Part III · Observers of ObserversChapter XII
The Coxeter Graph
How do observers who keep different clocks meet?
Two registers at one clock meet through the exchange rule: when their tops are a shared rod and its antipode, each appends the other’s top, and the occurrence enters both pasts at once. Registers at different clocks have no such rule. To compare two charts at different clocks one must choose a transport; every choice is curved, and the two promotion connections and disagree on every loop. Nothing on the finite sky selects one. The lift selects a third, its parallel transport, but this chapter does not need it.
This chapter finds a relation between clocks on which the difficulty cancels. It is a classical object, the cubic graph on twenty-eight vertices that Coxeter described, and on its edges a pair of observers can exchange records by a rule that uses no transport at all, the same whichever connection one would have chosen.
(i) Call anchored observers and complementary when the triples of points outside and outside are disjoint. This relation is the Coxeter graph, cubic and distance-regular on 28 vertices with intersection array . On the celestial line two observers are adjacent exactly when their pairs are disjoint and harmonic; in the crystal read modulo seven, exactly when their report axes are orthogonal.
(ii) Adjacent observers have . When their tops are or , let each append the antipode of its own top at its own clock. This exchange is the same for every choice of transport between the charts; with an explicit completion it is trace preserving and -covariant, both appended letters are received ones, and the age-dressed chirality and No Return hold at both ends.
(iii) The Coxeter graph is the invariant, connected relation between different clocks with the fewest edges. Permission for such observers to meet is an added clause.
Status
Parts (i) and (iii) are exact finite geometry, and the graph itself is classical. Part (ii) is an exact construction whose ingredients are all native: the appends, the records, the writer’s parity and the dictionaries. The one thing it adds is the domain on which it acts. The program’s exchange rule authorizes meetings at equal clocks only, and mathematical canonicity does not authorize an occurrence; so the chapter proves that a clean cross-clock meeting exists and where it lives, and it does not prove that the world already contains it.
In the lift the observers are geodesic edges of a hyperbolic three-manifold. Whether the edges of Coxeter neighbours actually cross there, at right angles, has not been checked.
Observers as triangles
An anchored observer is a clock and a vantage line not through it. The four points of leave three over,
and these three are never collinear. Conversely every non-collinear triple arises from exactly one observer, because its complement contains exactly one line, so the 28 observers are the 28 non-collinear triples of the Fano plane. Joining two triples when they are disjoint gives the graph Coxeter called “my graph”: equivalently, the graph on the 3-subsets of a 7-set, joined when disjoint, with the seven lines of one Fano plane removed.
The neighbours of the base observer
For the triple is . A neighbour’s triple must lie in and must not be the line 246, so it is , or ; their complements contain the lines 356, 347 and 257, with leftover points 7, 5 and 3. The three neighbours are , and , with celestial pairs , and .
In each case and , and the shared point of the two vantage lines is the sum of the clocks, a point being a vector of added digit by digit:
The complementarity graph
The distance is decided by incidence alone. For and with and , ask whether each clock lies on the other’s vantage line:
If or , but not both, the distance is four: the three other observers of the same clock and the three other observers of the same vantage line. Complementarity is not the cost of a move. Between distinct observers the program’s elementary moves, re-anchoring and a change of clock along a fixed vantage line, are exactly the distance-four relation, which as a graph of its own has shells 1,6,15,6 and is not distance-regular; neither metric is a physical distance between registers.
The complementarity relation on the 28 anchored observers is the Coxeter graph. It is cubic, with 42 edges, girth seven and diameter four, and distance-regular with intersection array ; from any observer the numbers at distances 0,1,2,3,4 are 1,3,6,12,6. Its automorphism group is , which acts distance-transitively; the relativity group has seven orbits on ordered pairs, of sizes 1,3,3,3,6,6,6 from a given observer, each inside one distance class.
Distance as a cross-ratio
For disjoint pairs and the cross-ratio is defined up to , and over it lies in one of the classes , , ; with it is simply . The neighbours give , and , all ; the six pairs at distance two give , the six at distance four , and the distance is three when the pairs share a point. Built from cross-ratios alone, with no chart, the graph “disjoint and harmonic” is the Coxeter graph.
On the complex celestial sphere two pairs with cross-ratio are the endpoints of two geodesics of hyperbolic space crossing at right angles, exactly when their trace-free matrices are orthogonal for the determinant form. Modulo seven those matrices are the observers’ report axes in the crystal, so Coxeter adjacency is orthogonality of report axes in space read at seven. There the fifty-seven lines of are eight nilpotent ones (the sky points), twenty-eight split ones (the observers) and twenty-one non-split ones (the half-turn axes), as over the thirteen lines are one clock’s four reports, six letters and three axes.
