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Comment The unique threshold bound state of the SU(N) BFSS model
Extension to D0-branes on D4-branes. We carried the strategy of this proof (massive index, strong-coupling profiles, exterior estimate, parity, count) over to the ADHM quantum mechanics of k D0-branes on N_f D4-branes, with the pp-wave deformation of Kim, Lee and Yi (2002) in place of the BMN one. The results, with complete proofs, verification scripts with saved outputs, a Lean 4 project and three blind cross-checks, are in app/APP-261030-0000.
- For k = 1 and one D4-brane the model has exactly one normalizable zero-energy state, Spin(5) × SU(2)_R invariant and even. Existence was proved by Erdős, Hasler and Solovej (2004), and an index argument for uniqueness was sketched by Sethi and Stern (1996). This is the first complete proof of uniqueness.
- For k = 1 and two coincident D4-branes, whose Higgs branch is the singular cone C²/Z₂, there is exactly one such state, a flavour singlet.
- For one D4-brane and every k, the deformed vacua are isolated and number the sum over j of p(j)p(k−j), which is the number of strong-coupling strata, and the index of the deformed supercharge on torus invariants equals this number. The conjecture of one bound state for every k remains open for k ≥ 2.
Three points may interest readers of this record. First, the coercive inequality with constant 2/3 is algebraic for the BMN algebra, but the Kim–Lee–Yi algebra has no analogue: no non-negative combination of its anticommutators bounds the rotation term by H. This proof uses coercivity only in Lemma 2.3, Lemma 3.3, eq. (3.23), Proposition 7.1 and Lemma 7.3, and there weaker forms with additive constants suffice, which we prove dynamically for one D0-brane. Second, the exterior-estimate constant (d−4)/(2√2) obtained from the Hardy inequality vanishes for a four-dimensional centre space, which occurs for D0-branes bound to coincident D4-branes, and a separation of variables with the angular Dirac operator gives (d−2)/(2√2) in this paper's normalization. Third, in adapting the steps used at k = 1 (Lemma 2.3, Section 3, and Sections 4 to 7) to the D0-D4 model we found no error in them. We did not check the inputs specific to BFSS.
The D0-D4 proofs were produced with Claude (Anthropic). Their algebraic identities were checked by computer algebra, and the supersymmetry algebras, the Sethi–Stern identities, the algebraic inputs of the energy bounds and the counting lemma are proved in Lean 4. The two inputs of the count at two D4-branes and the absence of algebraic coercivity were re-derived blindly by GPT-6-Astra, in agreement, and the analytic steps were checked by AI reviews. They have not been refereed.
Sources at the release commit of app/APP-261030-0000: the paper, which states the results and cites Erdős–Hasler–Solovej and Sethi–Stern, the Lean project, the blind cross-checks and the bibliography. The complete proofs are in the note that the paper cites, in the same repository.