tower

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

The program’s model of a register’s memory at a fixed age, words of fixed length with moves among them, extended to composites of registers. Composites of odd radius are read as matter and those of even radius as radiation.

Plate W.16Words of length three on two axes, read as binary numbers. Free moves join

2m

and

2m+1

, depth-two rewrites join 1,2 and 5,6, and the depth-three rewrite joins 3 and 4: together they join every

u

u+1

, the dyadic adding machine cut off at the word’s length.

At age $k$ it acts on all $6^k$ words of length $k$ . Besides re-anchoring, which changes the vantage, it has two kinds of move: a free move replaces the newest letter by an adjacent one, and a memory rewrite of depth $j$ , anchored at a vantage $v$ , joins $P\,A\,r(B)^{j-1}$ and $P\,B\,r(A)^{j-1}$ for adjacent letters $A$ , $B$ , where $r(x)$ is the letter on the axis of $x$ that contains $v$ . Free moves join siblings and a rewrite of depth $j$ joins histories whose common ancestor is $j$ steps back, so the walk is branchial in type, though it is not the branchial graph. Every word of the tower is the log of one register in the world of growing records, given partners at the three other reports.

Its composites are coarser observers built from crowds of registers, and a composite’s radius is its scale in the tower. A composite cannot audit which member realized a coarse move, so for odd radius each hop carries two blind sums and for even radius none: matter’s hop has modulus $\tfrac12$ , radiation’s is the bare hop, and the residue of the broken balance is the class constant.

Proposition(Memory counts in binary) proved

Take words whose letters lie on two of the three axes, write 0 for a letter on the first and 1 for a letter on the second, and read the newest letter as the lowest digit. Then every free move and every memory rewrite of the tower changes the number by exactly one. A free move of the newest letter adds or subtracts one with no carry, and a rewrite of depth $j$ adds one with a carry through $j-1$ digits. For words of length up to seven, every pair of numbers $u$ and $u+1$ is joined, except where adding one would overflow the word.

Proof

Letters on different axes are always adjacent, so a free move between the two axes flips the lowest digit. With $A$ on the first axis and $B$ on the second, the depth- $j$ rewrite joins $P\,A\,r(B)^{j-1}$ and $P\,B\,r(A)^{j-1}$ , which read $\dots0\,1^{j-1}$ and $\dots1\,0^{j-1}$ in binary, oldest digit first: a number ending in $j-1$ ones, and the number one larger. The last statement was checked exactly for all three pairs of axes.

As mathematics

A graph on the $6^k$ words of length $k$ over the six letters, with free moves, memory rewrites and re-anchorings as edges, and the generator, a Hermitian operator built from them. With positive symmetric conductances on its edges, the stationary frequency of the moves whose endpoints first differ at position $s$ , counted from the oldest letter, is proportional to the total conductance of the $12\cdot6^s$ such edges. With equal weights five moves in six touch only the newest letter, the infinite memory is an element of the six-adic integers, and rate and jump size are inverse: the level law of a hierarchical process of Vladimirov type with exponent one, which does not select a measure.

Its memory has two parts at two primes. What each record is, a report, a letter or an axis, is one digit at the prime above three, where the thirteen lines of $\mathfrak{sl}_2(\F_3)$ are four nilpotent, six split and three non-split. How records accumulate is a free group on the three axes, and on two axes a binary counter, which is 2-adic; 2 is not a place of the lift.

Its name in another field	Bridge
on two axes, the dyadic adding machine $x\mapsto x+1$ on $\Z_2$ , cut off at the word’s length	built
a hierarchical process of Vladimirov type, exponent one	type
a register’s memory at fixed age; matter at odd radius, radiation at even	a reading

A second sense

Draft one has three other towers. The covering tower of Part II (Chapter V) is the tree over the crystal over $K_4$ ; the tower of the lift’s finer levels (Chapter XIII) is its congruence quotients at $\mathfrak p^2$ and $\mathfrak p^3$ ; and the tower of scales (Chapters XII and XVIII) is the tree at the prime 7 whose every vertex is a scale with an eight-point sky of its own. The glossary sets the first two apart. In its theorems Seams calls the scale structure the scale tree and keeps the word tower for its own tower of floors.

Built from: registerletterrecord
Builds: class constant
In the volume: IThe Founding SentenceIIBlind ObserversIVWorld, Kernel, ObserverVITime as CountVIIThe Branchial TreeXXThe Commuting SquaresXXIIILight, Vacuum and HandednessXXIVA Number Nature Could Refute