The ZFC inequalities of Cichoń's diagram

In ZFC, the ten cardinals $\mathrm{add}(\mathcal{N})$, $\mathrm{cov}(\mathcal{N})$, $\mathrm{non}(\mathcal{N})$, $\mathrm{cof}(\mathcal{N})$, $\mathrm{add}(\mathcal{M})$, $\mathrm{cov}(\mathcal{M})$, $\mathrm{non}(\mathcal{M})$, $\mathrm{cof}(\mathcal{M})$, $\mathfrak{b}$, and $\mathfrak{d}$ satisfy the standard Cichoń-diagram inequalities: the elementary ideal arrows, the null-to-meagre arrows, the cross-ideal arrows, the eventual-domination arrows stated below, and all their transitive consequences. No independence or completeness assertion is part of this theorem.

The diagram displays exactly the seventeen generating inequalities in [F1]–[F4]. An arrow from $a$ to $b$ means $a\le b$; a directed path gives a transitive consequence. Crossings of curves are not additional nodes. The eight ideal invariants use the null and meagre ideals on $\mathbb{R}$, and $\mathfrak{b}$ and $\mathfrak{d}$ use eventual domination, as in the Facts above.

The figure illustrates the assembly in proof step 1.1; the rigorous proof uses the cited inequalities and order transitivity independently of the figure. It asserts neither independence of omitted arrows nor completeness of the relations shown.