passive_edges_eq
passive_edges_eq establishes that the passive edge count on the Q3 cube equals 11. Researchers assembling the alpha framework for fine-structure constant corrections cite this when certifying the cube combinatorics. The proof is a one-line reflexivity that follows from the definition of passive_edges as total edges minus active edges.
claimIn the Q_3 cube combinatorics the passive edge count satisfies $|E_{passive}| = 11$.
background
The module develops higher-order voxel-seam corrections to α^{-1} in Recognition Science. Q_3 is the structure from SpectralEmergence that forces SU(3)×SU(2)×U(1) gauge content, exactly three particle generations, and 24 chiral fermion flavors. Passive edges are defined as Q3_edges minus active_edges, with the total edge count supplied by the cube combinatorics lemmas in the same module.
proof idea
The proof is a one-line term that applies reflexivity directly to the definition passive_edges := Q3_edges - active_edges, which evaluates to 11 by the prior Q3_edges_eq and active_edges computations.
why it matters in Recognition Science
This supplies the passive edge count to the alphaFramework certificate that assembles the series α^{-1} = α_seed − f_gap + Σ δ_n. It fills the first-order correction slot in the module and connects to the D = 3 spatial dimensions and eight-tick octave from the forcing chain. The open question it touches is the explicit computation of δ_2.
scope and limits
- Does not derive active_edges or Q3_edges from first principles.
- Does not compute any δ_n for n > 1.
- Does not prove convergence of the full correction series.
- Does not incorporate the CODATA target as a hypothesis.
Lean usage
example : passive_edges = 11 := passive_edges_eqformal statement (Lean)
75theorem passive_edges_eq : passive_edges = 11 := rfl
proof body
Term-mode proof.
76
77/-- Number of wallpaper groups (Fedorov 1891). -/