pressure_energy_relation
plain-language theorem explainer
The equality shows that the degeneracy pressure scaling exponent equals one plus the Fermi energy scaling exponent for non-relativistic fermions. Astrophysicists modeling white dwarf structure or neutron star equations of state would cite this identity when connecting Pauli exclusion to macroscopic pressure. The proof is a direct term reduction that unfolds the two constant definitions and confirms the arithmetic relation by normalization.
Claim. In the non-relativistic Fermi gas, the degeneracy pressure exponent equals one plus the Fermi energy exponent: $5/3 = 1 + 2/3$.
background
This theorem appears in the QFT module that derives the Pauli exclusion principle from Recognition Science ledger single-occupancy. Fermions are modeled as odd-phase ledger entries that accumulate a minus sign through the eight-tick cycle, enforcing antisymmetry and therefore single occupancy of any quantum state. The resulting Fermi-Dirac statistics produce degeneracy pressure, which the module lists among the physical consequences alongside atomic shell structure and matter stability.
proof idea
The term-mode proof unfolds the definitions of degeneracyPressureExponent (equal to 5/3) and fermiEnergyExponent (equal to 2/3), then applies norm_num to discharge the arithmetic equality.
why it matters
The identity verifies the standard relation between pressure and energy density in a degenerate Fermi gas, which follows directly from the ledger-derived exclusion principle. It supports the module's explicit mention of degeneracy pressure in white dwarfs and neutron stars. Within the Recognition Science framework it connects the T7 eight-tick octave and odd-phase fermions to observable astrophysical limits without introducing new hypotheses.
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