candidate_4
plain-language theorem explainer
Defines the fourth numerical candidate for the ~0.001 additive correction to α⁻¹_RS: (ln φ)²/(2·102). Analysts comparing cube-geometry expressions against the CODATA gap would cite it. The body is a pure real definition, not a proved identity.
Claim. The fourth candidate correction term is $\delta_4 = (\ln\varphi)^2 / (2 \cdot 102)$, where $\varphi$ is the golden ratio and $102$ is the seam denominator from the existing $\alpha^{-1}$ formula.
background
The module studies the residual gap between the Recognition Science inverse fine-structure value $\alpha^{-1}_{\mathrm{RS}} = 4\pi\cdot 11 - w_8\ln\varphi + 103/(102\pi^5) \approx 137.0349$ and CODATA $\alpha^{-1} \approx 137.035999$. The required additive shift is $\delta_2 \approx +0.00110$.
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 integer $102$ already appears as the seam denominator in the $103/(102\pi^5)$ term; $\varphi$ is the self-similar fixed point forced at T6.
Sibling candidates include $1/(102\pi^2)$ and related first-order forms. This fourth candidate squares $\ln\varphi$ and divides by twice the seam denominator, framed as a second-order self-similar coupling through curvature channels.
proof idea
Not a proof: a one-line real definition. The expression is $(\mathrm{Real.log},\varphi)^2/(2\cdot 102)$ with $\varphi$ from the Constants hierarchy. No lemmas are applied; numerical proximity to $\delta_2$ is left to sibling bound lemmas and the module's candidate evaluation section.
why it matters
Closes one slot in the structured search for a first-principles $\delta_2$ that would bring $\alpha^{-1}_{\mathrm{RS}}$ inside the CODATA band (primer target roughly $(137.030, 137.039)$ before fine correction). The interpretation ties the gap to second-order $\varphi$-ladder curvature rather than a new parameter, consistent with constraints (A1)–(A4).
No downstream theorems currently consume this definition (used_by is empty); it feeds the local CorrectionAnalysis / analysis narrative beside candidate_1–candidate_3. It does not itself discharge the 8 ppm gap; it only supplies a geometrically motivated number to compare against $\delta_2 \approx 0.00110$.
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