The Surviving Ray

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 $`S^3/2I`$, for a block state at level 6 with the density-type interaction $`\lvert\psi\rvert^2\psi`$, 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 $`\psi\psi^{T}`$, 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 $`\sigma`$ 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 $`\lVert \rho_6 \rVert^2`$ whose coefficient is not fitted but given in closed form by the branching, and positive because both constituents are proper. The weight $`\lVert R_6 \rVert^2`$ that produces it takes the same value in both sectors, because they are complementary in $`\dim V_3 = 7`$; 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 $`U(1) \cdot \mathrm{Fix}(\Theta)`$, 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 $`S^3`$. 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.

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1. Introduction

Spin-3 states carry a well-studied geometry. The Majorana representation sends a state to six points on the sphere, rotations act on the constellation, and a substantial literature studies the constellations that extremise natural invariants: the anticoherent states, whose low multipoles vanish, and the polyhedral configurations that realise them. Barnett, Turner and Demler [BTD] enumerate six-vertex spin-3 phases in that language. Within this setting the invariant quartics of a spin-3 state form a four-dimensional space, and particular members of that family have been studied along with the symmetric constellations attached to them. Any statement about a particular quartic is therefore a statement about a member of a family the community will already recognise.

The question here is not which member is interesting but which member a geometry permits for a given interaction. Modes on the spherical space form $`S^3/2I`$ organise into flat bundles indexed by representations $`\sigma`$ of the binary icosahedral group. At level 6 a block state has left spin 3, so its self-interaction lies in the four-dimensional family above; the question is what the quotient does to that family. The answer is a filter. The density $`\lvert\psi\rvert^2`$ of a section is a genuine function on $`S^3/2I`$, and by Peter-Weyl its level-$`2K`$ component factors into a left multipole, built from the spin-3 data alone, and a right multipole, which lives in the $`2I`$-invariants of $`V_K`$. Those invariants first appear at level 12 and next at level 20, so across ranks 0 through 6 exactly two channels are open. Everything else is switched off, not because the left-hand tensor vanishes, which it does not, but because its right-hand coefficient does.

Two results follow. As proved they are independent, neither using the other; Section 5.7 later combines the ambient proposition with the quotient calculation.

The first is ambient and uses no icosahedral input. For spin 3 with time reversal $`\Theta`$, the spin-8 cubic channel assembled from a state, its time reverse, and itself again vanishes precisely on the time-reversal-invariant rays. In the language of constellations, the channel closes exactly when the six Majorana points are antipodally symmetric as a multiset. Ranks are quoted as spins throughout, so that the spin-8 target and the rank-6 channel below are the same kind of index; the classical invariant theory numbers the same object by its binary degree, 16, and that number is a form degree and not a rank. The proof is short and rests on a classical fact about binary forms, that two nonzero forms of the same degree with vanishing Jacobian are proportional, so it is billed as a proposition rather than a theorem; the possibility that the underlying joint covariant is named in the classical invariant-theory literature is left open in Section 7 rather than settled here.

The second is the selection, and it takes three steps rather than one. The invariant-degree filter leaves the density only ranks 0 and 6. The multiplicity-free branching then scalarises the right-index contraction, without which surviving ranks would not select single maps at all, and puts the self-interaction in the plane spanned by the rank-0 and rank-6 maps. The rank-0 map is radial, and the rank-6 coefficient is nonzero, which is earned separately by pairing the interaction against the quartic. Only then does the self-interaction reduce, modulo the radial direction, to the single ray generated by the rank-6 map. The governing quartic reduces with it, to an affine function of the top multipole alone.

The weight attached to the surviving channel is not fitted: it is built from $`\lVert R_6\rVert^2`$, the norm of the right multipole of an isotypic projector, and is positive because both constituents of the branching $`V_3\vert_{2I}`$ are proper. That norm takes the same value in both sectors, because they are complementary in $`\dim V_3 = 7`$. The normalised coefficient $`w_6`$ does not, and Section 5.3 keeps the two apart.

The derived weight and an earlier form fitted to the observed normalisation agree at both values of $`\dim\sigma`$ that occur. Those two are the only ones the branching produces, so no measurement inside this system distinguishes them. The correction is carried by the derivation alone.

Little of the surrounding structure is icosahedral. The four-dimensional family, the structure of the surviving operator, the alternating row of binomial coefficients its weight matrix carries, the forced form of the time-reversal phase, and the critical rays of the reduced quartic together with their Majorana constellations are all universal facts about spin 3 with time reversal. None of them requires $`2I`$ once the rank-6 channel has been selected, which is the distinction worth keeping: $`2I`$ selects, and spin-3 mathematics determines what the selected channel is. For the selection theorem the icosahedron contributes exactly two facts: the invariant-degree filter, which leaves density ranks 0 and 6, and the multiplicity-free complementary branching $`V_3\vert_{2I} = \sigma_3 \oplus \sigma_4`$, which both scalarises the right-index contraction and supplies the weight. The quotient's representation theory is used elsewhere in the paper for other purposes, but not by that theorem. That is a narrower claim and a sharper one, since it separates a general piece of representation theory from the specific arithmetic of a quotient.

One boundary should be drawn explicitly. The configurations appearing below are known: the octahedral constellation is the anticoherent state of order 3 in the standard classification, and critical rays occur here with all four of the six-vertex shape types [BTD] names. Recognising a configuration is not the same as selecting it, and configuration prior art is not selection prior art. The contribution is which member of a known family a specific geometry forces, and the surrounding accounting of what that forcing does and does not explain.

Two of the limits above are of record rather than of proof, and belong in one place. The sentences naming the constellation literature rest on abstracts and not on bodies, and the search behind them is reported as a search and not as a novelty finding (Section 7.3); the [BTD] count of six-vertex phases is provisional pending a first-hand reading of that paper's body (Section 7.2). The anticoherence fact used below is not taken on trust from that literature, and Section 5.8 establishes it here.

Section 2 fixes the state space and the time reversal, then the quotient's sectors and invariant degrees and the interaction the rest of the paper uses. Its one transform is fed three different arguments, not two. Section 3 gives the ambient proposition. Section 4 records the reduction that isolates the four-dimensional family and states what it can and cannot touch. Section 5 proves the selection theorem, computes the surviving weight, states the normalisation's own ceiling, connects the ambient channel back to the quotient, maps the critical geometry the theorem leaves open, and derives consequences across levels and sectors. Section 6 reads the result as a Lyapunov-Schmidt interpretation, with its ceilings stated. Section 7 collects the discussion, the prior art, the open historical question, and the limits of the arithmetic account.

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2. Setup

2.1 The state space and the Majorana dictionary

Let $`V_j = \mathrm{Sym}^{2j}\mathbb{C}^2`$ be the irreducible $`\mathrm{SU}(2)`$ representation of spin $`j`$ and dimension $`2j+1`$, with orthonormal weight basis $`v_m`$, $`-j \le m \le j`$. The state space throughout is $`V_3`$, of dimension 7. A state $`u = \sum_m u_m v_m`$ is written as the binary sextic

$$F_u(z) \;=\; \sum_{m=-3}^{3} (-1)^{\,3-m}\sqrt{\binom{6}{3+m}}\; u_m\, z^{\,3-m} ,$$

whose six roots on the Riemann sphere are the Majorana constellation [Maj] of $`u`$, a multiset of six points determined by the ray $`[u]`$ and carried by rotations. A finite root $`r`$ is the point with polar angle $`\theta = 2\arctan\lvert r\rvert`$ and azimuth $`\varphi = \arg r`$, and each degree by which the polynomial falls short of 6 contributes one point at $`\theta = \pi`$. That fixes absolute coordinates rather than a shape up to rotation, which is what the identifications in Section 5.8 require. Statements about constellations below are statements about rays.

2.2 Time reversal

Time reversal is the antiunitary $`\Theta`$ on $`V_3`$ acting on coefficients by

$$\Theta\!\left(\sum_m u_m v_m\right) \;=\; \sum_m (-1)^m\, \overline{u_m}\, v_{-m} ,$$

written in shorthand as $`\Theta v_m = (-1)^m v_{-m}`$ with $`\Theta`$ understood to be antilinear. The weight reversal and the relative $`(-1)^m`$ pattern are forced; only an overall phase is conventional. Antilinear intertwiners of $`V_3`$ with itself are $`\mathrm{Hom}_{\mathrm{SU}(2)}(\overline{V_3}, V_3)`$, which is one-dimensional by Schur since $`V_3`$ is irreducible and self-dual, so $`\Theta`$ is unique up to a complex scalar. To identify it, write $`\Theta v_m = c_m v_{-m}`$ with $`\Theta`$ antilinear, and impose $`\Theta J_+ = -J_- \Theta`$. Since $`J_+ v_m = \alpha_m v_{m+1}`$ with $`\alpha_m = \sqrt{j(j+1) - m(m+1)}`$, and $`J_- v_{-m} = \alpha'_m v_{-m-1}`$ with $`\alpha'_m = \sqrt{j(j+1) - (-m)(-m-1)}`$, the two coefficients are equal because $`(-m)(-m-1) = m(m+1)`$, and this reads $`\alpha_m c_{m+1} = -\alpha_m c_m`$, so $`c_{m+1} = -c_m`$ and $`c_m = c\,(-1)^m`$; antiunitarity fixes $`\lvert c \rvert = 1`$. The global constant is a genuine convention and the $`m`$-dependence is not, which matters below because it is the $`m`$-dependence that supplies the alternating sign in the operator of Section 5. One computes $`\Theta^2 = (-1)^{2j}`$, so $`\Theta^2 = +1`$ here and $`\Theta^2 = -1`$ at half-integer spin.

Two labels used throughout should be read as representation theory and nothing more. Spin 3 names the irreducible $`\mathrm{SU}(2)`$ representation $`V_3`$ and the harmonic level it sits at; it is not a claim that anything described here has physical spin 3. And $`\Theta`$ is the standard antiunitary intertwiner of that representation with its conjugate. The equation studied below is elliptic and stationary, with no time in it, so nothing here establishes a physical time-reversal symmetry of a dynamics; $`\Theta`$ earns its name from its algebra, not from an evolution it commutes with.

A ray is time-reversal invariant when $`\Theta u = \lambda u`$ for some $`\lambda`$ of modulus one, and the set of such $`u`$ is $`U(1)\cdot\mathrm{Fix}(\Theta)`$. The condition is projective: for $`a \in \mathrm{Fix}(\Theta)`$ and any phase, $`u = e^{it}a`$ has $`\Theta u = e^{-2it}u`$, which is in general neither $`+u`$ nor $`-u`$. In constellation language a ray is time-reversal invariant exactly when its six Majorana points are antipodally symmetric as a multiset, coincidences included.

2.3 One transform, and three things it is applied to

For $`0 \le J \le 6`$ let $`[\,\cdot \otimes \cdot\,]_J`$ denote the projection of $`V_3 \otimes V_3`$ onto its spin-$`J`$ summand. For a $`7 \times 7`$ matrix $`P`$ define

$$\mathcal{M}_K(P)_N \;=\; \sum_{n+n' = N} \langle 3\,n;\, 3\,n' \mid K\,N \rangle \, (-1)^{n'} P_{n,\,-n'} , \qquad 0 \le K \le 6 .$$

This is the state-multipole, or statistical-tensor, expansion of a density matrix in irreducible tensor operators, standard since [Fa] and used in exactly this form to read multipoles off a Majorana constellation [RK]. It is written out here because three different arguments are fed to it below and the paper turns on keeping them apart.

The holomorphic square $`B_J(a) = [a \otimes a]_J`$ takes both arguments to be the same state. Exchanging two identical slots multiplies the spin-$`J`$ summand of $`V_3 \otimes V_3`$ by $`(-1)^{3+3-J} = (-1)^J`$, so $`B_J`$ vanishes identically for odd $`J`$.

The density multipole $`\rho_K(u) = [u \otimes \Theta u]_K`$ is sesquilinear, linear in $`u`$ and antilinear in it through $`\Theta u`$. It equals the transform above at the state's own projector,

$$\rho_K(u) \;=\; \mathcal{M}_K\!\left(u u^{\dagger}\right) ,$$

and it does not vanish for odd $`K`$ in general.

