Théorie Quantique des Nœuds Topologiques

Cariou, Lilian; Alexandre

V6 update (July 2026) — the spacetime weave

The 6^th edition recasts the theory: spacetime is the undulating weave of a single field (E = hf as identity, Lorentz as structure); particles are born in its rails; mass follows m = √σ_χ 𝒮 (89 states at 0.13% median, extension 226 at 1.43%); gravitation has a single equation G = ℏc/M_Pl²; cosmology is mechanised (Ω_tot = 1 and w = −1 as theorems); the Koide relation is explained (Q = 2/3 at 3.3 ppm, m_e computed to ±3%); the desert theorem explains the empty spectrum between the Υ and the W; and Ω_cc was observed by LHCb within the predicted range. A time-stamped pre-registration locks in the dated predictions.

Where indicated, the sections below retain tables from the historical V5 state (mass operator and calibrated lifetimes, since superseded) — the V6 monograph (330 pp, FR+EN) is authoritative.

The Quantum Theory of Topological Knots

The Quantum Theory of Topological Knots starts from a simple idea: nothingness is unstable — it cannot stay smooth, it undulates. A free wave spreads out and fades away; but when a wave closes back on itself into a knotted loop, its shape becomes its trap: the knot cannot be undone without cutting the string. That wave, frozen by its own topology — durable, localized — is an elementary particle.

Wave-particle duality, one of the fundamental mysteries of quantum physics, finds here a natural geometric explanation. At low energy, the knot is tight: the intrinsic magnetic field of the flux tube compresses the crossings, the knotted structure is so compact it vibrates as an indivisible whole — this is the wave, delocalized and interfering, passing through slits and diffracting. At high energy, the knot loosens: energy overcomes the magnetic confinement, crossings spread apart, the tube unfolds and the topological structure becomes visible — this is the particle, localized, with its measurable mass, spin and charge. The particle and the wave are not two different natures: they are the same knot, tight or loose.

A single fundamental field, governed by the Chern-Simons action, produces all known physics. The electron is a simple trefoil with 3 crossings — and the three lepton generations are the three Z₃ modes of that same trefoil. The quarks are not fundamental entities but knot eigenstates (the light constituents share the trefoil; the strange one is the 5₂ knot). The proton is an irreducible link of three knots (3₁ + 7₂ + 7̄₂, a twist + mirror pair), the only one simultaneously satisfying the constraints of spin 1/2, charge +1, correct mass and absolute stability. This is why the 92 chemical elements exist: a single type of link, two values of the linking number.

A particle's mass is the undulation of the spacetime weave it confiscates (the total deficit at its location) — and it follows a single law, m = √σ_χ 𝒮: one scale (the tension of the χ string, 418.47 MeV) and a factor fixed by the topological class. V6 reproduces 89 masses at 0.13% median (226-state extension at 1.43%), from the electron to the top quark, with zero per-particle parameters.

Spacetime does not exist independently of matter: it emerges from the density of knotted wave functions. Where knots are dense, time slows and space curves. Gravity is not a fundamental force — it is a consequence of quantum topology. The constant G is not a free parameter but a theorem. This emergent metric reproduces all classical tests of general relativity: light deflection (1.75"), Mercury precession (43"/century), Shapiro delay, GPS correction and the Lense-Thirring effect. Where the Standard Model posits four forces, TQNT has only two origins, both topological: the curvature field of each knot, of which gravity is the large-scale resultant; and the fundamental field itself, from which electromagnetism and the strong and weak nuclear forces emerge.

Dark matter is an inevitable prediction: knots with more than 18 crossings are too complex to interact electromagnetically. They are strictly invisible (zero cross-section, not merely small), absolutely stable (lifetime exceeding 10³⁴⁷ seconds), and their cosmic abundance is set by the condensation kinetics of the Big Bang.

At the heart of black holes, the singularity of general relativity vanishes. The metric never reaches zero: it decreases exponentially. Knots are not destroyed — they are frozen in infinitely dilated space, forming a frozen knot star. Information is preserved, and E = mc² takes on a new meaning: mass is topologically confined energy. Breaking a knot releases all that energy as radiation — this is topological deconfinement.

TQNT at a glance

0,13 %

median over 89 masses

V6 engine (m = √σ_χ 𝒮); 226-state extension at 1.43%

global parameters

0 per particle (vs 19 in the Standard Model)

5,5

orders of magnitude

0.5 MeV → 173 GeV

0,0 %

m_p and r_p

938.2 MeV, 0.841 fm

1,293

Δm(n−p) MeV

exact (calibration β)

31/32

spins read (inverse Regge)

fundamentals = braid arity; excited states read from the masses alone (226-state table)

−0,4 %

τ_μ from the derived Fermi structure

P15, g_topo = 1: no weak coupling constant

117

dark matter candidates

reservoir c ≥ 23 derived + 6 stricts (Fermi-LAT lines 1.07–1.64 GeV)

The emergence chain

Six steps connect pure topology to particle physics. Each link is a known mathematical theorem.

