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 $|\Theta_\chi| \le \theta_{\max}$. This signal is a discrete counterpart of the orientation datum of the Q11 oriented-frontier note; 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 frontier backend does not supply $\sigma_L$: it is the surviving stub of the Lorentzian chiral lift. This note proves, under the hypothesis [H-WS] (the residual reflection $R_b$ acts on the chiral carrier as the scalar $-1$, so $[\gamma_5, R_b] = 0$; 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 equal arrow- 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 $\omega_\chi(e) = \zeta_q^{\Delta A_c(e)}$ — 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 and under [H-WS], not unconditionally.
Core contributions
- Parity obstruction, under [H-WS]: $\sigma_L \neq f(T(e))$. 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 under [H-WS]: 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 $\rho_\chi = 1$ by this definition; $\mathcal{N}_A = \theta_{\max} \neq 0$, $q$-stable to $0.4\,\%$ ($\theta_{\max} \cdot q = 2.114, 2.122, 2.118$ at $n_3^{\mathrm{obs}} = 2$ for $q \in \{61, 101, 151\}$), not $q$-invariant, while the per-shell rationals are $q$-independent over the first six shells for $q \in \{61,101,151,211,307\}$ (verified domain).
- $[\mathrm{H\text{-}orient}]$ in the inherited lift, under [H-WS]; residual obstruction. The result establishes the frontier amplitude datum in the inherited Heisenberg central-phase lift and under [H-WS], not unconditionally. The only remaining obstruction besides [H-WS] — an additional spin-Galois phase outside the central character, in particular a $\sqrt{5}$ or ADE-spin factor orthogonal to $\zeta_q$ — is open in this note. The AOG-Note shows, taking the Lorentzian chiral lift as an input it does not construct ([H-Spin] of Q14), and under [H-WS], its modelling premise [M], its boost premise [B] and the central-phase convention inherited from this note, that this lift induces no action on $\sqrt{5}$.
- Numerical validation of the backend. Raw average exactly zero, symmetrised anti-bias control exactly zero, $\theta_{\max} \cdot q$ stable to $0.4\,\%$ on $q \in \{61, 101, 151\}$, recomputed from the CHO repository alone. 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
Proved under [H-WS]. 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 $q$-stable to $0.4\,\%$.
Open. Whether the chiral spin-lift carries only the inherited central character, or an additional spin-Galois phase (a $\sqrt{5}$ or ADE-spin factor orthogonal to $\zeta_q$), is the single remaining obstruction besides [H-WS]. The AOG-Note shows, taking the Lorentzian chiral lift as an input ([H-Spin] of Q14) and under [H-WS], its modelling premise [M], the boost premise [B] and the inherited central-phase convention, that the lift induces no action on $\sqrt{5}$.
Not derived here. The split $u \neq 0$ itself. That conclusion would further require a transport of the oriented signal to the $J_\Pi$-odd generation sector. The only transport stated in PRS 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. CHO supplies the frontier amplitude datum, conditionally; 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 margin of a supplied two-form; no amplitude mechanism is selected), and the AOG-Note (spin-stratum type-rigidity: under [H-Spin] (as input), [H-WS], [M], the boost premise [B] and the inherited central-phase convention, its chiral-lift corollary shows that the Lorentzian chiral lift induces no action on $\sqrt{5}$).
References
Jérôme Beau. Chiral Orientation of the Directed Heisenberg Frontier. Working paper, 2026. 10.5281/zenodo.20735907