The Oriented Frontier Observable

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 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.

Scope statement. This note tests only whether the recursion produces a non-zero, bias-free oriented signal. No normalisation $\mathcal{N}_A$, no dictionary value, no $\varepsilon$ enters.

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 counterpart of the BCH term $\tfrac{1}{2}[X, Y]$ and hence of the dynamical $J_3$.
Recursive orientation. The orientation is the recursive arrow $\partial_\tau = \log g$ from the projection origin to increasing BFS depth, not an extracted weight.
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 the chiral weight $\gamma_5$ 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.

Position in the programme

This note belongs to the fermionic matter sub-programme (Presentation Note 6). It supplies the falsification protocol for the recursive existence of the $J_3$-odd cascade signal underlying the Q14 §6 inter-generation splitting mechanism. Together with the AAR note (BCH reduction; observable to be measured) and the PRS note (operator-side Schur form), 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