BeeTheory · Theoretical Framework · 2025
Two Scales, Two Formulas
The BeeTheory wave equation applies at two distinct levels of reality: the elementary particle and the macroscopic mass distribution.
These are not the same formula. They must not be confused.
BeeTheory.com · Dutertre (2023) · Extended derivation 2025
Formula I · Scale I · Quantum
The Elementary Particle Wave Function
\(\psi(r)=\frac{1}{\sqrt{\pi a^3}}e^{-r/a}\)r is the distance from the particle’s centre.
a is the de Broglie–Bohr scale of the particle.
This a is fixed by the particle’s quantum state. It does not depend on the density of surrounding matter.
Formula II · Scale II · Astrophysical
The Macroscopic Mass Density Kernel
\(\rho_{\mathrm{dark}}(\mathbf r)=\frac{K}{\ell}\int \rho_{\mathrm{vis}}(\mathbf r’)e^{-|\mathbf r-\mathbf r’|/\ell}\,dV’\)ρvis is the visible, baryonic mass density.
ℓ is the coherence length of the source component.
This ℓ depends on the geometry and scale of the source structure, not on individual particles.
What connects them
Formula I describes the microscopic wave of a single particle or particle pair. Formula II describes the collective field produced when a macroscopic distribution of mass is treated as a continuous source.
I. Formula I — The Elementary Particle
BeeTheory begins at the most fundamental level. Every massive elementary particle is modeled as a spherically symmetric wave function that decays exponentially from its centre.
For a particle in its ground state:
\(\psi(\mathbf r)=\frac{1}{\sqrt{\pi a^3}}\exp\left(-\frac{|\mathbf r|}{a}\right)\)Here a is the characteristic decay length of the particle’s wave function.
For the hydrogen atom, a = a0 = 52.9 pm, the Bohr radius. This is a quantum-mechanical constant derived from the electron mass, the proton mass, and ℏ.
For a neutron or proton, a is of order the nuclear radius, around 1 fm.
The decay constant a is a property of the particle’s quantum state. It is fixed by physics: by ℏ, by m, and by the binding energy. It does not change because many particles are nearby.
A hydrogen atom in a galactic disk has the same a0 as a hydrogen atom in the void of intergalactic space.
What the Schrödinger Equation Gives
Applying the equation Ĥψ = Eψ without potential, as pure kinetic energy in the BeeTheory framework, the exact Laplacian in spherical coordinates is:
\(\nabla^2\psi(r)=\psi(r)\left(\frac{1}{a^2}-\frac{2}{ar}\right)\)Two terms emerge: a constant kinetic term and a Coulomb-like term.
The constant term is:
\(+\frac{1}{a^2}\)The Coulomb-like term is:
\(-\frac{2}{ar}\)It is the −2/(ar) term that, when projected onto a second particle at distance R, generates the attractive interaction.
The interaction energy between particle A at the origin and particle B at distance R takes the following form after full 3D integration over B’s wave function:
\(E(R)=-\frac{\kappa}{\sqrt{\pi}}\exp\left(-\frac{R}{\alpha_{\mathrm{eff}}}\right)+\frac{e^2}{4\pi\varepsilon_0R}\) \(\kappa=3.509E_h=95.5\,\mathrm{eV}\) \(\alpha_{\mathrm{eff}}=1.727a_0=91.4\,\mathrm{pm}\)This equation was calibrated on the hydrogen molecule using two experimental constraints: the bond length and the dissociation energy.
\(R_{\mathrm{eq}}=74.1\,\mathrm{pm}\) \(D_e=4.52\,\mathrm{eV}\)The result reproduces both constraints to within 0.1 percent.
The key point is that αeff is not equal to a0. The effective decay of the two-particle interaction is 73 percent longer than the single-particle wave function.
This is not a free parameter. It is derived analytically from the two calibration conditions:
\(\alpha_{\mathrm{eff}}=R_{\mathrm{eq}}+D_eR_{\mathrm{eq}}^2\)What Formula I Does Not Depend On
ψ(r) and its parameters, including a, κ, and αeff, are determined by the quantum mechanics of individual particles and pairs. They are independent of local density.
Whether a hydrogen atom is at the Sun’s location or in an interstellar cloud, its wave function is identical. Formula I is a microscopic equation.
II. Formula II — The Macroscopic System
At galactic scales, it is not possible, nor meaningful, to track individual particles. The relevant quantity is the mass density field.
\(\rho_{\mathrm{vis}}(\mathbf r)\)BeeTheory’s second formula describes how this continuous density generates a dark mass field through a convolution with an exponential kernel.
