Chiral Orientation of the Directed Heisenberg Frontier
Year: 2026
Type: article
Abstract: The angular generation split of the projected Dirac residue ${E_\Pi}$ is carried, on the directed Heisenberg frontier, by a chirally dressed cocycle signal $\Theta_\chi = \langle \sigma_L\,(2\pi/q)\,{\Delta A_c}\rangle_{\partial^+}$. The raw cocycle average and the maximal-locking bound ${\theta_{\max}}$ are computable from the arithmetic backend, but the signal itself depends on the chiral weight $\sigma_L$, whose origin lies in the Lorentzian spin-lift and not in the arithmetic frontier. We prove that $\sigma_L$ cannot be any function of the real cascade generator $T(e)$: edge inversion and the residual reflection $\varphi$ both act as $T\mapsto -T$, so every $f(T)$ has identical arrow-parity and $\varphi$-parity, whereas $\sigma_L$ must be arrow-even and $\varphi$-odd. The resolution is that $\sigma_L$ is the Weyl representation type, ${\mathbf{2}}$ versus ${\bar{\mathbf{2}}}$: group inversion preserves the representation (arrow-even) while complex conjugation exchanges it ($\varphi$-odd). In the inherited Heisenberg central-phase lift the chiral phase is the linear cyclotomic character $\omega_\chi(e)={\zeta_q}^{{\Delta A_c}(e)}$, with no Gauss, Maslov, or $\sqrt{5}$ component; on the nonzero-cocycle support the natural branch satisfies $\sigma_L(e)={\operatorname{sign}} \Im\,{\zeta_q}^{{\Delta A_c}(e)} = {\operatorname{sign}}{\Delta A_c}(e)$, giving ${\rho_\chi} = 1$ and ${N_A} = {\theta_{\max}} \neq 0$. This establishes ${[\mathrm{H\text{-}orient}]}$ in the inherited Heisenberg central-phase lift, not unconditionally: the only remaining obstruction is an additional spin-Galois phase outside the central character, in particular a $\sqrt{5}$ or ADE-spin factor orthogonal to ${\zeta_q}$. The result is a conditional derivation of the frontier amplitude datum; it does not derive $u\neq 0$, which still requires the Schur transport of the polarisation onto the ${J_\Pi}$-odd generation sector.
Author: Beau, Jérôme
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165039080
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