Neutrino Hierarchy Solver
Predict the three neutrino masses and their hierarchy from the silver-ratio MFT potential — with no parameters fitted to neutrino data. The hierarchy ratio Δm²₃₂/Δm²₂₁ is determined by the silver ratio δ = 1+√2 alone: δ⁴ − 1 = 16 + 12√2 ≈ 32.97, matching the observed PDG value 32.58 to 1.2%. The absolute scale comes from a one-loop self-energy with the same gravitational coupling β that fits galactic rotation curves and passes the Cassini Solar-System bound.
Best-fit β = 1.016 × 10⁻⁴ gives Δm²₂₁ to 0.58% and Δm²₃₂ to 0.62% against PDG 2024 — three independent measurements of β (galactic, Solar-System, neutrino) all agreeing to 1.6% across 16 orders of magnitude in length scale.
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Connection to the Family-of-Three theorem
The Family-of-Three theorem (P3) says three modes are admissible in any sextic-confining potential. The neutrinos are the neutrino-sector incarnation of F3 — one neutrino per critical point of V(φ):
- ν₁ ↔ V(0) = 0 — the linear vacuum critical point. Mass = 0 in this mechanism.
- ν₂ ↔ V(φ_b) = 1/(3δ) — the barrier critical point.
- ν₃ ↔ V(φ_v) = δ/3 — the nonlinear-vacuum critical point.
The mass-squared splitting at each is proportional to V²(φᵢ) — the square of the potential energy at that critical point. The ratio of splittings comes out exactly as δ⁴ − 1 = 16 + 12√2, the thirteenth manifestation of the silver ratio in MFT. See the Family-of-Three Visualizer for the lepton-sector incarnation (where F3 produces the e, μ, τ Q-ball modes).
One coupling β across 16 orders of magnitude
The gravitational coupling β used to set the absolute neutrino-mass scale here is the same value that:
- Passes the Cassini Solar-System bound on PPN parameters (10 AU scale, ωBD > 40,000). [P13]
- Fits all six spiral galaxy rotation curves on the Galactic Solver at Σχ²/dof = 1.17 (kpc scale). [P5]
- Reproduces both observed neutrino mass-squared splittings to under 1 % (10⁻¹⁹ m scale, this page). [P8]
Three independent measurements of β across roughly 16 orders of magnitude in length scale, agreeing to within 1.6 %. The neutrino fit pins the value to β = 1.016 × 10⁻⁴ — currently the most precise of the three measurements. This is the strongest empirical evidence for MFT's monistic claim that the same elastic medium underlies particle physics and gravity.
Three sister solvers tell the same story together: the Q-Ball Spectrum (charged leptons + quarks + bosons) and this page (neutrinos) are two different applications of the same silver-ratio sextic potential at two different couplings. The Galactic Solver applies the same potential and the same β at galactic scale. The Family-of-Three Visualizer provides the structural reason all three sectors have exactly three modes.
About this calculation
In the Standard Model, neutrino masses are added by hand: the SM originally had massless neutrinos, and after the discovery of neutrino oscillations in 1998, three additional Yukawa-like couplings (or six, in the Majorana case) were grafted onto the model to give the observed mass-squared splittings. The mixing angles and mass values are fitted, not derived; no version of the SM explains why the hierarchy ratio is what it is, or why there are three neutrinos and not four.
MFT predicts both. The number three follows from the Family-of-Three theorem; the hierarchy ratio is fixed by the silver ratio δ = 1+√2; the absolute masses use the same gravitational coupling β as the galactic solver. No parameters are fitted to neutrino data.
The two-mechanism story
Two physical mechanisms — both derived from the MFT action — produce the neutrino mass spectrum:
-
Hierarchy ratio (closed-form algebra):
The mass-squared splitting at each critical point φᵢ of the sextic
potential is proportional to V²(φᵢ) — the square of the potential
energy. This V² mechanism arises from second-order gravitational
back-reaction through the F(φ)R coupling in the MFT action. At the
silver-ratio condition λ₄² = 8 m² λ₆, the potential values lock at
V(0) = 0, V(φ_b) = 1/(3δ), V(φ_v) = −δ/3, and the ratio of splittings
simplifies to:
Δm²₃₂ / Δm²₂₁ = δ⁴ − 1 = (1+√2)⁴ − 1 = 16 + 12√2 ≈ 32.97
Observed (PDG 2024): 32.58 ± 0.3. Agreement to 1.2 %, parameter-free. -
Absolute masses (one-loop with universal screening):
The conformal coupling F(φ) = 1+βφ shifts the Einstein-frame mass at
each critical point by δm²(φᵢ) = 6β²V(φᵢ). The neutral mode propagates
through the bulk medium with a universal screening mass M²_s = V''(φ_v) +
V''(0) = 4δ + 1 = δ(δ+2) — the same for all three neutrinos.
