Relative Transcendence and the Diagonal Generation of the Continuum

We produce a cumulative, ordinally indexed construction of the real continuum from a finite initial primitive, without presupposing its completion. Binary supports are organized by asymptotic growth classes with algebraic closure over the values of affine modification of these supports and prior construction. These form a diagonal whose real-valued output is utilized as the system of growth coordinates to refine the binary supports for the next diagonal. A uniform escape theorem shows that while the cumulative field remains countable there exists values with support-gaps in the Liouville regime that are relatively transcendental over all prior construction. Thus, no countable stage can be terminal. The construction produces exact values that occur in every finite binary cell. Successive algebraic closures yield nested transcendence density, where algebraically new values persist at arbitrarily fine binary resolution. The exact outputs form coherent Moore-Smith refinement nets whose birth stages are cofinal, while their containing binary cells have diameter tending to zero. The real valued limit is identified with the continuum and is obtained without adjoining Cauchy limits or a applying a completion operator. Version History: The Current manuscript (Version 1.4, October 11, 2026), supersedes earlier drafts circulated under the title The Constructive Emergence of the Continuum.