When can a connection be represented as a graph, and what survives aggregation?
Element · Relation · Boundary
Choose a finite set E of identified elements. For each relation type k, define Rₖ as a subset of E × E. Attributes A may hold dates, sources and states. This creates a directed labelled graph: a mathematical data structure, initially silent on whether recorded relations actually exist in the world.
Effect · Time · Feedback
Limited theorem: a finite graph and a partition P of its nodes define a quotient graph. Assume nonempty, disjoint blocks covering E. Proof: each e belongs to exactly one block, so q(e) is unique. Place an edge from U to V exactly when an original edge connects an element of U to one of V. This defines a unique subset of P × P, hence a relation and a graph. Only structural representability has been proved.
G = (E, R)
R′ = {(q(a), q(b)) : (a,b) ∈ R}
G′ = (P, R′)This preserves existence of connections between blocks, not automatically edge counts, weights, relation types, timing or causality. Types and weights need separate tested rules.
Limits of the claim
ADL order is not automatically a partial order. Feedback is not a logical contradiction. Recursive representation does not prove scale-independent natural laws. The construction uses established graph and relation ideas; its ADL value is a precise boundary on what is formally shown.
Test
For dynamics, test q(x) = q(y) for internally different states. If the same transition produces q(F(x)) ≠ q(F(y)), this summary lacks a unique autonomous update. This counterexample must not disappear by renaming order.
Source
Sources support the mathematical or technical foundations. Their connection to ADL is the interpretation disclosed here.