Energy Landscape Visualizer
The MFT sextic potential has just one free shape parameter: ρ = λ4² / (m² λ6). Slide it around. At ρ = 8 exactly, the Symmetric Back-Reaction Theorem (P2) is satisfied, the field-space ratio φv/φb equals the silver ratio δ = 1+√2, and the hierarchy ratio V²(φ_v)/V²(φ_b) − 1 locks at δ⁴ − 1 = 32.97 — matching the observed neutrino mass-hierarchy ratio to 1.2%. Move ρ off 8 and watch the prediction break.
Every other MFT solver lives at ρ = 8. This page is the parameter-space context for those calculations — what the potential looks like, and what would happen if the silver-ratio condition were violated.
V(φ) at ρ = 8.000
silver-ratio conditionTwo senses of "energy landscape"
MFT discusses two related but distinct objects, both sometimes called "energy landscapes." It's worth being clear which one this page shows:
- This page: the bulk potential V(φ) = m²φ²/2 − λ4φ⁴/4 + λ6φ⁶/6 as a function of the contraction-field amplitude φ, parametrised by ρ. Closed-form algebra. The shape of V(φ) is what determines whether solitons exist at all and, if so, what their mass hierarchy is.
-
The constrained Q-ball energy landscape:
E(φ_core; Q = Q0) — the total energy of a Q-ball
soliton as a function of its central amplitude, at fixed Noether charge.
This is what
mft_energy_landscape.pyandenergy_landscape_doublewell.pycompute (~3 minutes each on a workstation, impractical in the browser). At ρ = 8 it has exactly three critical points (muon, barrier, tau) — see the green "Independent verification" panel on the F3 Visualizer where this is shown.
Why ρ = 8 specifically
The condition ρ = 8 is not a fitted choice. It's the unique value at which Σ(φ_b) = Σ(φ_v) — the gravitational back-reaction strengths at the barrier and at the nonlinear vacuum are equal. The Symmetric Back-Reaction Theorem (P2 of the corpus) forces this, and the silver ratio δ = 1+√2 is the unique fixed point of the resulting back-reaction map.
The live readout in the sidebar shows how dramatically things break when you move off 8: a 10% deviation in ρ already shifts the hierarchy ratio by ~60-100% from its silver target. This is why the F3 theorem produces exactly three modes only at this specific ρ, why the Q-ball spectrum reproduces the lepton mass ratios only at this ρ, and why the neutrino hierarchy matches PDG only at this ρ. The four sister solvers are doing physics at one point in the parameter space this page lets you tour.
The four sister solvers all live at ρ = 8. The Q-Ball Spectrum computes lepton, quark, and boson masses at this point. The Neutrino Hierarchy gives Δm²₃₂/Δm²₂₁ = δ⁴ − 1 here. The Family-of-Three Visualizer shows the constrained Morse-index analysis at this point, with stress tests across ±10% perturbations confirming the topological robustness of the result. The Galactic Solver applies the same potential at galactic scale. This page shows you the parameter-space context where they all sit.
About this calculation
The MFT contraction-field potential is a sextic polynomial: V6(φ) = m²φ²/2 − λ4φ⁴/4 + λ6φ⁶/6. It has three free coefficients (m², λ4, λ6), but two of them set the units of φ and V. Only one dimensionless number controls the shape of the potential:
ρ = λ4² / (m² λ6)
Every physically meaningful claim about the potential's shape — whether it has a barrier, where the critical points are, what the field-space ratio is, what the hierarchy ratio is — depends only on ρ. This page lets you slide ρ through its full physically interesting range and watch what happens.
The five regimes
As ρ varies, V(φ) goes through five qualitatively distinct regimes:
- ρ < 4 — discriminant negative. No critical points other than φ = 0. Single well, no barrier, no nonlinear vacuum. Soliton solutions don't exist. Incompatible with MFT.
- ρ = 4 — discriminant zero. Inflection point at φ = √(λ4/2λ6). Barrier just emerging — the boundary case.
- 4 < ρ < 8 — barrier exists but the double-well is asymmetric. The back-reaction strengths Σ(φ_b) ≠ Σ(φ_v). The hierarchy ratio is below δ⁴ − 1 (and even passes through zero around ρ ≈ 6, where V_b² = V_v²).
