seam_ratio_from_topology
plain-language theorem explainer
The seam ratio equals the ratio of the seam numerator (denominator plus Euler closure) to the seam denominator (cube faces times wallpaper groups) at three dimensions. Researchers deriving the fine structure constant correction in Recognition Science would cite this to fix the 103/102 topological factor. The proof is a direct unfolding of the seam ratio definition followed by rewriting with the specialized D=3 evaluations and norm casting.
Claim. The curvature integrand equals the ratio of the seam numerator to the seam denominator evaluated at three spatial dimensions, where the seam numerator is the seam denominator plus the Euler closure contribution and the seam denominator is the product of the number of cube faces and the number of wallpaper groups: $103/102$.
background
The Curvature Space Derivation module shows that the curvature correction integrates over a five-dimensional configuration space: three spatial dimensions forced by T8, one temporal dimension from the eight-tick cycle, and one dual-balance dimension from ledger conservation. Each angular integration contributes a factor of π, producing the π^5 volume in the denominator of δ_κ. The seam denominator is defined as cube_faces d times wallpaper_groups and equals 102 at D=3 by the theorem seam_denominator_at_D3. The seam numerator adds the Euler closure of 1 and equals 103 at D=3 by seam_numerator_at_D3. The seam ratio is introduced as the curvature integrand measuring topological stress between spherical and cubic boundaries per unit configuration-space volume.
proof idea
The proof unfolds the definition of seam_ratio, rewrites using the theorems seam_numerator_at_D3 and seam_denominator_at_D3 that specialize the general expressions to D=3, and applies norm_cast to equate the resulting real numbers.
why it matters
This theorem fixes the numerical prefactor 103/102 that enters the curvature correction δ_κ = -103/(102 π^5) appearing in the fine-structure formula α^{-1} = 4π·11 - f_gap - 103/(102π^5). It closes the topological accounting step that follows the eight-tick phase definition and the spatial-dimension forcing T8, supplying the concrete ratio required by the AlphaDerivation module. The result anchors the curvature term inside the Recognition Science constants derivation without introducing new hypotheses.
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