curvature_term_eq
plain-language theorem explainer
Curvature term in the alpha derivation from cubic geometry equals -103 over 102 pi to the fifth. Physicists seeking first-principles values for fundamental constants cite this for the explicit correction factor. The proof is a direct term simplification that substitutes the seam numerator and denominator at D equals 3.
Claim. The curvature correction term arising from voxel seam topology in the cubic lattice equals $ -103/(102 pi^5) $.
background
The Alpha Derivation module constructs the inverse fine-structure constant from the cubic ledger geometry forced by the Meta-Principle. It begins with the cube Q3 having 8 vertices, 12 edges and 6 faces in D=3. One edge is active per tick, leaving 11 passive edges that contribute to the geometric seed of 4 pi times 11. The curvature term arises in Part 7 from the voxel seam topology. Seam numerator equals 103 and seam denominator equals 102, where 102 comes from 6 faces times 17 wallpaper groups and the extra 1 from Euler characteristic. The term is defined as minus the ratio of these integers over pi to the fifth power to account for the five-dimensional measure. Upstream theorems seam_numerator_at_D3 and seam_denominator_at_D3 fix these integers by native_decide, while curvature_fraction_num and curvature_fraction_den alias them.
proof idea
The term proof applies simp only to the curvature_term definition, the fraction num and den, the seam theorems at D3, and the natural number cast. This reduces the left side directly to the right side constant.
why it matters
It supplies the curvature correction value used in the closing theorem alphaInv_derived_eq_formula, which equates the derived alpha inverse to the target expression involving geometric seed minus f gap minus curvature term. This completes the curvature assembly in the module and supports the Recognition Science prediction of alpha inverse near 137.03. It instantiates the D=3 spatial dimensions from the forcing chain.
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