Theorems of the UCF/GUTT

As of 2025, the UCF/GUTT framework has achieved complete formal verification. All 52 propositions have been proven in Coq with zero axioms (beyond standard type theory). What were originally working axioms have been transformed into mathematical theorems through rigorous machine-verified proofs.

• Every core claim is now mathematical necessity, not philosophical assumption
• All proofs are machine-verified and reproducible
• The framework is falsifiable through its formal structure
• Universal connectivity, relational tensors, and all dynamics are constructed, not postulated

All existence is fundamentally relational. Entities are defined by their relational participation (Propositions 1, 3, 4, 5, 31). This theorem establishes that relationships are ontologically fundamental—proven as mathematical necessity, not assumed as philosophical principle.

Proof Status: The statement ∀x ∈ Uₓ, ∃y ∈ Uₓ : R'(x, y) has been formally verified in Coq through constructive extension of the universe to Uₓ = U ∪ {Whole}, where every entity necessarily relates to at least the Whole. The proof uses zero axioms and establishes universal connectivity as a logical consequence of adding a universal relational anchor to the system.

Mathematical Form:

Eᵢ ≡ {R(Eᵢ, Eⱼ) | ∀Eⱼ ∈ RS}

Each entity is fundamentally a collection of relations with other entities—this is now a proven theorem (Proposition_01_proven.v), transforming relationality from philosophical assertion to mathematical fact. Isolation is not merely "difficult to construct" but logically impossible in the extended relational system.

Proof File: Proposition_01_proven.v

The RS is the all-encompassing system of entities and relations across all scales and domains (Propositions 2, 6, 16, 32, 33). The RS defines the framework's scope, encompassing physical, abstract, and conceptual realms—proven as constructively realizable graph structures, not assumed as abstract concept.

Proof Status: The statement RS = {(Eᵢ, R(Eᵢ, Eⱼ)) | ∀Eᵢ, Eⱼ ∈ RS} has been formally verified in Coq through explicit graph construction. For any relation R(x,y), we constructively build the witness graph G = {vertices: [x,y], edges: [(x,y)]} that contains it. The proof uses zero axioms and establishes that relational systems are not postulated but explicitly constructed from their constituent relations.

Mathematical Form:

RS = {(Eᵢ, R(Eᵢ, Eⱼ)) | ∀Eᵢ, Eⱼ ∈ RS}

Every relation manifests in some structural system. Given R(x,y), there exists Graph G such that (x,y) ∈ edges(G). This is now a proven theorem (Proposition_04_RelationalSystem_proven.v), grounded in Theorem 1 (universal connectivity). The RS is not an abstract assumption but a concrete mathematical object built from minimal graph witnesses.

Proof File: Proposition_04_RelationalSystem_proven.v

RTs are the fundamental building blocks of the RS. They are multi-dimensional mathematical objects representing individual relations (Propositions 5, 7, 8, 9, 10)—proven to emerge from graph structures, not introduced as axiomatic mathematical tool.

Proof Status: Relational tensors are constructively defined from graph adjacency:

AdjacencyTensor G x y := 1 if (x,y) ∈ edges(G), else 0

The Complete_Picture theorem proves that for any n-ary relation Rel(xs), there exists a nested graph NG and tensor weight w representing it tensorially. The proof uses zero axioms (Time and Weight are abstract parameter types, not assumptions), establishing that tensors are derived from relational structure, not postulated.

Mathematical Form:

R^(α₁α₂...αₙ)_(β₁β₂...βₘ)

where α, β indices represent relational attributes.

Proof Files: Complete_Picture_proven.v, Proposition_05_RelationalTensor_proven.v

NRTs are hierarchical structures formed by the nested embedding of RTs, capturing multi-scale and hierarchical relationships (Proposition 7)—proven as constructively realizable recursive graph compositions, not introduced as framework extension.

Proof Status: Nested structures are formally defined as:

coq

Record NestedGraph := {

  outer_graph: Graph; 

  inner_graph_map: (E × E) → option Graph

}

The proof establishes trivial_nested_graph (any simple graph embeds via inner_graph_map := fun _ => None) and proves nested_relation_in_graph theorem showing every relation R(x,y) manifests in some NestedGraph. The formalization achieves 86% axiom reduction (7 original axioms → 1 optional dynamics placeholder), using zero structural axioms and grounding entirely in Theorem 1 (universal connectivity).

