FBT02A: The S-Gate as a Strict Affine Central Charge

We formulate the residual S-sector of the Fracture–Berry–Tension framework as a strict affine central sector while keeping the geometric, Hamiltonian, loop-algebraic, prequantum, Berry, and representation-theoretic layers mathematically distinct. The regular geometric input distinguishes the canonical abstract contrast quotient Qrel = T3/ΔU(1) from a chosen actual acting relative subtorus Krel ⊂ T3. On the symmetric branch, Krel = KA2 = ker(z1z2z3), with primitive relative lattice of A2-type. On a free and proper regular branch, Krel ↪→ X(6)Regπ −→ B(4)Reg. After choosing a Krel-invariant connection A, one has TX(6)Reg = HA ⊕ V, ΩB|V ×V = 0, ΩB = ωH + η, with no independent vertical symplectic block. A primitive integral vector q = (q1, q2) in a local integral cycle frame selects a residual circle ΣS ⊂ Krel. In a connection-adapted Hamiltonian chart, Aa = dϕa + αa, Aa(Vb) = δab , η = dJa ∧ Aa. Hence, for VS = qaVa, one has ιVSΩB = ιVS η = dμS, μS = −qaJa. This belongs to the classical Hamiltonian layer. The current target is a separate upstream input: gnc = span{X, Y, Z} ∼= su(2). The selected residual circle supplies the source of Lgnc = C∞(ΣS, gnc), while the affine central generator arises from the basic-normalised Kac–Moody cocycle, ωKM(X, Y ) = i/2π∫︂ 2π0⟨︃X,dYdθS⟩︃basdθS. The corresponding strict central extension satisfies [Jan, Jbm] = fabcJcn+m + nδabδn+m,0KS. In a representation, ρ(KS) = kaff id . For basic-normalised integrable positive-energy loop-group representations, kaff ∈ Z≥0. The central revision of v3.5 is that the affine-to-prequantum bridge is no longer left as a bare matching hypothesis on the primitive branch. Let H be a primitive basic-normalised SU(2) coroot satisfying ⟨H,H⟩bas = 2, and define the real first-harmonic Cartan loops Xc(θS) = H cos θS, Xs(θS) = H sin θS. With ωKM := −iωKM, FBT02B proves ωKM(Xc,Xs) = 1. Thus the first-harmonic affine plane contains a primitive unimodular integral symplectic lattice. Ifa = e1 − e3, b = e2 − e3 is the positively oriented primitive basis of the symmetric FBT0B relative lattice, the firstharmonic affine realisation Φharm(a) = Xc, Φharm(b) = Xs satisfies Φ∗harmωKM = Ebas. At affine level k, Φ∗harmω(k)KM = kEbas. Consequently FBT02B constructs a positive prequantum line L(k)pre −→ KA2 with c1(L(k)pre) = k ηK. Accordingly, on the primitive first-harmonic branch, kpre = kaff. The common positive integer is denoted kS := kaff = kpre > 0 on a nontrivial positive-energy branch. The Berry-ray convention is separate. FBT05A uses the tautological line O(−1), which is the dual of the positive prequantum line in the present sign convention. Hence L(k)Berry =(︂L(k)pre)︂∨, and therefore c1(L(k)Berry) = −k ηK. Thus, on the positively oriented primitive first-harmonic branch, kBerry = −kS, |kBerry| = kS = kaff = kpre. The minus sign is the tautological-ray duality sign; it is not a negative affine level. More generally, ωKM(︁H cos(nθS),H sin(nθS))︁= n. Therefore the n-th harmonic gives kBerry| = nkaff, while the first harmonic n = 1 is precisely the primitive affine branch. The strict FBT02A result still does not derive kS = 24. That numerical selection remains delegated to FBT28B, which conditionally obtains 24 from polarised full-torus quantisation and the regular structural-module hypothesis HkS (τhex) ∼= C[Kstr]. Finally, the full hexagonal modular stabiliser fixes no nonzero primitive cycle, so a globally fixed S-circle and full hexagonal symmetry cannot be imposed simultaneously without symmetry reduction, local-system transport, a cycle orbit, or a full-torus formulation. On an integral-affine branch, an oriented global S-circle requires the orientation-preserving descent condition of FBT01E. The strict architecture is therefore VSη−→ μS, ΣS −→ θS −→ dS, LgncωKM −→ KS −→ kaffFBT02B−→ kpre −→ kBerry. On the primitive first-harmonic branch, kaff = kpre = kS, kBerry = −kS, |kBerry| = kS.