curvature_correction_derived
plain-language theorem explainer
This definition supplies the explicit curvature correction δ_κ as the negative seam ratio divided by the five-dimensional phase-space volume. Researchers deriving the fine-structure constant in Recognition Science cite it to insert the geometric mismatch term into the alpha formula. The expression is assembled directly from the seam primitives evaluated at D=3 and the forced configuration-space dimension of 5.
Claim. The curvature correction is defined by $δ_κ := -s_n / (s_d ⋅ π^5)$, where $s_n$ is the seam numerator (102 + 1 = 103) and $s_d$ is the seam denominator (6 × 17 = 102) for three spatial dimensions, with the exponent 5 arising from the effective dimension of the ledger configuration space.
background
In the Recognition framework the curvature correction measures the mismatch between the discrete cubic lattice and smooth spherical geometry. The seam denominator is the product of cube faces and wallpaper groups; for D=3 this equals 102. The seam numerator adds the Euler closure term, yielding 103. The configuration space for the integral has five effective dimensions: three spatial (forced by T8), one temporal from the eight-tick cycle, and one from the dual-balance conservation constraint. Each angular integration contributes a factor of π, producing the π^5 volume factor.
proof idea
The definition is a direct algebraic expression that inserts the seam numerator and denominator (evaluated at D=3) together with the configuration-space dimension into the ratio and prefixes a minus sign. No additional lemmas are invoked inside the definition; the numerical values and the exponent 5 are supplied by the referenced seam primitives and the configSpaceDim constant.
why it matters
This definition supplies the concrete term that appears in the alpha derivation as the curvature correction. It is invoked by the equality theorem curvature_correction_eq_formula, which confirms the numerical value -103/(102 π^5), and by the uniqueness results showing that only the exponent 5 matches the derived form. The construction rests on the forced three spatial dimensions (T8) and the eight-tick temporal structure, together with the five-dimensional ledger phase space required for the curvature integral. It closes the gap between the geometric seam ratio and the phase-space volume in the fine-structure formula.
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