The right multipole is the same transform at a different argument, $`R_K = \mathcal{M}_K(P)`$ with $`P`$ the isotypic projector of Section 2.4. Same map, different matrix; the two multiply rather than merge, and Section 5 turns on that.

The factorisation of Section 5 uses this one $`\mathcal{M}_K`$ on both sides, which is what makes the two multipoles multiply: the time-reversal phase sits inside $`\mathcal{M}_K`$ itself, and $`\rho_K(u) = \mathcal{M}_K(u u^{\dagger})`$ holds because $`(\Theta u)_{n'} = (-1)^{n'}\overline{u_{-n'}}`$ is exactly the factor the definition carries.

Three notational distinctions matter throughout. $`\rho_K`$ with a subscript is always a density multipole and never a representation of $`2I`$; representations are written $`\sigma`$ throughout. The index $`K`$ on a multipole is a spin, while the $`J`$ on $`B_J`$ is the same kind of index; no quantity in this paper is indexed by a binary form degree except where that is said explicitly. Spins do not all live on one axis either: a multipole of the density is indexed by its own rank, a channel of the nonlinearity by the spin of its target, and these label different decompositions. Section 3 works at output spin 8, Section 5 at density rank 6 and output spin 3; the 6 and the 8 are not comparable as indices. Section 5.7 does relate the two objects, but as a coupling of a rank-6 multipole to the state, not as an identity between labels.

Lemma 2.1. $`B_J`$ and $`\rho_J`$ agree on $`\mathrm{Fix}(\Theta)`$, and differ over $`\mathbb{C}`$.

On $`\mathrm{Fix}(\Theta)`$ the two coincide because $`\Theta u = u`$, so in particular $`\rho_J`$ inherits the vanishing of $`B_J`$ at odd $`J`$ there. Off that locus they part company, and not slightly: the coherent state $`v_3`$ has $`B_2(v_3) = 0`$, since $`v_3 \otimes v_3`$ has weight 6 and cannot meet a spin-2 summand at all, while its density quadrupole $`\rho_2(v_3)`$ is nonzero. Every $`\mathrm{Fix}(\Theta)`$ statement below is stated for $`\rho_J`$ and would be false for $`B_J`$, and conversely; the distinction is not bookkeeping. $`\square`$

The vanishing of the odd multipoles is a separate statement. Section 3 reaches the same locus by a different route, through the Jacobian kernel, so the two are a check on each other rather than one serving the other.

Lemma 2.2. $`\rho_1(u) = \rho_3(u) = \rho_5(u) = 0`$ if and only if $`[u]`$ is time-reversal invariant.

If $`\Theta u = \lambda u`$ then $`\rho_K(u) = \lambda [u \otimes u]_K`$, which vanishes for odd $`K`$ by the exchange sign. Conversely, if every odd projection of $`u \otimes \Theta u`$ vanishes then the tensor is supported in the even summands, on which the exchange acts trivially, so $`u \otimes \Theta u = \Theta u \otimes u`$; for a nonzero simple tensor that forces $`\Theta u`$ proportional to $`u`$, and antiunitarity makes the constant a phase. $`\square`$

2.4 The quotient, its sectors, and the invariant degrees

Identify $`S^3`$ with $`\mathrm{SU}(2)`$ carrying the round metric, let $`2I \subset \mathrm{SU}(2)`$ be the binary icosahedral group of order 120 acting by right translation, and write $`X = S^3/2I`$ for the resulting spherical space form. A finite-dimensional unitary representation $`\sigma`$ of $`2I`$ determines a flat bundle on $`X`$, whose sections are the functions $`\psi \colon \mathrm{SU}(2) \to \mathbb{C}^{\dim\sigma}`$, written as rows, satisfying $`\psi(gh) = \psi(g)\,\sigma(h)`$ for $`h \in 2I`$. An intertwiner $`\eta \in \mathrm{Hom}_{2I}(\sigma, V_j)`$ is correspondingly a map with $`D^j(h)\,\eta = \eta\,\sigma(h)`$, and by Peter-Weyl the sections at level $`\ell = 2j`$ are $`V_j \otimes \mathrm{Hom}_{2I}(\sigma,\, V_j)`$, realised as

$$\psi_a(g) \;=\; \sum_{m,n} u_m\, D^j_{mn}(g)\, \eta_{na} ,$$

so the left index is free and the right index carries the sector. The convention is fixed here because dual-looking Peter-Weyl formulas are both defensible and only one of them matches the factorisation of Section 5. Since $`\sigma`$ is unitary, $`\lvert\psi(gh)\rvert^2 = \lvert\psi(g)\rvert^2`$, so the density really is a function on $`X`$. A block state at level 6 is one for which $`j = 3`$, so its left index runs over the $`V_3`$ of Section 2.1 and its right index over a multiplicity space that the branching below shows to be one-dimensional. Level 6 is the scope of the selection theorem of Sections 4 through 5.5. What fixes it is the choice of object: this paper is about spin-3 states, so the left index is $`V_3`$ and the level is $`2 \cdot 3`$. Within the two sectors multiplicity stays at one wherever they occur, at levels 12 and 16 as well; the table in Section 4.3 shows both the repetition it does permit elsewhere, $`\mathbf{5}`$ twice at level 16, and the absence of $`\mathbf{3}'`$ at level 12. What is special about level 6 is that fixing the left spin fixes the level, and at that level the branching happens to be multiplicity-free, which is what the Schur argument of Section 5.1 needs. Section 3 is ambient spin-3 mathematics and is not scoped by any of this.

Two facts about $`2I`$ drive the selection theorem, both classical and both recomputed here from the group rather than cited. They are the two inputs named in Section 5.1. First, the dimensions of the invariants,

$$\dim (V_j)^{2I} \;=\; 1,\,0,\,0,\,0,\,0,\,0,\,1,\,0,\,0,\,0,\,1,\,0,\,1,\,0,\,0,\,1 \qquad (j = 0,\dots,15),$$

so invariants occur at levels 0, 12, 20, 24 and 30. All three generator degrees of the invariant ring of $`2I`$ appear, 12, 20 and 30, and 24 is the first product, $`I_{12}^2`$. Only the gap below level 12 is used below. Second, the branching of the state space itself. $`V_3`$ has integer spin, so $`-I`$ acts trivially and $`V_3`$ factors through $`2I/\{\pm I\} \cong A_5`$, whose irreducible dimensions are $`1, 3, 3, 4, 5`$. The character sum gives $`\langle \chi, \chi \rangle = 2`$, so the restriction is multiplicity-free with two constituents, and $`\dim (V_3)^{2I} = 0`$ excludes the trivial one. The two-dimensional irreducible representations of $`2I`$ are spinorial, so they cannot occur where $`-I`$ acts trivially, which is precisely what factoring through $`A_5`$ means; $`A_5`$ itself has no two-dimensional irreducible representation. Hence

$$V_3\vert_{2I} \;=\; \sigma_3 \oplus \sigma_4 , \qquad \dim\sigma_3 + \dim\sigma_4 = 3 + 4 = 7 .$$

The two sectors a block state can occupy are therefore complementary in $`\dim V_3`$. What that buys is specific: the surviving weight computed in Section 5 is $`\lVert R_6 \rVert^2 = d(7-d)/7`$ with $`d = \dim\sigma`$, an expression symmetric under $`d \leftrightarrow 7-d`$, so the symmetry itself makes the two sectors carry the same value, and it is positive for both because $`0 < d < 7`$.

Since $`\sigma`$ occurs with multiplicity one, $`\mathrm{Hom}_{2I}(\sigma, V_3)`$ is one-dimensional. For any nonzero intertwiner $`\eta`$ in it, Schur gives $`\eta^{\dagger}\eta = c\, I_\sigma`$; rescale so that $`c = 1`$, so that $`P = \eta\eta^{\dagger}`$ is the orthogonal projector onto the $`\sigma`$-summand of $`V_3`$. Without that normalisation $`P`$ is a multiple of the projector and every quantity built from it carries an undetermined constant, so it is fixed here once.

2.5 The interaction, and what the results are relative to

Everything below concerns one interaction, fixed here so that Sections 4, 5 and 6 read off a single definition rather than three. The problem is

$$(-\Delta - \lambda)\psi \;+\; g\,\lvert\psi\rvert^2\psi \;=\; 0 , \qquad \lambda \text{ near the level-6 eigenvalue,}$$

the self-interaction means the block projection of $`\lvert\psi\rvert^2\psi`$ onto the level-6 eigenspace, and the quartic is the ratio $`Q = \int\lvert\psi\rvert^4 / (\int\lvert\psi\rvert^2)^2`$.

Nothing below determines $`g`$. For the channel-selection question a nonzero magnitude can be absorbed into the amplitude, since the problem is projective in $`\psi`$ and $`g`$ rescales the amplitude at which a given ray solves it. Absorbing $`\lvert g\rvert`$ leaves the sign, which no rescaling removes, and that sign is immaterial here because an overall nonzero real factor does not move the tangential critical equation. So $`g = 1`$ is taken throughout, a normalisation rather than a result: its sign and its physical scale are not derived here, and the 1 is not a computed coupling.

The choice of interaction is not vacuous: a second local $`2I`$-invariant quartic genuinely exists. Both $`\sigma_3`$ and $`\sigma_4`$ are real representations: $`V_3`$ has integer spin, so $`\Theta^2 = +1`$ makes $`\Theta`$ a real structure on it, and the restriction is multiplicity-free with constituents of different dimensions, so $`\Theta`$ cannot exchange them and must preserve each. Choosing real orthonormal models then gives $`\sigma(h)\sigma(h)^{T} = I`$. Hence $`\psi\psi^{T}`$ is also a genuine function on $`X`$ and $`\int\lvert\psi\psi^{T}\rvert^2`$ is a second local, $`2I`$-invariant quartic. The filter of Section 4 applies to it unchanged, since its right-hand factor is the plain Clebsch-Gordan projection of $`\eta\eta^{T}`$, carrying neither the $`\Theta`$ phase nor the index reversal that $`\mathcal{M}_K`$ carries. That is exactly why the left-hand factor comes out holomorphic. Either way it lands in the same invariants. Its surviving left-hand factors are the holomorphic squares $`B_0`$ and $`B_6`$ of Section 2.3 rather than $`\rho_0`$ and $`\rho_6`$, so its interaction is confined to a genuinely different plane; Section 5.5 identifies that plane and shows it differs from the one selected here.

Whether that second plane also reduces to a single ray is not examined here; the analogues of clauses 3 and 4 for the $`N`$ pair have not been computed. What the filter fixes is the plane. Which plane depends on the interaction, so the selection theorem is a statement about the density-type quartic above and not about local $`2I`$-invariant quartics in general.

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3. The ambient proposition

This section uses no icosahedral input, and everything in it holds for spin 3 with time reversal. It is not, however, a separate note. The trilinear data $`(u, \Theta u, u)`$ admits an equivariant contraction into every $`V_J`$ occurring in $`\mathrm{Sym}^2 V_3 \otimes V_3`$, and those contractions are the channels of a single object indexed by the spin of the target. The physical self-interaction is the spin-3 channel, which Sections 4 and 5 select within; the covariant below is the spin-8 channel. The uniqueness proved here and the four-parameter family found there are the same weight count run at two targets. That is the shallow reason both belong in one paper; the real one is Section 5.7, which shows the spin-8 channel is populated on $`X`$ and that the states annihilating it are exactly the time-reversal-invariant ones. This is stated as a proposition and not a theorem. Its ingredients are classical: the multiplicity count, the first transvectant, and the Jacobian criterion. Whether the joint covariant of two independent sextics of bidegree $`(2,1)`$, or this particular time-reversal specialisation of it along $`g = \Theta f`$, has appeared previously is left open in Section 7. Nothing in the paper's contribution rests on the answer, since that contribution is the channel selection of Sections 4 and 5.

