The sol-gel transition (SGT), upon which the infinite cluster (IC) of thermoreversibly bonded particles (gel fraction) appears against a background of a set of finite clusters (sol fraction), is quantitatively considered with due regard for large and complicated (mesoscopic) cycles inevitably present in the IC. To this end we present a new approach based on a concept of the monomer identity breaking and density functional description. We strictly derive, via a proper choice of basic structural units of the gel fraction, all statements usually supposed to be just Flory (Stockmayer) assumptions. A further analysis of the IC structure reveals some new IC structural units (those involved into mesoscopic cycles) overlooked in both Flory and Stockmayer approaches and to be described by a new order parameter characteristic only of the gel phase. As a result, the SGT is found to transform from a geometric phenomenon to a genuine first-order phase transition always followed by a phase separation into sol and gel phases. The free energy, total conversion, volume fractions of the gel fraction, and dangling monomers as well as other structural quantities are calculated as functions of a reduced monomer density and analyzed for all the existing models. The Flory approach is found to be superior to the Stockmayer-Tanaka one and satisfactorily describe some of the dense weak gel properties but fail (even qualitatively) in a quite extended vicinity of the SGT.