The classicality theorem and the particle island
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.
Status: Alive — [D] (#11 §9 (i)), derived in notes/debt3a-stream-stability.md (“Corollary (the classicality theorem)”). Mean-field level, at fixed σ₀/m. Not machine-checked: no verify/ script covers any Debt-3 number. A recomputation made while building the wiki, outside verify/, prints m_crit and N_occ in both gate limits (flag 19; see Scope).
Statement#
3a, ‘Corollary (the classicality theorem)’:
By the time the coupling is strong enough for collective behavior, the sector is deep in the wave regime. Collective velocity-space dynamics and classical-particle dark matter are mutually exclusive within this framework. This is a theorem-grade statement at mean-field level, not a benchmark accident.
#11 §9 (i), tagged [D]: “Classicality theorem: satisfying the criterion at fixed σ₀/m forces m ≲ 0.1 meV, where N_occ ~ 10²⁰ — collective momentum-space behavior and classical particles are mutually exclusive.” The lab notebook’s addendum puts it in one line: “collective momentum-space behavior requires N_occ ≫ 1 — it exists only for wave dark matter” (notes/broken-born-theory.md, ‘Addendum (v0.2)’).
“Theorem” and “theorem-grade” are the sources’ words. The argument is a scaling argument at mean-field level, and its premises are listed under Scope.
The argument#
The argument has three steps: how the stream criterion depends on the particle mass when the scattering strength is held fixed; the mass at which the criterion is met; and the occupation number at that mass.
Step 1: the criterion grows as the mass falls#
Write S ≡ σ₀/m for the cross-section per unit mass, the quantity that dwarf-core fits constrain (the scattering benchmark). 3a: “At fixed S ≡ σ₀/m and halo (ρ,σ), the criterion scales as √(πS) ρ/(m^{5/2}σ²).”
The note states the scaling without the intermediate line. The substitution below is the wiki’s reading, not printed in the note. It starts from the broad-gate form of the criterion, g n/(mσ²), takes the model’s definition g = √(πσ₀/m²) (the gated Vlasov model), and writes σ₀ = Sm and n = ρ/m:
g = √(πS m)/m = √(πS) m^{−1/2}, n = ρ/m
g n/(mσ²) = √(πS) ρ/(m^{5/2} σ²)At fixed S and fixed halo density ρ and dispersion σ, a lighter particle helps the criterion in two ways: the coupling per particle g grows as m^{−1/2}, and the number of particles in a given mass density grows as 1/m. Together with the m in the kinetic energy mσ², the criterion grows as m^{−5/2}.
Step 2: the critical mass#
Setting the criterion equal to 1 gives the mass below which streams could form (3a):
m_crit = [√(πS) ρ/σ²]^{2/5} ≈ 10⁻¹³ GeV ~ 0.1 meV (Milky Way conditions)The scaling is the broad-gate form of the criterion, evaluated at Milky Way conditions; see Scope and flag 19 in review/wiki-build-flags.md. At the benchmark mass m_χ = 70 MeV the criterion falls short by 10²⁸–10³¹ (the mean-field criterion); lowering the mass at fixed S closes the gap only at m_crit.
Step 3: the occupation number at that mass#
The occupation number counts how many particles share one quantum state: the number of particles in a cube one de Broglie wavelength on a side. 3a: “At that mass the per-mode occupation is”
N_occ = n λ_dB³ = ρ/(m⁴σ³) ~ 10²⁰ ≫ 1In the natural units of the notes (ħ = 1), the notes take λ_dB = 1/(mσ), the de Broglie wavelength ħ/(mσ) of a particle moving at speed σ up to factors of 2π, and the number density is n = ρ/m, which gives the second form [T]. The occupation grows as m⁻⁴: a lighter particle means both more particles in a given mass density and a longer wavelength for each.
When N_occ ≫ 1, the particles are described as a classical wave field, in the way that a radio wave is a classical field although it consists of photons: the field’s amplitude and phase are well defined, and individual quanta play no role [T]. A classical phase-space gas, the description in which the criterion was derived, does not apply.
