May 2026
The Free projection mass $M_{\text{Free}} = (w_1 + w_2) \times M_0^C$ depends only on the sum of the winding numbers. All lattice points with $w_1 + w_2 = n$ therefore lie on a diagonal line in the lattice and share the same Free mass — forming a "shell." We catalogue these Free shells, show that nature populates each shell with particles of different types (mesons, baryons, strange, non-strange), and identify the iso-Free lines as a natural organising principle complementary to the iso-Hopf circles (constant norm).
While the Hopf projection $M_{\text{Hopf}} = \sqrt{w_1^2 + w_2^2} \times M_0^C$ produces circles of constant mass (iso-norm circles $w_1^2 + w_2^2 = N$), the Free projection produces diagonal lines:
Each integer $n$ defines a "Free shell" — a set of lattice points $(1, n{-}1), (2, n{-}2), \ldots, (\lfloor n/2 \rfloor, \lceil n/2 \rceil)$ that all produce the same Free mass. On the lattice grid, these appear as anti-diagonal lines running from $(0, n)$ to $(n, 0)$.
The most densely populated shells (Free-projection assignments):
| $n$ | Free mass (MeV) | Particles (Free projection) |
|---|---|---|
| 2 | 139.5 | π± |
| 11 | 767.5 | ρ(770) |
| 16 | 1116.2 | Λ |
| 19 | 1325.7 | Ξ⁻ |
| 20 | 1395.4 | Σ*(1385) |
| 21 | 1465.2 | ρ(1450) |
| 22 | 1534.9 | Δ(1620) |
| 23 | 1604.7 | Δ(1620)* |
| 24 | 1674.7 | π₂(1670), N(1675), Σ(1670), Λ(1670), Ω⁻ |
| 25 | 1744.4 | — |
| 28 | 1953.7 | Δ(1930) |
| 29 | 2023.4 | f₄(2050) |
| 30 | 2093.1 | Λ(2100) |
| 36 | 2511.7 | f₆(2510) |
Shell $n = 24$ (Free mass = 1674.7 MeV) hosts a remarkable collection:
| Point | Particle | $J^P$ | $S$ | $I$ | Type |
|---|---|---|---|---|---|
| (1, 23) | π₂(1670) | $2^{-}$ | 0 | 1 | meson |
| (1, 23) | N(1675) | $\frac{5}{2}^{-}$ | 0 | ½ | baryon |
| (1, 23) | Σ(1670) | $\frac{3}{2}^{-}$ | 1 | 1 | baryon |
| (3, 21) | Λ(1670) | $\frac{1}{2}^{-}$ | 1 | 0 | baryon |
| (12, 12) | Ω⁻ | $\frac{3}{2}^{+}$ | 3 | 0 | baryon |
A single Free shell at 1675 MeV hosts a meson ($\pi_2$), a non-strange baryon (N), a singly-strange baryon (Σ), another singly-strange baryon (Λ), and the triply-strange Ω⁻. Five particles spanning strangeness $S = 0$ to $S = 3$, spin $J = \frac{1}{2}$ to $J = \frac{5}{2}$, and both meson and baryon types — all sharing the same Free mass. The Free shell is a mass degeneracy that cuts across all conventional quantum number boundaries.
The two iso-mass structures have complementary geometry on the lattice:
| Property | Free shell ($w_1+w_2 = n$) | Hopf circle ($w_1^2+w_2^2 = N$) |
|---|---|---|
| Geometry | Diagonal line | Quarter-circle |
| Degeneracy | $\lfloor n/2 \rfloor$ points per shell | Depends on $N$ factorisation |
| Mass spacing | Uniform: $\Delta M = M_0^C = 69.77$ MeV | Non-uniform: $\propto 1/\sqrt{N}$ |
| Physical role | Baryons (predominantly) | Mesons (predominantly) |
| Selection rule | All integers allowed | Fermat: some $N$ forbidden |
The Free shells are equally spaced in mass ($\Delta M = 69.77$ MeV), creating a comb-like structure. This uniform spacing is a direct consequence of the taxicab metric being linear. In contrast, Hopf circles crowd together at high mass ($\Delta M \propto 1/\sqrt{N}$), producing a quasi-continuous spectrum for heavy particles.
The Free mass comes in integer multiples of $M_0^C$. This "mass quantum" is entirely determined by $m_e$ and $\alpha$ — no additional parameters. The $n = 2$ shell gives the pion (139.54 MeV = $2 \times 69.77$), the $n = 16$ shell gives the Λ (1116.2 MeV = $16 \times 69.77$), and the $n = 24$ shell gives the Ω⁻ (1674.3 MeV = $24 \times 69.77$). The entire baryon mass ladder is quantised in units of $M_0^C$.