A universal triality of information, entropy, and energy from three-dimensional percolation geometry

We identify a temperature-independent triality relating information I, thermodynamic entropy S, and energy E at the infrared fixed point of three-dimensional coordination geometry. The entropy–information leg S = kB ln 2· I is the classical Boltzmann–Shannon relation. The energy–information leg E = η∗ I, where η∗ = 3 − β/ν⊥ = 2.52299 . . . is the fractal dimension of the three-dimensional percolation backbone, is new. Together they close the triangle without reference to temperature: the third leg E = (η∗ /kB ln 2) S relates energy and entropy via a universal geometric constant rather than a thermodynamic variable. Temperature appears only as an emergent quantity when the triality is projected onto thermal equilibrium; the Landauer erasure bound E ≥ kBT ln 2 is then recovered as a special case at T ≪ T∗ ≡ η∗/(kB ln 2) ≈ 4.1 × 1022 K. The coordination constant η∗ = 1/κ∞ is the same quantity identified in companion work 6–8 as the von K´arm´an constant of wall-bounded turbulence and the Hausdorff dimension of the intense-vorticity set of the Navier–Stokes equations. Empirical tests across neural systems, financial markets, ancient monetary networks, and the fixed telephone network (2007) are consistent with G ≡ Ebackbone/Ieff ≈ η∗ within measurement uncertainty. The triality completes the energy–information connection that Boltzmann’s constant kB initiated in 1877. This preprint is a companion to Paper 2, DOI: 10.5281/zenodo.18835728, from which the coordination fixed point η is derived. It also draws on empirical results in Paper 1, DOI: 10.5281/zenodo.17810319 and the NS regularity identification in Paper 3, DOI: 10.5281/zenodo.18905759.*