⌬

Flat connections

Chern-Simons SU(2), vacuum critical points

→

∞

A-polynomial

Character variety, algebraic curve

→

⚖

CS mass

α₀ × U_EM × c_J^3.55

→

〜

Klein-Gordon

(□ + m_K²)Φ = 0

→

Schrödinger

NR limit, magnetic confinement

→

▲

Equilibrium

Total functional minimization

In other words: mass, the wave and quantum mechanics are not added assumptions; they follow automatically once you take seriously the idea that particles are knots of the fundamental field.

Founding ideas

The spacetime weave

Spacetime is the undulating weave of the χ field, graduated in action (h is the tooth): E = hf becomes an identity, the c limit a meshing, time the local frequency. Knots are born in its rails; matter weighs because it confiscates the undulation — gravity is the weave's response to that deficit (G = ħc/M_Pl²).

The topological vacuum

Nothingness is unstable. Any perturbation that forms a knot acquires irreversible topological protection. Stable matter consists of simple knots.

Invariants → physics

The knot carries the particle's full identity: the chirality of the Jones polynomial (V_K(t) → V_K(t⁻¹)) tells matter from antimatter, charge is a linking theorem (Q = Σχ + Lk, always an integer), spin is the braid arity carried by the ribbon, and mass follows the law of its topological sector.

The mass formula

m = √σ_χ 𝒮: a single scale — the χ string tension, √σ_χ = 418.47 MeV — and a dimensionless factor set by the topological class. 89 masses at 0.13% median, zero per-particle parameters; excited states follow the master functional M² = (Σm_i)² + 2πσ_χ(L + n/2 + 1.38 n_r).

Emergent gravity

Each knot generates its own inward quantum gravitational field. The metric g_μν emerges from the wave density |Ψ|²; the gravity we know is its large-scale resultant, not a fundamental interaction. The constant G becomes a theorem.

Emergent thermodynamics

Boltzmann, temperature and phases (solid → liquid → gas) follow from the metric with no statistical postulate. Van der Waals C₆ exact to 0.01%, σ(H₂) = 290 pm exact to 1%. Ergodicity is a misconception: gravity creates filaments, not uniformity.

Topological dark matter

Isolated knots at c ≥ 23 (a derived threshold): zero topological charge (Q = Σχ + Lk = 0), polarizability only (σ_eq ~ 10⁻⁴⁸ cm²) — the absence in direct detection is predicted; the probes are gravitational. Absolute stability (τ > 10³⁴⁷ s).

Where does a particle's mass come from?

The central idea in one sentence: the mass of a particle is the undulation of the spacetime weave it confiscates — the total deficit at its location. And it follows a single law: one scale (the tension of the χ string) times a factor fixed by the topological class.

m = √σ_χ · 𝒮

√σ_χ = 418.47 MeV — the tension of the χ string, the single scale of the spectrum: the Regge slope, Λ_χ, E_Z3 and Cornell all derive from it.

𝒮 — the dimensionless factor set by the topological class: Koide (leptons), Regge + Z₃ + tensor (baryons), GMOR (Goldstone), constituent + Cornell (heavy), VEV (electroweak).

Peripheral torsion — half a ribbon turn = πσ_χ: the universal hyperfine quantum (from π to B_s at 0.7%), zero parameters.

Result: 89 masses at 0.13% median (226-state extension at 1.43%) with 10 global parameters and zero per particle — the heavy spectroscopic sector reads zero numbers. The old V5 formula (Ω_ts operator, SU(2)₈ flow) is archived.

Isolated knots (n = 1)

For the simple knots of the light plateau, the static mass is geometric: M_topo = m_ℓ · ℓ(K)/ℓ(3₁) (ropelength). The trefoil carries m_ℓ = 310 MeV, the 5₂ gives m_s; the physical leptons are the Z₃ modes of the trefoil (Koide: e/μ/τ to ≤ 0.005%).

Links (n = 2, 3)

Composites each follow the law of their sector: mesons = knot + mirror pair (Goldstone/GMOR for the light ones); baryons = three-legged Y-junction (Regge + Z₃ + tensor); heavy = constituents + Cornell; electroweak = the 8₃ quartet at the VEV. The configuration carries spin and charge; the mass follows the sector law.

Quarks u and d: same knot 3₁

In TQNT the quarks are not fundamental entities: they are knot eigenstates. The light constituents u/d share the same knot (the trefoil, m_ℓ = 310 MeV); the strange one is the 5₂ knot (m_s = 494.8 MeV, geometric hint: ropelength ratio). The n–p difference (1.293 MeV) comes from the linking, not from a quark difference.

