  For an input graph $G$, an additive spanner is a sparse subgraph $H$ whose
shortest paths match those of $G$ up to small additive error. We prove two new
lower bounds in the area of additive spanners:
  1) We construct $n$-node graphs $G$ for which any spanner on $O(n)$ edges
must increase a pairwise distance by $+\Omega(n^{1/7})$. This improves on a
recent lower bound of $+\Omega(n^{1/10.5})$ by Lu, Wein, Vassilevska Williams,
and Xu [SODA '22].
  2) A classic result by Coppersmith and Elkin [SODA '05] proves that for any
$n$-node graph $G$ and set of $p = O(n^{1/2})$ demand pairs, one can exactly
preserve all pairwise distances among demand pairs using a spanner on $O(n)$
edges. They also provided a lower bound construction, establishing that that
this range $p = O(n^{1/2})$ cannot be improved. We strengthen this lower bound
by proving that, for any constant $k$, this range of $p$ is still unimprovable
even if the spanner is allowed $+k$ additive error among the demand pairs. This
negatively resolves an open question asked by Coppersmith and Elkin [SODA '05]
and again by Cygan, Grandoni, and Kavitha [STACS '13] and Abboud and Bodwin
[SODA '16].
  At a technical level, our lower bounds are obtained by an improvement to the
entire obstacle product framework used to compose "inner" and "outer" graphs
into lower bound instances. In particular, we develop a new strategy for
analysis that allows certain non-layered graphs to be used in the product, and
we use this freedom to design better inner and outer graphs that lead to our
new lower bounds.
