The λ-transition of elemental sulfur occurring at about 159°C has long been associated with the conversion of cyclic S8 rings (c-S8) to amorphous polymer (a-S) via a ring opening polymerization. Here we demonstrate, with the use of both density functional and Hartree-Fock quantum mechanical calculations, the existence of an energetically accessible, locally stable, hypervalent state of S that can form branch sites in the polymer. The significance of this finding is that the λ-transition is best described as a gelation transition. The geometry of the tetravalent S atom is trigonal bipyramidal, with a lone pair occupying one of the three equatorial sites, it lies in a local energy minimum about 31 kcal/mol above the normal divalent state, and so is accessible both thermally and photochemically. Because the branched structure is formed endothermically, Le Chatelier's principle confirms that a percolation network can form on heating the element. The reactions that form branched structures are reversible, implying that the gel is fluxional. It decomposes at higher temperatures as chain scission competes with branching. The hypervalent structure provides an essential insight into the chemistry of elemental sulfur.