The stability index of the Carlotto–Schulz minimal hypertorus Xⁿ_(CS) in S2n: what Perdomo's conjecture (n³+9n²+11n+3)/3, and 27 at n=2, actually rests on — with an erratum for the printed shape operator
Abstract
For every integer n ≥ 2 Carlotto and Schulz constructed an embedded minimal hypersurface Xⁿ_(CS):Sn-1 × Sn-1 × S1 → S2n, generated by a closed profile curve whose initial radius r₀ and period T are known only numerically. Perdomo (arXiv:2508.09104, arXiv:2606.16980) proved that its stability index is at least n²+4n+3 and conjectured, on numerical evidence for 2 ≤ n ≤ 100, that it equals (n³+9n²+11n+3)/3 for n>2 and 27 for n=2. We take that argument apart and record, with machine-checked proofs, exactly what it rests on. (1) An erratum: the principal curvature κₜ in the profile direction printed in both papers is 1/(n-1) times the true one; the printed triple of principal curvatures has trace (2n-4)(cot rsinα-csc rcosαcot2θ) ≠ 0, so it cannot belong to a minimal hypersurface once n>2; the correct value is κₜ=α'+cot rsinα, the two agree at n=2, and the papers' own numerics must have used the correct one. (2) The eigenfunction f=(csc²rsecθcscθ)ⁿ⁻¹ of Perdomo's "unexpected multiplicity" lemma is exactly e⁻ᵇ with b=tfrac(n-1)2log(EG), the reciprocal of the Riemannian volume density, and the lemma collapses to the single identity b"=|A|²-(n-1)(1/E+1/G), true for every n with the corrected κₜ and false from n=3 with the printed one. (3) Writing the stability spectrum as the union of the spectra of Sturm–Liouville blocks Sᵢⱼ with multiplicities mᵢmⱼ, and taking the number neg(i,j) of negative eigenvalues of Sᵢⱼ as an abstract function antitone in each index, the sign facts the papers verify numerically pin the index to the conjectured polynomial, with both hypothesis sets satisfiable and the values ± 1 refuted; but the itemised list for n>2 is one fact short: a second antitone table satisfies every item of it with index larger by n(n²+n-2), and the fact that closes the gap, λ¹₃₂>0, is supplied by the source outside the list. The analytic core — the sign facts themselves — is not proved here, and we say why. All results are verified in Lean 4 against Mathlib; the source's printed numbers were recomputed independently, including its whole 2 ≤ n ≤ 100 index sweep, and we report the two places where its tables do not match that recomputation.
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Source snapshot 2026-09-07 03:53 UTC
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Claim ledger
Stated results
CS1known data2026-09-07
Compute-first gate: the Laplace data of Sⁿ⁻¹ the source uses – alphaᵢ = (i-1)(n+i-3) and mᵢ = C(n+i-2,i-1) - C(n+i-4,i-3) – give the printed values alpha₁..alpha₅ = 0, n-1, 2n, 3(n+1), 4(n+2) and m₃ = (n²+n-2)/2, m₄ = n(n+4)(n-1)/6, m₅ = n(n-1)(n²+7n+6)/24, for every n
CS2known data2026-09-07
Compute-first gate: an independent reimplementation reproduces every number the source prints – Table 1's (r₀, T/4), the n = 2 list of negative eigenvalues with multiplicities, all eight sign facts and their printed intervals, the 500-point delta₁1 scan of Figure 1, Table 2, and the conjectured index for all 2 <= n <= 100
This ledger entry is reported in prose and is not bound to a Lean theorem.CS3correction2026-09-07
Erratum: the principal curvature kappaₜ printed in section 2 of arXiv:2606.16980v1 (and, unchanged, in section 2 of the live v2 of part I arXiv:2508.09104) is 1/(n-1) times the true one – the printed triple has weighted trace (2n-4)(cot r sin a - csc r cos a cot 2th)!= 0, so it cannot be the shape operator of a minimal hypersurface once n > 2; the correct value is kappaₜ = a' + cot r sin a, and the two agree at n = 2
CS4candidate2026-09-07
Lemma 3.1 of arXiv:2606.16980v1, corrected and verified in one identity: its eigenfunction f = (csc² r sec th csc th)ⁿ⁻¹ is exactly exp(-b) with b = (n-1)/2 log(EG) – the reciprocal Riemannian volume density – so the lemma is the single scalar identity b" = |A|² - (n-1)(1/E + 1/G), true for every n with the corrected kappaₜ and false from n = 3 with the printed one
CS5routine2026-09-07
The conditional index theorem for n > 2: from the source's sign facts plus the Rayleigh antitonicity of part I, every block outside 1..4² contributes nothing and 3 * index = n³ + 9n² + 11n + 3, with non-vacuity and both negative controls
CS6routine2026-09-07
The n = 2 case as a separate cell: with the source's n = 2 sign list (where S₅1, S₁5 still contribute and S₆1, S₁6 do not) the stability index of the Carlotto-Schulz hypertorus in S⁴ is 27, and 26 and 28 are refuted
CS7candidate2026-09-07
The source's ITEMISED n > 2 sign list does not determine the index: a second antitone table satisfies every item of it and gives an index larger by n(n²+n-2); the fact that closes the gap is lambda¹₃₂ > 0, which the source supplies outside that list
CS8prose2026-09-07
deltaᵢj = deltaⱼi, which the source states as a numerical observation, is a theorem: t -> -t conjugates Sᵢj to Sⱼi, because (r(-t), pi/2 - th(-t), pi - a(-t)) solves the same profile system with the same initial data
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
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- Machina Mathematica
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- Korea Superintelligence Labs
- Source context
- arXiv:2606.16980v1, Oscar Perdomo, *The stability index and Yau's conjecture for Carlotto–Schulz minimal hypertori, part II* (submitted 15 Jun 2026, math.DG). The abs page and the arXiv API were queried live on 2026-09-07: v1 is the only version. Read from the author's own e-print, /backup/arxiv-src/papers/2606/2606.16980.gz, file OnTheMinimalCarlottoSchulzPartII.tex.
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- 2026-09-07 03:53 UTC
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