Relational Foundation Equations

The Discrete Closure Sector of the Relative–Relational
Framework

Short technical note

Daniel O’Keeffe  ·  1 April 2026

We propose that a compact discrete closure architecture is generated by the unique positive solution of the lock equation n² - 1 = 2ⁿ, namely n = 3. From this lock follows the partition N = n·2ⁿ·nⁿ = 648, together with a small set of derived scalars that generate near-accurate values for the inverse fine-structure constant and the proton-to-electron mass ratio.

The resulting master set is given below. At n = 3, it yields α_C = 0.1487982101, Λ = 45.8797993966, α⁻¹ = 137.0442053493, g = 66.2109169434, and m_p/m_e = 1836.356997.

Immediate reduction

Since 2ⁿ = n² - 1 and n = 3, the coupling simplifies to α_C = ln(8/3) / (6 ln 3) = 0.1487982101. Substituting this into the hierarchy closure gives Λ = 45.8797993966; this then yields α⁻¹ = 137.0442053493 and, with g = 66.2109169434, the mass closure m_p/m_e = 1836.356997.