane are one fact.

The table as a gluing rule

Read the octonion table in its cyclic form, $e_xe_{x+1}=e_{x+3}$ with labels in $\Z/7$, as a rule for gluing triangles: one generator $a_x$ for each imaginary unit and one relation $a_xa_{x+1}a_{x+3}=1$ for each oriented line. Cartwright, Mantero, Steger and Zappa found the conditions under which such a presentation defines a group acting simply transitively on the vertices of a building of type $\tilde A_2$, a two-dimensional complex of triangles whose vertex links are projective planes. The table’s triples satisfy them, with the line map $\lambda(x)=x+\{1,2,4\}$, the quadratic residues modulo 7, and each pair $(x,\lambda(x))$ is an antiflag of the Fano plane: an observer.

So the table glues in one handedness, and its lattice lives over the field of Laurent series in characteristic two, not over the 2-adic numbers. That is no accident: a group that acts transitively on the vertices of such a building and rotates their three types, with stabilizers of order 21 acting faithfully on the links, can contain a simply transitive normal subgroup only if the building is the one of $\F_2(\!(t)\!)$. The table’s lattice is normal in its symmetry, and the 2-adic lattice of the next step has no such normal subgroup. The building is identified by an explicit representation over $\F_8(t)$ and a covering argument; the abelianization, the generators and the orientations were checked by exact computation.

From the signed table to Mumford’s form

The table’s signs reach the 2-adic side by another road. Let $u=e_0+\dots+e_6$. Its square is $-7$, so left multiplication by $u$ is a square root of $-7$, a Gauss sum, and $J_*=L_u/\sqrt7$ is the democratic complex structure of Chapter XV. The Frobenius group $F_{21}$ of the table’s symmetries, the maps $x\mapsto ax+b$ with $a$ a square, fixes $u$.

Mumford’s lattice, a subgroup of index 21 in Mumford’s group, acts simply transitively on the vertices of a building whose links are Fano planes, as the table’s lattice does, and it is a different group. Its building is over $\Q_2$ rather than $\F_2(\!(t)\!)$, its abelianization is $\Z/2\times\Z/6$ against the table’s $\Z/2\times\Z/2\times\Z/6$, and by the rigidity theorem of Kleiner and Leeb no subgroups of finite index in the two are isomorphic. Both are completions of the Fano plane, infinite lattices whose every vertex sees the plane. The table builds one directly; its signs, through $\sqrt{-7}$, lead to the other.

The sky at three places

Reduce Mumford’s group at the prime $\sqrt{-7}$. Its kernel $\Phi$, the elements that reduce to the identity, is torsion-free, and the quotient of the building by $\Phi$ is a complex of eight vertices, fifty-six edges and fifty-six triangles, one triangle for every triple of the eight vertices. It is the sky, as a set with the action of the group of order 168, and the building is its universal cover. At each vertex the Fano link reads the twenty-eight observers as the twenty-eight pairs of sky points: seven are edges from the vertex and back, and twenty-one are the far sides of triangles. So one arithmetic group, reached from the signed octonion table, carries the Fano plane at the prime 2 and the sky at the prime 7.

The real place completes the assembly. The eight sky lines of Chapter XV are the octonion basis lines under $J_*$, with no phase between them: the line of 1 is a light direction’s sky state, and the seven imaginary units are the clocks of that direction’s seven observers. The labels, the groups and the orientation of every triple of sky points agree at the real place, at 2 and at 7. At the real place Mumford’s group is the unitary group of the three complex dimensions orthogonal to the light direction, and its centre is the circle generated by $J_*$. Colour is the same construction at one clock, the stabilizer of the clock’s unit rather than of $u$; the two meet in an $\SU(2)$ whose finite part is the Frobenius symmetry of order three, and together they generate $G_2$.

The interior on every site

An infinite lattice of observers’ networks needs the interior to travel from site to site, and on Mumford’s building it does so natively. The table’s own lattice behaves differently: its translations act through colour, which the program’s standing condition that colour must not act on space excludes, while no transport of Mumford’s group lies in any colour group. So the physical internal lattice is Mumford’s.

Two signs record the spinor. An observer’s loop out and back multiplies matter by $-1$, as a full turn does to a spinor. And matter’s one-way hop between sky points squares to $-7$: it is multiplication by the sum of the units again.

Three kinds of site

Invert 7 as well as 2. At the prime 7 the unitary group of Mumford’s form has a tree, eight edges at every vertex, and at the vertex of Mumford’s lattice the eight neighbours are the eight lines of a plane over $\F_7$, permuted as the sky’s group permutes the sky. The group with both primes inverted acts on the building at 2 times this tree, with three kinds of site, and the double life of the group of order 168 (Chapter X) sits at one point of one arithmetic group.

