The Quartet

The Fock space of an observer’s letters

At the clock $p=7$ the six letters, along the three axes, are $\gamma=(L_1,L_3,L_2,L_6,L_4,L_5)$, real antisymmetric with $\gamma_a\gamma_b+\gamma_b\gamma_a=-2\delta_{ab}$. Paired along the axes with the scalar $i$ of $\C\otimes\Oct$, $a_0=\tfrac12(\gamma_0+i\gamma_1)$, $a_1=\tfrac12(\gamma_2+i\gamma_3)$ and $a_2=\tfrac12(\gamma_4+i\gamma_5)$ satisfy the canonical anticommutation relations. Their common kernel is spanned by $(1+ie_7)/\sqrt2$, the states $\lvert n_0n_1n_2\rangle$ form a basis, and $N$ takes the values 0,1,2,3 with multiplicities 1,3,3,1.

The product of the six letters is the clock $L_7$, and $(-1)^N=iL_7=-D$. The shared octonion unit is $(\lvert000\rangle+\lvert111\rangle)/\sqrt2$, neither the vacuum nor a state of definite charge; with the clock it spans the line whose complexification is $\operatorname{span}\{\lvert000\rangle,\lvert111\rangle\}$, and that line is the lepton’s. Each letter flips the occupation of the mode on its own axis, so a commit, which writes one letter and applies it to the fiber, moves along one edge of the cube of occupation masks, changes $N$ by one and reverses $D$; this is why $(-1)^{\mathrm{age}}D$, with age the number of records written, is conserved.

One SU(4), and the fourth colour

For a real antisymmetric $6\times6$ matrix $M$, $s(M)=-\tfrac12\sum_{a<b}M_{ab}\gamma_a\gamma_b$ lifts rotations of the letter space to the fiber, and its image, the span of the fifteen bivectors, is the Lie algebra of $\Spin(6)\cong\SU(4)$, which preserves the two occupation sectors and acts irreducibly on each. The program reached this algebra three times: from a register’s changes of vantage, from its words, and from the part of the octonions’ $\Spin(7)$ that commutes with $D$. They are not one action: on words the generators act as $M$ and on the fiber as $s(M)$, with Casimirs 5 and $\tfrac{15}{4}$, and $\exp(2\pi M)=I$ while $\exp(2\pi s(M))=-I$, so the words carry $\SO(6)$ and the fiber its spin cover. Nor are the quartet’s four lines the four reports: the reports are places, and the lines are displacements with a zero.

So the Fock states read as follows: $\lvert000\rangle$, charge 0 and $B-L=-1$, the neutrino in Furey’s reading; three states of charge $\tfrac13$ and $B-L=-\tfrac13$, an anti-down triplet; three of charge $\tfrac23$ and $B-L=\tfrac13$, an up triplet; and $\lvert111\rangle$, charge 1 and $B-L=1$, a positron. The ratio $-1:\tfrac13$ is the Pati–Salam ratio. This quartet is one observer’s: its $\SU(4)$ is the spin group of the six letters orthogonal to that observer’s time, distinct from the $\Spin(6)_u$ in which the relativity group lies, and a change of clock carries one observer’s $\SU(4)$, colour group and quartets to another’s.

Records keep none of the $\SU(4)$: the letters generate $M_8(\C)$, so a fiber symmetry of the unchanged commit is a scalar. The obstruction is stronger than that. The operators commuting with colour that also reverse $D$ act only between the two singlets and vanish on the six coloured states, so no complete instrument on the unchanged fiber can have every branch both $D$-reversing and colour-neutral: a complete record-writing instrument must write colour into its outputs, keep a live colour frame, or change the chirality law.

