Relational Cosmology of Active Hyperbolic Voids (AHV): Divergence and Saturation Regimes

At the relational level, divergence is characterized by low structural persistence of connectivity, configurational openness, and intense reorganization, while saturation is characterized by increasing persistence, organization, and stabilization. Within emergent geometry, divergence manifests predominantly through negative curvature and geodesic divergence, with Active Hyperbolic Voids constituting its principal cosmological manifestations. Their interfaces constitute emergent geometric transition regions toward increasing saturation, where critical conditions are associated, within the physical domain, with persistent material structures and cosmic organization.

The approximate cosmological proportions 68%, 27%, and 5% are retained only as phenomenological observational references rather than as quantities mathematically derived from the ontology of AHV. They are associated, respectively, with cosmological manifestations of divergence, transitions toward saturation, and stable baryonic matter. Within this framework, AHV does not require dark matter and dark energy as independent fundamental cosmological components, nor primordial inflation or an initial physical singularity as necessary stages or conditions of cosmological evolution.

AHV currently constitutes a conceptual and pre-mathematical cosmological model. Its scientific evaluation depends on the development of a consistent mathematical formulation capable of beginning from connectivity, characterizing the divergence and saturation regimes, deriving emergent geometry, and producing independent quantitative and falsifiable predictions that can be observationally distinguished from those of the ΛCDM paradigm.

Introduction

Contemporary cosmology has achieved remarkable observational precision through measurements of the Cosmic Microwave Background (CMB), the large-scale structure of the Universe, gravitational lensing, and the history of cosmic expansion. Within the standard cosmological framework, dark matter and dark energy are introduced as major cosmological components, while primordial inflation is commonly invoked to address specific early-Universe problems, and extrapolations of classical cosmological descriptions raise the problem of an initial singularity.

Current physical theories describe nature through specific fundamental theoretical structures. In Quantum Field Theory, particles are interpreted as excitations of quantum fields. In General Relativity, gravitational phenomena are described through spacetime geometry. These theories achieve remarkable predictive success within their respective domains, but their standard formulations do not attempt to derive these structures from the deeper relational ontology proposed by AHV.

The Active Hyperbolic Voids (AHV) model investigates the possibility of a more fundamental ontological description of reality. Rather than assuming spacetime geometry, fields, or particles as primitive entities, the model adopts a single ontological postulate: pure relational activity constitutes the fundamental ontological primitive of reality.

Pure relational activity is not interpreted as a substance, physical field, collection of particles, geometry, or any pre-existing physical structure. Its first structured manifestation is connectivity. Connectivity organizes through two fundamental regimes: divergence and saturation. From the organization and interaction of these regimes, geometry emerges. Within the resulting emergent geometric and physical domain, space, time, matter, radiation, gravitation, and observable cosmological structures progressively become manifest.

At the relational level, divergence and saturation constitute distinct modes of organization of connectivity. Divergence is characterized by low structural persistence, configurational openness, and intense reorganization, whereas saturation is characterized by increasing structural persistence, organization, and stabilization of connectivity.

Within the emergent geometric domain, these relational regimes acquire distinct geometric manifestations. Divergence manifests predominantly through negative curvature and geodesic divergence, with Active Hyperbolic Voids constituting its principal cosmological manifestations. Transitions toward saturation manifest through increasingly persistent geometric configurations and curvature gradients, particularly at AHV interfaces. Within the emergent physical domain, these saturation-related conditions are associated with matter condensation and the organization of persistent cosmological structures.

Geometry is therefore not assumed to be a fundamental constituent of reality. It emerges from the organization of connectivity through the divergence and saturation regimes and subsequently makes possible the manifestation of spatial, temporal, and physical structure.

Based on this conceptual architecture, AHV proposes an ontological reinterpretation of several cosmological phenomena. The observational phenomena conventionally interpreted as cosmic expansion, the Cosmic Microwave Background, gravitational phenomena, the large-scale distribution of matter, and the cosmological proportions conventionally associated with the dark sector are interpreted as observable manifestations arising within an emergent geometric and physical domain ultimately grounded in the organization of connectivity through divergence and saturation.

The objective of AHV is not to replace established cosmological observations but to provide an alternative ontological framework within which their physical interpretation may be reconsidered. At its current stage, AHV constitutes a conceptual and pre-mathematical proposal whose scientific evaluation depends on the development of a consistent mathematical formulation, the derivation of independent quantitative predictions, and comparison with astronomical observations.

Connectivity as the First Structured Manifestation of Pure Relational Activity

Within AHV, connectivity does not designate relations among previously existing physical entities. It constitutes the first structured manifestation of pure relational activity and the first level at which relational organization becomes structurally describable.

Connectivity is not embedded in pre-existing space and does not presuppose particles, fields, geometry, or other physical entities. Rather, it may organize with different degrees of structural persistence, configurational openness, and organization.

