Prime-gap monotonicity: does $\{n:d_{n+1}\ge d_n\}$ have density $1/2$, and are there infinitely many $n$ with $d_{n+1}=d_n$? (Erdős #218)

Let $d_n=p_{n+1}-p_n$ be the $n$th prime gap. Erdős conjectured that the set of $n$ with $d_{n+1}\ge d_n$ has natural density $1/2$ (and symmetrically $\{n:d_{n+1}\le d_n\}$ has density $1/2$), and that there are infinitely many $n$ with $d_{n+1}=d_n$ (two consecutive gaps equal). All three quantities are directly computable from a prime table. GOAL (computational): for $N$ as large as feasible, compute the three proportions $\rho_{>}(N),\rho_{=}(N),\rho_{<}(N)$ of $n\le N$ with $d_{n+1}>d_n$, $=$, and $<$ respectively, exhibit their convergence toward $(1/2,0,1/2)$ (up to the density-zero equality set), and tabulate the count $E(N)=|\{n\le N:d_{n+1}=d_n\}|$ as evidence of infinitude.

Acceptance. PARTIAL / PRIMARY (finite, machine-checkable): the proportions $\rho_>(N),\rho_=(N),\rho_<(N)$ and the equal-gap count $E(N)$ for $N$ as large as feasible (state $N$), delivered with (i) a runnable segmented-sieve script computing consecutive gaps, and (ii) a convergence table/plot for $\rho_>(N)\to1/2$ and the growth of $E(N)$, cross-checked against A064113. A measured, slow approach to $1/2$ with quantified bias is a citable result. FULLY RESOLVES: a proof that the density is $1/2$ or that $d_{n+1}=d_n$ infinitely often. State $N$ and the sieve used.