Towards Standard Model Mixing Parameters from the A₃ Jones Subfactor: Candidate Derivations in Two-Sided Closure Theory (TSCT)

We present candidate derivations of Standard Model fermion mixing parameters from the A₃ Jones subfactor within Two-Sided Closure Theory (TSCT). The paper establishes two results with explicit proof status: (i) CONDITIONAL KOIDE THEOREM. The Koide formula K = 2/3 for charged lepton masses — observed since 1982, never previously derived — follows from the A₃ Fourier Recurrence Condition, which in turn follows from Yukawa democracy in the unrendered sector (the natural absence of a preferred DFT mode before the A₃ structure is imposed) and the Pimsner–Popa exact attenuation of the oscillatory Jones-complement mode by 1/δ = 1/√2 in amplitude. The three algebraic steps: correct DFT gives |ĥ₁|/|ĥ₀| = B/2; FRC forces B = δ = √2; K = (1/3)(1 + B²/2) = 2/3. The Jones index [M:N] = 2 of the A₃ subfactor is directly encoded in the lepton mass spectrum. Remaining proof obligations: Yukawa democracy (O2a) and Pimsner–Popa saturation for the oscillatory mode (O2b). (ii) SOLAR ANGLE CANDIDATE. The two-sided A₃ closure formula (δ + δ⁻¹)² / (‖d‖² ‖d⁻¹‖²) = 9/20 gives the PMNS solar mixing angle θ₁₂ = arctan √(9/20) ≈ 33.87° (observed: 33.4° ± 0.75°). The number 9/20 is rational, arising from pure A₃ Jones subfactor data — quantum dimensions (1, √2, 1), Jones index 2 — with zero free parameters. A structural lemma connecting the TSCT generation quadratic form to this two-sided product is the remaining proof obligation (O1). Four further mixing observables are discussed with explicit proof status:- Atmospheric angle θ₂₃ ≈ 45°: conditional on exact A₃ μ–τ symmetry - CP phase δ_CP = −π/2: structural conjecture (minimisation not constructed)- Reactor angle θ₁₃ ~ 8.5°: open (shear coefficient not computed) - Cabibbo angle θ_C ≈ 12.92°: conditional on m_d/m_s = 1/20 None of these four should be regarded as derived results in the present paper. The paper is an honest map to a complete Standard Model derivation, identifying bounded proof obligations rather than claiming completion.The two genuine results — conditional Koide and the rational 9/20 solar angle candidate — are the strongest algebraic connections between the Jones subfactor structure and particle physics phenomenology currently known. This is part of the Two-Sided Closure Theory (TSCT) series.Companion paper: Einstein's Field Equations from Algebraic Closure (doi: [insert Paper I DOI]). Keywords: Standard Model, Koide formula, PMNS matrix, CKM matrix, Jones subfactors, A₃ subfactor, solar mixing angle, Pimsner–Popa theorem, Two-Sided Closure Theory, modular flow, Yukawa sector, neutrino mixing.