The lepton as the zero displacement

Two facts pull against each other: colour singles out the lepton vector and fixes it, while the spinor transport fixes no line of the fiber at all. The resolution is that the transport does a different job from relabelling. At clock 7 the lift $U$ of the report transposition $1\leftrightarrow5$, $3\leftrightarrow4$ keeps $\operatorname{span}(1,e_7)$ under its relabelling part, and moves it to the colour line $\operatorname{span}(e_2,e_6)$ by its shift, right multiplication by $e_2$: the whole move is the shift. The same transport records what the order of a history does: around a decagon, a loop of zero tally, it acts on the base clock’s quartet with eigenvalues $\{1,1,\omega,\omega^2\}$, the class of the discrete colour rotation $\operatorname{diag}(1,\omega,\omega^2)$, not the colour centre, and half of the decagons move the octonion 1. What a transport that keeps the product does around a decagon is open.

The program takes the octonion product, with its unit, as physical, so the lepton is the zero displacement in every observer’s frame, the line through the shared unit in that observer’s complex structure. The double cover moves it only because it is spin, and rotations that act as spin on the four lines carrying the labels cannot leave the labels alone. In nature rotations never change lepton number or colour, because spin and internal labels sit on separate factors; the program’s matter takes the same form.

Spin on its own factor, and the gauge group

Put matter on the observer’s report qubit tensored with its fiber. The report qubit is the observer’s own spinor, and also the lift’s, with the lift’s hand: at each light direction its report state is the line the lift’s motions about that direction leave fixed (Chapter XV). On $\C^2\otimes\C^4$ a clock’s rotations act by spin on the first factor, through the binary octahedral group, and by product-preserving relabellings on the second, exactly up to the colour sign changes. The lepton’s line tensored with the report qubit is a spin-one-half doublet on which only spin acts, the commits stay covariant when the record carries the product’s signs, and across all observers the relabellings compose correctly up to colour signs. Over all observers the spin factor is the bundle of Chapter XV; the lepton doublets carry it exactly, the colour lines up to colour signs, and which of its two forms is physical is not decided.

When two observers meet, the comparison of their fibers also has its product-preserving form (Chapter XII). The record’s native comparison, a product of two left multiplications, moves the unit, and its permutation of the clocks is not a collineation. A meeting fixes its product-keeping comparison up to colour, and whatever that comparison does around a loop is pure colour, never a lepton turned into a quark; the flat comparison, unique up to colour gauge, is the gauge links’ vacuum unless they are very strongly coupled, within a scale and between scales.

The program adds continuous internal links between neighbouring observers’ frames, with a plaquette action. Their group is the Standard Model’s, $(\SU(3)\times\SU(2)\times U(1))/\Z_6$, which is the determinant-balanced group $K$ of Chapter V and what the clock leaves of the symmetry of one point of the Cayley plane (Chapter XVII). Pati and Salam’s $\SU(4)$ is not adopted as a gauge group: with the lepton fixed by every relabelling there is no symmetry between lepton and quarks to gauge or to break, and what does mix them natively is never a relabelling, but a commit’s letter, the spinor transport’s shift or the meeting comparison. Colour’s $\SU(3)$, the product’s stabilizer of the clock, is the native candidate for the links’ colour factor. $B-L$ is an exact operator on the fiber, but a commit changes $N$ by one, so whether the law conserves $B-L$ is not settled.

Weak isospin: absent, and added

The colour-only commutant contains a near miss: the Pauli matrices on the two singlets $\lvert000\rangle$ and $\lvert111\rangle$ commute with colour and form an $\SU(2)$, but it mixes a lepton of $B-L=-1$ with one of $B-L=+1$, reverses $D$ and acts on no quark. A Kramers structure gives only a choice of block, and the observer’s spin generators projected onto one quartet fail to commute with $B-L$ and with every commit. The missing resource is multiplicity. Adjoin a two-state space independent of colour and, with $P_\pm=\tfrac12(I\pm D)$, put $T_{L,j}=P_-\otimes\tfrac12\sigma_j$, $T_{R,j}=P_+\otimes\tfrac12\sigma_j$ and $Y=\tfrac12(B-L)\otimes I+T_{R,3}$. Left isospin acts on $\mathbf{4}$ and right isospin on $\overline{\mathbf{4}}$, and the sixteen states, $(\mathbf{4},\mathbf{2},\mathbf{1})\oplus(\overline{\mathbf{4}},\mathbf{1},\mathbf{2})$, decompose under $\SU(3)\times\SU(2)_L\times U(1)_Y$ as one Standard Model family, with hypercharges $\tfrac16$ for the quark doublet, $-\tfrac12$ for the lepton doublet, $-\tfrac23$ and $\tfrac13$ for the antiquarks, 1 for the positron and 0 for a neutral singlet. Here $\SU(4)$ and $\SU(2)_R$ organize the module; they are not gauge groups of the program.