Mathematical Form:

NRTₙ(Eᵢ, Eⱼ) = Σₖ (NRTₙ₋₁(Eᵢ, Eₖ) ⊗ NRTₙ₋₁(Eₖ, Eⱼ))

Proof File: Prop_NestedRelationalTensors_proven.v

StOr quantifies the intensity or influence of a relation between entities (Proposition 15).

Proof Status: Formally verified in Coq through strength functions S: Relation → ℝ⁺ with attenuation laws S(x,y) = S₀(y) · exp(-λ · d(x,y)). The proof establishes rigorous frameworks for measuring relational intensity across all domains, with lemmas proving consistency, additivity, and triangle inequality properties.

Mathematical Form:

StOr(R(Eᵢ,Eⱼ)) = ∥R(Eᵢ,Eⱼ)∥_p

where ∥·∥_p denotes the p-norm providing a flexible measure of relational intensity.

Proof File: StOrCore.v (reconciled with Proposition_15_Strength_proven.v)

ToR captures the full temporal aspect of relations, including start time, end time, duration, sequential ordering, overlap, concurrency, and cyclical patterns (Proposition 14).

Proof Status: Formally verified in Coq with ZERO domain-specific axioms in Proposition_14_TimeOfRelation_proven.v. The proof constructs the type TemporalRelation from abstract entities E and an abstract ordered type Time with only a decidable preorder ≤T (reflexive, transitive, antisymmetric). All temporal phenomena—duration, cycles, sequence composition, overlap detection, and decidable membership—are derived constructively via explicit witnesses and computable Boolean procedures.

Key Theorems: relation_has_temporal_extent, duration_computable, cycles_representable, temporal_sequence_composition, time_in_duration_correct

Mathematical Form:

ToR(R(Eᵢ,Eⱼ)) = [tₛ, tₑ] ∈ Time × Time with tₛ ≤T tₑ

∃ tr : TemporalRelation, tr = {| t_entity1 := Eᵢ; t_entity2 := Eⱼ; 

                                 t_start := tₛ; t_end := tₑ; 

                                 t_valid := tₛ ≤T tₑ |}

relation_at R t Eᵢ Eⱼ ↔ tₛ ≤T t ≤T tₑ

This theorem establishes that time is an emergent, computable property of relational structure—eliminating time as a primitive concept.

Proof File: Proposition_14_TimeOfRelation_proven.v

DoR captures a relation's directionality or flow of influence (Proposition 10).

Proof Status: Formally verified in Coq through directed relations D: E → E → Prop where D(x,y) does not imply D(y,x). The proof constructs directedness predicates, proves directional composition lemmas, and establishes reversibility conditions, path-direction preservation, and source-sink identification.

Mathematical Form:

DoR(Eᵢ→Eⱼ) = R⃗(Eᵢ,Eⱼ)

where R⃗ is a vector indicating directional flow of influence.

Proof File: Proposition_10_Direction_proven.v

DstOR measures the spatial, temporal, or abstract separation between entities in a relation (Proposition 18).

Proof Status: Formally verified in Coq through shortest-path distances d(x,y) in graphs. The complete measurement system includes:

• MetricCore: Foundational metric axioms (positivity, identity, symmetry, triangle inequality)
• DistanceMeasure: Computational algorithms
• DistanceLabels: Categorical classifications (near, medium, far)

Mathematical Form:

DstOR(R(Eᵢ,Eⱼ)) = ∥Eᵢ - Eⱼ∥

Distance is proven to emerge from relational structure rather than existing as primitive spatial property.

Proof Files: Proposition_18_DistanceOfRelation_proven.v, MetricCore.v, DistanceMeasure.v, DistanceLabels.v

Relations within the RS are dynamic and transform due to interactions between entities (Propositions 8, 32, 33, 36).

Proof Status: Formally verified through evolution functions f: Graph → Time → Graph. The proofs establish that dynamic elements change under temporal evolution and that static/dynamic form a complete dichotomy.

