The Neutrino Energy Density Paradox: Testing Internal Consistency of Hot Big Bang Thermodynamic

The standard Hot Big Bang (HBB) model predicts thermal production of relic neutrinos with number density n_ν ≈ 336 cm⁻³ at decoupling temperature T_dec ≈ 1 MeV. We present a mathematical analysis examining the internal consistency of the axiom system {number density, mean energy, critical density} under the hypothesis that neutrino energy remains unchanged after decoupling. Using elementary arithmetic and De Morgan’s logic, we show that this axiom system leads to an energy density exceeding the observed critical density by a factor of approximately 2.5×10⁵, demonstrating internal inconsistency. The standard cosmological model resolves this through metric redshift E ∝ 1/a, which reduces neutrino energy by the required factor. However, this mechanism has not been directly verified for neutrinos. We propose a multi-method observational program to test redshift universality: (1) re-analysis of SN1987A neutrino data, (2) future core-collapse supernovae at z > 0.01, (3) Diffuse Supernova Neutrino Background spectral analysis, and (4) direct Cosmic Neutrino Background detection. We estimate that a “golden event” (supernova at z > 0.01 with >500 detected neutrinos) has approximately 30% probability of occurrence during 2027-2035, potentially enabling direct comparison of z_ν versus z_γ with significance exceeding 5σ.