A counterexample to preservation of ratio log-concavity

A sequence of positive reals is ratio log-concave when q_{t+1}^3 q_{t-1} >= q_t^3 q_{t+2} --
that is, when its consecutive ratios are log-concave. It has been asked whether the hypergeometric
self-convolution preserves that property on the first half of its range, for inputs arising as the
normalised elementary symmetric means of a real-rooted polynomial.

It does not.

The witness

b = { 8, 400, 4000, 40000, 300000 }

All positive, so prod (1 + b_i z) is real-rooted by inspection. Its normalised elementary means
are ratio log-concave over their entire index range, with margins 7.14e26, 8.61e41, 5.82e55.
The self-convolution fails at t = 2 and t = 4, and the failure at t = 2 involves only
p_1 .. p_4, every index at or below m = 5. The exact margin there:

p_3^3 p_1 - p_2^3 p_4  =  -4771851289457589610298637645802556084678425575424 / 2187

Check it yourself

python verify.py

No dependencies beyond the Python standard library, and no code from the work that produced the
result -- so confirming the claim requires trusting none of it. Under a second.

Scope

Over 193 inputs satisfying the hypothesis, 48 break the conclusion. Every counterexample found has
a root ratio above 10^4, which is why samplers drawing inputs of comparable magnitude had not
reached the failure region.

The special case of squares of an arithmetic progression is unaffected and holds in every
instance tested -- 3168 progression-size pairs, plus 41 chosen deliberately in the regions a
systematic sweep could not reach. What the counterexample removes is the derivation of that case
from a general preservation theorem.

Verified four independent ways in exact rational arithmetic, one of them sharing no library with
the other three.