A Third Mixing Matrix?
Maximal Mixing from a New Hyperbolic Manifold

The HFG selection principle now applies to three closed arithmetic hyperbolic 3-manifolds with H₁ = ℤ/5. The first two gave the PMNS and CKM mixing matrices. The third gives something different — and potentially more surprising.

The Three Manifolds

Mixing angle vs. volume (in units of v₀ = 0.9814) for the three arithmetic manifolds. The mixing regime increases non-monotonically with volume: small (CKM), large (PMNS), near-maximal (m206). Shaded bands mark the three regimes.

What Makes m206(1,2) Special

The manifold m206(1,2) was found by scanning the first 300 closed hyperbolic 3-manifolds with H₁ = ℤ/5. Among them, only four had resonance angles outside the known PMNS (−180°) and CKM (+90°) zones. m206(1,2) stood out immediately: its short geodesics cluster near θ* = −60°, and it carries an exact algebraic symmetry absent from both PMNS and CKM.

The cusped parent manifold m206 has trace field exactly ℚ(√−3) — the Eisenstein field with discriminant −3. Its cusp shape is i√3, a purely imaginary Eisenstein number. The Dehn filling m206(1,2) deforms this to a quartic field (discriminant ≈ −9.18×10¹⁸), but the Eisenstein ancestry is preserved in the resonance angle −60° = −π/3, the argument of a primitive 6th root of unity.

The Torsion Taxonomy

The three resonance angles correspond to three different torsion orders of the holonomy eigenvalue's unit part:

The three resonance angles on the complex unit circle. PMNS at −180° (real negative, order 2), CKM at +90° (imaginary, order 4), m206 at −60° (Eisenstein, order 6). Gray diamonds mark predicted but unconfirmed sectors. The Eisenstein lattice (dashed lines) reflects the arithmetic ancestry of the cusped parent m206.

The Mixing Matrix

The best H₁-balanced triple near θ* = −60° is {aab, aaa, aBB} with H₁ classes (3, 3, 4), summing to 0 mod 5. The Borel/QR factorization of the product holonomy matrix gives a unitary K factor with mixing angle in the range 68°–81° depending on word ordering. The standard deviation across orderings is 4.8° — stable but not as sharply canonical as PMNS or CKM. The CP-phase analog is approximately −2.5°, nearly CP-conserving.

The chain from resonance angle to factorization method to mixing regime for each manifold. The factorization method for θ* = −60° is currently an open question.

Physical Interpretation

Near-maximal mixing (θ ~ 45°–90°) does not appear in the Standard Model flavor sector, but it is a feature of several BSM scenarios. Sterile neutrinos mixing maximally with active neutrinos could produce θ ~ 74°. Dark matter coupling through a mixing matrix allows maximal mixing. Mirror matter in a left-right symmetric extension naturally predicts maximal mixing between sectors — exactly the ℤ/2 parity structure we observe.

We are not claiming any of these identifications. We note that the geometric prediction — near-maximal mixing, Eisenstein arithmetic, ℤ/2 parity — is consistent with these scenarios, and that future experiments will either support or falsify this picture.

The Falsifiable Prediction

The three resonance angles −180°, +90°, −60° correspond to torsion orders 2, 4, 6. If there is a fourth sector, the framework predicts torsion order 3 (θ* = +120°) or order 8 (θ* = +45°). No clean candidate was found in the first 300 census manifolds — but the census has thousands more.

Mixing angle vs. torsion order for the three confirmed sectors (filled circles) and two predicted ones (open diamonds). The confirmed pattern spans small, large, and near-maximal mixing at torsion orders 2, 4, and 6. Orders 3 and 8 are open predictions.