Overview
The Q11 oriented-frontier diagnostic produces two computable quantities — the raw cocycle average $\langle \Delta A_c \rangle_{\partial^+} = 0$ and the maximal-locking bound $\theta_{\max} = \langle (2\pi/q)\,|\Delta A_c| \rangle_{\partial^+}$ — and one missing datum: the chiral weight $\sigma_L \in \{\pm 1\}$ that turns the bound into an amplitude through $\Theta_\chi = \langle \sigma_L\,(2\pi/q)\,\Delta A_c \rangle_{\partial^+}$, with $|\Theta_\chi| \le \theta_{\max}$.
The frontier backend does not supply $\sigma_L$: it is the surviving stub of the Lorentzian chiral lift. This note proves $\sigma_L$ 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 equal arrow- and $\varphi$-parity, whereas $\sigma_L$ must be arrow-even and $\varphi$-odd. The resolution 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 $\omega_\chi(e) = \zeta_q^{\Delta A_c(e)}$ — 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 outgoing-frontier overlap $\rho_\chi = 1$ and promoting the bound to $\mathcal{N}_A = \theta_{\max} \neq 0$. This establishes $[\mathrm{H\text{-}orient}]$ in the inherited central-phase lift, not unconditionally.
Core contributions
- Parity obstruction: $\sigma_L \neq f(T(e))$. Edge inversion $U(e^{-1}) = U(e)^{-1}$ and the residual reflection $\varphi = R_b \circ J_\Pi$ both send the $J_\Pi$-odd cascade generator $T(e) \mapsto -T(e)$, so every function of $T$ has equal arrow- and $\varphi$-parity; the admissible $\sigma_L$ is arrow-even and $\varphi$-odd. The global identification $\sigma_L = \operatorname{sign} \Delta A_c$ on all directed edges is therefore excluded — it makes $\sigma_L$ arrow-odd and breaks the reversal control $\Theta_\chi^{(\partial^+)^{-1}} = -\Theta_\chi^{\partial^+}$.
- Resolution: Weyl representation type. The chiral weight $\sigma_L$ is the representation label $\mathbf{2}$ versus $\bar{\mathbf{2}}$. Group inversion preserves the representation (arrow-even) and complex conjugation exchanges it ($\varphi$-odd) — the unique label distinguished by the conjugation that separates $\varphi$ from arrow inversion. For a real lift the representation is self-conjugate and the chiral contrast vanishes, matching the chiral symmetry of the real cascade.
- Inherited Heisenberg central-phase lift. The edge 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 yields $\sigma_L(e) = \operatorname{sign} \Delta A_c(e)$ locally, the global $\sigma_L$ being the representation label that survives the arrow/$\varphi$ parity constraints. Outgoing-frontier overlap $\rho_\chi = 1$; $\mathcal{N}_A = \theta_{\max} \neq 0$, $q$-invariant on $q \in \{61, 101, 151\}$.
- $[\mathrm{H\text{-}orient}]$ closed in the inherited lift, residual obstruction now discharged. The result establishes the frontier amplitude datum in the inherited Heisenberg central-phase lift. The only remaining obstruction — an additional spin-Galois phase outside the central character, in particular a $\sqrt{5}$ or ADE-spin factor orthogonal to $\zeta_q$ — is discharged by the chiral-lift corollary of AOG-Note: the same spin-stratum type-rigidity shows the Lorentzian chiral lift induces no action on $\sqrt{5}$, so $[\mathrm{H\text{-}orient}]$ holds and $\mathcal{N}_A \neq 0$ unconditionally within the present spin stratum.
- Numerical validation of the backend. Raw average exactly zero, symmetrised anti-bias control exactly zero, $\theta_{\max} \cdot q$ stable on $q \in \{61, 101, 151\}$. Backend and bound validated; $\sigma_L$ itself is supplied only by the chiral lift, not by the backend.
What this note settles, and what it does not
Settled. The chiral weight $\sigma_L$ cannot be a function of $T(e)$; it is the Weyl representation type. In the inherited central-phase lift the frontier amplitude datum $\mathcal{N}_A = \theta_{\max}$ is non-zero and $q$-invariant.
Closed within the present spin stratum. The result holds in the inherited Heisenberg central-phase lift. The single caveat — an additional spin-Galois phase outside this central character, a $\sqrt{5}$ or ADE-spin factor orthogonal to $\zeta_q$ — is discharged by the chiral-lift corollary of AOG-Note: the Lorentzian chiral lift induces no action on $\sqrt{5}$, so $[\mathrm{H\text{-}orient}]$ holds unconditionally within the present spin stratum, the chiral closure and the ADE-gate no-go being opposite cuts of one spin-stratum type-rigidity.
Not derived here. The split $u \neq 0$ itself. That conclusion still requires the Schur transport of the polarisation onto the $J_\Pi$-odd generation sector (cf. PRS, A4-Note). CHO supplies the frontier amplitude datum; it does not bypass the operator step.
Position in the programme
This note belongs to the fermionic matter sub-programme (Presentation Note 6). It closes the chiral-weight question left open by the Q11 oriented-frontier diagnostic and connects with the PRS reduction (Schur transport), the A4-Note (Born–Infeld saturation mechanism for the amplitude), and the AOG-Note (spin-stratum type-rigidity — its chiral-lift corollary discharges the residual obstruction of CHO and closes $[\mathrm{H\text{-}orient}]$ unconditionally within the present spin stratum).
References
Jérôme Beau. Chiral Orientation of the Directed Heisenberg Frontier. Working paper, 2026. 10.5281/zenodo.20735907