Two superconducting pairing models with a Kitaev spin-liquid glue: microscopic kernel and parity alternation on odd-fold bond stars

We construct two superconducting pairing models on doped Kitaev honeycomb, whose microscopic kernels are derived rather than assumed. Both are driven by a bond-local glue neutral between the even- and odd-parity pairing channels; the chirality–parity linkage is set by bond-star geometry, not by the microscopic glue. (i) Model~1(odd-parity sector): the kernel follows from the flux-sector selection rule of the pure model, fixing its component-diagonal, nearest-neighbor separable, rank-one-per-bond form, with $w_{\alpha }=J_K^2{\chi }_{\alpha }\propto r_{\alpha }{m}_{\alpha }$, where $m_{\alpha }$ follows from the Hellmann–Feynman theorem and $r_{\alpha }\simeq 0.24$ is computed numerically. The spin-vertex state is a unitary triplet with a bond-locked anisotropic $\mathbf{d}$ vector; its gap is a Gram sum of bond sines, so nodal directions are fixed by geometry. (ii) Model~2(even): the kernel is fixed by the $C_3$ little-group selection rule of a zone-corner chiral phonon, up to a Cooper-channel extension whose momentum constraint we state explicitly. The charge-vertex pairing is a $d+id$ singlet: the parity decomposition $f_{\pm }=g_{\pm }+p_{\pm }$ projects onto $g_{\pm }$, so chirality is slaved to the phonon polarization, with $C=\pm 4$ and ${\kappa }_{xy}\mathrm{/}T=2{\kappa }_0$. Reversing the pump helicity reverses the Chern number, Kerr rotation, and thermal Hall sign. (iii) General theorem: on odd-fold bond stars, the angular-momentum ladder of a single-star chiral form factor alternates in parity. Every chiral state selected by such a bond-local form factor is a geometrically enforced singlet–triplet composite with universal amplitude ratios ($n=3$: $d\oplus p$; $n=5$: $g\oplus p$ or $d\oplus f$); coherent doubled-star combinations can restore parity purity. The Chern number of the mixed state follows the dominant on-shell component across a topological boundary. The spin glue drives $C=\pm 2$; the charge kernel $C=\pm 4$.