The gated scattering law σ_T(v)

The gated scattering law gives the momentum-transfer cross-section σ_T of conjugate matter, the measure of how strongly its particles scatter off one another, as a function of the relative velocity v of a colliding pair. Because the velocity gate turns the interaction off at large relative velocity, self-scattering is strong in dwarf galaxies, where dark matter moves at tens of kilometres per second, and absent in galaxy clusters, where it moves at thousands; between the two, σ_T bends down at a knee near the gate velocity v*. Below the knee the scattering goes equally in all directions, unlike the forward-peaked scattering of the usual models with a light force carrier. The lab notebook first derived a sharp exponential cliff at v*; once the gate’s mechanism was found, the program kept only the knee, and the shape of its tail became an open question that observations could answer.

Status: Alive — [D], with the shape [O] (papers/broken-born-checkpoint-v2.md §8) — derived in the Born approximation in the lab notebook (#1 §5.1) and downgraded from cliff to knee in calc note #6 §4 (the Debt-1 calc notes); not machine-checked. The v0.1 exponential cliff is superseded. The law is tested by row 1 and row 2 of the predictions table and read by row 3.

The law#

The checkpoint, #11, states it in one sentence of §8:

“σ_T(v) = σ₀F(v²/v*²): robust knee at v*; tail (Gaussian vs rational) diagnoses the vertex completion; isotropic below the gate versus Yukawa forward-peaking.”

σ_T(v) = σ₀ · F(v²/v*²)

• σ₀ is the cross-section at low velocity, below the gate. Its value is fixed by the cores of dwarf galaxies at the scattering benchmark.
• F is the gate function of the amplitude M(k, q) = 4πα̃m_χ/(q² + m_φ²)·F(k²/μ²), now written in the relative velocity. Its shape is open (Open: the gate shape (OP4)).
• v* is the gate velocity, where the knee sits. The lab notebook defines it as v* ≡ √2μ/m_χ, with μ the gate’s momentum scale and m_χ the mass of a χ particle (#1 §5.1). #11 does not restate the definition; the paper on conjugate special relativity uses it (#13 §4; the invariant-length shell). The definition belongs to v0.1’s gate e^{−k²/μ²}; a suspected factor of √2 in μ between that gate and calc note #6’s overlap form, which matters once μ is read as the fiber width, is recorded for a ruling in review/wiki-build-flags.md item 17 (two conventions). The flag notes that no stated number changes; this page quotes the sources as written.

Three claims are packed into the sentence: a knee whose existence and location are robust; a tail whose shape is open and diagnostic; and isotropy below the gate.

Background: the transfer cross-section [T]#

Dark-matter self-scattering is measured by the momentum-transfer cross-section,

σ_T = ∫ dΩ (1 − cos θ) dσ/dΩ

which weights each collision by the fraction of forward momentum it removes. A grazing collision at small angle θ counts for little and a head-on rebound counts fully. This is the quantity that controls how fast scattering carries heat through a halo, and it is quoted per unit mass, σ_T/m, usually in cm²/g [T]. In the Born approximation the differential cross-section dσ/dΩ is proportional to the square of the amplitude [T].

For self-interacting dark matter (SIDM) with a light mediator, the Yukawa factor 1/(q² + m_φ²) is large at small transfer q, so scattering is concentrated at small angles, forward-peaked, and the cross-section falls as the velocity rises: “Yukawa SIDM falls as v⁻⁴ with a Coulomb-log correction” (#1 §5.1). For cold dark matter (CDM) there is no self-scattering at all. These are the two standard expectations against which row 1 is set: “CDM none; Yukawa SIDM power-law + Coulomb log” (#11 §13).

How the lab notebook derived it#

The law first appears in the lab notebook’s v0.1, as a boxed result of §5.1, tagged [D]:

“In Born approximation with a heavy transfer mediator (m_φ ≳ m_χ, contact-like in x), the momentum-transfer cross-section from the boxed amplitude is”

σ_T(v) = σ₀ e^{−v²/v*²},   v* ≡ √2 μ/m_χ,   σ₀ = 16πα̃²m_χ²/m_φ⁴

The boxed amplitude is the v0.1 one, with the Gaussian gate e^{−k²/μ²} and k = m_χv_rel/2 (#1 §3). The notebook gives the result without intermediate steps. Two features of it can be read off the amplitude [T]:

• The flat part. For slow particles the transfer q is at most 2k = m_χv, far below m_χ; with m_φ ≳ m_χ the factor 1/(q² + m_φ²) is then nearly the constant 1/m_φ². The interaction is contact-like in position, and the amplitude depends on the scattering angle only weakly, through q; the gate depends on k, which the collision energy fixes, not on the angle. σ₀ collects the constants, including 1/m_φ⁴ from the squared propagator.
• The gate, squared. The cross-section involves the square of the amplitude, so the gate enters squared. The definition v* ≡ √2μ/m_χ is the one for which the squared Gaussian gate, e^{−2k²/μ²} with k = m_χv/2, reads e^{−v²/v*²}.

