report

Part I, The Sentence and the Kernel · defined in Chapter I, The Founding Sentence

One of the four classes of past that the smallest register whose records matter can tell apart. On the observer’s Minkowski space it is a light-like direction.

Plate W.11The four reports as a regular tetrahedron of null directions on the Bloch sphere of the report qubit, its celestial sphere; the midpoints of the edges, the pair sums less

I/2

, are the six vertices of an octahedron.

The class of a word $w$ is the pair of the blind bit $s(w)=(-1)^{\#b(w)}$ and the consequence bit $\ell(w)$ , which is $-1$ exactly when $w$ ends in $b$ ; the register reads its own classes, $s$ by the empty continuation and $\ell$ by the probe ratio. The permutations of the reports that respect the sign and the probe ratio form $D_4$ , and the dressing $s\mapsto s\ell$ of the third clause completes them to the report group $S_4$ . On the Fano plane, relative to a clock $p$ , the reports are the four lines missing $p$ .

Give the reports vectors $e_0,\dots,e_3$ . The $S_4$ -invariant forms $\alpha(\sum x_i)^2+\beta\sum x_i^2$ are Lorentzian for $-4\alpha<\beta<0$ , and the reports are null exactly at $\beta=-\alpha$ , where the alphabet’s time and space units coincide. That point is concrete: the reports of a qubit’s tetrahedral measurement, $E_i=\tfrac14(I+r_i\cdot\sigma)$ , are pure effects, null for the determinant, and their sum $I$ is the observer’s time. Chapter V adds that the reports are places, permuted transitively, none of them an origin.

Proposition(Four reports) proved

The histories of the minimal consequential register fall into exactly four future-indistinguishability classes. The class of a word $w$ is the pair $(s(w),\ell(w))$ , where $s(w)=(-1)^{\#b(w)}$ and $\ell(w)=-1$ exactly when $w$ ends in $b$ . The register reads its own classes: the empty continuation reads $s$ , and the probe ratio $s(wp)/s(w)$ reads $\ell$ .

Proof

The rules act on $(s,\ell)$ alone, so words with equal pairs give equal readings after every continuation, by induction on its length: there are at most four classes. All four pairs occur; the words $a$ , $b$ , $ba$ and $bb$ reach $(+,+)$ , $(-,-)$ , $(-,+)$ and $(+,-)$ . Words with different signs are separated by the empty continuation, and words with equal signs and different switches by $p$ , which reads $s\ell$ . Finally $s(wp)/s(w)=s\ell/s=\ell$ .

Proposition(The report form) proved

The $S_4$ -invariant quadratic forms on $\R^4$ are $Q_{\alpha,\beta}(x)=\alpha(\sum x_i)^2+\beta\sum x_i^2$ , with eigenvalue $4\alpha+\beta$ on the line of $\sum e_i$ and $\beta$ on the three-dimensional space of report differences. For $\alpha>0$ the form is Lorentzian, of signature $(1,3)$ , exactly when $-4\alpha<\beta<0$ . The reports are null exactly when $\beta=-\alpha$ , the unique member with $Q(e_i+e_j)=-Q(e_i-e_j)$ .

Proof

The permutation module is the trivial line plus the irreducible standard representation, so an invariant form is a scalar on each, read off from the matrix $\alpha J+\beta I$ . Then $Q(e_i)=\alpha+\beta$ , $Q(e_i+e_j)=4\alpha+2\beta$ and $Q(e_i-e_j)=2\beta$ .

As mathematics

With the clock $c$ fixed, a report is a line of $\mathrm{PG}(2,2)$ missing $c$ , a vertex of the complete quadrilateral $K_4$ ; as a line of the plane it belongs to the object of the seven lines, with stabilizer $S_4^b$ . The vertices of $K_4$ , the four points of $\Proj^1(\F_3)$ and the four effects of the qubit’s symmetric informationally complete measurement are incarnations of one rigid object of $A_4$ , and the effects become four null vectors $n_k=(1,r_k)$ with $\sum_kn_k=(4,0,0,0)$ and Minkowski products $n_j\cdot n_k=\tfrac43$ for $j\ne k$ .

In the report lattice the reports of the base tetrahedron are the primitive null matrices $N_a=\xi_a\xi_a^\dagger$ for the cusps $\infty$ , 0, 1, $1+\omega$ . They form a $\Z$ -basis of $\mathrm{Herm}_2(\Z[\omega])$ , and $\det\bigl(\sum_ax_aN_a\bigr)=\tfrac12\bigl[(\sum_ax_a)^2-\sum_ax_a^2\bigr]$ is the null report form.

Its name in another field	Bridge
a Myhill–Nerode class of the minimal register	built
a vertex of $K_4$	built
a point of $\mathrm{AG}(2,\F_2)$ , with $S_4\cong\mathrm{AGL}(2,\F_2)$	built
a point of $\Proj^1(\F_3)$ , with $S_4\cong\PGL(2,3)$	built
an effect $\tfrac14(I+r\cdot\sigma)$ of the qubit’s tetrahedral measurement	built
a primitive null vector $N_a$ of $\mathrm{Herm}_2(\Z[\omega])$	built
a light-like direction	a reading