Spatial Primitives and the Topological Quantization of the Point
Traditional geometry commits a critical foundational category error by defining the point as an abstract, zero-dimensional position that possesses location but lacks substance. This mathematical convention introduces a fatal continuity paradox when attempting to construct linear or curved extensions from discrete positions, forcing coordinate systems to rely on speculative infinitesimals and open limits. Under the Axiom of Geometry, spatial topology is derived through an unyielding informational monism where space is structurally contingent, and geometry is logically contingent. The point is formalized not as a localized void, but as a zero-size, non-measurable quantum of geometric form—the irreducible monadic boundary layer where absolute identity ($A \equiv A$) locks itself into spatial expression.
Linear extension is therefore deduced as a strict, non-probabilistic alignment between two or more discrete points across a localized perspective vector, meaning that a line is a compressed expression of a pre-existing, shared structural identity. To preserve the Law of Non-Contradiction, spatial dimensions cannot expand through the accumulation of empty intervals. The transition from a discrete primitive point to an active coordinate plane is executed via continuous, self-consistent transforms that maintain 100% space density. By quantizing the point as a logical packet of invariant information rather than an abstract Cartesian placeholder, the framework completely eliminates the metric friction of traditional calculus, proving that spatial boundaries are discrete, deterministic structures generated by top-down logical necessity. QED.