Identification of the Reduced Barrier Cycle Operator

T86 establishes the reduced barrier cycle operator for the Q5 transport framework. Let\[A=E_d=iσ_y,^2=-I,\]and let\[ B\]denote the full Q5 barrier crossing operator containing the orientation-reversal component \(Θ\). Using the canonical phase projection\[Πₚₕₐₛₑ,\]introduced in T26-T29, the theorem identifies the reduced barrier operator\[Bred=Πₚₕₐₛₑ( B),\]under the conditional assumption that the barrier crossing acts as a quarter-turn on the reduced defect plane. Under this assumption,\[{Bred=e(π/4)A.}\] Using the exponential identity\[eθ A=Icosθ + Asinθ,\]The theorem proves\[Bred^4=-I,red^8=I.\] The reduced barrier operator therefore, produces the spinor state cycle\[ψₙ₊₄=-ψ_n,ψₙ₊₈=ψ_n,\]giving a \( Z_8\) state period and a \( Z_4\) observable period through projective sign identification. T86 is structurally important because it isolates the reduced coarse barrier operator explicitly within the Q5 framework. Earlier theorems established:- the rotational generator \(A\),- the barrier crossing structure,- and the projection architecture. T86 now combines these ingredients into a single reduced operator form. The theorem also sharply isolates the remaining open step. The algebraic structure\[e(π/4)A\]correctly reproduces the required \( Z_8\) cycling, but the geometric origin of the quarter-turn angle\[π/4\]has not yet been derived from first-principles Q5 barrier geometry. Accordingly, T86 proves:\[Πₚₕₐₛₑ( B)=e(π/4)A\]conditional on the quarter-turn assumption, while T87 is designated as the theorem responsible for deriving the angle geometrically. The theorem further clarifies that the identification applies only after projection onto the reduced defect plane. The full barrier operator\[ B\]contains orientation-reversal structure and scalar/symmetric dressing removed by\[Πₚₕₐₛₑ.\] Thus, the clean exponential form\[e(π/4)A\]It is a reduced-sector statement rather than a claim about the full raw barrier operator globally. T86 therefore serves as the algebraic bridge connecting:- the barrier crossing structure from T47-T49,- the \( Z_8\) spinor cycling from T68,- and the dual-scale generator structure established in T85. Status: solid for\[e(π/4)A\]producing\[Bred^4=-I,red^8=I,\]and for the resulting \( Z_8/ Z_4\) cycling structure; solid for the projection properties of\[Πₚₕₐₛₑ;\]conditional for the identification\[Πₚₕₐₛₑ( B)=e(π/4)A,\]which depends on deriving the quarter-turn angle\[π/4\]from Q5 barrier geometry in T87.