FBT01D: Balanced Tension Shell and the Central S-Gate

This bridge note formulates a balanced tension-shell condition linking three logically distinct structures in the Fracture–Berry–Tension framework: the non-central gate Casimir of FBT01B, the positive affine level introduced by FBT02A, its first-harmonic prequantum realisation in FBT02B, the conditional level selection of FBT28B, and the negative dualphase Morse sign established in FBT15A. FBT01B constructs the four-gate algebra on a gate-compatible regular relative-phase sector, ggate ∼= su(2) ⊕ u(1), where the non-central gates X,Y, Z are horizontal lifts of a base su(2)-type triad and the residual S-gate is a primitive vertical generator. Strict closure of the non-central triad requires the chosen relative-phase connection to obey FA(X,Y) = FA(Y, Z) = FA(Z,X) = 0. On such a gate-compatible branch, the non-central sector carries the Casimir-type invariant CT = X2 + Y2 + Z2. The geometric carrier of this statement is the fixed-cycle-frame principal branch of FBT0B. The abstract contrast quotient Qrel = T3/ΔU(1) is distinguished from an actual acting relative torus Krel ⊂ T3. On a free and proper regular sector the latter defines the unique orbit base Krel ,→ X(6)Regπ −−→ B(4)Reg. FBT01D uses this principal branch only. It neither assumes nor proves that its selected Sdirection descends through an optional affine torus family with nontrivial cycle monodromy. The corresponding global S-line preservation criterion belongs to FBT01E. At the classical Hamiltonian level, the selected residual phase direction admits a momentmap component μS = ⟨μrel, VS⟩. This classical charge is not the affine level. It is the Hamiltonian precursor of the Berry–Chern or current-algebra data whose quantized period defines kS = 1/2πZΣSΩS. FBT28B v2.3 [13] combines the FBT02B primitive first-harmonic identification kS = kaff = kpre with a universal theta/metaplectic family and the compactified discriminant-line matching DkS∼= λ12H , Dk∼= λk/2H . It thereby conditionally selects the branch on which kS = 24, rather than defining this value as a primitive normalization. Independently, FBT15A identifies a negative-definite dual-phase Hessian block in the real Morse problem. If Hphase > 0, then the phase contribution is −⟨θ,Hphaseθ⟩. Defining the positive phase magnitude s2S := ⟨θ,Hphaseθ⟩, the central Morse/readout square is S2Morse := −s2S < 0 for every nonzero phase displacement. A balanced readout branch is then defined by CT + S2Morse = 0, or equivalently CT = s2S = kSQ2S. On the calibrated unit branch kS = 24, Q2S = 1, this reduces to CT = 24. The balanced shell is not part of the bare gate algebra and is not an energy conservation law. It is a gate–Morse signature compatibility condition imposed on a selected readout branch. The relation to FBT01A is complementary rather than sequential. The present paper supplies the gate–Morse signature closure CT + S2Morse = 0, whereas FBT01A supplies the independent Liouville mixed-volume saturation identity Ω∧3B/3! = ωH ∧ η∧2/2 on a regular isotropic relative-phase fibration. The first controls quadratic-signature balance; the second controls symplectic volume nondegeneracy. They may constrain the same candidate physical branch, but neither is identified with, nor derived from, the other. In a classical coherent-state shadow, the balanced shell has the Lorentzian-like form x2 + y2 + z2 − s2S = 0. This is a signature seed at the gate-to-readout level, not a completed spacetime metric or causal cone.