The Binary Screen: Even, Odd, Mixed, and Null Readouts in Two-Sided Closure Theory

All five Standard Model mixing observables from one number. The Binary Screen is the first nontrivial rendered boundary in Two-Sided Closure Theory (TSCT). Three physical axioms — finite capacity, non-triviality, minimality — force Jones index [M:N]=2, giving the A₃ subfactor with one canonical amplitude: a₊ = 3/√20 From this single number, five mixing angles emerge through different readout functions — no free parameters in the algebra: tan θ₁₂ = a₊ → θ₁₂ ≈ 33.85° (solar, obs: 33.4°±0.75°, 0.6σ) sin θ₂₃ = a₊ → θ₂₃ ≈ 42.13° (atmospheric, obs: 42.3°±3.0°, 0.1σ) sin θ₁₃ = a₊·a₋ → θ₁₃ ≈ 8.63° (reactor, obs: 8.54°±0.15°, 0.6σ) δ_CP = ±π/2 → δ_CP = ±90° (CP phase, obs: -87°±38°, 0.1σ) sin θ_C = a₋ → θ_C ≈ 12.92° (Cabibbo, obs: 13.0°±0.1°, 0.8σ) The second amplitude a₋ = 1/√20 and their product a₊·a₋ = 3/20 complete the picture. The corrected atmospheric angle uses the same amplitude a₊ as the solar angle — different trig function, same source. The sign grading Γ and crossing operator S anticommute; their product I = ΓS satisfies I² = -1, providing a natural complex structure. The PMNS (e,3) entry U_{e3} = I·a₊a₋ = ±i·3/20 yields both the reactor angle and CP phase from one oriented residual coupling. All five observables within 1σ. Zero free parameters in the algebra.All results conditional on named readout lemmas — nothing overclaimed. Paper V of the TSCT series.Companion papers I–IV on Zenodo.