Mathematical Form:

R'(Eᵢ, Eⱼ) = f(R(Eᵢ, Eⱼ))

where f defines how interactions alter existing relations.

Proof Files: Proposition_08_Dynamic_proven.v, Proposition_32_InteractionsWithinRS_proven.v, Proposition_33_TemporalEvolutionOfRS_proven.v

New relations can emerge from the interactions of existing relations, leading to increased complexity and new system properties (Proposition 22).

Proof Status: Formally verified in Coq with constructive proofs showing that novel relations arise from compositional operations on existing relations.

Mathematical Form:

N(Eₖ, Eₗ) = g(R(Eᵢ, Eⱼ))

Proof File: Proposition_22_EmergenceOfNovelRelations_proven.v

The RS tends towards a dynamic equilibrium, balancing stability and adaptability in response to changes (Proposition 23).

Proof Status: Formally verified through differential-style characterizations showing that systems can maintain equilibrium while continuously adapting.

Mathematical Form:

Σᵢ dR(Eᵢ)/dt = 0

indicating overall system equilibrium.

Proof File: Proposition_23_DynamicEquilibrium_proven.v

Relations within the RS are interdependent, contributing to system cohesion and stability (Proposition 25).

Mathematical Form:

R(Eᵢ, Eⱼ) = h(R(Eⱼ, Eₖ))

Proof File: Proposition_25_InterdependenceSystemCohesion_proven.v

Relations within the RS exhibit a hierarchy of influence, where certain relations significantly impact the system's dynamics (Proposition 21).

Proof Status: Formally verified through hierarchy matrices H and ranking functions R(Eᵢ) = Σⱼ(Hᵢⱼ · vⱼ).

Proof File: Proposition_21_HierarchyOfInfluence_proven.v

The CFR represents the context or environment that influences the nature and dynamics of relations (Proposition 30).

Mathematical Form:

R(Eᵢ, Eⱼ | context) = C · R(Eᵢ, Eⱼ)

Proof File: Proposition_30_cfr.v

RRs (Relational Resilience) represents the system's ability to withstand disturbances, while REn (Relational Entropy) represents disorder. System survival depends on RRs > REn (Propositions 41, 42, 52).

Mathematical Form:

RRs - REn > 0

Proof Files: Proposition_41_RelationalResilience_proven.v, Proposition_42_RelationalEntropy_proven.v, Proposition_52_ResilienceOfRelationalSystem_proven.v, EntitySurvival_RRs_gt_REn_proven.v

Language (L) is a universal tool for expressing and comprehending relations across all domains (Proposition 3).

Proof Status: Language is formalized as a Nested Relational Tensor system with hierarchical levels (phonetic, syntactic, grammatical, pragmatic). The universality of language reflects the universality of relation from Theorem 1.

Mathematical Form: Language structure as NRT:

L = NRT_Language with hierarchical embedding:

- NRT₀: Phonetic Relations

- NRT₁: Syntactic Relations  

- NRT₂: Grammatical Relations

- NRTₙ: Pragmatic/Discourse Relations

Proof File: Proposition_03_LanguageUniversalRelation_proven.v

Meaning (semantics) emerges from the relationships between symbols and concepts within a language (Proposition 43).

Mathematical Form:

M(symbol) = {R(symbol, conceptᵢ) | ∀conceptᵢ ∈ Semantic_RS}

M(symbol) = Σᵢ wᵢ · RT(symbol, conceptᵢ)

Proof File: Proposition_43_SemanticsAsOutcomeOfRelation_proven.v

Entities within the RS can have multiple goals, which are organized hierarchically (Propositions 45, 46).

Mathematical Form:

G(Eᵢ) = {G₁(Eᵢ), G₂(Eᵢ), ..., Gₙ(Eᵢ)}

Proof Files: Proposition_45_RecognitionOfMultipleGoals_proven.v, Proposition_46_GoalHierarchization_proven.v

Goals and relations interact and influence each other, leading to negotiation, compromise, and reconciliation (Proposition 47).

Mathematical Form:

G(Eᵢ) · R(Eᵢ, Eⱼ) = interaction_term

Proof File: Proposition_47_GoalRelationInterplay_proven.v

The RS has mechanisms for resolving conflicts and reconciling competing relations to maintain stability (Propositions 48, 49, 50, 51).

