The missing triangle: Maxwell, Shannon and Boltzmann in adaptive coherence

Theoretical framework and exploratory cross-scale operationalization proposing the EICI (Energetic–Informational Coherence Index) as a candidate formal criterion connecting three foundational scientific frameworks: Maxwell's field equations (E — structured energy), Shannon's information theory (I — signal fidelity), and Boltzmann's statistical thermodynamics (F — entropic cost of order). Prigogine's dissipative structures and Mandelbrot's fractal geometry are identified as frameworks that approached but did not fully specify this criterion. EICI = Eₙ × Iₙ / Fₙ is a normalized relational index — not a dimensionally homogeneous physical equation — defining three system states: Δ+1 (coherence), Δ=0 (reorganization threshold), Δ−1 (collapse). Preliminary cross-scale operationalization across five domains (cellular, cognitive, organizational, national, civilizational). Formal mathematical supplementary (S1) included: normalization procedure, bifurcation analysis, Shannon channel capacity connection, Boltzmann entropy connection, Prigogine derivation agenda, Sobol sensitivity formalization. Conceptual triangle figure included. Part of K-Model cluster: 10.17605/OSF.IO/WST24.

This working paper proposes the EICI framework as the formal criterion that connects Maxwell (E), Shannon (I), and Boltzmann (F) as joint predictors of adaptive coherence — a connection that Prigogine's dissipative structures and Mandelbrot's fractal geometry each approached but did not explicitly formalize. EICI = Eₙ × Iₙ / Fₙ is presented as a candidate hypothesis, not a proven physical law. The paper explicitly separates three epistemological levels: formal (structural hypothesis), empirical (exploratory cross-scale operationalization), and philosophical (interpretive framing). Planetary biosignature application reserved for future dedicated paper. Supplementary S1 provides formal mathematical development including bifurcation ODE, critical slowing down (τ → ∞), Shannon SNR mapping, Boltzmann entropy export condition, and Sobol sensitivity definitions. Author: C.J. Pérez Pulido, ISHEA Institute, Bologna. Deposited May 2026 as child component of K-Model parent project WST24 (August 2025).