One Speed

Many speeds in one chart

The lift of Chapter XIII, the hyperbolic manifold of rest frames whose eight ends are the points of the finite sky, relocates the speed question. In the chart of one clock, the program’s free record dynamics, streaming at the record speed $v$ and turning with the spinor transport that the lift forces on the fiber, has exactly isotropic first-order cones, but two of them, with speeds $v/3$ and $v/(3\sqrt3)$, and a gapped pair whose dispersion is not relativistic at small momentum. Every record moves at $1/\sqrt3$ of the speed of light relative to its clock’s cube, no mode of the arena outruns that, and only the cusps, the light directions, move at the speed of light. Streaming the fiber’s own null states along the cube’s eight corners does not help: it gives two anisotropic bands, not a Weyl operator.

The resolution is kinematic. In the lift a letter is a unit-mass momentum written as a pair of null spinors, $T_{ab}=gg^\dagger$ with $g$ in the Bianchi group, and the reports are the null momenta; both are orbits of one group, so on-shell kinematics has one light cone by construction. The two cone speeds and the record speed belong to streaming in one clock’s frame, off shell, and the record speed is the speed of a letter in the rest frame of its reports, not a limiting speed. One velocity for all species is therefore a question about dynamics on the Lorentz orbits, amplitudes rather than streaming, and the program’s gravity question sits in the same place. A colour field placed on the arena would add one more speed: every choice of nearest neighbours there picks a rest frame, and with one weight on all of the arena’s smallest loops the field’s waves move at $c/\sqrt{10}$. Only weights chosen for the purpose would give $c$, one linear condition among the four kinds of smallest loop, which no native weighting supplies and no symmetry protects.

Running cannot make one speed

In a declared one-loop model of two Dirac flavours coupled to one critical boson, after Roy, Juričić and Herbut, the relative speeds converge if and only if the accumulated coupling $\int_0^\infty q(\ell)\,d\ell$ diverges. An attenuation rate $\gamma=Ae^{-B\ell}$ leaves a fraction $e^{-A/B}$ of the initial mismatch, and a flavour with zero coupling keeps its own speed. The criterion is the one-loop flow of Chadha and Nielsen, and of Anber and Donoghue, written in the accumulated coupling: sectors coupled to one gapless field relax toward a common speed at a rate proportional to the coupling, and a chain of couplings suffices.

The flow needs a mediator that reaches every species without a gap, and the program’s matter has none. Colour reaches only coloured states and is gapped by confinement; the weak bosons reach only doublets; the photon misses the neutrinos; the right-handed neutrino couples to no link at all; and gravity reaches everything, but with a finite accumulated coupling. With the measured strengths the running shrinks a mismatch by a factor of order one, while the bound from the neutron-star merger holds gravity’s speed to light’s within about one part in $10^{15}$, a difference no running below the Planck scale can change. A common speed must therefore already hold at the cutoff: it is a property of the law, not of its flow. The program’s own candidate mediator, a neutral core between registers, has a fixed gap and responds statically when weakly coupled, no native critical field has been constructed, and the joint criticality of Chapter XIX, in which light and matter become massless together, is a criticality in rates and defines no speed.

What the finite sky cannot fix

The finite structures fix the light directions and make each sector’s cone isotropic. The relativity group is transitive on the eight sky points, and in one clock’s cube frame the only invariant quadratic forms on energy and momentum are combinations of $E^2$ and $\lvert p\rvert^2$, which cannot fix the ratio between the two. The smallest nontrivial real representation of $\PSL(2,7)$ has dimension six, so no finite symmetry of the sky acts on a four-dimensional space of momenta, and the octonion lattice and Mumford’s lattice have no Lorentzian real place.

The lift’s group does fix it. Acting on Hermitian matrices by $X\mapsto gXg^\dagger$, the Bianchi group $\SL(2,\Z[\omega])$ has a real subgroup $\SL(2,\Z)$ that keeps two quadratic forms, and one non-real element, the null rotation $\bigl(\begin{smallmatrix}1&\omega\\0&1\end{smallmatrix}\bigr)$, leaves only the determinant, the Minkowski form. Even the elements that the finite sky cannot see do the same. What remains is one number per sector, its limiting speed, which lives at the real place of the lift’s group alone; covariance of the cutoff law under that group, or under its part invisible on the sky, would make all of them one.

The law as a shadow

The program’s built dynamics is covariant only in one clock’s frame. Its turns are exactly the reduction, modulo the prime above 7, of a transport by elements of the lift’s group that is covariant in this sense, so the law loses covariance only where it reduces to the finite sky and where it streams in one clock’s frame. A covariant law would keep the part the sky cannot see, and would change velocities only through momentum-conserving processes.