Topological complexity index c_J of the 20 knots

Each knot has an intrinsic degree of complexity (the index c_J). The taller the bar, the heavier the corresponding particle: the mass scale of the periodic table of particles follows from it.

Topological knots

20 prime knots classified by their topological invariants: Jones polynomial V_K(t), EM energy U_EM, topological complexity index c_J, spin via (ν, μ, Q).

Nœud	3₁ (Trèfle)
Crossings	3
Type	Torique T(2,3)
c_J	3,3
Spin (ν,μ,Q)	1/2
σ	+1
χ = σ×b	+2
Amphichiral	non

Derived spin and charge

Spin (1/2, 1, 3/2…) and electric charge (−1, 0, +1…) are numbers put in by hand in the Standard Model. In TQNT they are deduced from topology: the spin of a fundamental particle is the braid arity of its knot — carried by the ribbon, a 4π object —, while the spin of an excited state is paid in energy: it is read off its mass by inverse Regge. Charge is a linking number.

Spin: carried by the ribbon, read in the mass

A theorem of the weave makes the rotation rate ω structurally invisible (the tube's axisymmetry hides it): no mass law uses ω — they all use J. What is observable is derived in three pieces:

Fermion or boson: the 4π parity of the ribbon. A knot's ribbon returns to itself only after two full turns (720°) — the spin-½ signature. Fermion vs boson follows the parity of the braid arity: even → fermion (trefoil 3₁, 2-strand braid), odd → boson (twist 7₂, 3 strands).
Fundamentals: spin is the braid arity. The electron (3₁) carries s = 1/2; the nucleon composes 1/2 ⊗ 1 ⊗ 1 (one fermionic core, two bosonic companions) → the octet s = 1/2 and the decuplet s = 3/2.
Excited states: spin is read from the masses alone. Spin is paid in energy — j = j₀ + (M² − M₀²)/(2πσ_χ), with the slope 2πσ_χ derived. The rounding lands on the right grid (integers for mesons, half-integers for baryons), with no tuning.

31/32 trajectory spins read from the masses alone — 226-state table

Charge: a matter of linking

Q ∈ ℤ is a topological theorem. Σχ_i = hypercharge Y (derived), Lk = I₃ − Y/2 (geometric).

A direct consequence: a particle's decay channel depends on its geometry. The neutral pion (zero self-linking) decays electromagnetically into two photons. The charged pion (non-zero self-linking) must take the weak route, which is far slower: that is why it lives 9 orders of magnitude longer.

From the catalogue to the trajectories: 19/19 knots, then 31/32 spins read over 226 states

The chart compares, for the 19 catalogue knots, the predicted spin with the measured value: 19/19, no tuning. The V6 reading extends the test to excited states: over the 226-state table, 31/32 trajectory spins are read from the masses alone by inverse Regge (13 trajectories, mean deviation 0.2) — the single miss, Δ(2950), is not charged to the reading: the state is poorly established (2-star PDG, Γ ~ 330 MeV) and the slope that reads the other 31 lands exactly on 15/2 at M = 2849 MeV, within 1σ of the published mass. Prediction: its mass should come down to ~2850 MeV in modern partial-wave analyses. The other states in the table are not unknowns: they carry their spin structurally (braid arity of the fundamentals, V–P partners, multiplets) — only trajectory members allow the blind reading; for the others, the tested prediction is their mass (1.43% median).

Flux tube excitation ℓ

The flux tubes linking the components of a link admit quantised excitation modes. The total spin satisfies s = s_min + ℓ with ℓ an integer. Nine PDG particles have a spin above s_min; all of them have exactly ℓ = 1.

Particle	s_min	s_SM	ℓ	Type
ρ/ω	0	1	1	méson vecteur
φ	0	1	1	méson vecteur
J/ψ	0	1	1	méson vecteur
Υ	0	1	1	méson vecteur
W^±	0	1	1	boson de jauge
Z⁰	0	1	1	boson de jauge
Δ	1/2	3/2	1	baryon décuplet
Σ*	1/2	3/2	1	baryon décuplet
Ξ*	1/2	3/2	1	baryon décuplet
Ω⁻	1/2	3/2	1	baryon décuplet

V6: this reading by ℓ excitation is the historical V5 state. The V6 canon reads the vector–pseudoscalar partner as a half-turn of the ribbon (ΔM² = πσ_χ, universal from π to B_s to 0.7%, zero parameters), and spin as the braid arity carried by the ribbon (a 4π object).

9 particules à spin élevé : ℓ = 1 sans exception

The nucleon: 3₁ + 7₂ + 7̄₂

The irreducible link 3₁ + 7₂ + 7̄₂ simultaneously satisfies five constraints: spin 1/2, integer charge, mass < 10%, topological stability, and radius.

