Momentum clumping (Debt 3) on The Broken-Born program

Obituary: velocity clumping, the founding image

Mon, 01 Jan 0001 00:00:00 +0000

Velocity clumping was the program’s founding picture of dark matter: a smooth dark-matter halo would break up into a few streams, each made of particles moving with nearly the same velocity, so that dark matter clumps in velocity instead of in position. The lab notebook described it as the Jeans problem, the gravitational instability that makes lumps in space, with the roles of position and velocity exchanged. The Debt-3 calculation tested it and found it impossible for particle dark matter, fatal where it would be mandatory for wave dark matter, and merely optional elsewhere. The claim is dead as dynamics and stays dead.

Vlasov stability and Landau damping

Mon, 01 Jan 0001 00:00:00 +0000

The Vlasov equation describes a gas whose particles interact only through a smooth, averaged force field, so that the gas is fully specified by how densely it fills each region of position and velocity. Its stability theory asks whether a small ripple in that density grows or dies away. In a plasma a ripple usually decays even without collisions (Landau damping), while in a self-gravitating gas a ripple longer than a critical length grows into a lump (the Jeans instability). Where collisions do act, they drive any distribution toward the smooth thermal shape (Boltzmann’s H-theorem). The program used this textbook machinery in Debt 3 to test whether dark matter could clump in velocity.

The gated Vlasov model

Mon, 01 Jan 0001 00:00:00 +0000

The gated Vlasov model is the simplified description of dark matter that Debt 3 used to test velocity clumping. Dark matter is treated as a smooth gas in position and momentum. Besides gravity, each particle feels an attraction only from neighbours whose momentum lies within a narrow window around its own, the velocity gate, and the attraction’s range in space is shrunk to a point. Whether a small ripple in this gas grows then becomes an eigenvalue problem on momentum space, which the calculation brackets between a broad-gate limit and a narrow-gate limit instead of solving in general. The model also retired the earlier plan to race the growth of streams against orbital stirring, because below the threshold there is no growth to race.

The mean-field criterion and its 10²⁸–10³¹ shortfall

Mon, 01 Jan 0001 00:00:00 +0000

The mean-field criterion is the condition under which a gas of dark-matter particles with the gated attraction would break up into velocity streams on its own: the attraction energy among particles that share a velocity has to match the kinetic energy of their random motion. Debt 3 derived it in two limits of the gated Vlasov model and evaluated it at the scattering benchmark, where the coupling is already fixed by fitting the cores of dwarf galaxies. There the attraction falls short by a factor of about 10²⁸ in dwarf-galaxy cores and 10³¹ in the Milky Way’s halo near the Sun. Collisions cannot make up the difference, because they only drive the gas toward a smooth thermal distribution. The result rules out velocity clumping for particle dark matter, and the result itself stands.

The classicality theorem and the particle island

Mon, 01 Jan 0001 00:00:00 +0000

The classicality theorem says that dark matter coupled strongly enough to clump in velocity must be so light that it behaves as a wave, not as a collection of separate particles. It follows from the mean-field criterion: at the scattering strength that fits dwarf-galaxy cores, the criterion can be met only for masses below about 0.1 meV, and at such masses about 10²⁰ quanta share each quantum state, which is the regime of a classical wave. The program therefore splits dark matter into two regimes: a particle island, with χ masses of MeV to GeV, where dark matter scatters but cannot clump, and a wave island of far lighter masses. The theorem is derived at mean-field level and stands.

The net theorem: clumping is nowhere mandatory

Mon, 01 Jan 0001 00:00:00 +0000

The net theorem is the summary result of Debt 3: in every part of the program’s parameter space that survives, dark matter clumping by velocity is never a required phenomenon. For particle dark matter it cannot happen. For wave dark matter it was required only under one assumption about the coupling, Scenario A, which observation rules out; with the coupling left free, Scenario B, it is merely optional. The theorem closes the test of the program’s founding image with a negative result. It covers collective dynamics only, so whether a single topological object could be localized in momentum is a separate, untested question.

Obituary: velocity-space granularity in halos (old row 3)

Mon, 01 Jan 0001 00:00:00 +0000

Velocity-space granularity was the lab notebook’s prediction that the dark matter of a relaxed galaxy halo, such as the Milky Way’s, is split into of order ten broad streams, each with its own velocity and coherent over kiloparsec distances, and that their gravity leaves correlated ripples along thin streams of stars. It was row 3 of the notebook’s prediction table and one of its three flagship discriminants. It rested on the founding image of velocity clumping, so when Debt 3 showed that clumping is impossible for particle dark matter and never mandatory for wave dark matter, the row was withdrawn permanently, on both islands. The current table’s row 3 is a different prediction, the island discriminant.

Obituary: Fornax friction suppression (old row 4)

Mon, 01 Jan 0001 00:00:00 +0000

Fornax friction suppression was the lab notebook’s prediction that dark matter in the cores of dwarf galaxies condenses into a superfluid-like state in which dynamical friction, the drag that makes a massive body sink toward the centre of a galaxy, is suppressed. It would have explained why the globular star clusters of the Fornax dwarf galaxy have not sunk to its centre, the Fornax timing problem, and it was row 4 of the notebook’s prediction table. The condensed cores came from the same velocity-space condensation as the founding image of velocity clumping, so when Debt 3 killed that picture the row was withdrawn with it, permanently.

Open: the topological loophole

Mon, 01 Jan 0001 00:00:00 +0000

The topological loophole is the route that Debt 3 leaves open for localization in momentum space: not a collective clumping of many particles, which the net theorem rules out as a requirement, but a single object, such as a topological soliton, whose own structure would localize it in momentum. Debt 3’s calculations cover collective dynamics only, so they do not test this case. The candidate that occupied the loophole, superheavy solitons of a hidden gauge sector, was retired on 2026-10-19 after failing review on its abundance arithmetic, so the loophole is logically open but has no candidate. The checkpoint does not say consistently whether it remains a work item.