Possibility-Space Geometry: A Minimal Synthetic Foundation for Classical Mathematics

Euclidean geometry is derived from a minimal modal foundation: one axiom (a point exists), two definitions (a specification, and a geometric object as the possibility-space of a specification), and one constraint (self-consistency). Two points carrying a configuration force, by inference alone, a third point at equal distance from both, so that the triangle is present in the possibility-space of any two-point specification prior to any construction. The triadic structure is 2+1 at its foundation, and the triangle is the unique convex polygon fixed by diagonalization. From this 2+1 structure, the relation between two points carries a triadic primitive structure (points, configuration, orientation), and Euclid's Elements follows in all thirteen books by Euclid's own arguments operating on the admissible geometric objects that specifications produce. The possibility-space at every point is recursively generated in three directions — scale, lateral, dimensional — exhaustive for the operations the foundation makes available. The singularity at zero inter-point distance is resolved by a forced projective blow-up, which attaches to the collapsed point the space of directions of approach. That space carries an involution fixed by the framework's own data, whose fixed points are ±1; the scale axis carries its own reciprocal involution, under which its two limits are conjugate, so that the axiom, the blow-up, and the scale-limit are one structural object — a point with attached infinite coherent possibility-space, identified throughout. No further axioms are introduced. Submitted April 2026 to Transactions of the American Mathematical Society