Light, Vacuum and Handedness

Three origins of a letter

The kernel graph’s four reports 0,1,2,3 are joined by the six letters 01, 23, 02, 13, 03, 12. Each letter shares a report with four others and is disjoint from one, its antipode, so on the octahedron of letters the antipode is the opposite vertex and the octahedron has diameter two. A register’s live window here is a pair $(a,t)$, with $t$ the newest letter, beside a fiber $\C^8$ on which the letters act as six anticommuting unitaries, and every write multiplies the fiber by one letter.

Two registers anchored at $v$ and $w$ share the rod $\ell=vw$ and are ready when their newest letters are $\ell$ and $\bar\ell$. In the complete recorder $A$ appends, $B$ appends, or a ready pair exchanges: an own append exports the oldest letter and appends the equal-amplitude superposition of the five allowed letters, with weight five, and the exchange has weight $\lambda$; dividing by $z=10+\lambda[\text{ready}]$ makes one trace-preserving instrument. The projector onto windows whose two letters are antipodal is the reception flag, since no own append can put a window there.

One equilibrium for every observer

The proof of uniqueness is finite: from fixed base words every Pauli operator on the two fibers is realized by a closed path of own appends, padded to a common length by identity loops, one of odd length that begins with an exchange, and twirling over the Paulis shows that a power of the channel contracts every traceless operator. At $\lambda=0$ the two private recorders keep a second peripheral eigenvalue $-1$, on the mode $D_AD_B$: every occurrence flips one chirality. An exchange writes at both sites at once, and reception is what lets the recorder settle.

The equilibrium forgets its initial state but not the law. The pair of newest letters satisfies detailed balance with probability proportional to $z$, so the newest letter is $\ell$ with probability $(60+\lambda)/(360+2\lambda)$, $61/362$ at $\lambda=1$, against $60/362$ for each letter off the shared rod. The rate must agree between descriptions, because the probability of a ready pair, $(10+\lambda)/(180+\lambda)$, is strictly increasing, but no value is selected; if observers at different clocks may meet along the Coxeter graph, a pair meeting there under a fixed comparison is a relabelled copy of this recorder, and its equilibrium and reception law transfer at the same rate.

What the equilibrium is not

So no ground state is fixed by one own commit, and with exchanges the equilibrium’s reception weight stays below 0.052 at $\lambda=1$ and 6 and below the ground-state bound for every $\lambda$ up to about 20.3. It is not a cooled state either, and it is not the shared line of the octonions: the unit, fixed by all twenty-eight colour groups, reads $(\lvert000\rangle+\lvert111\rangle)/\sqrt2$ in an observer’s occupation basis, while that observer’s vacuum is $(c+iq_x)/\sqrt2$. What the unit gives every observer is the same slot: modulo the observer’s clock the octonion units fall into four classes, which add like displacements, and the unit’s class is their zero, the observer’s lepton line, fixed by every relabelling and containing its vacuum. And with the oldest-letter qubit read as weak isospin, its equilibrium coherence $x_\lambda\neq0$ would make the vacuum coherent across charges, so that qubit is not weak isospin.

The vacuum is relational, a reading: its signature, reception, is written only by exchanges, a private writer never produces it, and an energy criterion weights it wrongly. The test is the stationary state of the complete recorder on the infinite crystal, with a clock-changing occurrence that carries it to itself. Neither exists yet.

Light on the crystal

Space is the crystal of tallies, the maximal abelian cover of $K_4$. For a finite history of carriers moving along its bonds, the signed crossing tally $E$ and the density $n$ satisfy $B\,E=n-n(0)$, with $B$ sending an oriented bond to its target minus its source: Gauss’s law from crossing tallies. Making it a gauge constraint needs a link carrier on each bond. The program has adopted such links as an addition, the gauge-link clause: continuous internal links between neighbouring observers’ frames, with a plaquette action and the Standard Model’s group. A native realization of the link is still missing. With an added energy $\tfrac{U_E}2\sum_eE_e^2$ and ring term $-\tfrac K2\sum_c(R_c+R_c^\dagger)$ on the shortest loops, the long-wavelength symbol is $\omega^2A=16\,U_EK(\lvert k\rvert^2I-kk^{\mathsf T})A$: two transverse modes on an isotropic cone of speed $4\sqrt{U_EK}$.

The isotropy is the tight frame; everything dynamical is added, and varying $K$ changes the speed of light without touching any hop of matter. The native charge is only a parity: every letter anticommutes with $D$, so records carry charge away as a temporal balance of a $\Z_2$ charge, not a spatial Gauss law for a $\mathrm U(1)$ charge.