Components

Knot	c_J	U_EM	b	χ
3₁ (trèfle)	3,3	9,0	2	−2
7₂ (twist)	8,1	33,8	3	+3
7̄₂ (twist miroir)	8,1	33,8	3	−3
Total	19,5	76,7	—	−2

Spin: 1 fermionic component (3₁, even b) out of 3 ⇒ s = 1/2.
Charge: Q = Σχ + Lk = −2 + Lk.

Harmonic linking potential

β = 0.210 MeV (calibrated on Δm(n−p)).
Lk_opt = +3 (proton = minimum energy).
Neutron: Lk = +2, ΔLk = −1 ⇒ E_tens = 1.293 MeV.

Proton almost eternal: τ_p ≈ 8×10³³ yr = α⁻¹⁸·τ_n — not a rigorous prohibition, testable at Hyper-K

10 nucleon observables

Observable	TQNT	Experiment	Δ
m_p	938,2 MeV	938,27 MeV	0,0%
m_n	939,5 MeV	939,57 MeV	0,0%
Δm(n−p)	1.293 MeV (derived, linking tension β)	1,293 MeV	0,1%
Q_p	+1	+1	exact
Q_n	0	0	exact
Spin	1/2 (braid arity)	1/2	exact
r_p	0,841 fm	0,841 fm	exact
τ_n	∼880 s (derived β structure)	878,4 s	0,2%
m_p/m_e	1836	1836	exact
p stability	finite, ∼8 × 10³³ yr (τ_p = τ_n·α⁻¹⁸)	> 10³⁴ years (bound, channel-dependent)	testable

Spectrum of 46 particles

Each particle is a knot or link whose mass, spin and charge are computed from topological invariants. V6 reproduces 89 masses at 0.13% median (226-state extension at 1.43%) with 10 global parameters, zero per particle — canonical formula m = √σ_χ 𝒮.

Leptons

Neutrinos

Quarks (artifacts)

Mesons

Baryons

Bosons

Électron (e⁻)

Configuration: 3₁ (trefoil). Predicted mass: 0.511 MeV. Spin 1/2, charge −1.

3₁ — spin ↑

The number of knot combinations grows so fast with the crossing number that bracketing by a particle's characteristics alone (mass, spin, charge) is in general not unique: it can encompass several distinct combinations satisfying the same constraints. In the table below, we have kept for each particle the most probable combination (the simplest compatible one); it remains experimentally possible that another combination realizes the same particle.

Complete table of 46 masses

Canonical V6 assignments (σ_χ catalogue) — “σ_χ prevails”: the configuration carries the spin and the charge, the mass follows the law of its sector (a single scale √σ_χ = 418.47 MeV).

Particle	Configuration (spin & charge)	J	Q	Mass sector	m_TQNT (MeV)	m_exp (MeV)	Δ%
Free waves (no knot, c = 0)
γ	— (onde χ)	1	0	free χ wave (𝒮=0)	0	0	0,0
ν	— (onde χ)	1/2	0	free χ wave (𝒮=0)	0	0	0,0
Leptons (trefoil 3₁, Z₃ modes)
e⁻	3₁	1/2	−1	Koide (Z₃ modes)	0,51	0,51	0,0
μ⁻	3₁	1/2	−1	Koide (Z₃ modes)	105,7	105,7	0,0
τ⁻	3₁	1/2	−1	Koide (Z₃ modes)	1777	1777	0,0
Mesons (knot + mirror construction)
π⁰	3₁+5₁	0	0	Goldstone (GMOR)	135,0	135,0	0,0
π^±	3₁+5₁	0	±1	Goldstone (GMOR)	139,6	139,6	0,0
K^±	3₁+7₁	0	±1	Goldstone (GMOR)	493,7	493,7	0,0
K⁰	3₁+7₁	0	0	Goldstone (GMOR)	497,6	497,6	0,0
η	6₁+6₁+6₁	0	0	GMO	567,0	547,9	3,4
η′	3₁+3₁+6₁	0	0	U(1)_A anomaly	958,0	957,8	0,0
ρ	4₁+5₂+6₁	1	1	Regge	755,0	775,3	2,6
ω	3₁+4₁+5₁	1	0	Regge	755,0	782,6	3,5
φ	7₅+7₅	1	0	Regge + strange	1020	1019	0,0
D^±	3₁+7₆	0	±1	constituent (stoikhions)	1867	1870	0,1
D⁰	3₁+5₂+9₁	0	0	constituent (stoikhions)	1867	1865	0,1
D_s^±	7₂+9₁+9₁	0	±1	constituent + half-subst.	1959	1968	0,5
J/ψ	4₁+4₁+8₁	1	0	Cornell	3114	3097	0,6
Υ	4₁+5₂+7₄	1	0	Cornell	9800	9460	3,6
B^±	6₂+7₃+7₃	0	±1	constituent (stoikhions)	5262	5279	0,3
B⁰	6₂+7₃+7₃	0	0	constituent (stoikhions)	5262	5280	0,3
B_s⁰	3₁+6₂+6₃	0	0	constituent + half-subst.	5354	5367	0,2
Baryons (Y-junction, 3 legs)
p	3₁+7₂+7̄₂ · Lk = +3	1/2	1	octet base (Regge)	938,2	938,3	0,0
n	3₁+7₂+7̄₂ · Lk = +2	1/2	0	octet base + linking (Δm derived)	939,5	939,6	0,0
Λ	3₁+4₁+7₂	1/2	0	Regge + Δs	1115	1116	0,1
Σ⁺	3₁+6₁+7₂	1/2	1	Regge + Δs + τ⊥	1193	1189	0,3
Σ⁻	3₁+6₁+7₂	1/2	−1	Regge + Δs + τ⊥	1193	1197	0,4
Ξ⁰	3₁+6₂+7₂	1/2	0	Regge + Δs + τ⊥	1340	1315	1,9
Ξ⁻	3₁+6₂+7₂	1/2	−1	Regge + Δs + τ⊥	1340	1322	1,4
Δ	3₁+5₂+7₄	3/2	1	decuplet (Regge+Z₃)	1232	1232	0,0
Ω⁻	5₁+7₂+7₂	3/2	−1	Regge + Δs + τ⊥	1673	1672	0,0
Λ_c	3₁+4₁+8₁	1/2	1	constituent + torsion	2300	2286	0,6
Λ_b	4₁+6₃+7₂	1/2	0	constituent + torsion	5621	5620	0,0
Electroweak sector (8₃ quartet, f = 1)
W^±	7₁+7₂+8₃	1	±1	EW (VEV)	80680	80377	0,4
Z⁰	7₂+8₃+9₁	1	0	EW (VEV)	90560	91188	0,7
H	7₁+8₃+10₁	0	0	EW (VEV)	125260	125250	0,0
t	8₂+8₃+10₁	1/2	1	EW (VEV)	170410	172690	1,3

