In-medium hyperon potentials and the quarkyonic hyperon onset: charged $\mathrm{\Sigma }$'s in $\beta$-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 $\text{muB}=M_Y$ to $2M_Y-M_N$ and suppressing residual softening by $1\mathrm{/}{\text{Nc}}^3$ 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, ${\text{muB}}^{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}=(2M_Y-M_N)+2U_Y-\text{UN}$, carries $U_Y$ at weight 2, with $d{n}_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}}\mathrm{/}d{U}_Y\simeq 0.3\text{nz}$ per $10\text{MeV}$, twice the leverage. (ii) A self-consistent neutron potential enters at weight $-2$, so protection needs $\text{UN}(n_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{t}})\lesssim +96\text{MeV}$. (iii) With leptons in $\beta$ equilibrium, the ${\mathrm{\Sigma }}^{-}$ ($dds$) onset becomes $\text{mue}\ge 258\text{MeV}+\text{US}-\text{UN}$, never reached inside a $2\text{Msun}$ core: ${\mathrm{\Sigma }}^{-}$ switches from first hyperon to forbidden and the $\mathrm{\Sigma }$ ordering inverts. (iv) The $\text{kY}\ge \text{kbu}$ continuation gives TOV softening below $0.05\text{Msun}$ in the FKM ansatz family and below $0.025\text{Msun}$ 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 $\text{kbu}$ are omitted. (v) In an interacting low-density sector calibrated to $\text{Mmax}=2.12\text{Msun}$, core strangeness is controlled by $(\text{UL}(\text{nz}),c_{\mathrm{\Lambda }})$: at $\text{UL}(\text{nz})=-28\text{MeV}$, the maximum-mass star is hyperon-free once the supra-saturation $YNN$ turn-over exceeds a few-MeV threshold $c_{\mathrm{\Lambda }}^{\ast }$. Projected SBND+DUNE FSI precision pins $\text{UL}(\text{nz})$ but leaves $c_{\mathrm{\Lambda }}$ – to which neutrino data are blind – decisive: with the heavy-ion prior, $P(\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{n}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e})=0.90$, versus $0.89$ from priors alone, prior-dominated rather than measured. The sharpest observable is differential: $d\text{Mmax}\mathrm{/}d{U}_Y$ is an order of magnitude smaller than in mean-field models, discriminating the two resolutions.