The Minimal E_23 Rendered Boundary Quotient: A TSCT Derivation of D_1=13, D_2=29, and Their Physical Readouts

This note constructs the minimal rendered boundary quotient E_23 at the k=2/k=3 TSCT interface and derives both the mathematical structure (D_1=13, D_2=29) and its physical readouts (sin²θ_W, Ω_m, Ω_Λ) in a single framework. The rendered first-order boundary tangent is: T^ren_23 ≅ (TL_3 ⊗ SP/Q) ⊕ (TL_2' ∩ TL_3), giving D_1 = 5·2+3 = 13. The unique trace-preserving bimodular conditional expectation has image Im(E_23) ≅ M_2(ℂ)^⊕5 ⊕ M_3(ℂ), giving D_2 = 5·4+9 = 29. Under the Rendered Coupling Trace Rule (g_i² ∝ Tr(P_i)): sin²θ_W = 3/13. Under the Second-Order Trace Partition Rule (Ω_i = dim(A_i)/dim(A^ren)): Ω_Λ = 20/29, Ω_m = 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(TL_3)=C_3, dim(SP/Q) from P²=P, and dim(TL_2'∩TL_3)=3, giving the same D_1=13, D_2=29 by entirely different algebra.