Reduction of the Angular Amplitude Observable: the J_3 Split as Accumulated Oriented Symplectic Area

This internal note fixes the numerical observable for the absolute normalisation of the angular generation split of Q14, 6, before any Weil–BFS computation is written. A short Baker–Campbell–Hausdorff reduction identifies the per-step $J_\Pi$-odd $J_3$ component of the metaplectic cascade generator with the oriented symplectic area swept per step — the central, Carnot-degree-2 increment of ${\mathrm{Heis}}_3$ — and not with a free shear parameter. The quantity to be measured is therefore the accumulated oriented $J_3$-odd area along the Weil–BFS cascade, normalised by the projected capacity ${\widehat{I}}(n)$; at the intrinsic saturation rank $n_3^{\mathrm{obs}}$ this equals $\varepsilon_{\mathrm{Weil}}(n_3^{\mathrm{obs}})$, to be compared with the dictionary value $\varepsilon = \tfrac{1}{10}$ without fitting. In the O12 construction the two sectors are the two terms of the fingerprint exponent: the frequency $B_c$ carries the radial capacity $\sigma_c$, the central phase $A_c$ carries the angular split. A direct evaluation on symmetric BFS shells returns $\Theta_{\mathrm{raw}}=0$ identically: the symmetric capacity data fix the radial sector but cannot select an oriented angular branch, so the orientation prescription is an irreducible input prior to $\mathcal{N}_A$.