Promotion is a branching at distance three
Clock promotion takes each observer to twelve targets, all at Coxeter distance three, two for each new clock, and on the celestial line the two targets for one new clock keep opposite ends of the observer’s pair. A native promotion cannot choose between them: the elements of that fix the promoted letter form a group of order two whose nontrivial element exchanges the two targets.
So a clock change is a two-way branching wherever it occurs. Made into a complete quantum operation from the translations between charts and the fiber transports that keep the octonion unit and each clock’s unit, and so carry lepton plane to lepton plane, its reversible covariant weightings form one line, with , and neither trace preservation, covariance, parity, Kramers compatibility nor No Return fixes . If the permanent record of which map was used must reveal nothing about the vantage the observer started from, the four orbits of maps carry equal weight and : a selection by a principle about the record, valid within this family only. Counting every update equally gives the same weights, and that counting is itself a choice. Two further facts need clauses the law does not contain: a promotion counted as a commit changes the recorder’s stationary state at every positive promotion rate, so only an uncounted change of frame leaves it alone; and two registers promoted together by one map, an added synchronization, have a unique fixed state on their 84 pair sectors, with weight at each clock.
From , promoting the rod 2 leads to or . The element fixes the clock 1, the vantage 246 and the rod 2, and maps the line 347 to 356, so it exchanges the two targets. On the celestial line it is , which exchanges and . A rule that picked one target from the data alone would have to be invariant under this element, and it cannot be.
A meeting that needs no comparison
Across clocks a literal exchange cannot be mutual reception: for register at clock with top receiving , and at clock with top receiving , reception would need and , which together give . Some translation between the charts is needed, and translations are exactly what the curvature theorem makes ambiguous. On Coxeter edges the ambiguity disappears: for adjacent and , with , all four connections and both routes through an intermediate observer give the same readiness condition,
The exchange touches only the letters of each register’s own clock, so neither record needs to name a translation; its covariance was checked on all 168 group elements and all 84 ordered edges. Completed by a null branch, or by the recorder’s own menu, it discards no outcome, writes one letter at each register and keeps at both clocks, whose ages need not agree. At distance three a translated exchange can be built from either connection, and the two versions differ; Coxeter edges cannot tell the connections apart, and that is why they need neither.
Take at and at , so . At clock 1 the antipode of is , at clock 7 it is , and the two ready configurations act as
where is the older letter, exported to the record. Every appended letter is the antipode of its receiver’s own top, so each is a received letter; the occurrence exchanges the two ready configurations, and neither clock changes.
The meeting with the product kept
The fibers are carried across a meeting by operators of another kind than a comparison of charts, with the shared rod and left multiplication by . Each permutes the clocks by a three-cycle on the line that fixes the other four, which no collineation does, and each moves the octonion unit and keeps no lepton plane. Part IV takes the octonion product with its unit as physical, so matter’s labels must be compared across a meeting by a relabelling, an automorphism of the octonions that permutes the units with signs, and by one that carries the lepton plane of one observer, spanned by 1 and the unit of its clock, onto the lepton plane of the other. With the gauge links of Chapter I the colour part of that comparison is a link field on this graph, whose plaquettes are its twenty-four heptagons. Wilson’s plaquette action charges a heptagon , where is the colour part of its holonomy, so the flat class is the classical vacuum.
The matter alone would choose differently. Fill the lower band of matter’s colour hopping between the observers, one register to a mode, which is Fermi statistics, assumed for registers rather than derived; this is a band of a hopping between rest frames, not a sea of negative energies. Its energy on the flat class is in units of the hopping, colour flux through the heptagons lowers it to at the lowest found, and Lieb’s principle settles nothing here. Because the graph’s shortest cycles have length seven, every background gives matter’s colour hopping the same first six spectral moments, and the seventh is , with the total Wilson cost: the plaquette action is the seventh moment of the matter’s own hopping. Bounding the filled-state energy by its moments, the flat class is the vacuum whenever the plaquette weight exceeds 3.70 in units of the hopping, so that with the weight written every coupling with below 1.6 lies above the bound, while the most favourable flux found breaks even at 0.047. So the meeting’s vacuum is flat unless the links are coupled very strongly.
Let and be neighbours, with .
