Alpha - The Emergent Fine-Structure Constant

We present a conditional shell-geometric selection mechanism for the fine-structure coupling \(\alpha\) within the Relator framework on the paired spaces \(\mathbb{C}\) and \(\mathbb{R}^{3}\). Starting from the kinematical lock \(R\omega=c\), the construction fixes the minimal pinned electron branch, its matching-shell geometry, the scalar cavity--shell slowdown block \(\mathcal{D}_{\mathrm{C}}(\alpha)\), the vector shell constant \(\Lambda_{\rm geom}\), and the universal single-log coefficient \(\mathcal{C}_{\mathrm{log}}=1/3\). The Alpha Lock Point then selects the coupling through the compatibility condition between the scalar shell law and the vector shell geometry. No measured value of \(\alpha\), no QED anomaly coefficient, and no fitted parameter enters this primary closure.

The resulting finite-rank realization places the emergent coupling in the immediate neighborhood of the accepted fine-structure benchmark, while keeping all benchmark comparisons external to the derivation. As a secondary consequence, the same scalar branch, lifted through the ALP map and the operational orbital slowdown map, generates a pure-photonic electron \(g-2\) series. Its first benchmarked coefficients provide a stringent cross-check against the universal QED sequence, and its higher coefficients, beginning with the twelfth-order pure-photonic term \(A_{1}^{(12)}\), are falsifiable predictions of the specified no-free-parameter shell realization.

Update:  Complete Physical Derivation and Mathematically Rigorous Proof (Full Version)