representation-theory · Representation theory · 7 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
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
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
probes2
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-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
generated 2026-10-25nothing on this page is maintained by hand