geometric_seed_eq_solidAngle_times_11
plain-language theorem explainer
The geometric seed in the alpha derivation equals eleven times the solid angle of the unit sphere in three dimensions. Researchers deriving the fine-structure constant from Recognition Science would cite this to confirm that the 4π factor is fixed by isotropic D=3 geometry. The proof is a direct unfolding that reduces the multiplicative factor to the passive edge count of the 3-cube via reflexivity, followed by simplification and ring normalization.
Claim. Let $S$ be the solid angle of the unit sphere in three dimensions and let $G$ be the geometric seed; then $G = S × 11$.
background
In the Solid Angle Exclusivity module the geometric seed is defined as the product of the solid angle of the three-dimensional unit sphere and the passive edge count. The solid angle is the surface measure $4π$, which is the unique isotropic measure on the sphere in D=3. The passive edge count is obtained from the hypercube geometry: for dimension d=3 the cube has 12 edges, one of which is active per tick, leaving 11 passive edges that dress the interaction.
proof idea
The proof unfolds the definitions of geometric_seed and solidAngle. It then proves that geometric_seed_factor equals 11 by unfolding passive_field_edges, cube_edges, and active_edges_per_tick, which reduces immediately to reflexivity. The equality is completed by simplification and algebraic rearrangement.
why it matters
This theorem confirms that the geometric seed factor of 11 arises directly from the passive edge count in the D=3 hypercube, feeding into the alpha derivation. It supports the framework claim that 4π is the unique solid angle for isotropic coupling in three dimensions, as required by the eight-tick octave and D=3 forced by the forcing chain.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.