Coarse Spinor Cycling and Fine Phase Accumulation

T85 establishes the dual-scale generator structure underlying coarse spinor cycling and fine phase accumulation in the Q5 transport framework. Starting from the rotational generator\[A^2=-I,\]The theorem defines the coarse barrier-cycle operator\[B=e(π/4)A.\] Using the exponential identity\[eθ A=Icosθ + Asinθ,\]The theorem proves\[B^4=-I,^8=I.\] Thus, the coarse barrier operator exhibits a \( Z_8\) spinor cycle with sign reversal at the half-cycle. Separately, the reduced near-identity phase transport operator\[Tₚₕₐₛₑ=I+1{160}A+O(1{160^2})\]satisfies, by T80,\[Tₚₕₐₛₑ^k(k/160)A.\] The theorem, therefore establishes that the same generator \(A\) supports two compatible transport structures:\[B^n=e(nπ/4)A\]and\[Tₚₕₐₛₑ^k(k/160)A.\] The first defines coarse \( Z_8\) spinor cycling at step size\[π/4,\]while the second defines fine accumulated phase transport at the step size\[1/160.\] T85 is structurally important because it resolves the apparent tension between coarse barrier cycling and fine transport accumulation. The theorem shows that these are not competing clocks and do not contradict one another. They are structurally distinct operators generated by the same rotational algebra. The coarse barrier cycle governs finite crossing symmetry:\[B^8=I.\] The fine transport operator governs the gradual accumulated phase evolution:\[μ_k=sin(k/160)+O(1/160).\] The theorem therefore unifies the coarse spinor structure from T47-T49/T68 with the continuous transport structure developed in T80-T84. A key consequence is that coarse \( Z_8\) cycling does not reset fine accumulated transport. Eight fine transport steps accumulate only\[8/160=1/20,\]far from a full \(2π\) phase cycle, so fine accumulation persists independently of the coarse barrier period. The theorem also clarifies that both structures arise from the same Lie-generator algebra:\[A^2=-I.\]The coexistence of coarse cycling and fine transport is therefore a structural consequence of the common generator rather than a coincidence of notation. Status: solid for\[B^4=-I,^8=I,\]and for the dual exponential structures generated by \(A\); solid for the compatibility of coarse \( Z_8\) cycling and fine \(1/160\) phase accumulation given the T80 continuous-limit theorem; conditional for the physical identification of \(B\) and \(Tₚₕₐₛₑ\) with the Q5 barrier and reduced transport operators through the T47-T49 and T80-T82 chains.