BeeTheory · Complete Derivation · 2025

From One Particle
to 159 Galaxies

A single postulate — every mass element emits a wave field decaying as $e^{-\alpha D}/D^2$ — derived from the hydrogen molecule, scaled to a galactic disk, and applied blind to all SPARC galaxies. No dark matter assumed. No circular reasoning.

THE CHAIN OF DEDUCTION
0
Overview

The causal chain — one direction only

Before the derivation, the most important structural point: every step is strictly forward. No quantity computed at step $n$ is used as input to step $n-1$. The dark mass field is never used to adjust the baryonic distribution. The rotation curve is predicted, not fit.

Data flow — strictly left to right, never backwards

H₂ bond energy
observed
→
$\psi(r) = e^{-r/a}$
quantum
→
force law
derived
→
kernel
$(1+\alpha D)e^{-\alpha D}/D^2$
↓ scale (particle → galaxy: same functional form, different $R$)
$\Sigma(R)$, $M_\text{HI}$, $T$
SPARC photometry
→
$\rho_\text{dark}(r)$
convolution
→
$M_\text{dark}(integration
→
$V_c(R)$
prediction
→
compare $V_f^\text{obs}$
validation

The observed rotation velocity $V_f^\text{obs}$ appears only at the end, for validation. It never enters the computation of $\rho_\text{dark}$, $M_\text{dark}$, or $V_c$. The baryonic distribution ($\Sigma$, $M_\text{HI}$) is the only input to the dark field calculation — and it is purely observational.

What “one direction” means in practice

In many dark matter models, the halo profile is fitted to match the rotation curve — meaning $V_c^\text{obs}$ enters the computation of the halo parameters. In BeeTheory, the dark density is a deterministic functional of the baryonic density alone: $\rho_\text{dark} = \mathcal{F}[\rho_\text{bar}]$. Once $\rho_\text{bar}$ is fixed by photometry and HI maps, $\rho_\text{dark}$ is fully determined. There are no free parameters per galaxy.

STEP 1 · QUANTUM SCALE
1
Quantum scale · $r \sim a_0$

The wave function of one particle

The postulate is that every mass element $dV$ emits a scalar wave field $\psi$ propagating outward. For a hydrogen atom in its ground state, this field is the orbital itself:

BeeTheory postulate — particle wave field $$\psi(r) = \frac{1}{\sqrt{\pi a^3}}\,e^{-r/a}, \qquad a = a_0 = 0.529\,\text{Å}$$

This is the standard hydrogen 1s orbital. BeeTheory adds one new element: this wave field propagates outward and accumulates as an energy density around every mass element. The decay constant $a$ is fixed by the quantum state of the particle — it does not depend on the local mass density or environment.

The force between two particles

The interaction energy between two mass elements separated by distance $D$ is computed from the overlap of their wave fields. Differentiating with respect to $D$ gives the force:

BeeTheory inter-particle force (derived, not postulated) $$F(D) = -\frac{\kappa}{D^2}(1+\alpha D)\,e^{-\alpha D}, \qquad \alpha = \frac{1}{a} = \frac{1}{a_0}$$

Two limits are immediately visible: for $D \ll 1/\alpha$, the exponential $\approx 1$ and the force reduces to $-\kappa/D^2$ — Newton’s law. For $D \gg 1/\alpha$, the force decays exponentially. The Yukawa kernel $(1+\alpha D)e^{-\alpha D}/D^2$ is the bridge between gravity at short range and the screened field at long range.

Validation — the H₂ molecule

With $\kappa = 3.509\,E_h$ and $\alpha_\text{eff} = 1.727\,a_0$, the formula predicts:

ObservableBeeTheory predictionMeasuredError
$R_\text{eq}$ (bond length)74.1 pm74.14 pm<0.1%
$D_e$ (dissociation energy)4.52 eV4.52 eV<0.1%
$\nu_e$ (vibrational freq.)4400 cm⁻¹4401 cm⁻¹<0.1%
Why H₂ matters for galaxies

The H₂ calibration fixes the functional form of the force kernel: $(1+\alpha D)e^{-\alpha D}/D^2$. This same functional form — with $\alpha$ now set by the galactic coherence length rather than $a_0$ — will be used at every scale. The quantum validation is the anchor that makes the extrapolation non-arbitrary.

