group-theory · Groups & symmetry · 8 records · all concepts
runs5
037Trained to grokking on multiplication mod p, which characters does a small network learn — the additive characters of Z/p, or the multiplicative (Dirichlet) characters, i.e. Fourier modes in discrete-log coordinates on (Z/p)^x?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
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
probes3
P-031Grokking multiplication on the units of Z/561 — which characters appear when the group is not cyclic and no discrete log exists?probe
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-046Is "accidental symmetries are the wildly ramified ones" more than a reading — does ramification theory predict an integrality relation between the charge an operator violates and its dimension?probe
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