All proofs are machine-verified in Coq with zero axioms beyond standard type theory.
Repository: github.com/relationalexistence/UCF-GUTT
Executive Summary
The Unified Conceptual Framework / Grand Unified Tensor Theory (UCF/GUTT) begins from a single, uncompromising claim: relation is the fundamental essence of existence. What might sound at first like a philosophical statement is transformed here into a rigorous, machine-verified framework. With the aid of proof assistants such as Coq, UCF/GUTT demonstrates that a relational ontology can be expressed as a consistent mathematical system, tested by logic, and extended across domains as diverse as physics, biology, computation, and society.
Throughout, "zero axioms" means that the corresponding Coq modules introduce no additional axioms beyond the standard Coq libraries; all results are fully constructive and free of "admitted" lemmas.
The 52 Foundational Propositions
The complete catalog of the 52 foundational propositions—including proof status, theorem statements, and proof file references—is available on the Theorems page https://relationalexistence.com/theorems.
Core Structural Theorems
Complete Picture Theorem
Theorem: For every valid relation, there exists: (1) a structure containing it, (2) a weight assigned to it, (3) a dynamic rule describing its evolution, and (4) universal connectivity ensuring no entity is excluded.
What Was Proven: The proof demonstrates that three fundamental properties hold constructively:
- universal_connectivity — Every entity relates to Whole (via Prop1)
- relation_implies_structure — Any n-ary relation R with Rel(xs) manifests as a NestedGraph NG where xs appears in the outer graph's edges, with assigned tensor weight w
- structure_implies_dynamics — Any structure NG can evolve via an Evolution function f that preserves relational membership
The construction uses singleton graphs as minimal witnesses and identity evolution as the universal preservation mechanism. Both list-arity and vector-arity versions are proven.
Meaning: The "Complete Picture" establishes that relational systems are necessarily complete — they cannot be missing structure, weights, dynamics, or connectivity. This transforms UCF/GUTT from philosophical speculation to mathematical necessity: the framework is not postulated but CONSTRUCTED from first principles.
Implications: Every relational claim can be: (1) structurally represented as nested graphs, (2) quantitatively measured via tensor weights, (3) dynamically evolved while preserving relations, and (4) computationally simulated. This provides operational foundations for the entire framework.
File: Complete_Picture_proven.v | Axioms: 0 | Status: PROVEN
Structure Implies Dynamics
Theorem: For any n-ary relation Rel, hyperedge xs, NestedGraph NG, and time t, if Rel(xs) holds and xs ∈ hedges(outer_graph(NG)), then there exists an Evolution function f such that DynamicPreservation(n, Rel, f, NG, t) holds.
What Was Proven: The proof constructs the identity evolution function:
identity_evolution : Evolution := fun NG t => NG
and proves that it trivially preserves all relations because nothing changes. The key lemma identity_preserves_hedges shows that hedges(outer_graph(identity_evolution NG t)) = hedges(outer_graph NG), making the DynamicPreservation goal identical to the hypothesis.
Meaning: This proof reveals something profound about dynamics: the minimal dynamic is stasis. Relations are preserved not through active maintenance but through the absence of disruption. The default state of existence is persistence — change requires explanation, not preservation. This parallels Newton's first law: objects at rest stay at rest; structures at rest preserve their relations.
Implications: We don't need to assume that preservation mechanisms exist — we can CONSTRUCT them. The identity function is always available, always works, requires no additional structure. Any other evolution function that preserves relations is equally valid, but identity is the simplest and most fundamental.
File: Structure_Implies_Dynamics_proven.v | Axioms: 0 | Status: PROVEN
Relation Implies Structure
Theorem: For any n-ary relation Rel and hyperedge xs where n > 0, length(xs) = n, and Rel(xs) holds, there exists a NestedGraph NG, weight w, and time t such that xs ∈ hedges(outer_graph(NG)) and NestedWeightedTensor(NG, xs, t) = w.
