Technical Note XXXVIII
The current state of BeeTheory — a mathematical account of gravity, and where it stands among gravitational theories
A single regularized wave function reproduces Newton’s law as the T₂ term of its Laplacian — validated on the 8 planets to twelve decimal places — and, extended to the collective wave field of visible matter, reproduces the rotation curves of 101 disk galaxies (20 calibration + 81 blind) with a median error of 17% using only three universal parameters: λ = 12.7, c = 0.16, ℓ_floor = 3 kpc. No separate dark matter is invoked.
1. What BeeTheory asserts, mathematically
BeeTheory models each elementary particle as a regularized exponential wave function:
The regularization \( \sqrt{r^2+a^2} \) keeps the Laplacian finite at the origin while preserving exponential decay at large \(r\). Applying the spherical Laplacian decomposes \( \nabla^2\psi/\psi \) into three terms:
At macroscopic distance (\( r \gg a \), true beyond the atomic scale):
- \( T_1 \to 1/a^2 \): a constant — no gradient, no force.
- \( T_2 \to -2/(a\,r) \): exactly the spatial form of the Newtonian \(-1/r\) potential.
- \( T_3 \to a/r^3 \): decays \(r^2\) times faster than \(T_2\), negligible.
Newton’s potential is not an axiom of BeeTheory — it emerges as the \(T_2\) component of the Laplacian of a regularized wave function. With the coupling \( K = G\,M_1 M_2\,a/2 \), the interaction energy reads \( U = -G\,M_1 M_2 / r \), exactly Newton’s law.
2. The point-mass chain: Cavendish, the Sun, the Earth
The same construction climbs cleanly through scales. For two macroscopic spheres, the collective wave field is equivalent to a point mass at the centre with amplitude scaled by \(N\) (atom count); the pair force scales as \(N^2\), and the effective constant G_eff = K_BT/m_atom² ≈ 6.67×10⁻¹¹ matches the measured Newton constant. For the Sun–Earth system there are no free parameters: the predicted potential energy agrees with Newton to twelve decimal places, and the same holds across the eight planets, spanning thirteen decades of energy. For the Earth, the PREM density profile accumulates ψ internally and yields the \(T_2\) tail externally, giving \(g = 9.81\,\text{m/s}^2\) at the surface.
3. From points to galaxies
For a galaxy the source is no longer a point but a continuous density \( \rho_{vis}(\mathbf{r}’) \). Each mass element carries its own wave function; the collective field is the superposition. The distinctive physical statement: the collective wave field of visible matter extends spatially beyond the visible distribution, and the gradient of its outer tail produces the signature attributed to dark matter.
| Parameter | Value | Physical role |
|---|---|---|
| λ | 12.696 | Wave coupling strength |
| c | 0.163 | Geometric scaling with R_d |
| ℓ_floor | 3.00 kpc | Universal wave-field coherence length |
Validation
| Metric | Calibration (20) | Blind (81) |
|---|---|---|
| Median |err| | 16.0% | 17.3% |
| Mean signed err | −4.3% | −0.9% |
| Within ±30% | 85% | 74% |
| Within ±50% | 95% | 99% |
| Bias vs R_d / M_vis | — | None |
The blind sample matches calibration to within one percentage point of median error, with no residual bias over four decades in visible mass. This is the operational signature of a model that has captured genuine structure rather than overfit to its calibration set.
4. Relevance and standing among gravitational theories
An even-handed appraisal: what is solid, what is fragile, and how BeeTheory sits relative to the main alternatives.
Strengths
- The blind test is the real asset. Calibrating on 20 galaxies and then predicting 81 independent ones, unadjusted, with essentially identical performance, is exactly the methodological discipline many alternative models never submit to.
- Three parameters, four decades, no bias. The absence of residual trends versus R_d and M_vis is genuine — a model that merely fits usually leaves such a correlation behind.
- A point-mass regime that is parameter-free. At solar-system scale the construction is deterministic and reproduces Newton to twelve decimals.
Open weaknesses
- From quantum amplitude to physical force. The Schrödinger equation governs a probability amplitude, not a force-bearing field. The step from “wave peaks approach” to “an attractive force exists” is interpreted, not yet derived from a dynamical equation. Until that bridge is explicit, \(T_2 \propto 1/r\) is a match of form rather than a derivation of gravity.
- The solar / galactic discontinuity. The solar regime is parameter-free; the galactic regime needs three fitted parameters (with λ ≈ 6× the Milky-Way value). The continuity between the two regimes is not yet demonstrated.
- ℓ_floor is introduced to repair a failure. A constant 3 kpc length, independent of mass and not yet derived from \(a_0\) or any other constant, spans some thirty orders of magnitude from the Bohr radius. Calling it a “coherence length” is an interpretation pending a derivation.
- The decisive test is still outstanding. Comparison with the Radial Acceleration Relation (McGaugh 2016) — reproducing \(g_{obs}(g_{bar})\) point by point — has not been performed. This is the strictest constraint on any modified-gravity proposal.
Positioning against the alternatives
- ΛCDM (dark matter). BeeTheory shares MOND’s philosophy — no invisible halo. But ΛCDM rests on far more than rotation curves: the CMB, gravitational lensing, the Bullet Cluster, large-scale structure. BeeTheory is silent on these for now; matching disk galaxies is necessary but far from sufficient.
- MOND / RAR. The direct and most demanding competitor. MOND reaches comparable-or-better SPARC residuals with a single parameter and reproduces the RAR by construction. BeeTheory’s potential advantage is a proposed physical origin for the effect — convincing only once the quantum-to-force bridge is closed.
- String theory & Loop Quantum Gravity. Not genuinely comparable: these are quantum-gravity programmes, while BeeTheory is a galactic-dynamics model with a quantum motivation. The comparison is best left aside, as it tends to flatter neither side fairly.
BeeTheory is a methodologically honest, statistically non-trivial phenomenological model of disk-galaxy rotation, whose deeper theoretical claim — deriving gravity from quantum mechanics — currently rests more on a match of form than on a complete derivation. Its scientific value will be decided by two tests it has not yet passed: the RAR point by point, and a theoretical bridge from \(a_0\) to ℓ_floor.
BeeTheory.com — Current state & appraisal of wave-based gravity · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026