Transverse positivity for the special spinning family

The integer $d$ satisfies $1\le d<k\mathrm{/}2$; also $k^2>2kd$. Thus the weighted hypotheses in [F2] hold. The subspace $P$ of vectors constant on each cluster is invariant under the lifted Hessian: clique terms vanish, and each cross block has constant row sum equal to its weight, so the induced action is exactly the weighted Hessian. The lifted Hessian is real symmetric, hence $W=P^{\perp }$, the subspace with zero sum in every cluster, is invariant too. These spaces are orthogonal and exhaust all vectors.

Write $\alpha$ for the phase parameter. On $W$ each clique contributes $kI$. The $a$ and $c$ blocks are complete bipartite graphs because their weights equal $k$; their cross actions vanish on cluster-zero sums, leaving the diagonal contribution $k\cos \alpha +k\sin \alpha$. The $b$ coefficients vanish because their phase differences are $\pi \mathrm{/}2$. The $d$ blocks have signed coefficient $-\cos \alpha$ and each vertex has $d$ such neighbours. Let $B$ denote the unsigned adjacency matrix of just these blocks. Thus the restriction is $H{\mathrm{\mid }}_W=(k+k\cos \alpha +k\sin \alpha -d\cos \alpha )I+\cos \alpha B{\mathrm{\mid }}_W$. The symmetry of $B$ and $2\mathrm{\mid }{x}_u{x}_v\mathrm{\mid }\le x_u^2+x_v^2$ give $\mathrm{\mid }{x}^{T}Bx\mathrm{\mid }\le d\mathrm{\parallel }x{\mathrm{\parallel }}^2$ by summing over its edges.

Consequently $x^{T}Hx\ge [k+k\sin \alpha +(k-2d)\cos \alpha ]\mathrm{\parallel }x{\mathrm{\parallel }}^2>0$ for nonzero $x\in W$. On $P$, [F2] gives positivity except for the global constant mode. The orthogonal invariant splitting in step 1.1 eliminates cross terms, proving positivity on the full complement of constants and showing that there are no other zero modes.