Brändén and Huh showed that Lorentzian polynomials unify Hodge-Riemann relations in combinatorics: their supports are M-convex, and every M-convex set supports a Lorentzian polynomial. Baker, Huh, Kummer, and Lorscheid later proved that, for every q>0, the projectivized space PLJ of Lorentzian polynomials with support J is homeomorphic to the thin Schubert cell GrJw(Tq)
Nearby in the stack
of weak representations of
J
over the generalized triangular hyperfield
Tq
. We study the quantitative relation between Lorentzian polynomials and representations over triangular hyperfields. For every matroid
M
, we prove that some
q>0
depending on
M
satisfies
GrMw(Tq)⊆PLM⊆GrMw(T2)
. Thus
PLM
lies between two thin Schubert cells, each homeomorphic to it. More generally, for every M-convex set
J
, some
q>0
depending on
J
satisfies
NGrJw(Tq)⊆PLJ⊆NGrJw(T2)
, where
N
denotes normalization. We also study
q(M):=sup{q>0:GrMw(Tq)⊆PLM}
. For
q(n):=q(U2,n)
, we prove
q(4)=2
and
q(5)=log23
, with matching upper and lower bounds of order
1/n
; hence
q(n)=Θ(1/n)
, so in particular no universal positive lower bound for