The canonical mass law $m = \sqrt{\sigma_\chi}\,\mathcal{S}$

The computational core of TQNT: a single scale — the tension of the χ string — and sector laws whose coefficients are theorems or conditional derivations. This chapter unrolls the law, sector by sector, with statuses and dated predictions.

1. Statement and status

The canonical mass law of TQNT (postulate P21, V6 form) reads:

A single dimensionful scale — $\sqrt{\sigma_\chi} = 418{,}47$ MeV, the tension of the $\chi$ string — times a dimensionless factor $\mathcal{S}$ set by the state's topological class. There is no one-formula-per-knot: there is one scale and sector laws. Score: 89 states at 0.13 % median; the 226-state extension holds at 1.43 %.

The scale is doubly constrained: the 2026 recalibration showed that the lepton sector (through $\mu_{\rm Koide} = \tfrac34\sqrt{\sigma_\chi}$) and the baryon octet base demand the same $+0.2$ % shift, agreeing to $7\times10^{-5}$ — pinning the former Regge slope (known to ±2 %) thirty times better.

2. The single scale and its derivatives

Every scale of the spectrum derives from $\sqrt{\sigma_\chi}$:

The physical content is the peripheral twisting of the weave: a bound state winding its ribbon by a full turn costs $\Delta M^2 = 2\pi\sigma_\chi$ (the Regge slope, re-read); a half-turn (flipping the ribbon — the same $4\pi$ fact as spin) costs $\pi\sigma_\chi = 0{,}5501$ GeV². That half-turn is the vector–pseudoscalar hyperfine quantum: it links every $V/P$ pair from $\pi/\rho$ to $B_s/B_s^*$ at 0.7 % median, zero parameters.

3. Leptons: the Koide relation

The three charged generations are the three $Z_3$ eigenmodes of the trefoil circuit. The parametrisation $\sqrt{m_k} = \sqrt{m_0}\,(1 + r\cos(\delta + 2\pi k/3))$ is the discrete Fourier transform on $Z_3$; the cube-root-of-unity identities give identically:

verified to 3.3 ppm on the pole masses. All the empirical content concentrates in $r = \sqrt2$ — derived (conditionally on one declared identification: amplitude $=$ channel weight): the doublet of the two circulation senses gives two channels, and $\sqrt2 = d_{1/2}(k{=}2)$, the quantum dimension — the same $\sqrt2$ measured independently in the heavy baryons ($0{,}4202\pm0{,}004$ for $\sqrt2-1$). The phase decomposes as $\delta = \pi/12 - \varepsilon$ (one step of the minimal 24-step circuit; the electron quasi-nodal), and the scale $\mu_{\rm Koide} = \tfrac34\sqrt{\sigma_\chi}$ reproduces $e$, $\mu$, $\tau$ to ≤ 0.005 %. A retarded-Green chain on the ideal trefoil tube even computes $m_e$ to +1.4/+2.2 % with no parameter (see the Koide article).

4. Baryons: Regge base and recoupling ladder

The nucleon sets the base ($m_p = 938{,}2$ MeV, exact); heavy-baryon excitations are closed multiples of the half-turn, counted by the recoupling ladder:

Each coefficient is a conditional theorem: the base 1 (Y-junction triality, winding $=1$), the flip (the $V$–$P$ quantum), the ×3 per generation (the $Z_3$ step, measured $2{,}98\pm0{,}03$), the excess $\sqrt2-1$ (quantum-dimension theorem), the $\tfrac12$ (junction binding). Result: 15 heavy baryons at 0.23 % median, with no parameter or anchor per state — and dated predictions ($\Xi_{cc}^* = 3659$ MeV, $\Omega_b^* = 6067$ MeV).

5. Goldstone and the heavy sector

• Goldstone: the pion comes out of the GMOR relation with $E_{Z_3} = 139{,}49$ MeV — $m_\pi$ to $-0.06$ %.
• Heavy: stoikhion cores (the building-block knots $5_2$, $7_2$, $9_1$…; $m_\ell$, $m_s$, $m_c$, $m_b$) + the Cornell potential with $\sigma = \sigma_\chi$; half-substitution $\ell\to s$ in mesons (shared tube) vs full in baryons (own legs); 57 states at ~0.3 %.
• Metaxions: the threshold states (X(3872), T$_{cc}$, P$_c$…) are two whole hadrons pinned just below their threshold — counted with their half-turns, they fall below 1 %.

6. The electroweak sector

The asthenions (W, Z, H, top — four links sharing the $8_3$ knot) live at the VEV scale, itself derived:

i.e. $+28$ ppm from $v(G_F)$ — a conditional derivation (open-crossing on the $m_{\rm base}$ anchor), which removes the electroweak scale from the list of calibrated parameters.

7. Count and statuses

The full spectrum rests on 10 calibrated global parameters: 9 for the mass sector (the scale $\sqrt{\sigma_\chi}$, the stoikhion masses $m_\ell$/$m_s$/$m_c$/$m_b$, $f_\pi$, the radial ratio 1.38, $\tau_\perp$, the heavy hyperfine) plus the linking tension $\beta_\otimes$ (n–p) — never one per particle; the $k$ and $m_\ell$ identifications, if retained, bring the count down to nine. Every claim carries an explicit status: theorem (the Koide form, $Q\in\mathbb{Z}$), conditionally derived ($r=\sqrt2$, $v$, the recoupling ladder), calibrated (the 10), identified (recognised mechanisms without a pinned value).

8. Falsifiability

The law is refutable on dated numbers: $\Xi_{cc}^* = 3659$ MeV (vs ~3690 for quark models), the $B_0^*/B_1'$ centroid at 5665 MeV, the $\Sigma_b(1P)$ quintet 6076–6128, the fifth $\Omega_b(1P)$ at 6297–6304 MeV. Antecedent on record: $\Omega_{cc}$ announced by LHCb in 2026 at ~3727 MeV, inside the predicted 3715–3788 window. And any assignment agreement must pass the null test (how many random assignments would do as well?) before being presented as evidence.

9. Annex: the former $\Omega_{ts}$ operator

Earlier versions derived a single per-knot formula, $m(K)/m_e = \Phi_n\Phi_p\|\mu\tau^{3/4}\|_a\|\upsilon\|_1^{3/4}$ (the “$\Omega_{ts}$ operator”, one-loop Chern–Simons flow at SU(2)$_8$), calibrated through $\alpha_0$ on the electron — 118/122 masses at 0.68 % median. It is archived: the catalogue curves it rested on turned out to be topologically wrong (14/19), and the null test revealed too much assignment freedom. The exhaustive scan of chapter 5 remains its methodological archive; the full chapters are in the monograph's docs/Archives_recherches/.

10. Implementation

The canonical engine is reproducible from the open repository:

# spectre complet (89 états, médiane 0,13 % ; extension 226)
python3 -m simulation.experiments.spectrum_sigmachi_full
# recalibration de l'échelle (double contrainte leptons/octet)
python3 -m simulation.experiments.recalibration_sqrt_sigma
# l'excès sqrt(2)-1 des baryons lourds (3 déterminations)
python3 -m simulation.experiments.derive_indice_racine2