Where Heat Balances the Vacuum: How continued fractions count the equilibria of a twisted thermal torus

Summary: A massless boson field in D space-time dimensions, with one spatial direction a circle of length ℓ and Euclidean time a circle of length β = ħc/kT, has two classical energies: the negative Casimir energy of the circle and the positive thermal energy. Space and time circle together form a flat torus with modulus τ = θ + iy, y = β/ℓ; the real part θ is a twist of the time circle, physically an imaginary velocity of the heat bath. The energy is E ∝ −f with f(τ) = ∂_y(y Z(τ, D/2)), where Z(τ, s) = Σ′ |m + nτ|^(−2s) is the Epstein zeta function of the lattice ℤ + ℤτ, equivalently the real analytic Eisenstein series. We call a zero of E a boson node: a temperature at which heat and vacuum balance exactly. We prove: (i) a directional form f = Σ′ |v|^(−2s) (1 − D sin²φ_v), its SL₂(ℤ)-covariance, and that no node lies on the orbits of the square and the hexagonal lattice; (ii) without twist the node is unique in every dimension D ≥ 3, with x₀(D) → (D − 1)^(1/D), and kTℓ/ħc = 0.6578618… in D = 4; (iii) with twist there is exactly one node with y ≥ 1 for D ≤ 12; (iv) below y = 1, every reduced fraction p/q with 0 < q²|θ − p/q| ≤ κ̂_D adds a pair of nodes, in every dimension D ≥ 3; (v) for 3 ≤ D ≤ 6 there are no other nodes, up to a band of width below 0.01 in q²|θ − p/q| around the threshold. Hence the continued fraction of the twist decides how many equilibria exist: if infinitely many partial quotients are at least 6 (for D ≤ 16), there are infinitely many nodes, and this holds for almost every twist. Twists whose partial quotients are all at most 3, among them the golden ratio, have exactly one node for 3 ≤ D ≤ 6. The finite parts of the proofs are done in ball arithmetic. Two primers explain the physics for number theorists and the number theory (continued fractions, Ford circles, the modular group) for physicists; labels separate proved, computer-assisted, cited, numerical and open statements. The companion code (Python scripts with mpmath and python-flint/Arb ball arithmetic, the five pre-registered tests with their registrations and SHA-256 files, a runner that checks every step against its expected outcome, reference logs, the figure scripts and a reference checker) is included as P13_companion_code.zip; the figures used by the LaTeX source are in P13_figures.zip. - Epstein zeta function - real analytic Eisenstein series - Chowla–Selberg formula - Casimir effect - finite temperature - thermal torus - modular group - continued fractions - Ford circles - Diophantine approximation - Borel–Bernstein theorem - shifted boundary conditions - imaginary chemical potential - interval arithmetic - ball arithmetic - computer-assisted proof - pre-registration - 11M36 (Selberg zeta functions and regularized determinants; Epstein zeta functions) - 11F03 (modular and automorphic functions) - 11J70 (continued fractions and their generalizations) - 11K50 (metric theory of continued fractions) - 81T55 (Casimir effect in quantum field theory) - 82B05 (classical equilibrium statistical mechanics, general) - 65G30 (interval and finite arithmetic)