Recursive Trees of Reflective Theories with Controlled Branching: Π¹₁-Completeness and Stratification by Branching Width

We construct an explicit recursive tree I whose nodes are labelled with recursively enumerable theories extending Peano arithmetic (PA). Each theory is obtained by iterating uniform reflection and adding a Rosser sentence — a true Π₁ sentence independent of the parent theory. The branching degree of the tree is controlled by a primitive recursive function that takes values 2, 3, 4, 5, 6 and is constant (=2) along the leftmost infinite path, guaranteeing that the restricted trees Iₘ (paths never exceeding degree m) are non‑empty for all m ≥ 2. We prove: 1. Π¹₁-completeness. The set I of all infinite paths through I is Π¹₁-complete. This is shown by embedding a universal Π¹₁ binary tree into I; the embedding preserves infinite paths and is effectively continuous. 2. Stratification by branching width. For each m ∈ 2, 3, 4, 5, the set Iₘ of paths that never visit a node of degree > m is Π⁰₁-complete. The hierarchy I₂ ⊊ I₃ ⊊ I₄ ⊊ I₅ ⊊ I is strict and cannot be recursively reduced. 3. Effectivity. All constructions are primitive recursive and formalisable in RCA₀; the Π¹₁-completeness proof requires ACA₀ (to handle the Π¹₁-completeness of WF). The work provides a natural proof‑theoretic realisation of a complete Π¹₁ set, stratified by a simple combinatorial parameter (branching width). It requires no transfinite ordinal notations, no Kleene's O, and only elementary recursion theory and proof theory (Rosser sentences, uniform reflection). Open problems include: whether binary branching suffices for Π¹₁-completeness (conjecture: no) ; the fine structure of absolutely undecidable sentences along all paths; generalisation to ZFC and large cardinals.