On the Missed Structures Beyond the Real Line
Dominguez-Digat, Antonio
Abstract
Version 2: This version corrects errors, improves exposition, and updates terminology and figures to better reflect the article’s scope, while preserving the original preprint’s central thesis and role within the MGQC research program. Abstract The real line provides a complete and internally consistent representation of magnitude under linear order; however, this completeness may be achieved through a strong structural restriction. Under the interpretive framework adopted here, the real line may be read as preserving only two extreme orientational outcomes, thereby identifying a richer directional continuum with a rigid linear model. The present article makes explicit some of the structures suppressed by this identification. Orientation is introduced through an angular parameter , and collapse is described as the identification of a continuum of orientational states with two extreme directions associated with and . A distinction is drawn between fully oriented states, which correspond to classical real numbers, 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 . Absolute value is reinterpreted as enforcing a collapse-like identification linked to this threshold. 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 General 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.