IndisputableMonolith.Mathematics.GraphTheoryDepthFromRS
This module derives graph-theoretic properties for the Q3 structure within the Recognition Science framework. Researchers working on discrete geometry or the forcing chain to D=3 would cite it for explicit counts and relations. The module consists of sibling definitions and theorems that compute vertex, edge, and face numbers then verify the Euler relation by direct substitution.
claimThe central object is the Euler characteristic of the Q_3 graph: $V - E + F = 2$, realized by the concrete values $V=8$, $E=12$, $F=6$.
background
The module belongs to the Mathematics domain of Recognition Science, which obtains all physics from a single functional equation through the T0-T8 forcing chain. It imports Mathlib and introduces the sibling objects GraphTheorem, q3Vertices, q3Edges, q3Faces, q3EulerChar, q3Euler_eq_2, q3ChromaticNumber and GraphTheoryDepthCert to encode the three-dimensional graph that realizes T8.
The supplied module doc comment states the explicit computation Q₃ Euler: V - E + F = 8 - 12 + 6 = 2, providing the concrete numbers that close the graph-theoretic step.
proof idea
The module is a collection of small definitions and equality theorems rather than a single complex proof. Each count (vertices, edges, faces) is introduced by definition; the Euler relation is obtained by direct substitution into the characteristic formula, with no tactic scripts beyond reflexivity or rfl.
why it matters in Recognition Science
The module supplies the graph-theoretic content required by GraphTheoryDepthCert and thereby supports the T8 derivation of D=3 spatial dimensions in the forcing chain. It furnishes the concrete Euler characteristic that any downstream result invoking the eight-tick octave or the three-dimensional lattice must presuppose.
scope and limits
- Does not derive the Q3 graph from the J-cost or Recognition Composition Law.
- Does not treat arbitrary graphs or higher-dimensional polytopes.
- Does not compute physical constants or mass ladders from the graph.
- Does not prove chromatic-number results beyond the bipartite case listed among siblings.