The Oriented Frontier Observable: a Recursive Q11 Diagnostic for the Angular Generation Split

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 counterpart of the Baker–Campbell–Hausdorff term $\tfrac{1}{2}[X,Y]$ and hence of the dynamical $J_3$ . The orientation is the recursive arrow $\partial_\tau=\log g$ from the projection origin to increasing BFS depth, not an extracted weight. The first test is not an amplitude value but the existence of an oriented signal, $\langle\Delta A_c\rangle_{\partial^+S_m}\neq 0$, which may fail if a residual automorphism re-pairs the frontier; the discriminating anti-bias control is the symmetrised-frontier cancellation $\langle\Delta A_c\rangle_{\partial^+S_m\cup\partial^-S_m}=0$. No normalisation $\mathcal{N}_A$, no dictionary value, and no $\varepsilon$ appear in this note: it tests only whether the recursion produces a non-zero, bias-free oriented signal.