The streaming has a second limit, which concerns matter rather than speed. Its turns carry the interior by the spinor transport, so the charge they keep is the light direction’s, $J_*$, and not the clock’s quartet that an observer counts as matter, and its turn operator pairs no energy with its negative. Carried instead by matter’s own transport, the relabellings that keep each observer’s clock, with spin turned by the rotations that keep each record’s form, the lepton sector of the turns has a single cone speed, a ninth of light’s, with spin locked to momentum as for a Weyl fermion. The form of that transport was found after the planned one failed, and the arena still picks a frame, so this is isotropy and one speed for leptons, not relativity.

The obstruction familiar from causal sets does not forbid this. Bombelli, Henson and Sorkin show that a sprinkling invariant under the whole connected Lorentz group admits no equivariant choice of a direction; a law covariant only under a lattice acting on velocity space is outside their hypothesis. Their argument still yields one consequence here: such a lattice has no invariant probability measure on the space of rest frames, so a covariant law picks no rest frame, and velocities can change only through processes that conserve momentum.

The missing interaction

The program’s contact rule supplies no momentum-conserving process beyond the exchange of two letters: a meeting records what two registers share, or swaps them. The natural form of a transfer is two letters trading reports, $\{a,b\},\{c,d\}\to\{a,c\},\{b,d\}$, which conserves momentum report by report. On its own the trade is not colour-covariant, since it singles out the three spatial axes, which colour’s symmetry mixes. The only colour-covariant contact containing it is a single colour-neutral vertex, in which passing through, exchange and the four trades enter with equal weight. On antisymmetric pairs of letters it depends only on the energy and the momentum transfer, and on the lift trades occur at exactly three energies, two of them joining neighbouring cubes, each carrying one cross-ratio of its four reports as its phase.

The vertex has been built and adopted, reversibly, as a clause of the law, and in the octonions it is plain. Read the six letters as imaginary units: the vertex is the component of the two letters’ product along the clock’s lepton unit, the one imaginary unit that is not a colour, so two coloured displacements fuse into the lepton unit and re-emerge along any axis. Distinct units anticommute, so the vertex is antisymmetric under exchange without a rule that makes it so; the antisymmetry belongs to the vertex’s channel and not to the registers, since the product of two letters also has a symmetric part, the channel of the trap, and the vertex acts whatever the statistics of the registers. It passes every test the law sets for a contact: every observer’s record fixes the order of its timelike-separated events at every length of history, which holds for any contact that both participants record, so it is colour and not this test that selects the vertex; no register’s private moves depend on its partner; colour commutes with it exactly; and its correlations violate Bell’s inequality, with $S=2.32696232735\ldots$, a value certified twice in exact interval arithmetic, by two independent methods.

Across the lift

On the lift a trade’s four reports are four light directions. Every set of four light directions carries exactly one colour singlet, shared by its three ways of pairing them, and a trade keeps its four light directions, so it never leaves that singlet. The trades at the two higher energies join letters that no cube holds together, and they turn out to be the meetings of Chapter XII: each of the forty-two meetings between observers is the one meeting pairing of a set of four light directions that is not half of a cube, and these are exactly the sets on which the higher-energy trades act. In the gauge links’ flat vacuum the comparison along a meeting is canonical. So at every energy the vertex is defined without a choice, depends only on the energy and the momentum transfer, and is covariant under the lift’s whole group: momentum moves across the lift through a covariant process, not only through the turns, and the lattice that carries the gauge links and the support of that process are one structure.

Two limits remain. The vertex is covariant only when letters carry their octonion signs, as the program’s adopted transport gives them, and the seven line cubes of Chapter X, whose only singlet pairs a letter with itself, lie outside its reach. They are exactly the cells on which the octonion table’s four-fold form lives, where the product of the four light directions is a real number; whether they need a contact of their own is open. The vertex also leaves the interior’s fiber unchanged. Dressed by the steps that write its letters into the fibers, it does reach the fiber’s lepton line, as an exchange of $B-L$, but in its covariant form the pair of right-handed neutrinos of Chapter XVII is exactly left alone by that exchange, so the vertex is not the force that makes the pair condense.

Where the speed lives

Collected, the account places each part of the light cone. The light directions are the eight points of the finite sky, and every sector shares them. Each sector’s cone is isotropic in one clock’s frame. The speed itself is one real number per sector, invisible to every finite and two-adic structure, and the lift’s group, already its part that the sky cannot see, keeps only the cone common to all.

Two arithmetic groups share the finite sky at the prime 7. At the real place the lift’s group is the Lorentz group, while Mumford’s group is compact; they share the sky but not its completion. Each acts on a tree whose neighbours at a vertex are the eight points of the sky, and the two trees are alike as trees, but not with their symmetries: the lift’s group permutes the sky at every vertex by its even maps, while Mumford’s group, extended to invert 7, also applies the orientation-reversing ones. So spacetime and internal symmetry meet only at the finite sky. Mumford’s group with both 2 and 7 inverted also holds the double life of the group of order 168 at a single vertex, whose stabilizer moves its neighbours at 2 as the collineations of the Fano plane and its neighbours at 7 as the Möbius maps of the sky. The group whose real place carries the cone is the lift’s; the other, compact there, is the interior’s parent, described in Chapter XVIII.