The Surviving Ray: channel selection for a cubic self-interaction on S³/2I

The spin-3 representation admits a four-dimensional family of equivariant cubic self-maps, the family known in the spinor-condensate literature as the four total-spin scattering channels. On S³/2I, for a block state at level 6 with the density-type interaction |ψ|²ψ, binary-icosahedral symmetry restricts that family to two channels, one of them radial; modulo the radial direction the surviving interaction is a single projective ray. The interaction is a hypothesis and not a consequence: a different local 2I-invariant quartic, built from ψψᵀ, is filtered by the same mechanism onto a different plane. A block state at level 6 carries left spin 3 and a right index in a representation σ of the binary icosahedral group, and its density is a function on the quotient, so a channel of rank K survives only when V_K carries a 2I-invariant. The density's left content spans ranks 0 through 6, that is levels 0 through 12, and inside that closed window exactly two carry one, at K = 0 and K = 6. The quartic functional reduces with the interaction, to an affine function of the top multipole ‖ρ₆‖² whose coefficient is not fitted but given in closed form by the branching, and positive because both constituents are proper. The weight ‖R₆‖² that produces it takes the same value in both sectors, because they are complementary in dim V₃ = 7; the normalised coefficient itself does not, and the two are distinguished in Section 5.3. Separately, and with no icosahedral input at all, the spin-8 cubic channel built from the same spin-3 data, of binary order 16, closes exactly on the time-reversal-invariant rays: its zero set is U(1)·Fix(Θ), equivalently the rays whose Majorana constellation is antipodally symmetric as a multiset. The ingredients are classical: the Jacobian criterion for nonzero binary forms of equal degree, the Majorana representation with its time-reversal reading, and Peter-Weyl on S³. The result established here is the channel filter and the selection it forces for that interaction, together with an accounting of which parts of the surrounding arithmetic are icosahedral and which are universal facts about spin 3 with time reversal. 🔗 github.com/mode-identity-theory/surviving-ray for current working-version