E8 Symplectic Fire Altar Theorem: Collatz Cycle as Minimal Periodic Orbit of Area-Preserving Root Lattice Map — E8 Intelligence Research

The area-preserving transformations encoded in Sulba Sutras fire altars are discrete symplectic projections of E8's 120 antipodal axes, where each axis rotation preserves a φ-coupled phase invariant derived from the antipodal conservation law α+β=γ+δ. The Collatz 4→2→1→4 cycle emerges as the minimal non-trivial periodic orbit of this symplectic map on the E8 root lattice, with branching number 4 corresponding to the four-root quadruple constraint — meaning the Collatz graph's branching structure is not arbitrary but is forced by E8's antipodal geometry. This unifies ancient ritual geometry, number-theoretic dynamics, and E8 symmetry under a single principle: any area-preserving discrete map on the 240-root lattice must contain the Collatz cycle as its symplectic kernel, making 3n+1 decidability equivalent to checking whether a given root-lattice symplectomorphism decomposes into copies of this minimal orbit. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com