The σ-Constant: A Universal Algebraic Invariant for Energy Propagation in d-Dimensional Simplicial Lattices

We introduce the algebraic constant σ ≈ 1. 22074, the unique positive real root of x⁴ − x − 1 = 0, as the native energy-propagation invariant for four-dimensional simplicial lattices. We show that σ occupies the same structural role in four dimensions that the golden ratio φ occupies in two dimensions and the plastic constant ρ occupies in three dimensions, forming a complete dimensional hierarchy of algebraic organizing constants cd defined by xᵈ = x + 1. We derive the per-hop energy decay factor 1/σ ≈ 0. 819 from the characteristic polynomial of a geometrically motivated recurrence, the Tetradic recurrence T (n) = T (n−3) + T (n−4), which captures the sparse coupling structure of the A₄ root lattice's four-simplex geometry. We prove that σ is irreducible over Q, characterize its root structure and power algebra, and demonstrate the Semantic Octave property: σ⁴ = σ + 1 implies a four-step self-referential energy conservation law matching the lattice's diameter. We validate σ-decay empirically using independent, clean-room simulations: first, by optimizing multi-hop recall accuracy in an associative memory network built from first principles; and second, by computing the exact infinite-lattice heat-kernel decay via Fourier analysis over the A₄ Brillouin zone, which recovers 1/σ at an algebraically determined timescale t₀ = 2σ/ (σ−1). The result generalizes: for any d-dimensional simplicial lattice, the constant cd satisfying cᵈ = c + 1 is the principled, geometry-native energy decay base.