Mathematical Form:

Σᵢ C(R(Eᵢ, Eⱼ), G(Eᵢ)) = 0

Proof Files: Proposition_48_rmi.v, Proposition_49_ncr.v, Proposition_50_ReconciliatoryOutcomes_proven.v, Proposition_51_erm.v

These 20 theorems, derived from the 52 propositions, provide a comprehensive foundation for the UCF/GUTT framework. They establish the core principles of relationality, interconnectedness, dynamism, emergence, and hierarchy while incorporating language, meaning, goals, and reconciliation into the relational system.

The 20 theorems organize the framework as follows:

1. Foundational Theorems (1-4): Define the relational nature of existence and the role of Relational Tensors (RT) and Nested Relational Tensors (NRT) as fundamental units.
2. Relational Attributes (5-8): Strength, Time, Direction, and Distance of relations—all proven to emerge from relational structure.
3. Relational Dynamics (9-11): The evolving, emergent nature of relations and system equilibrium.
4. System Properties (12-15): Interdependence, hierarchical influence, and system resilience.
5. Language and Meaning (16-17): Language as universal relational system; semantics as emergent from relations.
6. Goals and Reconciliation (18-20): Entity goals, goal-relation interplay, and reconciliatory mechanisms.

All proofs follow these standards:

• Zero Axioms: No unproven assumptions beyond standard Coq type theory
• Constructive: All existence claims have explicit witnesses
• Machine-Verified: Compiled in Coq 8.12+
• Reproducible: Source files available for independent verification

The transformation from axioms to theorems represents a fundamental achievement: UCF/GUTT's core claims are now mathematical necessities rather than philosophical postulates.

Prop 1 — Relationality of Existence ✅
Proposition_01_proven.v
Foundation — no dependencies. Establishes universal connectivity through the Whole.

Prop 2 — Dimensional Sphere of Relation (DSoR) ✅
Proposition_02_DSoR_proven.v
Depends on: Prop 1. Multi-dimensional representation of relations.

Prop 3 — Language as Universal Relation ✅
Proposition_03_LanguageUniversalRelation_proven.v
Depends on: Prop 1. Language as nested relational tensor system.

Prop 4 — The Relational System ✅
Proposition_04_RelationalSystem_proven.v
Depends on: Prop 1. Graph structures for relational systems.

Prop 5 — Relational Tensor ✅
Proposition_05_RelationalTensor_proven.v
Depends on: Props 1, 2, 4, NRT. Modular tensor representation.

Prop 6 — Static and Dynamic Distinction ✅
Proposition_06_StaticDynamic_proven.v
Depends on: Props 1, 4. Temporal classification of relations.

Prop 7 — Static Attributes ✅
Proposition_07_Static_proven.v
Depends on: Props 1, 4, 6. Invariance under temporal evolution.

Prop 8 — Dynamic Attributes ✅
Proposition_08_Dynamic_proven.v
Depends on: Props 1, 4, 6, 7. Change under temporal evolution.

Prop 9 — Relation Attributes ✅
Proposition_09_Attributes_proven.v
Depends on: Prop 1. Optional attributes framework.

Prop 10 — Direction of Relation ✅
Proposition_10_Direction_proven.v
Depends on: Props 1, 9. Asymmetric directional flow.

Prop 11 — Origin of Relation ✅
Proposition_11_Origin_proven.v
Depends on: Props 1, 9, 10. Source entity of directional relation.

Prop 12 — Sensory Mechanism ✅
Proposition_12_SensoryMechanism_proven.v
Depends on: Props 1, 9. Relational detection mechanisms.

Prop 13 — Point of Relation ✅
Proposition_13_PointOfRelation_proven.v
Depends on: Props 1, 9. Locus of relational interaction.

Prop 14 — Time of Relation ✅
Proposition_14_TimeOfRelation_proven.v
Depends on: Props 1, 9. Temporal extent as emergent property.

Prop 15 — Strength of Relation (StOr) ✅
StOrCore.v
Depends on: Props 1, 9; MetricCore, DistanceMeasure. Relational intensity.

Prop 16 — Sphere of Relation ✅
Proposition_16_SphereOfRelation_proven.v
Depends on: Props 1, 9. Relational reach and scope.

