Angular Amplitude Reduction — The $J_3$ Split as Accumulated Oriented Symplectic Area

J. Beau, Independent Researcher, France

Status

Working note (preprint), v1.0. DOI: 10.5281/zenodo.20601228

Abstract

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 $\mathcal{N}_A$.

Position in the programme

This note belongs to the fermionic matter sub-programme (Presentation Note 6). It is a companion note to Q14, refining the §6 open deliverable on the inter-generation splitting amplitude. It identifies, before computation, the precise Weil–BFS observable to be measured to test the dictionary value $\varepsilon = 1/10$. Together with the projective-residue-schur note (Schur form of $E_\Pi$, A4 stratification, finite/Lorentzian separation) and the q11-oriented-frontier note (front diagnostics), it sets up the locked frame in which the remaining quantitative step of the fermionic sector can be carried out.

Compilation

bash compile.sh

Runs pdflatex → bibtex → pdflatex → pdflatex on tex/AngularAmplitudeReduction.tex and produces out/AngularAmplitudeReduction.pdf.