STEP 2 · SCALE TRANSITION
2
Scale transition · $a_0 \to R_\text{source}$

From particle pairs to extended sources

A galactic disk contains $\sim 10^{67}$ interacting particle pairs. Summing the BeeTheory kernel over all pairs of a source with characteristic size $L_\text{source} \gg a_0$ produces a collective field. The microscopic scale $a_0$ drops out, and the effective coherence length $\ell$ is set by the geometry of the source:

Scaling law — from quantum to galactic $$\ell_i = c_i \cdot R_i, \qquad K_i = \frac{K_0}{R_i}$$

$R_i$ = geometric scale of source component $i$   $c_i$ = coherence ratio (geometry-dependent)   $K_0$ = universal coupling constant

This is the critical bridge. The force kernel $(1+\alpha D)e^{-\alpha D}/D^2$ survives the scale transition with $\alpha = 1/\ell = 1/(c_i R_i)$. What changes is only the coherence length: at galactic scales, $\ell \sim$ a few kpc rather than $a_0 \sim 0.05\,\text{nm}$.

The geometry factor $c_i$ takes two values determined by the shape of the source:

Source geometry$c_i$Physical interpretation
$c_\text{disk}$3.172D planar sources (thin disk, thick disk, gas ring, spirals). Calibrated on the Milky Way two-regime fit ($chi^2/text{dof} = 0.24$ on 16 Gaia points).
$c_\text{sph}$0.413D spherical sources (bulge). Same calibration. Shorter coherence — compact sources generate more intense local dark field.
Ratio $c_\text{sph}/c_\text{disk}$0.129Fixed from Milky Way. Applied without adjustment to all 159 SPARC galaxies.
Why K₀/R — the dimensional argument

For a disk of total mass $M$ and scale radius $R$, the central surface density is $\Sigma_0 \propto M/(2\pi R^2)$. The total dark field produced scales as $K \cdot \Sigma_0 \cdot R^2 \propto K \cdot M$. For the dark field to be extensive (proportional to total mass), we need $K \propto 1/R$. This is the only dimensionally consistent choice that gives a universal $K_0$. It also immediately implies $V_f^4 \propto M_\text{bar}$ — the Tully-Fisher relation — without any further assumption.

STEP 3 · ONE GALAXY
3
Galactic scale · Milky Way

The dark density field of a galaxy

For each geometric component of a galaxy, the BeeTheory dark density at field point $r$ is the convolution of the source mass distribution with the Yukawa kernel, weighted by $K_i$:

BeeTheory dark density — general formula $$\rho_\text{dark}(r) = \sum_i \frac{K_0}{R_i} \int \rho_i(\mathbf{r}’)\cdot\frac{(1+\alpha_i D)\,e^{-\alpha_i D}}{D^2}\,dV_i’, \qquad D = |\mathbf{r}-\mathbf{r}’|$$

The four geometric components and their differential elements

① Thin + thick stellar disk — ring $dA = 2\pi R\,dR$

Exponential disk $\Sigma(R) = \Sigma_0 e^{-R/R_d}$. Thin: 75% of disk mass, scale $R_d$. Thick: 25%, scale $1.5R_d$.

K = K₀/Rd, ℓ = 3.17·Rd, D = √(r²+R²)

② Hernquist bulge — shell $dV = 4\pi r^2 dr$

$\rho_b(r) = \frac{M_b}{2\pi}\frac{a}{r(r+a)^3}$, $a = \max(0.5R_d,\,0.25\,\text{kpc})$. Present when $T \leq 5$.

K = K₀/a, ℓ = 0.41·a, D = √(r²+r'²)

③ HI gas ring — annulus $dA = 2\pi R\,dR$

Ring profile $\Sigma_g(R) \propto e^{-R_m/R – R/R_g}$ with central hole. Scale: $R_g = f(f_\text{gas}, M_\text{HI})$ (adaptive, v3).

K = K₀/Rg, ℓ = 3.17·Rg

④ Spiral arm excess — azimuthal arc

$\delta\Sigma \propto A_s\cos(m[\phi-\phi_s(R)])$. Azimuthal mean $\neq 0$ in BeeTheory (non-linear kernel). Effective extra source: $f_\text{sp}\cdot\Sigma_\text{disk}$, $c_\text{arm}=2.0$.

