In-medium hyperon potentials and the quarkyonic hyperon onset: charged $Σ$'s in $β$-equilibrium and the neutrino connection

Quarkyonic matter resolves the neutron-star hyperon puzzle statistically: neutrons fill low-momentum $d$-quark phase space, shifting the $S=-1$ threshold from $\muB=M_Y$ to $2M_Y-M_N$ and suppressing residual softening by $1/\Nc^3$ in the Fujimoto--Kojo--McLerran (FKM) mechanism. We dress FKM's IdylliQ model with in-medium potentials, constrained by hypernuclear data and neutrino-induced hyperon FSI, and find: (i) the dressed onset, $\muB^{\rm onset}=(2M_Y{-}M_N)+2U_Y-\UN$, carries $U_Y$ at weight 2, with $dn_{\rm onset}/dU_Y\simeq0.3\,\nz$ per $10\MeV$, twice the leverage. (ii) A self-consistent neutron potential enters at weight $-2$, so protection needs $\UN(n_{\rm onset})\lesssim+96\MeV$. (iii) With leptons in $\beta$ equilibrium, the $\Sigma^-$ ($dds$) onset becomes $\mue\ge258\MeV+\US-\UN$, never reached inside a $2\,\Msun$ core: $\Sigma^-$ switches from first hyperon to forbidden and the $\Sigma$ ordering inverts. (iv) The $\kY\ge\kbu$ continuation gives TOV softening below $0.05\,\Msun$ in the FKM ansatz family and below $0.025\,\Msun$ for the realistic interacting star, $4$--$8$ below hadronic models. In the family this is a ceiling; generally it is a floor, since hyperons above $\kbu$ are omitted. (v) In an interacting low-density sector calibrated to $\Mmax=2.12\,\Msun$, core strangeness is controlled by $(\UL(\nz),c_\Lambda)$: at $\UL(\nz)=-28\MeV$, the maximum-mass star is hyperon-free once the supra-saturation $YNN$ turn-over exceeds a few-MeV threshold $c^*_\Lambda$. Projected SBND+DUNE FSI precision pins $\UL(\nz)$ but leaves $c_\Lambda$ -- to which neutrino data are blind -- decisive: with the heavy-ion prior, $P({\rm hyperon\mbox{-}free\ core})=0.90$, versus $0.89$ from priors alone, prior-dominated rather than measured. The sharpest observable is differential: $d\Mmax/dU_Y$ is an order of magnitude smaller than in mean-field models, discriminating the two resolutions.