Technical Note XLI

From point to density — the wave-mass profile is the enclosed mass of an exponential kernel

BeeTheory.com · Wave-based quantum gravity · 21 May 2026

Result first

The enclosed wave-mass profile used throughout the galactic model, \( M_{wave}(not an ansatz. It is, to machine precision (error < 10⁻¹⁶), the exact enclosed mass of a single radial exponential wave density ( rho_{wave}(s) propto e^{-s/ell} ). The polynomial factor \((1+x+\tfrac{x^2}{2})\) is forced — it is the order-2 partial sum of \(e^{x}\) produced by the spherical integral. The whole galactic prescription therefore reduces to one statement: each visible mass element projects an exponential wave halo of range ℓ, and the total wave field is the convolution of the visible density by that kernel.

1. The question

BeeTheory derives Newton’s law from a single particle’s wave function: the (T_2) term of the Laplacian gives the (-1/r) potential, validated on the planets to twelve decimals. For an extended source — a galaxy — the visible matter is a continuous density ( rho_{vis}(mathbf{r}’) ), not a point. The natural extension treats each volume element \( dm’ = \rho_{vis}(\mathbf{r}’)\,dV’ \) as an elementary source carrying its own wave halo, and superposes them.

The question this note answers precisely: what wave density does a single source project, and does its spherical enclosed mass reproduce the profile already in use?

2. The exponential kernel

Let a point source of visible mass \(M\) project a spherically symmetric wave density of range \(\ell\):

\[ \rho_{wave}(s) = C\,e^{-s/\ell}, \qquad s = \text{distance to the source.} \]

The normalization is fixed by requiring the total projected wave mass to be ( lambda M ):

\[ \int_0^\infty 4\pi s^2\,\rho_{wave}(s)\,ds = \lambda M, \qquad \int_0^\infty s^2 e^{-s/\ell}\,ds = 2\ell^3 \] \[ \Rightarrow\; C = \frac{\lambda M}{8\pi\,\ell^3}. \]

3. The enclosed mass — closed form

Integrating this density inside a sphere of radius \(r\), with \(x = r/\ell\):

\[ M_{wave}(

The integral \( \int_0^r s^2 e^{-s/\ell}ds = 2\ell^3 – \ell^3\,e^{-x}\,(x^2 + 2x + 2) \) gives, after simplification:

\[ \boxed{\;M_{wave}(

This is exactly the profile used in the 3-parameter model. The factor \( (1 + x + x^2/2) \) is not chosen — it is the order-2 Taylor partial sum of \( e^{x} \) that the \( s^2 \) volume measure necessarily produces.

4. Numerical verification

The closed form was checked against direct numerical integration, and the total mass against the conservation requirement:

r / ℓ	Numerical ∫	Closed form	Error
0.5	0.01438768	0.01438768	2.6×10⁻¹⁷
1.0	0.08030140	0.08030140	4.2×10⁻¹⁷
3.0	0.57680992	0.57680992	1.1×10⁻¹⁶
8.0	0.98624603	0.98624603	0.0
∞	1.00000000	λ (= 1)	conserved

5. From one source to a density: the convolution

If every visible element projects this same kernel, the total wave density at a point \( \mathbf{r} \) is the convolution of the visible density by \(K\):

\[ \rho_{wave}(\mathbf{r}) = \int \rho_{vis}(\mathbf{r}’)\,K(|\mathbf{r}-\mathbf{r}’|)\,dV’, \qquad K(s) = \frac{\lambda}{8\pi\,\ell^3}\,e^{-s/\ell}. \]

As a consistency check, convolving a near-point source (a narrow Gaussian of width \( \varepsilon = 0.05\,\ell \)) reproduces the spherical enclosed mass to better than 1%, the residual vanishing as \( \varepsilon \to 0 \):

r / ℓ	Convolution (3D)	Closed form	Rel. error
0.5	0.014294	0.014388	0.7%
1.0	0.080072	0.080301	0.3%
3.0	0.576530	0.576810	0.0%
5.0	0.875243	0.875348	0.0%

6. Why this matters

Significance

The galactic model is no longer “a baryonic term plus an empirical wave term. The wave term has a single, transparent origin — an exponential kernel of range ℓ, applied to the visible density exactly as a sphere is the sum of its atoms. This converts a postulate into a derived consequence, and isolates the only remaining freedom.

Three consequences follow:

The profile is forced, not fitted. Given an exponential kernel, the \([1-(1+x+x^2/2)e^{-x}]\) shape is the unique enclosed-mass law. No degree of freedom hides in it.
The same logic spans scales. Point → sphere (Cavendish) → continuous density (galaxy) is now one operation — a convolution — whose spherical limit returns the verified planetary case. The point-to-density step is mathematically closed.
The freedom is reduced to ℓ. Everything is now carried by the kernel range \( \ell \). In the spherical/point limit \( \ell \) is universal. The model writes \( \ell = c\,R_d + \ell_{floor} \) with \( c = 0.16 \) nearly negligible — which raises the sharp, testable question of whether a single universal range \( \ell_{floor} \) suffices, collapsing the model from three parameters to two.

What is NOT yet shown

This note establishes the kernel and its spherical limit. It does not yet show that convolving a realistic disk density (Freeman profile + extended gas) with a single universal ( ell_{floor} ) reproduces observed rotation curves. That disk calculation — and the cluster-scale test — are the next steps and are not settled here.

BeeTheory.com — From point to density: the exponential kernel · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026