We show that phase-resolved ultra-high-energy (UHE) emission from LS $\mathrm{I}+61°303$ defines a critical energetic boundary linking the compact-object engine, circumstellar interaction depth, and dense-matter structure. Combining the GeV-TeV-UHE spectral transition with an orbital acceleration-transparency sieve and branch-resolved stellar sequences for 116 certified EoS, we find that the measured 25–100 TeV component remains energetically viable for all 116 canonical-mass EoS and therefore does not discriminate among them. EoS discrimination arises only in the deliberately more demanding pure-hadronic $>1\mathrm{TeV}$ stress-test construction: at 2.0 kpc and $1.4M_{\odot }$, this boundary gives $I_{45,\mathrm{crit}}=1.5504$, with 82 of 116 EoS satisfying the energetic condition at ${\dot{P}}_{\mathrm{r}\mathrm{e}\mathrm{f}}=3\times {10}^{-15}\mathrm{s}{\mathrm{s}}^{-1}$. A separately applied tidal diagnostic, ${\mathrm{\Lambda }}_{1.4}\le 580$, is satisfied by 43 EoS, with nine satisfying both conditions. These models span several microscopic classes but occupy a narrow structural region, indicating structural overlap rather than selection of a unique EoS family. A separate GeV power-budget test requires ${\dot{P}}_{\mathrm{rot}}\gtrsim 5.65\times {10}^{-14}{f}_{\mathrm{\Omega }}\mathrm{s}{\mathrm{s}}^{-1}$ at 2.0 kpc. For $f_{\mathrm{\Omega }}\sim 1$, the source lies well above the UHE crossing and the EoS discrimination disappears. The UHE observation therefore identifies the engine-environment regime in which differences in neutron-star structure can become observationally discriminating, rather than defining a universal dense-matter selector.

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• Equations of state of nuclear matter
• Gamma ray astronomy
• Nuclear astrophysics
• Nuclear matter in neutron stars
• Binary stars
• Neutron stars & pulsars
• Monte Carlo methods