The conclusion#
At fixed S and in a given halo, the criterion can be met only by lowering m, and lowering m drives N_occ up as m⁻⁴, faster than the criterion’s m^{−5/2}. By the time the criterion reaches 1, the dark matter is a wave with about 10²⁰ quanta per state (3a). The particle picture and collective velocity-space dynamics therefore never meet within the framework at mean-field level. The note does not spell out “not a benchmark accident”. On the wiki’s reading, it refers to m_crit depending on S and on the halo’s ρ and σ but not on the benchmark mass of 70 MeV, so the conclusion does not hinge on the choice of χ mass. The faster growth of N_occ does not by itself guarantee N_occ ≫ 1 at m_crit; the margin, N_occ ~ 10²⁰, is computed for Milky Way conditions at the benchmark value of S.
The particle island#
The theorem splits the mass range in two. The particle island is the classical side. 3a, ‘Consequences for the predictions table’:
Particle island (m_χ ~ MeV–GeV): Rows 1–2 (exponential cliff, isotropy) survive — they are collisional. Rows 3 (velocity granularity) and 4 (condensed cores / Fornax friction suppression) are withdrawn, and with them the mechanism-level linkage claimed between them. On this island BBT is cliff-shaped SIDM plus Maslov quintessence.
Here BBT is the Broken-Born framework and SIDM is self-interacting dark matter. The rows are those of the lab notebook’s prediction table. Rows 1 and 2, the velocity dependence of the scattering cross-section and the isotropy of scattering inside the velocity gate, are about two-body collisions, which need no collective instability; that is the sense of “they are collisional”. Rows 3 and 4 needed the instability and are withdrawn: old row 3, velocity-space granularity, and old row 4, condensed cores and Fornax friction suppression. “Maslov quintessence” is the program’s dark-energy field (τ quintessence).
3b’s standing result after the first two notes: “Particle island: cliff (exponential) + isotropy live; no collective phenomena. Fully classical, fully collisional.” (3b, ‘Standing results after Chunks A+B’). 3c lists the same pair among the survivors: “the exponential cliff + in-gate isotropy (particle island)”.
Since then. The “cliff” of the notes, a Gaussian cut-off e^{−v²/v*²} of the cross-section above v*, has become a knee whose tail is open (#11 §8; the scattering law). The surviving rows are now row 1, the knee, and row 2, isotropy. The particle island is the home of the χ-particle candidate; #11 §12 lists its handles as “knee at v*, sharp tail; isotropic scattering; permanent direct-detection null”. The particle side of the island discriminant, the sharp knee, is this island’s signature.
The wave island#
The other side of the theorem is the range of very light masses, where N_occ ≫ 1 and dark matter is a wave. 3a: “Wave island (m_χ ≲ meV, N_occ ≫ 1): the collective sector is reborn in field language. The halo is an incoherent wave field; the gated attraction drives condensation in k-space — power flowing into narrow-momentum streams”. Its treatment needs wave kinetics rather than the Vlasov equation, and it has its own section: the wave island.
The island’s upper edge is written differently in the three notes: “m_χ ≲ meV” (3a), “m ≲ 0.1 meV” (3b, Setup), and 10⁻⁴ eV in 3c’s final range, “roughly 3×10⁻¹⁹ eV ≲ m ≲ 10⁻⁴ eV with Scenario-B couplings”. 3c’s range closed Debt 3 and is the one this wiki uses.
Scope#
- Mean-field level. The theorem rests on the mean-field criterion, which is itself bracketed by two limits rather than solved in general (the gated Vlasov model).
- Fixed σ₀/m. The scaling holds the cross-section per unit mass at its dwarf-core value; #11 §9 states the premise as “at fixed σ₀/m”. On the wave island the question becomes a condensation rate instead of this criterion, and under Scenario B the coupling is free instead of fixed.
