alphaInv_derived_eq_formula
The derived inverse fine-structure constant from cubic ledger geometry equals 4 pi times 11 minus the gap weight plus the curvature correction 103 over 102 pi to the fifth. Researchers deriving fundamental constants from discrete geometry cite this result to close the alpha derivation. The proof is a single simplification that substitutes the established equalities for the geometric seed and curvature term.
claim$alpha^{-1}_{derived} = 4 pi cdot 11 - (f_{gap} + (-103/(102 pi^5)))$ where $f_{gap}$ is the gap weight from the eight-tick projection and the curvature term follows from voxel seam topology.
background
The module derives alpha inverse from the geometry of the cubic ledger in three dimensions. The fundamental cell is the cube Q3 with 8 vertices, 12 edges and 6 faces. During one atomic tick one edge is active while the remaining 11 passive edges couple to the vacuum geometry, yielding the geometric seed 4 pi times 11 via the solid-angle factor. The curvature term arises from the seam topology: numerator 103 from Euler closure plus one, denominator 102 from 6 faces times 17 wallpaper groups. The gap weight f_gap equals w8 from the eight-tick octave times log phi.
proof idea
The term proof applies simp to unfold alphaInv_derived and substitute the two upstream equalities geometric_seed_eq and curvature_term_eq. No further tactics are required once those rewrites are in scope.
why it matters in Recognition Science
This theorem completes the first-principles derivation of alpha inverse inside the Recognition Science framework, confirming that the geometric seed from D=3 cube geometry, the seam-derived curvature correction, and the eight-tick gap weight assemble into the target formula. It directly implements the module's main result that 4 pi originates from Gauss-Bonnet on vertex deficits of Q3 and that the passive-edge count is 11. The declaration sits at the end of the alpha assembly sequence and supplies the explicit closed-form expression referenced by Constants.Alpha.
scope and limits
- Does not evaluate the numerical value of alpha inverse.
- Does not derive f_gap from first principles inside this module.
- Does not incorporate higher-order terms beyond the listed curvature correction.
- Does not address experimental bounds or measurement precision.
formal statement (Lean)
247theorem alphaInv_derived_eq_formula :
248 alphaInv_derived = 4 * Real.pi * 11 - (f_gap + (-(103 : ℝ) / (102 * Real.pi ^ 5))) := by
proof body
Term-mode proof.
249 simp only [alphaInv_derived, geometric_seed_eq, curvature_term_eq]
250
251/-! ## Part 9: Summary Theorems (Closing the Gap) -/
252
253/-- The number 11 is not arbitrary: it is the passive edge count of Q₃. -/