curvature_term
plain-language theorem explainer
The curvature term supplies the correction -103/(102 π^5) subtracted in the first-principles derivation of the inverse fine-structure constant. Researchers assembling α from cubic ledger geometry cite this when closing the alpha assembly step. The definition is a direct algebraic combination of the seam numerator over the seam denominator scaled by π to the fifth power from the five-dimensional measure.
Claim. The curvature term is defined as $ -103/(102 π^5) $, where the exponent five arises from the integration measure over three spatial dimensions, one time dimension, and one dual-balance dimension.
background
The Alpha Derivation module constructs the fine-structure constant from the cubic unit cell Q_3 in three dimensions. This cell has twelve edges, with one active per recognition tick and eleven passive field edges. The seam numerator is 103 and the denominator is 102, built from six faces times seventeen wallpaper groups plus one for Euler characteristic closure. Upstream, curvature_fraction_num and curvature_fraction_den are the seam numerator and seam denominator evaluated at D=3. The π^5 factor originates from the five-dimensional integration measure consisting of three spatial, one temporal, and one dual-balance dimension.
proof idea
The definition directly substitutes the seam numerator 103 and denominator 102 into the negated ratio divided by π^5. It applies the prior definitions of curvature_fraction_num and curvature_fraction_den with no additional reduction steps.
why it matters
This term completes the curvature correction inside alphaInv_derived, which equals geometric_seed minus (f_gap plus curvature_term). It matches the independent curvature correction formula in the CurvatureSpaceDerivation module. Within the Recognition framework it realizes the D=3 geometry and supplies the five-dimensional measure contribution to the alpha band derivation.
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