nu_monotone_xy_heisenberg
plain-language theorem explainer
The theorem asserts that the correlation-length exponent ν increases when moving from the O(2) XY universality class to the O(3) Heisenberg class in the Q3-derived bootstrap values for three dimensions. Researchers comparing critical exponents across symmetry groups in condensed-matter physics would cite this ordering. The proof is a one-line wrapper that unfolds the two class definitions and performs direct numerical comparison.
Claim. In the O(N) universality classes obtained from subgroups of Aut(Q₃) in three dimensions, the bootstrap values satisfy ν_XY < ν_Heisenberg, where ν_XY = 0.67169 for N=2 and ν_Heisenberg = 0.71164 for N=3.
background
The module maps O(N) universality classes to subgroups of the automorphism group of the three-dimensional cube Q₃ and supplies reference bootstrap values at D=3: O(2) XY with ν=0.67169, η=0.03810 and O(3) Heisenberg with ν=0.71164, η=0.03784. The two upstream definitions are concrete records: the XY class stores the pair (N=2, ν=0.67169, η=0.03810) while the Heisenberg class stores (N=3, ν=0.71164, η=0.03784). These instances supply the numerical data whose ν components are compared.
proof idea
The proof is a one-line wrapper that unfolds the definitions of the XY and Heisenberg bootstrap classes and applies numerical comparison to their ν fields.
why it matters
The result records the expected monotonic rise of ν with symmetry rank N from 2 to 3 inside the Q₃ automorphism framework at D=3. It supplies an explicit ordering datum that any larger comparison of universality classes can invoke, consistent with the module's bootstrap reference list. No downstream theorems currently depend on it.
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