PAPER-FBT01D: Balanced Tension Shell and the Central S-Gate

This bridge note formulates a balanced tension-shell condition linking the non-central gate Casimir of FBT01B with the central S-gate normalisation of FBT02A and the dualphase Morse-negative sign of FBT15A. FBT01B establishes the four-gate algebra ggate ∼= su(2) ⊕ u(1), where the non-central gatesX, Y, Zform an su(2)-type horizontal triad and the residual gateSis central and vertical. The non-central triad carries the quadratic Casimir-type invariantCT = X2 + Y2 + Z2. FBT02A/FBT28B identifies the S-gate as a strict affine central charge with positive level kS = 24. FBT15A, independently, shows that the dual-phase sector enters the real Morse problem through a negative-definite two-dimensional Hessian block. The present note records the controlled bridge among these three facts. The negative sign of the S-sector square is not treated as a bare algebraic fact about an abstract central generator. It has a two-level origin. First, structurally, the S-gate is the central readable output of the dual-phase reduction generated by the non-central su(2)-type triad. Hence it is naturally represented as a phase-output generator, S = i󰁳kS QS, so that S2 = −kSQ2S. Second, at the Morse/readout level, this algebraic phase sign is realised by the negativedefinite dual-phase Hessian block of FBT15A. Under the phase-type Hessian condition of FBT15A, the dual-phase block contributes−〈θ,Hphaseθ〉 to the quadratic expansion of ℜScl, with Hphase > 0. Hence the S-sector Morse/readout square is S2Morse = −s2S = −〈θ,Hphaseθ〉< 0 for every nonzero phase displacement θ. With central level normalisation, s2S = kSQ2S, kS = 24. A balanced readout branch is then defined by CT + S2Morse = 0, or equivalently CT = s2S = kSQ2S. In the unit S-readout normalisation Q2S = 1, this becomes CT = 24. The balanced shell condition is not part of the bare gate algebra. It is a readout-level compatibility condition connecting the non-central horizontal tension radius, the central S-gate level, and the dual-phase Morse sign. Its classical coherent-state shadow has the Lorentzian-like form x2 + y2 + z2 − s2S = 0, but this should not be interpreted as an already-derived spacetime metric. It is a gate-toreadout bridge condition.