FBT02B: Affine Heisenberg Prequantisation of the Relative–Phase Torus

FBT02A introduces a basic-normalised affine Kac–Moody central extension with integer level kaff = k ∈ Z>0. FBT02C shows that this affine level cannot be identified directly with the two-dimensional Berry–Chern number of the FBT0B relative-phase fibre by equating the rank-one finite SU(2) coroot lattice with the rank-two relative-phase lattice. The missing ingredient is a genuine two-dimensional affine realisation. The present paper supplies such a realisation directly from the first Fourier harmonic of the basic affine Cartan current. Let H be a primitive basic-normalised SU(2) coroot satisfying ⟨H,H⟩bas = 2, and define the real first-harmonic pair Xc(θ) = H cos θ, Xs(θ) = H sin θ. After removing the conventional central factor i, the real Kac–Moody two-cocycle is ωKM(X, Y ) = 1/2πZ 2π0X(θ),dYdθ(θ)basdθ. A direct computation gives ωKM(Xc,Xs) = 1. Hence the lattice Γ1 = ZXc ⊕ ZXs is a primitive unimodular integral symplectic lattice: for λ = mXc + nXs, μ = pXc + qXs, one has ωKM(λ, μ) = mq − np. The physical symmetric relative-phase lattice of FBT0B is ΛK = Z(e1 − e3) ⊕ Z(e2 − e3) ≃ A2. Choosing its positive primitive basis a = e1 − e3, b = e2 − e3, we define the first-harmonic affine realisation Φharm(a) = Xc, Φharm(b) = Xs. The pullback of the basic affine cocycle is then exactly the primitive alternating form of the relative lattice:Φharm∗ωKM = Ebas. At affine level k, Φharm∗ω(k)KM = kEbas. This replaces the former Affine–Heisenberg Restriction Condition by an explicit theorem. The integer k appearing in the relative Heisenberg multiplier is therefore not inserted independently: it is the affine level of FBT02A evaluated on the primitive first-harmonic plane. The first-harmonic plane exponentiates inside the loop group to an R2-type Abelian subgroup rather than to a compact two-torus. The physical torus is obtained instead by quotienting the affine plane by its primitive integral lattice, T(1)aff = V1/Γ1 ≃ T2. Via Φharm, this torus is identified with the marked physical relative torus KA2 = (ΛK ⊗ R)/ΛK. The level-k central multiplier therefore defines a Heisenberg extension and an associated prequantum line Lk −→ KA2withc1(Lk) = kηK. FBT05A uses the tautological ray line O(−1). Relative to the positive prequantum generator, the effective Berry line is therefore the dual line, L(k)Berry = L∨k , so that on the positively oriented first-harmonic branch c1(L(k)Berry) = −kηK, kgeom = −kaff. The convention-independent level statement is |kgeom| = kaff. More generally, the n-th harmonic pair H cos(nθ), H sin(nθ) has affine pairing n. Thus the previously abstract realisation degree becomes the harmonic multiplicity: |νaff| = n. The first harmonic n = 1 is precisely the primitive affine branch. The paper therefore supplies the prequantum line itself, rather than a global Hilbert-space orbit, and establishes the constructive chain FBT02A affine level −→ first-harmonic affine plane −→ primitive Heisenberg lattice−→ prequantum line −→ FBT05A projective Berry readout. FBT28B [6] imports the primitive first-harmonic equality proved here and, under its universal theta/metaplectic-family and compactified discriminant-line matching hypotheses, conditionally selects k = kS = 24. On that selected branch, the theta space H24,τ = H0(Eτ ,L24) has dimension 24. The finite theta group acts through the level-24 Schrödinger representation, while oriented cycle-frame changes act through the Weil representation. After quotienting the unavoidable central phases, these actions combine into a canonical projective lift G24 : K(L24) ⋊ SL(2,Z) −→ PU(H24,τ ). For a quantisable FBT0B transition cover, local unitary lifts satisfy on triple overlaps GijGjkGki = eiϑijk1, GijGjkGki = 1. Thus the level-24 theta spaces glue canonically as a projective Hilbert bundle. The scalar cocycle is invisible to rays, the Fubini–Study QGT, and gauge-invariant scalar contractions, but it obstructs a genuine U(24) vector-bundle lift unless its Čech class is trivialised. The dual theta transition has the inverse scalar multiplier. Consequently, although the individual theta spaces may remain only projective, the paired transition on H∨24 ⊗ H24 has a strict cocycle. Relative Serre duality identifies the required dual sector intrinsically as R1π∗􀀀(L(24)pre )−1 ⊗ ωπ≃ (π∗L(24)pre )∨. Hence the Serre evaluation pairing and every equivariant contraction with one dual theta factor and one theta factor are unaffected by the projective U(1) obstruction. This removes the projective descent obstruction for the common Dirac-density programme, but it does not by itself construct the Dirac identification, the holomorphic density, or its thimble extension. The value 24 is therefore specialised here as a conditional import from FBT28B. It is not derived by the prequantum construction itself.