Dark matter pearls: the knot star TQNT allows — and the data now fence in

A miniature degenerate dwarf: 10⁵⁴ knots within 21 km · topological Chandrasekhar ceiling M_Ch = 0.50 M_☉ · fraction bounded at ≤ 1% of dark matter · a window still wide open below 5.3×10⁻¹¹ M_☉.

What is a pearl made of?

In TQNT, dark matter is a reservoir of isolated prime knots of high complexity (c ≥ 23, a derived threshold), dominated by its edge: knots of 1.7 to 1.9 GeV, electrically neutral — their topological charge cancels exactly (Q = Σχ + Lk = 0: the knot's twist is compensated by its ribbon linking); their only electromagnetic channel is the tube's polarisability (σ_eq ∼ 10⁻⁴⁸ cm², seven decades below the detectors). These knots are fermions: a collapsing cloud cannot contract forever, topological Fermi pressure holds it up. The result is a pearl — a scaled-down white dwarf (a few tens of kilometres), with knots in place of electrons.

degenerate Fermi gas (n = 3/2 polytrope): R ∝ M^−1/3 · M_Ch = M_Pl³/m² = 0.50 M_☉ [0.45–0.56]

1.3×10⁵⁴knots in the nominal pearl (2×10⁻³ M_☉)

21 kmits radius — compact-star density (6×10¹⁴ kg/m³)

0.50 M_☉topological Chandrasekhar ceiling: beyond it, collapse — no pearls

Explore the internal structure — computed live

The slider varies the pearl's mass; everything else is computed with the exact Lane-Emden n = 3/2 polytrope equations (the same as the monograph's script, reservoir bricks at 1.8 GeV) — not interpolated values. Note the counter-intuitive behaviour: the lighter the pearl, the bigger it is (R ∝ M^−1/3), and the more massive, the faster its knot gas jiggles.

Mass 2.0×10⁻³ M☉

Radius R21.1 km

Number of knots N1.3×10⁵⁴

Central density ρ_c6.2×10¹⁴ kg/m³

Fermi velocity v_F/c0.020

Line luminosity L_γ (if E content)7.3×10²² ph/s

Observational statusf ≤ 1% (OGLE)

Schematic cross-section: each dot stands for ∼10⁵² knots; the jiggle traces the Fermi velocity. The orange halo appears as the pearl approaches the M_Ch ceiling (collapse).

🌀 What if you dived in?

You would touch nothing. The only contact channel is the tube's polarisability (σ ∼ 10⁻⁴⁸ cm²): a nucleon can cross the pearl billions of times without a single scattering. No surface, no impact — 10⁵⁴ knots crossed like vacuum.
But gravity is white-dwarf grade, within 21 km: ∼6×10⁷ g⊕ at the surface, escape velocity ∼5,000 km/s (1.7% of c), compactness GM/Rc² ≈ 1.4×10⁻⁴.
An extended body never arrives in one piece: Roche limit at ∼86,000 km for rock; the debris, accelerated to 5,000 km/s and focused behind the pearl, collide — a UV/X flash without any contact at all.
A small compact object crosses in 8 ms and exits barely slowed (a gravitational slingshot). Nothing is swallowed: no horizon — whatever enters with energy comes back out.
Dropped at rest at the surface, it becomes a perpetual pendulum. Nothing slows it: it falls straight down, sweeps through the centre at ∼3,600 km/s, comes out the other side, stops exactly at the opposite surface — and falls back. Inside, gravity grows linearly with radius (a harmonic field): the round trip takes 37 ms always, whatever the amplitude — ∼1,600 crossings of the star per minute, indefinitely.
The interior tide is an isotropic compression: ∼3×10⁴ s⁻² per metre of span — across the two ends of a 2 m object, the crushing reaches ∼6,000 times Earth's gravity (each half pushed toward the middle with ∼3,000 times its weight; a fighter pilot withstands 9 g). And clocks at the centre run ∼0.02% slow.
The realistic case is reversed: it is the pearl that crosses a star or a gas cloud — it comes out dressed in a thin hot baryonic envelope. Seeding a star, however, is ruled out in today's disc: halo velocity (∼200 km/s) kills Bondi-Hoyle accretion — fewer than one event per galaxy in ten billion years, even at the f = 1% ceiling (computation in the repository). The channel would only remain viable in the young Universe, when gas and dark matter moved together.
A falsifiable curiosity: an object that nevertheless harboured a pearl core would be signed from within — the dark well heats ∼6,500 km of core by a factor ∼2, and pp fusion scales as T⁶: a brown dwarf could ignite below the standard mass limit (order of magnitude, degeneracy not treated).

