Angular Amplitude Reduction: the Oriented Shear Product of the sl(2) Step and the Central-Phase Observable of the Weil Cascade

This internal note fixes the numerical observable for the absolute normalisation of the angular generation split of Q14, 6, and reports the Weil–BFS computations of it (Section subsec:capacity-audit). A short Baker–Campbell–Hausdorff reduction in $\mathfrak{sl}_2$ identifies the per-step $J_\Pi$-odd $J_3$ component of the metaplectic cascade generator with the oriented product $ts$ of the two shear parameters, produced by the $\mathfrak{sl}_2$ commutator $[E,F]=H$ (Q14, Proposition 6.5), and not with a free shear parameter. In the $\mathrm{SL}(2)$-on-$V$ model of Q14 the computation uses only the $\mathfrak{sl}_2$ relations and no Heisenberg bracket; in the oscillator (Weil) realisation the $\mathfrak{sl}_2$ generators are quadratic in the Heisenberg generators, and no map between the two brackets is supplied here. The observable proposed for the Weil–BFS cascade is the accumulated oriented central-phase increment $A_c$ of the Heisenberg lift, normalised by the projected capacity ${\widehat{I}}(n)$. That $A_c$ carries the $J_3$ signal is an input of this note, not a consequence of the reduction: the increment $ts$ and the Heisenberg cocycle have the same form, which is an analogy and not an identification. Under this input, at the intrinsic saturation rank $n_3^{\mathrm{obs}}$ the normalised increment 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 $A_c$ is, under that input, the carrier of 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 $\mathcal{N}_A$.