Part V · Dynamics and LimitsChapter XX

The Commuting Squares

Plate XX.1One graph and its covers: the tree of ordered histories, the crystal of tallies, the kernel graph, the seven clocks on the sky, and the lift below.

XX.1
XX.2
XX.3
XX.4
XX.5
XX.6

Do the levels agree when a history is evolved and then described?

The program has proved many exact things about separate constructions: a crystal built from report changes, a finite celestial geometry of observers, a record law with interference, a recorder with a shared equilibrium. One question decides whether these describe one world: whether a single growing, sequential world, with its kernel retained, can carry the observer structure, the spatial quotient of histories, branchial interference and rulial changes of description as compatible constructions. Compatibility is more than co-presence. Evolving a history and then describing it should agree with describing it and then evolving the description, exactly or to a stated extent, and a failure names the retained state the description needs.

Parts II and III supply a skeleton on which the question becomes concrete. One graph, the kernel graph $K_4$ , lies under every level, and each level is a cover of it or a relation among its copies. On that skeleton the question splits into three squares, one for space, one for interference and one for changes of chart; a fourth square, on the lift, commutes exactly and decides where the massless fields couple.

The central result

Compatibility of the levels is three squares over the kernel graph. The chart square commutes exactly for the program’s word rules: every clock dictionary intertwines them, and a closed loop of clock changes returns each history up to a permutation of the three rods, with all of $S_3$ occurring. The tally square commutes for amplitudes but not for states: a sharp crystal address changes the finite register’s statistics at order $t^4$ , and the permanent record does not fix the address, so space needs one retention clause. Under that clause the kernel square holds in one direction: two alternatives interfere only if their permanent receipts agree and their difference loop lies in the commutator subgroup of $\pi_1(K_4)$ , on which the transposition transport, whose structure the lift’s parallel transport realizes, acts through the cyclic group of order three. The converse is untested.

Status

The chart square is exact for the word rules, and passive: it re-describes histories and says nothing about an occurrence that changes one register’s clock while its partners keep theirs. The tally square commutes on amplitudes and fails on states at order $t^4$ ; the retention clause that keeps the crystal is an addition. The kernel square is proved in the direction that forbids interference, conditional on that clause, and open in the direction that would produce it; the only instance computed is the trivial loop.

The square on the lift commutes by the naturality of the cup product, a theorem of topology, and restriction to the light directions is injective for the lift’s massless fields in the degrees used. It decides which couplings exist, not how strong they are.

One graph, four covers

Plate XX.1One graph and its covers: the tree of ordered histories, the crystal of tallies, the kernel graph, the seven clocks on the sky, and the lift below.

Fix one clock: its four reports are the vertices of $K_4$ , its six letters the edges, and a history of frame changes is a path. The branchial level keeps the whole ordered path, up to backtracks: its home is the universal cover, the trivalent tree, with deck group $\pi_1(K_4)$ , free of rank three. The spatial level keeps only the signed tally: Sunada’s $K_4$ crystal, with periods $H_1(K_4;\Z)\cong\Z^3$ . The kernel of the tally map, the commutator subgroup, is the set of loops the tally forgets. The rulial level relates the seven clocks’ copies by $\PSL(2,7)$ . Time is not among these levels; it is the count of committed occurrences.

When $\delta$ is invertible the square always commutes, and the content is whether the evolution $U'$ is again a rule of the program. When $\delta$ forgets, $U'$ exists only if $U$ never separates what $\delta$ identifies, which for Markov chains is the lumpability condition of Kemeny and Snell; its failure means the description depends on something it has dropped.

Definition(a compatibility square)

Let $U$ be an evolution on a space of histories $\mathcal{H}$ and $\delta\colon\mathcal{H}\to\mathcal{H}'$ a description. The square for $(U,\delta)$ commutes if there is an evolution $U'$ on $\mathcal{H}'$ with $\delta\circ U=U'\circ\delta$ , exactly or with a stated error.