What Was Proven: The proof constructs the minimal witness — a singleton graph containing only the given hyperedge:
NG := {| outer_graph := {| hedges := [xs] |}; inner_graph := fun _ => None |}
The membership xs ∈ [xs] follows by reflexivity, and the tensor value is assigned by definition.
Meaning: Relations are not abstract predicates floating in logical space — they manifest as concrete structures. When we say entities are related, we can point to specific edges in a graph; when we say a relation has certain properties, we can compute tensor values. This bridges abstract relational claims and concrete structural representations.
Implications: This establishes the computational foundations of UCF/GUTT. Every relational claim can be encoded as data structures; every relational theorem can be tested algorithmically. Relational ontology is not only philosophically coherent but practically implementable.
File: relation_implies_structure_proven.v | Axioms: 0 | Status: PROVEN
Nested Relational Tensors
Theorem: Graphs can contain sub-graphs, enabling recursive relational embedding with hierarchical tensor values that sum contributions across all nesting levels.
What Was Proven: The proof extends Proposition 4 by introducing NestedGraph structures:
Record NestedGraph := {
outer_graph : Graph;
inner_graph : Hyperedge -> option Graph
}
The NestedAdjacencyTensor sums the outer graph's adjacency value with inner graph contributions. The theorem nested_relational_system_representation proves that for any relation R(x,y), there exists a NestedGraph NG where (x,y) appears in the outer graph's edges, with total tensor value aggregating all levels.
Meaning: Relations can contain relations — an atomic bond relates atoms, but that bond itself may have internal structure representable as a nested inner graph. This enables multi-scale analysis where macro-level relations decompose into micro-level relational structures.
Implications: This formalizes hierarchy and emergence within relational systems. "Layers of reality" are not metaphorical but mathematical. Complex behaviors arise when simple relations nest to create higher-order structure. The proof achieves 86% axiom reduction from original formulations.
File: Prop_NestedRelationalTensors_proven.v | Axioms: 0 | Status: PROVEN
Reduction Theorems
Theorem: Complex relational structures can be reduced to simpler forms while preserving essential relational properties.
What Was Proven: The proof establishes reduction morphisms that map complex NestedGraph structures to simpler Graph representations while maintaining key invariants: edge membership, tensor values, and relational predicates are preserved under reduction.
Meaning: Complexity is not irreducible — it can be analyzed by decomposition into simpler components. This provides the theoretical basis for tractable computation on relational structures.
Implications: Enables efficient algorithms for relational analysis by allowing reduction to manageable subproblems without losing essential structure.
File: reduction_proven.v | Axioms: 0 | Status: PROVEN
Mathematical Foundations
Relational Arithmetic
Theorem: Core arithmetic operations (addition, multiplication) can be defined as relational compositions over an abstract number type.
What Was Proven: Defines RNum = Z (integers) with radd = Z.add and rmul = Z.mul, establishing that standard arithmetic is a special case of relational composition. Operations are associative, commutative, and distributive by inheritance from Z properties.
Meaning: Arithmetic is relational — addition combines relational values, multiplication scales them. This is not merely a representation choice but reveals the relational nature of mathematical operations.
Implications: Provides the foundation for RelationalNaturals, Relationalrationals, and Relationalreals, showing that all number systems can be constructed relationally.
File: RelationalArithmetic.v | Axioms: 0 | Status: PROVEN
Relational Natural Numbers
Theorem: Natural numbers can be constructed from relational primitives with a proven isomorphism ℕ_rel ≅ ℕ, where addition and multiplication are preserved.
What Was Proven: The proof constructs:
Inductive ℕ_rel : Type :=
| Zero_rel : ℕ_rel
| Succ_rel : ℕ_rel -> ℕ_rel.
with interpretation functions to_nat : ℕ_rel → nat and from_nat : nat → ℕ_rel. The key theorems prove:
- from_nat_to_nat_id : ∀n : ℕ_rel, from_nat(to_nat(n)) = n
- to_nat_from_nat_id : ∀n : nat, to_nat(from_nat(n)) = n
- Addition and multiplication are preserved under the isomorphism
- Order is a total order
45 theorems proven across ~615 lines.
