Black Hole Missing Intermediate Sizes in DRUMS

Field Ontology

The DRUMS framework models spacetime as a coherent superfluid field. The macroscopic wavefunction of the condensate is given by:

The superfluid velocity field emerges from the gradient of the phase:

This is the same fundamental medium that gives rise to emergent gravity, quantized CMB modes, and the cosmic web. In the context of black hole physics, the superfluid field provides the underlying structure for all gravitational objects.

Black Holes as Quantized Topological Defects

In DRUMS, a black hole is not a singularity in spacetime but a quantized vortex in the superfluid. The circulation of the superfluid velocity around the vortex core is quantized:

This quantization condition enforces discrete angular momentum states:

Thus, black holes in DRUMS are not continuous in their properties but come in discrete families labeled by the integer winding number \(n\). The smallest stable vortex corresponds to the black hole with the lowest allowed circulation (the stellar‑mass black hole), while higher windings produce larger, more massive black holes.

Energy–Size Scaling

The energy of a vortex structure in the superfluid scales with the phase gradient. For a vortex of radius \(R\), the energy is:

Using the quantization condition, the phase gradient scales as \(\nabla \theta \sim n/R\). For a spherical vortex of radius \(R\), the volume scales as \(R^3\). Thus:

This gives a linear scaling of energy with radius for fixed winding number \(n\):

For a given \(n\), the energy increases linearly with size. However, stability imposes additional constraints on which radii are allowed.

Stability Condition

Stability of a vortex requires a balance between the kinetic energy of the superfluid flow and the quantum pressure that arises from density gradients. The quantum pressure term is:

For a stable vortex configuration, the gradients in the phase field must be compatible with the density distribution. Equilibrium occurs when:

This condition forces a relationship between the winding number \(n\) and the vortex radius \(R\). Solving yields discrete, quantized stable radii:

where \(\ell_0\) is the fundamental coherence length of the superfluid. The stable radii form a discrete, equally spaced sequence: \(R_1, R_2, R_3, \dots\).

Instability of Intermediate Scales

A black hole whose radius does not satisfy the quantization condition \(R = n \ell_0\) is not a stable phase winding. Such configurations experience energy gradients:

This forces the structure to evolve rapidly toward the nearest stable state. The system does not permit metastable intermediate configurations. Any intermediate‑sized black hole will quickly collapse or grow to a stable quantized size. The timescale for this transition is:

For black hole scales, this transition occurs rapidly compared to cosmic timescales, making intermediate states unobservable.

“Intermediate‑mass black holes are not forbidden by any fundamental principle — they are simply unstable, like a pencil balanced on its tip.”

Bifurcation Behavior

Unstable intermediate structures bifurcate in one of two directions:

• Collapse to lower \(n\): The structure sheds energy and decreases its winding number, becoming a smaller stable black hole.
• Growth to higher \(n\): The structure absorbs energy and increases its winding number, becoming a larger stable black hole.

This behavior is governed by energy minimization. The energy difference between successive stable states is:

The system will always evolve toward the lower‑energy configuration accessible via available energy exchange with its environment. The bifurcation is not a choice but a dynamical necessity: the intermediate state is a saddle point in the energy landscape, and any perturbation will push it off toward one of the two stable attractors.

Observational Gap Emergence

Because only discrete radii are stable:

there exists a natural absence of intermediate sizes. The allowed radii are spaced by \(\ell_0\), leaving gaps between them. Transitions occur rapidly compared to observational timescales:

Thus, any intermediate‑sized black hole that might form transiently will have evolved to a stable size before we can observe it. The observed population of black holes will therefore cluster at the discrete stable radii, with a gap between the stellar‑mass family and the supermassive family.

Mass Scaling

The mass of a black hole relates to its radius via integration of the superfluid density within the vortex core:

Substituting the quantized radii \(R_n \sim n \ell_0\) gives discrete mass bands:

The mass scales as \(n^3\), while the radius scales as \(n\). This cubic scaling means that the mass gap between successive stable black hole families grows rapidly. The transition from \(n=1\) to \(n=2\) represents a factor of 8 in mass — a large jump. The observed gap between stellar‑mass black holes and supermassive black holes is therefore not an anomaly but a direct consequence of the cubic mass scaling of quantized vortices.

Final Interpretation

The absence of intermediate‑mass black holes arises naturally in DRUMS due to:

• Quantized vortex topology — black holes are topological defects with integer winding number \(n\),
• Discrete stability radii — only radii \(R_n = n \ell_0\) support stable phase winding,
• Energetic instability between allowed states — intermediate configurations are saddle points, not local minima,
• Rapid transitions between quantized configurations — evolution timescales are short compared to cosmic ages.

Thus, the observed mass gap is not anomalous but a direct consequence of the underlying superfluid structure. In this reading, every black hole is a quantized vortex in the superfluid medium. The missing intermediate masses are not missing because they never form — they are missing because they are unstable and evolve away before they can be observed. The existence of two distinct families of black holes — stellar‑mass and supermassive — is not a coincidence but a signature of the quantization of circulation in the superfluid universe.

This interpretation unifies the black hole mass gap with other DRUMS anomalies. The same quantization that explains the discrete CMB multipole suppression and the quantized galaxy mass function also explains why black holes come in two distinct families separated by a gap. The universe does not need exotic physics or unknown formation channels to explain the gap. It needs only the superfluid and its topological constraints.