arena

Part I, The Sentence and the Kernel · defined in Chapter IV, World, Kernel, Observer

The positions a register reaches when each record adds its letter’s rest frame to a running sum of report counts. It is the crystal’s lattice read as positions.

Plate W.14One body-centred cubic lattice, read twice: as report counts, the letters’ steps

\pm2e_i

at the cube corners and the single reports at the centres; as the crystal’s periods, square loops at the corners and triangles at the centres.

Chapter IV builds it as a reading, not a clause. If each record adds its letter’s rest frame to the register’s report count, the register traces a worldline of unit steps of proper time, one per record. In the frame in which the four reports are symmetric every letter has the same time component, so coordinate time is $\sqrt{3/2}$ times the count for every register, and the letters’ spatial parts are $\pm1/\sqrt2$ along three orthogonal axes: a register walks on a cubic lattice at $1/\sqrt3$ of the speed of light. The arena is flat and has a preferred frame. The lift’s transport does not bend it: carrying each letter’s frame to the next by the lift’s spinor transport develops any path into exactly its report count.

It looked like a rival to the crystal of tallies, since the report group acts on the arena with mirrors and on the crystal’s periods by rotations. In the lift the two are one lattice read in two ways, as positions and as circulations, and the Hodge star is the only isometry between the readings that commutes with all twenty-four relabellings of the reports, up to one sign. The conclusion is conditional on the lift’s symmetries being the symmetries of space, which the axioms do not yet record.

Proposition(One lattice, two readings) proved; items 2 and 3 computed

(1) The spatial parts of the report counts form a body-centred cubic lattice. A count made of an even number of reports lies on the cubic lattice of corners, where the letters’ steps are; a count made of an odd number lies at the cube centres, where the single reports are. The crystal’s periods form the same lattice, with its square loops at the corners and its triangles at the centres.

(2) The Hodge star induces an isometry between the two lattices. It sends each report to the triangle opposite it and each letter’s step to a square loop of the crystal.

(3) On the report qubit’s own symmetry the isometry commutes with relabellings of the reports only up to the sign of the permutation. In the lift a clock’s relabellings act on the report counts by proper rotations, the isometry commutes with all twenty-four, and by Schur’s lemma it is the only such isometry, up to the one sign that the orientation convention fixes.

Proof

For (1): in suitable units the reports’ spatial parts are four alternate corners of a cube, $(1,1,1)$ , $(1,-1,-1)$ , $(-1,1,-1)$ and $(-1,-1,1)$ . Their integer combinations are the integer triples whose three coordinates have the same parity, odd exactly when the number of reports used is odd, and two reports sum to a letter’s step $\pm2e_i$ . The crystal’s periods are the same set: the square $0\to1\to2\to3\to0$ has period $(0,-2,0)$ , and the four triangle periods are the odd points $(\pm1,\pm1,\pm1)$ . Items (2) and (3) are exact finite computations over the twenty-four relabellings.

As mathematics

The even sublattice $\Lambda_{\mathrm{even}}=\{\sum n_aN_a:\sum n_a\ \text{even}\}$ of the report lattice $\Lambda=\mathrm{Herm}_2(\Z[\omega])$ , of index 2, with the six records $T_x=N_a+N_b$ as bonds. Each record has $\det T_x=1$ . In the rest frame of the centre $D=\sum_aN_a$ every record has time component $\sqrt{3/2}$ and a spatial part $S_x$ with $\langle S_x,S_x\rangle=-\tfrac12$ , $S_{x^c}=-S_x$ for the complementary pair, and $\langle S_x,S_y\rangle=0$ otherwise.

The arena subdivides: $b\Lambda_{\mathrm{even}}$ has index $b^4$ , and a coarse bond $bT_x$ is a sum of $b$ records in exactly one way. But every nonzero vector of $\Lambda$ has an infinite orbit under $\SL(2,\Z[\omega])$ , so every mesh on it selects a rest frame, and for the reports of the base tetrahedron the stabilizer is the binary tetrahedral group $2T$ . Its orbits on the twenty-eight observers have sizes 4,6,6,12, so no map from the arena to the observers is both translation-invariant and $2T$ -equivariant: positions carry no observers’ frames.

Its name in another field	Bridge
$\Lambda_{\mathrm{even}}\subset\mathrm{Herm}_2(\Z[\omega])$ , with the records as bonds	built
the period lattice $H_1(K_4;\Z)$ of the $K_4$ crystal, through the Hodge star	built
positions	a reading

Built from: recordreport countscrystal
In the volume: IVWorld, Kernel, ObserverVIIISpace as a TallyXIIThe Coxeter GraphXXIIOne SpeedXXIVA Number Nature Could Refute