Quarter-Turn Forcing of the Reduced Barrier Operator

T87 establishes that the reduced Q5 barrier operator is not merely compatible with a quarter-turn exponential structure, but is forced into that form by the combined requirements of orientation reversal, transverse axis exchange, and \ (Z₈\) spinor closure. Let = Ed = iᵧ, ² = -I, let\ B=e^ (/4) A the full Q5 barrier crossing operator, where\ the orientation-reversing boundary action. The theorem analyzes the reduced barrier action on the defect plane through the canonical phase projection\₏₇₀ₒ₄, in T26-T29. The key structural observation is that a plain orientation reversal\ insufficient to describe the barrier crossing. While it reverses sign, \ -, does not exchange the transverse reduced axes. The full barrier crossing requires both: - orientation reversal, - and transverse axis exchange. Within the reduced defect-plane interpretation, the crossing acts schematically as -X, a nontrivial rotational component in addition to sign reversal. The theorem then studies the exponential family^ A=I + A. \ Imposing the \ (Z₈\) spinor closure condition gives\ (e^ A) ⁸ = I, \ = m4, Z. \ Among these admissible angles, the theorem identifies\ = 4 the primitive positive generator producing the required single-step transverse exchange while preserving the \ (Z₈\) closure structure. Larger values correspond to higher-step composite rotations rather than primitive single-crossing transport. Accordingly, the reduced barrier operator is forced to beₑ₄₃=₄^ (/₄) ₀. \ Applying the canonical phase projection then yields\₏₇₀ₒ₄ (₁) =₄^ (/₄) ₀. \ The theorem further verifies: ₑ₄₃⁴=-I, ₑ₄₃⁸=I, the \ (Z₈\) state cycle and the associated sign reversal at half-period. T87 is structurally important because it upgrades the quarter-turn angle\/4 a consistency assumption to a forced consequence of the reduced barrier architecture. The theorem also carefully isolates the remaining open boundary. The reduced forcing argument operates within the established T47-T49 barrier interpretation: \ B = e^ (/4) A. , the orientation-reversal operator\ not yet been derived directly from the T17 six-segment kernel geometry itself. Thus T87 proves the reduced quarter-turn forcing result within the established barrier framework, while leaving the deeper derivation17 the remaining unresolved structural step. Together, T86 and T87 form a paired result: - T86 identifies the reduced operator form assuming the quarter-turn structure. - T87 proves that the quarter-turn structure is forced by reduced barrier requirements. The theorem, therefore, closes the reduced coarse-scale barrier identification arc linking: - the T47-T49 barrier architecture, - the T68 \ (Z₈\) spinor cycling, - and the T85 dual-scale generator framework. Status: - solid for the exponential forcing structure, - solid for the \ (Z₈\) closure and primitive quarter-turn selection, - solid for the reduced identification\₏₇₀ₒ₄ (B) =e^ (/4) A the established T47-T49 framework, - conditional only on the unresolved derivation of\ from the T17 six-segment kernel geometry.