Connectivity continuously organizes and reorganizes through two complementary regimes. At the relational level, the divergence regime is characterized by low structural persistence, configurational openness, and intense reorganization. The saturation regime is characterized by increasing structural persistence, greater organization, and stabilization of connectivity.

Negative curvature and geodesic divergence do not belong to the relational definition of divergence. They appear only within the emergent geometric domain, where they constitute the predominant geometric manifestations of the divergence regime. Likewise, curvature gradients and increasingly persistent geometric configurations belong to the emergent geometric manifestation of transitions toward saturation.

The interaction between divergence and saturation gives rise to emergent geometry. Within this geometry, space, time, matter, radiation, gravitation, and observable cosmological structures progressively become physically manifest.

Divergence and saturation therefore do not constitute independent substances, physical components, or pre-existing spatial domains. They are the two fundamental modes through which connectivity organizes.

The Meaning of Connectivity

Connectivity constitutes the central organizing principle at the first structured level of AHV, without replacing pure relational activity as the ontological primitive of the model.

It does not describe connections among independently existing objects. Rather, it designates the first structured manifestation through which relational organization becomes distinguishable and may acquire different degrees of persistence, openness, and structural organization.

Observable physical phenomena do not emerge directly from connectivity. Connectivity first organizes through the divergence and saturation regimes. From the organization and interaction of these regimes, geometry emerges. Physical phenomena subsequently become manifest within the resulting emergent geometric and physical domain.

Within this domain, different phenomena are associated with different relational conditions. Within the emergent physical domain, saturation-related conditions are associated with increasingly persistent material configurations and the organization of cosmic structures. Divergence-related regions are associated with quantum vacuum fluctuations and approximately isotropic radiation within the emergent physical domain.

Gravitational phenomena likewise do not constitute direct manifestations of connectivity. They are interpreted through the curvature and geometric organization of the emergent domain, whose underlying structure ultimately reflects the organization of connectivity through divergence and saturation.

Observable physics is therefore not described by AHV as arising from interactions among ontologically independent fundamental objects. Physical properties and structures become manifest within emergent geometry as consequences of the deeper relational architecture of the model.

From Connectivity to Emergent Physics

Within AHV, connectivity constitutes the first structured manifestation of pure relational activity and the level at which the divergence and saturation regimes become distinguishable.

The ontological transition toward observable physics follows a definite sequence. Connectivity organizes through divergence and saturation. Their interaction gives rise to geometry. Within emergent geometry, spatial and temporal structures become physically distinguishable and physical phenomena subsequently become manifest.

At the relational level, divergence corresponds to low structural persistence, configurational openness, and intense reorganization. Within emergent geometry, these conditions manifest predominantly through negative curvature and geodesic divergence. At the cosmological level, Active Hyperbolic Voids constitute the principal manifestations of this regime.

Saturation, by contrast, corresponds at the relational level to increasing persistence, organization, and stabilization of connectivity. Within emergent geometry, transitions toward saturation manifest through increasingly persistent geometric configurations and curvature gradients, especially at AHV interfaces. Within the physical domain, critical saturation-related conditions are associated with matter condensation and persistent cosmic structures.

Matter, radiation, gravitation, and cosmological structures therefore do not constitute additional fundamental ontological levels. They are physical manifestations belonging to the emergent domain established after the emergence of geometry.

The Meaning of Relation in AHV

Within the AHV model, the term relation does not designate a connection established between previously existing physical objects. In traditional philosophical and physical descriptions, relations are often conceived as relations between entities that already possess independent existence. AHV reverses this ontological priority.

Pure relational activity constitutes the ontological primitive. It is not a relation between previously existing things, because no physical objects, spatial positions, fields, or geometric structures are assumed to exist at this fundamental level.

Connectivity constitutes the first structured manifestation of this activity. Connectivity itself is not spatial, since space has not yet emerged at this ontological level. It may continuously organize and reorganize with different degrees of structural persistence and configurational openness.

Through this organization, divergence and saturation become distinguishable as the two fundamental regimes of connectivity. At the relational level, divergence is characterized by low structural persistence, configurational openness, and intense reorganization, whereas saturation is characterized by increasing persistence, organization, and stabilization.

From the organization and interaction of these regimes, geometry emerges. Only within this emergent geometric domain do spatial structure, temporal order, matter, radiation, gravitation, and cosmological structures become physically manifest.

Thus, AHV does not require pre-existing physical entities in order to define its fundamental relational ontology. Physical entities themselves belong to the emergent domain.

Ontological Foundation

The AHV model is founded upon a single ontological postulate: pure relational activity constitutes the fundamental ontological primitive of reality.

Pure relational activity is not interpreted as a substance, physical field, collection of particles, geometry, or any pre-existing physical structure. Nor is it derived within AHV from a deeper ontological level. It constitutes the foundational hypothesis upon which the conceptual architecture of the model is developed.

Its first structured manifestation is connectivity. Connectivity organizes through the divergence and saturation regimes. From the organization and interaction of these regimes, geometry emerges.