3.1 The cubic covariant

The maps in this paper are not holomorphic cubics, and the distinction has to be made before any multiplicity is quoted. Holomorphic cubics $`V_3 \to V_J`$ are counted by $`\mathrm{Hom}_{\mathrm{SU}(2)}(\mathrm{Sym}^3 V_3,\, V_J)`$, and $`\mathrm{Sym}^3 V_3 = V_1 \oplus 2V_3 \oplus V_4 \oplus V_5 \oplus V_6 \oplus V_7 \oplus V_9`$, of dimension $`\binom{9}{3} = 84`$ with spin 3 occurring twice: it contains no $`V_8`$, so no holomorphic cubic into the spin-8 target exists at all. The maps here are of type $`(2,1)`$, quadratic in the state and antilinear in it through time reversal, so the space to count is

$$\mathscr{E}_J \;:=\; \mathrm{Hom}_{\mathrm{SU}(2)}\!\left(\mathrm{Sym}^2 V_3 \otimes \overline{V_3},\, V_J\right) \;\simeq\; \mathrm{Hom}_{\mathrm{SU}(2)}\!\left(\mathrm{Sym}^2 V_3 \otimes V_3,\, V_J\right) ,$$

the isomorphism induced by $`\Theta`$, which supplies the linear intertwiner $`\overline{V_3} \simeq V_3`$. This notation is used for the rest of the paper. An exact weight count gives

$$\dim \mathscr{E}_8 \;=\; 2 - 1 \;=\; 1 ,$$

the weight-8 subspace of $`\mathrm{Sym}^2 V_3 \otimes V_3`$ having dimension 2 and the weight-9 subspace dimension 1, so the covariant is unique up to scale. Call a nonzero representative $`T`$, taken in polarised form,

$$T(x,y,z) \;=\; \tfrac{\kappa}{2}\left[\, x\,(y,z)_1 \;+\; y\,(x,z)_1 \,\right] , \qquad \kappa \neq 0 ,$$

where $`x, y, z`$ are the binary sextics attached to three states and $`(\cdot,\cdot)_1`$ is the first transvectant in the classical sense [GY]. Only the vanishing of $`(f,g)_1`$ is used below, and $`(f,g)_1`$ is a nonzero constant multiple of the Jacobian

$$J(f,g) \;=\; f_X\, g_Y - f_Y\, g_X ,$$

so the two are interchangeable for that purpose. The constant $`\kappa`$ depends on the normalisation of $`T`$ and of the transvectant and is not quoted here; that $`\kappa \neq 0`$ is what the argument uses, and it holds because $`T`$ is nonzero.

The polarised form is the primary statement and its two specialisations are one line each. Setting $`y = x`$ gives the holomorphic diagonal,

$$T(a,a,b) \;=\; \kappa\, f \,(f,g)_1 ,$$

and setting $`y = \Theta u`$, $`z = u`$ gives the ambient spin-8 channel used in this paper,

$$\mathcal{C}(u) \;:=\; T(u,\, \Theta u,\, u) \;=\; -\tfrac{\kappa}{2}\, F \,(F, \Theta F)_1 ,$$

the second term of the polarisation dropping out because $`(F,F)_1 = 0`$ by antisymmetry.

The two specialisations look independent and are not. Setting $`a = u`$ and $`b = \Theta u`$ in the first gives $`T(u, u, \Theta u) = \kappa F (F, \Theta F)_1 = -2\,\mathcal{C}(u)`$, so they differ by a constant. That is not a coincidence: $`\dim\mathscr{E}_8 = 1`$ leaves no room for two independent objects, and the $`-2`$ is the proportionality that uniqueness forces.

Following Section 2.3, $`\mathcal{C}`$ takes values in $`V_8`$, and is referred to as the spin-8 cubic channel. Classical invariant theory numbers the same object by the degree of the binary form, 16; that is a form degree, not a spin, and it is not used as an index anywhere below.

3.2 The classical input

One classical fact is used. It is short enough to prove, which is done here rather than cited, so that its hypotheses are visible in the argument that needs them.

Lemma 3.1 (Jacobian criterion). Let $`f, g`$ be nonzero binary forms of the same degree $`n > 0`$ over a field of characteristic zero. If $`J(f,g) = 0`$ then $`f`$ and $`g`$ are proportional.

Euler's relation gives $`Y f_Y = n f - X f_X`$, and likewise for $`g`$, so that

$$Y \cdot J(f,g) \;=\; n\left(f_X\, g - f\, g_X\right)$$

as a polynomial identity. Hence $`J(f,g) = 0`$ forces $`f_X g = f g_X`$, that is $`\partial_X(g/f) = 0`$ wherever $`f \neq 0`$. So $`g/f`$ does not depend on $`X`$, and being homogeneous of degree 0 it is constant. $`\square`$

The hypotheses are used in order and each is visible: characteristic zero so that the positive degree $`n`$ is nonzero in the field and can be cancelled after Euler, whose identity itself needs no such hypothesis, $`f \neq 0`$ to divide, and equal degree for the last step, since $`g/f`$ is homogeneous of degree zero only when the degrees agree. The two forms in Section 3.3 are both sextics, so the degree hypothesis is automatic there; Section 5.7 needs a weaker unequal-degree form, developed where it is used. Nonvanishing is supplied by the next remark.

Because $`\Theta`$ is injective and the dictionary from states to sextics is linear, $`F \neq 0`$ and $`\Theta F \neq 0`$ whenever $`u \neq 0`$. So Lemma 3.1 applies to the pair $`(F, \Theta F)`$ at every nonzero state.

3.3 The kernel, and the proposition that follows from it

Lemma 3.2. For $`a \neq 0`$, the kernel of the linear map $`b \mapsto T(a,a,b)`$ is exactly $`\mathbb{C}a`$.

By the diagonal specialisation, $`T(a,a,b) = \kappa f (f,g)_1`$, and $`\kappa \neq 0`$ with $`f \neq 0`$, so the kernel is cut out by $`(f,g)_1 = 0`$. By Lemma 3.1 this holds exactly when $`g`$ is proportional to $`f`$, that is when $`b \in \mathbb{C}a`$. $`\square`$

Proposition 3.3. For $`u \neq 0`$, $`\mathcal{C}(u) = 0`$ if and only if $`[u]`$ is time-reversal invariant. Equivalently,

$$Z(\mathcal{C}) \;=\; U(1)\cdot\mathrm{Fix}(\Theta) .$$

Apply Lemma 3.2 at $`a = u`$, $`b = \Theta u`$. Its kernel condition reads $`T(u,u,\Theta u) = 0`$ if and only if $`\Theta u \in \mathbb{C}u`$, and by Section 3.1 that left side is $`-2\,\mathcal{C}(u)`$. Antiunitarity makes the constant a phase, so $`\Theta u = \lambda u`$ with $`\lvert\lambda\rvert = 1`$, which by Section 2.2 is exactly membership in $`U(1)\cdot\mathrm{Fix}(\Theta)`$. $`\square`$

So the proposition is a corollary rather than a parallel result: one classical criterion, applied once, yields both statements. The slot distinction that Section 2.3 insists on is a distinction between $`B_J`$ and $`\rho_K`$, and it remains; what does not survive is the idea that these two zero sets are reached independently.

3.4 The constellation reading

One standard fact is used: time reversal acts on the Majorana constellation as the antipodal map of the sphere. Everything else follows in a line. The constellation is the root multiset of the sextic $`F`$, so by the fundamental theorem of algebra it determines $`[u]`$ and is determined by it; and $`\Theta u`$ proportional to $`u`$ says exactly that the antipodal map fixes that multiset, that is, that the six points are antipodally symmetric, coincidences included. Combining with Proposition 3.3:

The spin-8 cubic channel closes exactly on the rays whose Majorana constellation is antipodally symmetric.

Section 2's Lemma 2.2 reaches the same locus by a different route, through the vanishing of the odd density multipoles, and the agreement is a useful check rather than a second proof: one argument runs through the Jacobian criterion on binary forms, the other through the exchange symmetry of a simple tensor.

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4. The reduction

Section 3 was ambient. From here the quotient enters, and this section covers the first of the two ways it does so: it decides which levels a density is allowed to have. The second, the multiplicity-free branching that scalarises the right-index contraction and fixes the weight, enters in Section 5.

4.1 The density is bandlimited

Let $`\psi`$ be a block state at level 6, so its left index runs over $`V_3`$. Its density $`\lvert\psi\rvert^2`$ is a function on $`X = S^3/2I`$, and by Section 2.4 its level-$`2K`$ component lies in

$$V_K \otimes \left(V_K\right)^{2I} ,$$

the left factor built from the spin-3 data and the right factor from the sector. A level therefore survives only when $`\left(V_K\right)^{2I} \neq 0`$.

The density's left content is $`V_3 \otimes \overline{V_3}`$, which spans ranks 0 through 6, that is levels 0 through 12. That window is closed: the density cannot reach beyond level 12, so what happens at level 20 and above is irrelevant to it. Within the window, the dimensions recorded in Section 2.4 leave exactly two survivors, at levels 0 and 12.

Lemma 4.1. The density of a block state at level 6 has the form

$$\lvert\psi\rvert^2 \;=\; c_0 \;+\; d_{12} ,$$

a constant plus a level-12 component. No intermediate level occurs.

This is the first of the paper's two icosahedral inputs, the invariant-degree filter. It gives five vanishings and is silent both on whether the surviving ranks select single maps and on whether the surviving weight is nonzero. Both of those come from the second input, the multiplicity-free branching of Section 2.4: it scalarises the right-index contraction in Section 5.1 and supplies the weight in Section 5.3. Lemma 4.1 is a statement about the coefficients of the density's expansion, not about the tensors that carry them. The level-$`2K`$ component of the density factorises, and Lemma 4.1 says the right-hand factor vanishes for $`K = 1,\dots,5`$. It does not say the left-hand factor does, and in general the left-hand factor does not: the density multipoles $`\rho_K(u)`$ of Section 2.3 are generically nonzero across the whole range, and at the octahedral state $`\rho_4`$ is the largest of them. A vanishing coefficient is not a vanishing tensor.

4.2 The four-dimensional family

The other side of the reduction is what there is to select from: the type-$`(2,1)`$ self-maps of $`V_3`$, which by Section 3.1 form the space $`\mathscr{E}_3`$. The weight count gives

$$\dim \mathscr{E}_3 \;=\; 16 - 12 \;=\; 4 ,$$

the weight-3 subspace having dimension 16 and the weight-4 subspace dimension 12. Both counts in this paper are of that form and both can be checked on the page.

This family is not new. A spin-3 contact interaction is specified by four scattering lengths, one for each total-spin channel $`S = 0, 2, 4, 6`$, and the resulting mean-field energy is the corresponding four-parameter combination [DH], [KU]. That is the same four-dimensional space, in the $`N_J`$ indexing below. Nothing in this paper claims the family; what is claimed is what the quotient does to it.

There is also a structural derivation, and it comes with a basis. Since $`\mathrm{Sym}^2 V_3 = V_0 \oplus V_2 \oplus V_4 \oplus V_6`$, each summand once, and since $`V_3`$ occurs exactly once in $`V_J \otimes V_3`$ for every $`J \le 6`$,

$$\mathscr{E}_3 \;=\; \bigoplus_{J = 0,2,4,6} \mathrm{Hom}_{\mathrm{SU}(2)}\!\left(V_J \otimes V_3,\, V_3\right) ,$$

four summands, each one-dimensional. A generator of the $`J`$-th is

$$N_J(u) \;=\; \left[\, [u \otimes u]_J \otimes \Theta u \,\right]_3 , \qquad J = 0, 2, 4, 6 ,$$

built from the holomorphic square of Section 2.3, which is why only even $`J`$ occurs: the odd $`N_J`$ vanish identically by the exchange sign, the same fact as $`B_J = 0`$ for odd $`J`$.

The maps the selection is naturally phrased in are indexed instead by the density's ranks. For $`0 \le K \le 6`$ put

$$A_K(u)\,v \;=\; \left[\, \rho_K(u) \otimes v \,\right]_3 , \qquad M_K(u) \;=\; A_K(u)\, u ,$$

so $`A_K(u)`$ is a linear operator on $`V_3`$ with the density frozen at $`u`$, and $`M_K`$ is the cubic self-map obtained by feeding the same state back in. The distinction is not cosmetic and Section 5 will need it.

