PAPER-FBT01B: The Four Tension Gates: Structural Origin, Gate Algebra, and Geometric Roles

Abstract

The four tension gates{X, Y, Z, S}provide the minimal generator-level skeleton of the Fracture–Berry–Tension (FBT) framework. Their role is not to replace later curvature, scalar-readout, gauge, or dynamical constructions, but to furnish a common structural layer from which later developments draw their admissible generators and geometric constraints.This paper isolates and organizes three basic questions: (1) the structural origin of the four gates from the first-principle tension triad and the compressed six-dimensional symplectic geometry; (2) the gate algebra ggate ∼= su(2) ⊕ u(1); (3) the geometric separation between the non-central horizontal sector and the observable residual vertical sector. The emphasis of the present paper is structural. We identify what is established at the level of gate construction and geometric organization, while leaving scalar readouts, mixed densities, gauge-sector realizations, and dynamical equations to separate papers. In particular, the scalar readout layer developed in FBT01A[3] is not part of the present gatealgebra construction. The only interface needed here is the observation that the residual vertical gate S may be contracted with a mixed horizontal–vertical formη ∈ H∗ ∧ V ∗to produce an S-projected horizontal readoutηS := ιSη ∈ H∗.Thus the S-gate selects a residual observable vertical direction, while mixed densities belong to the later readout layer. The result is a consolidated reference for the four-gate layer within the FBT series: the non-central gates{X, Y, Z}form a horizontal su(2)-type triad, while the S-gate is a central residual vertical mode, giving the minimal gate architecturesu(2) ⊕ u(1).The paper also records the Casimir-type quadratic invariant CT = X2 + Y2 + Z2 of the non-central gate triad, which later papers may use as the algebraic prototype of quadratic readout stiffness.