The limits, channel by channel (95%)

A pearl is invisible in light — but not in gravity: it deflects the light of background stars (microlensing), and its amphichiral content would produce narrow γ lines (1.07–1.64 GeV). Four campaigns bound the sector:

Channel	Window [M_☉]	f_max	Source
Subaru HSC (M31)	[5.3×10⁻¹¹, 5.6×10⁻⁶]	min 0.61% at 1.2×10⁻⁹	Smyth+ 2020, clean pixel digitisation
OGLE high-cadence (LMC/SMC)	[1.4×10⁻⁸, 0.013]	1%	Mróz+ 2024, ApJL 976, L19
20-year OGLE-IV (LMC)	[1.8×10⁻⁴, 6.3]	1%	Mróz+ 2024, Nature 632
18-year Fermi-LAT γ lines	above 7×10⁻³	26% × (M/2×10⁻³)^−8/3 → 10⁻⁷ at M_Ch	our line search (UL ∼17 photons/line) — if nominal E content

The combined envelope f_max(M) — hover to read values. Golden zone: open window. Purple zone: no pearls (beyond M_Ch). Red star: the historical nominal (5%), excluded ×5.

f ≤ 1% continuously from 1.4×10⁻⁸ M_☉ up to the M_Ch ceiling. The two OGLE campaigns hand over without a gap, and the γ channel crushes the top of the domain (massive pearls are dense, hence over-luminous in lines: Γ ∝ M^11/3, an exact result of the degenerate gas).

The deepest dip is at 1.2×10⁻⁹ M_☉: 0.61%. That is Subaru HSC on M31 — one night, 7 hours, 190 exposures of the same galaxy. The curve used here is our pixel digitisation of the Smyth+ 2020 benchmark figure (realistic stellar sizes), cross-checked against the community PBHbounds compilation.

The open window: M < 5.3×10⁻¹¹ M_☉. Below the HSC edge, no constraint: light pearls (N ≲ 3×10⁴⁶ knots) could carry all of the pearled fraction. And below 6×10⁻¹² M_☉ the pearl grows larger than its own Einstein radius (r_p > R_E): microlensing switches off structurally — an extended lens no longer focuses.

Beyond 0.50 M_☉: no pearls, by theorem. That is the topological Chandrasekhar M_Pl³/m² with the canonical reservoir bricks (1.8 GeV): Fermi pressure fails, the pearl collapses. All limits above it are moot.

⚖ The 4 August audit — honesty first

The 12th refutation of the TQNT registry (the nominal 5% fraction at 2×10⁻³ M_☉) had been recorded on two channels: OGLE (×5) and our γ lines (×13). Redoing the computations, we found a numerical bug in the internal-structure solver: any pearl requested below 6.4×10⁻³ M_☉ received the same over-dense "floor star" — the nominal line flux was inflated ×69.
Corrected: the true flux passes below the Fermi limit (×5.6 margin). The γ arm was not a refutation — the 12th refutation stands, on the OGLE channel alone (×5). The timeline is declared in the monograph (two channels on 30 July, reduced to one on 4 August).
Favourable corollary: the six narrow dark matter lines drop by the same factor — now ×550–900 below the 2015 limits: the predicted absence strengthens.
Reserves: the γ channel assumes the pearls' nominal amphichiral content; the spread between the two HSC digitisations (Smyth vs Croon) at the window edges is the declared systematic of the stellar-size treatment.

The compact landscape: two populations, six orders of magnitude apart

In November 2025, Powell and collaborators detected by VLBI a dark object of about one million solar masses in the B1938+666 lens — decomposed by the companion analysis into a point (2–5×10⁵ M_☉) dressed with an extended envelope, a phenomenology the authors deem incompatible with standard cold dark matter. That object sits ×2×10⁶ above M_Ch: it is not a pearl. Its canonical TQNT reading is Class F — a primordial defect dressed by accretion, whose abundance is pinned at f_F ∼ 10⁻⁵–10⁻⁴ by B1938 and the GD-1 stellar stream.

✓ A binding prediction

The TQNT compact landscape is bimodal and disjoint: pearls (≤ 0.50 M_☉, f ≤ 1%) and Class F defects (10⁵–3×10⁵ M_☉) — nothing in between made of stacked knots. Any dark compact population detected between ∼1 and ∼10⁴ M_☉ would have to be read in the catalogue's other classes, or would refute the taxonomy.

What next? The data calendar

Probe	Status (August 2026)	What can change
Rubin/LSST	Year 1 started 2026-06-29; public alerts (brokers) since February; DR1: June 2028	eventually f ≲ 10⁻³ over [10⁻³, 1] M_☉ — the great squeeze of the middle band
OGLE high-cadence	published (2024) — integrated here	already holds the band at 1%
DECam "AMPM" (LMC, 1-min cadence)	2026 pilots; one candidate at ∼10⁻⁷ M_☉ ("Phoebe"), already contested	if such a population were confirmed at f ≤ 1%, its natural TQNT reading would be… a pearl
Fermi-LAT (lines)	binding 18-year UL (∼17 photons/line)	excluding the true nominal would take a ×28 lower UL: out of reach — microlensing takes over

✓ In one sentence

TQNT allows knot stars — miniature degenerate dwarfs, 10⁵⁴ knots within twenty kilometres, capped at half a Sun — and the 2024-2026 data confine them to less than 1% of dark matter over most of their domain… except below 5×10⁻¹¹ M_☉, where everything remains to be discovered.

🔖 References

L. A. Cariou, "Dark Matter Pearls" (Zenodo) — doi:10.5281/zenodo.20559267; limits defined on 2026-08-04 (draft limites_perles_dm + script, TQNT repository). External data: Mróz+ 2024 (Nature 632; ApJL 976, L19), Smyth+ 2020 (PRD 101, 063005), Powell+ 2025 (Nat. Astron. 9, 1714).

The TQNT page → The pearls paper (Zenodo) → The TQNT monograph →