So the family has two natural indexings, and they are the two objects Section 2.3 warns must be kept apart: the canonical basis $`N_J`$ is indexed by the ranks of the holomorphic square, and the set $`M_K`$ by the ranks of the density. They are related by recoupling. The $`M_K`$ are seven vectors in a four-dimensional space and are therefore linearly dependent, which is worth remembering before drawing any conclusion from the index $`K`$ alone. Since $`M_0`$ turns out to be the radial direction, the selection theorem needs only that $`M_6`$ is not radial, which is what makes the surviving ray a ray rather than nothing. That is proved in Section 5 from material the theorem needs anyway, and is not assumed here. The seven $`M_K`$ together span $`\mathscr{E}_3`$, verified by exact computation. That is a separate fact and is not used below.

4.3 The spine

These are the two targets named at the head of Section 3, and the contrast between them is what gives the paper its shape. Put the counts side by side.

$$\dim \mathscr{E}_8 = 1 , \qquad \dim \mathscr{E}_3 = 4 .$$

The spin-8 target is rigid: its type-$`(2,1)`$ covariant space is one-dimensional, so there is no nontrivial projective choice for a geometry to make within it. A geometry might still decide whether that channel appears, or with what coefficient; what it cannot do is choose among projectively distinct maps, because there is only one ray. The spin-3 target is four-dimensional, so channel selection has genuine content there.

The same method answers the corresponding availability question, whether the quotient supplies a compatible spin-8 target slot at all. Both $`V_3`$ and $`V_8`$ have integer spin and so factor through $`A_5`$, and the branching is

The two constituents of $`V_3`$ are $`\mathbf{3}'`$ and $`\mathbf{4}`$, and both occur in $`V_8`$ with multiplicity one, so both sectors occurring at level 6 also occur at level 16 and the spin-8 target is available in either. That is what the table establishes and all that is claimed here: it says the slot exists, not that the particular covariant has nonzero coefficient into it, which would need a right-index contraction for $`\mathcal{C}`$ of the kind Section 5 carries out for the density. The slot is at least reachable: by Lemma 4.1 the density has levels 0 and 12 only, level 0 times level 6 returns level 6 alone, and level 12 times level 6 spans levels 6 through 18. So the entire level-16 output comes from $`d_{12}\psi`$, and the only obstruction is whether the right-hand factor vanishes. That is settled in Section 5.7, which needs the weight computed in Section 5.3 and so cannot be argued here: it does not vanish, and the channel is populated in both sectors. The level-12 row is the one Lemma 4.1 turns on, the trivial representation appearing there being the density's surviving channel, so the bandlimit and the output channel can be read off a single table. This is why the two halves of the paper have different characters. Proposition 3.3 is a statement about a canonical object and is true for any spin-3 system with time reversal. The selection theorem is a statement about which member of a genuine family a particular quotient permits, and it has no content at all without the family being larger than one.

4.4 What the reduction cannot touch

Lemma 4.1 constrains the density's spectrum and nothing else. It says nothing about $`\mathscr{E}_3`$ itself, which remains whatever spin-3 representation theory makes it: the dimension, the canonical basis $`N_J`$, the operators $`A_K`$ and their weight structure, the maps $`M_K`$, an overcomplete family, and the relations among them are all fixed before $`2I`$ is mentioned, and are the same for any system carrying spin 3 with time reversal. What the quotient supplies is a set of coefficients, most of them zero.

Stated as a slogan: the family is universal, and the quotient selects the nonradial ray and its weight. Section 5 carries out that choice and computes the one coefficient that survives.

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5. The selection theorem

5.1 Statement

Theorem 5.1. Let $`\psi`$ be a block state at level 6 on $`X = S^3/2I`$, in the sector $`\sigma`$, write $`d = \dim\sigma \in \{3,4\}`$, and take the self-interaction and the quartic to be those of Section 2.5, namely the block projection of $`\lvert\psi\rvert^2\psi`$ and the ratio built from $`\int\lvert\psi\rvert^4`$. Then:

1. the self-interaction lies in $`\mathrm{span}\{M_0, M_6\} \subset \mathscr{E}_3`$;
2. $`M_0`$ is radial, $`M_0(u) = -\lVert u\rVert^2 u/\sqrt{7}`$;
3. $`M_6`$ is not radial;
4. modulo the radial direction the self-interaction spans the single projective ray $`[M_6]`$, its coefficient being nonzero by the argument in Section 5.5; on the unit sphere the tangential critical problem is then governed by $`M_6`$ alone;
5. on the unit sphere the governing quartic is $`Q = w_0/7 + w_6 \lVert\rho_6\rVert^2`$, with

$$\frac{w_6}{w_0} \;=\; \frac{7-d}{13\,d} \;>\; 0 .$$

Clause 1 needs one step beyond Lemma 4.1, which constrains the density and not the self-interaction. The level-$`2K`$ component of the density multiplies $`\psi`$; the left factor couples to give $`A_K(u)u = M_K(u)`$, and the right factor contracts against $`\eta`$ to a scalar, by Schur, since the multiplicity is one. So each surviving level contributes a multiple of its own $`M_K`$ and the vanishing levels contribute nothing, which with Lemma 4.1 is clause 1. Clauses 2 through 5 are proved below.

The theorem uses two icosahedral inputs and no others, and naming them exactly matters here, since the scalarisation just used is one of them. They are the invariant-degree filter, which leaves density ranks 0 and 6 and is Lemma 4.1; and the multiplicity-free complementary branching $`V_3\vert_{2I} = \sigma_3 \oplus \sigma_4`$, which does two jobs, scalarising the right-index contraction here by Schur and supplying the weight in Section 5.3. Multiplicity one and complementarity are two consequences of that one branching statement, which is why the count is two and not three.

5.2 The quartic, and the Peter-Weyl factorisation

Two preliminaries. First, clause 2. The rank-0 coupling contracts $`u`$ against $`\Theta u`$ to a scalar,

$$\rho_0(u) \;=\; [u \otimes \Theta u]_0 \;=\; -\frac{\lVert u\rVert^2}{\sqrt{7}} , \qquad\text{so}\qquad M_0(u) \;=\; [\rho_0(u) \otimes u]_3 \;=\; -\frac{\lVert u\rVert^2}{\sqrt{7}}\, u ,$$

which is clause 2. Within $`\mathscr{E}_3`$, "radial" and "proportional to $`M_0`$" are the same condition: a radial equivariant map has the form $`\lambda(u)u`$ with $`\lambda`$ an invariant of type $`(1,1)`$, and $`\mathrm{Hom}_{\mathrm{SU}(2)}(V_3 \otimes \overline{V_3}, \mathbf{1})`$ is one-dimensional, so $`\lambda \propto \lVert u\rVert^2`$ and the map is a multiple of $`M_0`$.

Second, the quartic itself, which has two forms. The homogeneous one is

$$\widetilde{Q}(u) \;=\; \sum_K w_K \lVert \rho_K(u)\rVert^2 ,$$

and the functional actually extremised is its projective normalisation

$$Q([u]) \;=\; \frac{\widetilde{Q}(u)}{\lVert u\rVert^4} , \qquad u \neq 0 ,$$

which descends to rays and agrees with $`\widetilde{Q}`$ on the unit sphere. Since $`\lVert\rho_0(u)\rVert^2 = \lVert u\rVert^4/7`$, clause 5 is the value of $`Q`$ there.

The scale of the $`w_K`$ is not a convention either, because the denominator can be carried explicitly. With Haar measure normalised and $`\eta^{\dagger}\eta = I_\sigma`$,

$$\int \lvert\psi\rvert^2 \;=\; \frac{1}{7}\lVert u\rVert^2\, \mathrm{tr}\,P \;=\; \frac{d}{7}\lVert u\rVert^2 ,$$

so $`(\int\lvert\psi\rvert^2)^2 = (d^2/49)\lVert u\rVert^4`$ and there is no undetermined factor anywhere.

By the convention fixed in Section 2.4 a section is $`\psi_a(g) = \sum_{m,n} u_m D^3_{mn}(g)\,\eta_{na}`$. Expanding the density and using $`D^3 D^3 = \sum_K \langle\cdot\rangle\langle\cdot\rangle D^K`$ separates the two indices: the level-$`2K`$ component of $`\lvert\psi\rvert^2`$ is

$$d_{2K} \;=\; \rho_K(u) \otimes R_K(P) , \qquad P = \eta\eta^{\dagger} ,$$

with both factors the same transform $`\mathcal{M}_K`$ of Section 2.3, applied to $`uu^{\dagger}`$ on the left and to $`P`$ on the right. Peter-Weyl orthogonality, $`\int D^K_{MN}\overline{D^{K'}_{M'N'}} = \delta/(2K+1)`$, gives the numerator, while the normalisation above gives the denominator:

$$A(u) \;:=\; \int\lvert\psi\rvert^4 \;=\; \sum_K \frac{\lVert\rho_K(u)\rVert^2\,\lVert R_K(P)\rVert^2}{2K+1} , \qquad B(u) \;:=\; \int\lvert\psi\rvert^2 \;=\; \frac{d}{7}\lVert u\rVert^2 .$$

Hence

$$Q([u]) \;=\; \frac{A(u)}{B(u)^2} \;=\; \frac{1}{\lVert u\rVert^4}\sum_K w_K \lVert\rho_K(u)\rVert^2 , \qquad w_K \;=\; \frac{49}{d^2}\cdot\frac{\lVert R_K(P)\rVert^2}{2K+1} ,$$

and $`\widetilde{Q}(u) = \sum_K w_K\lVert\rho_K(u)\rVert^2 = (49/d^2)A(u)`$. With the measure fixed that way there is no free constant left anywhere, and in particular $`w_0 = (49/d^2)(d^2/7) = 7`$ is derived, not chosen. The measure is the one choice being made, and it is worth saying what depends on it: $`Q = A/B^2`$ scales inversely with the measure, so the individual $`w_K`$ and the normalisation of Section 5.6 inherit that choice, while the ratio $`w_6/w_0`$, the selection theorem and the critical set do not.

Lemma 4.1 in this language says $`R_K(P) = 0`$ for $`K = 1,\dots,5`$, which is a statement about $`P`$ and leaves $`\rho_K(u)`$ untouched.

5.3 The weight

Lemma 5.2. $`\lVert R_6(P)\rVert^2 = d(7-d)/7`$, and $`\lVert R_0(P)\rVert^2 = d^2/7`$.

Three steps. First, $`\mathcal{M}_K`$ obeys a Parseval identity, $`\sum_K \lVert \mathcal{M}_K(P)\rVert^2 = \lVert P\rVert_F^2`$ for any $`7\times 7`$ matrix $`P`$. Second, $`P`$ is an orthogonal projector by the normalisation fixed in Section 2.4, so $`\lVert P\rVert_F^2 = \mathrm{tr}\,P = d`$. Third, $`R_0(P)_0 = -\mathrm{tr}(P)/\sqrt{7}`$, so $`\lVert R_0\rVert^2 = d^2/7`$. Lemma 4.1 removes every other term from the Parseval sum, leaving

$$\lVert R_6 \rVert^2 \;=\; d - \frac{d^2}{7} \;=\; \frac{d(7-d)}{7} . \qquad \square$$

This is where the second icosahedral input does its second job. It has already entered, in Section 5.1, where multiplicity one scalarised the right-index contraction; here the same branching supplies the weight. The expression $`d(7-d)/7`$ is positive exactly when $`0 < d < 7`$, and Section 2.4 supplies that: $`V_3\vert_{2I}`$ is multiplicity-free with two constituents, of dimensions 3 and 4, so both are proper and neither $`d`$ is 0 or 7. The expression is also symmetric under $`d \leftrightarrow 7-d`$, so the two sectors carry the same value,

$$\lVert R_6\rVert^2 \;=\; \frac{3 \cdot 4}{7} \;=\; \frac{12}{7} \quad\text{in both},$$

for a reason rather than by coincidence: they are complementary in $`\dim V_3`$. Combining with Section 5.2,

$$\frac{w_6}{w_0} \;=\; \frac{\lVert R_6\rVert^2/13}{\lVert R_0\rVert^2/1} \;=\; \frac{7-d}{13\,d} \;>\; 0 ,$$

which is clause 5. Since $`w_0 = 7`$ was derived rather than chosen, the absolute weight follows too: $`w_6 = 7(7-d)/(13d)`$, namely $`21/52`$ at $`d = 4`$ and $`28/39`$ at $`d = 3`$.

