Papers · William Alexander Aldred

The ε_eff Turnover Is Exactly 3/16 ε_Pl — and It Is Inert

by William Alexander Aldred — Independent Researcher

Abstract

For a relativistic degenerate Fermi gas coupled to Einstein–Cartan torsion through the incoherent spin prescription, the effective energy density εₑff = ε(n) − ε_tors(n) does not rise monotonically to its zero crossing. It turns over first. This paper computes that turnover exactly. In the massless (ultra-relativistic) limit the balance density is n = (9/(8√π)) n_Pl, at which the bare energy density is ε = 81/64 ε_Pl. The maximum of εₑff occurs earlier, at n_turn = (2/3)³/² n* = 8.182971×10¹⁰³ m⁻³, and its height is εₑff,max = 3/16 ε_Pl exactly — not 0.1875 ε_Pl as a numerical approximation, but the rational (4/27)(81/64). An independent computation for the massive gas, solving the exact Fermi-gas integral for electrons, protons, and neutrons, returns the same position and the same height to the precision shown: the turnover is mass-independent for every carrier with m ≪ m_Pl, with finite-mass corrections proportional to (m/m_Pl)². The result is nevertheless physically inert for the published Aldred pin. The pin lives on the dust branch, whose crossing is at n_A = 5.801909×10⁸⁴ m⁻³; the Fermi turnover sits a factor 10¹⁹.¹⁴⁹ higher in number density, and the torsion scale 𝒯_A = 1.346746×10¹⁶ m⁻¹ is re-derived here from the dust crossing alone, with the Fermi turnover entering nowhere. Nor is the turnover a limiting-density state: on its own branch εₑff reaches zero only at the super-Planckian bare density 81/64 ε_Pl, where classical Einstein–Cartan theory has no jurisdiction. The turnover is therefore real, exact, and inert — an exact diagnostic that closes the last unworked computation of the spin homework and that supplies exact fractions for a qualifier the Aldred Limit paper already carries.

The program

• The Aldred Limit
• Two Bounds and an Integration Constant
• Supplementary Memos for “Two Bounds and an Integration Constant”
• The Constant Changes Address
• The Crossing Charges Nothing
• No Derivation in the Frame
• Static Defects Carry No Nieh–Yan Charge
• Creation Events Are Nieh–Yan Charged
• There Is No Twisted Vacuum
• Propagation Is Not Enough
• Axiom H
• One Flaw, Two Residues
• The ε_eff Turnover (this page)
• The Generation Map Has a Fixed Point
• Three Fences Any Λ Mechanism Must Pass