Chiral Orientation of the Directed Heisenberg Frontier

The directed Heisenberg frontier carries a chirally dressed cocycle signal $\Theta_\chi = \langle \sigma_L\,(2\pi/q)\,{\Delta A_c}\rangle_{\partial^+}$, a discrete counterpart of the orientation datum (in the sense of ) of the angular generation split of the projected Dirac residue ${E_\Pi}$; it is not the $J_3$ coefficient of the metaplectic step, which in the $\mathrm{SL}(2)$ model of Q14 is produced by the $\mathfrak{sl}_2$ commutator $[E,F]=H$ (Q14, Proposition 6.5). The raw cocycle average and the maximal-locking bound ${\theta_{\max}}$ are computable on the Heisenberg frontier of, but the signal itself depends on the chiral weight $\sigma_L$, whose origin lies in the Lorentzian spin-lift and not in the arithmetic frontier. We prove, under the hypothesis [H-WS] of (the residual reflection $R_b$ acts on the chiral carrier as a scalar, so $[\gamma_5,R_b]=0$, which is all that this note uses; the value $-1$ of the scalar is a second part, used only in; supplied by no source), that $\sigma_L$ cannot be any function of the real cascade generator $T(e)$: edge inversion and the residual reflection $\varphi$ both act as $T\mapsto -T$, so every $f(T)$ has identical arrow-parity and $\varphi$-parity, whereas $\sigma_L$ must be arrow-even and $\varphi$-odd. The resolution, under [H-WS], is that $\sigma_L$ is the Weyl representation type, ${\mathbf{2}}$ versus ${\bar{\mathbf{2}}}$: group inversion preserves the representation (arrow-even) while complex conjugation exchanges it ($\varphi$-odd). In the inherited Heisenberg central-phase lift the chiral phase is the linear cyclotomic character $\omega_\chi(e)={\zeta_q}^{{\Delta A_c}(e)}$, with no Gauss, Maslov, or $\sqrt{5}$ component; on the nonzero-cocycle support the natural branch satisfies $\sigma_L(e)={\operatorname{sign}} \Im\,{\zeta_q}^{{\Delta A_c}(e)} = {\operatorname{sign}}{\Delta A_c}(e)$, giving, by this definition, ${\rho_\chi} = 1$ and ${N_A} = {\theta_{\max}} \neq 0$. This establishes ${[\mathrm{H\text{-}orient}]}$ in the inherited Heisenberg central-phase lift and under [H-WS], not unconditionally: the only remaining obstruction besides [H-WS] is an additional spin-Galois phase outside the central character, in particular a $\sqrt{5}$ or ADE-spin factor orthogonal to ${\zeta_q}$. The result is a conditional derivation of the frontier amplitude datum; it does not derive $u\neq 0$, which would further require a transport of the oriented signal to the ${J_\Pi}$-odd generation sector; the only transport stated in is the hypothesis [H-Tr] for the $\mathfrak{sl}_2$ opening, supplied by no source, and the identification of the frontier signal with that opening is a further input.