The chart square

Plate XX.2The chart square: relabelling by a collineation commutes with evolution, and only a loop of clock changes leaves a mismatch in

S_3

The square commutes exactly: re-describing a history at another clock and then evolving it agrees with evolving it and then re-describing. What does not vanish is the mismatch around a loop. A composite of three promotions returning an observer to itself lies in its stabilizer $S_3$ , and over the $2{,}016$ rooted three-clock triangles each promotion assignment generates all of $S_3$ : the curvature of Chapter XI seen from the histories.

For the tally the combinatorial half is formal, since a graph isomorphism lifts to the maximal abelian covers uniquely up to a deck translation, and the loop mismatch then acts on the crystal’s periods as a proper rotation. What has not been checked is the operational square: an active change of clock for a register that carries a retained address.

Theorem(the finite principle of relativity for the word rules) proved

Let $U_g$ relabel every letter of a word by $g$ . For every $g\in\GL(3,2)$ and every anchored observer, the private generator $H$ , the contact generator $K$ and the depth-two writer $D$ satisfy

\begin{gathered}H_{gp,gL}\,U_g=U_g\,H_{p,L},\qquad D_{gp,gL}\,U_g=(R_g\otimes U_g)\,D_{p,L},\\K_{gp,gL,gM}\,(U_g\otimes U_g)=(U_g\otimes U_g)\,K_{p,L,M},\end{gathered}

where $R_g$ relabels the fresh departure. The exchange rule and re-anchoring are equivariant in the same way. Consequently every finite history of fixed-clock occurrences at clock $p$ maps to a history at clock $gp$ , with its event identities, causal order and read records preserved and every letter-valued field translated by $g$ .

The tally square

Plate XX.3In

K_4

the direct route

0\to2

and the detour

0\to1\to2

end at one report; in the crystal they end in cells differing by the period

c_1

Follow one register with state $\lvert v,R,w,\xi\rangle$ : a report, an address $R\in\Z^3$ , a word and a fiber vector. Lifted to the crystal, the generator uses the program’s coefficients unchanged: hops and memory rewrites leave the address alone, and a re-anchoring translates it by a voltage. Summing over the address absorbs every translation, so $\Sigma H_{\mathrm{lift}}=H\Sigma$ and amplitudes descend exactly. But the finite probability is $\lvert\sum_RA_R\rvert^2$ and the lifted one $\sum_R\lvert A_R\rvert^2$ , and they differ by the cross terms between alternatives with different tallies.

The permanent record does not fix the address: under the working record a solitary commit records no vantage, and filling the triangles of $K_4$ kills $H_1$ without deleting any history. The crystal is kept only by an added clause: retain the abelian circulation of the live vantage path as an unbounded coherent coordinate, incremented reversibly by re-anchoring and unchanged by commits. Composites, whose own vantage graph is $K_3$ , inherit no three-dimensional crystal without a binding map.

Proposition(states do not descend) computed

Start one register at report 0 with top 01 and a sharp address, and evolve by the depth-one generator at $h=\rho=1$ , normalized by 7. The lifted and finite probabilities first differ at order $t^4$ : for the return state, $P_{\mathrm{lift}}(0,01)-P_{\mathrm{base}}(0,01)=-t^4/4802+O(t^6)$ , and a solitary commit written in the moving frame already sees the change, its letter outcomes 01 and 23 shifting by $\mp t^4/7203+O(t^6)$ . The finite statistics are recovered at zero Bloch momentum, or from a translation-uniform preparation on a finite periodic quotient.

The kernel square

Plate XX.4The kernel square as a map of extensions: the loops space forgets are sent onto the cyclic group of order three.

The two-layer law says that alternatives differing only in content may reconverge and interfere, and alternatives differing in recorded structure never do: a cross term carries the overlap of the two records. In a fixed vantage two words exchanged by a memory rewrite evolve under $-X$ ; the pulse $U=(I+iX)/\sqrt2$ gives $U^2=iX$ , so a register never returns to its word, while recording the intermediate axis makes the return probability $\tfrac12$ . With the tally retained, a cross term between two histories from one sharp address also needs equal windings, and two paths reach the same crystal vertex exactly when their difference loop lies in the commutator subgroup.

