1 · The essentials
These manuscripts study a model of quantum particles moving on the surface of a round sphere. In the setting considered, the particles have a special lowest-energy arrangement called the Laughlin state. The first question is whether a fixed minimum amount of energy is needed to leave that state, even as the system grows. A Lean-formalized result establishes such an energy separation for all sufficiently large systems. The main manuscript then shows that sufficiently weak, bounded changes to the energy at different locations preserve a unique lowest-energy state and a fixed minimum energy separation. The allowed strength of these changes does not shrink as the particle count grows, and the result applies uniformly across all profiles within the stated bound. The Lean formalization identified in the source covers the underlying energy-separation result, not the subsequent stability claim.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a building whose lowest floor is separated from every higher floor by a minimum step up. The first result supplies a minimum step height that persists as the building grows; the stability result says that sufficiently small changes to the layout do not eliminate that separation or split the lowest level into several equally low choices.
Why it matters
The companion manuscript resolves the spherical fermionic Laughlin spectral-gap conjecture for the full interaction specified in the source. The main manuscript addresses robustness: the energy separation is not restricted to the model without an added potential. Its uniformity means that neither growing particle number nor changing the bounded potential profile forces the stated guarantees to weaken.
2 · Going further
The setting is the fermionic Laughlin Hamiltonian at filling on expanding round spheres, with coefficient one for each pair projector. For particles, the main manuscript uses Laughlin flux . The spectral-gap question asks whether the energy separation above the Laughlin ground state has a positive lower bound independent of system size, once the system is sufficiently large.
The Lean-formalized companion, “A Fock-space inequality and the Laughlin spectral gap,” establishes a gap of at least for all sufficiently large systems under this normalization. It also gives the stronger inequality for some fixed , on the entire lowest-Landau-level Fock space for all sufficiently large flux , independently of particle number. This is the uniform unperturbed Fock-space gap that the main manuscript uses.
“Uniform Stability of the Spherical Laughlin Gap” shows in the manuscript that adding sufficiently weak bounded real scalar one-body potentials, projected to the lowest Landau level, preserves a unique ground state and a uniform spectral gap. Both the gap bound and the perturbation threshold are uniform over all sufficiently large particle numbers and all potential profiles with supremum norm at most one. The abstract identifies reliance on the unperturbed Fock-space gap, but does not describe further steps of the stability argument; its stability constants are existential rather than explicitly supplied.
Useful notions
- Quantum energy levels and ground states
- Spectral gaps
- Fermions and particle number
- Lowest Landau level
- Operator inequalities and Fock space
What was known before / what is new
Before
The source identifies the unperturbed question as the spherical fermionic Laughlin spectral-gap conjecture for the full interaction. Within this manuscript family, the uniform unperturbed Fock-space gap supplies the foundation for the stability result. The supplied abstracts do not describe earlier partial results or a broader historical sequence.
New
The Lean-formalized companion establishes the unperturbed gap bound of at least and the stronger Fock-space inequality with a fixed . The main manuscript shows stability of a unique ground state and a uniform gap under the specified weak bounded scalar potentials. Its threshold is independent of sufficiently large particle number and of the choice of potential profile within the stated norm bound.
Limits and assumptions
The conclusions concern the full fermionic interaction on round spheres, with coefficient one for each pair projector, and the stability statement uses filling and flux . The size guarantees apply only to sufficiently large systems. Stability is restricted to sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level, with profiles of supremum norm at most one; the source does not supply explicit stability constants. The identified Lean formalization is the companion gap and Fock-space result, while stability is shown in the main manuscript without a stated Lean formalization. The source does not identify further open problems or establish extensions beyond these hypotheses.
Mini glossary
- Ground state —
- A state with the lowest energy allowed by the quantum model.
- Spectral gap —
- The energy separation between the ground-state energy and the higher energy spectrum.
- Uniform bound —
- A bound that remains fixed across the system sizes or potential profiles specified in the result.
- Fock space —
- A framework that includes states with different particle numbers in a single space.
- Supremum norm —
- The bound on the largest absolute value of a potential profile.
- One-body potential —
- An added energy contribution acting on individual particles rather than on particle pairs.
Keywords : Laughlin spectral gap · Fock-space inequality · Bounded scalar disorder
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Uniform Stability of the Spherical Laughlin Gap
05/10/2026We prove that the fermionic Laughlin V1 Hamiltonian at filling 1/3 on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. With coefficient one for each pair projector and Laughlin flux , the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one. The result uses the uniform unperturbed Fock-space gap and gives existential constants.
A Fock-space inequality and the Laughlin spectral gap
24/09/2026We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V1 interaction. With coefficient one for each pair projector, the gap above the Laughlin state at filling 1/3 on the round sphere is at least 1/25 for all sufficiently large systems. More generally, for some fixed on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.