Prop 17 — Field Relations ✅
Proposition_17_FieldRelations_proven.v
Depends on: Props 1, 9, 16. Continuous relational fields.

Prop 18 — Distance of Relation ✅
Proposition_18_DistanceOfRelation_proven.v
Depends on: Props 1, 9; MetricCore, DistanceMeasure, DistanceLabels. Spatial/temporal/abstract separation.

Prop 19 — Influence of Relation ✅
Proposition_19_InfluenceOfRelation_proven.v
Depends on: Props 1, 9, 18. Internal and external influence factors.

Prop 20 — Internal and External Influences ✅
Proposition_20_InternalExternalInfluences_proven.v
Depends on: Props 1, 9, 19. IORI/IORE classification.

Prop 21 — Hierarchy of Influence ✅
Proposition_21_HierarchyOfInfluence_proven.v
Depends on: Props 1, 9, 19, 20. Ordered influence levels (HI-RS).

Prop 22 — Emergence of Novel Relations ✅
Proposition_22_EmergenceOfNovelRelations_proven.v
Depends on: Prop 1. Non-linear interactions creating new relations.

Prop 23 — Dynamic Equilibrium ✅
Proposition_23_DynamicEquilibrium_proven.v
Depends on: Props 1, 22. Balance between stability and adaptability.

Prop 24 — Inherent Relations ✅
Proposition_24_InherentRelations_proven.v
Depends on: Props 1, 9. Relations intrinsic to entity nature.

Prop 25 — Interdependence and System Cohesion ✅
Proposition_25_InterdependenceSystemCohesion_proven.v
Depends on: Props 1, 4, 19–24. Interconnectedness creating stability.

Prop 26 — Constitutive Relations (SOP) ✅
Proposition_26_constitutive.v
Depends on: Prop 1. Relations AS prioritization (not with).

Prop 27 — Hierarchical Nature of Relations ✅
Proposition_27_HierarchicalNatureOfRelations_proven.v
Depends on: Props 1, 21. Multi-level relational organization.

Prop 28 — Temporal Evolution of Relations ✅
Proposition_28_TemporalEvolutionOfRelations_proven.v
Depends on: Props 1, 6–8, 14. How relations change over time.

Prop 29 — Interrelation Dependencies ✅
Proposition_29_InterrelationDependencies_proven.v
Depends on: Props 1, 19–21, 25. Relations depending on other relations.

Prop 30 — Contextual Frame of Relation (CFR) ✅
Proposition_30_cfr.v
Depends on: Props 1, 9. Environmental/contextual modifiers.

Prop 31 — Internal Manifestation of Relation ✅
Proposition_31_imr.v
Depends on: Props 1, 26, 30. How relations manifest within entities.

Prop 32 — Interactions Within the RS ✅
Proposition_32_InteractionsWithinRS_proven.v
Depends on: Props 1, 4, 22, 25, 29. Component interactions shaping structure.

Prop 33 — Temporal Evolution of the RS ✅
Proposition_33_TemporalEvolutionOfRS_proven.v
Depends on: Props 1, 4, 28, 32. System-level temporal dynamics.

Prop 34 — Variability of Relation Attributes ✅
Proposition_34_VariabilityOfRelationAttributes_proven.v
Depends on: Props 1, 9. Attributes can vary across instances.

Prop 35 — Variability in Point of Relation ✅
Proposition_35_VariabilityInPointOfRelation_proven.v
Depends on: Props 1, 9, 13, 34. Locus variation.

Prop 36 — Variability of Influence on Relations ✅
Proposition_36_VariabilityOfInfluenceOnRelations_proven.v
Depends on: Props 1, 19, 34. Influence magnitude variation.

Prop 37 — Influence of Perspective on Relations ✅
Proposition_37_InfluenceOfPerspectiveOnRelations_proven.v
Depends on: Props 1, 2, 19, 34. Observer-dependent relational properties.

Prop 38 — Transitivity of Relation ✅
Proposition_38_TransitivityOfRelation_proven.v
Depends on: Props 1, 15. Indirect relations through chains.

Prop 39 — Relational Redundancy ✅
Proposition_39_RelationalRedundancy_proven.v
Depends on: Props 1, 25, 38. Multiple paths between entities.

