The sequential-circuit reading of self-similar groups — latch, register, carry, counter — and why "which types exist" is a 3×3 grid, not a line
Why this note exists. The owner heard "reads the first bit, acts on the rest" and said that's a latch — and it is, exactly, not by analogy. Automaton groups are groups generated by sequential circuits acting on bit streams. This note fixes the dictionary so the two vocabularies stop drifting past each other, and then asks the question the dictionary makes askable.
The dictionary
| engineering | on the tree / in the group | where the row breaks |
|---|---|---|
| bit stream, one bit per clock | an infinite binary string; the boundary of the rooted tree = ℤ₂; the tree level is the clock | — |
| combinational circuit (no state) | Hensel's lemma: the first digit decides, every later digit is forced | only where f′ ≢ 0 mod p; for x² at p = 2 you need three digits first |
| sequential circuit / Mealy machine | a generator: read a bit, maybe flip it, hand the rest to a section — b = (a, c) is a transition table | — |
| state register | the set of sections an element can reach; for a contracting group, the nucleus | — |
| register width | nucleus size — 3 states for the odometer, 5 for Grigorchuk, unbounded for Aleshin | Basilica has a small register and exponential growth: width alone is not growth |
| carry chain | the sections at successive levels | — |
| carry dies out | contraction — sections eventually shorter than the word; Grigorchuk's length-halving | — |
| T-flip-flop ripple counter | the odometer a = (1, a)σ: toggle the low bit, carry = recurse — incrementing ℤ₂ | — |
| how many stages are toggling at depth n | activity (Sidki): bounded, polynomial, exponential | verify the attribution before citing |
| bistable with a noise margin | a minimum bisection with a wall — run 040's seam, 39,000 edges at level 5 | — |
| degenerate bit (no margin) | a free spin — run 042's 45,000 cosets with two neighbours each side of the cut | flips for free; not memory |
| write line | left multiplication by a group element: a graph automorphism carrying one seam to another | — |
| register states | the orbit of the seam; the quantized overlap ladder (steps ≈ 0.16) is the shift by one unit | P-044 measures the orbit |
| capacity (bits) | log₂ of the number of distinct minimum classes | — |
Two memories, then, and both had been measured before either was named: the machine's memory (the nucleus — what the dynamics must remember to process a digit) and the host's memory (the seam orbit held by extensive walls, with the group action as the write line). The first is the mechanism of intermediate growth; the second is the classical-memory reading of runs 030 and 040.
Why "which types exist" is a grid
Two trichotomies, independent:
- Activity (circuit): bounded / polynomial / exponential — how much of the register is still toggling at depth n.
- Growth (group): polynomial / intermediate / exponential — Gromov's, with polynomial = virtually nilpotent and the gap conjecture (nothing between polynomial and exp(√n)) open.
They do not coincide. What is known, from memory and to be verified before any of it is cited: bounded activity ⇒ amenable (Bartholdi–Kaimanovich–Nekrashevych); linear activity ⇒ amenable (Amir–Angel–Virág); polynomial activity conjectured amenable; contracting ⇒ amenable is Nekrashevych's conjecture. Occupied cells with the five groups of P-043:
| polynomial growth | intermediate | exponential | |
|---|---|---|---|
| bounded activity | odometer | Grigorchuk, IMG(z²+i) | Basilica |
| polynomial activity | ? | ? | (examples exist) |
| exponential activity | — | ? | Aleshin / free |
The question the owner asked — why are these the only types — is the empty cell in bold: can a circuit whose carry never settles grow slower than exponentially? If the register coordinate means what N-034 and P-043 say it means, no: unbounded activity should force exponential growth. I do not know whether that is a theorem, a conjecture, or false. It is the one line in this note that is a claim rather than a dictionary entry, and it is flagged as such.
Back-links (2026-09-29)
Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.
The register reading was falsified clause by clause by run 048 (16 seeds, 16 distinct orbits; the ladder is not a symmetry structure) and run 050 (the frozen core's orbits are distinct too). The rows above predate those scores.
- Run 040 — unanticipated (2026-09-28, measured).
lab/runs/040-t0-barrier-extensive.md: Clause 1 CONFIRMED (Δ₄ = 47, Δ₃ = 0, Δ_rand(1024) = 12). - Run 042 — unanticipated (2026-09-28, measured).
lab/runs/042-valley-wall-is-local-global.md: Twin clause CONFIRMED on all eight twin pairs — a twin difference of ~24,700 cosets is 22,100–22,400 connected components, the largest 15–20 cosets, and 100% of them are zero-modes. - Run 044 — falsified (2026-09-28, measured), via the builds_on thread.
lab/runs/044-quantum-walk-on-the-cell.md: FALSIFIED on the boundary clause — the Grigorchuk level-10 Schreier graph walk is ballistic under the sealed Laplacian, β = 0.973 ≥ 0.95, with the return floor (1.5/N), decay (γ = 0.82) and participation (0.999) of the path control. - Run 048 — falsified (2026-09-28, measured), via the builds_on thread.
lab/runs/048-seam-orbit-as-register.md: FALSIFIED on every clause, controls all passing (L_h label-preserving involutions;
79 source lines, no addenda.