Defining Information: Order from the Prime Number Structure of the Riemann Zeta Function

Abstract Physics invokes “information” in contexts ranging from quantum measurement to black hole evaporation, yet no consensus exists on what information physically is. Shannon entropy quantifies surprise; von Neumann entropy quantifies mixedness; Landauer’s principle assigns an energy cost; Englert’s distinguishability measures which-path knowledge; Wheeler’s “it from bit” proposes a program. All provide operational measures. None identify a physical referent. This paper proposes a definition: information is order — the integer structure generated by the prime numbers through multiplication, as encoded in the Euler product of the Riemann zeta function. The quantitative measure is the Benford devi- ation δB, which tracks how faithfully a physical system’s leading-digit distribution reflects the integer-generated baseline given by Benford’s Law. Drawing on prior re- sults connecting ζ(s) to the Schwarzschild metric, Bose–Einstein condensate phase transitions, and causal set theory, we show that δB tracks quantum-to-classical transitions continuously, that Benford’s Law emerges as the least-action distribu- tion under scale invariance, and that the Second Law of Thermodynamics can be reframed as the natural return to Benford conformance. Shannon entropy, Lan- dauer’s principle, Englert’s duality relation, and Wheeler’s program all emerge as consequences of the definition rather than axioms. The proposal is falsifiable: if δB fails to track quantum-to-classical transitions in new experimental systems, the framework fails. We present quantitative evidence from Bose–Einstein condensate simulations, Kretschmann scalar profiles of black hole interiors, and a nine-model quantum gravity comparison.