Q-Ball Spectrum Solver

About this calculation

In the Standard Model, fermion masses come from Yukawa couplings: each charged lepton, each quark, gets its own free coupling constant fitted to observation. That's roughly 13 independent fitted parameters across the lepton and quark sectors, plus three more for the gauge bosons.

MFT replaces all of those with one universal shape-fixing condition on the contraction-field potential — λ₄² = 8 m₂ λ₆ (the silver-ratio condition, derived in P2 of the corpus from a back-reaction theorem) — plus a single Coulomb coupling Z per sector. Three of the four Z values come from the medium's structure (derived); one is conjectured. The discrete soliton spectrum of the resulting equation reproduces every charged-particle mass ratio across the Standard Model, with one anchor mass per sector setting the overall energy scale.

The four sectors

Each Standard Model fermion or boson family is identified with a specific soliton in the spectrum, distinguished only by its Coulomb coupling Z and its angular momentum ℓ:

• Charged leptons (e, μ, τ) — derived Z = m² = V″(0) = 1, ℓ = 0. The Coulomb coupling equals the curvature of the potential at the linear vacuum.
• Up-type quarks (u, c, t) — derived Z = 1, ℓ = 0. Same coupling as leptons; same field-space regimes. The (u, c, t) and (e, μ, τ) families occupy identical positions in the potential.
• Down-type quarks (d, s, b) — derived Z = λ₄/(2λ₆) = 2, ℓ = 0. The Vieta average of the two nontrivial critical-point field values.
• Gauge bosons (W, Z, H) — conjectured Z = 9/5 = 1.8, ℓ = 1 for vectors (W, Z), ℓ = 0 for the Higgs. The 9/5 is conjectured from SO(3) mode counting and verified numerically at 0.07 % on m_Z/m_W. The Weinberg angle emerges as sin²θ_W = 1 − (E_W/E_Z)².
• Photon (γ) — theorem Z = 0, ℓ = 1. At Z = 0 the Q-ball equation reduces to the hydrogen Schrödinger equation, which has no bound state; therefore E → 0 exactly. The photon's masslessness is a derived consequence of the equation, not an input.

Why three families, not four or two

The Family-of-Three Stability Theorem (P3 of the corpus) gives a structural answer. The constrained energy functional E(φ_core) at fixed Noether charge has exactly three critical points: the linear vacuum minimum, a barrier maximum, and the nonlinear vacuum minimum. These correspond to three Morse-stability classes:

• n = 0 mode (stable) — the lightest particle in each sector (electron, up, down, W).
• n = 1 mode (also stable) — the second-generation particle (muon, charm, strange, Z).
• n = 2 mode (metastable) — the third-generation particle (tau, top, bottom, Higgs). Decays before the medium can fully reorganise.

Higher-n modes are multiply unstable and don't survive as physical particles. A fourth charged lepton would require a second barrier in the potential — ruled out experimentally by LEP bounds (M₄ > 100 GeV would need a barrier height of >833 MFT units, far above the observed potential structure).

The lepton sector tells a particularly clean story

Within each sector, the three families occupy three distinct regimes of the potential. For charged leptons (most clearly):

• Electron — sits at φ_core ≈ 0.02, deep in the linear vacuum. The dominant physics is the Coulomb well; the nonlinear λ₄, λ₆ terms are negligible.
• Muon — sits at φ_core ≈ 0.71, about 93 % of the way to the barrier at φ_b = 0.7654. Still in the linear regime, but the nonlinear terms start to matter.
• Tau — sits at φ_core ≈ 1.93, past the barrier in the nonlinear vacuum (φ_v = 1.85). The local contraction field has crossed the barrier, and the soliton is sextic-stabilised.

This is why the tau is rare: forming one requires the local field to cross the barrier, which costs energy comparable to twice the tau mass. The pair- and single-production thresholds shown above (0.4 % errors) follow directly from the barrier-crossing geometry — they're not independent fits.

Free parameter count vs. the Standard Model

Spelling it out, end-to-end:

• Standard Model: 9 charged fermion masses (3 leptons + 6 quarks) require 9 independent Yukawa couplings, all fitted. Plus the three gauge boson masses (three more parameters: g, g′, v).
• MFT: One shape condition (λ₄² = 8 m₂ λ₆, derived in P2), fixing the ratio λ₄/λ₆ = 4. One Z value per sector (three derived from the medium's structure, one conjectured). One anchor mass per sector for the overall energy scale.

That's it. Every other charged-particle mass — and the Weinberg angle, and the photon's masslessness, and the equal decuplet baryon spacings on the hadronic side — emerges from solving the same Q-ball equation in different regimes.

Sources

The Q-ball spectrum sector is verified by six scripts in the corpus, plus the full P4 paper:

• mft_qball_lepton_masses.py — the canonical lepton solver. Reproduces (e, μ, τ) to within 1 % from one calibration (m_e). Includes the tau production threshold prediction.
• mft_quark_sector.py — up-type quark sector. Demonstrates that (u, c, t) occupy identical field-space regimes to (e, μ, τ) at the same Z = 1.
• mft_vector_bosons.py — gauge boson sector with ℓ = 1 vectors. Predicts m_Z/m_W and m_H/m_W, derives the Weinberg angle and photon masslessness.
• mft_cross_sector.py — cross-sector universality test. Same potential, four sectors, three derived Z values plus one conjectured.
• mft_uniqueness_test.py — confirms each lepton soliton branch has exactly one energy minimum (each particle is unique on its branch, not cherry-picked from a menu).
• mft_energy_landscape.py — the constrained energy landscape E(φ_core) at fixed Q. Verifies the three-critical-point structure underlying the Family-of-Three theorem.

Full physics is in P4 of the corpus (Charged Lepton Masses from a Single Q-Ball Equation) and the cross-sector synthesis in P8 (Microphysics from the Silver Ratio). Corpus DOI: 10.5281/zenodo.19343255. See Particles for the broader MFT context.