Constants Alpha Derivation Per Face Solid Angle Eq

The per-face solid angle

The solid angle subtended by a cube at its center is a classical quantity: 4π steradians, the full sphere. The declaration per_face_solid_angle in the Recognition Science library establishes the direct consequence that each of the cube's six faces subtends one sixth of that total, or 2π/3 steradians. This is a theorem about Euclidean geometry, proved by dividing the total solid angle by the number of faces. The proof relies on the Gauss-Bonnet theorem, which links the total curvature of a surface to its topology.

The library defines the cube in three dimensions with 8 vertices, 12 edges, and 6 faces. The total solid angle is computed as the number of vertices times the angular deficit at each vertex, where the deficit is the difference between a full turn (2π) and the sum of the three right angles meeting at a corner. This gives 4π, and the per-face value follows by division. The declaration is a theorem in the machine-checked library, meaning its statement is verified by the proof assistant's kernel.

In Recognition Science, this geometric fact was once used as a step toward assembling a seed for the inverse fine-structure constant, 1/α ≈ 137.036. The seed combined the total solid angle 4π with a count of 11 passive edges, giving 4π·11. The library now marks this identification as retired. A kernel proof shows the count 11 does not match the gauge degrees of freedom, and a no-go theorem demonstrates that no normalization-blind condition can pin the coupling at all. The exact value of 1/α is a boundary datum, not a derived consequence of the framework.

What the declaration establishes, then, is a clean geometric fact: the per-face solid angle of a cube is exactly 2π/3. What it does not establish is any connection to the fine-structure constant. The geometric seed built from this quantity is convicted, not open, and the derivation question is closed by the no-go result. The page is a record of a retired identification, not an active derivation.

/-- Per-face solid angle: by cubic symmetry, each of the 6 faces
    subtends equal solid angle 4π/6 = 2π/3 from the cube center. -/
noncomputable def per_face_solid_angle : ℝ :=
  solid_angle_Q3 / (cube_faces D : ℝ)

theorem solid_angle_Q3_eq : solid_angle_Q3 = 4 * Real.pi := gauss_bonnet_Q3

theorem solid_angle_Q3_eq : solid_angle_Q3 = 4 * Real.pi := gauss_bonnet_Q3

/-- The alpha seed factorizes into solid angle × passive channels,
    both derived from Q₃ cube geometry with zero imported constants. -/
theorem alpha_seed_structural :
    geometric_seed = solid_angle_Q3 * (passive_field_edges D : ℝ) := rfl

What this page does not claim

The per-face solid angle declaration does not derive the fine-structure constant. The count 11 does not represent gauge degrees of freedom. The exact value 1/α is not a first-principles consequence of the framework.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/AlphaDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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