partial_alpha
plain-language theorem explainer
partial_alpha supplies the N-term truncation of the infinite series for the inverse fine-structure constant as seed minus gap weight plus the sum of the first N correction terms. Analyses of the residual discrepancy between the RS prediction and CODATA would cite this when bounding the series approximants. The definition evaluates directly as an algebraic partial sum over the supplied deltas sequence.
Claim. The partial sum of the series for the inverse fine-structure constant up to order N is given by $alpha_s - f_g + sum_{n=1}^N delta_n$, where $alpha_s$ is the geometric seed, $f_g$ the gap weight, and $delta_n$ the nth voxel-seam correction on $Q_3$.
background
Recognition Science derives alpha inverse from a geometric seed on the cube Q3, a gap weight from the phi-ladder, and an infinite series of higher-order voxel-seam corrections. The module formalizes the cube combinatorics of faces, wallpaper groups, and face-wallpaper pairs on Q3 to generate the delta_n terms, with the full series written as alpha inverse equals alpha seed minus f gap plus the sum from n equals 1 to infinity of delta n. Upstream results supply the structure of Q3 forcing SU(3) times SU(2) times U(1) and three generations (SpectralEmergence), the convexity of J-cost minimization (PhysicsComplexityStructure), and the phi-forcing structures (PhiForcingDerived).
proof idea
The definition is a direct one-line algebraic expression that subtracts the gap weight from the seed and adds the finite sum of the first N terms drawn from the deltas sequence.
why it matters
This definition is invoked inside the AlphaPrecisionHypothesis structure to state that the partial sums converge to the CODATA value of alpha inverse. It fills the series framework in the module for addressing the open 8 ppm residual between the RS additive and exponential formulas and the measured 137.035999206(11). The construction sits inside the alpha band (137.030, 137.039) and the T5 J-uniqueness landmark of the forcing chain.
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