That $`R_K(P) = 0`$ for $`K = 1,\dots,5`$ is not a separate fact: $`P`$ is $`2I`$-invariant, so its multipoles lie in $`(V_K)^{2I}`$, and Lemma 4.1 says those vanish.

5.4 The surviving operator, and the map it generates

Write $`\Lambda_m = (-1)^{3+m}\binom{6}{3+m}`$, the alternating sixth row of Pascal's triangle. At weight states the frozen-density operator of Section 4.2 is an outer product,

$$A_6(v_i)\, v_j \;=\; c\, \Lambda_i \Lambda_j\, v_j , \qquad c \;=\; -\frac{1}{\binom{12}{6}}\sqrt{\frac{\dim V_3}{\dim V_6}} ,$$

diagonal in the second slot. Feeding the same state back in gives the cubic map, which carries $`\Lambda`$ squared:

$$M_6(v_i) \;=\; c\, \Lambda_i^2\, v_i , \qquad \Lambda^2 \;=\; (1,\, 36,\, 225,\, 400,\, 225,\, 36,\, 1) .$$

The outer product belongs to the operator and the square to the map; they are different objects.

The two tensor factors of $`\Lambda\otimes\Lambda`$ have different origins, which is worth recording because the alternation is the whole content. The multiplication slot alternates by itself; the density slot does not, and takes its alternation from the phase in $`\Theta`$:

$$\langle 3\,m;\, 3\,{-m} \mid 6\,0\rangle \;=\; \tfrac{\sqrt{231}}{462}\,(1,\,6,\,15,\,20,\,15,\,6,\,1) , \qquad \langle 6\,0;\, 3\,m \mid 3\,m\rangle \;=\; \tfrac{\sqrt{429}}{858}\,\Lambda .$$

The first is strictly positive, in the fixed Condon-Shortley convention, and in closed form $`\langle j\,m;\, j\,{-m}\mid 2j\,0\rangle = \frac{(2j)!}{\sqrt{(4j)!}}\binom{2j}{j+m}`$. So it is not the case that both Clebsch-Gordan evaluations produce an alternating row. The density row comes out with unsigned magnitudes and time reversal supplies its signs; the multiplication row displayed above is already alternating. The two factors of $`\Lambda`$ therefore reach $`\Lambda^2`$ by different routes, which is the point of the next paragraph.

What the alternation buys is that $`\Lambda`$ is a sixth finite difference:

$$\sum_m \Lambda_m\, m^k \;=\; 0 \quad (k = 0,\dots,5) , \qquad \sum_m \Lambda_m\, m^6 \;=\; 720 .$$

The unsigned row annihilates no even moment, giving $`64, 96, 384`$ at $`k = 0, 2, 4`$. Both rows are even in $`m`$, so both kill every odd moment by parity; that is a shared symmetry carrying no information, and the separation is entirely in the even moments.

5.5 That $`M_6`$ is not radial, and the quartic

Clause 3 is one line from Section 5.4. By Section 5.2, radial and proportional to $`M_0`$ are the same condition in $`\mathscr{E}_3`$, so suppose $`M_6 = \alpha M_0`$. Evaluating both at the weight state $`v_i`$ gives $`c\,\Lambda_i^2 = -\alpha/\sqrt{7}`$ for every $`i`$, so $`\Lambda^2`$ would be constant. It is $`(1, 36, 225, 400, 225, 36, 1)`$. Hence $`M_6`$ is not radial. The same row is why the second interaction of Section 2.5 lands in the genuinely different plane $`\mathrm{span}\{N_0, N_6\}`$: $`N_0`$ annihilates every weight state $`v_i`$ with $`i \neq 0`$ and does not annihilate $`v_0`$, while no combination of $`M_0`$ and $`M_6`$ does that, since this row is not constant off the centre.

That is not yet clause 4, which also needs the self-interaction to contain $`M_6`$ with nonzero coefficient. Write the self-interaction as $`t_0 M_0 + t_6 M_6`$, which clause 1 permits. It is the block projection of $`\lvert\psi\rvert^2\psi`$, and the block projection is self-adjoint with $`\psi`$ in its range, so pairing it with $`\psi`$ returns $`\int\lvert\psi\rvert^4`$, that is $`\widetilde{Q}`$ up to the normalisation of Section 5.2. Carrying that normalisation explicitly, the pairing is $`(d/7)\,\langle u, \mathcal{N}(u)\rangle = A(u)`$, the $`d/7`$ being the block inner product inherited from $`\int\lvert\psi\rvert^2 = (d/7)\lVert u\rVert^2`$. For a fixed sector $`\sigma`$, $`t_0`$ and $`t_6`$ are constants determined by the right contractions, and they are not assumed related. If $`t_6`$ vanished, the self-interaction would be $`t_0 M_0`$, so $`A(u)`$ would be proportional to $`\langle u, M_0(u)\rangle = -\lVert u\rVert^4/\sqrt{7}`$, hence constant on the unit sphere. It is not: by clause 5, $`\widetilde{Q} = (49/d^2)A`$ is $`w_0/7 + w_6\lVert\rho_6\rVert^2`$ with $`w_6 > 0`$, and a positive multiple of $`A`$ is constant only if $`A`$ is, and $`\lVert\rho_6\rVert^2`$ takes the values $`1/\binom{12}{6}`$ and $`400/\binom{12}{6}`$ at $`v_3`$ and $`v_0`$. So $`t_6 \neq 0`$, and with clauses 1 and 2 that is clause 4. $`\square`$

The same non-constancy appears on the quartic side as an identity worth recording,

$$\binom{12}{6}\, \lVert \rho_6(v_m)\rVert^2 \;=\; \Lambda_m^2 ,$$

which is an identity rather than a sample: at a weight state $`\rho_6`$ has only its zonal component, so $`\lVert\rho_6\rVert^2 = \binom{6}{3+m}^2/\binom{12}{6}`$. The $`\binom{12}{6}`$ here is the same constant as in $`c`$ above and for the same reason, the normalisation of the stretched Clebsch-Gordan coefficient, and the $`\Lambda^2`$ carried by the cubic map and the $`\Lambda^2`$ appearing here are one fact seen twice.

Both coefficients of the self-interaction are available, not only the nonvanishing of the second. Pairing it with $`u`$ and evaluating at $`v_3`$ and $`v_0`$ gives two linear equations whose solution, written $`\mathcal{N}`$ from here on, is

$$\mathcal{N}(u) \;=\; \frac{d}{7}\lVert u\rVert^2 u \;-\; \frac{7-d}{\sqrt{91}}\, M_6(u) ,$$

the radial part carrying the expected sign. Solved from two states, it holds at the octahedral and hexagonal rays as well, so it is determined rather than fitted.

Relating the two sides away from weight states is a separate identity:

$$\nabla_{\bar u} \lVert \rho_K(u)\rVert^2 \;=\; c_K\, M_K(u) , \qquad c_K \;=\; (-1)^{K+1}\, 2\sqrt{\frac{\dim V_K}{\dim V_3}} ,$$

so the quartic weight and the cubic weight differ by a fixed nonzero factor per channel, and it is not the case that the same coefficient appears in both. Every $`c_K`$ is nonzero, so no channel is lost this way. Section 7.5 obtains the same relationship structurally, as an equivariant isomorphism from the invariant quartics onto $`\mathscr{E}_3`$; the identity here is that correspondence written out rank by rank. The two are not interchangeable as stated: the $`M_K`$ are seven vectors in a four-dimensional space and so are linearly dependent, and reading nonvanishing of every $`c_K`$ as injectivity would need the spanning fact of Section 4.2, which is not used here. This holds as an identity in $`u`$ and $`\bar u`$, carried as fourteen independent variables rather than checked at finitely many states. Nothing in Theorem 5.1 depends on it: clause 3 comes from $`\Lambda^2`$ and clause 4's bridge from the pairing argument above, neither of which needs a per-channel gradient. It is stated here because the per-channel factor is easy to assume away.

One last calculation completes clause 4's second sentence, that on the unit sphere the tangential critical problem is governed by $`M_6`$ alone. Since $`Q = A/B^2`$,

$$\nabla_{\bar u} Q \;=\; \frac{1}{B^2}\nabla_{\bar u}A \;-\; \frac{2A}{B^3}\nabla_{\bar u}B ,$$

and $`B(u) = (d/7)\lVert u\rVert^2`$, so $`\nabla_{\bar u}B = (d/7)\,u`$ and the second term is radial. The first is proportional to the projected self-interaction $`\mathcal{N}`$. Projecting onto $`u^{\perp}`$,

$$\Pi_{u^{\perp}} \nabla_{\bar u} Q \;\propto\; \Pi_{u^{\perp}} \mathcal{N}(u) \qquad (\lVert u\rVert = 1) ,$$

so the denominator contributes only the radial, Lagrange-multiplier term. With clause 1 and $`M_0`$ radial, the tangential equation involves $`M_6`$ alone, and by the bridge above with a nonzero coefficient. That is clause 4 in full, and none of it uses the per-channel identity.

5.6 The normalisation, and what it does not settle

The theorem is complete; one ceiling belongs here before the discussion, on the constant its weight is built from. The sector normalisation is $`N = \binom{12}{6}/w_6`$, and Section 5.3 gives it in closed form:

$$N \;=\; \binom{12}{6}\,\frac{\dim V_6}{\dim V_3}\,\frac{d}{7-d} \;=\; \binom{13}{6}\,\frac{d}{7-d} , \qquad 1287 \text{ at } d = 3 , \quad 2288 \text{ at } d = 4 .$$

The first form is canonical because it is what the derivation produces: the left Clebsch-Gordan constant, times the Peter-Weyl dimension ratio, times the sector. The binomial identity $`\binom{12}{6}\cdot 13/7 = \binom{13}{6}`$ is noticed afterwards and is not a new primitive.

Two ceilings belong here rather than in the discussion. First, an earlier form of this normalisation, $`N = 143\,d^2`$, is exactly correct on both sectors and always will be, since $`d(7-d) = 12`$ at $`d = 3`$ and $`d = 4`$. What the derivation shows is that it is not the primitive form: the quadratic shape is a consequence of the two sectors being complementary in 7. Second, and more sharply, $`d = 3`$ and $`d = 4`$ are the only values that occur, so no measurement inside this system separates the two formulas. They agree on every available case. The correction is carried entirely by the derivation above and has no empirical content of its own. Agreement on every available case invites the assumption that the choice between the two formulas was measured; it was not.

5.7 The spin-8 channel on the quotient

Section 4.3 left one question open, whether the spin-8 channel whose target slot exists in both sectors actually carries a nonzero coefficient there. With Lemma 5.2 in hand it closes.

This section's tool is the unequal-degree form of Lemma 3.1. Without equal degree the statement genuinely fails, and a two-line witness shows what replaces it: $`f = X^2`$ and $`g = X^3`$ have $`J(f,g) = 0`$ and are not proportional, while $`f^{\deg g} = g^{\deg f} = X^6`$, which is the weaker conclusion available at unequal degrees. The same Euler computation with degrees $`m = \deg f`$ and $`n = \deg g`$ gives $`Y \cdot J(f,g) = n f_X g - m f g_X`$. Homogeneity alone settles nothing here, since $`f^{n}`$ and $`g^{m}`$ both have degree $`mn`$ whatever $`J`$ does, and a degree-zero homogeneous rational function need not be constant, as $`X/Y`$ shows. The vanishing is used through the derivative instead: $`\partial_X(f^{n}/g^{m}) = f^{\,n-1}(n f_X g - m f g_X)/g^{\,m+1}`$, which is zero when $`J(f,g) = 0`$. So $`f^{n}/g^{m}`$ is independent of $`X`$, and being homogeneous of degree zero it is constant, exactly as at equal degree in Section 3.2. Since $`\mathbb{C}[X,Y]`$ is a unique factorisation domain, a proportionality $`f^{n} \propto g^{m}`$ constrains the root multisets: if $`f`$ has a root of multiplicity $`\mu`$ then $`g`$ has the same root with multiplicity $`\mu n/m`$. The witness above checks it: $`f = X^2`$ has a double root and $`g = X^3`$ a triple one, and $`\mu n/m = 2\cdot 3/2 = 3`$.

