The Neutrino CP Phase from Geometry: Charge-Locked Charged-Lepton Corrections in an E8-Derived E6 Model

In the TTOE framework the flavor sector is described by a w0-based ansatz within an E6 grand-unified cascade whose structural value for the leptonic CP phase is δ_PMNS = 180° (CP conservation), while global oscillation data prefer δ_PMNS ~ 197°; the joint T2K+NOvA analysis still permits CP conservation at present precision, so the ~17° displacement is a tension, not an exclusion — but it is the sector's only one, and this paper asks whether the framework can produce it. The setting is the geometric layer that the framework acquired with the flag-manifold resolution of the three-generation problem: three 27's whose U(1)² charges are the weights of the internal 3, six E6-singlet "flavons" whose charges are the six roots (a Froggatt–Nielsen chart with no chosen charges), and a single renormalizable inter-family trilinear. Three no-go theorems are proved first: (i) the trilinear alone forces hollow symmetric Yukawa textures, whose singular values obey σ₂ ≥ σ₁/2 — excluded by the observed hierarchies by factors 135/26/8 in the up/down/lepton sectors; (ii) no combination of the three geometric Higgs doublets escapes the bound; (iii) Kaluza–Klein doublet admixture cannot rescue it, since every geometric–KK mixing requires a vev by U(1)² charge conservation, leaving the light doublet geometric to one part in 10⁴. The geometric layer therefore cannot replace the flavor ansatz; it can only perturb it — and that perturbation is precisely what the CP tension needs. A charged-lepton correction with derived 1–2 dominance (the mass denominators m_μ/m_τ do the work) of size one flavon insertion, θ^e_12 ≈ 3° with near-maximal phase, moves δ from 180° to 196.8° with all three mixing angles within 1σ — succeeding exactly where a pre-registered Majorana-phase route had failed by destroying θ₁₂. Two locks are then tested. The Cabibbo lock: applying the sibling correction to the down sector of the reconstructed CKM benchmark of the framework (reproduced digit-exact from its repository) shows that any sibling strength above one tenth of the leptonic one is excluded at every phase. The charge-arithmetic lock then explains why the required suppression is structural rather than tuned: enumerating the E6 cubic invariant by SU(5)×U(1)_χ×U(1)_ψ charges yields exactly four terms, and the only vertex that rotates chiral fermions at first order in GUT vevs is ⟨ν^c⟩·5̄(16)·5(10): it rotates the lepton doublets (PMNS-active) and the right-handed downs (CKM-blind), and it cannot touch the quark doublets or e^c because the required 1·10·X term admits no charge solution. The mechanism is lopsided by arithmetic; its mediators are the classic E6 exotics with mass λ⟨θ⟩, and the geometric coupling cancels between vertex and mediator, leaving θ^e ~ ⟨ν^c⟩/⟨θ⟩ ≈ 0.05. Three falsifiable consequences follow: with δ confirmed in [196°, 213°] the mixing angles must remain frozen at the ansatz fit within stated bands (JUNO decides); no induced CKM or first-row unitarity anomaly may accompany the phase; and δ outside [~142°, 218°] kills the mechanism outright. What is derived, what is conditional (the near-maximal phase; the vev ordering ⟨ν^c⟩/⟨θ⟩), and what is not claimed (a first-principles value of 197°) are tabulated explicitly.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-08-27
DOI
https://doi.org/10.5281/zenodo.22130732
Primary Topic
Neutrino Physics Research
Type
preprint
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preprint

The Neutrino CP Phase from Geometry: Charge-Locked Charged-Lepton Corrections in an E8-Derived E6 Model

E.U.O.
Zenodo (CERN European Organization for Nuclear Research)
Neutrino Physics Research
preprint

The Neutrino CP Phase from Geometry: Charge-Locked Charged-Lepton Corrections in an E8-Derived E6 Model

E.U.O.
preprint en

Abstract

In the TTOE framework the flavor sector is described by a w0-based ansatz within an E6 grand-unified cascade whose structural value for the leptonic CP phase is δ_PMNS = 180° (CP conservation), while global oscillation data prefer δ_PMNS ~ 197°; the joint T2K+NOvA analysis still permits CP conservation at present precision, so the ~17° displacement is a tension, not an exclusion — but it is the sector's only one, and this paper asks whether the framework can produce it. The setting is the geometric layer that the framework acquired with the flag-manifold resolution of the three-generation problem: three 27's whose U(1)² charges are the weights of the internal 3, six E6-singlet "flavons" whose charges are the six roots (a Froggatt–Nielsen chart with no chosen charges), and a single renormalizable inter-family trilinear. Three no-go theorems are proved first: (i) the trilinear alone forces hollow symmetric Yukawa textures, whose singular values obey σ₂ ≥ σ₁/2 — excluded by the observed hierarchies by factors 135/26/8 in the up/down/lepton sectors; (ii) no combination of the three geometric Higgs doublets escapes the bound; (iii) Kaluza–Klein doublet admixture cannot rescue it, since every geometric–KK mixing requires a vev by U(1)² charge conservation, leaving the light doublet geometric to one part in 10⁴. The geometric layer therefore cannot replace the flavor ansatz; it can only perturb it — and that perturbation is precisely what the CP tension needs. A charged-lepton correction with derived 1–2 dominance (the mass denominators m_μ/m_τ do the work) of size one flavon insertion, θ^e_12 ≈ 3° with near-maximal phase, moves δ from 180° to 196.8° with all three mixing angles within 1σ — succeeding exactly where a pre-registered Majorana-phase route had failed by destroying θ₁₂. Two locks are then tested. The Cabibbo lock: applying the sibling correction to the down sector of the reconstructed CKM benchmark of the framework (reproduced digit-exact from its repository) shows that any sibling strength above one tenth of the leptonic one is excluded at every phase. The charge-arithmetic lock then explains why the required suppression is structural rather than tuned: enumerating the E6 cubic invariant by SU(5)×U(1)_χ×U(1)_ψ charges yields exactly four terms, and the only vertex that rotates chiral fermions at first order in GUT vevs is ⟨ν^c⟩·5̄(16)·5(10): it rotates the lepton doublets (PMNS-active) and the right-handed downs (CKM-blind), and it cannot touch the quark doublets or e^c because the required 1·10·X term admits no charge solution. The mechanism is lopsided by arithmetic; its mediators are the classic E6 exotics with mass λ⟨θ⟩, and the geometric coupling cancels between vertex and mediator, leaving θ^e ~ ⟨ν^c⟩/⟨θ⟩ ≈ 0.05. Three falsifiable consequences follow: with δ confirmed in [196°, 213°] the mixing angles must remain frozen at the ansatz fit within stated bands (JUNO decides); no induced CKM or first-row unitarity anomaly may accompany the phase; and δ outside [~142°, 218°] kills the mechanism outright. What is derived, what is conditional (the near-maximal phase; the vev ordering ⟨ν^c⟩/⟨θ⟩), and what is not claimed (a first-principles value of 197°) are tabulated explicitly.

Zenodo (CERN European Organization for Nuclear Research)
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Neutrino Physics Research
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