Theoretical Cartography 1: Weighted Hypergraph Formalization of Theoretical Space—Conservation Flow, Cohesive Closure, and Proof Relevance

A mutually connected theoretical system can itself become a mathematical object.This paper formalizes theoretical space as an axiom-annotated weighted hypergraphT = (V,E,w,A): nodes are concepts, definitions, theorems, and conjectures; hyperedges are derivation chains; weights are Hartley information; and annotations recordthe axiom subset on which each node depends. The formalization rests on two axioms—conservation of information gain and loss and computability. On this basis, we give anedge-by-edge criterion for conservation admissibility, constructively prove that everyfinite acyclic derivation hypergraph admits at least one conservation-admissible weight,establish the three axioms—extensivity, idempotence, and monotonicity—for the cohesive closure operator, and prove that closed sets form a lattice. We further introduce aproof layer P, distinguish forgetful loss within a single hyperedge from conclusion-levelredundancy among multiple hyperedges sharing a conclusion, and point out that thetwo are determined by data at different layers and cannot substitute for each other.All structural assertions are verified by exhaustive enumeration and linear algebra ona 12-node running example, and the conservation ledger discriminates between twocandidate weight schemes. We close by registering four open problems: encoding dependence, root-layer conventions, higher-order rigorization of the proof layer, and scalestress testing.