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April 26, 2026Open Access

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

Key Points

The research aims to derive four parameter-free predictions for the PMNS leptonic mixing matrix using advanced mathematical concepts.
Derived predictions from the level-5 metaplectic modular structure using the Rogers dilogarithm identity.
Compared theoretical predictions with existing measurements from NuFIT and JUNO, verifying results numerically.
Utilized independent methods to confirm results up to 100 decimal places.
Predicted values for θ₂₃, θ₁₂, and θ₁₃ were consistent with measured values, showing deviations of 0.07°, 0.04°, and 0.09°, respectively.
Predicted δ_CP at 233.3° aligns within 1σ of prior measurements.
Demonstrated a unique geometric point selection in the Rice-Mele model, supporting theoretical assertions.

Abstract

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: RM = 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) = RM·φ⁻³ → θ₁₃ = 8. 71° θ₁₃ = 8. 62° → |Δ| = 0. 09° δCP = 2π·RM → δCP = 233. 3° δCP = 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 RM selects a unique geometric point in theRice-Mele condensed matter model at v = w/φ, via the Zak-phase identity γZak (Δ*) = 2π·RM, 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, RM) → (ρ₄, O₁₂, RM; modular weights 1, 2, 1) → (q₂₃, q₁₂, q₁₃, 2π·RM) → UMR → M_ν. Open element: explicit identification of the modular-form components ofΓ' (5) and Γ (5) realizing qᵢj as specific ratios. Numerical verificationcode (Python/mpmath and C/GCC) is included. Falsifiable commitments: (i) θ₂₃ remains in the lower octant (< 45°) ; (ii) δCP converges toward 233° as DUNE and Hyper-Kamiokande accumulatedata; (iii) precision improvements on θ₁₃ converge toward 8. 71°.

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