Along the tree the interior no longer travels honestly. At sky sites and at clock sites the fiber’s transports act only up to sign, so on the parent the spinor sign cannot be avoided: matter there carries half-integer spin, whose statistics needs more (Chapter XVI). And the kernel at one site is geometry at the next. The element that an observer’s loop sends to $-1$ at a light-direction site acts at the neighbouring sky site as a half-turn of the sky, an orientation-reversing map that fixes two of the eight points. No action of matter on the whole parent descends to the sky’s group, so a law written on the parent must carry the part of it that the sky cannot see.

Where the parents meet

Both parents have a tree at 7 whose link at a vertex is the sky. As bare trees they are alike, and the seam between their links is unique, because the eight-point sky is a rigid object. With their symmetries they differ. Admitting determinant $-1$ adds odd maps on the spacetime side too, but only at the root: what separates the parents is where the orientation-reversing maps live, at the root of the spacetime tree and inside the root’s kernel on the interior’s. So spacetime, at the real place of the lift’s group, and the interior, at the compact real place of Mumford’s, meet only at the finite sky.

The orientation-reversing maps belong to the interior’s parent alone. Acting on the fiber’s signed basis they are antilinear: they exchange the quartet and antiquartet for $J_*$ and carry colour to the colour of the mirror table, whose lines are $x+\{0,4,6\}$. The parent cannot use them on matter. A charge conjugation of the fiber that implements them exists, but on an edge of the tree between a light-direction site and a sky site one element acts as fermion parity at the first and as that charge conjugation at the second, so no action of the parent on matter restricts to both. A law for matter on either parent is a connection, whose holonomy lives on the squares that combine a loop at one site with a step along the tree, never a representation of the parent.

Which completion matter’s labels follow is then a question for seam theory. The labels, which say which octonion unit is which clock, form a rigid object: between any two of its incarnations there is exactly one seam. On the lift’s tree every hidden loop only changes which clock is which, so the labels are carried along that tree in exactly one way. On Mumford’s tree the element that is fermion parity at one site acts one step deeper as a reflection of the sky, which turns the clocks into the lines and the octonion table into its mirror, and no seam joins a table to its mirror. So matter’s labels follow the spacetime completion and not the interior’s, a theorem given only that the octonion product is physical; inside the interior’s parent they travel along its building at two, at the sites of the light direction’s kind, and nowhere else. On the spacetime side, one table per scale follows the tree exactly once every deeper frame’s table points its light direction back along the tower of scales, and such a direction can always be chosen, since the lift’s levels have a coarsest scale, the lift itself, at which it can be rooted. The parent does not supply the choice: it is a choice of base scale.

One lattice where they meet

The site of Mumford’s arithmetic at which the group of order 168 holds both its lives is a classical object, Klein’s lattice: three dimensions over the integers of $\Q(\sqrt{-7})$, unimodular, with the group of order 168, up to sign, as its symmetry. Elkies showed that Klein’s quartic curve is built from it, the curve’s Jacobian being an elliptic curve with complex multiplication tensored with this lattice; Allcock and Kato showed that its group and Mumford’s are the two densest arithmetic lattices acting on the building at two; lattice theorists know it as the Hermitian Barnes lattice. The lattice and its symmetries are classical. What is new is the dictionary: a clock is a cube, an axis is a half-turn, a report is a diagonal, and the observers are vectors of one lattice that both parents share.

One step further out the two trees differ, and the difference is classical. On the lift’s tree the second step adds the Lorentz algebra read modulo seven, the crystal of Chapter XIII, a space of vectors. On Mumford’s tree it adds a doublet, a plane over the field with seven elements on which the sky’s double cover acts as on spinors, with the turn through $2\pi$ acting as $-1$; the vector space is the doublet’s symmetric square, as vectors are built from spinors in the continuum. So, one step out, spacetime forgets vectors and the interior forgets spinors. The doublet is not the complex qubit the finite sky cannot carry, but Weil’s construction turns it into exactly the spin quartet of Chapter XV together with Klein’s three-dimensional representation, the representation on this lattice; and the lattice, read at seven, is the spacetime tree’s forgotten vector space, uniquely up to scale. What one parent forgets one step out, the meeting site remembers at its first step.

