On the Missed Structures Beyond the Real Line

Abstract

Version 3: This version clarifies the distinction between sectorial collapse and the continuous auxiliary orientation index, while preserving the original preprint’s central thesis and role within the MGQC research program. It also refines the treatment of quasi-oriented states, quasi-zero, quasi-positive and quasi-negative tendencies, and the structural interpretation of absolute value. 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 θ, and collapse is described as the sectorial identification of a continuum of orientational states with two extreme directions associated with θ=0 and θ=π. 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 θ=π/2, and a continuous auxiliary orientation index is incorporated as a graded measure of signed tendency prior to collapse. Absolute value is reinterpreted as enforcing a collapse-like identification at the level of terminal scalar outcome rather than as the source of magnitude itself. 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.

Topics & Concepts