The Binary Screen: Minimal Rendering and the Origin of the Solar Angle, Cabibbo Angle, and Koide Formula in Two-Sided Closure Theory

The primary source in Two-Sided Closure Theory (TSCT) is the first nontrivial algebraic boundary between rendered and unrendered reality: the Binary Screen. Three physical rendering axioms force the Jones index [M:N]=2: (i) Finite capacity: [M:N] < ∞ (ii) Non-trivial: N ≠ A(W), otherwise no screen metric (iii) Minimal: least rendering cost By Jones' index theorem (1983), the allowed indices form the sequence 1, 2, 2.618, 3, ... Index 1 is trivial. Index 2 is the unique minimum. Index 2 forces the A₃ subfactor with δ=√2 and d=(1,√2,1). Because TSCT is two-sided, the canonical reciprocal amplitudes are: a₊ = (δ+δ⁻¹) / (‖d‖·‖d⁻¹‖) = 3/√20, a₊² = 9/20 a₋ = (δ-δ⁻¹) / (‖d‖·‖d⁻¹‖) = 1/√20, a₋² = 1/20 a₊·a₋ = 3/20 Both 9/20 and 1/20 are rational numbers — pure A₃ algebra, zero free parameters. Three conditional results from named readout lemmas: (i) Solar angle: tan θ₁₂ = a₊ → θ₁₂ ≈ 33.85° (obs: 33.4°±0.75°, 0.6σ)(ii) Cabibbo angle: sin θ_C = a₋ → θ_C ≈ 12.92° (obs: 13.0°±0.1°, 0.8σ)(iii) Koide formula: K=2/3 ⟺ C(h)=0, where C(h)=‖h₊‖²-‖h₋‖² and h=(√mₑ, √m_μ, √m_τ). Three-line proof. The Fourier Recurrence Condition of earlier papers is a coordinate corollary of C(h)=0. The Pimsner-Popa saturation and Yukawa democracy routes are subsumed. 9/20 and 1/20 are not independent — they are symmetric and antisymmetric faces of the same Z₂ decomposition: (δ+δ⁻¹)² - (δ-δ⁻¹)² = 4 = ‖d‖² Paper IV of the TSCT series.Companion papers: I (gravity), II (mixing), III (primer), V (full PMNS). Companion DOIs: Paper I: 10.5281/zenodo.20282969 Paper II: 10.5281/zenodo.20283404 Paper III: 10.5281/zenodo.20284625 Paper V: 10.5281/zenodo.20329022