Overview
This working 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 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 $\varepsilon = 1/10$ without fitting.
Core contributions
- BCH reduction. The per-step $J_\Pi$-odd $J_3$ component of the metaplectic cascade generator equals the oriented symplectic area swept per step (central Carnot-degree-2 increment of $\mathrm{Heis}_3$), not a free shear parameter.
- The observable. Accumulated oriented $J_3$-odd area along the Weil–BFS cascade, normalised by the projected capacity $\widehat{I}(n)$; at $n_3^{\mathrm{obs}}$ this equals $\varepsilon_{\mathrm{Weil}}(n_3^{\mathrm{obs}})$ and is to be compared to the dictionary $\varepsilon = 1/10$ without fitting.
- O12 fingerprint factorisation. 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.
- Symmetric BFS null result. 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. The orientation prescription is an irreducible input prior to $\mathcal{N}_A$.
Position in the programme
This note belongs to the fermionic matter sub-programme (Presentation Note 6). Together with the PRS note (operator-side Schur form, finite/Lorentzian separation) and the Q11OF note (frontier diagnostic), it sets up the locked frame in which the remaining quantitative step of the fermionic sector — the cascade normalisation $\mathcal{N}_{\mathrm{casc}}$ — can be carried out.
References
Jérôme Beau. Reduction of the Angular Amplitude Observable: the $J_3$ Split as Accumulated Oriented Symplectic Area. Working paper, 2026. 10.5281/zenodo.20601228