A growth axis on one tree — five self-similar groups on the binary tree, the nucleus as the coordinate, and the claim that nucleus size predicts which instruments move

The question, sharpened (revised the same morning). Every result in the intermediate-growth lineage comes from one group. Which are about growth, which about relators, and is there a single tree-side coordinate that predicts which is which? The rooted binary tree's boundary is ℤ₂, so a self-similar group is a group of 2-adic dynamics: each generator reads the first bit, possibly flips it, and hands the rest to a section. Hensel's lemma is the case where the section is unique — one digit decides, the rest is forced. For a contracting group the sections of every element eventually fall into a finite set, the nucleus; the nucleus is the "state the machine must remember after reading a digit", Hensel's latch with more than one state. Grigorchuk's proof of intermediate growth is this latch: descending one level halves word length. So the coordinate is the nucleus — its size, and whether the recursion shortens words — and the claim is that it sorts our instruments into two families.

The occupants — all on the same tree, all built from a wreath recursion (every recursion below is from memory and MUST be verified against a primary source before the builder is written; the builder reports the source it used):

Optional sixth: the lamplighter ℤ/2 ≀ ℤ as a two-state automaton (Grigorchuk–Żuk), exponential, amenable, pure-point spectrum. And one G_ω with a different periodic pattern, kept from the first draft, as the "same lineage, different relators" point.

The coordinate, computed not assumed. For each group the builder computes the nucleus by Nekrashevych's closure (sections of generators and their products until the set stops growing; cap at word length 12 and report "not contracting to length 12" for Aleshin), and the contraction ratio max |section| / |g| over words of length ≤ 12. Those two numbers are the x-axis.

The measurable claim. The seven instruments split into two families, and the split is predicted in advance:

Relator-driven — set by the presence in the recursion of a generator x with a trivial section next to an involution a, which gives (a x)⁴ = 1 and hence the square decomposition of N-034. Grigorchuk has it (d = (1, b)). IMG(z² + i) has it (c = (b, 1), and b, c are involutions by the recursion — (ac)² = (b, b), (ac)⁴ = 1). Basilica and Aleshin do not. The odometer's Cayley graph is a cycle, degenerate. Prediction: unique-neighbour expansion is identically zero for Grigorchuk and IMG(z² + i) and strictly positive at every small size for Basilica and Aleshin; free-spin density on the minimum bisection (run 042's 4.3%) and Kuramoto vortex trapping (run 035) follow the same split. These do not care about growth.

Growth-driven — set by the growth class, in the order odometer → Grigorchuk → IMG(z² + i) → Basilica → Aleshin: the edge-expansion ratio at N/8 against a random 4-regular graph of the same N (run 036's 0.78 for Grigorchuk; predicted → 1 for Aleshin, ≈ 0 for the odometer); the T = 0 barrier between minimum bisections as a fraction of the cut (run 040's 27% at N ≈ 1000; predicted below 12% for Aleshin, like the random glass, and above 20% for both intermediate groups); the share of the minimum bisection in the lowest non-trivial spectral band (0.70 for Grigorchuk; predicted 0.81 = 8/π² for the odometer's cycle, below 0.4 for Aleshin). The cogrowth constant κ (run 033) belongs here too, and the two intermediate groups are predicted to differ from each other on every growth-driven number while agreeing on every relator-driven one.

The nucleus claim itself: groups with a nucleus (contracting) occupy the amenable side and show extensive barriers; the group without one shows the random glass's numbers. If Basilica — a nucleus, exponential growth — lands with Aleshin on the growth-driven instruments and with Grigorchuk on nothing, the coordinate is growth, not the nucleus; if it lands between, the nucleus is doing work of its own.

