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 $\mathcal{N}_A$.