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

Abstract

We formulate the residual S-sector in the Fracture–Berry–Tension (FBT) framework as a strict affine central-charge sector. The central point is that the physically relevant S-direction is not merely a finite-dimensional commuting u(1) factor, but the genuine onedimensional center of an affine extension of a reduced su(2) current algebra associated with the observable dual-phase sector. Starting from the dual-phase T2-geometry, we quotient the unobservable diagonal phase and identify the surviving observable phase direction θS. Along this reduced phase circle we construct the loop algebra Lsu(2) and prove that it admits the canonical affine central extension [Jan, Jbm] = fabcJcn+m + k n δabδn+m,0 S. We then show that, under the prequantum normalization inherited from the parent dualphase torus, the affine level is fixed by the normalized Berry flux k =1/2π󰁝Σ2Ω_phase. The strict result of the paper is therefore the identification of the S-gate as a geometrically normalized affine central generator of the reduced dual-phase current algebra. Possible macroscopic interpretations are briefly indicated only at the end. In later effective papers, this affine level may also be used as the microscopic normalization source of coarse-grained S-sector readouts. In particular, any readable phase capacity or S-sector stiffness should be understood as normalized by the same Berry-flux level k, rather than by an arbitrary external constant.