May 2026
We present the four complete baryon families on the Gaussian lattice: the N* tower (16 resonances, $J = 1/2$ to $11/2$), the Δ tower (10 resonances, $J = 1/2$ to $11/2$), the Λ tower (10 resonances, $J = 1/2$ to $9/2$), and the Σ tower (7 resonances, $J = 1/2$ to $7/2$). Each tower forms a coherent path through the lattice, with spin excitations progressing outward from the ground state. We identify systematic patterns in the lattice trajectories and show that the ratio of ground-state masses across families follows from the lattice geometry.
The nucleon family is the most complete tower on the lattice, spanning from the proton at 939 MeV to N(2600) at $J = 11/2$:
| Particle | $J^P$ | $(w_1, w_2)$ | $\theta$ | Proj. | Pred. | Meas. | Error | ★ |
|---|---|---|---|---|---|---|---|---|
| N(939) | $\frac{1}{2}^+$ | (9, 10) | 48.0° | Hopf | 938.8 | 939.6 | −0.09% | ★★★★ |
| N(1440) | $\frac{1}{2}^+$ | (5, 20) | 76.0° | Hopf | 1437.7 | 1440 | −0.16% | ★★★★ |
| N(1520) | $\frac{3}{2}^-$ | (12, 18) | 56.3° | Hopf | 1509.5 | 1515 | −0.36% | ★★★★ |
| N(1535) | $\frac{1}{2}^-$ | (9, 20) | 65.8° | Hopf | 1536.2 | 1530 | +0.41% | ★★★★ |
| N(1650) | $\frac{1}{2}^-$ | (11, 21) | 62.3° | Hopf | 1652.2 | 1655 | −0.17% | ★★★★ |
| N(1675) | $\frac{5}{2}^-$ | (1, 23) | 87.5° | Free | 1674.7 | 1675 | −0.02% | ★★★★ |
| N(1680) | $\frac{5}{2}^+$ | (12, 21) | 60.3° | Hopf | 1685.3 | 1685 | +0.02% | ★★★★ |
| N(1700) | $\frac{3}{2}^-$ | (8, 23) | 70.8° | Hopf | 1698.5 | 1700 | −0.09% | ★★★ |
| N(1710) | $\frac{1}{2}^+$ | (5, 24) | 78.2° | Hopf | 1711.4 | 1710 | +0.08% | ★★★ |
| N(1720) | $\frac{3}{2}^+$ | (9, 23) | 68.6° | Hopf | 1724.7 | 1720 | +0.27% | ★★★★ |
| N(1875) | $\frac{3}{2}^-$ | (19, 19) | 45.0° | Hopf | 1874.5 | 1875 | −0.03% | ★★★ |
| N(1880) | $\frac{1}{2}^+$ | (7, 26) | 74.9° | Hopf | 1879.6 | 1880 | −0.02% | ★★★ |
| N(1900) | $\frac{3}{2}^+$ | (8, 26) | 72.9° | Hopf | 1900.5 | 1900 | +0.03% | ★★★ |
| N(2190) | $\frac{7}{2}^-$ | (19, 25) | 52.7° | Hopf | 2190.4 | 2190 | +0.02% | ★★★★ |
| N(2220) | $\frac{9}{2}^+$ | (22, 23) | 46.3° | Hopf | 2220.5 | 2220 | +0.02% | ★★★★ |
| N(2600) | $\frac{11}{2}^-$ | (22, 30) | 53.7° | Hopf | 2598.6 | 2600 | −0.05% | ★★★ |
Mean error: 0.12%. All 10 ★★★★ resonances matched. The tower spans a mass range of 1660 MeV and a spin range of 5 units.
| Particle | $J^P$ | $(w_1, w_2)$ | $\theta$ | Proj. | Pred. | Meas. | Error | ★ |
|---|---|---|---|---|---|---|---|---|
| Δ(1232) | $\frac{3}{2}^+$ | (12, 13) | 47.3° | Hopf | 1233.2 | 1232 | +0.10% | ★★★★ |
| Δ(1600) | $\frac{3}{2}^+$ | (8, 21) | 69.1° | Hopf | 1573.3 | 1570 | +0.21% | ★★★ |
| Δ(1620) | $\frac{1}{2}^-$ | (1, 22) | 87.4° | Free | 1604.7 | 1600 | +0.29% | ★★★★ |
| Δ(1700) | $\frac{3}{2}^-$ | (14, 20) | 55.0° | Hopf | 1703.3 | 1700 | +0.19% | ★★★★ |
| Δ(1905) | $\frac{5}{2}^+$ | (10, 25) | 68.2° | Hopf | 1879.8 | 1880 | −0.01% | ★★★★ |
| Δ(1910) | $\frac{1}{2}^+$ | (15, 22) | 55.7° | Hopf | 1855.3 | 1860 | −0.25% | ★★★★ |
| Δ(1920) | $\frac{3}{2}^+$ | (9, 26) | 70.9° | Hopf | 1919.9 | 1920 | −0.01% | ★★★ |
| Δ(1930) | $\frac{5}{2}^-$ | (1, 27) | 87.9° | Free | 1953.7 | 1950 | +0.19% | ★★★ |
| Δ(1950) | $\frac{7}{2}^+$ | (18, 21) | 49.4° | Hopf | 1930.1 | 1930 | +0.01% | ★★★★ |
| Δ(2420) | $\frac{11}{2}^+$ | (24, 25) | 46.2° | Hopf | 2418.6 | 2420 | −0.06% | ★★★★ |
Mean error: 0.13%.
