May 2026
A striking feature of the Gaussian integer lattice framework is that a single lattice point $(w_1, w_2)$ can host multiple physically distinct particles. Unlike conventional models where each quantum state maps to a unique label, the lattice naturally accommodates degeneracies — particles sharing the same address but distinguished by $J^{PC}$, quark content, or projection type. We catalogue all multi-occupancy points on the lattice up to $w = 30$, identify three distinct classes (dual pairs, same-projection siblings, and triple occupancy), and argue that multi-occupancy is not an artefact but a feature: the lattice point encodes the mass, while the additional quantum numbers arise from the photon mode's internal structure.
In the standard quark model, particles are distinguished by their quark content, spin, parity, and charge conjugation quantum numbers $J^{PC}$. Two particles can share the same mass only by coincidence. In the Gaussian lattice framework, the situation is fundamentally different: the lattice point $z = w_1 + iw_2$ determines the mass through two projections (Hopf and Free), but does not uniquely fix $J^{PC}$. Multiple particles can therefore share a lattice address, provided they differ in internal quantum numbers.
This is analogous to atomic physics, where a single energy level $n$ can host multiple states distinguished by $\ell, m_\ell, m_s$. Here, the lattice point is the "principal quantum number" and the additional labels ($J$, $P$, $C$, $I$, $S$) correspond to the mode structure on the torus.
The most fundamental form of multi-occupancy: two particles at the same lattice point but different projections. These are the "Plato's Cave shadows" — one lattice structure casting two distinct shadows.
| Point | Hopf particle | Hopf mass | Free particle | Free mass | Mass ratio |
|---|---|---|---|---|---|
| (1, 1) | Δ width (99 MeV) | 99.0 | π± (140 MeV) | 139.5 | $\sqrt{2}$ |
| (5, 6) | η (548 MeV) | 544.9 | ρ (775 MeV) | 767.5 | $\sqrt{2}$ |
| (8, 8) | ω (783 MeV) | 789.4 | Λ (1116 MeV) | 1116.2 | $\sqrt{2}$ |
| (12, 12) | Σ⁺ (1189 MeV) | 1184.0 | Ω⁻ (1672 MeV) | 1674.3 | $\sqrt{2}$ |
In all confirmed dual pairs, the Free/Hopf mass ratio is exactly $\sqrt{2}$, arising from $(w_1+w_2)/\sqrt{w_1^2+w_2^2}$ at $w_1 = w_2$. This ratio is the geometric hallmark of the Clifford torus.
Particles sharing both the lattice point and the projection type, distinguished only by $J^{PC}$:
| Point | Proj. | Particle 1 | $J^{PC}$ | Particle 2 | $J^{PC}$ |
|---|---|---|---|---|---|
| (3, 18) | Hopf | f₂(1270) | $2^{++}$ | K₁(1270) | $1^{+}$ |
| (14, 15) | Hopf | K₂*(1430) | $2^{+}$ | f₁(1420) | $1^{++}$ |
| (12, 13) | Hopf | a₁(1260) | $1^{++}$ | b₁(1235) | $1^{+-}$ |
| (11, 15) | Hopf | π(1300) | $0^{-+}$ | η(1295) | $0^{-+}$ |
| (15, 19) | Hopf | Λ(1690) | $\frac{3}{2}^{-}$ | Ξ(1690) | $?^?$ |
| (14, 20) | Hopf | f₀(1710) | $0^{++}$ | Δ(1700) | $\frac{3}{2}^{-}$ |
| (16, 28) | Hopf | N(2250) | $\frac{9}{2}^{-}$ | Ω(2250) | $?^?$ |
| (17, 29) | Hopf | Λ(2350) | $\frac{9}{2}^{+}$ | f₂(2340) | $2^{++}$ |
The (16,28) case is particularly striking: a non-strange nucleon excitation N(2250) shares a lattice point with the triply-strange Ω(2250). In the quark model these particles have no structural connection; on the lattice they are siblings.
The densest lattice point hosts three distinct particles:
| Point | Particle | $J^P$ | Mass (MeV) | Strangeness | Projection |
|---|---|---|---|---|---|
| (18, 18) | K₂(1770) | $2^{-}$ | 1773 | 1 | Hopf |
| K₃*(1780) | $3^{-}$ | 1776 | 1 | Hopf | |
| Σ(1775) | $\frac{5}{2}^{-}$ | 1775 | 1 | Hopf |
All three particles at (18,18) are singly-strange, negative-parity, and within 6 MeV of each other. The symmetric point $w_1 = w_2 = 18$ has norm $648 = 2^3 \times 3^4$, which is richly composite, and all three particles use the Hopf projection. The triple degeneracy arises because the lattice point provides the mass ($\sqrt{648} \times M_0^C = 1776.1$ MeV), while $J = 2, 3, \frac{5}{2}$ come from different mode polarisations on the torus.
A feature that has no analogue in the quark model: mesons and baryons sharing a lattice point.
| Point | Meson | Baryon | Same Projection? |
|---|---|---|---|
| (12, 13) | a₁(1260), b₁(1235) | Δ(1232) | All Hopf |
| (12, 18) | f₂′(1525) | N(1520) | Both Hopf |
| (14, 20) | f₀(1710) | Δ(1700) | Both Hopf |
| (17, 29) | f₂(2340) | Λ(2350) | Both Hopf |
| (18, 18) | K₂(1770), K₃*(1780) | Σ(1775) | All Hopf |
The meson/baryon distinction is not determined by the lattice point. Both particle types access the same lattice addresses. In the CPT framework, what the Standard Model calls "mesons" and "baryons" are different internal mode structures (polarisations, knot types) of the same confined photon at the same lattice point — they are different shadows of the same object, not fundamentally different objects.
With 105 particles assigned to ~95 distinct points on a lattice with ~465 available points (up to $w=30$), random chance would predict very few multi-occupancy events. The expected number of doubly-occupied points from a uniform random assignment is approximately $\binom{105}{2}/465 \approx 11.8$. We observe approximately 15 multi-occupied points, which is modestly above expectation — but the pattern of co-habitation (mesons with baryons, strange with non-strange, different spins) is far more structured than random overlap would produce.
| Class | Count | Defining feature | Example |
|---|---|---|---|
| I: Dual pairs | 4 confirmed | Same point, different projection (Hopf/Free) | (8,8): ω / Λ |
| II: Siblings | ~10 | Same point, same projection, different $J^{PC}$ | (16,28): N(2250) / Ω(2250) |
| III: Triple | 1 | Three particles at one point | (18,18): K₂, K₃*, Σ |
| Meson–baryon | 5 | Both types at same address | (12,13): a₁, b₁, Δ |