The right-hand factor of the level-16 output is built from $`R_6(P)`$, which lies in $`(V_6)^{2I}`$. That space is one-dimensional by Section 2.4, and its generator has twelve simple roots, which the orbit structure forces and the picture only suggests. The zero divisor of a $`2I`$-invariant binary form is $`2I`$-invariant, hence a union of orbits of $`A_5`$ on $`\mathbb{P}^1`$, and those orbits have sizes 12, 20, 30 and 60, with stabilisers the cyclic groups of orders 5, 3, 2 and 1. A divisor of degree 12 must therefore be the size-12 orbit taken once, so its roots are twelve distinct points, the icosahedron's vertices. Lemma 5.2 gives $`\lVert R_6(P)\rVert^2 = 12/7 \neq 0`$, so $`R_6(P)`$ is a nonzero multiple of that invariant; write it $`I_{12}`$.

The coupling $`V_6 \otimes V_3 \to V_8`$ is unique up to scale and is the first transvectant, the degrees $`12`$ and $`6`$ giving order $`12 + 6 - 2 = 16`$. So the question is whether $`v \mapsto (I_{12}, f_v)_1`$ can kill a nonzero $`v`$. It cannot. If $`(I_{12}, f)_1 = 0`$ with $`f \neq 0`$, the unequal-degree form gives $`I_{12}^{\,6} \propto f^{\,12}`$, so each root of $`I_{12}`$ occurs on the left with multiplicity $`6\mu`$ and on the right with multiplicity $`12\nu`$, forcing $`\mu = 2\nu`$. A root of $`I_{12}`$ has $`\mu \ge 1`$, so $`\nu = 0`$ would give $`\mu = 0`$, while $`\nu \ge 1`$ gives $`\mu \ge 2`$. Both contradict twelve simple roots.

So $`v \mapsto (I_{12}, f_v)_1`$ is injective, and composing it with the nonzero intertwiner $`\eta`$ leaves the right-hand factor nonzero in both sectors. One sector suffices for the norm: $`P_{\mathbf{3}'} + P_{\mathbf{4}} = I`$ and $`\mathcal{M}_6(I) = 0`$, so $`R_6(P_{\mathbf{3}'}) = -R_6(P_{\mathbf{4}})`$ and the two norms are equal for that reason rather than by separate computation. The left-hand factor is $`[\rho_6(u)\otimes u]_8`$. Since $`\dim\mathscr{E}_8 = 1`$ leaves no other equivariant map, it is some multiple of $`\mathcal{C}(u)`$, say $`\alpha\,\mathcal{C}(u)`$; that $`\mathcal{C}`$ is itself nonzero does not yet make $`\alpha`$ nonzero, so the constant is evaluated rather than inferred. At $`u = v_3`$, $`\rho_6(v_3) = -\tfrac{\sqrt{231}}{462}\,v^{(6)}_0`$ and coupling to spin 8 gives $`[\rho_6(v_3)\otimes v_3]_{8,3} = \tfrac{\sqrt{273}}{1092} \neq 0`$. Hence $`\alpha \neq 0`$. The level-16 component of the nonlinearity therefore does not vanish identically, and Proposition 3.3 becomes a statement about $`X`$ rather than an ambient one:

Among block states at level 6, those whose cubic nonlinearity has no level-16 component are exactly the time-reversal-invariant ones.

That also settles what Section 3 is doing in this paper. It is not an independent note sharing a state space, and not merely the same weight count at a second target: the ambient proposition describes a channel that the quotient leaves open, and the states it distinguishes are states of $`X`$.

5.8 The critical rays of the reduced quartic

Clause 5 of Theorem 5.1 is what this subsection resumes, after the normalisation ceiling and the spin-8 payoff: the quartic is affine in $`\lVert\rho_6\rVert^2`$ alone, so its critical rays are those of $`\lVert\rho_6\rVert^2`$ on the unit sphere in $`V_3`$, and since $`w_6 > 0`$ the maximiser of one is the maximiser of the other. Both are questions about spin 3 with time reversal and neither mentions $`2I`$. Four values, with $`\binom{12}{6}\lVert\rho_6\rVert^2`$ quoted as an integer:

Those four rays are critical, which the table alone does not show.

Lemma 5.3. Each of the four rays above is a critical point of $`\lVert\rho_6\rVert^2`$ on the unit sphere.

Let $`H`$ be the stabiliser of $`[u]`$ in $`\mathrm{SO}(3)`$, acting on $`u`$ by a character $`\chi`$. Since $`\lVert\rho_6\rVert^2`$ is $`\mathrm{SO}(3)`$-invariant its gradient is equivariant, so the tangential gradient at $`[u]`$ lies in the $`H`$-fixed part of the projective tangent space, which is $`\mathrm{Hom}_H(\chi, V_3)`$ modulo $`\mathbb{C}u`$. For each of the four rays that quotient is zero, because the $`\chi`$-isotypic subspace of $`V_3`$ is one-dimensional and $`u`$ already spans it. Hence the tangential gradient vanishes. $`\square`$

For the two weight states the cyclic group of rotations about the quantisation axis suffices: the projective tangent weights are $`\pm 1, \pm 2, \pm 3`$ at $`[v_0]`$ and $`-1, \dots, -6`$ at $`[v_3]`$, and none of them is zero. For the octahedral ray a subgroup of order 8 already isolates it, and for the hexagonal ray the dihedral group of order 12 does. No orientation is assumed: in each case the stabiliser is computed rather than posited.

Two further critical rays do not appear in that table. Each lies on a fixed locus of dimension one rather than at an isolated fixed point, so Lemma 5.3 does not cover them as stated and needs a second clause.

Lemma 5.3(b). Suppose the $`\chi`$-isotypic space of $`H`$ is two-dimensional, so the fixed locus is a projective line. If $`\lVert\rho_6\rVert^2`$ is stationary along a real curve in that line and a further symmetry makes it even in the transverse coordinate, the ray is critical.

The transverse derivative vanishes by that evenness, and those two real directions exhaust the tangential fixed space, so the whole tangential gradient vanishes. Both rays below satisfy it, and both can be checked against the relation

$$\langle u, M_6(u)\rangle \;=\; \frac{2}{c_6}\,\lVert\rho_6\rVert^2 \;=\; -\frac{\sqrt{91}}{13}\,\lVert\rho_6\rVert^2 ,$$

which follows from the gradient identity by Euler and holds at every state, critical or not. Where $`M_6(u) = \lambda u`$ the constant is therefore read off the value rather than quoted beside it.

The pentagonal pyramid. On the line $`u = \cos t\, v_2 + \sin t\, v_{-3}`$,

$$\lVert\rho_6\rVert^2 \;=\; -\tfrac{125}{132}\sin^4 t \;+\; \tfrac{10}{11}\sin^2 t \;+\; \tfrac{3}{77} ,$$

stationary in the interior at $`\sin^2 t = 12/25`$, where $`\lVert\rho_6\rVert^2 = 9/35`$ and $`\binom{12}{6}\lVert\rho_6\rVert^2 = 1188/5`$, the first such value that is not an integer. The rotations of order 5 about the quantisation axis act on $`\mathrm{span}\{v_2, v_{-3}\}`$ by a single character, so that line is a fixed locus. Rotation about the same axis carries the line to itself and advances the relative phase of the two components at five times its own angle, so $`\lVert\rho_6\rVert^2`$ is constant transverse to the real curve and Lemma 5.3(b) applies along it; the criterion confirms it, with $`M_6(u) = -\tfrac{9}{455}\sqrt{91}\,u`$. Its Majorana polynomial is $`z(a + bz^5)`$ with $`a, b \neq 0`$, so the constellation is one point at a pole and five in a ring, six distinct points: a non-degenerate pentagonal pyramid. The endpoints $`t = 0`$ and $`t = \pi/2`$ are stationary too, but they are the weight states $`v_2`$ and $`v_{-3}`$, already covered by Lemma 5.3; the interior point is the one this line contributes.

The trigonal prism, and with it the octahedron. Take $`H = \langle R_z(2\pi/3), R_x(\pi)\rangle \cong D_3`$, with $`R_x`$ the in-plane axis through a vertex column: it acts by $`-1`$ on both $`v_3 + v_{-3}`$ and $`v_0`$, so the $`\chi`$-isotypic space is $`\mathrm{span}\{v_3+v_{-3},\, v_0\}`$ and the fixed locus is the projective line $`\mathbb{P}\,\mathrm{span}\{v_3+v_{-3},\, v_0\}`$. A second family of in-plane axes, turned by $`30^\circ`$ from these, acts by $`+1`$ on $`v_3+v_{-3}`$ and does not fix this line. The distinction is easy to state wrongly. With three columns, the line through one of them runs between the other two on the far side, so through a column and between two columns pick out the same three lines; what separates the two families is the $`30^\circ`$ turn, and they give opposite answers. The chart $`u = v_3 + z\,v_0 + v_{-3}`$ covers that line except at one point, the zonal ray $`[v_0]`$ at $`z = \infty`$, which is treated below. On the chart, writing $`z = x + iy`$,

$$\lVert\rho_6\rVert^2 \;=\; \frac{100\lvert z\rvert^4 - 20x^2 + 148y^2 + 463}{231\,(\lvert z\rvert^2+2)^2} ,$$

whose critical set on the chart is five points: $`z = 0`$, the hexagon at $`463/924`$; $`z = \pm\sqrt{23/10}`$, where $`\lVert\rho_6\rVert^2 = 200/903`$ and $`\binom{12}{6}\lVert\rho_6\rVert^2 = 8800/43`$; and $`z = \pm i\sqrt{5/2}`$, where the value is $`24/77`$. The expression is even in each of $`x`$ and $`y`$ separately, so each is stationary in both directions and Lemma 5.3(b) applies along either axis. The rotation $`R_z(\pi/3)`$ acts on the chart by $`z \mapsto -z`$, so the two signs in each pair are one orbit and the five points are three orbits.

The point the chart omits is critical as well, and it is not a new ray. It lies on this locus because $`v_0`$ spans one of the two $`\chi`$-isotypic directions, and it is critical by Lemma 5.3, which already covers it: it is the zonal ray of the table above, at $`\lVert\rho_6\rVert^2 = 100/231`$, that is $`400/924`$. The reason to argue it that way rather than by differentiating in the complementary chart is that the derivative there settles nothing. Since $`v_0`$ is critical on the whole sphere, the gradient vanishes along every line through it, including lines that have nothing to do with $`D_3`$. Counting both charts, the locus carries six critical points in four orbits, and the hexagon's $`463/924`$ is the largest value on it.

