May 2026
When a positive integer $N$ admits more than one representation as a sum of two squares, multiple lattice points share the same norm and therefore the same Hopf mass. We call such points norm partners. We enumerate all norm-partner pairs relevant to the hadron lattice (norms up to 900), identify which are populated by known particles, and show that norm partners have different interference parameters $\mu$ — meaning they occupy the same Hopf circle but at different phase angles. This provides a classification of "near-degenerate" hadrons that is distinct from isospin or flavour symmetry.
By Fermat's two-square theorem, a positive integer $N$ is representable as $a^2 + b^2$ if and only if all prime factors of $N$ that are $\equiv 3 \pmod{4}$ occur to even powers. The number of essentially different representations (with $0 \leq a \leq b$) is determined by the number of prime factors $\equiv 1 \pmod{4}$.
The smallest norms with two representations:
| Norm $N$ | Factorisation | Representations |
|---|---|---|
| 25 | $5^2$ | $(0,5), (3,4)$ |
| 50 | $2 \times 5^2$ | $(1,7), (5,5)$ |
| 65 | $5 \times 13$ | $(1,8), (4,7)$ |
| 85 | $5 \times 17$ | $(2,9), (6,7)$ |
| 100 | $2^2 \times 5^2$ | $(0,10), (6,8)$ |
| 125 | $5^3$ | $(2,11), (5,10)$ |
| 130 | $2 \times 5 \times 13$ | $(3,11), (7,9)$ |
Among the 105 assigned hadrons, we identify the following norm-partner relationships:
| Norm | Point A | Particle A | $\theta_A$ | Point B | Particle B | $\theta_B$ |
|---|---|---|---|---|---|---|
| 50 | (1,7) | K± | 81.9° | (5,5) | f₀(500) | 45.0° |
| 185 | (4,13) | — | 72.9° | (8,11) | η′(958) | 54.0° |
| 225 | (3,6)×5 | — | — | (9,12) | — | — |
| 325 | (1,18) | — | 86.8° | (6,17) | — | 70.6° |
| 340 | (2,18) | — | 83.7° | (8,16)×(3,18) | — | — |
| 365 | (2,19) | — | 84.0° | (13,14) | — | 47.1° |
Most norm-partner norms below 400 have at most one occupied point. The norm-50 pair is the most significant because it involves a stable particle (K±), as discussed extensively in [1].
Norm partners sit on the same Hopf circle (same $|z|$) but at different phase angles. The angular separation $\Delta\theta$ between partners quantifies how differently the charge compression wave wraps around the torus for the same total energy:
For the norm-50 pair: $\Delta\theta = 81.9° - 45.0° = 36.9°$ — a dramatic angular separation. One point is maximally symmetric, the other maximally asymmetric. They share the same Hopf mass but have completely different Free masses (697.7 vs 558.2 MeV), interference parameters (0.50 vs 0.22), and compression ratios (0.707 vs 0.884).
Norm partners are geometrically distinct despite having identical Hopf masses. They represent fundamentally different winding configurations on the Clifford torus — the same "energy" distributed between the two cycles in different proportions. The physical distinction between norm partners must arise from the phase angle $\theta$ and its associated quantum numbers.
As the lattice extends to higher $w$ values, norm-partner pairs become increasingly common. For norms up to $N = 900$ (covering the lattice to $w = 30$), there are approximately 60 norms with two or more representations. At very high norms, some values have three or more representations, creating "norm triplets" — three lattice points with identical Hopf masses. These represent a rich degeneracy structure that may be relevant for understanding highly excited resonances.
The η′ sits at $(8,11)$ with norm 185. This norm also has the representation $(4,13)$: $4^2 + 13^2 = 16 + 169 = 185$. The point $(4,13)$ is currently unoccupied. Its properties would be:
The existence of this unoccupied norm partner at $(4,13)$ is a prediction: there should be a state near 949 MeV with the phase characteristics of the $(4,13)$ point. Whether this is a known resonance or an unobserved state remains to be determined.
| Result | Status |
|---|---|
| Norm partners arise from multi-representation of $N$ as $a^2+b^2$ | Exact (number theory) |
| Norm-50 is the only partner norm among stable hadrons | Exact |
| Partners differ in $\theta$, $\mu$, Free mass, and factorisation | Verified |
| $(4,13)$ is an unoccupied partner of η′ at $(8,11)$ | Prediction |