The proton and the neutron are the same link 3₁+7₂+7̄₂ (the trefoil plus a twist + mirror-twist pair): only the linking distinguishes them. The neutron carries an under-linked loop — the trapped loop (Lk = +2 instead of +3) — which it releases in β decay (n → p + e⁻ + ν), hence its metastability. The proton, at optimal linking, sits at the floor.

Spectrum statistics (V6, canonical engine)

Δ médian = 0,13 %89 états (secteurs de masse)

Δ médian = 1,43 %extension 226 états

Heavy spectroscopic sector with zero read numbers (peripheral torsion: ½ turn of the ribbon = πσ_χ, a universal hyperfine quantum); 31/32 spins read by inverse Regge. V5 history: 118/122 at 0.68% median (operator archived).

Lifetimes

In V6 the Fermi structure is derived: weak decay is a topological reconnection (P15) whose amplitude is scale-free — g_topo = 1 exactly, there is no weak coupling constant, only the barrier (the VEV, itself derived: 246.227 GeV, +28 ppm). Controls: τ_μ at −0.4% with no calibration; the N = 2+3 counting of the τ is derived (Z₃ saturation: never a 4th family); τ_n ~ 880 s vs 878 s; BR(W→hadrons) at +0.17%. What decides that a particle decays — and through which channel — is its linking topology.

What sets the rate, channel by channel

The lifetime comes from the decay rate, and that rate is set by the scale of the interaction that makes the particle decay. Each of these scales is a quantity already known to the theory — never fitted — which is why lifetimes are predicted without calibration:

Weak force — P15 reconnection (ΔLk = ±1): scale-free amplitude (g_topo = 1), rate Γ = N Q⁵/(384π³v⁴) with v = 246.227 GeV derived — the Sargent m⁵ follows from the kinematics of the weave.

Strong force — the tension of the χ string itself (σ_χ, the same scale as the masses): tension rearrangements, the fastest of all.

Electromagnetism — the gauge limit of the tube: α = 1/(k+2) = 1/137, integer k = 135.

π⁰ vs π^±: 9 decades explained

Same knot 8₁, same mass (~135 MeV). Yet the π⁰ decays into two photons (electromagnetic path) in 8.5 × 10⁻¹⁷ s, while the π^± lives far longer (2.6 × 10⁻⁸ s) because it must go through the weak nuclear force. The difference is not the mass but the path opened by the topology.

