The net theorem: clumping is nowhere mandatory

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.

Status: Alive — [D] in #9 §7 (“Three results [D]”); #11 §9 states it as result (iii) without a separate tag, resting on (i) [D] and (ii) [D + T]. Collective dynamics only, at the level of the calc notes’ mean-field and wave-kinetic estimates. Not machine-checked: no verify/ script covers Debt 3.

Statement#

As first stated, in the note that closed Debt 3, where BBT is the Broken-Born framework (notes/debt3c-ufd-endpoint.md, ‘Verdict and closure of Debt 3’):

Nowhere in the viable parameter space of BBT is velocity-space clumping a mandatory phenomenon. It is forbidden for classical-particle χ, fatal where it would have been mandatory (Scenario A), and merely optional elsewhere (Scenario B, ten tunable orders of magnitude — a degeneration flag, not a prediction).

The checkpoint’s form, #11 §9 (iii): “Net theorem: velocity-space clumping is nowhere mandatory — forbidden (particles), fatal (committed wave), decorative (free wave).” Its technical abstract: “collective momentum-space clumping is proven non-mandatory everywhere and falsified where it would have been mandatory” (#11 §1).

“Mandatory” is the operative word. A program’s prediction counts only if the program requires it; a phenomenon the program merely permits, at a tunable strength, tests nothing. The theorem says that velocity clumping is never in the first class.

The three legs#

The parameter space divides first by mass, through the classicality theorem, and then, on the wave side, by what fixes the coupling. Each part gets one verdict.

Forbidden: classical particles [D]#

On the particle island, dark-matter particles with masses of MeV to GeV, the stream instability needs the attraction energy of co-moving particles to reach their kinetic energy. At the scattering benchmark, where the coupling is fixed by dwarf-galaxy cores, the mean-field criterion fails by 10²⁸–10³¹. Below threshold there is no growth, and collisions only drive the gas toward a Maxwellian. Meeting the criterion at fixed cross-section per mass would need m ≲ 0.1 meV, where about 10²⁰ quanta share each state and dark matter is a wave (#11 §9 (i); on the regime in which Debt 3a evaluates that mass, see flag 19 in review/wiki-build-flags.md). 3a’s summary: “Collective velocity-space dynamics and classical-particle dark matter are mutually exclusive within this framework.”

Fatal: the committed wave [D + T]#

On the wave island, the gated attraction condenses the wave field at a rate Γ_self. Streams need this rate to beat orbital stirring, Γ_self ≳ Ω = 1/t_dyn, with t_dyn the dynamical time of the system. Under Scenario A, the same gated channel makes the cores of classical dwarf galaxies, which fixes the coupling; “committed” in #11’s phrase means this. Then streams lose to stirring “everywhere by 10²–10⁴ — except ultra-faint dwarfs” (3b). In the ultra-faint dwarf galaxies condensation is mandatory: for Segue 1, “rate ratio ~660; condensation ~10⁴ yr” (#11 §9 (ii)). Because the interaction is attractive, the condensate collapses whenever it exceeds a critical mass and ejects part of its mass as fast waves, so the galaxy’s dark matter is drained. Observed ultra-faint dwarfs are dense and old, so Scenario A is “falsified by observed UFD density and longevity”, with UFD an ultra-faint dwarf, and “the light-mass escape is closed independently [15†, Lyman-α]” (#11 §9 (ii)). 3c’s robustness check: “shrinking v* makes it worse …; growing v* leaves R unchanged; κ < 1 is by definition Scenario B. No parameter direction rescues it.” (3c, ‘The data’).

Decorative: the free wave#

Under Scenario B, dwarf cores come from something else, “fuzzy-end quantum pressure at m ~ 10⁻²² eV, or baryonic feedback”, and the coupling is free (3b). The first option lies below the wave island’s lower edge of 3×10⁻¹⁹ eV, which 3c draws from the fuzzy-dark-matter bounds; neither note addresses this, and the question is unresolved. The stream condition then holds over a window of the mean-field parameter X ≡ gn/(mσ²): “MW numbers, m = 0.1 meV: window X ∈ [5×10⁻¹¹, 1] — ten orders of magnitude. Verdict B: streams are tunable, not mandatory.” (3b, where MW is the Milky Way). Streams can be switched on or off by choosing the coupling, so their presence or absence tests nothing. 3b: “on this branch the granularity prediction is no longer a novel requirement of the program, only an option — a degeneration flag.” #11 §15 lists the “tunable wave-island collective sector (quarantined)” among the program’s degeneration risks (Lakatosian methodology).

Why the legs cover the viable space#

The mass split is the classicality theorem’s: collective behaviour exists only where dark matter is a wave. On the wave island, the coupling is either fixed by the cores of classical dwarfs or it is not, and 3c closes the gap between the two cases by definition: “κ < 1 is by definition Scenario B”, where κ is Scenario A’s normalization of the condensation rate to dwarf cores. “Viable” excludes the light-mass branch of Scenario A, m ≲ 5×10⁻²¹ eV, which bounds on fuzzy dark matter already rule out (3c, ‘The attractive endpoint’; bounds on fuzzy dark matter). The surviving wave island is “roughly 3×10⁻¹⁹ eV ≲ m ≲ 10⁻⁴ eV with Scenario-B couplings” (3c).

