A six-term Laurent expansion for α−1 from the SO(3,3) matrix model on the Poincaré homology sphere:

We present a six-term Laurent series for the inverse ne-structure constant in the smallparameter ε = 1/|2I∗| = 1/120, with coecients drawn from the ring Zφ, e, 1/120, whosevalue matches the CODATA 2022 measurement α−1 = 137.035 999 177(21) to twelve significantdigits. The series is the empirical content of an IKKT-type reduction of the bosonicSO(3,3) matrix model on the Poincaré homology sphere P3 = S3/2I∗. We separate theformula's content into three derivation tiers. Tier A rigorously derived from the matrixmodelconstruction contains the perturbative-exactness theorem that forces c1 = 0, theBurnside-projection mechanism that removes evanescent Hessian eigenvalues with universalprojection factor 2/3 (veried independently in this work), the multiplicative-invisibility theoremthat constrains the form of any spectral correction, the pentagonal-dominance resultand the threefold-class invisibility, the icosahedral integer quotients 10 and 4, and the classaveragedtrace structure on End(D6) yielding exactly φ2 and 1/φ2 on the vertex classes.Tier B structural motivation with post-hoc numerical identication contains the specic functional choice 1/φ2 (rather than φ2) for the leading coecient c−1, the constantterm c0 = −2/φ3, the spectral correction δ = log(2/3)/30, and consequently the valuesc2 = φ(e + φ/e) and c3 = −eφ/2. Tier C open contains the choice of c4 from threestructurally motivated candidates lying within current experimental uncertainty. The TierB identications are not independent open questions: they are four facets of a single openanalytic step the IKKT one-loop eective action on fuzzy P3, decomposed by conjugacyclass of A5. We document explicitly an obstruction internal to this open question (Ÿ4.7): thesimplest version of the spectral correction proposed in the companion literature is not simultaneouslyconsistent with the parallel target identications for ρ2, ρ3 and ρ4 at the level ofthe underlying class-resolved shifts. The obstruction admits three distinct resolutions, withcorrespondingly dierent implications for the structural status of the formula. We commit,prior to the next generation of α measurements at the 10−13 level, to the prediction c4 = e2on parsimony grounds, and identify experimental outcomes that would refute it.