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 θC = a₋ → θC ≈ 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