May 2026
When a particle family produces excited states at increasing mass, the successive excitations do not simply march along a single lattice trajectory. Instead, they alternate between Hopf and Free projections and scatter across phase angles, creating distinctive excitation patterns. We document these patterns for the ρ meson series, the ω/φ vector recurrences, the nucleon tower, and the scalar sector. The most striking finding is Hopf–Free alternation: ground states and their first radial excitations often use different projections, suggesting that torus excitation involves switching between the Euclidean and taxicab geometries.
The ρ meson family (isovector, $J^{PC} = 1^{--}$) has three well-established members:
| State | Mass (MeV) | $(w_1, w_2)$ | Proj. | $\theta$ | $\mu$ |
|---|---|---|---|---|---|
| ρ(770) | 775.3 | (5, 6) | Free | 50.2° | 0.496 |
| ρ(1450) | 1465 | (10, 11) | Free | 47.7° | 0.499 |
| ρ(1700) | 1720 | (9, 23) | Hopf | 68.6° | 0.388 |
The first two ρ states are both Free and both near-diagonal: (5,6) → (10,11). This is a perfect doubling pattern: $(10,11) = 2 \times (5,6) - (0,1)$. The winding numbers approximately double. The third state switches to Hopf at an asymmetric point. The pattern suggests: radial excitation first doubles the winding numbers (staying Free), then switches projection.
The doubling of (5,6) to (10,11) is remarkable because it preserves both:
The Free mass doubles exactly: $(5+6) \times M_0^C = 767.5$ MeV → $(10+11) \times M_0^C = 1464.9$ MeV. The ratio $M_{\rho'}/M_\rho = 1464.9/767.5 = 1.909 \approx 2$. This is not a coincidence — it is the signature of a torus mode doubling its winding number.
| State | Mass | $(w_1, w_2)$ | Proj. | $\theta$ |
|---|---|---|---|---|
| ω(782) | 782.7 | (8, 8) | Hopf | 45.0° |
| ω(1420) | 1410 | (11, 17) | Hopf | 57.1° |
| ω(1650) | 1670 | (17, 17) | Hopf | 45.0° |
The ω series stays Hopf throughout. The ground state is at (8,8) — perfectly symmetric. The first excitation moves off-diagonal to (11,17), then the second returns to the diagonal at (17,17). This oscillation in symmetry is:
Interestingly, (17,17) is approximately double (8,8): $17/8 = 2.125$. The winding-doubling pattern appears again, but this time within the Hopf projection.
| State | Mass | $(w_1, w_2)$ | Proj. | $\theta$ |
|---|---|---|---|---|
| π± | 139.6 | (1, 1) | Free | 45.0° |
| π(1300) | 1300 | (11, 15) | Hopf | 53.7° |
The pion's first radial excitation switches from Free to Hopf. The winding numbers jump from (1,1) to (11,15) — a large leap in both $w_1$ and $w_2$. The ratio $M_{\pi'}/M_\pi = 1300/140 = 9.3$, far larger than the $\sim$2× doubling seen in the ρ series. This suggests the pion occupies a uniquely low point on the lattice (norm 2, the smallest non-trivial norm), and its first excitation is not a simple doubling but a jump to a qualitatively different region.
| State | Mass | $(w_1, w_2)$ | Proj. | $\theta$ |
|---|---|---|---|---|
| η | 547.9 | (5, 6) | Hopf | 50.2° |
| η(1295) | 1294 | (11, 15) | Hopf | 53.7° |
| η(1475) | 1475 | (11, 18) | Hopf | 58.6° |
The η series stays Hopf and drifts slowly off-diagonal. The first excitation to (11,15) is again approximately a winding doubling: $(11,15) \approx 2 \times (5,6) + (1,3)$. The approximate ratio $(11+15)/(5+6) = 26/11 = 2.36$ is close to the $\sim$2× pattern.
| State | $J^P$ | $(w_1, w_2)$ | Proj. | $\theta$ |
|---|---|---|---|---|
| Λ | $\frac{1}{2}^+$ | (8, 8) | Free | 45.0° |
| Λ(1405) | $\frac{1}{2}^-$ | (9, 18) | Hopf | 63.4° |
| Λ(1520) | $\frac{3}{2}^-$ | (6, 21) | Hopf | 74.1° |
| Λ(1670) | $\frac{1}{2}^-$ | (3, 21) | Free | 81.9° |
| Λ(1690) | $\frac{3}{2}^-$ | (15, 19) | Hopf | 51.7° |
The Λ tower shows clear Hopf–Free alternation:
The ground state (Free) and first negative-parity excitation (Hopf) switch projections. The second negative-parity excitation returns to Free. This alternation between the two projections resembles a standing wave bouncing between two modes — the torus equivalent of parity alternation in atomic physics.
The confirmed dual pairs [1] form a ladder of increasing mass:
| Lattice point | Hopf particle | Free particle | Ratio |
|---|---|---|---|
| (1, 1) | (Δ width, 99 MeV) | π± (140 MeV) | $\sqrt{2}$ |
| (5, 6) | η (548) | ρ (775) | $\sqrt{2}$ |
| (8, 8) | ω (789) | Λ (1116) | $\sqrt{2}$ |
| (12, 12) | Σ⁺ (1184) | Ω⁻ (1674) | $\sqrt{2}$ |
This ladder has its own progression: the lattice points $(1,1) \to (5,6) \to (8,8) \to (12,12)$ march outward along the near-diagonal, with approximate norm ratios $2 : 61 : 128 : 288$. Each rung couples a meson to a baryon (or a width to a mass), always at the universal ratio $\sqrt{2}$.
The dual-pair ladder connects the lightest meson (π) to the heaviest ground-state baryon (Ω⁻) through four rungs, each locked at ratio $\sqrt{2}$. This is a global organising structure that links the meson and baryon sectors through the geometry of the Clifford torus.
| Pattern | Example | Status |
|---|---|---|
| Winding doubling ($w \to 2w$) | ρ: (5,6)→(10,11); ω: (8,8)→(17,17) | Confirmed |
| Hopf–Free alternation | Λ: Free→Hopf→Hopf→Free→Hopf | Observed |
| Phase oscillation (on/off diagonal) | ω: (8,8)→(11,17)→(17,17) | Observed |
| Projection switching at 3rd excitation | ρ: Free→Free→Hopf | Observed |
| $\sqrt{2}$ dual-pair ladder | (1,1)→(5,6)→(8,8)→(12,12) | Confirmed |