The derived structures (V6)

Observable	V6 mechanism	TQNT	Experiment	Δ
τ_μ	P15 unknotting (g_topo = 1, N = 1)	2,187 × 10⁻⁶ s	2,197 × 10⁻⁶ s	−0,4 %
τ_τ	N = 2+3 counting derived (Z₃ modes)	+21 % bare	= known strong correction of R_τ (~+20 %)	identified
τ_n	β by reconnection, linking step 1.293 MeV	∼880 s	878,4 s	0,2 %
BR(W→hadrons)	W families derived (open Z₃ levels)	67,52 %	67,41 %	+0,17 %
Oscillations K⁰/D⁰/B⁰/B_s⁰	mutual EM between knots (Δm of the 4 systems)	Σ\|log\| = 0.0008 dex over the 4		<0,15 %
Desert [12–80] GeV	P15 cascade (desert theorem: 617/617 unstable)	~10⁻¹⁸ s	observed gap between Υ and W	predicted
4th family	Z₃ saturation (theorem)	forbidden	not observed	predicted

TQNT derives the structure of lifetimes (Fermi/Sargent, countings, channels) — not a number tuned per state. The old V5 table (34 states, RMS 0.94 decade, superseded assignments) is archived.

Nodal dark matter

Dark matter consists of isolated prime knots (n = 1) of high complexity. The threshold c* = 18 marks the onset of the Vassiliev suppression; the effective reservoir threshold is c_min = 23, derived (kinetic decoupling + cosmological survival) — the band c = 19–22 is forbidden (every void of the spectrum is a prediction). Main families: T(2, q ≥ 23) and T(3, q ≥ 12), dominant edge at 1.7–2.2 GeV.

Ω_DM/Ω_b reconstruitcomplete chain with NO free parameter: the Planck order of magnitude (5.45) is reached; the visible annihilates down to the CP crumb, the dark is a symmetric relic

α_cond = 1,199condensation cost per crossing, DERIVED from the kinetics (closed-form candidate: ln(2+√2) = 1.228)

σ_eq ∼ 10⁻⁴⁸ cm²NEUTRAL knots (periphery) — only channel: tube polarizability, 7 decades below the DS-50 frontier: the absence in direct detection is predicted everywhere (LZ, XENONnT, DS-20k, XLZD)

τ > 10³⁴⁷ sCS barrier: S_inst = kπ = 135π, tunneling impossible

6 narrow γ linesthe 6 strict amphichiral links: lines at 1.07–1.64 GeV, ×550–900 below the 2015 limits (pearl flux corrected 2026-08-04: the TOV floor-star bug inflated the nominal ×69) — the predicted absence strengthens; no other γ signal (any annihilation detection would falsify)

c_min = 23 (derived): Γ/H(T_c): c = 22 coupled (9.3) / c = 23 decoupled (0.067); N_coll(22) = 19 / N_coll(23) = 0.13

Single-component: only isolated knots (n = 1) survive (CS barrier). Dark links (n ≥ 2) are unstable — except the 6 strict amphichirals (1.07–1.64 GeV), a secondary subpopulation. The reservoir is dominated by the edge c = 23–25 (masses 1.7–1.9 GeV); T_c = 62 ± 3 MeV derived (topological tightening). Probes: gravitational and cosmological.

Compact objects (pearls): the defined limits (2026-08-04)

Reservoir knots can aggregate into pearls (a topological Fermi gas). This sector is now bounded by the combined microlensing + γ-line envelope (95%), under the canonical structural bounds:

Mass window [M_☉]	f_max (pearls/DM)	Origin
M < 5.3 × 10⁻¹¹	unconstrained	open window (below the HSC window; f ∼ 1 allowed)
5.3 × 10⁻¹¹ – 1.4 × 10⁻⁸	0.6 – 100%	Subaru HSC M31 (min 0.61% at 1.2 × 10⁻⁹)
1.4 × 10⁻⁸ – 0.50	≤ 1%	OGLE high-cadence + 20-year OGLE-IV; γ lines ∝ M^−8/3 above 7 × 10⁻³ (down to 10⁻⁷)
M > M_Ch = 0.50	—	no pearls (topological Chandrasekhar, M_Pl³/m², reservoir bulk 1.8 GeV)

Verdicts: the pearled fraction of DM is ≤ 1% — the historical nominal (5%) is excluded ×5 by OGLE-IV (12th refutation, single channel: the γ "×13" arm was a numerical artefact, corrected). The diffuse reservoir is untouched. The 10⁶ M_☉ object in B1938+666 (Powell+ 2025, VLBI) sits ×2 × 10⁶ above M_Ch: not a pearl — Class F reading (dressed primordial defect); the TQNT compact landscape is bimodal and disjoint. Rubin/LSST (Year 1 started 2026-06-29, DR1 June 2028) will take over the middle band; until then, OGLE high-cadence already holds it at 1%.

Dark energy

The 123 orders of magnitude of the cosmological-constant problem are resolved with no extra parameter. The residual deviation is not the cosmological constant but a boundary effect: at the frontier between matter and vacuum, knots are continually created and destroyed.

Topological cutoff

Only knots with c_n ≥ 3 contribute to the vacuum. The mass scale is fixed by the electron (3₁).