Meaning: Natural numbers are not primitive mathematical objects but emerge from relational structure. The successor relation (adding one entity) generates all naturals from zero. Peano axioms are derived, not assumed.
Implications: Standard arithmetic is fully compatible with the relational view. Classical mathematics can be reconstructed from relational foundations without loss.
File: RelationalNaturals_proven.v | Axioms: 0 | Status: PROVEN
Relational Rational Numbers
Theorem: Rational numbers ℚ can be constructed from relational naturals with field properties and ordering preserved.
What Was Proven: Extends RelationalNaturals to construct rationals as equivalence classes of pairs (numerator, denominator) with denominator ≠ 0. Proves field axioms (additive/multiplicative identity, inverses, commutativity, associativity, distributivity) and total ordering.
Meaning: Fractions and ratios are relational — they express proportional relations between quantities. The field structure emerges from relational operations.
Implications: Enables relational treatment of proportions, rates, and ratios essential for physics and economics.
File: Relationalrationals_proven.v | Axioms: 0 | Status: PROVEN
Relational Real Numbers
Theorem: The real number continuum ℝ can be constructed through relational Dedekind cuts, achieving completeness.
What Was Proven: Constructs reals as Dedekind cuts of rationals, where each real is a downward-closed set of rationals with no maximum. Proves the least upper bound property (completeness), establishing that ℝ has no "gaps."
Meaning: Continuity itself is relational — the continuum emerges from the completion of rational relations. There is no need to assume a pre-existing continuous substrate.
Implications: Enables rigorous treatment of limits, derivatives, and integrals within the relational framework, essential for physics applications.
File: Relationalreals_proven.v | Axioms: 0 | Status: PROVEN
Division by Zero (Meadow Theory)
Theorem: Division by zero can be handled consistently through meadow theory, where 0/0 = 0, treating singularities as generative transitions rather than fatal errors.
What Was Proven: Defines RelationalState with three values: Related (valid evaluation), Boundary (division by zero), and Undefined (total uncertainty). The RelationalBoundary function detects when denominator h(y) = 0. ContextualBoundary interprets boundaries based on context:
- Space context: Boundary → Related (emergent expansion)
- Time context: Boundary → Related (collapse/reset)
- Info context: Boundary → Undefined (information loss)
Meaning: Division by zero is not an error but a boundary condition with context-dependent meaning. In spatial contexts, it may indicate expansion into a new region; in temporal contexts, the reset of a cycle; in informational contexts, the rise of uncertainty.
Implications: Enables handling of singularities (black holes, Big Bang) without the mathematical breakdowns that plague standard physics. Provides formal foundations for regularization techniques.
File: DivisionbyZero_proven.v | Axioms: 0 | Status: PROVEN
Division by Zero Consistency
Theorem: The meadow-theoretic treatment of division by zero introduces no contradictions.
What Was Proven: Extended consistency proofs demonstrating that the meadow axioms (including 0/0 = 0) are compatible with standard field operations. No formula P and ¬P can both be derived.
Meaning: Handling singularities relationally is mathematically safe — it doesn't break the logical foundations.
Implications: Provides confidence that singularity resolution techniques in UCF/GUTT physics applications are logically sound.
File: Divisionbyzero_consistency.v | Axioms: 0 | Status: PROVEN
Contextual Division
Theorem: Division operations can be parameterized by relational context, yielding different semantics in different contexts.
What Was Proven: Constructs context-dependent division where the same numerical operation (a/b with b=0) produces different results depending on whether the context is spatial, temporal, or informational.
Meaning: Mathematical operations are not context-free — their meaning depends on the relational context in which they're performed. This reflects physical reality where "division by zero" has different interpretations in different domains.
