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

HFG Dispatch · June 2026

We have identified a third arithmetic hyperbolic 3-manifold following the same selection principle as PMNS and CKM — but with a new resonance angle and near-maximal mixing. This is a geometric prediction for a beyond-Standard-Model sector.

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.

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The Three Manifolds

Fig 2 — Mixing Regime Continuum Mixing angle vs. volume (in units of v₀ = 0.9814) for the three arithmetic manifolds. The mixing regime — small, large, near-maximal — increases non-monotonically with volume. The dashed line connects the three known points; the shaded bands mark the three mixing regimes.

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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 that the other two manifolds do not.

That symmetry is tr(ρ(a)) + tr(ρ(b)) = 0 exactly — the traces of the two generators of the fundamental group sum to zero. Equivalently, their dominant holonomy eigenvalues satisfy λ_b = −λ_a exactly. This is a genuine Z/2 anti-conjugation symmetry of the holonomy representation, absent in both PMNS and CKM manifolds.

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

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The Torsion Taxonomy

The three resonance angles correspond to three different torsion orders of the holonomy eigenvalue:

Order 2: θ* = −180° → λ = −|λ| (real negative) → PMNS
Order 4: θ* = +90° → λ = i|λ| (purely imaginary) → CKM
Order 6: θ* = −60° → λ = e^{−iπ/3}|λ| (Eisenstein) → NEW

These are the first three non-trivial torsion orders of the eigenvalue’s unit part. Each one selects a different arithmetic structure, a different factorization method, and a different mixing regime. The pattern raises an obvious question: does an order-3 manifold (θ* = +120°) exist? An order-8 manifold (θ* = +45°)? The scan found candidates, but none with the clean resonance structure of the three identified here.

Fig 4 — Eisenstein Geometry The complex unit circle with the six 6th roots of unity marked. The three resonance angles of the known manifolds (−180°, +90°, −60°) are highlighted. The Eisenstein lattice (dashed lines at 0°, 60°, 120°) reflects the arithmetic of the cusped parent m206 whose trace field is ℚ(√−3) with discriminant −3.

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The Mixing Matrix

The best H₁-balanced triple near θ* = −60° is {aab, aaa, aBB} with H₁ classes (3, 3, 4), summing to 10 = 0 mod 5. The Borel/QR factorization of the product holonomy matrix gives a 2×2 unitary K factor with mixing angle in the range 68°–81° depending on the ordering of the words. The standard deviation across orderings is 4.8° — stable, but not as sharply determined as the PMNS or CKM cases. The CP-phase analog is approximately −2.5°, nearly CP-conserving.

What factorization method corresponds to θ* = −60°? For PMNS (−180°), Borel is natural because the eigenvalues are real negative. For CKM (+90°), Iwasawa is natural because they are purely imaginary. For −60°, the eigenvalues are Eisenstein units — a third case that may require a third decomposition. This is currently open.

Fig 3 — Torsion → Factorization → Mixing The chain from resonance angle to factorization method to mixing regime for each manifold. The factorization method for θ = −60° (m206) is currently an open question — Borel and Iwasawa are natural for real-negative and imaginary eigenvalues respectively, but the Eisenstein case requires further analysis.*

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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. If active neutrinos mix maximally with a sterile sector, the mixing angle could be ~74°. Current experiments (IceCube, MicroBooNE) constrain sterile neutrino mixing but have not ruled out all parameter space.

Dark matter interactions. If dark matter couples to Standard Model particles through a mixing matrix, maximal mixing is allowed. The Z/2 symmetry tr(a) = −tr(b) could correspond to a matter-antimatter or visible-dark parity.

Mirror matter. Left-right symmetric extensions of the Standard Model predict a mirror sector with its own mixing matrix. Maximal mixing between sectors is a natural prediction of exact parity symmetry — exactly the Z/2 structure we observe.

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

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The Falsifiable Prediction

If a BSM sector is discovered with mixing angle near 74° and the discrete symmetry tr(a) = −tr(b), the geometric selection principle predicts it should correspond to the arithmetic of m206(1,2). If no such sector is found — or if one is found with a different mixing angle — this prediction is falsified.

More specifically: the three resonance angles −180°, +90°, −60° correspond to torsion orders 2, 4, 6. If there is a fourth SM or BSM sector, the geometric framework predicts it should correspond to torsion order 3 (θ* = +120°, Eisenstein conjugate) or order 8 (θ* = +45°). The scan found no clean candidate for these in the first 300 manifolds — but the census has thousands more to check.

Fig 5 — The Fourth Sector Prediction Mixing angle vs. torsion order for the three confirmed sectors and two predicted ones (open diamonds). If the pattern continues, torsion order 3 (θ = +120°) and order 8 (θ* = +45°) should correspond to manifolds with intermediate mixing angles. No clean candidate was found in the first 300 census manifolds — the prediction is open and falsifiable.*

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All computations are reproducible via github.com/drmlgentry/hyperbolic-flavor-scan. The SnapPy manifold is OrientableClosedCensus[209], filing m206(1,2).

Marvin L. Gentry, ND
Seattle, Washington · June 2026
hyperbolicflavorgeometry.org

Next dispatch: Scanning for a fourth manifold — the complete torsion spectrum.