We derive an exact identity: the integral of 1/(φ − t) ln φ from 0 to 1 equals 2, where φ = (1 + √5)/2 is the golden ratio. This identity generalizes to an infinite family of resonance conditions indexed by integers n ≥ 2, each selecting a unique algebraic constant xₙ satisfying xₙⁿ − xₙⁿ⁻¹ − 1 = 0. For n = 2, 3, 4, the constants are Pisot–Vijayaraghavan numbers; for n ≥ 6, they are not; n = 5 is a Salem number at the boundary. We prove this PV classification via Rouché's theorem and show that n = 2 uniquely maximizes a habitability ratio H(n) = (xₙ − 1)/n. Numerical computation of diffraction spectra for substitution chains generated by the family confirms the spectral prediction: PV members exhibit pure point diffraction (sharpness ratio > 6:1 over non-PV). Direct computation of the wandering exponent ω = ln|λ₂|/ln λ₁ verifies the Luck criterion input: ω 0 for non-PV. We identify quantum dimension in topological quantum field theory as the natural bridge variable; the Fibonacci anyon fusion rule independently forces d = φ, matching the n = 2 resonance condition exactly. A UMTC admissibility check shows φ is the only member of the family realizable as a quantum dimension, establishing a third independent basis for the structural privilege of duality. Interpretive applications to aspiration dynamics and humanistic mathematics are developed.
Mudholkar et al. (2026) studied this question.