Matched N, not matched level. The quotients G/St(n) have wildly different sizes (the odometer's is ℤ/2ⁿ; Aleshin's approaches |Aut(T_n)|), so every instrument runs at matched vertex count — the level at which the Cayley graph of G/St(n) has N in [512, 4096], the odometer at N = 1024 exactly. Where a group's coset trick (contracting a Klein-four ⟨b,c,d⟩) has no analogue, the instruments run on the Cayley graph itself; for Grigorchuk both are run so the two are calibrated against each other. Leaf Schreier graphs on 2ⁿ vertices are recorded as a free extra at every level (they are the boundary graphs — Hanoi's is the Sierpiński gasket of R7).

Build (Claude Code, a new lab chat; run after run 039 releases the M5). One builder, R27_tree_groups/: (1) a wreath-recursion interpreter — a group is a table {generator: (section₀, section₁, flip)} — that produces the permutation action on level n and the finite quotient by BFS, exactly as grig.py/grig_build.c do for Grigorchuk, verified by reproducing Grigorchuk's orders 128, 4096, 4,194,304 and the odometer's 2ⁿ; (2) nucleus and contraction ratio; (3) Cayley graph and, where ⟨b,c,d⟩-type Klein subgroups exist, the coset graph, both in the .edges format the instruments read; (4) the seven instruments unchanged: uniq.c, profile.py, the bisection (anneal or refine), barrier.py, the free-spin count from patches5.py, dense spectral bands (bands_l5.py level-4 mode), cogrowth.py; Kuramoto (kuramoto.py) as the eighth. Report one table: five groups × (nucleus size, contraction ratio, N, and the eight numbers), with the random 4-regular control at each N.

Sealed as a run once the builder reproduces Grigorchuk's known numbers — that reproduction is the control, sealed with the predictions above turned into closed constants.

Corrections from the build, 2026-09-28 (Lab_Tree_Model, R27, before any non-Grigorchuk group was measured). Two of the five recursions above are wrong as written: Aleshin's is in the output-letter convention — the source (Vorobets–Vorobets §3, input-indexed) gives a = (c, b)σ, b = (b, c)σ, c = (a, a), and the probe's version is a different automaton; Basilica's has its two letters swapped relative to Bartholdi–Virág (same group, relabelled tree). The builder used the sources. And the square prediction for IMG(z² + i) was wrong in its mechanism: (σβ)⁴ = 1 holds (Bux–Pérez Lemma 20), but N-034's squares need the Klein-four contraction as well as the relator, and IMG has no Klein-four subgroup among its generators (its pairs generate dihedral groups of order 8 and 16). So the run sealed on this probe (043) predicts the square obstruction is Grigorchuk's alone. Nuclei as computed: 3, 5, 4, 7, none. Degrees are not matched (2, 4, 3, 4, 6); the controls are random d-regular at each group's degree.

Back-links (2026-09-29)

Dated pointers added by the codifier from lab/backlinks.py; nothing above this heading was changed.

Substantially answered by runs 043 and 047: the ordering follows the nucleus coordinate (amenable vs not), not the growth class (N-038).

• Run 031 — confirmed (2026-09-27). lab/runs/031-grig5-bisection.md: CONFIRMED — scored 2026-09-27 (codifier).
• Run 036 — confirmed (2026-09-27, measured). lab/runs/036-expansion-profile-coset-graph.md: CONFIRMED — ρ_Φ = 104/134 = 0.776.
• 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 041 — unanticipated (2026-09-28, measured). lab/runs/041-unique-neighbour-expansion.md: Clause 1 withdrawn (control band miscalibrated;
• 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 043 — falsified (2026-09-28, measured). lab/runs/043-growth-axis-on-one-tree.md: FALSIFIED — the relator clause's falsifier fires: Basilica's unique-neighbour minimum at N/8 is 1 of its control's 101 at best-of-20 and 0 of 94 at best-of-40, u/u_random ≤ 0.10, so the square obstruction of N-034 is not specific to the Klein-four contraction;
• Run 047 — unanticipated (2026-09-28, measured), via the builds_on thread. lab/runs/047-growth-axis-in-transport.md: UNANTICIPATED on the ordering clause, with every per-group clause hit and the class separation confirmed outright.