Attainable forms of lower spectra

Let $d\in\mathbb{N}$ and $\varphi\colon(0,1)\to[0,d]$. We prove there exists a set $F\subset\mathbb{R}^d$ whose lower spectrum $\operatorname{dim}^{\theta}_{\mathrm{L}} F$ satisfies $(1-\theta)\operatorname{dim}^{\theta}_{\mathrm{L}} F = \varphi(\theta)$ for all $\theta\in(0,1)$ if and only if for all $\lambda,\theta\in(0,1)$, \begin{equation*} \varphi(\theta) \leq \varphi(\lambda\theta) - \theta \varphi(\lambda) \leq (1-\theta) d. \end{equation*} We also obtain a similar classification result for $\underline{\operatorname{dim}}^{\theta}_{\mathrm{L}} F$. In contrast to the case for Assouad spectra, it is insufficient to consider homogeneous (or uniform) sets. Instead, we follow the approach introduced by Orgov\'anyi--Rutar in arXiv:2510.07013 and proceed via a more general classification result for appropriate two-scale branching functions.