The real axis carries the eclipsed pairs of triangles, Majorana polynomial $`z^6 - sz^3 + 1`$ with $`s = \sqrt{20}\,x`$ real, and the imaginary axis the staggered ones, where $`s`$ is purely imaginary. No rotation relates the two: a rotation preserves shape, and these families are not congruent. What $`R_z(\pi/6)`$ does is carry this line to the other $`\chi`$-isotypic line $`\mathrm{span}\{v_3-v_{-3},\, v_0\}`$, which is the same fact as its conjugating the $`-1`$ family of axes onto the $`+1`$ family: applied to a prism it returns a prism, turned by $`30^\circ`$, which on a configuration with a three-fold axis is indistinguishable from a turn of $`90^\circ`$ the other way. Both rays are identified exactly, and the quadratic does it. In $`w = z^3`$ the Majorana polynomial is $`w^2 - sw + 1`$, whose two roots multiply to $`1`$, so the triangles sit at reciprocal radii; since $`\lvert z\rvert \mapsto 1/\lvert z\rvert`$ sends $`\theta`$ to $`\pi - \theta`$, they sit at heights $`\pm h`$ for a common $`h`$ whatever $`s`$ is. The argument of $`w`$ separates the two axes. At $`x^2 = 23/10`$, $`s = \sqrt{46}`$ and both roots are real and positive, so $`\arg w = 0`$ and each triangle has vertices at azimuths $`0^\circ, 120^\circ, 240^\circ`$: the two are aligned, which makes the ray a trigonal prism and not an antiprism, at heights $`\pm 0.5585\ldots`$. At $`y^2 = 5/2`$, $`s = i\sqrt{50}`$ and the two roots are purely imaginary of opposite sign, with arguments $`+\pi/2`$ and $`-\pi/2`$; dividing the difference of $`\pi`$ by three, the azimuths differ by $`60^\circ`$: the two are staggered, and there $`h = 1/\sqrt{3}`$ exactly. A staggered pair of equilateral triangles at heights $`\pm 1/\sqrt{3}`$ is the regular octahedron, which is the same statement as its full multipole row there, $`1/7, 0, 0, 0, 6/11, 0, 24/77`$. Its reappearance is a check rather than a coincidence: an octahedron has a three-fold axis through opposite faces, so it has to occur in this locus. The tabulated state is exact by the same standard, with no root-finding: its sextic is a constant times $`z(z^4+1)`$, of degree 5, so one root sits at the far pole, one at the origin and four at the fourth roots of $`-1`$, spaced $`90^\circ`$ apart on the equator.

The hexagonal value is the largest of the four, and it is not this paper's. Romero, Klimov, Goldberg, Leuchs and Sanchez-Soto give the closed form $`\varrho^2_{2S} = \tfrac12 + \binom{4S}{2S}^{-1}`$ for the top multipole of the NOON state at integer spin $`S`$ [RK]; at $`S = 3`$ that is $`463/924`$, and their NOON state is this paper's hexagon up to a rotation. Their normalisation is the Parseval one used here, so the numbers are directly comparable.

The question of whether it is the global maximum of $`\lVert\rho_6\rVert^2`$ is also theirs, and they pose it for general $`S`$: they ask what the largest attainable highest-order multipole is, sample $`6 \times 10^4`$ random constellations, and find none exceeding the NOON value. The table does not settle it and four exact values do not determine a global extremum. What this section adds at that ray is not the value but its position: it is one point of an exactly determined critical set on each of two symmetry loci. The three time-reversal-invariant rays among them are the zonal, octahedral and hexagonal ones, and by Lemma 2.2 their odd density multipoles vanish, which is visible in the octahedral case as anticoherence of order exactly 3: its multipoles are $`1/7, 0, 0, 0, 6/11, 0, 24/77`$, so ranks 1, 2 and 3 vanish and rank 4 does not.

5.9 Three corollaries

Theorem 5.1 is a statement about one level and one interaction. Three consequences follow from it using the same representation-theoretic machinery, and they bracket it from below, across and above. They are not free: the first needs a character computation the theorem did not, and a rank-zero evaluation at general spin. Throughout this subsection write

$$\widehat{r}_6([u]) \;:=\; \frac{\lVert\rho_6(u)\rVert^2}{\lVert u\rVert^4} ,$$

which is the projective form of the top multipole and agrees with $`\lVert\rho_6\rVert^2`$ on the unit sphere of $`V_3`$. The distinction matters in the third corollary, where the normalisation imposed is on the section and not on the fibre representative.

Corollary 5.4. Nothing projective below level 6. For a single-sector block state with the density-type interaction of Section 2.5, the projected self-interaction is radial at every level $`\ell < 6`$. Level 6 is the first level at which $`S^3/2I`$ permits a non-radial cubic self-interaction at all.

Two pieces of notation first, since the statement ranges over levels and Section 2.3 fixed its transform on $`V_3`$ alone. Replacing $`3`$ by $`j`$ there is not quite enough: the sign $`(-1)^{n'}`$ inside $`\mathcal{M}_K`$ is real only at integer $`n'`$. Write instead

$$\mathcal{M}^{(j)}_K(P)_N \;=\; \sum_{n+n' = N} \langle j\,n;\, j\,n' \mid K\,N \rangle \; \varepsilon_j\,(-1)^{\,j+n'}\, P_{n,\,-n'} ,$$

with $`\rho^{(j)}_K`$ and $`M^{(j)}_K`$ built from it as in Section 2.3, and write the time-reversal phase as

$$\Theta_j v_m \;=\; \varepsilon_j\,(-1)^{\,j-m}\, v_{-m} , \qquad \lvert\varepsilon_j\rvert = 1 ,$$

which is defined at every $`j`$, integer or half-integer, because $`j - m`$ is an integer in both cases. The exponent $`m`$ used in Section 2.2 is not, which is why the convention is written this way here. As there, $`\varepsilon_j`$ is a free global phase and only the $`m`$-dependence is forced; Section 2.2's choice at $`j = 3`$ is $`\varepsilon_3 = -1`$, since $`(-1)^m = -(-1)^{3-m}`$.

Below level 6 the restriction $`V_j\vert_{2I}`$ is irreducible: the character sum gives $`\langle\chi,\chi\rangle = 1`$ at $`2j = 1,\dots,5`$ and first gives 2 at $`2j = 6`$, where the restriction becomes $`\mathbf{3}' \oplus \mathbf{4}`$. So at every level below 6 the sector fills $`V_j`$ and its isotypic projector is the identity. The identity is invariant under all of $`\mathrm{SU}(2)`$, not merely under $`2I`$, so $`\mathcal{M}_K(I)`$ is an $`\mathrm{SU}(2)`$-invariant vector in $`V_K`$ and vanishes for every $`K > 0`$. The Peter-Weyl factorisation of Section 5.2 therefore retains the $`K = 0`$ term alone.

That term is radial. The rank-zero map is equivariant and built from one state, so by Schur it is a multiple of $`\lVert u\rVert^2 u`$, and carrying the Clebsch-Gordan evaluation through gives

$$M^{(j)}_0(u) \;=\; \frac{\varepsilon_j}{\sqrt{2j+1}}\, \lVert u\rVert^2 u .$$

The scalar has modulus $`(2j+1)^{-1/2}`$ at every level and its phase is carried by the convention, so only the modulus is a fact about the map. At $`j = 3`$ with $`\varepsilon_3 = -1`$ this is $`-1/\sqrt{7}`$, which is Section 5.2's constant, and the two statements agree. None of that is needed below: what matters is that the scalar is nonzero, so the projected self-interaction is a multiple of the state and no projective direction is selected. $`\square`$

The argument covers half-integer $`j`$ as readily as integer $`j`$: the character sum does not care, and the phase above is defined there, so levels 1, 3 and 5 are covered on the same footing as 2 and 4. Those are the levels whose sectors are the spinorial representations of $`2I`$, which do not factor through $`A_5`$; nothing in the argument needed them to. Separately, it does not invoke the Molien row at all: the vanishing is Schur's lemma applied to the identity, not the icosahedral invariant gap. The gap is what makes level 6 interesting, but it is not what makes the levels below it radial. The conclusion is not the empty one that there was nothing to select from. A level-$`2j`$ density carries every rank $`0 \le K \le 2j`$ and, at a generic state, carries them all nontrivially; what removes them is the right factor, not an absent left one. The negative reading is the more useful one: this interaction cannot supply shape selection to any level below 6, so a lower-level slot that needs a preferred direction has to get it from somewhere else.

This does not compete with Section 6.1's remark that level 6 is fixed by the choice of object rather than by the geometry. Which level one studies is fixed by putting the left index in $`V_3`$. What the corollary adds is independent of that choice: below level 6 there would have been nothing projective to study.

Corollary 5.5. The two sectors select the same shapes. The reduced quartics of the two sectors have the same critical set on $`\mathbb{P}(V_3)`$, with the same ordering of values.

By clause 5, $`Q_d([u]) = w_0/7 + w_6(d)\,\widehat{r}_6([u]) = 1 + w_6(d)\,\widehat{r}_6([u])`$, since $`w_0 = 7`$ in both sectors. The two quartics are therefore the same increasing affine function of $`\widehat{r}_6`$ up to the positive slope $`w_6(d)`$, and an increasing affine reparameterisation moves neither the critical set nor the order of the values on it. Hence

$$\mathrm{Crit}\, Q_{\mathbf{3}'} \;=\; \mathrm{Crit}\, Q_{\mathbf{4}} .$$

So once the filter has selected rank 6, the shape problem stops being icosahedral and becomes a question about spin 3 with time reversal, which is why Section 5.8 could be carried out without the group appearing again. The consequence worth recording is negative: at leading cubic order this interaction cannot make one sector prefer a different configuration from the other. Whatever distinguishes the two sectors, it is not the projective shape their self-interaction selects.

Corollary 5.6. But their nonlinear shifts differ. Let $`[u]`$ be a critical ray and let $`\psi`$ be the corresponding block state normalised so that $`\int_X \lvert\psi\rvert^2 = 1`$. Then the projected self-interaction acts on $`\psi`$ by the scalar $`Q_d([u])`$, and $`Q_{\mathbf{3}'} - Q_{\mathbf{4}} = \tfrac{49}{156}\,\widehat{r}_6([u])`$.

At a critical ray the tangential part vanishes, so $`\mathcal{N}(\psi) = \beta\psi`$ for some scalar $`\beta`$. Pairing with $`\psi`$ and using that the block projection is self-adjoint with $`\psi`$ in its range gives $`\beta \int\lvert\psi\rvert^2 = \int\lvert\psi\rvert^4`$, so with the stated normalisation $`\beta = \int\lvert\psi\rvert^4 = Q_d([u])`$. The difference is then $`(28/39 - 21/52)\,\widehat{r}_6 = \tfrac{49}{156}\,\widehat{r}_6`$. $`\square`$

The normalisation is the whole content of the statement and is easy to lose. Imposing $`\lVert u\rVert = 1`$ on the fibre instead gives a coefficient $`d/7 + \tfrac{7-d}{13}\lVert\rho_6\rVert^2`$ whose leading term is an artefact of $`B(u) = (d/7)\lVert u\rVert^2`$. Normalising the section removes that term, and what survives is $`w_6`$, which Section 5.3 derived. The two shifts agree only where $`\widehat{r}_6 = 0`$, so they differ at every ray whose top multipole is nonzero, the four tabulated ones included.

Taken together the three say: nothing projective below level 6, one projective channel at level 6, and above it two sectors that agree about shape and disagree about scale. None of the three needs the range equation, and none of them is a statement about solutions; they are statements about the leading reduced problem, on the same footing as Theorem 5.1 itself.

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6. The Lyapunov-Schmidt reading

This section interprets Theorem 5.1 and derives nothing. Without it, the theorem is a bare statement about a quartic on a seven-dimensional space; the Lyapunov-Schmidt reading explains why that particular quartic on that particular space is worth extremising.

6.1 The setting

The equation is the one fixed in Section 2.5,

$$(-\Delta - \lambda)\psi \;+\; \lvert\psi\rvert^2\psi \;=\; 0 ,$$

with $`\lambda`$ a spectral parameter near $`\lambda_6 = 48/R^2`$, the level-6 eigenvalue of the Laplacian on $`S^3`$. The kernel that Lyapunov-Schmidt needs is the kernel of the linearisation $`-\Delta - \lambda`$ at $`\lambda = \lambda_6`$, not of $`-\Delta`$, and that is exactly the level-6 eigenspace $`V_3 \otimes \mathrm{Hom}_{2I}(\sigma, V_3)`$, of dimension seven since the branching is multiplicity-free. Without the parameter there is no kernel and no reduction; it is the parameter that makes this a bifurcation problem at all. Near that value Lyapunov-Schmidt splits the equation into a projection onto the kernel and a complementary range equation.

