Black Hole Missing Intermediate Sizes in the DRUMS Framework
1. Field Ontology
The DRUMS framework models spacetime as a coherent superfluid field:
\[
\Psi(\mathbf{x},t) = \sqrt{\rho(\mathbf{x},t)} e^{i\theta(\mathbf{x},t)}
\]
Velocity arises from phase gradients:
\[
\mathbf{v} = \frac{\hbar}{m} \nabla \theta
\]
2. Black Holes as Quantized Topological Defects
Black holes correspond to quantized vortices with circulation:
\[
\Gamma = \oint \mathbf{v} \cdot d\mathbf{l} = n \frac{h}{m}
\]
This enforces discrete angular momentum states:
\[
L \sim n \hbar
\]
3. Energy–Size Scaling
The energy of a vortex structure scales as:
\[
E(R) \sim \rho \int (\nabla \theta)^2 dV
\]
For a vortex of radius \(R\):
\[
E(R) \sim \rho \frac{n^2}{R^2} \cdot R^3 = \rho n^2 R
\]
This gives a linear scaling:
\[
E \propto n^2 R
\]
4. Stability Condition
Stability requires balance between kinetic and quantum pressure:
\[
Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}}
\]
Equilibrium occurs when:
\[
\frac{n^2}{R^3} \sim \frac{\hbar^2}{m R^3}
\]
Which implies discrete stable radii:
\[
R_n \sim n \ell_0
\]
where \(\ell_0\) is a characteristic coherence length.
5. Instability of Intermediate Scales
Intermediate sizes require non-integer or unstable configurations:
\[
R \neq n \ell_0 \Rightarrow \text{no stable phase winding}
\]
Such configurations experience energy gradients:
\[
\frac{dE}{dR} \neq 0
\]
Driving rapid evolution toward nearest stable state.
6. Bifurcation Behavior
Unstable intermediate structures bifurcate via:
- Collapse to lower \(n\)
- Growth to higher \(n\)
This is governed by energy minimization:
\[
\Delta E \sim \rho (n+1)^2 R_{n+1} - \rho n^2 R_n
\]
7. Observational Gap Emergence
Because only discrete radii are stable:
\[
R \in \{R_1, R_2, R_3, ...\}
\]
There exists a natural absence of intermediate sizes.
Transitions occur rapidly compared to observational timescales:
\[
\tau_{transition} \ll \tau_{cosmic}
\]
8. Mass Scaling
Mass relates to radius via density integration:
\[
M \sim \rho R^3
\]
Thus discrete radii imply discrete mass bands:
\[
M_n \sim \rho (n \ell_0)^3
\]
9. Final Interpretation
The absence of intermediate-mass black holes arises naturally in DRUMS due to:
- Quantized vortex topology
- Discrete stability radii
- Energetic instability between allowed states
- Rapid transitions between quantized configurations
Thus, the observed mass gap is not anomalous but a direct consequence of the underlying superfluid structure.