Optimal Information Efficiency in the Primorial Sieve and its Convergence with a Physical Constant Ropt = 1/(6 log2 3) = ln 2/(6 ln 3) ≈ 0.105155

Abstract of the article We define a Return on Investment (ROI) functional for the primorial sieve of Eratosthenes, quantifying the marginal selectivity gain per additional bit of descriptive complexity. We prove that the transition from parity (modulus~2) to hexality (modulus~6) achieves the unique maximum ROI among all non‑trivial steps, with exact value Ropt=1/(6log⁡23)=ln⁡2/(6ln⁡3)≈0.105155. This constant is transcendental and emerges without any adjustable parameters. Surprisingly, the same number has been independently proposed as the vacuum informational impedance in the Modular Substrate Theory, where it arises from the Z/6Z global structure of the Standard Model gauge group and the holographic entropy bound. We examine the structural reasons for this convergence and discuss its implications for the discrete–continuum transition and the architecture of the quantum vacuum. All results are derived analytically; a finite verification up to the 100th prime is provided as supplementary material. LicenseAll materials in this repository are distributed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). This repository contains the supplementary materials accompanying the article "Optimal Information Efficiency in the Primorial Sieve and its Convergence with a Physical Constant"José Ignacio Peinador SalaEntropy (MDPI), 2026 (Manuscript ID: entropy-4443267).