Row 3: the island discriminant

The island discriminant is a proposed way to read from astronomical data whether dark matter is made of particles or of a classical wave field. The program’s gated self-interaction weakens above the gate velocity v* in both cases, but differently: for particles the cutoff is sharp, while for a wave field it falls off only as a power law, (v*/σ)⁴, where σ is the velocity dispersion of the halo. The program calls the two regimes the particle island and the wave island, and the shape of the knee in the self-interaction strength, the subject of row 1, would say which one dark matter occupies. The diagnostic came out of the calculation that killed velocity clumping (Debt 3). #11 counts it among its predictions not previously posed, and it selects a branch rather than killing the program. Its particle side assumes a sharp tail, which the program now treats as open.

Status: Alive, as a diagnostic — [D] — a corollary of the wave-island condensation rate in calc note #3 (notes/debt3b-wave-island.md, where the rate is tagged [D, heuristic coefficients]), carried in #11 §9’s wave-island result [D + T] and stated as row 3 of #11 §13; not machine-checked. Its particle side assumes a sharp tail, while the tail’s shape is open [O] (#11 §14 item 4). No source records a test.

The row#

Checkpoint #11 §13, row 3:

Two columns differ from the other rows. There is no standard expectation, because the question had not been asked: v1 wrote “— (question not previously posed)” (#9 §9). And there is no kill condition. Whatever shape the tail has, the row assigns dark matter to one branch or the other; it is “Diagnostic, not a kill: identifies the branch” in v1’s words (#9 §9).

The two islands#

The row’s two branches are the two regimes that Debt 3 found for conjugate matter χ, separated by mass (the Debt-3 calc notes).

• The particle island. Heavy χ, of MeV to GeV mass, is a gas of classical particles. Its self-interaction acts collision by collision, through the two-body cross-section of the gated scattering law. This is the home of the χ-particle candidate (the particle island).
• The wave island. Very light χ has so many quanta in each quantum state that the halo behaves as a classical wave field (occupation number). After Debt 3, “The surviving wave island is roughly 3×10⁻¹⁹ eV ≲ m ≲ 10⁻⁴ eV with Scenario-B couplings” (notes/debt3c-ufd-endpoint.md; the wave island).

The classicality theorem explains why there are two islands rather than one continuum of behaviour: satisfying the criterion for collective behaviour “at fixed σ₀/m forces m ≲ 0.1 meV, where N_occ ~ 10²⁰ — collective momentum-space behavior and classical particles are mutually exclusive” (#11 §9 (i)), N_occ being the occupation number. Flag 19 in review/wiki-build-flags.md notes that calc note #2 evaluates this corollary with its broad-gate formula at Milky Way conditions, which it treats elsewhere as narrow-gate; N_occ ≫ 1 either way, so the conclusion stands. The discriminant asks which of the two the data favour.

The two laws#

Calc note #3 states the discriminant as the first of two corollaries of its condensation rate, “before any numbers” (notes/debt3b-wave-island.md, ‘The rate’; quotations from the Debt-3 notes on this page are transcribed from their LaTeX):

“Island discriminant. The velocity suppression of self-interaction effects is e^{−v²/v*²} on the particle island but (v*/σ)⁴ on the wave island. The cliff’s sharpness distinguishes the islands observationally.”

The particle side#

On the particle island the suppression is the gate’s own tail. The amplitude M(k,q) = 4πα̃m_χ/(q² + m_φ²)·F(k²/μ²) carries a gate factor F in the incoming relative momentum k (#11 §8; the velocity gate), and the transfer cross-section follows as σ_T(v) = σ₀F(v²/v*²). Every collision faster than v* is cut off by F itself. With the lab notebook’s Gaussian gate the cutoff is e^{−v²/v*²}, the form calc note #3 quotes; the lab notebook called it “An exponential cliff, not a power-law tail” (#1 §5.1, transcribed from its LaTeX).

