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

Overview

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. 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 $\varepsilon = 1/10$ without fitting.

Scope statement. This page provides a structured summary. The authoritative technical reference is the preprint linked above.

Core contributions

$\mathfrak{sl}_2$ reduction. The per-step $J_\Pi$-odd $J_3$ component of the metaplectic cascade generator is the oriented product $ts$ of the two shear parameters, produced by the $\mathfrak{sl}_2$ commutator $[E,F]=H$ (Q14, Proposition 6.5), not a free shear parameter; the computation in the $\mathrm{SL}(2)$-on-$V$ model uses no Heisenberg bracket.
The observable. Accumulated oriented central-phase increment $A_c$ of the Heisenberg lift, normalised by the projected capacity $\widehat{I}(n)$; at $n_3^{\mathrm{obs}}$ it equals $\varepsilon_{\mathrm{Weil}}(n_3^{\mathrm{obs}})$ and is to be compared to the dictionary $\varepsilon = 1/10$ without fitting. That $A_c$ carries the $J_3$ signal is an input of the note, its resemblance to the shear product being an analogy and not an identification.
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 $A_c$ is, under that input, the carrier of 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 $\mathcal{N}_A$.
Capacity audit: the onset coefficient. A subsequent audit on the exact Heisenberg–BFS frontier fixes the onset central-phase coefficient: the directed outgoing-frontier mean is an exact rational sequence ($1/3, 5/9, 2/3, \dots$ per shell, $q$-independent over the first six shells for $q \in \{61,101,151,211,307\}$ only), whose first-shell (onset) value is $1/3$, hence the maximal-locking bound at the onset shell $\theta_{\max}^{(1)} = 2\pi/(3q)$ (exact for the standard generating campaign, generator-dependent). The $\widehat I$-normalised bound at $n_3^{\mathrm{obs}} = 2$ is $2.114/q$, $2.122/q$, $2.118/q$ for $q = 61, 101, 151$, that is $0.9$–$1.3\,\%$ above the onset value. This fixes the onset coefficient, not the dictionary value: since $\theta_{\max}^{(1)} \to 0$ with $q$, the comparison with $\varepsilon = 1/10$ is a required-normalisation diagnostic (the geometric $\mathrm{Sym}^2$ normalisation $3q/(20\pi)$ is the algebra at the onset shell), leaving $\varepsilon$ gated by the same ADE datum; within the input that $A_c$ carries the $J_3$ signal, no value of $\mathcal{N}_A$ turns $\Theta_{\mathrm{raw}} = 0$ into a split. The per-shell rationals are recomputed from the AAR repository alone.

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 of the compression remainder, conditional localisation of $u$) 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. Angular Amplitude Reduction: the Oriented Shear Product of the $\mathfrak{sl}(2)$ Step and the Central-Phase Observable of the Weil Cascade. Working paper, 2026. 10.5281/zenodo.20601228