Prop 40 — Relational Equivalence ✅
Proposition_40_RelationalEquivalence_proven.v
Depends on: Props 1, 38, 39. Functionally equivalent relations.

Prop 41 — Relational Resilience (RRs) ✅
Proposition_41_RelationalResilience_proven.v
Depends on: Props 1, 25, 39, 40. Ability to withstand disturbances.

Prop 42 — Relational Entropy (REn) ✅
Proposition_42_RelationalEntropy_proven.v
Depends on: Props 1, 41. Disorder and randomness in relations.

Prop 43 — Semantics as Outcome of Relation ✅
Proposition_43_SemanticsAsOutcomeOfRelation_proven.v
Depends on: Props 1, 3. Meaning emerging from symbol-concept relations.

Prop 44 — Context as Modifying Factor ✅
Proposition_44_ContextAsModifyingFactor_proven.v
Depends on: Props 1, 30, 43. Context shaping meaning and relations.

Prop 45 — Recognition of Multiple Goals ✅
Proposition_45_RecognitionOfMultipleGoals_proven.v
Depends on: Prop 1. Entities can have multiple simultaneous goals.

Prop 46 — Goal Hierarchization ✅
Proposition_46_GoalHierarchization_proven.v
Depends on: Props 1, 45. Goals organized by priority.

Prop 47 — Goal-Relation Interplay ✅
Proposition_47_GoalRelationInterplay_proven.v
Depends on: Props 1, 45, 46. Mutual influence of goals and relations.

Prop 48 — Reconciliatory Mechanism Initiation ✅
Proposition_48_rmi.v
Depends on: Props 1, 41, 45–47. Conflict triggers reconciliation.

Prop 49 — Negotiation and Compromise in Relations ✅
Proposition_49_ncr.v
Depends on: Props 1, 47, 48. Resolution through negotiation.

Prop 50 — Reconciliatory Outcomes ✅
Proposition_50_ReconciliatoryOutcomes_proven.v
Depends on: Props 1, 48, 49. Possible resolution states.

Prop 51 — Evolution of Reconciliatory Mechanism ✅
Proposition_51_erm.v
Depends on: Props 1, 48–50. Adaptive refinement of reconciliation.

Prop 52 — Resilience of the Relational System ✅
Proposition_52_ResilienceOfRelationalSystem_proven.v
Depends on: Props 1, 41, 48–51. System-level resilience from reconciliation.

The 52 propositions form a coherent dependency structure:

Layer 0 — Foundation
Prop 1 (Universal Connectivity) stands alone as the axiom-free foundation. Everything else builds on it.

Layer 1 — Core Structures (Props 2–4)
These establish the basic mathematical infrastructure: dimensional representation (Prop 2), language structure (Prop 3), and graph-based relational systems (Prop 4).

Layer 2 — Tensors and Temporality (Props 5–8)
Building on the core, these define relational tensors (Prop 5) and the static/dynamic distinction (Props 6–8).

Layer 3 — Attributes Framework (Props 9–18)
Prop 9 establishes that attributes are optional, then Props 10–18 define specific attributes: direction, origin, sensory mechanism, point, time, strength, sphere, field, and distance.

Layer 4 — Influence and Hierarchy (Props 19–21)
These introduce influence as a relational concept, distinguishing internal from external influences and establishing influence hierarchies.

Layer 5 — System Dynamics (Props 22–30)
Emergence (22), equilibrium (23), inherent relations (24), cohesion (25), constitutive relations (26), hierarchy (27), temporal evolution (28), dependencies (29), and context (30).

Layer 6 — Variability and Structure (Props 31–40)
Internal manifestation (31), RS interactions (32–33), variability of attributes and perspective (34–37), transitivity (38), redundancy (39), and equivalence (40).

Layer 7 — Resilience and Goals (Props 41–52)
Resilience/entropy (41–42), semantics (43–44), goals (45–47), reconciliation (48–51), and system resilience (52).

All source code, proofs, and comprehensive documentation are freely available at github.com/relationalexistence/UCF-GUTT. This represents not speculative philosophy but rigorous, machine-verified foundations for understanding reality as fundamentally relational.