The 3D one-loop integral (dim reg) gives I = 1/(32π M_s³), and the
masses come out as:
m_νᵢ = 3 β² |V(φᵢ)| × √I × (m_e/E_e)
Every factor in this formula is derived from the MFT action: β² is the gravitational coupling squared (measured), |V(φᵢ)| is the potential at critical point i, the prefactor 3/√(32π) is the one-loop dim-reg result, and the screening factor [δ(δ+2)]⁻³/⁴ is fixed by the silver ratio.
The headline numbers
At the canonical silver-ratio parameters (m₂ = 1, λ₄ = 2, λ₆ = 0.5) with β = 1.016 × 10⁻⁴:
- m_ν₁ = 0 eV — the linear vacuum has V(0) = 0 exactly. This is the lightest neutrino in the canonical mechanism.
- m_ν₂ = 8.65 meV — the barrier critical point.
- m_ν₃ = 50.4 meV — the nonlinear vacuum critical point.
- Σm_ν = 59 meV — well below the Planck cosmological bound of 120 meV. Falsifiable: future measurements pushing the bound below ~60 meV would rule out the canonical mechanism.
Mass-squared splittings:
- Δm²₂₁ = 7.49 × 10⁻⁵ eV² (observed 7.53 × 10⁻⁵, error 0.58 %)
- Δm²₃₂ = 2.47 × 10⁻³ eV² (observed 2.45 × 10⁻³, error 0.62 %)
For context: the Standard Model needs three independent fitted parameters to reproduce these two splittings (the three Majorana phases that enter m_ν₁², m_ν₂², m_ν₃²). MFT reproduces both to under 1 % with one parameter (β) measured independently in two non-neutrino regimes.
The thirteenth manifestation of the silver ratio
The δ⁴ − 1 prediction is striking because it's structurally parameter-free: it depends on the silver ratio δ alone, not on the absolute scale of any parameter. This is one of fourteen independent manifestations of δ in MFT, spanning six powers of δ and the mixed combination δ(δ+2). In the lepton sector, the corresponding manifestation is the field-space ratio φ_v/φ_b = δ. In the neutrino sector, it's the hierarchy ratio Δm²₃₂/Δm²₂₁ = δ⁴ − 1. Same δ, different physical observable, same theorem (the Symmetric Back-Reaction Theorem in P2 of the corpus).
The sensitivity plot above makes the falsifiability concrete: a 10 % deviation in λ₄² shifts the predicted ratio away from observation by ~64 %. The silver-ratio condition is the source of the precision, and any future tightening of the observed Δm²₃₂/Δm²₂₁ measurement either further confirms the silver-ratio condition or rules it out.
Honest scope and limitations
- Mass ordering: the V² mechanism gives normal ordering naturally (m_ν₁ < m_ν₂ < m_ν₃). If next-generation experiments establish inverted ordering, the canonical V² mechanism would need revision.
- Lightest neutrino mass: m_ν₁ = 0 in the canonical mechanism (V(0) = 0 exactly). Cosmological measurements or 0νββ-decay detection of m_ν₁ > 0 eV would require an additional contribution not present in the current corpus calculation.
- PMNS mixing matrix: the current calculation gives masses but not mixing angles or the CP phase. The PMNS matrix is one of three remaining open problems in the corpus (alongside cosmology and SU(3)). Mixing comes from the geometric overlap structure between three classes of solitons mediated by W-boson modes — a calculation the corpus has scoped but not executed.
- Other neutrino-mass mechanisms ruled out: the corpus explicitly tested four alternatives — static neutral solitons, Z = β Q-balls, phase fluctuations of the contraction field, and phase-channel binding — all four were ruled out (energies far off, by factors of 10⁹ in the worst case). The conformal-coupling + 1-loop self-energy mechanism on this page is the only one that survives.
Sources
The neutrino-mass calculation is verified by four scripts in the corpus:
mft_neutrino_hierarchy.py— the canonical closed-form derivation of the δ⁴ − 1 hierarchy ratio.mft_neutrino_masses.py— the absolute-mass calculation via 1-loop self-energy with universal screening; this page's solver derives from this script.mft_neutrino_quartic_gradient.py— systematic investigation of the quartic-gradient neutral-soliton alternative. (One of the four alternatives ruled out; preserved for completeness.)mft_neutrino_qball.py— the gravitationally-bound Q-ball variant. (Also ruled out as the primary mechanism; preserved for completeness.)
The full physics is in P8 of the corpus (Microphysics from the Silver Ratio) with cross-references to P2 (the silver-ratio condition λ4² = 8 m2 λ6), P3 (the Family-of-Three theorem), P5 (galactic β = 10⁻⁴), and P13 (Cassini bound on β). Corpus DOI: 10.5281/zenodo.19343255. See Particles for the broader MFT context.