Q11 Oriented Frontier — A Recursive Diagnostic for the Angular Generation Split

J. Beau, Independent Researcher, France

Status

Working note (preprint), v1.0. DOI: 10.5281/zenodo.20601245

Abstract

The symmetric-capacity obstruction establishes that the raw shell observable of the angular generation split vanishes identically, $\Theta_{\mathrm{raw}} = 0$: on a symmetric BFS shell the involution $g \mapsto g^{-1}$ pairs opposite central phases and erases the oriented $J_\Pi$-odd component before the radial recursion can act.

This note defines the honest replacement: a frontier transfer observable attached to the directed outgoing edges $g \to gs$ of the cascade, whose central increment is the Heisenberg cocycle $a(g), s_b$, the discrete counterpart of the Baker–Campbell–Hausdorff term $\tfrac{1}{2}[X, Y]$ and hence of the dynamical $J_3$.

The orientation is the recursive arrow $\partial_\tau = \log g$ from the projection origin to increasing BFS depth, not an extracted weight. The first test is not an amplitude value but the existence of an oriented signal, $\langle \Delta A_c \rangle_{\partial^+ S_m} \neq 0$, which may fail if a residual automorphism re-pairs the frontier; the discriminating anti-bias control is the symmetrised-frontier cancellation $\langle \Delta A_c \rangle_{\partial^+ S_m \cup \partial^- S_m} = 0$.

No normalisation $\mathcal{N}_A$, no dictionary value, and no $\varepsilon$ appear in this note: it tests only whether the recursion produces a non-zero, bias-free oriented signal.

Position in the programme

This note belongs to the fermionic matter sub-programme (Presentation Note 6). It is a companion diagnostic note to Q14, supplying the falsification protocol for the recursive existence of the $J_3$-odd cascade signal that underlies the Q14 §6 inter-generation splitting mechanism. Together with the angular-amplitude-reduction note (BCH reduction; observable to be measured) and the projective-residue-schur note (operator-side Schur form, finite/Lorentzian separation), it completes the diagnostic 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/Q11OrientedFrontier.tex and produces out/Q11OrientedFrontier.pdf.