The CP Phase Is a Manifold Invariant
For the past several months the Hyperbolic Flavor Geometry programme has stated: the leptonic CP-violating phase is given by δ = π + φ(aaB) + φ(baa), using the dominant eigenvalue branch of the holonomy representation, giving 195.91° against the PDG value of 197.0°.
A natural question arose. Is this number tied to the particular words aaB and baa? Could a different word choice give a different phase? Or does the value emerge from the manifold itself, independent of any word choice?
We now have the answer.
The Test
We enumerated all reduced words of length ≤ 6 in the fundamental group of the Meyerhoff manifold m003(−2,3), computed their holonomy eigenphases, and grouped them by their homology class in H₁(M) = ℤ/5. For each nontrivial class, we identified the first primitive geodesic whose eigenphase enters a window around the manifold's natural resonance — which turned out to be θ* = −180°, the condition that the eigenvalue is purely real and negative.
| H₁ class | First-in-window geodesic | Phase φ | Distance D = |φ + 180°| |
|---|---|---|---|
| 1 | aab | −167.362° | 12.638° |
| 2 | aaBABB | −159.192° | 20.808° |
| 3 | aaBAB | −151.857° | 28.143° |
| 4 | aaB | −176.731° | 3.269° |
The ℤ/5 group has two inverse pairs: (1,4) where 1+4 = 0 mod 5, and (2,3) where 2+3 = 0 mod 5. The PMNS pair is (1,4). Their sum of distances to the resonance is:
D(2) + D(3) = 20.808° + 28.143° = 48.951°
The pair (1,4) is 3.1× closer to the −180° resonance than the competing pair (2,3) — without reference to the PDG value.
The Conjugacy Collapse
The full enumeration found 216 candidate word triples satisfying the H₁ constraints. All of them — every single one — gave the same CP phase: 195.91°, with the same error of 1.09° from the PDG value. This is not a coincidence. After collapsing by conjugacy, the apparent multiplicity of 216 reduces to a single canonical geometric object:
length ≤ 3, H₁-correct
Class A: φ = −176.73° · Class B: φ = −167.36°
H₁ classes 1 and 4 = ±1 mod 5
PDG 197.0° · 1.09° error · 0 free parameters
The 216 apparent solutions are not 216 independent predictions. They are 216 representatives of the same canonical geometric pair — related by the symmetry group of the fundamental group of m003(−2,3).
What Is the −180° Resonance?
The condition φ = −180° means the dominant holonomy eigenvalue is a negative real number: λ = −|λ|. This forces the trace of the holonomy matrix to be real and negative — a distinguished locus in the space of PSL(2,ℂ) representations. It is an intrinsic arithmetic condition on the trace field K = ℚ(w), where w⁴ = w + 1 and disc(K) = −283.
The four geodesic classes in the table above are ordered by proximity to this locus. The pair (1,4) sits dramatically closer — the class-4 geodesic aaB has D = 3.27°, nearly at the resonance. This ordering is a property of the manifold's arithmetic structure, not of any fitting procedure.
The Unified Picture: PMNS and CKM
The same mechanism applies to the CKM manifold m006(−5,2), but with a different resonance angle. The CKM geodesics cluster near θ* = +90° — purely imaginary trace — rather than −180°. This corresponds to the Iwasawa factorization rather than the Borel factorization, which produces small quark mixing rather than large lepton mixing.
m003(−2,3): θ* = −180° → Borel factorization → large mixing → PMNS
m006(−5,2): θ* = +90° → Iwasawa factorization → small mixing → CKM
Both manifolds select the same inverse pair (1,4) — the generators of ℤ/5. Both show a ≈ 3× advantage over the competing pair. The resonance angle is different, the factorization method is different, the matrix is different — but the selection mechanism is the same.
Falsification
This is a quantitative, testable prediction. The Meyerhoff manifold's geometry forces a specific value: δ = 195.91°. If future experiments (Hyper-Kamiokande, DUNE) move the central value of the CP phase away from 195.91°, this selection principle is falsified. If they move toward it — if the error bar narrows and includes 195.91° — the principle is supported.
The current PDG central value is 197.0° ± 27°. The HFG prediction sits well within the current uncertainty. The experimental error will eventually decide.