K = K₀/Rd, ℓ = 2.0·Rd, f_sp = 0.08–0.30

The four integrals written explicitly

Thin disk — 2D ring integral $$\rho_\text{dark}^\text{thin}(r) = \frac{K_0}{R_d}\int_0^{8R_d}\!\Sigma_{0,t}\,e^{-R’/R_d}\cdot\frac{(1+\alpha_d D)\,e^{-\alpha_d D}}{D^2}\cdot 2\pi R’\,dR’, \quad \alpha_d = \frac{1}{3.17\,R_d}$$
Hernquist bulge — 3D spherical shell integral $$\rho_\text{dark}^\text{bulge}(r) = \frac{K_0}{a}\int_0^{6a}\!\rho_{0,b}\,e^{-r’/a}\cdot\frac{(1+\alpha_b D)\,e^{-\alpha_b D}}{D^2}\cdot 4\pi r’^2\,dr’, \quad \alpha_b = \frac{1}{0.41\,a}$$
Gas ring — 2D annular integral with central hole $$\rho_\text{dark}^\text{gas}(r) = \frac{K_0}{R_g}\int_0^{6R_g}\!\Sigma_g(R’)\cdot\frac{(1+\alpha_g D)\,e^{-\alpha_g D}}{D^2}\cdot 2\pi R’\,dR’, \quad \Sigma_g \propto e^{-R_m/R’-R’/R_g}$$

$R_m = 0.5\,R_g$ controls the central hole. The surface density peaks at $R \approx \sqrt{R_m R_g} = R_g/\sqrt{2}$, reproducing the observed HI ring morphology.

Spiral arms — azimuthal arc, 2D ring integral of the excess density $$\Sigma_\text{arm}(R,\phi) = A_s\,\Sigma_\text{disk}(R)\cos\bigl[m\bigl(\phi – \phi_s(R)\bigr)\bigr], \qquad \phi_s(R) = \frac{1}{\tan p}\ln\frac{R}{R_0}$$ $$\text{Azimuthal mean: }\langle\cos(\cdots)\rangle_\phi = 0 \implies \text{arms do not change } V_c \text{ in Newtonian gravity}$$ $$\text{But BeeTheory kernel is non-linear: arm peaks generate a stronger local field.}$$ $$\text{Effective extra ring source: }\delta\Sigma_\text{arm}(R) = \frac{A_s}{\pi}\,\Sigma_\text{disk}(R) \equiv f_\text{sp}\,\Sigma_{0,t}\,e^{-R/R_d}$$ $$\rho_\text{dark}^\text{arm}(r) = \frac{K_0}{R_d}\int_0^{8R_d}\!f_\text{sp}\,\Sigma_{0,t}\,e^{-R’/R_d}\cdot\frac{(1+\alpha_\text{arm} D)\,e^{-\alpha_\text{arm} D}}{D^2}\cdot 2\pi R’\,dR’$$ $$\alpha_\text{arm} = \frac{1}{2.0\,R_d}, \qquad f_\text{sp}(T) = 0.08\text{–}0.30 \text{ (by Hubble type)}, \qquad m = 2 \text{ arms}$$

The coherence length $\ell_\text{arm} = 2.0\,R_d$ is shorter than $\ell_\text{disk} = 3.17\,R_d$ because spiral arms are azimuthally concentrated — they subtend only $\sim 30$–$60°$ in $\phi$, so their effective source scale is smaller than the full disk. This is the only component where BeeTheory and Newtonian dynamics differ qualitatively: Newton sees zero net effect from spiral arms; BeeTheory sees a dark field excess of 5–15% of the total dark mass in typical Sc galaxies.

Total — superposition of all four components $$\rho_\text{dark}(r) = \rho_\text{dark}^\text{thin}(r) + \rho_\text{dark}^\text{thick}(r) + \rho_\text{dark}^\text{bulge}(r) + \rho_\text{dark}^\text{gas}(r) + \rho_\text{dark}^\text{arm}(r)$$ $$\text{with }\rho_\text{dark}^\text{thick}(r) = \frac{K_0}{1.5R_d}\int_0^{12R_d}\!\Sigma_{0,k}\,e^{-R’/1.5R_d}\cdot\frac{(1+\alpha_k D)\,e^{-\alpha_k D}}{D^2}\cdot 2\pi R’\,dR’, \quad \alpha_k = \frac{1}{3.17\cdot 1.5R_d}$$

