# How does "spreading-with-determinacy" compare with Cichon's diagram?

For $$\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$$, say that $$\mathbb{A}$$ spreads onto $$\mathbb{B}$$ iff there is some $$F:\omega^\omega\rightarrow\omega^\omega$$ such that $$F[A]\in\mathbb{B}$$ whenever $$A\in\mathbb{A}$$. Write "$$\mathbb{A}\trianglerighteq\mathbb{B}$$" for "$$\mathbb{A}$$ spreads onto $$\mathbb{B}$$."

Each cardinal characteristic in Cichon's diagram $$\mathfrak{x}$$ is defined as "The smallest cardinality of an element of $$\mathbb{X}_\mathfrak{x}$$" for some $$\mathbb{X}_\mathfrak{x}\subseteq\mathcal{P}(\omega^\omega)$$. I'm curious how the $$\mathsf{ZFC}$$-provable cardinality relationships between the $$\mathfrak{x}$$s compare to the $$\mathsf{ZF+DC+AD}$$-provable spreading relationships between the $$\mathbb{X}_\mathfrak{x}$$s. Specifically:

• Is it the case that, for all $$\mathfrak{x},\mathfrak{y}$$ in Cichon's diagram, $$\mathsf{ZFC}\vdash\mathfrak{x}\le\mathfrak{y}$$ implies $$\mathsf{ZF+DC+AD}\vdash \mathbb{X}_\mathfrak{x}\trianglelefteq \mathbb{X}_\mathfrak{y}$$?

• Is it the case that, for all $$\mathfrak{x},\mathfrak{y}$$ in Cichon's diagram, $$\mathsf{ZF+DC+AD}\vdash \mathbb{X}_\mathfrak{x}\trianglelefteq \mathbb{X}_\mathfrak{y}$$ implies $$\mathsf{ZFC}\vdash \mathfrak{x}\le\mathfrak{y}$$?

As positive evidence, Paul Larson showed that $$\mathsf{ZF+DC+AD}\vdash\mathbb{X}_\mathfrak{b}\triangleleft \mathbb{X}_\mathfrak{d}$$, so the "spreading-under-determinacy" picture agrees with (and in fact strengthens) the "cardinality-under-choice" picture in this instance.

I'm particularly interested in the behavior of either of the triples $$\mathsf{add}(\mathcal{B}),\mathsf{cov}(\mathcal{B}),\mathfrak{b}$$ or $$\mathsf{cof}(\mathcal{B}),\mathsf{non}(\mathcal{B}),\mathfrak{d}$$, where $$\mathcal{B}$$ is the $$\sigma$$-ideal of meager sets, since there is a $$\mathsf{ZFC}$$-theorem about each triple which isn't a consequence of any of the provable "simple inequalities" - I don't know how (if at all) I'd expect this more complicated relationship to be reflected on the "spreading"-side.