The notebook adds a caveat on the Born approximation at the benchmark coupling: “At α̃ ~ 1 the Born approximation is marginal (expect O(1) corrections); the shape claims above survive because they follow from the k- vs q-dependence, not from the coupling strength” (#1 §5.1).

The two signatures#

The lab notebook lists “Two structural signatures” that distinguish the law “from every x-local mediator model” (#1 §5.1), x-local meaning local in position:

1. “An exponential cliff, not a power-law tail. Yukawa SIDM falls as v⁻⁴ with a Coulomb-log correction; BBT falls off a cliff at v*. The discriminating band is v ~ 150–400 km/s (large spirals, small groups), where the two shapes diverge by orders of magnitude.” BBT is the notebook’s name for the framework, Broken-Born Theory.
2. “Isotropy inside the gate. Because the gate depends on k and the contact part is flat in q, scattering below v* is isotropic; light-mediator Yukawa scattering is forward-peaked. Energy- vs momentum-transport efficiencies differ, which feeds into halo relaxation morphology.”

The first signature was weakened when the gate’s mechanism was derived (below); the band of 150–400 km/s survives in the kill condition of row 1. The second survives unchanged as row 2. Its argument assumes a heavy mediator, since “the contact part is flat in q” only when m_φ is large compared with the transfers.

The downgrade to a knee#

Calc note #6 derived the gate from exchange of the momentum graviton X over the overlap of the χ quanta’s momentum-space wavepackets (the gate mechanism). A Gaussian gate then requires a Gaussian profile, and generic vertices give rational, power-law-tailed gates. The note draws the consequence for the scattering law, a downgrade of the flagship row:

“row 1’s ’exponential cliff’ weakens to ‘sharp knee at v*, with the tail shape (Gaussian vs rational) a diagnostic of the vertex completion.’ The knee’s existence and location are robust; its steepness is model-internal information, not a guaranteed exponential.” (notes/debt1a-ii-X-classification.md §4; the note writes v_* for v*)

The lab notebook’s v0.4 addendum records the amendment: “the exponential cliff weakens to a sharp knee at v_* with tail shape (Gaussian vs rational) diagnosing the vertex completion; the knee’s existence and location remain the robust prediction” (#1 ‘Addendum (v0.4)’). The knee’s existence follows from the gate depending on k, whatever F is; the location is set by μ. What changed is that the steepness, which v0.1 took from the assumed Gaussian, is no longer a prediction.

Row 1 changed with it, from the lab notebook’s table (#1 §9) to v1 (#9 §9) and the checkpoint (#11 §13):

v1 (#9 §9) already reads “Knee at v* ~ 10² km/s; tail shape = vertex diagnostic; zero cluster signal (particle χ)”. The history of the whole table is on the history of the predictions table.

The tests#

The checkpoint’s predictions table (#11 §13) has three rows on this law:

• Row 1, the knee. “σ_T(v) across halo masses”: requirement “Knee at v* ~ 10² km/s; zero cluster signal (particle χ)”; killed by “Robust self-interaction at v ≫ v*; knee excluded in the 150–400 km/s band”. #11 §15 lists row 1, with rows 4 and 7, as the program’s near-term falsification exposure. See Row 1: the knee in σ_T(v).
• Row 2, isotropy [D]. “Scattering anisotropy”: standard expectation “Forward-peaked (light mediators)”; requirement “Isotropic below the gate”; killed by “Morphology demanding forward peaking”. See Row 2: isotropic scattering below the gate.
• Row 3, the island discriminant. “Knee tail: sharp ⇒ particle; (v*/σ)⁴ ⇒ wave”, a diagnostic that selects between particle and wave χ rather than a kill condition. See Row 3: the island discriminant.

In #11’s comparison of the two remaining dark-matter candidates, χ particles carry “knee at v*, sharp tail; isotropic scattering; permanent direct-detection null”, and the X-condensate carries “knee shape wave-like” (#11 §12). The popular article #12 describes the knee and the smoothing of dwarf-galaxy cusps in the present tense, as if observed; in the program’s documents the core fit is the input that fixes σ₀, and the knee is a prediction (the popular articles).

On the wave island#

The law above is a two-body cross-section for classical particles. The Debt-3 calculations found that collective effects of the gated attraction can exist only if χ is light enough to behave as a classical wave, on what the notes call the wave island (the classicality theorem). There the velocity dependence takes a different form.