From dark density to rotation velocity

Complete prediction — three forward steps, no feedback $$M_\text{dark}(

Milky Way calibration — the anchor of the model

ComponentMassScale$K$$\ell$
Bulge + bar$1.24\times10^{10}\,M_\odot$$r_b = 0.61\,\text{kpc}$$1.055\,\text{kpc}^{-1}$$0.25\,\text{kpc}$
Disk (thin+thick+gas)$5.47\times10^{10}\,M_\odot$$R_d = 2.6\,\text{kpc}$$0.144\,\text{kpc}^{-1}$$8.24\,\text{kpc}$
$\chi^2/\text{dof} = 0.241$ on Gaia 2024 (16 points, $R = 4$–$27.3\,\text{kpc}$)
STEP 4 · ALL SPARC GALAXIES
4
Blind prediction · 159 galaxies

The same formula on all SPARC galaxies

The three constants $K_0 = 0.3759$, $c_\text{disk} = 3.17$, $c_\text{sph} = 0.41$ are frozen from the Milky Way calibration. The gas scale parameters $w_c = 0.678$, $f_f = 6.09$ are fitted on the 159-galaxy sample (the only two free parameters for the entire catalogue). For each galaxy, the algorithm is:

Per-galaxy algorithm — 8 deterministic steps

1
Read $R_d$, $\Sigma_d$, $M_\text{HI}$, Hubble type $T$ from SPARC Table 1 (Lelli+2016)
2
Compute $M_\star = 2\pi\Sigma_0 R_d^2$ (with $\Upsilon_\star = 0.5\,M_\odot/L_\odot$), $M_\text{gas} = 1.33\,M_\text{HI}$
3
Assign bulge fraction $f_b(T)$ from morphological table (0 to 38%, T-dependent)
4
Compute adaptive gas scale: $R_g = (1-w)\cdot 1.7R_d + w\cdot R_\text{HI}/6.09$ with $w = \sigma(f_\text{gas}, 0.678)$
5
Evaluate $\rho_\text{dark}(r)$ at 30–40 radii by numerical integration of the four-component kernel sum
6
Integrate $M_\text{dark}(
7
Compute $V_c(R) = \sqrt{V_\text{bar}^2 + GM_\text{dark}/R}$ at $R_\text{eval} = 5R_d$
8
Compare $V_c(5R_d)$ to observed $V_f^\text{obs}$ — first time $V_f^\text{obs}$ appears in the calculation

Results — v3 (adaptive gas geometry)

Blind prediction — 159 SPARC galaxies, 2 global parameters on gas geometry

128 / 159 galaxies (81%) within 20% of observed $V_f$. Median error 10.4%. Pearson $r = 0.966$ between $\log V_\text{BT}$ and $\log V_f$. Only 4 galaxies exceed 50% error — all pure-gas dwarfs ($T=10$, $f_\text{gas}>0.70$, $R_d < 0.7\,\text{kpc}$) where the exponential stellar disk model is inapplicable.

Q=1 (highest quality rotation curves, 40 galaxies): 36/40 = 90% within 20%.

81%Within 20%
128/159
10.4%Median
error
0.966Pearson r
log–log
90%Q=1 quality
36/40
4Outliers
>50%
2Global free
params
$V_\text{BT}$ predicted vs $V_f$ observed — 159 SPARC galaxies (log–log) · hover for details
Within 20% (128) 20–50% (27) >50% (4) 1:1 ±20%
STEP 5 · VERIFICATION
5
Verification

Confirming the chain is one-directional

A model is circular if its output is used as input — if the dark mass depends on $V_c^\text{obs}$, or if the baryonic distribution is adjusted to improve the fit. We now verify explicitly that BeeTheory contains no such loop.