The most complete strange baryon family:
| Particle | $J^P$ | $(w_1, w_2)$ | $\theta$ | Proj. | Pred. | Meas. | Error | ★ |
|---|---|---|---|---|---|---|---|---|
| Λ | $\frac{1}{2}^+$ | (8, 8) | 45.0° | Free | 1116.2 | 1115.7 | +0.04% | ★★★★ |
| Λ(1405) | $\frac{1}{2}^-$ | (9, 18) | 63.4° | Hopf | 1406.7 | 1405 | +0.12% | ★★★★ |
| Λ(1520) | $\frac{3}{2}^-$ | (6, 21) | 74.1° | Hopf | 1524.3 | 1519.5 | +0.32% | ★★★★ |
| Λ(1670) | $\frac{1}{2}^-$ | (3, 21) | 81.9° | Free | 1674.7 | 1674 | +0.04% | ★★★★ |
| Λ(1690) | $\frac{3}{2}^-$ | (15, 19) | 51.7° | Hopf | 1688.7 | 1690 | −0.08% | ★★★★ |
| Λ(1800) | $\frac{1}{2}^-$ | (15, 21) | 54.5° | Hopf | 1801.3 | 1800 | +0.07% | ★★★ |
| Λ(1820) | $\frac{5}{2}^+$ | (14, 22) | 57.5° | Hopf | 1811.6 | 1820 | −0.46% | ★★★★ |
| Λ(1830) | $\frac{5}{2}^-$ | (8, 25) | 72.3° | Hopf | 1833.3 | 1830 | +0.18% | ★★★★ |
| Λ(2100) | $\frac{7}{2}^-$ | (5, 25) | 78.7° | Free | 2093.1 | 2100 | −0.33% | ★★★★ |
| Λ(2350) | $\frac{9}{2}^+$ | (17, 29) | 59.6° | Hopf | 2351.7 | 2350 | +0.07% | ★★★ |
Mean error: 0.17%. All $J = 1/2$ through $J = 9/2$ states present. The Λ tower spans 1234 MeV and 4 units of spin.
| Particle | $J^P$ | $(w_1, w_2)$ | $\theta$ | Proj. | Pred. | Meas. | Error | ★ |
|---|---|---|---|---|---|---|---|---|
| Σ⁺ | $\frac{1}{2}^+$ | (12, 12) | 45.0° | Hopf | 1184.0 | 1189.4 | −0.45% | ★★★★ |
| Σ*(1385) | $\frac{3}{2}^+$ | (10, 10) | 45.0° | Free | 1395.4 | 1385 | +0.75% | ★★★★ |
| Σ(1660) | $\frac{1}{2}^+$ | (6, 23) | 75.4° | Hopf | 1660.0 | 1660 | +0.00% | ★★★ |
| Σ(1670) | $\frac{3}{2}^-$ | (1, 23) | 87.5° | Free | 1674.7 | 1670 | +0.28% | ★★★★ |
| Σ(1750) | $\frac{1}{2}^-$ | (2, 25) | 85.4° | Hopf | 1751.2 | 1750 | +0.07% | ★★★ |
| Σ(1775) | $\frac{5}{2}^-$ | (18, 18) | 45.0° | Hopf | 1776.1 | 1775 | +0.06% | ★★★★ |
| Σ(1915) | $\frac{5}{2}^+$ | (15, 23) | 56.9° | Hopf | 1912.1 | 1915 | −0.15% | ★★★★ |
Plus the high-spin entry:
| Particle | $J^P$ | $(w_1, w_2)$ | $\theta$ | Proj. | Pred. | Meas. | Error | ★ |
|---|---|---|---|---|---|---|---|---|
| Σ(2030) | $\frac{7}{2}^+$ | (19, 22) | 49.2° | Hopf | 2027.9 | 2030 | −0.10% | ★★★★ |
Mean error: 0.23%.
| Family | $S$ | $I$ | Ground state | # Resonances | Spin range | Mean error |
|---|---|---|---|---|---|---|
| N* | 0 | ½ | (9,10) Hopf, 939 MeV | 16 | ½ → 11/2 | 0.12% |
| Δ | 0 | 3/2 | (12,13) Hopf, 1232 MeV | 10 | ½ → 11/2 | 0.13% |
| Λ | 1 | 0 | (8,8) Free, 1116 MeV | 10 | ½ → 9/2 | 0.17% |
| Σ | 1 | 1 | (12,12) Hopf, 1189 MeV | 8 | ½ → 7/2 | 0.23% |
| Ξ | 2 | ½ | (7,12) Free, 1322 MeV | 4 | ½ → 5/2 | 0.11% |
| Ω | 3 | 0 | (12,12) Free, 1672 MeV | 2 | 3/2, ? | 0.08% |
The ratios of ground-state baryon masses are determined by the lattice geometry:
Measured: $1115.7/939.6 = 1.187$. Error: 0.14%.
Measured: $1672.4/939.6 = 1.780$. Error: 0.18%.
The Gell-Mann–Okubo mass formula relates baryon masses within a multiplet using SU(3) flavour symmetry and requires 2 free parameters per multiplet. The Gaussian lattice reproduces the same mass ratios from pure geometry — the lattice addresses of the ground states — with zero free parameters.
As spin increases within a family, the lattice point moves outward. The N* tower traces a path from (9,10) to (22,30):
The pattern is clear: low-spin excitations scatter to diverse phase angles, but high-spin states converge to the diagonal ($\theta \approx 45°$). This mirrors the finding in [1] that angular momentum requires symmetric winding.
| Achievement | Detail |
|---|---|
| Total baryon resonances assigned | 50 |
| Complete families | N*(16), Δ(10), Λ(10), Σ(8), Ξ(4), Ω(2) |
| Maximum spin matched | $J = 11/2$ (Δ(2420), N(2600)) |
| Ground-state mass ratios from geometry | Λ/N, Ω/N reproduced at 0.1–0.2% |
| Combined mean error | 0.15% across all 50 baryons |