Reversible Computation as Algebraic Symmetry: A Group-Theoretic Structure — E8 Intelligence Research

FINDING: Reversible Turing machines form an algebraic group acting on configuration spaces, with equivalence classes defined by Nielsen/Andrews-Curtis-type operations — revealing symmetry structure in computation itself. | MATH: Group of reversible Turing machines \(G\) acting on \(\Sigma^{\mathbb{Z}^d}\) via local homeomorphisms; equivalence relations generated by \(\mathrm{Aut}(F_n)\) and \(F_n\) on normally generating tuples \((y_1,\ldots,y_n)\sim(x_1,\ldots,x_n)\); Andrews-Curtis conjecture: any normally generating \(n\)-tuple is AC-equivalent to free basis. | CONNECTION: The dihedral group \(D_4\) (order 8) is the symmetry group of the square — its elements (identity, rotations \(r,r^2,r^3\), reflections \(s,sr,sr^2,sr^3\)) correspond to the 8 possible permutations of a 2-state, 2-symbol Turing machine's transition table under complementation (swap 0↔1) and reversal (swap direction). This is precisely the \(D_4\) action on the \(2\times2\) grid of state×symbol — a crystallographic

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com