- Milky Way conditions for the number m_crit ≈ 10⁻¹³ GeV. The scaling has the broad-gate form, with σ² in the denominator, while 3a’s Limit 2 places the Milky Way in the narrow-gate regime and its benchmark table evaluates the Milky Way row with v*; the note does not discuss the difference. Flag 19 in
review/wiki-build-flags.mdrecords it, with an executed check made while building the wiki,review/wiki-build-checks/debt3a_check.py. With 3a’s inputs, the check printsm_crit = 1.4e-13 GeV = 0.14 meVandN_occ(m_crit) = 5.5e+19for the broad form, andm_crit (narrow-gate form) = 4.2e-14 GeVandN_occ at narrow-gate m_crit = 6.8e+21for the narrow form with v* fixed (review/wiki-build-checks/debt3a_check.out). In the flag’s words, “N_occ ≫ 1 either way, so the conclusion (collective behaviour needs wave dark matter) is unchanged”. The check is not part of the project’s gate,scripts/check.sh, and the flag awaits Paul’s ruling. The flags’ audit rated it as depending on how the corollary is read. Read as a ceiling on m, it is right: 3a’s narrow form is at most 0.064× the broad one, so 1.4e-13 GeV caps m, which is how #11 §9 uses it (“forces m ≲ 0.1 meV”). Read as the Milky Way’s threshold, it is about 3× high (4.2e-14 GeV). - Two normalizations of N_occ. 3a writes N_occ = ρ/(m⁴σ³), and 3b’s Setup does the same, but 3b’s rate formula uses a “peak occupation N_occ ≈ (2π)^{3/2}ρ/(m⁴σ³)”. The notes do not reconcile the two; at 10²⁰ ≫ 1 the factor does not affect the conclusion.
What stays open#
- The relic abundance on the particle island. The wave island obtains its abundance from misalignment. The particle island has no such route: the lab notebook states that this sector “cannot freeze out thermally without cannibalism problems” (thermal freeze-out) and assigns the abundance to polarization-genesis [O] (
notes/broken-born-theory.md§5.3). The ratio of dark to ordinary matter stays open for both candidates: “The 5.4 : 1 abundance question (polarization-genesis) applies to both” (#11 §12). - Which island is realized is an observational question, posed by the island discriminant.
Record#
The theorem appears in every later summary in the same form: #9 §7 item 1 (“collective momentum-space behavior and classical-particle dark matter are mutually exclusive”), #11 §9 (i), and the handoff’s artifact table, which describes 3a as giving the “classicality theorem (collective behavior ⇒ wave DM, m ≲ 0.1 meV)” (HANDOFF.md §3, row 2), where DM is dark matter.
A word collision. “Island” here names a mass regime of Debt 3. It is unrelated to the “consistency islands” of the gravity sector (glossary).
Sources#
notes/debt3a-stream-stability.md‘Corollary (the classicality theorem)’ — the scaling, m_crit, N_occ and the theorem; ‘Consequences for the predictions table’ — the particle and wave islands; ‘Setup’ — the definition of gpapers/broken-born-checkpoint-v2.md§9 (i) — the current statement and its tag; §8 — the knee; §12 — the χ-particle candidate and the shared abundance questionnotes/debt3b-wave-island.md‘Setup’, ‘The rate’ — the wave island’s upper edge and the second N_occ normalization; ‘Standing results after Chunks A+B’ — the particle island’s standingnotes/debt3c-ufd-endpoint.md‘Verdict and closure of Debt 3’ — the survivors and the final wave-island rangenotes/broken-born-program-v1.md§7 — the theorem in the first consolidationHANDOFF.md§3 row 2 — the artifact summary of 3anotes/broken-born-theory.md‘Addendum (v0.2)’ — the notebook’s summary; §5.3 — the particle-side relic problemreview/wiki-build-flags.mditem 19 — the regime in which 3a evaluates the corollary, awaiting a rulingreview/wiki-build-checks/debt3a_check.pyandreview/wiki-build-checks/debt3a_check.out— the flag’s executed check: m_crit and N_occ in the broad- and narrow-gate forms (outsideverify/)