The Theory of Something: A Theorem Supporting the Conditions for Existence of a Physical Universe, from the Empty Set to the Biological Self

The theorem stating that "the conditions for existence of a physical universe including conscious perception are provided by the empty set", is based on a sequence of proved lemmas. The space composed of the empty set (as constitutive member), and the set theory and general topology (as logical reasoning system), provides existence to connected topological n-spaces. Then, an observable universe can exist as a sequence of intersections (shown to be Poincaré sections) of such topological spaces owning nonequal dimensions. The Jordan-Veblen theorem states that pathes connecting the respective interiors of two closed spaces own nonempty intersections with their frontiers. Mappings of -members from one into another section provide order relations in the sequences of Sections, thus generating a physical arrow of time. Any two sections are connected by a momentum-type structure. Provided the Jordan's points of one closed space are the preimage-of a sequence of mappings, since the interior of such spaces is compact and connected, sequences converge to fixed points which account for mental images. The set of-fixedpoints. of such a closed also contains: (i) a Brouwer's type fixed points accounting for self-identitication of the closed ; (ii) a fractal component. The whole provides the corresponding closed with characteristics of a conscious self, with fractalaided retrieval of stored information, that is non-localized memory. This completes the proof. Consequently, also the Planetary-ecosystem, whose components arc living and nonliving structures involved in a set of functions, owns the structure of a mathematical space demonstrably provided with topologies and shown to be compact, complete, and connected, except if discontinuities are artificially provoked. The Weierstrass theorem states that it can reach its supremum, with maximization of richness and complementarity in the distribution of species and habitat, while the Bolzano theorem requires continuity to be fulfilled for this evolution. Finally, the Heine-Borel-Lebesgue property of compact spaces needs that mutualism is fullilled as a necessary ecological rule. Finally: (i).Life is -supported by converging sequences of closed topologies fulfilling- mathematical conditions. The corresponding mappings own at least a surjective step on the way to fixed points of neuronal chainings, but not in the reciprocal direction: therefore, these sequences are not symmetric since the fixed points would not be the same in both directions. This provides the biological arrow of time with the property of irreversibility, with respect to the conscious self ; (ii) The mathematical conditions for functionality of biological beings and of their ecosystems may appear as a purely conceptual "driving pulse" which has sometimes been interpreted in terms of a project or a "vitalistic" force, but which is more clearly accounting for an anticipatory process.