Geometry therefore does not constitute a fundamental background. It is itself an emergent manifestation of the relational organization established through connectivity and its two fundamental regimes. Within the emergent geometric and physical domain, space, temporal order, matter, radiation, gravitation and observable cosmological structures become manifest.

Hyperbolicity is likewise not postulated as a primordial geometric property of reality. It constitutes an emergent geometric characteristic associated predominantly with the manifestation of the divergence regime. Negative curvature and geodesic divergence therefore belong to the geometric domain and do not define connectivity at the fundamental relational level.

The complete ontological sequence proposed by AHV is:

pure relational activity → connectivity → divergence and saturation regimes → emergent geometry → space, time, and emergent physical phenomena.

This sequence constitutes the organizing architecture of the model.

Conceptual Challenges

Every relational ontology faces a classical philosophical question: if physical reality is relational, relations between what? The question assumes that relations require independently existing entities as their prior terms.

AHV challenges precisely this assumption. Pure relational activity constitutes the ontological primitive rather than a relation established between pre-existing physical objects. Connectivity constitutes its first structured manifestation and organizes through divergence and saturation. From the interaction of these regimes geometry emerges, and physical entities subsequently become manifest within the emergent geometric and physical domain.

From this perspective, particles, matter, fields, spatial structures, and geometry are not fundamental constituents required in advance to support relational activity. They belong to progressively emergent physical descriptions.

A second conceptual challenge concerns the remarkable stability and regularity of physical laws. AHV does not yet provide a mathematical derivation of these laws. Within its conceptual framework, such regularities may be interpreted as persistent regularities of the emergent physical domain ultimately grounded in stable patterns of relational organization. Demonstrating this connection quantitatively remains a future requirement of the model.

A third challenge concerns observational cosmology. AHV retains the approximate cosmological proportions conventionally described as approximately 68%, 27%, and 5% only as phenomenological observational references. They are associated, respectively, with cosmological manifestations predominantly related to divergence, transition regions toward saturation, and stable baryonic matter.

These values are not presently derived mathematically from connectivity or from the divergence and saturation regimes. Their reproduction through a quantitative formulation constitutes an empirical requirement rather than an already established theoretical result.

Likewise, AHV does not regard dark matter and dark energy as independent fundamental cosmological components required by its ontology. It also does not require primordial inflation or an initial physical singularity as necessary stages or conditions of cosmological evolution. Whether the model can quantitatively reproduce the corresponding observational phenomena remains subject to future mathematical development and empirical testing.

Conclusion

The Active Hyperbolic Voids (AHV) model proposes a relational cosmology founded on a single ontological primitive: pure relational activity.

Its first structured manifestation is connectivity, which organizes through the divergence and saturation regimes. At the relational level, divergence is characterized by low structural persistence, configurational openness, and intense reorganization, while saturation is characterized by increasing persistence, organization, and stabilization of connectivity.

From the organization and interaction of these regimes, geometry emerges. Within the resulting emergent geometric and physical domain, space, time, matter, radiation, gravitation, and observable cosmological structures progressively become manifest.

Active Hyperbolic Voids constitute the principal cosmological manifestations of the divergence regime within emergent geometry. Their predominant negative curvature and geodesic divergence are geometric manifestations of the underlying relational organization rather than fundamental properties of reality. AHV interfaces, in turn, constitute characteristic emergent geometric manifestations of transitions toward increasing saturation, where curvature gradients and increasingly persistent geometric configurations are associated with the organization of matter and cosmic structure.

Space, time, matter, radiation, and gravitation are therefore not regarded as ontologically fundamental constituents. Nor do they emerge directly from connectivity. They belong to an emergent domain made possible after connectivity has organized through divergence and saturation and geometry has arisen from their interaction.

AHV takes the established cosmological observations successfully described within the ΛCDM framework as empirical references and proposes an alternative ontological interpretation of them. It does not introduce dark matter and dark energy as independent fundamental cosmological components and does not require primordial inflation or an initial physical singularity as necessary stages or conditions of cosmological evolution.

The model also preserves the approximate observational proportions conventionally represented by 68%, 27%, and 5% only as phenomenological references. These values are not currently derived from the ontology of AHV and therefore remain quantities that a future mathematical formulation must explain rather than predictions already produced by the model.

The central ontological architecture of AHV can be summarized as:

pure relational activity → connectivity → divergence and saturation regimes → emergent geometry → space, time, and emergent physical phenomena.

At its present stage, AHV should be understood as a conceptual and pre-mathematical cosmological framework. Its scientific viability depends on the development of a consistent mathematical formulation capable of beginning from connectivity, characterizing the divergence and saturation regimes, deriving emergent geometry, and subsequently producing independent, quantitative, and falsifiable predictions for observable physical and cosmological phenomena.

Only through such predictions and their comparison with astronomical observations will it be possible to determine whether the relational-emergent architecture proposed by AHV provides an empirically viable alternative cosmological description.