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 thatσ occupies the same structural role in four dimensions that the golden ratio φ occupies in twodimensions and the plastic constant ρ occupies in three dimensions, forming a complete dimensionalhierarchy 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 ofa geometrically motivated recurrence, the Tetradic recurrence T (n) = T (n−3) + T (n−4), whichcaptures the sparse coupling structure of the A4 root lattice’s four-simplex geometry. We provethat σ is irreducible over Q, characterize its root structure and power algebra, and demonstrate theSemantic Octave property: σ4 = σ + 1 implies a four-step self-referential energy conservation lawmatching the lattice’s diameter.We validate σ-decay empirically using independent, clean-room simulations: first, by optimizingmulti-hop recall accuracy in an associative memory network built from first principles; and second,by extracting the intrinsic diffusion profile of the A4 graph Laplacian heat kernel, which recovers1/σ to four decimal places. The result generalizes: for any d-dimensional simplicial lattice, theconstant cd satisfying cd = c + 1 is the principled, geometry-native energy decay base.