Phase-Resolved Ultra-High-Energy Emission Defines an Energetic Boundary for Compact-Object Engines and Dense-Matter Structure

We show that phase-resolved ultra-high-energy emission from LS 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 tail primarily constrains the circumstellar environment, whereas a more demanding pure-hadronic > 1 TeV interpretation reaches the physical moment-of-inertia distribution. At 2.0 kpc and 1.4 M sun, the latter gives I45,crit = 1.5504; 82 of 116 EoS satisfy the corresponding energetic boundary at Pdot ref = 3 x 10-15 s s-1. A separately applied tidal diagnostic, Lambda 1.4 <= 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 Pdot rot >= 5.65 x 10-14 f Omega s s-1 at 2.0 kpc. For f Omega 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.