The wave side#

On the wave island two-body scattering is not the right observable, and the self-interaction acts through wave kinetics: waves scatter off one another with a rate multiplied by the occupation number, and power collects at low relative momenta (wave kinetics). Calc note #3 computed that rate for the gated attraction (the gated condensation rate), tagged [D, heuristic coefficients]:

Γ_self ≈ c · g²ρ² min(σ, v*)⁴ / (m³σ⁶),      c = O(5–15)

with g the gated coupling, ρ the halo’s density, σ its velocity dispersion and m the χ mass. The two regimes: “Gate-open (σ ≲ v*): Γ ≈ c g²ρ²/(m³σ²). Gate-narrow: suppressed by (v*/σ)⁴ — a power law, not an exponential.” In a halo hotter than the gate, the gate removes most scattering partners but leaves those moving within about v* of each other, and that co-moving fraction shrinks only as a power of v*/σ (where the powers come from, a reading the wiki gives and the note does not write out). The “self-interaction effects” on this side are of this kinetic kind: calc note #3 calls Γ_self “the analog of SIDM heat transport (kinetic relaxation redistributes energy through the halo)” (‘What pins g? Two scenarios’), SIDM being self-interacting dark matter.

Reading the knee#

The two laws describe how the effects of self-interaction fall off above the gate velocity: steeply for particles, as a power law for waves. The row compresses this into “Knee tail: sharp ⇒ particle; (v*/σ)⁴ ⇒ wave”. It reads the tail of the knee that row 1 requires, an observable that row 1 states “across halo masses”, that is, across halos of different internal speeds.

Where it came from#

The discriminant was salvaged from the calculation that killed velocity clumping, the program’s founding image (Obituary: velocity clumping, the founding image).

1. Calc note #2 (Debt 3a). On the wave island “a cliff-analog survives as (v*/σ)³ suppression rather than an exponential” (notes/debt3a-stream-stability.md, ‘Consequences for the predictions table’), a first estimate.
2. Calc note #3 (Debt 3b). The rate above, with corollary 1. Its standing results: “Wave island: collective phenomena live but tunable, except Scenario A’s mandatory UFD signature; cliff softens to (v*/σ)⁴”, UFD meaning ultra-faint dwarf.
3. The v0.2 addendum. With the net theorem, old rows 3 and 4 were withdrawn and “A new row replaces them: cliff-sharpness island discriminant — self-interaction suppression is e^{−v²/v*²} for particle χ but (v*/σ)⁴ for wave χ, so the measured shape of the SIDM knee identifies the island” (#1 ‘Addendum (v0.2, post Debt-3)’, transcribed from its LaTeX).
4. Calc note #4 (Debt 3c). The closing note lists “the cliff-sharpness island discriminant (exponential vs (v*/σ)⁴)” among what “remains distinctively BBT’s own”, BBT being Broken-Born Theory, the notes’ name for the framework (‘Verdict and closure of Debt 3’).
5. v1. “Salvage: the island discriminant — self-interaction suppression is a sharp knee for particle χ but (v*/σ)⁴ for wave χ, so the knee’s shape identifies dark matter’s fundamental nature” (#9 §7 item 3), and row 3 of v1’s table.
6. #11. “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” (§9), and row 3 of §13.

The row took the place of the two withdrawn rows in the table’s numbering; the renumbering is on the history of the predictions table.

Its standing in the assessment#

#11 §15 counts “predictions not previously posed (rows 3, 11)” among the program’s progressive signs, in the vocabulary of Lakatos’s methodology, and its plain-language passage lists “one new experimental question (the knee-shape discriminant)” among what the program has produced. v1’s assessment had said the same: “plus one new experimental question (the knee-shape island discriminant)” (#9 §11). The handoff records it among the live results as “knee-tail shape = particle-vs-wave ‘island discriminant’ (new observable)” (HANDOFF.md §2). The popular article #12 states it in plain terms: “the sharpness of that knee differs according to whether dark matter is a gas of particles or a coherent quantum wave, so the knee’s measured shape identifies which it is” (‘Dark matter as the other polarization’; the popular articles).

The same §15 lists “gate tail not yet forced” among the degeneration risks. That item bears on this row’s particle side (below).

The data situation#

No source records a measurement of the knee’s tail, and no source names a survey or a set of halos that would read it. #11 §15 names rows 1, 4 and 7 as the near-term falsification exposure; row 3, a diagnostic, is not among them. In this wiki’s reading, since the row reads the tail of row 1’s knee, it has nothing to read until the knee itself is found.