- ρ = 8 (silver-ratio condition) — Σ(φ_b) = Σ(φ_v) holds exactly. φv/φb = δ = 1+√2 (the silver ratio). V(φ_b) = +1/(3δ), V(φ_v) = −δ/3. Hierarchy ratio = δ⁴ − 1 = 32.97.
- ρ > 8 — the double-well becomes increasingly skewed in the opposite direction: barrier shrinks, vacuum deepens. The hierarchy ratio overshoots δ⁴ − 1, badly, and grows roughly as ρ².
Why the silver-ratio condition is forced
The condition ρ = 8 — equivalently λ4² = 8 m² λ6 — is the Symmetric Back-Reaction (SBR) condition, derived in P2 of the corpus from a self-consistency requirement on the gravitational back-reaction map.
Briefly: when the contraction field has two distinct nontrivial critical points (the barrier and the nonlinear vacuum), each one sources its own metric back-reaction with strength Σ(φ) = V(φ)/V''(φ). The SBR theorem requires these strengths to be equal at the two critical points, otherwise the two-vacuum structure is inconsistent with the underlying gravitational action. Imposing Σ(φ_b) = Σ(φ_v) gives the unique condition λ4² = 8 m² λ6, and the silver ratio δ = 1+√2 falls out as the unique fixed point of the back-reaction map.
This is what makes the silver ratio appear in 14 independent manifestations across the corpus: the lepton mass ratios, the pion decay constant (fπ² = δ to 0.03%), the neutrino hierarchy (Δm²₃₂/Δm²₂₁ = δ⁴ − 1), the field-space ratio φv/φb = δ, and many more. They all trace back to this one condition on V(φ).
The hierarchy ratio is the most sensitive diagnostic
The live readout in the sidebar shows the hierarchy ratio Δm²₃₂/Δm²₂₁ = [V²(φ_v) − V²(φ_b)] / V²(φ_b) at the current ρ. Because this ratio depends on V4, it's extremely sensitive to the silver-ratio condition:
- ρ = 8.0 → 32.97 exactly (δ⁴ − 1)
- ρ = 7.92 (1% off) → ~30.0 (drift −9%)
- ρ = 7.20 (10% off) → ~11.8 (drift −64%)
- ρ = 8.80 (10% off in the other direction) → ~71.5 (drift +117%)
The PDG-observed value is 32.58 ± 0.3 — a 1.2% discrepancy from MFT's δ⁴ − 1 prediction. Tightening this measurement either tightens the silver-ratio claim further or rules it out. Watching the hierarchy ratio drift live as you slide ρ shows directly what falsifiability looks like in MFT: a 1% miss in ρ corresponds to a ~10% miss in the most precise physical observable this potential predicts.
What this page is, and isn't
This page is a closed-form algebra explorer for the bulk potential V(φ). The slider varies ρ; the JavaScript computes critical points and V values from the sextic formula; the canvas plots V(φ). No solitons, no Q-ball dynamics, no constrained extremization.
This page is not the constrained Q-ball energy landscape
E(φ_core; Q = Q0) — that's the calculation in
mft_energy_landscape.py and energy_landscape_doublewell.py,
which takes minutes per run on a workstation and is shown (with cached results)
in the green "Independent verification" panel on the
F3 Visualizer.
Sources
The closed-form formulas for V(φ_b), V(φ_v), φ_v/φ_b, and the hierarchy ratio at the silver-ratio condition are derived in P1 and P2 of the corpus. The Python scripts that compute the constrained energy landscape (which is the verification of the Family-of-Three theorem in field space) are separate calculations:
mft_energy_landscape.py— computes E(φ_core; Q = Q0) at the canonical silver-ratio parameters, and verifies that it has exactly three critical points (muon at φ_core ≈ 0.83, barrier at 1.72, tau at 1.99). Shown on the F3 page.energy_landscape_doublewell.py— alternative implementation of the same calculation. Independent verification of the three-critical-point result.
Full physics is in P1 (sign constraints on the sextic potential — K4 < 0 from Derrick's theorem on Q-ball solitons), P2 (the Symmetric Back-Reaction theorem deriving λ4² = 8 m² λ6), and P3 (the Family-of-Three theorem operating at the silver-ratio condition). Cross-references throughout to P4 (lepton masses), P5 (galactic halos), P8 (microphysics including neutrino masses). Corpus DOI: 10.5281/zenodo.19343255. See Foundations for the broader MFT context.