Galactic Rotation Curves Solver

About this calculation

In the Standard Model + General Relativity, galactic rotation curves require dark matter — an invisible particle species that's never been detected directly, with halo profiles fitted empirically (NFW, Burkert, Einasto). The three-parameter NFW profile is the standard, but its shape is not derived from any underlying microphysics.

MFT replaces dark matter with a configuration of the same elastic medium that constitutes ordinary space. Concentrations of baryonic matter drive the contraction field δ(r) deep into the nonlinear vacuum of the same silver-ratio sextic potential that produces the charged-lepton mass hierarchy in the particle sector. The energy stored in this contracted field gravitates, producing dark-matter-like halos. No new particle is introduced. No modification of gravity is required. The halo profile shape is derived from particle microphysics, not fitted.

The headline: one coupling across 16 orders of magnitude

The dimensionless gravitational coupling β ≈ 10⁻⁴ is the same value that:

• Passes the Cassini Solar-System bound on PPN parameters (10 AU scale, ω_BD > 40,000), measured at the limits of post-Newtonian gravity tests. [P13]
• Enters the one-loop gravitational self-energy that gives absolute neutrino masses to ~6% (10⁻¹⁹ m scale), with best-fit β = 1.016 × 10⁻⁴. [P8]
• Reproduces all six spiral galaxy rotation curves in this dataset (kpc scale) at Σχ²/dof = 1.17 with β = 10⁻⁴. [P5]

These are three independent measurements of the same physical parameter across roughly 16 orders of magnitude in length scale, agreeing to within 1.6% on the value of β. This is the strongest empirical evidence for MFT's monistic claim that the same medium underlies particle physics and gravity.

What this solver does

For one chosen galaxy, the solver:

1. Builds the baryonic mass model from literature (Hernquist bulge, exponential disk, gas disk, central black hole — parameters from standard rotation-curve papers).
2. Solves the nonlinear contraction-field BVP on r ∈ [0.1, 80] kpc: ∇²δ + (2/r) dδ/dr = (1/κ)[m²_g δ + λ_4_g δ³ + λ_6_g δ⁵ + β ρ_baryon(r)]. The gravitational sextic coefficients (m²_g, λ_4_g, λ_6_g) come from the master-to-gravitational mapping in P5; the silver-ratio condition λ_4_g² = 8 m²_g λ_6_g propagates exactly.
3. Computes the halo energy density from the solved field: ρ_MFT = ½ κ (dδ/dr)² + V_g(δ), and integrates to get M_MFT(r).
4. Fits two free parameters per galaxy: Υ* (stellar mass-to-light ratio, standard for any halo model) and ρ_scale (overall halo amplitude, the only galaxy-specific number).
5. Returns the rotation curve v(r) = √[G·M_total(r)/r] and the residuals (Δv/σ) shown above.

Six-galaxy fit quality

The published P5 paper applies this same calculation to six spiral galaxies spanning a factor of ~50 in baryonic mass:

• Four of six galaxies achieve χ²/dof < 1.
• All six are below 3.
• Total Σχ²/dof = 1.17 — competitive with MOND's typical performance on well-measured spirals (McGaugh 2004).
• In all six galaxies the field enters the nonlinear vacuum (max|δ|/δ_b ranges from 24× to 96×). The silver-ratio double-well structure is dynamically active at galactic scales, not a perturbative correction.

Comparison: NFW uses three free parameters per galaxy. MOND uses one (per galaxy) plus a universal acceleration scale a₀. MFT uses two per galaxy (Υ*, ρ_scale) plus a universal coupling β that's also measured in two non-galactic regimes. The halo profile shape is fully determined by β and the potential — only the amplitude varies per galaxy.

The silver-ratio link to particle physics

The same potential parameters that set the lepton mass hierarchy (m_e, m_μ, m_τ from the Q-Ball solver) also set the galactic halo profile shape. The connection is the silver ratio δ = 1+√2:

• The muon sits at 93% of the barrier in the microphysical potential.
• The tau sits past the barrier in the nonlinear vacuum.
• Galactic baryons drive the field deep past the barrier into the same nonlinear vacuum.
• The stiffness ratio 4δ governs both the τ-to-electron mass ratio and the sharpness of the outer halo transition.

Particle masses and galactic halo profiles are therefore two measurements of the same potential geometry, related by δ = 1+√2.

Honest scope and limitations

Three points the published P5 paper makes explicitly:

• Per-galaxy halo amplitude. ρ_scale varies by 1.55 dex across the sample — more than pure unit-conversion can explain. Either our simplified spherical baryon models miss structure, or the scalar sector couples to matter slightly differently per galaxy. Imposing one global ρ_scale raises Σχ²/dof by 2-3×.
• Simplified baryon models. We use sphericalised Hernquist bulges and exponential disks; real disks are planar. Better HI maps would likely reduce residuals — but MOND comparisons in the literature use similarly simple models and reach χ²/dof ~ 1-3.
• Cluster-scale systems are not yet accommodated. The BVP formulation as it stands does not reproduce the much larger halos needed for massive elliptical central galaxies in clusters (e.g. M87 in Virgo). The scalar-sector halo in its current form is a galaxy-scale phenomenon; cluster-scale physics may require additional ingredients. This is an open problem, flagged explicitly by the paper, not a claim it makes.

Sources

The galactic-rotation calculation cross-references several scripts and papers:

• mft_galactic_nonlinear.py — the canonical galactic BVP solver (this page is a browser adaptation). Reproduces the full six-galaxy fit at Σχ²/dof = 1.17.
• mft_microphysics.py — the one-loop neutrino-mass calculation that independently fixes β = 1.016 × 10⁻⁴.
• mft_compact_objects.py — the Solar-System screening calculation that confirms β ≲ 10⁻⁴ from Cassini bounds on PPN parameters.
• mft_grav_verification.py — verifies the master-to-gravitational mapping that derives the galactic-scale potential from the microphysical master potential.

Full physics is in P5 of the corpus (Galactic Rotation Curves from the Silver Ratio) with cross-references to P0 (gravitational field equations), P8 (microphysics, β from neutrino masses), and P13 (compact objects, β from Cassini). Corpus DOI: 10.5281/zenodo.19343255. See Gravity for the broader MFT context.