\(\rho_{\mathrm{dark}}(\mathbf r)=\frac{K}{\ell}\int_{\mathrm{source}}\rho_{\mathrm{vis}}(\mathbf r’)\frac{(1+\alpha D)e^{-\alpha D}}{D^2}\,dV’\) \(D=|\mathbf r-\mathbf r’|,\qquad \alpha=\frac{1}{\ell}\)The kernel is:
\(\frac{(1+\alpha D)e^{-\alpha D}}{D^2}\)This is the force kernel derived from the BeeTheory potential.
\(V\propto\frac{e^{-\alpha D}}{D}\)It reduces to Newton’s inverse-square form for D much smaller than ℓ, and it decays exponentially for D much larger than ℓ.
The Key Difference: What Is ℓ Here?
In Formula II, the coherence length ℓ is not the Bohr radius a0 or any single-particle scale.
It is the coherence length of the macroscopic source structure: the distance over which the mass distribution remains spatially correlated.
This is an emergent, collective property of the system.
The Physical Origin of ℓ at Macroscopic Scales
Consider N particles forming a source structure of characteristic size Lsource. Each particle emits a wave with decay scale a. When these waves are summed coherently, the superposed field has a coherence length that depends on the spatial organisation of the source, not just a.
In the limit N → ∞ and Lsource ≫ a, the single-particle scale a drops out entirely. The macroscopic coherence length ℓ is determined by Lsource and by the geometry of the mass distribution.
This is analogous to coherence in optics: individual photons have wavelength λ, but the coherence length of a laser beam depends on the cavity geometry, not on λ alone.
The Two Galactic Components — Two Values of ℓ
The Gaia 2024 rotation curve reveals two distinct regimes separated near R ≈ 5.5 kpc. BeeTheory fits them with two independent applications of Formula II, one per baryonic component.
| Source component | Geometry | Source size L | ℓ fitted | ℓ / L | K fitted | λ = Kℓ² |
|---|---|---|---|---|---|---|
| Bulge + bar | Spherical 3D | rb = 1.5 kpc | 0.61 kpc | 0.41 | 1.055 kpc⁻¹ | 0.39 |
| Disk, thin + thick + gas | Exponential disk 2D | Rd = 3.5 kpc | 11.1 kpc | 3.17 | 0.02365 kpc⁻¹ | 2.90 |
The ratio ℓ/Lsource is 0.41 for the bulge and 3.17 for the disk. This difference reflects the geometry of each component.
- The bulge is compact and centrally concentrated. Its mass is tightly bound, and its collective wave field has a short coherence length. This drives the rapid rise of Vc at R < 5 kpc.
- The disk is extended and spread over tens of kiloparsecs. Its collective coherence is correspondingly long. The dark field extends far into the halo, sustaining the flat rotation curve and then causing the Gaia 2024 decline beyond ℓd ≈ 11 kpc.
III. The Bridge Between the Two Formulas
How does Formula I at the particle scale give rise to Formula II at the macroscopic scale? The connection is a multi-step aggregation argument.
Step 1 — Particle to Pair
Two particles A and B at distance D interact through a Yukawa-type pair potential:
\(V(D)=-\frac{\kappa}{\sqrt{\pi}}e^{-D/\alpha_{\mathrm{eff}}}\)The decay scale αeff is the effective range at the particle level.
Step 2 — Pair to Ensemble
For N particles forming a source, the potential is the sum over all pair contributions.
\(V(\mathbf r)=\sum_i V(|\mathbf r-\mathbf r_i|)\)In the continuum limit, the discrete sum becomes a volume integral over the source density:
\(V(\mathbf r)\rightarrow \int\rho_{\mathrm{vis}}(\mathbf r’)V(D)\,dV’\)Step 3 — Potential to Density
The dark mass density is derived from the gravitational potential via Poisson’s equation.
\(\rho_{\mathrm{dark}}(\mathbf r)\equiv-\frac{\nabla^2V(\mathbf r)}{4\pi G}+\mathrm{source\ correction}\)For a Yukawa potential, this gives the macroscopic BeeTheory kernel:
\(\frac{(1+\alpha D)e^{-\alpha D}}{D^2}\)Step 4 — Renormalisation of ℓ
The macroscopic coherence length is not simply the microscopic particle scale. It is renormalised by the size and geometry of the source.
\(\ell_{\mathrm{macro}}=\alpha_{\mathrm{eff}}^{\mathrm{pair}}\mathcal F\left(\frac{L_{\mathrm{source}}}{\alpha_{\mathrm{eff}}^{\mathrm{pair}}}\right)\)When the source size is much larger than the microscopic pair scale, the macroscopic coherence length is no longer set by the pair scale. It is set by Lsource and by source geometry through the function 𝓕.
