A quartic product-prior counterexample to the general small-ball ratio conjecture

We give a proposed counterexample to the unresolved general statement in AIM Problem 506, equivalently Conjecture 4.12 of Ayanbayev, Klebanov, Lie and Sullivan. On $\ell^2$ we construct a countable product of scaled copies of the density $\rho(z)\propto e^{-z^4}$ and an explicit center $h$ with formal energy $Q(h)=1$, but \[ _{r0}\frac{(B_r(h))}{(B_r(0))}=0 e^{-1}. \] The density is positive, real analytic and log-concave, with finite Fisher information. A one-dimensional Gaussian-convolution inequality and a Laplace-transform averaging identity give a self-contained analytic disproof. We also give explicit bounds on a vanishing sequence of ball ratios, arbitrarily small-energy counterexamples, and a second example with polynomial coordinate scales. Exact-arithmetic certificates and separate numerical checks are supplementary, not prerequisites for the main theorem. No assertion that the full small-ball ratio limit exists is needed or made.