The Information Conservation conjecture for the essential prime and fractal zero sets

Let $P_{\text{ess}}$ be the essential fractal prime set, with information measure $\iota(P_{\text{ess}})=-\zeta(1/2)$, and let $Z_F$ be the fractal zero set constructed from the positive imaginary parts of the non-trivial zeros of $\zeta(s)$. Information Conservation conjecture. The fractal zero set $Z_F$ has information measure

$$\iota(Z_F)=\zeta(1/2)=-\iota(P_{\text{ess}}),$$

and therefore

$$\iota(P_{\text{ess}})+\iota(Z_F)=0.$$

The conjecture is motivated by the explicit formula relating primes and zeta zeros, the common Hausdorff dimension $1/2$ of the two fractal sets, the functional equation, and spectral dualities in noncommutative geometry; no proof or disproof is supplied.