Quantum Trees, Collatz Gaps, and Compact Automorphism Groups — E8 Intelligence Research

FINDING: Quantum automorphism groups of trees are characterized; Collatz spectral-gap proofs use functional analysis; tree almost-automorphism groups are compactly presentable. MATH: - Quantum automorphism groups of trees: characterized via quantum permutation groups acting on vertex sets, with tree structure encoded in adjacency matrix \(A\); quantum symmetry group \(QAut(T) ⊆ S_n^+\) acting on \(n\) vertices, preserving \(A\). - Collatz spectral gap: Constructed via branch-counting dynamics; the Collatz map \(T(n) = n/2\) (even), \((3n+1)/2\) (odd) is embedded in a Banach space with a transfer operator whose spectral gap \(γ > 0\) separates the leading eigenvalue 1 from the rest. - Compact presentability of tree almost automorphism groups: Neretin's group \(N\) (spheromorphisms of \(d\)-regular tree) is compactly presented — locally compact analogue of finite presentability, with generating set from finite subtrees and profinite stabilizers. CONNECTION: - Tre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com