The Minimal E_23 Rendered Boundary Kernel: A Compact TSCT Derivation of D_1=13, D_2=29, and the First Physical Readouts

This note consolidates the minimal E_23 rendered boundary construction into a single compact source document integrating the mathematical kernel, the readout rule discipline, and the physical predictions. Mathematical kernel: T^ren_23 ≅ (TL_3⊗S_{P/Q}) ⊕ (TL_2'∩TL_3), D_1=13; Im(E_23) ≅ M_2(ℂ)^⊕5 ⊕ M_3(ℂ), D_2=29. Physical readouts: sin²θ_W = 3/13 (squared stabiliser projection weight); tan θ_C = 3/13 (charged-current tilt — geometrically distinct from Weinberg despite the same ratio); Ω_m = 9/29, Ω_Λ = 20/29 (second-order block trace fractions); w = -1+1/(6π) (modular scalar-residue drain); a_H = cH_0/(2π) (horizon scale). The readout rule table is the critical discipline layer: it specifies why the same ratio 3/13 appears as sin² in one case and tan in another (different geometric operations on the same stabiliser share), preventing arbitrary reuse of structural numbers. The late-time tracial state with block chemical potential η=0 gives the minimal cosmological fractions; η≠0 is the non-minimal extension. DESI DR2 BAO geometry test: Δχ²=2.25 (inside approximate 1σ). SPARC RAR residual hierarchy: structured, not random.