Observable Invariance under Non-Rotational Transport Dressing

T82 establishes observable invariance under non-rotational transport dressing within the reduced Q5 phase-transport sector. Starting from the raw barrier transport operator\[Traw=I+ε A+Sₙᵣ+O(ε^2),ε=1/160,\]with\[A=iσ_y,^2=-I,\]The theorem proves that the T26-T29 phase-extraction functional satisfies\[L(Traw)=ε+O(ε^2)=L(Tₚₕₐₛₑ)+O(ε^2),\]where\[Tₚₕₐₛₑ=I+ε A+O(ε^2).\] The non-rotational dressing\[Sₙᵣ\{I,σ_x,σ_z\}\]is annihilated by the extraction functional:\[L(Sₙᵣ)=0.\] Thus, the observable phase coefficient is determined entirely by the rotational generator \(A\) and is invariant under scalar, grading, or symmetric transverse dressing at first order in \(ε\). The theorem is structurally important because it completes the T80-T82 transport arc:\[Traw{Pₚₕₐₛₑ}Tₚₕₐₛₑ{× k,\ ε→0}etA{L}sin t.\] T80 established the discrete-to-continuous phase-flow limit\[Tₚₕₐₛₑ^ktA=Icos t + Asin t.\] T81 established that the canonical T26-T29 projection kills the non-rotational dressing:\[Pₚₕₐₛₑ(Traw)=Tₚₕₐₛₑ.\] T82 now proves the observable consequence: the measurable phase readout is invariant under the dressing sector and depends only on the rotational transport component. The theorem therefore replaces any remaining implicit “raw purity” interpretation with observable-sector equivalence under canonical extraction. The non-rotational dressing changes the representation of the raw transport operator but not the observable phase coefficient. The gauge-redundancy interpretation introduced in the theorem is structurally important:\[Traw=N_ε · Tₚₕₐₛₑ,\]with\[N_ε=I+Sₙᵣ,\]defines a family of raw transport operators projecting to the same observable phase sector, analogous to gauge-equivalent representations sharing the same observable curvature structure. Status: solid for the first-order observable invariance\[L(Traw)=L(Tₚₕₐₛₑ)+O(ε^2)\]under the T26-T29 null-class structure; conditional on the assumption that the full dressing sector remains entirely within\[span\{I,σ_x,σ_z\},\]and for higher-order corrections beyond first order in \(ε\).