On the Missed Structures Beyond the Real Line

Version 4: This version consolidates the article within the MGQC foundational framework by refining the non-destructive interpretation of projection and clarifying the conventional use of the term “collapse.” It removes residual formulations that could suggest the real line acts as an active restricting agent, strengthens the distinction between terminal sign classes and the continuous auxiliary orientation index, and improves the treatment of quasi-zero, quasi-positive, and quasi-negative states. It also clarifies that the quasi-orientational domain is not to be identified with the ordinary Cartesian plane ℝ², while preserving the article’s original pre-axiomatic, structural, and metamathematical scope Abstract The real line provides a complete and internally consistent representation of magnitude under linear order; however, this completeness corresponds to a strong structural restriction. Under the interpretive framework adopted here, the real line may be read as corresponding to only two terminal orientational outcomes, thereby identifying a richer directional continuum with a rigid linear model. The present article makes explicit some of the structures that are not distinguished within this restrictive regime. Orientation is introduced through an angular parameter θ, and collapse is described, in a conventional representational sense, as the identification of a continuum of orientational states with two effective directions associated with θ = 0 and θ = π. A distinction is drawn between fully oriented states, which correspond to classical real outcomes, and quasi-oriented states, which precede numerical resolution. Within this framework, the quasi-zero is interpreted as an intrinsically infinite family of pre-sign states around the classical transition point; analogous orientational neighborhoods are proposed around every real magnitude. Quasi-positive and quasi-negative are introduced as orientational tendencies relative to the threshold θ = π/2. The article also distinguishes between the fundamental sectorial projection of linear representation and a continuous auxiliary orientation index, useful for expressing graded signed tendency prior to projection without thereby defining the projection rule itself. Absolute value is reinterpreted as enforcing a projection-like identification linked to the terminal sign classes. The framework is deterministic and pre-axiomatic and is intended as a conceptual bridge toward subsequent formal developments. This preprint forms part of the Model of Generalized Quasi-Coherence (MGQC) research program.The author publishes under the name Antonio Dominguez-Digat. Earlier records may appear under Antonio Domínguez, Antonio Dominguez, or Antonio Dominguez Digat.