Parameter-free prediction for the PMNS matrixfrom the level-5 modularstructure and the Rogers dilogarithm at the golden-ratio point

We derive four parameter-free predictions for the PMNS leptonic mixing matrixfrom the level-5 metaplectic modular structure and the Rogers dilogarithmidentity at the golden-ratio point. The central invariant is the regulator: R_M = 1 − 15·ln²(φ)/π² = 0.6480636743378963580533...obtained from the Rogers identity L(φ⁻²) = π²/15 and verified to 100 decimalplaces by independent methods. Here φ = (1+√5)/2 is the golden ratio. Predictions (no free parameters after fixing φ): Comparison with NuFIT 6.1 (Nov 2025, IC24+SK, Normal Ordering,www.nu-fit.org/?q=node/309): tan(θ₂₃) = 4·φ⁻³ → θ₂₃ = 43.36° θ₂₃ = 43.29° → |Δ| = 0.07° tan(θ₁₂) = 12·φ⁻⁶ → θ₁₂ = 33.80° θ₁₂ = 33.76° → |Δ| = 0.04° tan(θ₁₃) = R_M·φ⁻³ → θ₁₃ = 8.71° θ₁₃ = 8.62° → |Δ| = 0.09° δ_CP = 2π·R_M → δ_CP = 233.3° δ_CP = 212° (1σ: 176°–238°) → prediction within 1σ Comparison with the first JUNO measurement (59.1 days, arXiv:2511.14593): sin²θ₁₂ = 0.3092 ± 0.0087 → θ₁₂ ≈ 33.78° Prediction 33.80° → |Δ| < 0.02°. Cross-domain anchor: the same R_M selects a unique geometric point in theRice-Mele condensed matter model at v = w/φ, via the Zak-phase identity γ_Zak(Δ*) = 2π·R_M, Δ* = 0.44263293143506090037...certified by five independent procedures including interval arithmetic(Arb/Flint, certified width 7×10⁻³⁸). Deductive chain: M'₅ (level-5 metaplectic) → (A'₅ → A₅, R_M)→ (ρ₄, O₁₂, R_M; modular weights 1, 2, 1) → (q₂₃, q₁₂, q₁₃, 2π·R_M)→ U_MR → M_ν. Open element: explicit identification of the modular-form components ofΓ'(5) and Γ(5) realizing q_ij as specific ratios. Numerical verificationcode (Python/mpmath and C/GCC) is included. Falsifiable commitments: (i) θ₂₃ remains in the lower octant (< 45°);(ii) δ_CP converges toward 233° as DUNE and Hyper-Kamiokande accumulatedata; (iii) precision improvements on θ₁₃ converge toward 8.71°.