Derivation of the Barbero-Immirzi Parameter, Black Hole Entropy, and the Area Gap from the R12 Rendering Algebra

Abstract

The Barbero-Immirzi parameter γBI, the sole free parameter of Loop Quantum Gravity (LQG), has resisted derivation from first principles for three decades. We show that γBI = ln 2/(π√3) is a direct consequence of the R12 rendering algebra of
Array Cosmology (AC), where √3 = 2 cos(2π/12) is the nearest-neighbour coupling magnitude of the 12-axis loop and ln 2 is the Landauer erasure cost per bit. The minimum area quantum ∆Amin = 4 ln 2 l2Pemerges as four Volma strokes times one Landauer bit, and the Bekenstein-Hawking entropy S = A/(4l2P) is recovered exactly via a parity-pair counting argument.

The logarithmic correction coefficient

α = −3/2 is derived from three zero modes per Volma stroke. The framework
uses three axioms (12, π, ln 2) with zero free parameters and yields a falsifiable
prediction: ∆Amin = 4 ln 2 ≈ 2.773 l2P, distinguishable from the competing Hod
conjecture ∆Amin = 4 ln 3 ≈ 4.394 l2P by a factor of log2 3 ≈ 1.585.