Professor of Mathematics
Paata Ivanisvili
Department of Mathematics, University of California, Irvine
Vibe proofs
These are proofs I got from AI/SI in various experiments. Some came from several agents working continuously for many hours. I think this information might be valuable, so I decided to share it publicly rather than keep it on my computer. Some are polished and some are raw drafts.
How to cite: P. Ivanisvili, title, day/month/year, https://paatai.github.io/ai-proofs/#n
- refereed
- an independent AI/SI referee or auditor rated the proof valid
- partial result
- the note says what is proved and what is open
- raw draft
- has not been refereed
- polished
- I partly edited the draft
22 entries, newest first
- 22 A comparison inequality for Gaussian vectors and product functions partial result Sufficient conditions for the comparison inequality are refereed; in an explicit example the exact threshold is open on [2.175…, 2.1925). [PDF · TeX · code]
- 21 Correlated forward hypercontractivity with outer exponent α<1 refereed With equal parameters and equal exponents p≥1, the Ivanisvili–Kalantzopoulos covariance condition is necessary and sufficient for every α∈(0,1). With unequal parameters it is not sufficient for r=αp>2; whether it is sufficient for r∈(0,2]∖{1} is open. [PDF · TeX · code]
- 20 A non-degenerate Gaussian kernel without Gaussian maximisers: Lieb’s formula fails below the diagonal refereed An Arb-certified counterexample to Lieb’s formula below the diagonal (p>q, genuinely complex case): for the Mehler transform with z=0.72e0.7i from Lp to L4, p≥16, every Gaussian satisfies the inequality but a unimodular function violates it. Rated valid by an independent referee. [PDF · TeX · code]
- 19 Complex hypercontractivity for several correlated Gaussian functions below the diagonal raw draft Three parts: an exact criterion for imaginary parameters and every outer exponent; the exact imaginary-axis region for two ρ-correlated functions; and joint narrow-bump counterexamples for general complex z, with two failure points certified in ball arithmetic. Not refereed; off the axes only inner and outer bounds are proved, the region between them is open, and the remaining numerical facts are stated only as evidence. [PDF · code]
- 18 The local neighborliness threshold of the symmetric moment curve, and the Barvinok–Lee–Novik conjecture for k≤440 and all sufficiently large k raw draft Not refereed; the conjecture for intermediate k is open. [PDF · TeX]
- 17 On the Améndola–Lee–Meroni conjecture: the signature volume formula on all subpaths does not characterize cyclic curves raw draft Claims a counterexample to the conjecture. Not refereed. [PDF · TeX]
- 16 Carathéodory numbers of the Barvinok–Novik orbitopes, of all four-dimensional SO(2)-orbitopes, and of a further family raw draft Not refereed. [PDF · TeX]
- 15 The exact Rosenthal bound for independent zero-mean summands when 3≤p≤4, and in the two extreme regimes when 4<p<5 partial result Independent adversarial referee passes found no critical or major issues; the middle regime of 4<p<5 is open. [PDF · source and numerics]
- 14 No 421 equiangular lines in R44, no 94 in R20, no 74 in R19 and no 59 in R18 partial result N(44)≤420 and N(20)≤93 are rated valid by an independent auditor and a referee; N(19)≤73 and N(18)≤58 have not been refereed yet. [PDF · TeX · code]
- 13 Sharp L1 Poincaré inequality on the Hamming cube polished An explanation of the proof, based on Tomar’s preprint. [PDF]
- 12 A sharp reverse Khinchin-type inequality for biased bits: the optimal constant R(q) for every q∈(0,1) refereed Computer-assisted; rated valid by an independent referee and an auditor. Several agents worked on it over several days. The original output was 192 pages long. [PDF · proof code · certificates]
- 11 List-decoding thresholds of approximate polymorphisms refereed Shows that a lemma (5.6) in the CFMMS paper is false and that the upper threshold sU is not tight; rated valid by an independent auditor and a referee. [PDF · numerics]
- 10 The Keller–Klein Berry–Esseen conjecture: the reachable half for all Rademacher sums, and the full conjecture for a1≥0.07 partial result Computer-assisted (ball arithmetic); rated valid by an independent auditor and a referee. The full conjecture for a1<0.07 remains open. [PDF · certified computation]
- 9 AND testing with square-root loss: the optimal ε–δ dependence is Θ(√ε), and polynomial bounds for all approximate polymorphisms refereed Rated valid by an independent auditor and a referee. [PDF · numerics]
- 8 Ball’s integral inequality via weighted level sets refereed A short new proof of K. Ball’s 1986 inequality for ∫|sin t / t|s, s≥2; rated valid by an independent auditor. [PDF · one-page version]
- 7 The sin-omial integral: a one-page elementary proof refereed Two short proofs of (1/π)∫0π sinrx / (sins(sx/r) sinr−s((r−s)x/r)) dx = binom(r,s); both rated valid by an independent auditor. [PDF · second proof · audit]
- 6 The Sánchez-Ruiz entropy bound for all harmonic-oscillator eigenstates: a proof of his Eqs. (30) and (37) refereed Proof of Eq. (37) of Sánchez-Ruiz (1997); rated valid by an independent auditor. [PDF]
- 5 Short, computer-free proofs of three nonlinear-mean inequalities of Nazarov, Vasyunin and Volberg refereed Hand proofs of the I3, I4 and I6 inequalities of arXiv:2204.01336; an independent hostile audit found no errors. [PDF · longer version]
- 4 Counterexample to Sedykh and Shapiro conjecture refereed Written-up counterexample. [PDF]
- 3 Sharp linear hypercontractivity for ultraspherical measures polished Written-up proof. [PDF]
- 2 An explicit Bellman function for disjoint partitions: the sharp logarithmic gain for three families polished Written-up proof with verification code. [PDF · TeX · verification code]
- 1 Sharp Markov–Bernstein inequality in Gauss space for p≥2 polished Written-up proof. Prompted and discussed together with Alexandros Eskenazis. [PDF]