The Same Symmetry Twice

The bound levels of hydrogen are n^2-fold degenerate, which Fock traced to a hidden four-dimensional rotational symmetry. We show that for a recognizer carrying three binary distinctions this symmetry arises twice, from two independent constructions. From the dynamics: the field a posting walker sources on its lattice falls off as the inverse square, the inverse square is the unique power law conserving the Runge-Lenz axis, and the Poisson brackets of the conserved quantities close into two commuting rotation algebras with equal invariants fixed by the energy. From the algebra: the recognizer's unit-cost composites act on functions on their own state sphere from the left and the right, the new content at rank k has dimension (k + 1)^2, and the joint action is irreducible at every rank, by an elementary ladder argument given in full. One premise then joins the two constructions: that the quantized level space is reached from the rank block by a single equivariant bijection. Under it the identification is rigid, level n carries n^2 states, and the polarity count of a companion paper doubles this to the shell capacities 2, 8, 18, 32, ... The dynamical side rests on two modelling premises, stated and carried visibly; the algebraic side is unconditional.