All four anomaly traces vanish; for instance $\operatorname{Tr}Y=6\cdot\tfrac16+2\cdot(-\tfrac12)+3\cdot(-\tfrac23)+3\cdot\tfrac13+1+0=0$. On the upper component of each doublet $Q=\tfrac12+\tfrac12(B-L)=N/3$: Furey’s charge is the electric charge of the upper components. This is the family the forcing theorem builds by a different route, as $\Lambda^{\mathrm{even}}(W_3\oplus W_2)$. The fiber’s colour lines and the matching differences that span $W_3$ are both indexed by the three axes, and the report group permutes both through the same $S_3$, so at the level of lines the two colours are identified; as modules, identifying them is still an added step. Clifford multiplication by the weak directions, as in Furey’s construction, needs the full five-mode exterior algebra, on which pairing through the top exterior power with a weak vector as the Higgs gives exactly the terms $Qu^cH$, $LNH$, $Qd^cH^*$ and $Le^cH^*$, with no Majorana term for the singlet; none is realized by a native pulse on the fiber. The two-state space adjoined here is the one addition this chapter needs, and Chapter XVII reads it as a point of the Cayley plane.

What recording does to the native qubit

The multiplicity need not be a new carrier. At the anchor, the orientation of a two-letter word’s oldest letter, rod or antirod, is a qubit that commutes with the observer’s private dynamics: the thirty-six words factor as $\C^2_o\otimes\C^{18}$, and on fiber $\otimes$ oldest sign the qubit carries the family exactly, with the same generators. But the commit’s word factor reads it: $B_a^\dagger B_a=\lvert a\rangle\langle a\rvert_o\otimes I$, every coherence between the two signs is destroyed in the live output, and the sign leaves with the record, where No Return forbids using it again. The sign is also relative to the anchor: re-anchoring changes its convention on 24 of the 36 words, so its third component is frame data.

The fiber factor of each commit has an exact place in the Pati–Salam picture, read as kinematics of the module, not as a gauge symmetry. Every letter reverses $D$, so $W_a=\gamma_a\otimes I_2$ exchanges $\SU(2)_L$ with $\SU(2)_R$ and lies in the D-parity component of the normalizer, the discrete left–right symmetry of Kibble, Lazarides and Shafi. A single letter is not the textbook representative: it toggles one occupation, carries a lepton into a coloured state and does not reverse $B-L$; on the grading a letter is a translation, a nonzero displacement applied by an event of the record, not a relabelling. The product $C_0=\gamma_0\gamma_2\gamma_4$ of one letter from each axis is particle–hole conjugation, sending $N$ to $3-N$ and $B-L$ to $-(B-L)$, and each letter is D-parity dressed by an $\SU(4)$ rotation. The commit itself, irreversible, is not a parity.

The commit’s step on the lift completes the picture. A record is a rest frame of the lift and the next record a boost of it, so the spinor’s hand is the same on every record of a history. Every commit reverses $D$, in every sign convention for the octonion units, since each letter’s left multiplication squares to $-I$ and anticommutes with the clock’s; so the product of the hand and $D$ is not conserved, and $(-1)^{\mathrm{age}}D$ is. In the Standard Model’s all-left-handed form the left–right parity of the left–right models is D-parity composed with a reflection of space. The commit’s fiber factor is D-parity alone, acting internally at a fixed spacetime hand.