Handedness: one sign between two geometries

If the letters are placed in space by a fixed dictionary sending antipodal letters to opposite vectors, every report permutation acts with determinant equal to its sign, and a walk covariant under an odd permutation pairs a node of charge $q$ with one of charge $-q$ at the same quasi-energy, so every complete mirror-closed set of nodes has zero net charge. On the crystal the sign changes: the report tetrahedron transforms as the vector representation $V$ and the cycle space as $\Lambda^2V\cong V\otimes\operatorname{sign}$, so all twenty-four report permutations act as proper rotations, and isotropic charged crossings at $\Gamma$ and $R$ become possible.

The native dynamics makes Weyl particles of both hands: a walk built from the program’s contact step and private generator has an isolated node of charge $+1$, located by interval arithmetic, and its mirror image of charge $-1$, with anisotropy near 1300. In a spin-steered walk on the crystal the charged crossings carry net point charge $\pm4$ at eight quasi-energies, but the total over each level surface is the winding number $W_3$, and every native sector examined has $W_3=0$: no native net hand. The crystal behaves like a real chiral crystal, whose remaining hand is its enantiomer, and nothing native chooses the screw sense.

Light and hand in the lift

Degree one therefore holds helicities 1, $\tfrac32$ and 2 at the same eight light directions, the photon, the fiber’s spin module and the lift’s own geometric moduli, and every massless sector shares one light cone, the cone of the eight cusps; degree two holds the one-helicity partners. Light on the lift is one field with both circular polarizations at each light direction, and nothing else. How matter’s labelled fiber, $\mathbf{1}\oplus\mathbf{7}$ under the product’s relabellings, relates to the spin module is open.

The hand is native to the lift: it has no orientation-reversing isometry, its mirror image is the lift at the conjugate prime, and the lift’s hand and the spinor’s hand are one bit, the choice of prime, which the finite sky cannot see. The spin module’s quartet, $\mathbf{4}$ or $\overline{\mathbf{4}}$, is a second bit, independent of the first: the mirror lift carries the spin module to itself, the lift’s odd isometries, which exchange clocks with lines, act on it as charge conjugation, it reads the two classes of steps of Chapter XI, since on the fiber’s complex structure a step of the spinor transport has trace $(1\pm i\sqrt7)/2$ and the sign gives the class, and it is the same bit as the choice between the octonion table and its mirror, as the choice between the two prime factors of 2 among the integers of $\Q(\sqrt{-7})$ and, in the sky’s frame, as the reversal of a clock. The two bits are the sign changes of $\sqrt{-3}$ and of $\sqrt{-7}$. The first is a hand: the lift’s couplings see it, and matter’s spinor, which is the lift’s own, carries it. The second is an exact symmetry of everything native, so beyond the Cayley plane’s point the weak force’s hand needs a vacuum that breaks it and one naming, which quartet the vacuum keeps as weak, relative to which is called matter.

Where the massless fields meet

Every massless class is fixed by its data at the eight cusps, two numbers at each, one per helicity, and half of this boundary data extends, as the graph of an $8\times8$ matrix $L$, the boundary scattering matrix. For light the constant in $L$ is, up to a sixth root of unity, the cubic Jacobi sum $J(\chi,\chi)$ divided by 7, itself a prime of $\Z[\omega]$ over 7 by Gauss’s theorem: nothing transcendental enters. The fiber’s two quartets scatter with opposite amplitudes, a charge and not a chirality, since a rotation of the lift that exchanges the quartets carries the whole scattering to itself. The fiber’s constant is fixed as well, $y=(22-4\omega)/49$, so that in light’s coordinate its entries at the border are $2\sqrt{-3}$ times light’s, with the phase of the prime $3+\omega$; no rescaling by numbers of the lift’s field makes it real or imaginary, so it changes under the lift’s mirror image while the exchange of the quartets leaves it alone, the pattern, as a reading, of a phase that violates the combined symmetry of mirror and charge conjugation, not of the weak force.

At the first vertex helicities add: the fiber’s $-\tfrac32$ absorbs light’s $+1$ into the chiral octet at $-\tfrac12$, while $\tfrac32+1=\tfrac52$, equal to $-\tfrac12$ only modulo three, is forbidden by the continuous rotation. Chirality enters the first interaction as a selection rule, not a choice.

Gravity’s $-2$ meets the fiber’s $+\tfrac32$ into the same octet, meets light’s $+1$ at $-1$, and has no cubic self-coupling. Two photons entering at $e$ and $f$ multiply to $L_{ef}$ times the class of the observer $\{e,f\}$, but the product changes sign when the photons are exchanged, so a pair of identical photons has zero amplitude to merge: the observers are the channels through which massless quanta scatter between light directions, not places where two quanta fuse. For the fiber of degree one the signs reverse, as spin and statistics require of a phase-space pairing, but this does not give matter its statistics: matter’s chiral octet sits in degree two, and its one pairing invariant under the sky’s group is antisymmetric, as for every spin one-half. Each coupling is fixed only up to a constant, and comparing constants needs an energy that the lift’s topology does not supply.