Coherent Euclidean Structures with Internal Orientational Parameters

Abstract

Version 2: This version improves the formal presentation of the article, refines several proofs and definitions, and strengthens the structural interpretation of coherent Euclidean structures as a bridge between effective geometry and suppressed internal structure. It also clarifies the metric compatibility of the orientational operator, the role of structural projection, and the status of the regular regime as canonical rather than exhaustive, while preserving the article’s central thesis and its role within the MGQC research program. Abstract This article introduces a formal class of coherent Euclidean structures designed to distinguish geometrically effective directions from internal orientational degrees of freedom. In standard Euclidean modeling, all coordinates are typically treated uniformly as dimension-generating variables. By contrast, the framework proposed here separates a geometrically effective subspace from an internal subspace and studies the relation between them through a structural projection. A coherent Euclidean structure is defined as a finite-dimensional real inner-product space equipped with a distinguished effective subspace, a distinguished internal subspace, a projection onto the effective component, and an internal orientational operator. Within the regular axis-wise regime, each effective axis is paired with one internal orientational direction, giving rise to coherent models of type D1, D2, and D3. In these models, the total number of real parameters may exceed the geometric dimension without increasing the number of geometrically effective axes. The article proves that every regular coherent Euclidean structure of rank n is coherently isomorphic to the canonical model of the same rank and hence isometrically isomorphic, as a real vector space endowed with a compatible complex structure, to ℂⁿ. It also shows that geometric dimension and internal rank are invariants under coherent isomorphisms, that the structural projection induces equivalence classes of internally distinct states with the same effective image, and that the regular regime carries a metric-compatible complex structure. This framework does not replace quaternionic algebra and does not challenge classical classification theorems. Its purpose is narrower: to provide a formal geometric analogue of the distinction between visible classical output and suppressed internal structure already introduced in the theory of coherent collapse. In this sense, coherent Euclidean structures provide a formal bridge between quasi-magnitudinal collapse and later quasi-numeric constructions. 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.

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