Abstract
What does it mean for something to exist within a web of relations? We answer this question by developing a rigorous mathematical framework in which existence emerges not as a primitive concept but as a gauge-invariant topological invariant of relational structure. Starting from five foundational axioms grounded in graph theory and gauge theory, we define (i) relations as edges carrying both scalar weights and group-valued transitions; (ii) natural aggregates ("fruits") as low-conductance clusters detectable via spectral clustering; (iii) boundary signatures ("doors") as internal singularities arising from anomalous external contact; and (iv) existence as a gauge-invariant triple ([A_∞], Σ, e) combining optimal gauge class, door locus, and residual energy. The framework yields three complementary topological readouts: intrinsic cohomology revealing internal holes; relative cohomology capturing aggregate-singularity interface via explicit matrix rank formulas; and multi-scale persistence barcode structure of singularities. We prove all three axes satisfy a long exact sequence, encode a gauge-invariant portrait of the system, and are computable via finite linear algebra. Validation on five- and ten-node examples with U(1) gauge structure confirms theoretical predictions across all three axes.
The mathematical companion to the RelationWorld research thread —
this paper formalises what it means for an aggregate to exist in a
purely relational setting. The answer, in short: existence is a
gauge-invariant topological invariant computable from the weighted
graph alone.
- Five axioms (A1–A5) sufficient for deriving all subsequent
definitions and theorems.
- Six definitions characterising Fruit, Door, Kernel, Existence,
and the connecting maps between them.
- Three-axis cohomological framework:
- Axis 1 — intrinsic fruit cohomology H*(K_α; 𝔽)
- Axis 2 — pair relative cohomology H*(K_α, L_α; 𝔽) with explicit
matrix rank formulas
- Axis 3 — persistence barcode on the singular locus Σ
- All three axes related by a long exact sequence (Theorem 2).
- Explicit polynomial-time algorithms for each axis.
Provides the formal backbone of the RelationWorld
notes and feeds directly into the perception-theory
strand. Related to the Cheeger-spectral and Yang–Mills-flow chapters
of the theory book.