Threshold and positive-energy bound states of the BFSS matrix model

1 · The essentials

These manuscripts ask how many states with a finite total probability can exist at particular energies in the BFSS mathematical model. The main manuscript reports exactly one such state at zero energy for every finite model size $N \geq 2$ . This statement concerns the unchanged model after its overall motion has been removed. The manuscript presents this as settling the threshold-bound-state conjecture. A companion manuscript reports infinitely many such states at positive energies for size $N = 2$ , with energies reaching arbitrarily high values. According to the source, this conflicts with the original BFSS paper’s exclusion of additional bound states at that size. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine sorting a building’s rooms by floor height: the main manuscript says there is exactly one room at ground level, while the companion says that, for one particular building size, there are rooms at positive heights with no highest height. This image illustrates only the counting and energy levels; it does not explain what makes a quantum state normalizable.

Why it matters

The main manuscript addresses both existence and uniqueness: its claim is not merely that a zero-energy state exists, but that there is exactly one under the stated conditions. The companion changes the reported picture at positive energy for $N = 2$ , where the source says the original BFSS paper excluded normalizable positive-energy states. Together, the manuscripts distinguish a unique zero-energy state from a much larger collection of positive-energy states.

2 · Going further

The main problem concerns undeformed relative $SU (N)$ BFSS matrix quantum mechanics, where “relative” means that the center-of-mass motion has been removed. For every finite $N \geq 2$ , the main manuscript claims exactly one normalizable state at energy zero. Its abstract identifies this statement as the threshold-bound-state conjecture and presents the result as establishing that conjecture.

The companion studies the relative $SU (2)$ BFSS Hamiltonian. It claims infinitely many positive eigenvalues tending to infinity, each with a square-integrable eigenvector. The abstract also specifies how the operator is defined: by closing the gauge-invariant supercharge form. According to the source, the claimed positive-energy states contradict the exclusion stated in the original BFSS paper at $N = 2$ .

The two results concern different energy ranges and different scopes in $N$ : the zero-energy uniqueness claim covers every finite $N \geq 2$ , while the positive-energy claim is stated only for $N = 2$ . Neither abstract describes the main idea of its argument; the operator construction in the companion is the only methodological detail supplied. Consequently, the supplied text supports a summary of the claims, but not an explanation or assessment of their derivations.

Useful notions

Quantum states and normalizability
Hamiltonians and energy eigenvalues
Center-of-mass versus relative motion
The role of SU(N) in the stated model

What was known before / what is new

Before

The source identifies the zero-energy existence-and-uniqueness question as the threshold-bound-state conjecture. It also reports that the original BFSS paper excluded normalizable positive-energy states at $N = 2$ . No further history or earlier partial results are supplied.

New

The main manuscript claims a unique normalizable zero-energy state in the undeformed relative $SU (N)$ model for every finite $N \geq 2$ . The companion claims infinitely many normalizable positive-energy eigenstates for the relative $SU (2)$ model, with eigenvalues tending to infinity. The source presents the first claim as settling the conjecture and the second as refuting the original positive-energy exclusion at $N = 2$ .

Limits and assumptions

The main claim requires the undeformed model, removal of the center of mass, and finite $N \geq 2$ . The companion’s positive-energy claim is restricted to $N = 2$ and uses the Hamiltonian defined by closing the gauge-invariant supercharge form. The supplied abstracts give no argument details sufficient to assess either derivation. The main result is not formalized in Lean and has not been formally verified; the source supplies no formal-verification status for the companion. Positive-energy states for other values of $N$ , and any infinite- $N$ conclusion, are not addressed by the supplied text; it does not specify further open problems.

Mini glossary

Normalizable state: A state whose wavefunction has finite squared integral, allowing its total probability to be normalized.
Relative model: The model after its center-of-mass motion has been removed.
Hamiltonian: The operator whose eigenvalues give the energies of energy eigenstates.
Threshold bound state: In the claim discussed here, a normalizable state at zero energy.
Square-integrable eigenvector: An eigenvector with finite squared norm, expressing the normalizability required in the companion manuscript.

Keywords : BFSS matrix model · Threshold bound state · Positive-energy eigenstates

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 that the undeformed relative $SU (N)$ BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For $SU (2)$ , a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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The unique threshold bound state of the SU(N) BFSS model

24/09/2026

To be verified

For every finite N ≥ 2, we prove that undeformed $SU (N)$ BFSS matrix quantum mechanics has exactly one normalizable zero-energy state after removing the center of mass. This establishes the threshold-bound-state conjecture.

PDF GitHub folder

Positive eigenvalues of the relative SU(2) BFSS Hamiltonian

05/10/2026

To be verified

We prove that the relative $SU (2)$ BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.

PDF GitHub folder