A Static Carrier Construction for the Low-Energy Fine-Structure Constant in Recursive Interval Geometry

A static carrier construction for the low-energy, zero-momentum fine-structure constant is formulated within Recursive Interval Geometry (RIG). The carrier hierarchy is fixed before numerical evaluation by four requirements: discrete minimal closure, space-filling compatibility, boundary recursion, and nonzero boundary-state recursion. Each free boundary variable has two states, active or silent. The link has two freely variable endpoints, the triangle has three freely variable edges, and the octahedron has eight triangular faces with one whole-shell closing relation, leaving seven freely variable faces. Thus \( b_1=2 \), \( b_2=3 \), and \( b_3=7 \), and exclusion of the all-silent state gives \( D_1=3 \), \( D_2=7 \), and \( D_3=127 \). Adding the three level counts gives the skeleton \( N_{\mathrm{sk}}=137 \), while choosing one state from every level gives the joint boundary-state resolution \( \Omega=D_1D_2D_3=2667 \). The localized base ring organizes this recursive arithmetic skeleton; the exact physical state is its continuous geometric readout in the real metric completion of the depth-one principal/structural space. The structural sector is built from three dimension-filtered terms in this continuous readout. Pairing-sphere geometry gives an exact null locus for the oriented link readout; the centre-preserving antipodal branch supplies the phase holonomy \( \pi \) and a two-endpoint support loss \( -2 \). The octet-truss completion locks the pairing circle to four face-centred-cubic phases and gives a tetrahedral transmission-sharing contribution \( 127/2 \). Of the connector's four faces, one is occupied by attachment to the central octahedron and the closed-shell condition removes one further outward contribution, leaving the transmission coefficient \( 2/4=1/2 \). The geometric coefficients belong to the real readout, while \( \mathbb Z[\Omega^{-1}] \) carries the recursive arithmetic skeleton. The completed RIG norm defines the model output \( \alpha^{-1}_{\textrm{RIG}} \), and the electromagnetic interface postulate identifies it with the Thomson-limit inverse fine-structure constant. With the carrier and bridge rules fixed, this gives \( \alpha^{-1}_{\textrm{RIG}}=\sqrt{137^2+\left(\pi-\frac{2}{2667}+\frac{137+127/2}{2667^2}\right)^2}=137.035999176253\cdots \) The difference from the CODATA 2022 value is \( 5.45\times10^{-3} \) ppb. The construction addresses the static zero-momentum baseline, while QED supplies finite-momentum dynamics. The norm-to-coupling identification enters as a foundational interface postulate. Conditional on it, carrier, pairing, and closure geometry fix the \( \pi \), \( -2 \), and \( 127/2 \) terms without continuously fitted coefficients.