Band-center spectral density as a no-adjustable-constant design criterion for room-temperature topological spin textures

## Summary This work derives a parameter-free spectral design criterion Q² = 1/π ≈ 0.318 for identifying and optimizing room-temperature topological spin textures (skyrmions, bimerons, hopfions) in chiral magnets. The criterion follows from a four-step derivation chain: 1. **Arc-sine DOS:** The nearest-neighbor projected magnon channel has band-center spectral density ρ₀(0) = 1/π, with constant spectral current ρ·v_g = 1/π across the entire band. 2. **Born self-energy:** The on-shell second-order DMI vertex load has unit coefficient (sin²k = 1 on the ε = 0 shell), giving L_λ(0) = λ² exactly. 3. **Exchange-length scaling:** Coherent phase accumulation over ξ/a lattice sites gives Q² = λ²(ξ/a)² as the continuum spectral stress. 4. **Self-dual stability window:** The functional W(x) = (1−e⁻ˣ)(1−e⁻¹/ˣ), derived from Poisson nucleation statistics, is uniquely maximized at x = Q²/ρ₀(0) = 1 (proven via strict concavity and Jensen's inequality), yielding Q² = 1/π. The resulting design formula **D_opt = √(2Aμ₀Ms²/π)** contains no adjustable constants. ## Validation Tested on 9 chiral magnets from 3 structural families (B20, β-Mn, intercalated): - **Mann-Whitney p = 0.012** (RT vs sub-RT separation) - **Co₈Zn₈Mn₄** — the only material with controllable RT bimerons — has the smallest self-dual mismatch d_spec = 0.025 from the 1/π optimum - Leave-one-out: all p < 0.036 - Monte Carlo (50,000 trials): classification robust at 92% - 4 falsifiable predictions with explicit falsification conditions ## Key result The quality factor Q² = Bc/(μ₀Ms) − 1, computable from two experimentally accessible quantities, provides a single-number screening criterion for room-temperature topological spin textures. Materials with Q² ≈ 1/π have optimal controllability windows; materials with Q² > 0.5 lack first-order spectral coherence for sustained topological textures.