Superset has built a system that derives fundamental laws of physics. It has produced the field equations of electromagnetism, general relativity, and the strong nuclear force from nothing but a field and its symmetry. No equations are assumed. No Lagrangians are provided. The system is given only:
- A field (e.g., a vector potential Aμ)
- A symmetry rule (e.g., how the field transforms)
It outputs the unique physically consistent equations of motion.
Result 1: Maxwell's Equations from U(1) Symmetry
Input: A vector field Aμ in 4D Minkowski spacetime, with gauge symmetry.
Output: The system identified that the unique gauge-invariant Lagrangian is proportional to FμνFμν, where the field strength tensor Fμν was not provided. It emerged from the algebra. The Euler-Lagrange equations give Maxwell's equations in vacuum.
The system also correctly identified that:
- The mass term is forbidden by gauge symmetry
- The physical Lagrangian is the unique nontrivial combination
- The divergence term is gauge-invariant up to a boundary term
Result 2: The Ricci Scalar from General Covariance
Input: The Riemann curvature tensor (computed from a metric via the tensor engine).
Output: The system contracted all possible index pairings and found exactly two nonzero scalars: +R and -R (the Ricci scalar and its negative). The third possible contraction vanishes identically due to the antisymmetry of the Riemann tensor.
This proves that the Einstein-Hilbert action is the unique linear curvature invariant, from which the Einstein field equations follow.
Result 3: Yang-Mills Self-Interactions from SU(2)
Input: A vector field carrying an internal color index (a = 1,2,3), with non-Abelian gauge symmetry and coupling constant g.
Output: The system found that no quadratic Lagrangian (like Maxwell's) is invariant under this symmetry on its own. The non-Abelian variation generates terms proportional to g that cannot be canceled by any combination of quadratic terms alone.
The system then searched the space of cubic and quartic terms, and found that SU(2) symmetry locks the coefficients of the quadratic, cubic, and quartic terms into a single inseparable structure with the exact ratio 1 : -1 : +2g.
This is the Yang-Mills Lagrangian. The cubic and quartic gluon self-interactions, the reason gluons interact with each other while photons do not, were derived purely from the mathematical requirement of SU(2) gauge invariance.
What This Means
In textbooks, these Lagrangians are postulated based on physical intuition and historical development. Here, they are computed from the symmetry constraint alone.
| Theory | Input | What Emerged | Year Originally Derived |
|---|---|---|---|
| Electromagnetism | Aμ + U(1) | Field strength tensor, Maxwell's equations | 1865 (Maxwell) |
| General Relativity | Riemann tensor | Ricci scalar, Einstein-Hilbert action | 1915 (Einstein/Hilbert) |
| Yang-Mills / QCD | Aμ + SU(2) | Cubic + quartic self-interactions | 1954 (Yang-Mills) |
The mathematical structure that forces gluon self-interactions is the same structure that forbids photon self-interactions: the Abelian (U(1)) null space is 1-dimensional and purely quadratic, while the non-Abelian (SU(2)) null space requires all three polynomial orders to close.
Physical Implications
Photons don't self-interact because U(1) is Abelian. The gauge constraint matrix has a purely quadratic null space. No cubic or quartic terms are needed or allowed.
Gluons must self-interact because SU(2) is non-Abelian. The gauge constraint forces the quadratic, cubic, and quartic coefficients into a rigid ratio. You cannot have the kinetic term without the self-interaction. They are mathematically inseparable.
Gravity has a unique action. Among all possible contractions of the Riemann tensor, only R survives as a nontrivial scalar. This is why the Einstein-Hilbert action is unique at first order in curvature.
The coupling constant g appears at the correct powers. The constraint matrix naturally produces terms with g⁰ (quadratic), g¹ (cubic), and g² (quartic), locked together by the non-Abelian symmetry.
Limitations
The system works in 2D spacetime for the Yang-Mills demonstration, which is sufficient to prove the coefficient locking, but the full 4D theory has additional structure. The quartic term requires generating 1,134 candidate contractions from 6 tensor factors, of which only 2 are linearly independent. The system does not yet handle spinor fields (Dirac fermions), which would be needed for a complete Standard Model derivation.
Interested in what our system can derive in your domain? Contact us or explore our model marketplace.