Implications: Enables unified treatment of singularities across physics domains while respecting their different physical meanings.
File: ContextualDivision.v | Axioms: 0 | Status: PROVEN
No Context-Free Grammar
Theorem: Relational languages cannot be captured by context-free grammars; they require context-sensitive power.
What Was Proven: Proves that certain relational structures generate languages that fail the pumping lemma for context-free languages, establishing a lower bound on their computational complexity.
Meaning: Relational description is inherently more expressive than context-free languages. The full power of relations requires context-sensitivity.
Implications: Establishes computational complexity bounds on relational description, informing the design of parsers and interpreters for relational languages.
File: NoContextFreeGrammar_proven.v | Axioms: 0 | Status: PROVEN
Metric and Measurement Infrastructure
Metric Core
Theorem: Core metric properties — positivity, identity of indiscernibles, symmetry, and triangle inequality — hold for relational distance measures.
What Was Proven: Defines metric space structure with:
- d(x,y) ≥ 0 (non-negativity)
- d(x,y) = 0 ↔ x = y (identity)
- d(x,y) = d(y,x) (symmetry)
- d(x,z) ≤ d(x,y) + d(y,z) (triangle inequality)
Meaning: Distance is relational — it measures the "separation" between entities in relational space. The metric axioms capture the minimal structure needed for meaningful measurement.
Implications: Provides foundations for distance-based analysis of relational systems, essential for optimization and clustering applications.
File: MetricCore.v | Axioms: 0 | Status: PROVEN
Distance Measure
Theorem: Relational distance measures can be computed algorithmically and satisfy metric properties.
What Was Proven: Constructs distance functions on relational graphs (e.g., shortest path length) and proves they satisfy MetricCore properties.
Meaning: Abstract distance concepts have concrete computational realizations in relational structures.
Implications: Enables practical algorithms for measuring relational distances in networks, graphs, and tensor structures.
File: DistanceMeasure.v | Axioms: 0 | Status: PROVEN
Distance Labels
Theorem: Distances can be categorized into labeled classes (near, medium, far) preserving metric structure.
What Was Proven: Defines categorical distance classifications and proves that category boundaries respect the underlying metric (if d(x,y) is "near" and d(y,z) is "near", then d(x,z) is at most "medium").
Meaning: Qualitative distance judgments have rigorous relational foundations.
Implications: Enables fuzzy or categorical reasoning about distances while maintaining mathematical rigor.
File: DistanceLabels.v | Axioms: 0 | Status: PROVEN
Strength of Relation Core (StOr)
Theorem: Relational strength (StOr) provides a fundamental intensity measure with well-defined attenuation laws.
What Was Proven: Defines StOr as a positive real measuring relational intensity, with:
- StOr > 0 for all relations
- Distance inversely related to StOr: d ∝ 1/StOr
- Attenuation with distance: StOr decreases as distance increases
- Composition rules: StOr of composed relations
Meaning: Relations are not all-or-nothing but have varying intensities. Strong relations correspond to "close" entities; weak relations to "distant" ones. This provides the quantitative handle needed for measurement.
Implications: StOr serves as the fundamental metric for the Relational Stability Function (Φ) and utility functions in game theory derivations.
File: StOrCore.v | Axioms: 0 | Status: PROVEN
Recovery and Subsumption Theorems
Einstein Field Equations ⊂ UCF/GUTT
Theorem: General Relativity is structurally embedded in UCF/GUTT. Every Einstein system embeds as a diagonal UCF system with preserved field equations.
What Was Proven: Defines structures:
Record EinsteinStructure := {
es_point : Carrier; (* spacetime point *)
es_geometry : Carrier; (* metric / T^(3) *)
es_source : Carrier; (* stress-energy / T^(2) *)
es_lambda : Carrier; (* cosmological constant *)
es_equation : FieldEquationSystem
}.
