We show that SO(10) is the minimal natural ultraviolet completion of the SO(8) mesoscale vacuum of Modular Entropic Gravity (MEG), and derive from this completion the Weinberg angle sin²θ_W = 3/8 and, via standard renormalisation group running, the fine structure constant 1/α ≈ 137. The Principle of Least Entropic Stress (PLES) forces a single gauge coupling per generator. Applied to the leptoquark generators of the Pati-Salam subgroup SU(4) × SU(2)_L × SU(2)_R, this forces equal coupling to the left-handed sector 8_s and the right-handed sector 8_c of the SO(8) vacuum, requiring both to transform as parts of a single representation of the ultraviolet group. The SO(10) roots involving the first coordinate connect 8_s to 8_c, making the combined 16-dimensional representation irreducible. The requirement that the ultraviolet group provides the B−L generator (rank ≥ 5) and contains the D4 substructure of the SO(8) mesoscale vacuum points naturally and minimally to SO(10). The D4 triality-governed SO(8) structure survives as a distinguished mesoscale substructure within D5, governing the effective vacuum at lower energies. Within the SO(10) vacuum, the Standard Model hypercharge is Y = T_3R + (B−L)/2, native to the SO(10)/Pati-Salam embedding. The factor 1/2 arises from the group-theoretic structure of the hypercharge generator within that embedding. Applied to the full 16-dimensional spinor with the PLES coupling convention g² ∝ 1/Γ, the electroweak stiffness traces Γ_SU(2) = Tr_16(T_3L²) = 2 and Γ_Y = Tr_16(Y²) = 10/3 give the Weinberg angle sin²θ_W = 2/(2 + 10/3) = 3/8, the canonical grand unified prediction. Standard one-loop renormalisation group running from the ultraviolet scale gives sin²θ_W(M_Z) ≈ 0.231 and 1/α ≈ 137, consistent with observation. All 16 Standard Model hypercharge assignments are verified independently from Y = T_3R + (B−L)/2, with tracelessness confirmed. The paper identifies and corrects a coupling convention error in the companion MEG hypercharge paper (Devlin, 2026), which computed cos²θ_W = 1/4 using only the 8_s sector and labelled it sin²θ_W. The correct MEG prediction uses the full 16 of SO(10) with the coupling convention established in the MEG Maxwell's equations paper and gives sin²θ_W = 3/8. The hypercharge assignments Y_q = 1/6 and Y_ℓ = −1/2 from the companion paper are confirmed and incorporated. The ten prior MEG papers are correct within their domain as the effective SO(8) mesoscale vacuum. The SO(10) ultraviolet completion does not modify any of the gravitational phenomenology, galaxy rotation curve results, Dirac equation derivation, Maxwell equations derivation, or Standard Model flavour structure established in those papers. The relationship between the SO(8) mesoscale vacuum and the SO(10) ultraviolet vacuum is the standard Wilsonian relationship between an effective theory and its ultraviolet completion. Three open problems are identified: a complete exhaustive proof of the minimality of SO(10), a derivation of the factor 1/2 in the hypercharge formula from the D5 root lattice alone, and a MEG-native derivation of the GUT scale from the PLES functional. This paper is part of the MEG programme. Companion papers covering gravitational phenomenology, quantum mechanical ontology, Standard Model flavour structure, baryogenesis, the Maxwell-Boltzmann distribution, the Dirac equation, Maxwell's equations and Yang-Mills theory, the fine structure constant mechanism, and the hypercharge derivation are available at the DOIs listed in the references.