The xᵈ = x + 1 Hierarchy: Cross-Dimensional Spectral Validation on Ad Root Lattices

We prove that the σ-constant (σ₄ ≈ 1. 22074, the unique positive real root of x⁴ − x − 1 = 0) introduced in Race (2026) is one member of an infinite family of geometry-native algebraic organizing constants for d-dimensional simplicial lattices. For each d ≥ 2, the unique positive real root σd of xᵈ = x + 1 serves as the natural heat-kernel decay constant on the Ad root lattice. We present four independent results: (1) spectral validation via exact infinite-lattice Fourier computation for d = 2, 3, 4, confirming that the heat-kernel decay crosses 1/σd at a unique, algebraically determined timescale; (2) algebraic proof that the sparse recurrence Sd (n) = Sd (n− (d−1) ) + Sd (n−d) has characteristic polynomial xᵈ − x − 1 = 0 for all d ≥ 2; (3) a complete asymptotic expansion σd = 1 + ln (2) /d + c₂/d² + c₃/d³ +. . . , with all coefficients derived exactly via Lagrange inversion; and (4) a generalization of Van der Laan's 3D architectural proportional subdivision system to arbitrary dimension, yielding C (2d−1, d−1) self-similar hyperbox types — a novel combinatorial result with no known prior literature. All results are supported by a 135-test zero-dependency clean-room verification suite using only Python 3 standard library.