Awakening the Sleeping Giant: Rediscovering Archimedes' Density Method for Fingerprinting of Multicomponent Alloys

If the matter is the manifestation of the God-particle, then density is its soul, as one cannot exist without the other. Hence, density inherently becomes mother of all properties of the matter. The Archimedes' density method presents a mathematically unsolvable, computationally non-recursive, and undecidable NP-Hard problem for non-binary alloys. Resolving underdetermined linear equations, discretizing infinite Probable Iso-density Compositions (PICs), solving NP-Hard problems, and achieving certainty from probabilities are key to its application. Non-binary alloy densities reveal multiple PIC series interconnected through "True Composition" (TC) in the Isopycnic Region (IR) of Vast Alloys Space (VAS), replicating as Concordant Compositions (CCs) similar to centromere replication during cell division. The Principle of Vernier coincidence in multiple dimensions echoes CCs as unique alloy fingerprints in geometrical superimposition. We modified Archimedes' equations for additional metals and developed Density Decoding System (DDS) (www.densityfingerprinting.com) to compute PICs using Successively increasing Predefined Imaginary Numerical values (SPIN-values) and standard metal densities, generating a real-time database. A perfectly symmetric C-band breaks the asymmetry of Density Genome fractals, visualizing Density Fingerprints to conclusively determine "True Composition" up to octonary alloys with absolute accuracy in polynomial time. The precision range of alloy density limits the chaotic Butterfly effect, creating a fractal composition space that enables self-authentication of True Compositions. The study uncovers the coexistence of chaos and order, the butterfly effect, the emergence of order from infinity, and the fractal nature of the composition space. The research reveals the manifestation of quantum-like phenomena in the classical realm of alloy compositions, challenging our understanding of the nature of matter. Our work unveils density as the genetic code for non-living matter, leading to a novel alloy classification into binary and non-binary alloys. Furthermore, it introduces the Density Fingerprinting and Density Genome framework, which enables comprehensive materials analysis, characterization, design, and discovery, marking a paradigm shift in our understanding about density and its pivotal role in shaping the future of material science.