Tree suspensions and transfer functions for single degree Tur\'an spectra
For integers $1\le \ell<k$, let $\Pi^k_\ell$ denote the single-forbidden $\ell$-degree Tur\'an spectrum of $k$-uniform hypergraphs. We introduce transfer functions for this spectrum: explicit functions $f$ such that, for every $F$, there is another single $k$-graph $F^*$ with $\pi_\ell(F^*)=f(\pi_\ell(F))$. This gives a mechanism for producing new single-forbidden densities while retaining full control of the resulting value. Our transfer functions are realized by a new family of suspension-type operations, called tree suspensions. From these operations we obtain three explicit maps: one acting on $\Pi^k_\ell$ for every $1\le\ell<k$, a second acting when $\ell\ge k/2$, and a third acting in the ordinary Tur\'an case $\ell=1$. The common feature is a robust tree structure which gives the lower bound by a two-part construction and, in the regimes above, admits a matching embedding or Lagrangian upper bound. As a first application, the universal transfer function propagates accumulation points. Using the recent zero-accumulation results for $\ell\ge2$ together with the ordinary Tur\'an accumulation result of Conlon and Sch\"ulke, we prove that $\Pi^k_\ell$ has infinitely many accumulation points for every $k\ge3$ and every $1\le\ell<k$. This recovers, in particular, the known infinitude of accumulation points in the ordinary and codegree spectra. As a second application, combining two independent transfer functions forces algebraic degrees to grow. For every $k\ge3$ and every $\ell\in\{1,\lceil k/2\rceil,\ldots,k-2\}$, the spectrum $\Pi^k_\ell$ contains algebraic numbers of arbitrarily large degree over $\mathbb Q$. Thus the arithmetic complexity previously known for finite forbidden families already occurs in the single-forbidden spectrum, both for ordinary Tur\'an density and for a broad range of degree Tur\'an densities.