Core Definitions

The fundamental unit of existence in UCF/GUTT. A relation is a connection between entities that partially constitutes both entities involved. Relations are ontologically primary—they do not presuppose pre-existing entities but rather give rise to entities as stable relational patterns.

Relations have attributes including strength, direction, origin, and temporal characteristics. Relations can themselves enter into relations with other relations. No entity exists independent of relations.

This inverts the traditional substance-first ontology. Where classical approaches treat objects as fundamental and relations as secondary, UCF/GUTT treats relations as fundamental and objects (entities) as emergent.

Formal grounding: Proposition_01_proven.v

A dynamically bounded nexus of relations existing at a particular level of focus within a broader relational system. Rather than treating entities as fundamental objects that subsequently enter into relations, UCF/GUTT treats relations as primary—entities emerge as stable patterns within relational networks.

Relational Constitution: An entity is fundamentally constituted by its internal relations (attributes, properties, components) and external relations (connections to other entities within its sphere of influence). There is no entity-in-itself independent of relations.

Context-Dependent Boundaries: Entity boundaries are not fixed but shift depending on context and the observer's level of analysis. Where one entity ends and another begins is determined by the pattern and strength of relations at a particular scale.

Nesting and Emergence: Entities can exist within other entities through Nested Relational Tensors. An entity's properties and behaviors can emerge from the interplay of its internal relations and its position within the broader system.

Perspective: An entity's perspective is shaped by its relational position—its goals, sensory mechanisms, and comparisons to other entities in the system. This is not a metaphor but a structural feature of relational systems.

Scale Independence: "Entity" is a relative concept. An atom, a cell, a person, a solar system—all can be viewed as entities at their respective scales. The mathematical framework handles this uniformly through the same relational tensor structures regardless of domain. This does not mean all entities are equivalent or that scale doesn't matter. It means the formal machinery for describing relational structure applies consistently across scales.

Illustrative examples: A person as entity includes internal relations (physical constitution, cognitive states, accumulated experiences) and external relations (family, social groups, professional networks). Boundaries depend on whether we consider biological body, social identity, or sphere of influence. An ecosystem as entity includes organisms and non-living components as relationally coupled, with boundaries depending on whether focus is a pond, watershed, or biosphere. An atom as entity is a nexus of interacting fields and probabilities rather than a discrete object, with boundaries that blur into the surrounding quantum environment. These examples illustrate the concept but are not validated applications.

What this definition does not claim: The entity concept provides formal vocabulary for describing relational structure. It does not by itself predict specific properties of physical, biological, or social entities, replace domain-specific scientific models, or establish that all domains behave identically. Applications to particular domains require grounding the abstract relational structure in domain-specific dynamics, followed by validation against known results.

Formal grounding: Proposition_01_proven.v, Proposition_02_DSoR_proven.v, Prop_NestedRelationalTensors_proven.v, NRT_Structure_Uniqueness.v, Proposition_12_SensoryMechanism_proven.v

A bounded collection of entities and relations that can be analyzed as a coherent whole. A Relational System includes all relations among its constituent entities and specifies the scope of analysis.

Every RS exists within a larger RS (except "the Whole"). RS boundaries are context-dependent, not absolute. An RS can contain nested sub-systems.

When analyzing any domain—physical, biological, social—the first step is specifying the Relational System under consideration: which entities, which relations, at what scale.

Formal grounding: Proposition_04_RelationalSystem_proven.v, RelationalCore_Existence.v

A hierarchical tensor structure where relational tensors contain other relational tensors, enabling representation of multi-scale systems. NRTs formalize how entities at one scale compose into entities at larger scales while preserving relational information across levels.

NRTs maintain relational coherence across nesting levels. Properties at higher levels can emerge from lower-level relational dynamics. Structure uniqueness conditions constrain valid NRT configurations.

Standard tensors represent relations at a single level. NRTs capture how those relations themselves form patterns that constitute higher-level relations.

Formal grounding: Prop_NestedRelationalTensors_proven.v, NRT_Structure_Uniqueness.v, CrispDynamicsNRT.v

A measure of the intensity, robustness, or significance of a relation between entities. Strength is not merely quantitative but reflects how much the relation contributes to constituting the entities involved.

Strength can vary over time. Strength affects how changes propagate through a relational system. High-strength relations are more constitutive of entity identity.

Specific metrics for strength depend on domain. The formal framework establishes that relations have strength as an attribute; quantification requires domain-specific grounding.

Formal grounding: Proposition_09_Attributes_proven.v

The orientation or asymmetry of a relation. Not all relations are symmetric—A's relation to B may differ from B's relation to A in character, not merely strength.

Directed relations have source and target. Direction affects information and influence flow through systems. Some relations are bidirectional but asymmetric.

Causation, authority, perception, and dependency are typically directed. Spatial proximity and similarity are typically undirected.

