The paper develops the Life Origination Hydrate Hypothesis (LOH-hypothesis), according to which living-matter simplest elements (LMSEs, which are N-bases, riboses, nucleosides, nucleotides), DNA- and RNA-like molecules, amino-acids, and proto-cells repeatedly originated on the basis of thermodynamically controlled, natural, and inevitable processes governed by universal physical and chemical laws from CH4, niters, and phosphates under the Earth's surface or seabed within the crystal cavities of the honeycomb methane-hydrate structure at low temperatures; the chemical processes passed slowly through all successive chemical steps in the direction that is determined by a gradual decrease in the Gibbs free energy of reacting systems. The hypothesis formulation method is based on the thermodynamic directedness of natural movement and consists ofan attempt to mentally backtrack on the progression of nature and thus reveal principal milestones alongits route. The changes in Gibbs free energy are estimated for different steps of the living-matter origination process; special attention is paid to the processes of proto-cell formation. Just the occurrence of the gas-hydrate periodic honeycomb matrix filled with LMSEs almost completely in its final state accounts for size limitation in the DNA functional groups and the nonrandom location of N-bases in the DNA chains. The slowness of the low-temperature chemical transformations and their "thermodynamic front" guide the gross process of living matter origination and its successive steps. It is shown that the hypothesis is thermodynamically justified and testable and that many observed natural phenomena count in its favor.