(i) Up to sign, carries the complex structure of the clock to that of , that of to , and that of to , and it carries the octavian order of to that of , ‘s to ‘s and ‘s to ‘s. Every relabelling moves clocks and octavian orders alike, so none agrees with on both: this is the exact sense in which does not keep the product.
(ii) A relabelling that carries ‘s lepton plane onto ‘s carries to , and up to a colour transformation at there is only one. Around any closed path such comparisons compose to an element of colour at the path’s first clock: whatever they do around a loop is pure colour, and never turns a lepton into a quark.
(iii) The comparisons fixed by symmetry alone are not reversible. Exactly four of them carry to , the two exchanges of and and the two clock changes along and along , and for each, going there and back gives the meeting’s own involution, which fixes every point of the line and acts on both observers’ fibers as a colour transformation.
(iv) Comparisons that are flat, composing to the identity around every heptagon of the graph, exist. Any two differ only by a colour transformation at each observer, so up to colour gauge there is one.
Item (i) is a computation on all 84 ordered edges. For (ii), a relabelling carries to , so carrying onto forces . Two such relabellings differ by one that fixes , which lies in colour at , and along a closed path each step carries the current clock’s unit to the next, so the composite fixes the first clock’s unit. In (iv) a flat comparison is a change of frame, a relabelling at each observer from one reference, because the graph with its heptagons as faces is simply connected: a comparison trivial around every heptagon is trivial around every closed path. Two frames that both carry the reference lepton plane to each observer’s differ at each observer by a relabelling that fixes its clock, which is colour. Item (iii) and the existence in (iv) were checked by exhaustive computation.
Why this graph
The invariant symmetric relations between observers with different clocks are unions of ‘s orbits on pairs. There are two orbits of 42 edges: the Coxeter graph, which is connected, and the relation of sharing a vantage line, which falls apart into seven copies of . Distance two has 84 edges, and the two directed orbits at distance three are inverse to each other, so a symmetric relation must take both, with 168 edges. The Coxeter graph is the connected invariant relation between different clocks with the fewest edges, and its 42 edges lie two over each of the 21 pairs of clocks, so every clock meets every other.
The minimality is a finite classification; adopting “the least connected invariant relation” as a principle would be a new clause of its own. What is added is permission: the statement that two registers whose observer types are Coxeter neighbours may meet. The law as it stands has no such statement, and the chapter does not derive one. If it is granted, the pair recorder transfers, all seven clock worlds join into one occurrence structure, and the meetings are blind to the choice between , and the lift’s transport alike.
A meeting is a world event, one occurrence in two pasts, and the Coxeter graph says which observer types can share one without first agreeing on a chart. It is a graph of types, not of registers and not of physical space, and a change of clock is not a Lorentz boost. On the finite sky it is the harmonic relation between pairs of points, in the Fano plane the complementarity of triples, and in space read at seven the orthogonality of report axes. Seams finds the same graph intrinsic to the rigid object of size twenty-eight: an orbital graph, which every seam carries from one incarnation to another. In the lift an observer has a second role: the channel through which a massless quantum scatters between the two light directions of its pair, not a place where two quanta fuse. Whether those channels have anything to do with Coxeter meetings has not been examined.
Neighbouring scales are not separate cells. The seven observers through one light direction are seen as a single observer by a frame one scale deeper, two neighbouring frames share forty-nine observers, and the shortest round trips across scales, the 882 star squares of five meetings, are the plaquettes between scales: the fifth spectral moment of matter’s hopping is . With Wilson’s one weight the vacuum is flat between scales for every below 0.67, though matter alone again prefers flux, so the tower of scales is one gauge theory. It is not a renormalization ladder: each frame keeps its own bare coupling and its seven finer neighbours attach in parallel, so integrating them out only stiffens the coarser frame and the flow stops after one level, and on the crystal inside each meeting every heptagon is copied whole times, so the coupling does not run there either. The program’s positions, the report counts made of an even number of reports, do tile their loops, but colour’s frames belong to observers, which are classes of rest frames, so colour lives on states of motion, where physics stood before Yang and Mills, and runs on positions only if a colour frame is added at every position; given that addition the sign would be asymptotic freedom, every mesh picks a rest frame, and nothing native fixes the weights that would set the colour field’s speed. Across scales every loop returns each observer’s matter as matter, changed only by colour, and no charge conjugation remains. An arrow of scale, aligning each finer frame’s octonion table so that its light direction points back to the coarser scale, lets one table serve a whole step of scale; it can always be chosen, and it is a convention of bookkeeping that costs nothing.