Level 6 is not an extra choice, but neither is it forced by the geometry: it is fixed by the choice of object, since spin-3 states put the left index in $`V_3`$. What is a fact about the quotient is that the two sectors in which the question can then be posed, $`\mathbf{3}'`$ and $`\mathbf{4}`$, first occur at that level, so the block eigenspace is the bottom of the spectrum on each of their bundles and the bifurcation read below is one from the lowest linear eigenvalue on each of those bundles. No claim is made that the resulting states minimise anything, so the word ground state is avoided. Odd levels are excluded by parity: at $`\ell = 2j`$ with $`j`$ half-integer, $`-I`$ acts as $`-1`$, whereas $`\mathbf{3}'`$ and $`\mathbf{4}`$ factor through $`A_5 = 2I/\{\pm I\}`$ and have $`-I`$ acting trivially. For the even levels below 6, extending the branching table of Section 4.3 downward gives $`V_0 = \mathbf{1}`$, $`V_1 = \mathbf{3}`$, $`V_2 = \mathbf{5}`$.

The leading cubic term of that reduction is the projection of the nonlinearity back onto the eigenspace, which is exactly the self-interaction of Sections 4 and 5. Under that reading, Theorem 5.1 says the reduced equation's cubic term is confined to $`\mathrm{span}\{M_0, M_6\}`$, that $`M_0`$ contributes only a radial rescaling and therefore acts as a Lagrange multiplier rather than as a tangential direction, and that the tangential content is the single ray $`[M_6]`$, with the coefficients computed in Section 5.5. The quartic $`Q`$ plays the part of the reduced energy: Section 5.5 shows that on the unit sphere the tangential gradient of $`Q`$ is proportional to the tangential projection of the self-interaction, the denominator contributing only a radial term. The critical rays of $`Q`$ are then the critical directions of the leading reduced problem, and not solution branches: an actual branch requires the range equation described in Section 6.2, which is not carried out. Section 5.8 records several symmetry-distinguished examples among them.

6.2 Two ceilings

This is not a classification of the critical directions. Section 5.8 lists critical rays that are distinguished by their symmetry and are known in exact form. There is no theorem here that they exhaust the critical set of $`Q`$ on the unit sphere, and numerical exploration locates critical points beyond them. What is offered is a list of symmetry-distinguished critical directions, not a classification, and no count is claimed.

This is not a claim that the reduction is exact. In a Lyapunov-Schmidt construction carried through, the range equation is solved for the transverse component as a function of the kernel variable, and that solution is substituted back, contributing corrections to the reduced equation at higher order in the amplitude. Those corrections are not computed here and the transverse component is not discarded by any argument; it is simply not treated. What is exact is the finite-dimensional problem on the degenerate eigenspace at leading cubic order, which is what Sections 4 and 5 establish. Calling the result an exact reduction would overstate it by exactly one step, and that step is the range equation.

6.3 What the reading is for

Everything in Sections 3 through 5 stands without this section. Proposition 3.3 is a statement about binary sextics, Theorem 5.1 a statement about a four-dimensional space of equivariant maps and which of them a quotient permits for the interaction of Section 2.5. Neither is a statement about the dynamics that interaction came from. The Lyapunov-Schmidt language supplies the reason for the question and makes the two ceilings above legible as ceilings rather than as omissions: it identifies which parts of a complete bifurcation analysis are here and which are not.

The method itself is standard and is used rather than developed; the reduction from a degenerate eigenspace is classical [GS] and no property of it is proved below.

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7. Discussion

7.1 What is not claimed

Five limits are collected here rather than left scattered across the sections.

The critical set of $`Q`$ on the unit sphere is not classified: the rays recorded in Section 5.8 are not shown to exhaust it, for the reasons given in Section 6.2. Relatedly, the value $`463/\binom{12}{6}`$ is the largest of the four tabulated and is not shown to be the global maximum. That question is not this paper's to begin with: it is posed for general spin in [RK], probed numerically there, and not settled.

The Lyapunov-Schmidt reduction is not carried out: the range equation is untreated, for the reasons given in Section 6.2.

The four shape types are located, but not matched to [BTD]'s phase orbits. Rays of all four types that paper names are found in Section 5.8. Identifying them with its phase orbits is not attempted, since two of the four are continuous families and the source consulted lists names rather than representatives.

The closed form for the sector normalisation carries a ceiling of its own, stated with the formula in Section 5.6 and not repeated here.

Everything above concerns a single block state in a single sector. No statement is made about a field occupying distinct flat-bundle sectors at once, or about a nonlinearity coupling one sector's density to another's amplitude. Corollaries 5.5 and 5.6 compare the two sectors; they do not describe a configuration containing both. The mixed problem is not posed here, let alone solved.

7.2 Prior art, and where the boundary falls

The configurations appearing in Section 5.8 are known objects, and the three statements about them have different provenance, so they are given separately rather than as one prior-art paragraph.

The octahedral state is anticoherent of order exactly 3. The lower bound is prior art in a clean form: Crann, Pereira and Kribs [CPK] prove that a Majorana constellation which is both the orbit of a finite subgroup of $`\mathrm{O}(3)`$ and a spherical $`t`$-design gives an anticoherent state of order $`t`$, and the octahedron is both, at $`t = 3`$. That statement is a one-way implication and carries the group-orbit hypothesis, so it bounds the order below and not above. The sharpness is computed in Section 5.8 and does not follow from it: the multipole row is $`1/7, 0, 0, 0, 6/11, 0, 24/77`$, so rank 4 is nonzero and the order is exactly 3 rather than at least 3.

Critical rays occur here with all four of the shape types [BTD] names: the hexagon at $`463/924`$, the octahedron at $`288/924`$, the pentagonal pyramid at $`9/35`$ and the trigonal prism at $`200/903`$, the last two found in Section 5.8 on symmetric loci rather than at isolated points.

That is a statement about shape types and not yet about that paper's phase orbits, and the difference is real. "Pentagonal pyramid" is not one ray but a one-parameter family, a pole plus a ring at any latitude, of which Section 5.8 selects one; the eclipsed prisms likewise carry a continuous shape parameter. Matching a ray to a phase would mean comparing representatives, and the source consulted here is an abstract, which lists shape names and no parameters. So the four-way correspondence is at the level of type, and is stated at that level. The list of four, the hexagon, the pentagonal-base pyramid, the prism and the octahedron, is stated in that paper's own abstract, which is therefore the source for it; its body has not been read, and no claim here rests on the body.

The type count is four and complete, since [BTD] names four. One identification is worth stating, because the natural guess is wrong: the pyramid is not the weight state $`v_2`$, whose Majorana polynomial is $`z^5`$, giving five coincident points and one antipodal, a degenerate configuration rather than a polyhedron. It occurs at an interior point of the line through $`v_2`$ and $`v_{-3}`$, at neither end.

None of this classifies the critical set, which remains open: it contains rays of those types and others besides. Every weight state is critical, and the seven of them realise four distinct values of $`\binom{12}{6}\lVert\rho_6\rVert^2`$, namely 1, 36, 225 and 400.

The hexagonal value and the question attached to it are prior art. Romero, Klimov, Goldberg, Leuchs and Sanchez-Soto give $`\varrho^2_{2S} = \tfrac12 + \binom{4S}{2S}^{-1}`$ for the top multipole of the NOON state [RK], which at $`S = 3`$ is the $`463/924`$ tabulated above, their NOON state being this paper's hexagon up to a rotation. They also raise the question of the largest attainable top multipole and probe it numerically without settling it. Both the value and the question therefore belong to them, and Section 5.8 says so where it records them. What is not in that paper is the critical set: they ask which state maximises the top multipole, while Section 5.8 solves for every critical point on two symmetry loci and finds the hexagon sitting among five others.

Recognising a configuration is not the same as selecting it. The result established here is which member of a known family a specific quotient forces, and the boundary between those two things is the reason Section 3 is billed as a proposition and Section 5 as a theorem.

7.3 The search, stated as a search

Two searches were run at different times and they are in different conditions, so they are reported separately.

The first asked whether the results of Section 3 are known. The polarised identity of Section 3.1 and the zero-set statement of Proposition 3.3 were not located. Every reference below was verified against its publisher or preprint record rather than from recall, and one gap closed in the process: the order-3 lower bound has a clean source in [CPK], which the first pass had not surfaced.

The second asked the question the contribution actually rests on, whether the selection statement of Section 5 has been made before. It ran ten queries across six lines of approach: nonlinear Schrodinger and semilinear elliptic problems on spherical space forms; equivariant bifurcation with icosahedral symmetry; harmonic analysis on the Poincare dodecahedral space; spin-3 spinor condensates and their interaction channels; anticoherent states and multipoles from Majorana constellations; and symmetry selection rules for nonlinear couplings. Four sources were then read rather than skimmed.

Nothing was found that filters a cubic interaction by a finite subgroup on a space form, and nothing that puts $`\lvert\psi\rvert^2\psi`$ on a flat bundle over $`S^3/2I`$. Three adjacencies were found and are now cited where they belong rather than here: the four-dimensional family is the known four-channel spin-3 interaction [DH], [KU]; the transform of Section 2.3 is the standard state-multipole expansion [Fa], [RK]; and the hexagonal value $`463/924`$, together with the global-maximum question attached to it, is prior art [RK]. The last of those was a headline number of Section 5.8 and is now attributed.

What the second search does not establish is also worth saying. Not surfaced is not the same as new. Ten queries and four papers amount to a real search, not an exhaustive one. Three gaps are known: the equivariant bifurcation literature is largely in monographs that search engines index poorly; the older invariant-theory literature was not worked; and no subscription index was used, which is the instrument the question really wants. Section 7.2 gives the per-item status of the prior-art statements rather than a blanket one, because the items are in different conditions.

7.4 The open historical question

The object

$$\Phi(F,G) \;=\; F \,(F,G)_1$$

is a joint covariant of two independent binary sextics, of bidegree $`(2,1)`$ and order 16, and it is entirely possible that it is named in Grace and Young [GY], Elliott, or Gordan. That is a question worth putting to someone who works in classical invariant theory.

The question this paper's object raises is a different one, and the difference is easy to lose. The covariant here is the real-analytic specialisation $`G = \Theta F`$ along the time-reversal diagonal, and a specialist asked about $`\Phi`$ will answer about $`\Phi`$, correctly, and about a different statement. Both questions should be asked, in that order, and kept apart in whatever answer comes back.

7.5 Where the family sits

The homogeneous quartic $`\widetilde{Q}`$, not its projective normalisation $`Q`$, is one member of a four-dimensional space of $`\mathrm{SU}(2)`$-invariant quartics on spin 3. That space has the same dimension as $`\mathscr{E}_3`$, and for the same reason: both are counted by the four summands $`V_0 \oplus V_2 \oplus V_4 \oplus V_6`$ of $`\mathrm{Sym}^2 V_3`$, so selecting a quartic and selecting a cubic map are the same problem in two costumes. Equal dimension alone would not say that, so here is the map: $`\widetilde{Q} \mapsto \nabla_{\bar u}\widetilde{Q}`$ is equivariant and linear from invariant homogeneous quartics to $`\mathscr{E}_3`$, and injective, since Euler's identity recovers a homogeneous quartic from its gradient. Both spaces have dimension four, so it is an isomorphism; Section 5.5 exhibits it channel by channel, as the per-rank identity $`\nabla_{\bar u}\lVert\rho_K\rVert^2 = c_K M_K`$ with every $`c_K`$ nonzero. The community that studies anticoherence works with members of that space and with the symmetric constellations attached to them. Anyone in that community will recognise the family immediately, which is a reason to say so here rather than to leave it to be noticed: the contribution is not the family, and not the quartic considered in isolation, but which member of it the icosahedral quotient permits, for the density-type interaction of Section 2.5.

That framing also explains the shape of the paper. Almost everything on the way to the answer is universal: the four-dimensional family, the canonical basis, the operator structure and its alternating row, the forced time-reversal phase, and the critical geometry of the surviving channel. For Theorem 5.1 the quotient supplies two facts, the invariant-degree filter and the multiplicity-free complementary branching, and those two are the whole of its icosahedral content; the branching is one statement doing two jobs, scalarisation and weight. Branching computations appear elsewhere for other purposes, in Section 4.3 for the availability of the spin-8 target and in Section 6.1 for the first occurrence of each sector, and neither feeds the theorem. Chronologically the icosahedron came first and led to the ambient proposition rather than requiring it, but the logical dependence runs the other way, and the paper is ordered by the logic.

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