The Lattice Octave: A Characteristic-Delay Model and Growth-Rate Partition on Simplicial Lattices
We demonstrate that the algebraic polynomial family xd = x + 1 (for integer d ≥ 2), whose roots are known as generalized golden ratios, characterizes dual-channel boundary-bulk energy transport on d-dimensional simplicial lattices under a dominant characteristic-delay model. We formulate and prove the Face-Poset Delay Decomposition Theorem, establishing that the combinatorial face-poset topology of a regular d-simplex admits exactly two maximal transit channel classes: a boundary facet transit channel of characteristic delay d-1 hops, and a bulk interior transit channel of characteristic delay d hops. Under the dominant-delay approximation (in which sub-leading multi-hop path corrections are neglected), the unique positive real root cd > 1 of xd = x + 1 induces the growth-rate partition identity cd-(d-1) + cd-d = 1, balancing asymptotic energy transport between boundary and bulk. We explicitly distinguish three conceptual layers: (i) exact algebraic partition and topological channel exhaustion on the simplex face poset, (ii) dynamical mode transport governed by the discrete recurrence u(n) = u(n-(d-1)) + u(n-d) under the leading-order Hamiltonian delay model, and (iii) cross-lattice continuous heat-kernel validations showing that spatial decay rates across Ad (simplicial root), ℤd (hypercubic), and Dd (checkerboard) lattices cross 1/cd at matching timescales, with high-dimensional convergence governed by isotropic Gaussian diffusion. Related Work & Prior Art: The σ-Constant: A Universal Algebraic Invariant for Energy Propagation in d-Dimensional Simplicial Lattices (10.5281/zenodo.20350425) The xd = x + 1 Hierarchy: Cross-Dimensional Spectral Validation on Ad Root Lattices (10.5281/zenodo.20692936)