An Analytic Threshold for LESA-Driven Negative ELN Flux Directions in Core-Collapse Supernovae: Derivation and Population Census

In core-collapse supernovae (CCSNe), deleptonization normally favors $\nu_e$ over $\bar{\nu}_e$ emission. However, lepton-number emission self-sustained asymmetry (LESA) can make the energy-integrated emitted lepton-number flux negative along some directions. We derive a simple diagnostic for this transition and test it in 33 independent 3D CCSN simulations: 25 Princeton/Fornax models ($8.1$--$100\,M_\odot$) and 8 Garching models, including non-, slow-, and fast-rotating $15\,M_\odot$ cases. Of 23 non-black-hole-forming Princeton models, 22 cross the threshold, with median onset $t_c=225\,\mathrm{ms}$, IQR $162$--$264\,\mathrm{ms}$, and cross-model scatter $\mathrm{CV}=18.6\%$. Full-sky flux-sign searches show that the threshold identifies the anti-LESA-pole transition, distinguishing the global LESA-driven crossing from early localized turbulent crossings. The fast-rotating Garching $15\,M_\odot$ model, where rapid rotation suppresses the LESA dipole, is correctly classified as a non-crosser without using any rotation parameter. Both black-hole-forming Princeton models cross near $250\,\mathrm{ms}$ post-bounce and remain above threshold for $1807$ and $2463\,\mathrm{ms}$ before collapse. Thus, in the next nearby CCSN, the emitted $\bar{\nu}_e$ energy flux may exceed the $\nu_e$ flux along some lines of sight. Such directions may also correlate with sustained fast flavor instability, although testing this requires local phase-space distributions or dedicated linear stability analysis. The relevant quantity here is the energy-integrated emitted flux field, i.e. a luminosity difference per steradian, not a neutrino number flux.