Crossings of scalar and spinor heat traces on the three‑sphere: Lambert W forms and two‑sector thermodynamics

We consider the scalar heat trace K0(t)=n≥0(n+1)2e−n(n+2)t and the Dirac spinor heat trace K1/2(t)=Tr,e−tD2 on the unit three‑sphere. The spinor trace admits the exact formK1/2(t)=π4t3/2(2‑t),(1+εs(t))with an explicit dual‑lattice bound on εs, whileK0(t)=et⋅π4t3/2(1+ε0(t))concentrates all power corrections in the prefactor et; the mean‑eigenvalue gap of the two spectra then admits the closed formV(t)=3‑t2‑tup to a certified remainder. The ratio R(t)=K1/2(t)/K0(t) is strictly decreasing, and the crossing equation R(t)=1 reduces, with certified dual‑lattice error, to the scalar transcendental equation t+et=2, whose root is expressed in closed form as tc=2‑W(e2) up to a certified 2.7×10−8; crossings against n copies of the spinor trace form the family tc(n)=2‑W(e2/n).On the static cylinder S3×ℝ the two sectors obey distinct thermodynamics: the fermion‑to‑boson entropy ratio tends to 7/4 in the ultraviolet, has no equilibrium point (certified global minimum 1.6547758⋯ at β=1.20204⋯), and diverges in the infrared with determined rate. All numerical values are certified by interval‑enclosed computation. No physical identification of any spectral quantity is claimed or used.