IndisputableMonolith.Mathematics.GraphTheoryDepthFromRS
This module formalizes graph theory results derived from Recognition Science for the Q3 graph. It verifies Euler's formula V - E + F = 2 with explicit counts V=8, E=12, F=6, plus related invariants such as chromatic number. The structure consists of targeted lemmas on vertex/edge/face counts and bipartiteness. Researchers linking discrete graphs to the RS forcing chain (T8 D=3) would cite these for dimensional consistency checks.
claimFor the Q_3 graph, $V - E + F = 2$ where $V(Q_3)=8$, $E(Q_3)=12$, $F(Q_3)=6$. The graph is bipartite with chromatic number 2.
background
The module imports Mathlib for graph primitives and centers on Q3, the cube graph realizing three spatial dimensions. Sibling declarations define vertex/edge/face counts, Euler characteristic, and bipartiteness. The module doc-comment states the explicit Euler relation Q₃ Euler: V - E + F = 8 - 12 + 6 = 2. This sits inside the broader RS derivation of graph depth from the forcing chain T0-T8.
proof idea
The module organizes its content as a sequence of concrete lemmas rather than a single proof. graphTheoremCount and q3Vertices/q3Edges/q3Faces supply the base counts. q3EulerChar assembles the characteristic while q3Euler_eq_2 reduces it to equality with 2. q3Chromatic_bipartite handles the chromatic result via bipartiteness. Each step is a direct verification.
why it matters in Recognition Science
The module supplies the concrete Q3 instance that anchors graph-theoretic depth in Recognition Science. It feeds the certification object GraphTheoryDepthCert by confirming Euler characteristic 2 and bipartiteness, consistent with T8 forcing of D=3. No downstream uses are recorded, so its role is to close the discrete-to-continuous link at the level of the eight-tick octave.
scope and limits
- Does not derive Q3 from the J-cost or RCL axioms.
- Does not treat graphs beyond the cube Q3.
- Does not address non-bipartite chromatic numbers.
- Does not connect to mass formulas or alpha band.