Where Fermions Count Modulo Two: spin structures, the Euler factor at 2 and continued-fraction parity in the thermal Casimir energy of a twisted torus (Paper P20)

A free field on a two-dimensional torus of space length ℓ and inverse temperature β, with D − 2 further flat directions, has an internal energy that is a sum over closed windings. For bosons every winding counts with +1; for fermions it counts with a sign (−1)^(am+bn) fixed by the spin structure, that is, by whether the field is periodic or antiperiodic around space (a) and time (b). We show where number theory enters through these signs and why. (i) Each fermion sum is twice the boson sum of a sublattice of index two minus the boson sum. This one identity produces the factor 7/8 of Stefan–Boltzmann, the Euler factor 1 − 2^(1−s) at the prime 2, the three congruence subgroups of level two, and closed values with the L-functions of ℚ(i) and ℚ(√−3) at the symmetric shapes. (ii) For antiperiodic space the energy vanishes on the untwisted torus at β/ℓ = 1.2101889539 in four dimensions, that is at T ≈ 1893 K for ℓ = 1 µm; it is positive on the square torus, where it equals (π²/3)G with Catalan's constant G. (iii) For a twisted torus the energy at the temperature of a continued-fraction convergent p_k/q_k of the twist has the sign −(−1)^(a p_k + b q_k) whenever the next partial quotient is at least three, in every dimension D ≥ 3. Hence the zeros of the energy are counted by the parities of the convergents: the golden twist has infinitely many fermion zeros where the boson has exactly one, and almost every twist has infinitely many zeros in all three fermion structures. (iv) On the four-torus the same mechanism changes only the local factor at 2 for the lattice D₄ and for A₂ ⊕ A₂, but not for ℤ⁴, and no finite dimension guarantees that the shortest winding decides the sign. (v) At D = 2 the sums are logarithms of Jacobi theta functions, and the classical Bose–Fermi counting of the superstring is the same index-two identity. All physics in the paper is textbook physics; what is new is the number-theoretic account of where the signs come from and what they control. Every result carries a status label (proved, proved computer-assisted, classical, numerical, hypothesis, negative, open). The companion code reproduces all quoted numbers; computer-assisted proofs use interval arithmetic (Arb), and the numerical tests were preregistered.