Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

==== For State S the desired distributive law λ : State S ∘ O ⇒ O ∘ State S would be: ====

<syntaxhighlight>λ_A : (S -> (O A × S)) -> O (S -> (A × S))

</syntaxhighlight>

This requires pushing state inside the O structure. Two natural semantics:
# Shared-state semantics: meta-levels and base-level share one global S. That is, O computations access the same S. If the O structure is such that we can push s into each leaf and reconstruct, an appropriate λ may exist. For Tree, one could attempt: ``<code> λ (h : S -> (Tree A × S)) := Node MergeOp [ ... ] ???   </code>`` but in general coordinating the threaded state across branching Node requires design decisions (how nodes interleave, whether branches run in sequence or in parallel, whether to merge resulting states).
# Isolated-state semantics (practical): prefer StateT S O or O applied over State depending on semantics: - StateT S O A := S -> O (A × S) — meta-runner with global state threaded inside O. - O (State S A) := O (S -> (A × S)) — meta-structures each have stateful computations as leaves.

Hence, instead of proving existence of λ in general, the practical engineering solution is to stack monads with StateT or State transformer and choose composition according to desired semantics. This is the standard pattern in applicative/monad transformer design (Haskell-style). If you want a formal distributive law existence proof for a particular O and a chosen state semantics, give me O concretely (e.g., this Tree with merge semantics) and I’ll build/verify λ.