branchial

Part II, One Graph, Four Covers · defined in Chapter VII, The Branchial Tree

Wolfram’s word for the level of alternative histories. Here it is the tree of ordered histories.

Plate W.15The tree of reduced histories from report 0 to depth three; each vertex shows the report reached. Two histories that differ once stay different.

A history of frame changes is a walk on $K_4$ , and the reduced walks, which never retrace a letter at once, are the vertices of the universal cover of $K_4$ , the 3-regular tree, with deck group $\pi_1(K_4)\cong F_3$ . Two histories that differ once stay different. Within one observer the depth of the last common ancestor is an ultrametric, and Wolfram’s branchial graph is its first layer: at a fixed depth it joins alternatives with an immediate common predecessor. The unweighted graph forgets the report incidence that the quantum overlaps keep, so branchial adjacency is a statement about ancestry, not about distinguishability. Between observers the ultrametric fails, because causal pasts merge: four registers on a path already violate it.

Blindness enters at this level as the branchial clause B of Chapter II: an occurrence the observer cannot audit is one occurrence over every history consistent with its records, and its amplitude is the uniform sum over those histories, the siblings the records cannot tell apart. The program types its memory walk, its internal clock operators, its masses and the class constant as branchial. The memory walk is branchial in type, a sparse graph organised by depth of shared past, and it is not the branchial graph.

Definition(Histories, tree, branchial distance)

A history from report 0 is a walk $0=v_0,v_1,\dots,v_n$ on $K_4$ . It is reduced if $v_{i+1}\ne v_{i-1}$ for every $i$ . The reduced histories are the vertices of the tree $T$ , with an edge from each history to its one-step extensions. For two histories $h,h'$ of equal length, write $h\wedge h'$ for their longest common prefix, and set $d(h,h')=\beta^{-|h\wedge h'|}$ for $h\ne h'$ and $d(h,h)=0$ , for a fixed $\beta>1$ .

Proposition(Prefixes nest) proved

The branchial distance is an ultrametric: $d(h,h'')\le\max\{d(h,h'),\,d(h',h'')\}$ .

Proof

If $h$ and $h'$ agree for $a$ steps and $h'$ and $h''$ agree for $b$ steps, then $h$ and $h''$ agree for at least $\min\{a,b\}$ steps, because both agreements are prefixes of $h'$ and prefixes of one word are nested.

As mathematics

The universal cover $T$ of $K_4$ : the 3-regular tree, with $3\cdot2^{n-1}$ reduced histories of length $n\ge1$ from a report, on which $F_3=\pi_1(K_4,0)$ , freely generated by the three triangle loops through 0, acts by prefixing and reducing. The covering $T\to X$ onto the crystal has deck group $[F_3,F_3]$ , so two histories from one report have the same tally exactly when they differ by a commutator. The tally separates all histories of length at most four; the first it identifies have length five, and each such pair closes a decagon of the crystal.

Its name in another field	Bridge
the 3-regular tree, the universal cover of $K_4$	classical
Wolfram’s branchial graph	name only
an ultrametric space of histories	type

Built from: kernel graphanchor
In the volume: IIBlind ObserversIVWorld, Kernel, ObserverVFour Reports, Six LettersVIIThe Branchial TreeXIRulial RelativityXXThe Commuting SquaresXXIVA Number Nature Could Refute