Two hands

Each parent carries a flip of its own. The lift comes in two mirror images, one for each prime above seven, and its spinors have the matching hand; the interior’s octonion table on the sky has a mirror table, and choosing between them is the same choice as which of the quartet and its conjugate is called matter. Only one of the two flips is a hand. The lift’s mirror image is seen by the lift’s own structures: the fiber’s scattering constant, which the lift’s integral structure fixes, changes under it, and so do the orientations of the lift’s cells read against the table’s four-fold form, in every combination that does not depend on which of the two opposite unit vectors at each light direction is called positive. The interior’s flip is not seen. That choice of unit vectors is a convention, which the record absorbs as a phase, and once it is set aside, exchanging the table with its mirror and the quartet with its conjugate is an exact symmetry of the lift, of the sky and of matter’s interaction. An earlier reading took the table’s cells relative to the lift for a relative orientation of the two flips; it compared the table and its mirror each in its own convention, and in matched conventions the two agree.

So the program has one native hand, spacetime’s, and its matter is symmetric under charge conjugation: every coupling of the lift’s massless fields is unchanged by the rotations that exchange the quartets, and where it tells them apart it does so as a charge does. The fiber’s scattering phase changes under the mirror alone, the pattern of a phase that violates the combined symmetry of mirror and charge conjugation, not the weak force’s (a reading). The spacetime half of the weak force’s chirality is native: matter’s spinor is the lift’s own and carries its hand, every step of a history keeps it, and so every field of matter has the same spacetime hand, the Standard Model’s own form when all its fields are written as left-handed. The added point of the Cayley plane gives the arrangement in which the left weak group acts on the quartet and the right on its conjugate, the vacuum, an added condensate, chooses which survives, and no orientation of spacetime has been added. One choice remains, of charge-conjugation type: whether the vacuum keeps the weak group of the quartet called matter or of its conjugate. While the exchange of the quartets is an exact symmetry, that choice is a naming.

Matter’s law on the two parents

Can the two parents be made one structure, so that matter’s law runs along the interior’s sites and along spacetime’s scales at once? Spacetime’s parent, with its prime inverted, moves along the tower of scales, a tree whose every vertex is a scale with an eight-point sky of its own; the interior’s moves along its building at two. Mathematics has one way to join two such groups, the fiber product of the pairs that act on the sky in the same way, and Margulis’s arithmeticity theorem says that nothing more intimate exists here. Item (1) comes from spacetime’s side alone: closeness in the tower is seven-adic, so the symmetries fixing a scale come arbitrarily close to every symmetry of that scale, and discarding finitely many, as any group of joint symmetries must, leaves enough to turn the sky in every way. Item (3) rests on a theorem of Serre’s: every way of reducing spacetime’s group, with its prime inverted, onto the sky’s group is reduction at the other prime.

So matter’s law stays on spacetime, and the interior attaches at one scale. Spacetime carries matter across scales observer by observer: its loops through the tower bring every observer’s matter back as matter, changed only by a colour rotation at its own clock, uniquely up to colour and in agreement with every meeting. Only a shortcut fails. On one loop in eight a single relabelling of the table serves every observer at once; on the other seven any single relabelling reverses at least two of the seven clocks, and then twenty or more of the forty-two meetings would compare matter with antimatter, so the meetings rule it out. Nothing reverses charge as matter crosses scales, and the tower has no native source that changes lepton number. Colour does couple the scales: the shortest loops across scales, five meetings long, are among the plaquettes, so the tower is one gauge theory, but its coupling does not run along the tower, finer scales only stiffening coarser ones once, nor on the crystal inside each meeting, which copies loops rather than subdividing them. The program’s positions do subdivide, but the links join observers’ frames, not positions, so a running coupling would need a colour frame added at every position.

Matter’s labels can be carried from scale to scale in two ways. Holding the octonion product fixed, observer by observer, keeps every lepton a lepton and every clock’s orientation, and in each observer’s frame even the light direction’s charge $J_*$ is restored by a colour rotation. Carrying the product as frame data, by the spinor transport, keeps $J_*$ exactly and lets the table’s light direction move from scale to scale. The two differ by a half-turn in the plane of two clocks, and $J_*$ is not what an observer counts as matter. Nothing built from the sky, the observers and their clocks, $J_*$ or the spinor structure can tell the transported table from the one that is there; only the octonion unit can, and carried as frame data a lepton would come back as a quark at six of the seven clocks. The program made the product with its unit physical, so across scales the product is held fixed, observer by observer, and it costs nothing. Which scale is the base is a choice the parents do not supply. Spacetime’s two primes above seven divide the work, one carrying scale and the other the attachment. That this has the shape of an internal symmetry meeting spacetime symmetry only through a finite group, as Coleman and Mandula’s theorem would lead one to expect, is a reading.