Mathematics Graph Invariants From Config Dim Graph Invariant Count

Five invariants, one count

In graph theory, an invariant is a number or property that stays the same no matter how you draw the graph. The chromatic number, the fewest colors needed to color vertices so no two adjacent share a color, is the most famous. The clique number, the size of the largest set of mutually connected vertices, and the independence number, the largest set with no edges between them, are close cousins. Genus, the minimum number of holes a surface needs to hold the graph without edge crossings, and treewidth, a measure of how much a graph resembles a tree, round out a standard list of five.

These five appear together in a machine-checked library of formal theorems. The library defines a type called GraphInvariant with exactly those five constructors, then proves a theorem, graphInvariant_count, that the number of distinct invariants in this type is five. The proof is a direct computation, checked by the kernel, with no gaps and no added axioms. A separate certificate structure packages this fact for use elsewhere in the library.

In Recognition Science, this count is not an accident. The framework models physical structure as a discrete record of events, and its core result forces a specific combinatorial depth: five dimensions of configuration space. The library's declaration connects that depth to graph theory by showing that five canonical invariants, no more and no fewer, are exactly what the framework's counting principle licenses. The theorem itself is about the cardinality of a finite type; the interpretation as a physical claim is a separate step.

What the result does not claim is broader. It does not say these five invariants are the only ones that exist in mathematics, nor that they are the most important. It does not derive any property of a specific graph, like the chromatic number of the Petersen graph. It only establishes that, within this formal system, the named list of five invariants is complete and exactly five in number.

theorem graphInvariant_count : Fintype.card GraphInvariant = 5 := by decide

What this page does not claim

This does not claim the five invariants are the only ones in all of graph theory. This does not claim any specific graph's invariant values are derived here. This does not claim the physical interpretation of the count is proven; it is a framework model.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/GraphInvariantsFromConfigDim.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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