Momentum clumping (Debt 3)

Momentum clumping is the program’s founding picture of dark matter: instead of gathering into lumps in space, dark matter would gather into a few streams whose members share nearly the same velocity. Whether that happens was the program’s third debt, its name for a calculation still to be done, and three short notes carried it out with standard kinetic theory. The answer was no. For particle dark matter the attraction falls short by 28–31 orders of magnitude. For wave dark matter, the one scenario that made clumping mandatory predicts that the smallest galaxies lose their dark matter, and they have not; with the coupling left free, clumping is merely optional. This section holds the founding claim’s obituary, the calculation that killed it, and what survived.

Stands: the mean-field criterion, the classicality theorem, the gated condensation rate, Scenario B, quarantined, relic abundance by misalignment, the net theorem · Died: velocity clumping, the founding image, Scenario A, velocity-space granularity (old row 3), Fornax friction suppression (old row 4) · Open: the topological loophole · Superseded: racing stream growth against orbital stirring

Map#

Debt 3 began as item 3 of the lab notebook’s open problems: the growth rate of the stream instability against orbital phase mixing, which “determines whether Prediction 3’s granularity survives in Milky-Way-like halos” (notes/broken-born-theory.md §11). Three calc notes carried it out: notes/debt3a-stream-stability.md (“negative on the particle island, with a structural corollary”), notes/debt3b-wave-island.md (“rate derived, environments evaluated, verdict reached”) and notes/debt3c-ufd-endpoint.md (“Scenario A falsified; Debt 3 closed”). Checkpoint #11 §9 is the current summary, tagged [D] on the particle side and [D + T] on the wave side; no verify/ script covers any Debt-3 number. The history of the three debts places Debt 3 among the others. One page below is established physics and one is method; every other page is a claim of the program, alive, dead or open.

Obituary: velocity clumping, the founding image comes first, because the death is the subject of the section. It records the lab notebook’s picture of a halo breaking into discrete velocity streams, “the Jeans problem with the roles of x and v exchanged”, with its analogies to superconductivity and to Berry-curvature transport; the two predictions it carried; how Debt 3 tested it; and the three verdicts that killed it.

Vlasov stability and Landau damping is the established kinetic theory the test used: the collisionless Boltzmann equation, linear response about a Maxwellian, Landau damping, the static threshold of an attractive gas and its best-known case, the Jeans instability, and Boltzmann’s H-theorem. The gated Vlasov model is the method built on it: a Gaussian velocity gate on a contact attraction, whose linear response becomes an eigenvalue problem on momentum space, bracketed by a broad-gate and a narrow-gate limit. It also holds the superseded plan to race stream growth against orbital stirring, which the threshold made moot.

The mean-field criterion and its 10²⁸–10³¹ shortfall is the first result: the threshold for streams in both limits, g·n_gate ≥ m·min(σ, v*)², and its failure by 28–31 orders of magnitude once the coupling is fixed by dwarf-galaxy cores, with collisions excluded by the H-theorem. It also records why the handoff’s machine-check mark on these numbers has no script behind it (flag 16 in review/wiki-build-flags.md). The classicality theorem and the particle island follows from it by a scaling argument: meeting the criterion requires masses below about 0.1 meV, and at that mass about 10²⁰ quanta share each quantum state (on the regime in which Debt 3a evaluates these numbers, see flag 19 in review/wiki-build-flags.md), so classical particle dark matter scatters but cannot clump, and collective behaviour belongs to waves.

The wave island is the branch the theorem leaves open, with its own pages: the gated condensation rate, Scenario A (dead), Scenario B (quarantined) and the misalignment route to the relic density. The net theorem then states the outcome: velocity clumping is nowhere mandatory, “forbidden (particles), fatal (committed wave), decorative (free wave)”. Its salvage, in #11 §9, is the island discriminant, now row 3 of the predictions table, and misalignment production on the wave island.

Obituary: velocity-space granularity in halos (old row 3) and Obituary: Fornax friction suppression (old row 4) record the two predictions withdrawn with the founding image: macro-streams perturbing cold stellar streams in the Milky Way, and superfluid-like dwarf cores in which globular clusters do not sink. Open: the topological loophole comes last: a single topological object localized in momentum, which Debt 3 did not test and which no longer has a candidate; the checkpoint calls it both untested and closed, and does not reconcile the two.

Boundaries#