candidate_3
plain-language theorem explainer
Defines the third geometric candidate for the fine-structure correction δ₂ as 1/(102(π²−1)), i.e. active-edge count over seam denominator times reduced π². Anyone comparing cube-geometry ansätze against the ~0.00110 gap between α⁻¹_RS and CODATA would cite it. The body is a pure closed-form definition with A=1 and seam_denominator=102 substituted.
Claim. The third candidate correction is the real number $1/(102(\pi^2-1))$, equivalently $A/(\mathrm{seam\_denom}\cdot(\pi^2-1))$ with active-edge count $A=1$ and three-dimensional seam denominator $6\times 17=102$. The factor $(\pi^2-1)$ is the reduced-$\pi^2$ (non-spherical) curvature weight.
background
The module studies the residual ~8 ppm mismatch between the Recognition Science inverse fine-structure constant $\alpha^{-1}_{\mathrm{RS}}\approx 137.0349$ and CODATA $\alpha^{-1}\approx 137.035999$, which requires a positive additive correction $\delta_2\approx +0.00110$. Admissible corrections must be built from counting-layer integers and the fixed transcendentals $\pi,\varphi$, stay $O(10^{-3})$, introduce no free parameters, and admit a cube-geometry reading.
The seam denominator is the base curvature normalization: faces times wallpaper groups, equal to $6\times 17=102$ in $D=3$. The active-edge count per tick is $A=1$. Candidate 3 packages these with the reduced factor $(\pi^2-1)$ rather than bare $\pi^2$, interpreted as active-edge coupling through curvature channels with a non-spherical correction.
proof idea
Not a proof: a one-line real definition. Substitute $A=1$ and $\mathrm{seam_denominator}(3)=102$ into the schematic form $A/(\mathrm{seam_denominator}\cdot(\pi^2-1))$ to obtain $1/(102(\pi^2-1))$. No lemmas are invoked.
why it matters
Sits in the candidate battery (alongside candidates 1, 2, 4) that the Alpha Correction Analysis uses to hunt a first-principles source for $\delta_2$. The $(\pi^2-1)$ factor is the distinctive geometric claim: curvature channels on the cube are not pure spherical $\pi^2$ modes. Downstream the module compares numerical proximity of each candidate to the required $+0.00110$ window; this declaration supplies the third expression in that comparison. It does not yet close the $\alpha$ band $(137.030,137.039)$ by itself; it is raw material for that verification step.
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