Treatise: From the Distinguished Relation to Complete Mathematics A genetic reconstruction of all mathematics from a single distinguished relation, without postulates other than distinction itself

The paper proposes a radically minimalist foundation for mathematics: all its structures — from numbers and geometry to analysis, topology and algebra — are generated from a single act of distinction. A pre-geometric layer is introduced, in which only the relation “being distinguished” is primitive. It is shown how the right isosceles triangle △₁ₓ₁ (Infinium) necessarily arises, and from its mosaic multiplication unfold the natural, integer, rational, real and complex numbers, Euclidean and non-Euclidean spaces, analysis with a natural spectral gap, topological invariants and non-abelian symmetries. It is demonstrated that key millennium problems (the Riemann Hypothesis, P vs NP, the Navier–Stokes equations and others) receive a constructive resolution within △-ontology. A neurophysiological grounding is given: the triangular lattice is optimal from the standpoint of the brain’s energy economy. The final formula ∀ Math ≅ Proj(Topos(△₁ₓ₁)) asserts that all of mathematics is the collection of projections of the topos generated by the Infinium. Philosophical implications (mathematics as a rigidly determined unfolding of distinction, the nature of consciousness) and mathematical conclusions (unity of foundations, built-in regularization, quantization of measure) are discussed.