The Algebraic Closure of Iterated Straightedge Constructions

Iterated straightedge constructions are traditionally analyzed as geometric procedures generated by recursive incidence axioms. We develop an algebraic semantic framework that reformulates straightedge geometry as functional computation over projective rational spaces: a typed semantics that treats constructions as incidence-preserving programs. By interpreting points and lines through homogeneous coordinates in $\\KK^3$, we show that both primitive operations---joining two points and intersecting two lines---are uniformly realized by the bilinear vector cross product under the duality isomorphisms $\\bigwedge^2(\\KK^3) \\cong (\\KK^3)^\\ast$ and $\\bigwedge^2((\\KK^3)^\\ast) \\cong \\KK^3$ induced by the standard volume form. We prove the \\emph{Projective Rational Realization Theorem}: every finite non-degenerate straightedge construction directed acyclic graph (DAG) induces a well-defined projective rational map $F_{\\D}: (\\PP^2(\\KK))^{m_1} \\dashrightarrow \\prod_{v \\in \\mathrm{Out}(\\D)} \\mathcal{Y}_v$ between products of projective spaces, given by multi-homogeneous polynomial coordinates, regular on its admissible projective open domain, and equipped with an associative composition law closed under sort-compatible output--input identification. We further characterize the Zariski-closed indeterminacy locus descending to the projective configuration space, resolve projective coordinate ambiguities through a canonical primitive representative map, and establish the sort-aware equivariance of construction maps under the action of $\\mathrm{PGL}(3,\\KK)$. Finally, via a two-sorted incidence closure system characterized as a Knaster--Tarski least fixed point, we prove the algebraic realization theorem for the geometric straightedge closure: for any rational initial configuration $S_0 \\subseteq \\PP^2(\\QQ)$, the geometric point closure and the algebraic point closure coincide identically, $\\Pcl(S_0) = \\A_P(S_0)$. This provides a formal basis for the arithmetic, asymptotic, and dynamical study of straightedge closures.