Proof‑Theoretic Trees: Branching Width and Π⁰₁ Classes

We construct a primitive recursive tree 𝒯 whose nodes are labelled with recursively enumerable theories extending PA, obtained by iterating uniform reflection and adding Rosser sentences. The branching degree at each node is determined by the number of previous deviations from the leftmost path. We prove: - The set [𝒯] of infinite paths through 𝒯 is a Π⁰₁ class. Moreover, the complement of the halting problem is many-one reducible to [𝒯] via a recursive functional, establishing that [𝒯] is not effectively open (i.e., not lightface Σ⁰₁). - For each m ∈ {2,3,4,5}, the restriction [𝒯_m] to paths that never use a node of degree > m is also a Π⁰₁ class, and the inclusions [𝒯₂] ⊂ [𝒯₃] ⊂ [𝒯₄] ⊂ [𝒯₅] = [𝒯] are strict. The construction is effective and can be formalized in RCA₀ modulo external consistency assumptions. This provides a proof-theoretic realisation of a stratified family of Π⁰₁ classes, connecting progressions of theories with the combinatorial structure of trees.