This note constructs the minimal rendered boundary quotient E₂3 at the k=2/k=3 TSCT interface and derives both the mathematical structure (D₁=13, D₂=29) and its physical readouts (sin²θW, Ωₘ, Ω_Λ) in a single framework. The rendered first-order boundary tangent is: Tʳen₂3 ≅ (TL₃ ⊗ S/ₐ) ⊕ (TL₂' ∩ TL₃), giving D₁ = 5·2+3 = 13. The unique trace-preserving bimodular conditional expectation has image Im (E₂3) ≅ M₂ (ℂ) ^⊕5 ⊕ M₃ (ℂ), giving D₂ = 5·4+9 = 29. Under the Rendered Coupling Trace Rule (gᵢ² ∝ Tr (Pᵢ) ): sin²θW = 3/13. Under the Second-Order Trace Partition Rule (Ωᵢ = dim (Aᵢ) /dim (Aʳen) ): Ω_Λ = 20/29, Ωₘ = 9/29. Matter is the second-order self-action of the gauge/stabiliser sector; dark energy is the second-order self-action of the bridge/disclosure sector. Cross-validated by the independent Coxeter derivation: the same three structural ingredients (5, 2, 3) arise from dim (TL₃) =C₃, dim (S/ₐ) from P²=P, and dim (TL₂'∩TL₃) =3, giving the same D₁=13, D₂=29 by entirely different algebra.
David Manton Sparks (Sun,) studied this question.