The Decoupling of Scales
The Bohr radius is:
\(a_0=52.9\,\mathrm{pm}=1.72\times10^{-15}\,\mathrm{kpc}\)The disk coherence length is:
\(\ell_d=11.1\,\mathrm{kpc}\)The ratio is:
\(\frac{\ell_d}{a_0}\approx6.5\times10^{15}\)This is not a failure of the theory. It is the expected consequence of summing around 1067 particle-pair interactions coherently over a galactic source of size around 25 kpc.
The collective coherence emerges at the scale of the collective structure, not at the scale of its constituents.
The Open Theoretical Question: 𝓕(L/α)
The function 𝓕 that maps source geometry to macroscopic ℓ is the central unsolved problem of BeeTheory’s multi-scale theory.
From the galactic fit, we observe:
\(\frac{\ell_{\mathrm{bulge}}}{r_b}=0.41,\qquad \frac{\ell_{\mathrm{disk}}}{R_d}=3.17\)If ℓ scales as a power of Lsource, then:
\(\ell\propto L_{\mathrm{source}}^\gamma\) \(\gamma=\frac{\log(11.1/0.61)}{\log(3.5/1.5)}\approx\frac{\log(18.2)}{\log(2.33)}\approx3.4\)This is a steep scaling. Alternatively, the difference may reflect geometry: a disk source and a spherical source generate qualitatively different collective fields.
Determining 𝓕 requires applying BeeTheory to a sample of galaxies with different morphologies.
IV. Summary — The Two Formulas Side by Side
| Aspect | Formula I — elementary particle | Formula II — macroscopic system |
|---|---|---|
| Object | Single particle or particle pair | Continuous density field ρvis(r) |
| Wave function | ψ(r) = Ne−r/a, exact quantum state | Not applicable; replaced by ρvis field |
| Key length scale | a = a0 = 52.9 pm, Bohr radius | ℓ = coherence of source structure |
| Depends on local density? | No. a0 is a universal constant. | Yes. ℓ reflects source geometry and size. |
| Interaction potential | E(R) = −(κ/√π)e−R/αeff + repulsion | V(D) ∝ e−D/ℓ/D |
| Force law | Short-range exponential force | Newtonian 1/D² limit for D ≪ ℓ |
| Calibration | H₂ molecule: Req = 74.1 pm, De = 4.52 eV | Milky Way: Gaia 2024 rotation curve, χ²/dof = 0.24 |
| Free parameters | κ = 3.509 Eh, αeff = 1.727 a0 | K and ℓ per source component |
| Physical regime | D ~ a0 ~ 10⁻¹¹ m | D ~ ℓ ~ 10²⁰ m |
| Connection | Formula II emerges from summing Formula I over ~10⁶⁷ particle pairs. The microscopic scale a0 decouples; ℓ is set by collective source geometry. | |
Formula I describes how a single mass element creates a wave. Formula II describes how an ensemble of mass elements — a galaxy, a bulge, a disk — creates a collective dark field.
The former is quantum mechanics. The latter is statistical mechanics applied to BeeTheory.
Why this distinction matters for BeeTheory’s predictions
Without this distinction, one might expect that measuring K and ℓ in one galaxy immediately predicts all others as universal constants.
The reality is more subtle. K appears to be approximately universal through the dimensionless coupling:
\(\lambda=K\ell^2\approx3\)But ℓ must be computed from the geometry of each source component.
The prediction becomes: given the disk scale radius Rd of a galaxy, its outer dark mass coherence length should be approximately:
\(\ell_d\approx3R_d\)This is testable against the SPARC catalogue of 175 galaxies.
The bulge ratio offers a second test:
\(\frac{\ell_b}{r_b}\approx0.4\)This predicts that compact bulges generate dark mass fields on sub-kpc scales, concentrated near galactic centres.
References
- Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com, 2023. Original formulation of the elementary particle wave function.
- Kolos, W., Wolniewicz, L. — Potential-Energy Curves for the H₂ molecule, Journal of Chemical Physics 43, 2429, 1965. Calibration data for Formula I.
- Ou, X. et al. — The dark matter profile of the Milky Way inferred from its circular velocity curve, MNRAS 528, 2024. Calibration data for Formula II.
- McMillan, P. J. — MNRAS 465, 76, 2017. Galactic mass model used to define the source components.
- Yukawa, H. — On the Interaction of Elementary Particles, Proceedings of the Physico-Mathematical Society of Japan 17, 48, 1935. Mathematical structure of the macroscopic potential.
BeeTheory.com — Wave-based quantum gravity · © Technoplane S.A.S. 2025