An observer’s one charge

A commit reverses the chirality at every record, and the count of records changes its parity at every record too. An observer can tell apart nothing that commutes with everything its law does (Chapter II), and applied to one register that finds the register’s charges. Give the fiber one more two-state factor holding the parity of the count, $Z=(-1)^{\mathrm{age}}$, which a commit advances by $X$. A commit at the clock $e_p$ applies $L_a\otimes X$, and the observer can read its own count’s parity, $I\otimes Z$; what it can measure is the algebra these generate.

In the left-handed form of one family the charge that is $+1$ on $(\mathbf4,\mathbf2,\mathbf1)$ and $-1$ on $(\overline{\mathbf4},\mathbf1,\mathbf2)$ is the overall fermion number $F=3B+L$, and on a register’s fiber $D=-F$. So the register’s one charge is fermion number with its sign reversed at every record: matter’s quartet, read with the parity of the observer’s count, is the only charge the observer’s own law keeps. On the octonions alone the seven left multiplications generate the same Clifford algebra with a scalar product of the seven and one sector; the count stands where the clock’s own generator stood, as the count is the observer’s proper time. It does not supply a weak doublet, since the only doublet with weak isospin’s pattern among the program’s finite groups is spin, shared by the quartet and its conjugate; nor baryon and lepton number separately, whose conservation must come from the way registers are coupled; nor a world, since superselection in the full sense needs infinitely many degrees of freedom, and nothing in the native law creates or removes a register.

Mass needs pairs

An observer’s law conserves its one charge, and something more basic: the number of registers. A commit writes its record into a fresh cell of the tape, and a cell is not a register; an exchange, a contact or a meeting takes two registers to the same two; a composite is built from registers that already exist. All of matter carries one spacetime hand, the lift’s, so a Lorentz-invariant mass, which must reach the opposite hand, is a pair term: with the number of registers fixed there is no Lorentz-invariant mass, and every pair amplitude vanishes in a state of definite register number. For one-handed matter, mass and the creation of pairs are a single structure, and the program does not have it natively. Partners of opposite charge come from an added pair term, which conserves baryon and lepton number, unique if it respects the register’s own $\SU(4)$ and two, a lepton pair and a quark pair, if only colour is respected; or from Dirac’s sea, which would rest on several additions. Neither gives the opposite hand, since a hole transforms by the inverse transpose of the spinor matrices, the same hand. The opposite hand comes from Wigner’s construction, which the records carry natively, and the program carries the pair term as a working, reversible clause. Locality does not yet force it: registers are individuals with a fixed number, and on the report lattice the prime above three blocks the needed reflection at some separations, so CPT would be an input.

Register identities are a symmetry of the law, but the sign of a swap, the statistics, is not fixed, and the program assumes Fermi statistics as a working clause; inside one register the fiber’s three modes are a genuine exclusion principle. The spinor’s own antisymmetric pairing does not decide the sign, but pair creation does: a pair made in the spin singlet exists for fermions exactly when its interior part is symmetric, and the one interior pairing that keeps every observer’s chirality is the octonion norm, which is symmetric. A pair term the same for every observer would therefore fix Fermi statistics, but under colour alone it makes the lepton and quark pair terms equal, and any dressing that treats the generations alike then leaves $(m_s/m_b)/(m_\mu/m_\tau)$ at one, where nature gives about a third. For fermions, at a given clock, exactly two pair terms respect colour and matter’s number: one treats leptons and quarks alike, and the other, baryon minus lepton number read through the clock’s chirality, weights them as one to minus a third, with Georgi and Jarlskog’s factor. Keeping both, as the masses require, ties the pair term to the observer’s clock, and then nothing forces its interior part to be symmetric, so the statistics stays a working clause. The other condition is met: registers can be the quanta of a field if their records copy each other’s state rather than name each other. Nor does the count of records supply a mass. In Feynman’s checkerboard the turns reverse a spacetime chirality; a record keeps the hand and exchanges the quartets, a flip of charge-conjugation type, and the size of a mass belongs to the pair structure’s strength, not to how often a register writes.