What it withdrew#

3c: “Rows 3 and 4 of the predictions table are permanently withdrawn, on both islands. The Fornax mechanism goes with them.” These are the lab notebook’s rows, old row 3, velocity-space granularity in halos, and old row 4, suppressed dynamical friction in condensed dwarf cores. 3b had kept a “surviving descendant of old rows 3–4”, a signature in ultra-faint dwarfs; it died with Scenario A. #11 §13 lists both rows under “Withdrawn with cause” (history of the predictions table).

What survives#

#11 §9: “Salvage: the island discriminant (knee shape: sharp cutoff ⇒ particle χ; (v*/σ)⁴ ⇒ wave χ) and free relic abundance via misalignment on the wave island”.

  • The island discriminant, row 3 of the current predictions table. The self-interaction of particle dark matter switches off sharply above v*, while on the wave island the condensation rate falls as (v*/σ)⁴, a power law. The shape of the knee would therefore tell the two islands apart. The particle side assumes the Gaussian tail of the notes, which #11 §8 now leaves open.
  • Relic abundance by misalignment on the wave island.

3c’s longer list of what “remains distinctively BBT’s own” adds “the exponential cliff + in-gate isotropy (particle island)”, which #11 §8 has since turned into a knee with an open tail (row 1, row 2); “the dark-energy sector (thawing, no-phantom, turnaround — live at DESI DR3)”, with DESI the Dark Energy Spectroscopic Instrument and DR3 its Data Release 3 (whether the stated potential can turn negative, as the turnaround needs, is flag 27 in review/wiki-build-flags.md); and “the permanent nulls (direct detection, monopoles, linear dispersion)”, which are rows 7, 8 and 9 of the current table. The dark-energy item has since come under pressure from DESI’s Data Release 2 (DR2): the root CLAUDE.md records that “DESI DR2 favors a phantom crossing above z ≈ 0.4, against the framework’s strict w ≥ −1” and that “The dark-energy sector is under real pressure ahead of DR3” (the phantom crossing).

Scope#

  • Collective dynamics only. The theorem concerns the collective behaviour of many particles or of a wave field. “The Debt-3 loophole — topological momentum-space localization of individual objects — remains untested, but has no candidate” (#11 §12; the topological loophole).
  • The level of the estimates. The particle leg rests on a mean-field criterion bracketed by two limits. The wave legs rest on a condensation rate whose coefficients are “heuristic at O(1)–O(10) level” and await calibration against Levkov–Panin–Tkachev numerics (3b, header; wave kinetics); #11 §9 tags them “coefficients [O]”. The notes argue that the verdicts survive these uncertainties: the particle side is “Robust to every dropped O(1)” (3a), and in Segue 1 “There is no timescale refuge” even after allowing for the slower growth seen in Levkov–Panin–Tkachev-type kinetics (3c, ‘Sharpened input numbers’).
  • Not machine-checked. No verify/ script covers a Debt-3 number (the iron rule).

Later use#

The round-3 adjudication cited the Debt-3 obituary as a precedent for retiring the soliton candidate (review/round3-adjudication.md, ‘Decisions’).

Record#

  • 3c first stated the theorem and closed Debt 3: “Verdict: Scenario A falsified; Debt 3 closed.”
  • The lab notebook’s addendum v0.2 summarizes it: “Net: velocity-space clumping is nowhere a mandatory prediction of BBT.” It adds that “open problems 3 and 5 close (5 via misalignment production on the wave island)”; 3b limits the relic-abundance closure to the wave island, and #11 §12 keeps the ratio of dark to ordinary matter open for both candidates.
  • #9 §7 lists it as the third of “Three results [D]”, in the form “forbidden (particles), fatal (wave, committed coupling), decorative (wave, free coupling)”.
  • #11 §9 (iii) carries it into the current checkpoint without a separate tag.

Sources#

  • notes/debt3c-ufd-endpoint.md ‘Verdict and closure of Debt 3’ — the theorem as first stated, the withdrawals, the salvage list and the final wave-island range; ‘The data’ — the robustness checks; ‘The attractive endpoint’ — the light-mass branch; ‘Sharpened input numbers’ — no timescale refuge
  • papers/broken-born-checkpoint-v2.md §9 — the three results and the salvage; §8 — the open tail; §1 — the abstract’s summary; §12 — the loophole; §15 — the quarantined wave-island sector
  • notes/debt3a-stream-stability.md ‘Corollary (the classicality theorem)’ — the particle leg
  • notes/debt3b-wave-island.md ‘What pins g? Two scenarios’ — Scenarios A and B and the degeneration flag; ‘Standing results after Chunks A+B’ — the interim standing
  • notes/broken-born-program-v1.md §7 — the [D] tag on the three Debt-3 results
  • notes/broken-born-theory.md ‘Addendum (v0.2)’ — the notebook’s “Net” summary
  • review/round3-adjudication.md ‘Decisions’ — the Debt-3 obituary as precedent
  • CLAUDE.md ‘Since the handoff’ — the dark-energy sector “under real pressure” since the second data release
  • review/wiki-build-flags.md items 19, 27 — the regime of the classicality corollary; the sign of the dark-energy potential; both awaiting a ruling