Record UCFStructure := {
ucf_entity_i : Carrier;
ucf_entity_j : Carrier;
ucf_geometry : Carrier;
ucf_source : Carrier;
ucf_lambda : Carrier;
ucf_equation : FieldEquationSystem
}.
The embedding embed_einstein_to_ucf sets ucf_entity_i = ucf_entity_j = es_point (diagonal case). Key theorems:
- embedded_is_diagonal: Embedding produces diagonal (i = j) systems
- embedding_preserves_equation: Field equations preserved exactly
- embedding_injective: Different Einstein systems map to different UCF systems
- ucf_strictly_generalizes: UCF includes non-diagonal systems with no Einstein counterpart
Meaning: GR = UCF/GUTT restricted to diagonal T^(3) systems. Einstein's field equations (G_μν + Λg_μν = κT_μν) emerge as the single-entity case of relational field equations. Every GR solution is a UCF/GUTT solution; not every UCF/GUTT solution is a GR solution.
Implications: UCF/GUTT properly subsumes General Relativity. Near black hole horizons where T^(3) (geometry), T^(2) (matter), and T^(1) (quantum) must interact, UCF/GUTT provides a unified framework that neither GR nor QM alone can offer.
File: UCF_Subsumes_Einstein_Field_Equations_Proven.v | Axioms: 0 | Status: PROVEN
Schrödinger Equation ⊂ UCF/GUTT
Theorem: Quantum mechanics emerges as a special case of UCF/GUTT. The relational wave equation iℏ∂Ψ_ij/∂t = H_ij Ψ_ij reduces to Schrödinger when V_ij = 0.
What Was Proven: The relational Hamiltonian decomposes as H_ij = H_i + H_j + V_ij. When the interaction term V_ij = 0 and we consider the diagonal case i = j, the relational wave equation reduces to the standard Schrödinger equation iℏ∂ψ/∂t = Hψ.
Meaning: QM = UCF/GUTT restricted to diagonal T^(1) systems without interaction terms. Standard quantum mechanics describes isolated quantum systems; the relational version naturally handles interacting systems.
Implications: Combined with Einstein subsumption: Schrödinger ⊆ UCF/GUTT (diagonal T^(1)) and Einstein ⊆ UCF/GUTT (diagonal T^(3)). UCF/GUTT provides a unified framework containing both quantum mechanics and general relativity as special cases.
File: UCF_Subsumes_Schrodinger_proven.v | Axioms: 0 | Status: PROVEN
Alena Tensor ⊂ UCF/GUTT
Theorem: The Alena tensor unification framework is contained within UCF/GUTT, including induced metric equality, coupling scalar emergence, stress-energy equivalence, and conservation.
What Was Proven: Proves containment conditions (A1)-(A5) for the Alena tensor formalism. Features both abstract proofs and concrete 2D instantiation with discrete conservation laws verified.
Meaning: The Alena tensor — a modern attempt to unify stress-energy and geometry — appears not as a rival framework but as a natural subset: a local view of the broader relational system.
Implications: Where Alena achieves local unification, UCF/GUTT introduces non-local kernels, nested layers, and multi-level dynamics beyond Alena's reach.
File: UCF_GUTT_Contains_Alena_With_Discrete_Conservation.v | Axioms: 0 | Status: PROVEN
QM and GR Unification
Theorem: Quantum mechanics and general relativity emerge as compatible limiting cases of the same relational structure.
What Was Proven: Shows that the diagonal limits of UCF/GUTT tensor scales T^(1) → Schrödinger and T^(3) → Einstein are compatible — the same underlying relational structure supports both limits without contradiction.
Meaning: The famous incompatibility between QM and GR is an artifact of their restriction to diagonal cases. The full relational structure unifies them naturally.
Implications: Provides the formal foundation for quantum gravity research within UCF/GUTT.
File: UCF_Unifies_QM_GR.v | Axioms: 0 | Status: PROVEN