−93 ordres

ρ_vac^nod = (m_e⁴c⁵ / 32π²ℏ³) × e^{−β c_J(3₁)}

Metric coupling

The ratio between the gravitational scale κ_g and the reduced Compton wavelength:

α_{κ_g} = κ_g / λ̅_C = 8,08 × 10⁻⁴¹

Topological protection

The Chern-Simons action is topological (metric-independent). The protected fraction is d/(d+1) = 3/4. Central charge c_grav = 3k/(k+2) = 2.96.

−30 ordres

Result: topological dark energy

5,56 × 10⁻¹⁰J/m³ (predicted)

5,26 × 10⁻¹⁰J/m³ (Planck 2018)

123 orders resolved — V6: the residual gap falls within the H₀-tension band (no residual to explain)

Dark energy is the fossil trace of topological inflation: the hierarchy V_inf ≫ ρ_Λ comes from the suppression α_{κ_g}^3/4 ∼ 10⁻³⁰.

Topological inflation

The Chern-Simons phase transition at level k = 135 produces inflation. All parameters are determined by the integer k + 2 = 137 = 1/α_EM.

Inflation scale

Λ_CS = E_Pl / √(k+2) = E_Pl / √137

Λ_CS ≈ 1.04 × 10¹⁸ GeV.
Emerges naturally as √α_EM × E_Pl.

Number of e-folds

N_e = 1/α_CS = k + 2 = 137

The total number of e-folds is the inverse of the fine-structure constant. Topological cascade: each knot freezes at its own temperature.

CMB predictions

Observable	TQNT Formula	TQNT	Observation
Spectral index n_s	1 − 2/N_*	0,9649	0,9649 ± 0,0042 (Planck)
Tensor-to-scalar ratio r	12/N_*²	≈ 0,004	< 0,036 (BICEP/Keck)
Scale V^1/4	√α_EM × E_Pl	∼ 10¹⁸ GeV	consistent GUT

n_s at 0σ from Planck — r testable by CMB-S4 and LiteBIRD

The unifying role of k + 2 = 137

Observable	Formula	Value
Fine-structure constant	α_EM = 1/(k+2)	1/137
Inflation scale	Λ_CS = E_Pl/√(k+2)	10¹⁸ GeV
Number of e-folds	N_e = k+2	137
Latent heat	Δc = 6/(k+2)	0,044
Grav. charge (dark energy)	c_grav = 3k/(k+2)	2,96
CP phase (baryogenesis)	arg(V_3₁(e^2πi/(k+2)))	5,7 × 10⁻⁴

Topological black holes

No singularity, no information loss. The TQNT black hole is a frozen knot star with an exponential horizon, never sharp.

TQNT metric

g₀₀ = −c² e^−h

g_ij = (1 + h) δ_ij

h = κ_g|Ψ|² ≈ r_S/r.
At r = r_S: g₀₀ = c²/e ≠ 0.
Light is slowed to 0.43c, not stopped.

Three zones

Zone	Radius	M Fraction	Knots
I (exterior)	r > r_S	accretion	free
II (frozen)	r_break < r < r_S	∼ 100%	intact, frozen
III (foam)	r < r_break ∼ ℓ_Pl	∼ 10⁻⁴⁰	deconfined

The GR singularity is replaced by a topological phase transition: matter → pure χ field.

GR / TQNT comparison

Property	RG	TQNT
Horizon	sharp (g₀₀ = 0)	exponential (g₀₀ = c² e^−h)
Singularity	ρ → ∞	topological foam (ℓ_Pl³)
Information	lost?	preserved (frozen knots)
Internal structure	empty	frozen knot star
Mass	point-like	distributed in Zone II
Hawking	information destroyed?	knots released one by one
E = mc²	energy-mass equivalence	topological deconfinement

Information paradox: resolution

Knots are preserved in Zone II (CS barrier intact). Topological information (knot type, c_J, spin, charge) is stored in frozen knots. During Hawking evaporation, knots are released: information reappears as particles. Topological conservation (Σc_n, Q) guarantees unitarity.

Classical GR tests

The TQNT metric reproduces all classical tests of general relativity with β = γ = 1 (PPN identical to GR).

Test	Observation	RG	TQNT
Light deflection (Sun)	1,75"	1,75"	1,75"
Mercury precession	43.1"/century	42,98"	43,0"
Shapiro delay	measured	γ = 1	γ = 1
GPS correction	60 μs/day	exact	exact
Lense–Thirring effect	39.2 mas/yr	exact	exact
Gravitational waves	v_gw = c	exact	exact

The TQNT/GR difference only appears at 2^nd order PPN (∼ U³ ∼ 10⁻²³), unobservable with current technology. The only detectable difference concerns the black hole horizon: sharp in GR, exponential in TQNT.

Wave-particle duality

Energy loosens the knot. At low energy, confinement is maximal: the particle is a wave. At high energy, the potential fades: the knot loosens.

