geometric_seed
plain-language theorem explainer
The geometric seed is the product of the total solid angle of the cube boundary and the passive edge count from the D=3 ledger. Researchers deriving the fine-structure constant from cubic geometry cite this quantity as the initial factor in the alpha assembly. The definition is a direct multiplication of two upstream geometric quantities with no further reduction.
Claim. The geometric seed equals $4π$ times the number of passive field edges, where $4π$ is the total solid angle of the cube surface boundary and the passive count equals 11 for the three-dimensional case.
background
In the cubic ledger of Recognition Science the unit cell is Q₃ in D=3 dimensions. This cell has 12 edges; one is active per recognition tick, leaving 11 passive field edges. The boundary ∂Q₃ is topologically a sphere, so its total solid angle equals the Gauss-Bonnet total curvature 4π obtained from vertex angular deficits. The module derives α⁻¹ from this geometry without imported constants. Upstream, geometric_seed_factor is defined as passive_field_edges D and documented as the source of the factor 11. solid_angle_Q3 is defined as the total solid angle of ∂Q₃ and equals the Gauss-Bonnet curvature because ∂Q₃ ≅ S².
proof idea
This is a direct definition that multiplies solid_angle_Q3 by geometric_seed_factor. No lemmas are applied beyond the unfolding of the two upstream definitions; the body is a single multiplication expression.
why it matters
The definition supplies the leading term for alphaInv_derived and is invoked by the structural theorem alpha_seed_structural. It realizes the geometric seed required by the alpha derivation chain, incorporating the D=3 cube geometry and the 11 passive channels that arise from the eight-tick octave. The construction feeds the main result α⁻¹ = geometric_seed − (f_gap + curvature_term) and is cross-referenced in the cosmological constant derivation.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.