growth-of-groups · Growth of groups · 34 records · all concepts
runs22 · +1 retired
030On the intermediate-growth host Cay(G/St(4)) (n = 4096), is the domain-wall free-energy barrier controlled by the host's local, low-dimensional structure — or does the entropy of near-minimal cuts eat it?run
031Does the minimum bisection of Cay(G/St(n)) track the Schreier-fibre count |G_n|/2^n — a separator of order n/log n whose exponent climbs toward 1 — or does the exponent 0.76 measured between n = 128 and n = 4096 (run 030) hold?run
032Run 030 found two metastable families of near-minimal cuts on Cay(G/St(4)) (≤ 224 edges from METIS, 270 from disorder).run
033The Ihara primes of the host are its relators.run
035Expanders are globally synchronising (the Kuramoto flow has no stable equilibrium but synchrony — no twisted states, no U(1) loop-closing sectors).run
036Expander codes (Sipser–Spielman, Tanner) need unique-neighbour expansion of small sets; amenability kills expansion only asymptotically.run
038At level 4 the sign of the 176-cut's projection onto the λ₂ irreducible (the sibling-odd representation, dimension 8) is a balanced cut of exactly 192 = (3/4)·256, and the bands up to dimension 72 reproduce 176, so 11/16 = 3/4 − 1/16.run
039At level 4 the 176 minimum bisection of the ⟨b,c,d⟩-coset graph is exactly the sign of its projection onto the lowest 73 Laplacian eigenvectors — the constant, the λ₂ = 4−4cos(π/8) irrep (multiplicity 8, share 0.697), two bands it is orthogonal to (4−4cos(π/6), multiplicity 8, and 2−√2, multiplicity 16; together +0.0002), and the 4−4cos(π/5) band (multiplicity 40, +0.189) — with agreement 1.000 and the cut 176 on the nose.run
040Run 034's instrument (Franz–Parisi at T = 1) failed its control and was withdrawn, but its second look stood unsealed — at T = 0, greedy pair-swap paths between minimum bisections of the level-4 ⟨b,c,d⟩-coset graph climb ~50 edges above the endpoints (28% of the 176 cut), against ~2 on a random 4-regular graph of the same size and ~0 on the level-3 coset graph.run
041Run 036 measured edge expansion of the ⟨b,c,d⟩-coset graph at 0.78 of a random 4-regular graph's at s = N/8, and P-026 read the intermediate cell's code ceiling from it.run
042Two "two-flavor" splits have appeared on the intermediate-growth host.run
043Five self-similar groups on the binary tree — odometer (nucleus 3, polynomial growth), Grigorchuk (5, intermediate), IMG(z² + i) (4, intermediate), Basilica (7, exponential, amenable), Aleshin (not contracting, free, non-amenable) — measured by the instruments of runs 033–042 at matched vertex count.run
044The quantum reading of the intermediate cell (P-045) — the boundary as a quasicrystal, the bulk as a flat-band localizer — is either a transport signature or a decoration.run
045Run 044 sealed the graph Laplacian as the walk Hamiltonian, and on the Schreier graph the Laplacian cancels every loop — which is exactly where the graph's aperiodicity lives (which of b, c, d fixes a leaf follows the substitution).run
046Runs 044 and 045 walked the Grigorchuk Schreier graph under isotropic operators (the Laplacian, then the equal-weight Hecke operator) and found a ballistic path both times — every leaf carries exactly one loop, so equal weights make the loop potential constant and the operator periodic.run
047Run 046 found the Grigorchuk boundary walks like a quasicrystal under the anisotropic Hecke operator (β = 0.800) and run 044 found a random expander's walk saturates by t ≈ 2 and a path's is ballistic.run
048Run 040 produced seventeen level-5 minimum bisections at exactly 90,112 and run 042 sorted them into ten classes (twins at |q| ≈ 0.953 are one class — a balanced random half of the free cosets flipped), with the overlaps between classes on a ladder 0, 0.33, 0.48, 0.64, 0.80.run
050Run 048 found the level-5 minimum bisections are not one small orbit — 16 seeds, 16 orbits, ≥ 4,096 states each, no stabiliser within reach — and that the generator d carries the seam to its twin at zero barrier, so part of the free-spin layer is symmetry.run
064Probe P-048 argues that block renormalization fails to converge on the Sierpiński-type carpets because they are infinitely ramified (blocks share whole edges).run
065Run 064 found that the ⟨b,c,d⟩-coset tower is not an exact renormalization under φ(μ) = μ(8 − μ)/2 (18 of 24 new level-4 eigenvalues orphans).run
088Run 044's bulk clause was confirmed — on the Grigorchuk level-4 coset graph the quantum walker stays near its origin at long times (PR/N = 0.096, return floor 76/N) against a random 4-regular graph's PR/N = 0.99 — and P-045 reads it as a transport signature of the intermediate-growth cell.run
101On the sixteen stored minimum bisections of the level-5 ⟨b,c,d⟩-coset graph (cut 90,112), read on 2026-10-04, exactly 40,960 free cosets are independently flippable on every one, and they lie over the map to level 4 the same way on every one.run
034Run 032 identified the glassy walls of run 030 with the minimum-bisection landscape of the ⟨b,c,d⟩-coset graph, a spin glass by Fu–Anderson.run
probes6
P-025Topological protection conserves non-locality — the measure is the distance exponent, and the frontier has an open cornerprobe
P-026Primes upstairs, zeros downstairs — the growth trichotomy as a prime-number-theorem / Riemann-hypothesis dichotomy on the host graphprobe
P-036A Tanner code on the squared coset graph — the empirical code ceiling of the intermediate-growth host, now that N-034 rules out an expander code on the graph itselfprobe
P-043A 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 moveprobe
P-044The seam orbit as a register — capacity and write cost of the host's memory, from the group action instead of quenchesprobe
P-045A quantum walk on the intermediate-growth cell — is the boundary a quasicrystal and the bulk a flat-band localizer, or is the analogy decorative?probe
notes5
N-026The intermediate-growth cell is real — effective dimension grows without bound, computed on the Grigorchuk groupnote
N-034The coset graph is a union of squares, and squares kill unique-neighbour expansion — no expander code on the intermediate-growth host at any levelnote
N-035The sequential-circuit reading of self-similar groups — latch, register, carry, counter — and why "which types exist" is a 3×3 grid, not a linenote
N-036Two routes to zero unique-neighbour expansion on a Cayley graph, both through g² — the Klein-four squares of N-034 and the cosets of ⟨g²⟩ found on Basilica and Aleshinnote
N-037What runs 048–060 say together — order holds, magnitude misses; first-order models rank and correlations set values; the seal is the weakest linknote
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