Overview
The symmetric-capacity obstruction establishes that the raw shell observable of the angular generation split vanishes identically, $\Theta_{\mathrm{raw}} = 0$: on a symmetric BFS shell the involution $g \mapsto g^{-1}$ pairs opposite central phases and erases the oriented $J_\Pi$-odd component before the radial recursion can act.
This note defines the honest replacement: a frontier transfer observable attached to the directed outgoing edges $g \to gs$ of the cascade, whose central increment is the Heisenberg cocycle $a(g)\, s_b$ — the discrete oriented increment of the cascade. It is a counterpart of the orientation datum, and not the $J_3$ coefficient of the metaplectic step, which in the $\mathrm{SL}(2)$ model of Q14 comes from the $\mathfrak{sl}_2$ commutator $[E,F] = H$ (Q14, Proposition 6.5). The orientation is the arrow of the Cayley graph from the projection origin to increasing BFS depth, not an extracted weight; no identification of this arrow with an ordering derivative $\partial_\tau$ is made here (Q14 states, for the generator $\log g$ of the metaplectic step, that such an identification is a separate hypothesis that no source supplies).
Core contributions
- Symmetric-capacity obstruction. The involution $g \mapsto g^{-1}$ pairs opposite central phases on symmetric BFS shells, erasing the oriented $J_\Pi$-odd component; the raw shell observable vanishes identically.
- The frontier transfer observable. Attached to the directed outgoing edges $g \to gs$ of the cascade, with central increment given by the Heisenberg cocycle $a(g)\, s_b$ — the discrete oriented increment of the cascade; a counterpart of the orientation datum, not the $J_3$ coefficient of the metaplectic step, which, in the $\mathrm{SL}(2)$ model of Q14, comes from the $\mathfrak{sl}_2$ commutator $[E,F] = H$.
- Orientation. The orientation is the arrow of the Cayley graph from the projection origin to increasing BFS depth, not an extracted weight. No identification of this arrow with an ordering derivative $\partial_\tau$ is made; for the generator $\log g$ of the metaplectic step, Q14 states that such an identification is a separate hypothesis that no source supplies.
- Chiral lift of the residual reflection. Reduced to one named hypothesis, [H-WS]: $R_b$ (i) acts as a scalar on the chiral carrier, so that $[\gamma_5, R_b] = 0$; (ii) the scalar is $-1$, used only by the AOG-Note; supplied by no source.
- Existence test before amplitude. The first test is $\langle \Delta A_c \rangle_{\partial^+ S_m} \neq 0$, which may fail under residual frontier re-pairing. The discriminating anti-bias control is the symmetrised-frontier cancellation $\langle \Delta A_c \rangle_{\partial^+ S_m \cup \partial^- S_m} = 0$.
- Chiral weight identified (CHO companion). The companion CHO note proves, under [H-WS], that the chiral weight $\sigma_L$ cannot be any function of the cascade generator $T(e)$ — arrow inversion and the residual reflection act identically on it — and identifies it as the Weyl representation type $\mathbf{2}$ vs $\bar{\mathbf{2}}$. In the inherited Heisenberg central-phase lift the edge phase is the linear cyclotomic character $\zeta_q^{\Delta A_c}$, giving outgoing-frontier overlap $\rho_\chi = 1$ and promoting the bound to $\mathcal{N}_A = \theta_{\max} \neq 0$ in that lift, under [H-WS].
Position in the programme
This note belongs to the fermionic matter sub-programme (Presentation Note 6). It supplies the protocol testing whether the recursion produces a non-zero, bias-free oriented signal on the directed frontier, a counterpart of the orientation datum of the Q14 §6 inter-generation splitting analysis, not its $J_3$ coefficient. Together with the AAR note ($\mathfrak{sl}_2$ reduction; observable to be measured) and the PRS note (operator-side Schur form of the compression remainder), it completes the diagnostic frame in which the remaining quantitative step of the fermionic sector can be carried out.
References
Jérôme Beau. The Oriented Frontier Observable: a Recursive Q11 Diagnostic for the Angular Generation Split. Working paper, 2026. 10.5281/zenodo.20601245