A First-Principles Determination of the $SU(3)$ Gauge Coupling from an 11D--4D Effective Action

We derive a current-normalized reference 𝑆𝑈(3) gauge coefficient and construct its primitive geometricflux representative within an explicitly specified branch of an eleven-dimensional effective action. The parent information-projection theory supplies the higher-dimensional architecture and admissible variationalsetting. The strong-sector derivations are developed in the language of effective actions, Kaluza–Kleingeometry, conserved color currents, associated bundles and gauge holonomy.The first principal result is the local action–current normalization. All action sectors are combinedbefore the shared auxiliary fields are eliminated, and the transverse kinetic matrix is compared with thephysical current through𝐾𝐽 =𝑉−†𝐽𝑎𝐾redraw𝑉−1𝐽𝑎.A maximally mixed three-level reference state and the stated current scores give the relative-entropy Hessian 𝐷2Γsrc(0) = (ln3)𝐼8. Its plaquette realization produces the local Yang–Mills operator. The sameinternal profile enters the action, an independent Kaluza–Klein curvature evaluation and the current norm,so the common factor 7/8 cancels. With the specified source-action law and no additional unmatchedlocal term,𝐾CN𝐴 =𝐼8,𝑔CN3 =1, 𝛼CN𝑠 = 14𝜋 .The second principal result is a geometric and topological reconstruction through a distinct calculation.A correlated reference–color state with unchanged color marginal supplies a primitive associated line over𝑆𝑈(3)/𝑈(1)(1,1). We calculate its unit first-Chern pairing, construct a round color frame, and derive aspatial connection by minimizing the source hopping energy over all 𝑆𝑈(3) links. Foranorientedsphericaltriangle, its full Wilson matrix is𝑊 =𝑔𝑥𝑒𝑖𝑌Ω /2𝑔†𝑥,𝑌 = diag(1,1,−2),and its projected continuum curvature obeys −𝑠†𝑔F∗𝑠𝑔 = 𝑓𝐵 = dΩ/2. The primitive period, physicaltransport and generator norm thereby provide complementary checks of the action–current normalization.These routes share the declared source framework: topology fixes the flux unit and isotropy its distribution,while the action and physical current fix the strength. Joint stationary and finite-support realizations aredistinguished from stability and free-core selection.A separate six-mass BFM-MOM prescription gives 𝜇CN = 0.4167999616251471GeVwithinthestatedlogarithmic functional class and a one-loop threshold interpolation from coefficient 11 to 7. The geometricreconstruction is not a second loop term. Matching an additional microscopic parent contribution, finiteconversion to MS, confinement and a Yang–Mills mass gap remain distinct questions