Proof Graphs and Algorithm Capsules: A Corpus Study of Diagonalization Proofs from Cantor to Gödel to Goodstein

Mathematics encompasses three fundamentally distinct object types — algorithms, axiomatic systems, and proofs. These are conventionally treated as categorically separate in mathematical practice, education, and knowledge representation. We demonstrate that all three can be expressed as labeled directed graphs using Mermaid Markdown. This unified representation reveals structural properties that conventional prose and static diagrams obscure. The central empirical finding is the regularity of algorithm capsules — embedded procedural substructures within proofs — across mathematically distant domains. Analysis of the Mathematics Database corpus identifies a diagonalization family spanning three entries: Cantor diagonal proofs, the Gödel First Incompleteness proof graph, and the Kirby–Paris/Goodstein independence result. In each case the algorithm capsule is not incidental to the argument but is its structural core. This family resemblance is visible and measurable in the graph representation. It is not accessible from prose alone. The representation is implemented as three graph types. Algorithmic flowcharts capture procedural structure. Axiomatic dependency graphs represent logical dependency among axioms, definitions, and theorems. Proof graphs — a hybrid form introduced here — encode justification structure using a domain-specific eight-role node vocabulary: source, assumption, construction, assertion, inference, algorithm capsule, contradiction, and conclusion. Together the three graph types constitute the Mathematics Database, a publicly accessible machine-readable corpus spanning classical geometry, number theory, algebra, set theory, mathematical logic, and theoretical computer science. The methodology is a domain-specific application and extension of the Programming Framework, a general method for LLM-assisted process visualization. The mathematics case demonstrates that the framework's core claim extends from procedural processes to logical and justificatory structures: that formal structure is recoverable from natural language descriptions of complex systems, and is meaningful, measurable, and comparable once recovered.