Caveats and open questions#

• The particle side assumes a sharp tail. The e^{−v²/v*²} of the Debt-3 notes is the lab notebook’s postulated Gaussian gate. Calc note #6 then found that “tree-level exchange with generic vertices yields rational (power-law-tailed) gates; the Gaussian requires the Gaussian fiber profile” (notes/debt1a-ii-X-classification.md §4), and the v0.4 addendum made the tail “Gaussian vs rational” (#1 ‘Addendum (v0.4)’). #11 §8 keeps that open: “tail (Gaussian vs rational) diagnoses the vertex completion”. The gate’s shape is item 4 of #11 §14 (Open: the gate shape (OP4)). The row’s “sharp ⇒ particle” holds for the Gaussian case. If the particle gate is rational, both branches have power-law tails, and no source compares the two power laws or says whether the discriminant survives.
• Two different variables. The particle law is written in the collision speed v, the wave law in the halo’s dispersion σ. The sources do not convert one into the other or say how a measured knee would be compared with each.
• An uncalibrated coefficient. The rate’s coefficient c is “heuristic at O(1)–O(10) level”, awaiting calibration against Levkov–Panin–Tkachev numerics (calc note #3, header; Open: calibration of the coefficient c); #11 §9 marks the coefficients [O].
• Rows 1 and 2 in wave language. Calc note #2 asked that “Rows 1–2 must be re-derived in wave language (two-body Born scattering is no longer the right observable)”. The (v*/σ)⁴ law answers this for the knee’s tail only (Open: rows 1 and 2 in wave language).
• The X-condensate. #11 §12 gives the X-condensate the handle “knee shape wave-like” and χ particles “knee at v*, sharp tail”. The (v*/σ)⁴ law was derived for wave χ; no document derives a knee for a condensate of X or relates it to wave χ.

Where the row is used#

Rows 1–3 are the discriminators of row 11, particles or condensate: “Particles vs condensate, separable via rows 1–3 + coherence signatures” (#11 §13), which is open problem 9 of #11 §14. The companion rows are row 1, the knee, whose tail this row reads, and row 2, isotropy, which holds on the particle island.

Superseded: the exponential and the cube#

Two parts of the Debt-3 statement have been replaced.

• “Exponential vs (v*/σ)⁴”. The Debt-3 notes and the v0.2 addendum set an exponential particle side against the wave law and called the row the “cliff-sharpness” discriminant. Since the v0.4 addendum the particle side is a knee whose tail is open, and v1 and #11 write “sharp” instead of “exponential” (#9 §7; #11 §9, §13).
• (v*/σ)³. Calc note #2’s first estimate of the wave-side suppression was the third power. Calc notes #3 and #4, v1 and #11 use the fourth, and the fourth governs (Superseded: the cubic suppression of Debt 3a).

Sources#

• papers/broken-born-checkpoint-v2.md §13 row 3 — the row; §9 — the salvage, the classicality theorem and the rate’s tags; §8 — the open tail; §12 — the candidates’ handles; §14 item 4 — the open gate shape; §15 — “not previously posed”, the new experimental question and “gate tail not yet forced”
• notes/debt3b-wave-island.md ‘The rate’ — the rate, the (v*/σ)⁴ suppression and corollary 1; ‘What pins g? Two scenarios’ — the rate as the analog of SIDM heat transport; ‘Standing results after Chunks A+B’ — “cliff softens to (v*/σ)⁴”
• notes/debt3c-ufd-endpoint.md ‘Verdict and closure of Debt 3’ — the surviving wave island and the discriminant among the surviving content
• notes/debt3a-stream-stability.md ‘Consequences for the predictions table’ — the first (v*/σ)³ estimate and the rows 1–2 to-do
• notes/debt1a-ii-X-classification.md §4 — rational, power-law-tailed gates
• notes/broken-born-theory.md ‘Addendum (v0.2, post Debt-3)’ — the discriminant as the replacement row; ‘Addendum (v0.4)’ — the tail made open
• notes/broken-born-program-v1.md §7, §9, §11 — v1’s salvage, row 3 and assessment
• HANDOFF.md §2 — “new observable”
• popsci/universes-other-half.md ‘Dark matter as the other polarization’ — the popular statement
• review/wiki-build-flags.md item 19 — the regime in which calc note #2 evaluates the classicality corollary; conclusion unchanged