Family-of-Three Visualizer

About this calculation

The Standard Model contains exactly three charged leptons (e, μ, τ) with identical gauge quantum numbers but widely separated masses. Why three? In the SM, this number is implemented as an empirical input — encoded in the fermion content but not derived from any structural principle. A large literature tries to explain it using horizontal symmetries, textures, extra dimensions, or compositeness, but in every approach the number "three" is imposed somewhere in the construction.

MFT takes a different route. The Family-of-Three Stability Theorem (P3 of the corpus) shows that for a broad class of nonlinear scalar field theories with a confining sextic potential, at most three localised modes are dynamically admissible when physically motivated constraints are imposed. The number three is a topological consequence of the variational structure, not a fitted parameter.

The theorem in one sentence

Under the assumptions "sextic confining potential + fixed Noether charge + dilation projected out":

m_phys(u_n) = max(0, n − 1)

The physical Morse index of the n-th radial mode is exactly max(0, n−1). This makes only three modes admissible as observable particles:

• n = 0 (electron): stable. Index 0 — no decay channels.
• n = 1 (muon): also stable in the lepton sector. Index 0 — decays only via electroweak emission of a neutrino, not via the F3 instability.
• n = 2 (tau): metastable. Index 1 — exactly one decay direction.
• n ≥ 3: Index 2 or more. Multiply unstable, decays simultaneously along multiple channels, never observable as a long-lived particle.

How the index reduction works

The unconstrained problem on full function space gives the simple result m_un(u_n) = n (Sturm-Liouville oscillation theorem). MFT then imposes two physical constraints that each remove negative directions:

• Fixed Noether charge Q = Q₀. The total contraction of the elastic medium is conserved. This restricts perturbations to the tangent space of the constraint surface — but the charge gradient direction itself does not contribute a negative eigenvalue, so this constraint does not change the index.
• Dilation projected out. For each soliton there is one direction in function space corresponding to global radial dilation (u_n(r) → u_n(r/μ)). This direction is a redundancy of the description — like a gauge direction — and contributes one negative eigenvalue that we project out. This is the −1 in n − 1.

The result m_phys = n − 1 is therefore not a coincidence, it's the unconstrained index minus one (one negative direction = the dilation redundancy).

Why this is a topological result, not a fit

The theorem assumes only:

• A tachyonic quadratic region stabilised by a sextic tail (K₂ > 0, K₄ < 0, K₆ > 0).
• The barrier-existence condition K₄² > 4 K₂ K₆ (which gives the double-well structure).
• The two physical constraints above.

In MFT, these three structural inputs are all derived rather than assumed: K₂ > 0 from soliton localisation, K₄ < 0 from Derrick's theorem applied to Q-ball solitons in 3D (P1), K₆ > 0 from the requirement that the energy is bounded below, and the barrier condition from the Symmetric Back-Reaction Theorem (P2). The 14-configuration stress test above shows that the index pattern is preserved across ±10 % variations in each numerical parameter — confirming that the three-family count is a topological feature of the variational problem, not a fine-tuned coincidence.

What this calculation does NOT predict

The theorem explains why three, not how heavy. The position-space model on this page yields E₂/E₁ = 1.91, which is far from the observed m_τ/m_μ = 16.82. This is by design: F3 is topological. The mass ratios require the full nonlinear Q-ball dynamics in field space — which is what the Q-Ball Spectrum Solver computes. There, with the silver-ratio condition λ₄² = 8 m₂ λ₆, the same theorem yields m_τ/m_μ = 16.95 (0.8% from observation).

The two pages are two views of the same theorem: F3 says three modes are admissible; the Q-Ball solver shows which three modes the four Standard Model sectors actually fill (charged leptons, up-quarks, down-quarks, gauge bosons) and what their masses are.

Why this isn't a live solver

Computing the F3 dataset at full resolution (N = 3000, four modes, full Hessian eigendecomposition with constraint projection) takes minutes on a workstation and tens of minutes in a browser via Pyodide. The result is also a one-shot proof: at the canonical silver-ratio parameters, the Morse-index pattern is (0, 0, 1, 2) — and that's the theorem. The 14 stress tests above confirm there's nothing meaningful to vary that would change the qualitative answer.

So this page presents the published canonical result rather than recomputing it. To verify it yourself, run family_of_three_theorem.py from the corpus locally — the JSON output it produces is exactly what's visualised here. The canonical command is:

python family_of_three_theorem.py --Rmax 14.0 \
  --N 3000 --Z 1.1 --a 1.1451604 --k2 0.039 --k4 2e-06 \
  --k6 2e-07 --solve_up_to_n 3 --fluct_spectrum \
  --report --outfile_json theorem_results.json

Sources

The F3 theorem and its numerical verifications are spread across four scripts and three corpus papers:

• family_of_three_theorem.py — the canonical position-space solver. Computes the four soliton modes, the unconstrained spectrum, and the constrained Morse indices. The JSON output is exactly what this page visualises.

  python family_of_three_theorem.py --Rmax 14.0 \
    --N 3000 --Z 1.1 --a 1.1451604 --k2 0.039 --k4 2e-06 \
    --k6 2e-07 --solve_up_to_n 3 --fluct_spectrum \
    --report --outfile_json theorem_results.json

• mft_family_of_three_theorem_presentation.py — the full publication-ready run, including all 14 stress tests and the extension to n = 5.

  python mft_family_of_three_theorem_presentation.py --Rmax 14.0 \
    --N 3000 --Z 1.1 --a 1.1451604 --k2 0.039 --k4 2e-06 \
    --k6 2e-07 --solve_up_to_n 3 --fluct_spectrum \
    --report --outfile_json mft_family_of_three_theorem_results.json

• figures_family_of_three.py — figure generation for the published P3 paper.
• mft_energy_landscape.py — the independent verification via the Q-ball energy landscape (shown in the green panel above): E(φ_core; Q = Q₀) has exactly three critical points.

The full physics is in P3 of the corpus (The Family-of-Three Stability Theorem in MFT) with cross-references to P1 (the K₄ < 0 derivation), P2 (the silver-ratio condition λ₄² = 8 m₂ λ₆), and P4 (the Q-ball mass calculation that turns the three admissible modes into m_e, m_μ, m_τ). Corpus DOI: 10.5281/zenodo.19343255. See Particles for the F3 theorem in the broader MFT context.