Formal grounding: Proposition_10_Direction_proven.v

The scope or extent of an entity's relational reach—how many and what kinds of relations an entity participates in. DSoR captures the "relational footprint" of an entity within its system.

DSoR varies with perspective and level of analysis. Entities with larger DSoR have broader systemic influence. DSoR can change as relations form or dissolve.

An entity's boundaries are partly determined by where its DSoR diminishes below significance thresholds.

Formal grounding: Proposition_02_DSoR_proven.v

The temporal characteristics of a relation—when it exists, how long it persists, and how it changes over time. ToR captures that relations are not instantaneous but have duration and history.

Relations have formation, persistence, and dissolution phases. Temporal dynamics affect relational strength and character. Some relations are momentary; others persist across long timescales.

ToR addresses when relations exist and change. The deeper question of whether time itself is relational (emerges from relational dynamics) is addressed separately in spacetime propositions.

Formal grounding: Proposition_14_TimeOfRelation_proven.v

The source or genesis of a relation—what gives rise to a particular relational connection. Origin addresses how relations come into being within a system.

Relations can originate from entity interactions, systemic constraints, or emergence from lower-level dynamics. Origin affects relational character and stability. Understanding origin aids prediction of relational evolution.

Formal grounding: Proposition_11_Origin_proven.v

The means by which an entity registers or responds to relations with other entities. Sensory mechanisms determine what aspects of the relational environment an entity can "perceive" or be affected by.

Sensory mechanisms are themselves relational structures. They constrain and shape an entity's perspective. Different entities have different sensory access to the same relational system.

An entity's "view" of its relational system is always partial and shaped by its sensory mechanisms. Complete views require either omniscient perspective or integration across multiple limited perspectives.

Formal grounding: Proposition_12_SensoryMechanism_proven.v

Relations that persist without significant change over a given timescale. Static relations provide structural stability to relational systems.

Static does not mean permanent—it means stable relative to the timescale of analysis. What appears static at one scale may be dynamic at another.

Formal grounding: Proposition_07_Static_proven.v

Relations that change, evolve, or fluctuate over time. Dynamic relations drive system evolution and enable adaptation.

Dynamic relations can strengthen, weaken, change direction, or dissolve entirely. The interplay between static and dynamic relations determines system behavior.

Formal grounding: Proposition_08_Dynamic_proven.v

A state where relational changes balance such that the overall system structure remains stable despite ongoing local dynamics. DER captures how systems can be simultaneously active and stable.

Equilibrium is dynamic, not static—relations continue changing. Stability emerges from balanced rates of change. Perturbations can shift systems to new equilibrium states.

Formal grounding: Proposition_08_Dynamic_proven.v

The transition zone where an entity's relational influence diminishes to the point of non-significance. Boundaries are not sharp edges but gradients in relational strength and density.

Boundaries are context-dependent and observer-relative. The same system can have different boundaries at different scales. Boundary conditions affect how systems interact with their environment.

UCF/GUTT handles boundary cases (including division by zero scenarios) through contextual boundary operators rather than undefined operations.

Formal grounding: RelationalBoundaryContext.v, boundary_division.v, DivisionbyZero_proven.v, ContextualDivision.v

The arising of properties, behaviors, or structures at one level of a system that are not present in (or simply predictable from) the components at lower levels. In UCF/GUTT, emergence is a formal consequence of relational composition through NRTs.

Emergent properties are real features of the higher-level system. Emergence does not require mysterious causation—it follows from relational mathematics. What emerges depends on both component relations and their configuration.

The mathematical machinery for emergence (NRTs, multi-level dynamics) is established. Specific emergence claims for particular domains require additional validation.

Formal grounding: Prop_NestedRelationalTensors_proven.v(implicit in NRT structure)

Status: Explicit emergence propositions are in development.

The maximal Relational System containing all entities and relations. "The Whole" serves as the universal context within which all other Relational Systems are nested.

Every entity and relation exists within the Whole. The Whole has no external relations (there is nothing outside it). The Whole provides universal connectivity as a mathematical necessity.

"The Whole" is a formal/mathematical concept, not a metaphysical claim about the ultimate nature of reality. It functions as the maximal RS for purposes of the framework.

Formal grounding: Complete_Picture_proven.v

The following terms are used in UCF/GUTT discourse but await complete formal grounding. These concepts appear in theoretical development but should be treated as provisional until corresponding proof files are complete.

Relational Resilience (RRs): System capacity to maintain coherence under perturbation.

Relational Entropy (REn): Measure of disorder or uncertainty in relational configurations.

Emergent Readiness (α): System state indicating potential for new relation formation.

Relational Stability Function (Φ): Measure of configuration stability.

All cited proof files are available at the GitHub repository github.com/relationalexistence/UCF-GUTT.