curvature_correction_eq_formula
plain-language theorem explainer
The theorem establishes that the curvature correction derived from the five-dimensional configuration space equals -103 over 102 pi to the fifth. Researchers deriving the fine-structure constant in Recognition Science cite it to fix the explicit geometric correction term. The proof is a one-line term reduction that unfolds the derived definition and substitutes the seam numerator and denominator values fixed at three spatial dimensions.
Claim. The curvature correction derived from the five-dimensional configuration space equals $ -103 / (102 π^5) $.
background
In the Recognition Science framework the curvature correction appears in the fine-structure constant derivation as α^{-1} = 4π·11 - f_gap - 103/(102π^5). The configuration space dimension is defined as five, arising from three spatial dimensions forced by the eight-tick octave, one temporal dimension from the cycle evolution, and one dual-balance dimension from conservation constraints. Each angular integration contributes a factor of π, producing the π^5 denominator. The upstream definition curvature_correction_derived expresses the quantity as the negative seam ratio divided by π raised to the configuration space dimension. The seam numerator and denominator at D=3 are fixed at 103 and 102 by direct computation in AlphaDerivation.
proof idea
The proof unfolds curvature_correction_derived and configSpaceDim, then rewrites using seam_numerator_at_D3 and seam_denominator_at_D3 to insert the explicit values 103 and 102. Reflexivity closes the equality.
why it matters
This equality connects the abstract derived form to the explicit expression used in AlphaDerivation.curvature_term. It supports downstream uniqueness results showing that only the exponent five matches the derived value, confirming the five-dimensional integration measure. The result reinforces the forcing of D=3 spatial dimensions and the 3+1+1 decomposition of the configuration space.
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