The other covariant choice, the Klein four-group transport, is flat, and covariance on the kernel graph alone does not choose; the lift does, its parallel transport having the transposition transport’s structure. Under the retention clause and a sharp address, two alternatives contribute a cross term only if their permanent receipts agree and their difference loop lies in $[\pi_1(K_4),\pi_1(K_4)]$ , where the transports differ by an element of $A_3$ . The converse, that native alternatives differing by a nontrivial commutator loop, the shortest a decagon, actually interfere, is the open half.

Proposition(the kernel’s flavour holonomy) proved

Let $\varphi\colon\pi_1(K_4)\to S_3$ be the flavour transport, the covariant choice in which each rod of an axis acts as the transposition of the other two axes. Then $\varphi$ is onto; its sign factors through the tally $\pi_1\to H_1(K_4;\Z)\cong\Z^3$ ; and $\varphi$ maps the kernel of the tally, $[\pi_1,\pi_1]$ , onto $A_3\cong\Z/3$ . The commutator of two distinct triangles maps to a three-cycle.

Proof

The three triangles through a vertex map to the three transpositions, which generate $S_3$ . A homomorphism carries commutators onto commutators, so $\varphi([\pi_1,\pi_1])=[S_3,S_3]=A_3$ . The sign of $\varphi$ is a homomorphism to an abelian group and so factors through the abelianization $H_1$ .

A square on the lift

Plate XX.5The square on the lift: couple two massless fields and restrict to the light directions, or restrict and couple there; the right-hand restriction is injective.

The lift supplies one more square, and it commutes exactly. Its description is restriction to the sky: a massless field of the lift is described by its data at the eight light directions, the only places where observers receive it. Its evolution is a coupling: two massless fields multiply into a field of one degree higher, by the cup product followed by the map that combines their values. Multiplying and then restricting agrees with restricting and then multiplying, because restriction respects products; this is the naturality of the cup product, and the evolution on the side of the description is again a natural one, the cup product on each cusp torus.

For the lift’s massless fields the description also forgets nothing: restriction is injective in degree one, and in degree two for light, for the fiber’s Clifford module and for the helicity-two sector. So every coupling among them is decided at the light directions. What the square does not decide is strength: each coupling is fixed only up to a constant, and comparing constants needs an energy that topology does not supply.

What the squares decide

Plate XX.6The chart square commutes exactly; the tally square commutes on amplitudes and fails on states at order

t^4

The rulial square commutes, and its curvature is its whole content: one rule family, seven charts, an $S_3$ mismatch around every loop. The spatial square commutes where it cannot see the address and fails where it can, and the failure names the one clause space needs. The kernel square is proved in the direction that forbids interference and open in the other. The squares also sort the three relativities: between the rulial and the spatial there is curvature, the Thomas–Wigner rotation carried by the loops space forgets; between the branchial and the spatial only a forgetting map, which is flat; and at matter’s mass the branchial and the rulial meet through a modulus, not a curvature.

Results move between carriers only through explicit bridges that preserve what they need, and four are known to fail or be missing: the composites’ critical ratio $\sqrt2/2$ is not critical for the base register on the crystal; the composites’ vantage graph is $K_3$ , whose maximal abelian cover is a line; the recorder’s equilibrium, shared passively by every observer, changes when promotions are committed; and the sequential limit is proved for one restricted recorder, not for the all-observer world.

Open questionopen

Is there a native process in which alternatives differing by a nontrivial commutator loop of $K_4$ interfere, with the cross term weighted by the order-three holonomy of the flavour transport, while the retained address and the permanent receipts stay common to both? A positive answer would close the kernel square; a negative one would show that interference and holonomy, though located in the same subgroup, do not meet in the program’s dynamics.

In the volume’s architecture the squares are the joints between world, kernel and observers: the chart square joins observers to observers, the tally square joins the world’s histories to space, the kernel square joins what space forgets to what can interfere, and the square on the lift joins the massless fields to the observers who receive them. Space is not yet a function of the kernel; the active clock change for a register with an address is the next square to draw.

A limit taken of one description, on a family where its square fails, is not a limit of the world. The next chapter asks what survives when any of these is taken to infinite size.

Concepts: bridgestatuscoherence
Objects: the twenty-eightthe seven pointsthe sky