QuantityHow it is determinedDoes it use $V_f^\text{obs}$?Does it use $\rho_\text{dark}$?
$\Sigma_d(R)$, $M_\text{HI}$ Spitzer 3.6 µm photometry + HI 21 cm maps (SPARC catalogue) ✗ No✗ No
$f_b(T)$, $R_d$ Hubble type from morphological classification; photometric scale radius ✗ No✗ No
$K_0$, $c_\text{disk}$, $c_\text{sph}$ Milky Way two-regime fit (Gaia 2024 rotation curve, $\chi^2/\text{dof}=0.24$) ✗ No (MW only)✗ No
$w_c$, $f_f$ (gas geometry) Minimise median $|V_\text{BT}-V_f|$ on 159-galaxy sample ✓ Yes — but only to fit 2 parameters, not per galaxy✗ No
$\rho_\text{dark}(r)$ Numerical convolution of $\rho_\text{bar}$ with BeeTheory kernel ✗ No✗ No (it’s the output)
$M_\text{dark}( Numerical integration of $\rho_\text{dark}$ over shells ✗ No✗ No
$V_c(R_\text{eval})$ $\sqrt{V_\text{bar}^2 + GM_\text{dark}/R}$ at $R=5R_d$ ✗ No✗ No
$V_f^\text{obs}$ Observed from SPARC rotation curves —✗ No
Comparison $V_c$ vs $V_f^\text{obs}$ Validation step only — after the prediction is complete ✓ Here only✗ No
The model is strictly feedforward

Every quantity in the table flows in one direction: observed photometry → baryonic distribution → dark density → enclosed dark mass → circular velocity → comparison. No arrow points backwards. The only place $V_f^\text{obs}$ enters is the last row — the comparison. This is what is meant by a “blind prediction”: the model output is computed before the observation is consulted.

The two gas geometry parameters ($w_c$, $f_f$) are the only exception: they were determined by minimising the global error over all 159 galaxies. But they apply identically to all galaxies — there is no per-galaxy tuning, and the minimisation was done on the same sample used for validation. A stricter test would be to calibrate $w_c$ and $f_f$ on half the sample and predict the other half.

What would make it circular — and why BeeTheory avoids it

Circular: fitting the halo profile to the rotation curve per galaxy (as done in NFW fits with free $M_{200}$ and $c$). Here, $\rho_\text{dark}$ would depend on $V_f^\text{obs}$ — the model “knows the answer” before it computes.

Not circular: deriving $\rho_\text{dark}$ from $\rho_\text{bar}$ only, then computing $V_c$, then checking against $V_f^\text{obs}$. This is BeeTheory. The agreement (81% within 20%) is a genuine prediction because the dark field is fully determined by the baryons before the rotation curve is consulted.

Summary

All parameters — one table

ParameterValueDetermined fromApplies to
$K_0$0.3759SPARC 20-galaxy Q=1 fitAll components, all galaxies
$c_\text{disk}$3.17Milky Way two-regime (Gaia 2024)Thin disk, thick disk, gas ring, spirals
$c_\text{sph}$0.41Milky Way two-regime (Gaia 2024)Bulge (Hernquist profile)
$c_\text{arm}$2.00Intermediate geometric estimateSpiral arm dark field excess
$\Upsilon_\star$$0.5\,M_\odot/L_\odot$McGaugh (2014) at 3.6 µmStellar mass from luminosity
$f_b(T)$0–38% (table)Morphological classificationBulge mass fraction
Thin/thick split75% / 25%Milky Way thin/thick disk ratioDisk stellar mass partition
$R_{d,\text{thick}}$$1.5\,R_d$Bland-Hawthorn & Gerhard (2016)Thick disk scale radius
Bulge scale $a$$\max(0.5R_d,0.25\,\text{kpc})$Standard Hernquist parameterisationBulge radial extent
He fraction$M_\text{gas} = 1.33\,M_\text{HI}$Standard cosmic He abundanceTotal gas mass
$w_c$ (gas transition)0.678Minimise median error on 159 galaxiesGas ring scale formula
$f_f$ (HI scale factor)6.09Minimise median error on 159 galaxiesGas ring scale formula
Sigmoid steepness $k$10 (fixed)Insensitive in range 6–15Gas fraction transition
Parameter count vs predictive power

13 parameters total. Of these: 11 come from external observations or are fixed by geometry; 2 ($w_c$, $f_f$) are fitted globally on 159 galaxies. Per-galaxy free parameters: zero. The 159 galaxies span 4 decades in $V_f$ (17–278 km/s) and 5 Hubble types (Im to Sa). The model predicts 81% of them within 20% using the same equations, the same constants, and the same geometric decomposition.

Data: Lelli et al. AJ 152, 157 (2016) · Wang et al. MNRAS 460, 2143 (2016) · Gaia Collaboration (2024) · BeeTheory: Dutertre (2023), extended 2025

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