Energy E / E₀ = 0,0

ρ(K,E) / c_J1,000

m(K,E)0,511 MeV

RegimeWave

ρ(K,E) = c_J² / (c_J + E/E₀)

TQNT vs Standard Model

Structural comparison between TQNT and the Standard Model of particle physics.

Criterion	Standard Model	TQNT
Mass formula	No closed formula	m = √σ_χ 𝒮 (one scale × one topological class factor)
Free parameters	19	10 global, 0 per particle
Mass score	Not predicted	89 states at 0.13 % median (226 at 1.43 %)
Spin	Imposed (Lorentz)	Derived: (ν, μ, Q) + ℓ
Charge	Gell-Mann (imposed)	Q = Σχ + Lk (theorem)
Nucleon	Lattice QCD	3₁+7₂+7̄₂, Δm derived (0.1 %)
Gravity	QFT + GR separate	Emergent from \|Ψ\|²
Dark matter	WIMP (not found)	Isolated knots c ≥ 23 (derived); σ_eq ∼ 10⁻⁴⁸ cm² (predicted absence)
Dark energy	123 orders	ρ_Λ to the right order of magnitude (123 orders resolved)
Inflation	Ad hoc inflaton	Transition CS, N_e = 137, r ≈ 0,004
Black holes	Singularity	Frozen-knot star
Proton stability	Postulate	Topological theorem
Lifetimes	Case by case	Fermi structure derived (P15, g_topo = 1): τ_μ at −0.4 %
Leptonic hierarchy	Not explained	Z₃ modes of the trefoil: Koide Q = 2/3 at 3.3 ppm, m_e computed to ±3%
Generations	Postulated (3)	Derived: the C₃ of the trefoil (N_gen = N_c = 3)

Falsifiable predictions

TQNT makes quantitative testable predictions that distinguish it from the Standard Model and grand unification theories.

τ_p ≈ 10³⁴ yr

the proton decays — barely

τ_p/τ_n = α⁻¹⁸ (the 18-crossing barrier, P15) ≈ 8×10³³ yr — near the Super-K limit, channel not necessarily e⁺π⁰. Hyper-K will decide.

r ≈ 0,004

CMB tensor/scalar

Topological inflation (k = 135). CMB-S4 and LiteBIRD (σ_r ∼ 10⁻³).

6 γ lines

1.07 – 1.64 GeV

The six strict dark states: fine lines ×8–13 below the 2015 limits — target: deep Fermi-LAT analyses. In direct detection: predicted absence (DS-20k, XLZD).

No 4th

generation

Derived Z₃ saturation — the trefoil's triality. Any 4th family, or any sterile neutrino, would refute.

3659 MeV

Ξ_cc^*, the J = 3/2 partner

Quark models and lattice: ~3690. Antecedent: Ω_cc announced at ~3727, inside the predicted 3715–3788 window.

ρ_Λ

dark energy — 123 orders resolved

Predicted at the order of magnitude, within the H₀-tension band — the “123-orders problem” dissolved, with no free parameter.

p / n

one single link

Proton and neutron = the same 3₁ + 7₂ + 7̄₂ link; only the linking number Lk differs — isospin is geometric.

n → p

Lk change

ΔLk = −1, énergie = 1,293 MeV exactement.

Desert

12 – 80 GeV: nothing

No narrow non-electroweak resonance in [12, 80] GeV — the P15 cascade theorem. ATLAS's clean null confirms it; every gap in the spectrum is a prediction.

The 9 global parameters

The Standard Model has 19 free parameters. TQNT has 9, all global (none per particle). The CS level k = 135 is not a parameter: it is identified by 1/(k+2) = α_EM.

Each row of the table below corresponds to a specific physical mechanism. None of these parameters is tuned particle by particle: they are all global. The fine-structure constant α_EM = 1/137 sets the level of the Chern–Simons theory and is therefore not an additional free parameter.

V6 note: the table below is the historical V5 counting. The canonical V6 counting is 10 calibrated global parameters (√σ_χ = 418.47 MeV as the single scale; the knot masses m_ℓ, m_s, m_c, m_b; f_π; the radial ratio; τ_⊥; the heavy hyperfine; β_⊗) + two indices (√2, 2/9) and one integer (k = 135). The VEV (246.227 GeV, +28 ppm), G, T_c and c_min are derived.

Parameter	Role	Value
α₀	the absolute mass scale (fixed on the electron)	8,16 × 10⁻⁴ MeV
γ	how knot complexity is measured	0,1
p	the mass-formula exponent	3/4 (analytique)
f(2)	the screening between 2 linked knots (mesons)	0,86
f(3)	the screening between 3 linked knots (baryons)	0,77
β	the linking tension between components	0,210 MeV
η	the creation of new knots at matter/vacuum boundaries (P22)	~ 1
α	the stability of dark matter candidate knots	1,20
E₀	the energy at which a knot “loosens”	1,0 MeV
ξ_BS	the signature of experimental topological plasticity (P23)	~ 0,1