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

The four tension gates

{X, Y, Z, S}

provide the minimal generator-level architecture of the Fracture–Berry–Tension framework. This revision gives a conditional geometric realisation of that algebra on the single-carrier regular relative-phase architecture of FBT0B and the mixed symplectic geometry of FBT01A.

The abstract relative-phase quotient

Qrel = T3/ΔU(1)

does not automatically act on the coherent carrier. The active regular geometry instead uses a chosen acting subtorus Krel ⊂ T3:

Krel ↪→ X(6)Regπ −→ B(4)Reg, B(4)Reg = X(6)Reg/Krel.

An invariant principal connection A ∈ Ω1(X(6)Reg; krel) determines

TX(6)Reg = HA ⊕ V, HA = ker A, V = ker dπ.

The non-central gates are not derived from dimension counting. The paper assumes a faithful infinitesimal base action

ρH : su(2) ↪→ X(B(4)Reg)

whose distinguished generators are ¯XL, ¯XN, ¯XD, and lifts them horizontally:

X = (¯XL)HA, Y = (¯XN)HA, Z = (¯XD)HA.

Because brackets of horizontal lifts generally acquire a vertical curvature term, exact closure of the lifted triad requires

FA(X, Y) = FA(Y, Z) = FA(Z, X) = 0.

Under this gate-compatibility condition, the lifted gates satisfy

[X, Y] = 2Z, [Y, Z] = 2X, [Z, X] = 2Y.

The residual gate S is represented geometrically by a selected primitive integral direction ξS ∈ krel,Z, with S = ξ#S . For an invariant connection, fundamental vertical fields commute with invariant horizontal lifts. Hence

[S, X] = [S, Y] = [S, Z] = 0,

and on every fixed-frame regular gate chart

ggate(U) ∼= su(2)T ⊕ u(1)S.

The new structural result of this revision is internal to that established gate algebra. Under the adjoint action of the integrated non-Abelian subgroup SU(2)T , 

ggate(U)SU(2)T = RS ≃ u(1)S, dim ggate(U)SU(2)T = 1. 

Equivalently, normalized Haar averaging defines the invariant projection

Pinv(A) :=∫︂SU(2)TAdg(A) dg,

and, for A = aXX + aY Y + aZZ + aSS, one has

Pinv(A) = aSS.

Thus the central S-direction is not merely a chosen closed circle at the geometric-realisation stage; once the four-gate algebra is given, it is also characterized intrinsically as the unique mode that survives the full non-Abelian invariant projection.

This statement does not derive the four-gate algebra from more upstream phase, fracture, or pairing data, and it does not prove that an arbitrary upstream algebra must contain exactly one singlet. It proves uniqueness only inside the conditionally constructed su(2)T ⊕ u(1)S gate algebra.

FBT07D v0.4 supplies a conditional upstream candidate for the selected circle. If its retained thimble lattice is saturated and rank one, then the pre-central evaluation map

Fth : ˆ︁ Trel −→ Hom(Λth,U(1))

has a unique connected kernel Usurv ∼= U(1). With the FBT02B unimodular alternating form and a compatible oriented vertical realisation, there is an isomorphism

Φ07→01S : Usurv∼−→ U(1)S, dΦ07→01S(︁LieUsurv)︁= RS.

Hence, on that additional branch,

gSU(2)Tgate = dΦ07→01S(︁LieUsurv)︁
.
This interface does not prove the rank-one thimble hypothesis and does not derive the noncentral gate triad.
The paper also distinguishes the abstract Casimir CT = X2 + Y2 + Z2 from its local Hamiltonian symbol Ccl
T = L2+N2+D2, and records the controlled mixed-readout interface with FBT01A and FBT0C. If gate compatibility fails, vertical curvature corrections belong to the mixed readout layer rather than to the bare four-gate algebra.