Identification of the Reduced Barrier Cycle Operator

T86 establishes the reduced barrier cycle operator for the Q5 transport framework. Let=Ed=iᵧ, ²=-I, let\ B the full Q5 barrier crossing operator containing the orientation-reversal component \ (\). Using the canonical phase projection\₏₇₀ₒ₄, in T26-T29, the theorem identifies the reduced barrier operatorₑ₄₃=₏₇₀ₒ₄ (B), the conditional assumption that the barrier crossing acts as a quarter-turn on the reduced defect plane. Under this assumption, ₑ₄₃=₄^ (/₄) ₀. \ Using the exponential identity^ A=I + A, theorem provesₑ₄₃⁴=-I, ₑ₄₃⁸=I. \ The reduced barrier operator therefore, produces the spinor state cycle\₍+₄=-ₙ, ₍+₈=ₙ, a \ (Z₈\) state period and a \ (Z₄\) 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^ (/4) A reproduces the required \ (Z₈\) cycling, but the geometric origin of the quarter-turn angle\/4 not yet been derived from first-principles Q5 barrier geometry. Accordingly, T86 proves: \₏₇₀ₒ₄ (B) =e^ (/4) A 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 orientation-reversal structure and scalar/symmetric dressing removed by\₏₇₀ₒ₄. \ Thus, the clean exponential form^ (/4) A 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 \ (Z₈\) spinor cycling from T68, - and the dual-scale generator structure established in T85. Status: solid for^ (/4) Aₑ₄₃⁴=-I, ₑ₄₃⁸=I, for the resulting \ (Z₈/ Z₄\) cycling structure; solid for the projection properties of\₏₇₀ₒ₄; for the identification\₏₇₀ₒ₄ (B) =e^ (/4) A, depends on deriving the quarter-turn angle\/4 Q5 barrier geometry in T87.