• Calc note #2 (Debt 3a, the first of the Debt-3 calc notes): on the particle island, “Rows 1–2 (exponential cliff, isotropy) survive — they are collisional”; on the wave island, “a cliff-analog survives as (v*/σ)³ suppression rather than an exponential. Rows 1–2 must be re-derived in wave language (two-body Born scattering is no longer the right observable)” (notes/debt3a-stream-stability.md). Here σ is the velocity dispersion of the halo.
• Calc note #3 (Debt 3b): the gated condensation rate, when the gate is narrow, is “suppressed by (v*/σ)⁴ — a power law, not an exponential”, and “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” (notes/debt3b-wave-island.md, ‘The rate’).
• Calc note #4 (Debt 3c) lists among what remains “distinctively BBT’s own” “the exponential cliff + in-gate isotropy (particle island)” and “the cliff-sharpness island discriminant (exponential vs (v*/σ)⁴)” (notes/debt3c-ufd-endpoint.md).

The Debt-3 notes were written before the Debt-1 notes and so still speak of an exponential cliff. The fourth power of calc notes #3 and #4 is the one #11 uses; calc note #2’s third power was a first estimate (conflicts between sources). The re-derivation of rows 1–2 in wave language that calc note #2 called for is answered only in part by the (v*/σ)⁴ law; no document restates isotropy for waves. That to-do belongs with the gated condensation rate.

What is open#

• The tail. Whether F is Gaussian or rational is item 4 of #11 §14; #11 §15 counts “gate tail not yet forced” among the program’s degeneration risks. See Open: the gate shape (OP4).
• The particle side of the discriminant. Row 3 reads a sharp tail as particle χ. Calc note #6 calls the rational gates power-law-tailed, and the wave-island suppression is also a power law; no source compares the two or says whether the discriminant survives a rational tail for particles.
• The strength. Why σ₀ is as large as dwarf cores require while ordinary matter couples to X only at gravitational strength is Open: the dual-stress enhancement (OP5).

Superseded: the exponential cliff of v0.1#

In v0.1 the law was σ_T(v) = σ₀e^{−v²/v*²}, “An exponential cliff, not a power-law tail” (#1 §5.1). It rested on the postulated Gaussian gate, and the lab notebook drew its sharpest predictions from it:

• the v0.1 row 1, “cliff at v* ∼ 120 km/s; exactly zero in clusters forever”, killed by “Robust nonzero self-interaction at v > 3v*” (#1 §9);
• “Rows 1, 3, and 7 are the flagship discriminants: no mainstream framework predicts an exponential scattering cliff, gravitationally-detectable macro-granularity of the local velocity distribution, and a guaranteed-null direct-detection program” (#1 §9). Row 3 of that table, velocity-space granularity, has since died (Obituary: velocity-space granularity in halos);
• the benchmark numbers above the gate, “σ_T/m (1500 km/s) ≈ 30 e^{−156} ≈ 0” and “Cluster self-interaction is identically zero at any future sensitivity” (#1 §5.1), on the scattering benchmark.

The v0.4 downgrade replaced the cliff with the knee. Numbers computed from the exponential hold only if the gate’s tail is Gaussian, and none of them has been machine-checked. The cliff’s formulas, v* ≡ √2μ/m_χ and σ₀ = 16πα̃²m_χ²/m_φ⁴, come from the lab notebook and are not restated in #11; #13 §4 uses the definition of v*, with μ read as the fiber width (flag 17 in review/wiki-build-flags.md).

Sources#

• papers/broken-born-checkpoint-v2.md §8 — the current law; §13 rows 1–3 — the requirements and kill conditions; §12 — the candidates’ handles; §15 — “gate tail not yet forced”, falsification exposure
• notes/broken-born-program-v1.md §6, §9 — v1’s law and its row 1 with the knee
• notes/debt1a-ii-X-classification.md §4 and notes/broken-born-theory.md ‘Addendum (v0.4)’ — the downgrade from cliff to knee
• notes/debt3a-stream-stability.md ‘Consequences for the predictions table’ — rows 1–2 survive on the particle island, to be re-derived in wave language; the (v*/σ)³ estimate
• notes/debt3b-wave-island.md ‘The rate’ — the (v*/σ)⁴ suppression and the island discriminant
• notes/debt3c-ufd-endpoint.md ‘Verdict and closure of Debt 3’ — what remains distinctive after Debt 3
• notes/broken-born-theory.md §5.1 — the v0.1 law, σ₀, v*, the two signatures, the Born caveat, the cluster numbers; §9 — the v0.1 row 1 and the flagship discriminants; §3 — the v0.1 amplitude and k = m_χv_rel/2
• papers/conjugate-special-relativity.md §4 — v* = √2 μ/m_χ with μ read as the fiber width
• review/wiki-build-flags.md item 17 — the suspected √2 in μ between the two gate conventions, awaiting a ruling
• popsci/universes-other-half.md — the knee and core smoothing in the present tense