Fröberg’s conjecture for quintics and septics in four variables
partialconfidence 70%
VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
Let $k$ be a field of characteristic zero and let $S=k[x_1,x_2,x_3,x_4]$. We prove Fröberg's predicted Hilbert series for ideals generated by $r$ general forms of equal degree $d$ for every $r\geq1$ in each of the two cases $d=5$ and $d=7$. Relative to the classical cases $r\leq5$ and the equal-degree theorem through degree $d+2$ of Boij--Dannetun--Lundqvist, the generator-count ranges requiring new input are $6\leq r\leq11$ for quintics and $6\leq r\leq21$ for septics. The proof reduces each slice to finitely many endpoint ranks of Macaulay multiplication matrices. For quintics, ten exact endpoint computations based on twenty-one sparse forms suffice. For septics, a nested family of 120 integral forms supplies fifteen endpoint computations. In every endpoint certificate for these new ranges, an explicitly recorded maximal minor is nonzero modulo $2$, hence is a nonzero integer. The case $r=5$ is the classical strong Lefschetz instance; for quintics we also record a matching modular rank and Koszul bound. Zariski openness then gives the result over every characteristic-zero field. The unrestricted Fröberg conjecture remains outside the scope of the paper.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-5.6 Sol; Claude Fable 5; Grok 4.6
Disclosed in the abstract and in a closing section, "Disclosure of automated assistance". The abstract states that "The main results of this paper were obtained through a generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, and Grok 4.6", and the disclosure section lists what the workflow did: "the formulation of mathematical ideas, generation of conjectures and proof strategies, derivation and checking of intermediate steps, construction of examples and exact certificates, comparison of literature and candidate proof approaches, organization of arguments, LaTeX drafting, and revision of the final text". It adds that the workflow "decomposed the problem into smaller subproblems and used repeated self-critique and alternative derivations to test the quantifiers, the characteristic-zero scope, and the endpoint-rank reductions".
That credits the mathematics rather than tooling, which is what puts this in scope, and it goes well beyond editing: proof strategies and the exact certificates are the substance of this paper. It is filed as co-developed rather than AI-discovered because no single step is attributed to a named model, the account is of a workflow rather than of a system solving a stated subproblem, and two human authors direct it throughout.
Site-confirmed: this site reproduced the computation, twice over, on 28 August 2026.
First, independently of the authors' code. The paper prints all 21 quintic forms with every coefficient $+1$ and states the monomial order, so the Macaulay matrices can be rebuilt from the text alone. I did that and ran my own elimination: all ten rows of Table 2 reproduce exactly, in shape and rank, over $\mathbb F_2$, $\mathbb F_{101}$ and $\mathbb F_{1009}$. I then checked the step the paper leaves implicit - that each rank gives exactly Fröberg's predicted dimension. Recomputing the prediction myself, with $\dim(S/I)_j=\binom{j+3}{3}-\mathrm{rank}$, every endpoint agrees: $(6,9)\to10$, $(8,8)\to5$, $(11,7)\to10$, $(20,6)\to4$, and $0$ at each surjective endpoint.
Second, the authors' verifiers, published as arXiv ancillary files, both pass here: quintics 1.2 s, septics 0.9 s reporting 15 endpoints, 120 forms and 15 maximal minors. Neither uses floating point or randomness. The septic verifier genuinely recomputes rather than trusting stored numbers, which I confirmed by altering one exponent of one form: the run then failed at the matrix digest instead of passing.
Not established here: the reduction of each $r$-range to its endpoints is a mathematical argument that was not checked, and the 120 septic forms exist only inside the certificate, so for septics I ran their verifier rather than rebuilding independently. Unrefereed, with no human review.
Let $S=k[x_1,x_2,x_3,x_4]$ over any characteristic-zero field. For each $d\in\{5,7\}$ and every $r\ge1$, the paper proves that $r$ general degree-$d$ forms satisfy Fröberg’s predicted Hilbert series
$$
\operatorname{HS}_{S/(F_1,\ldots,F_r)}(t)
=
\left[\frac{(1-t^d)^r}{(1-t)^4}\right]_+.
$$
The genuinely new ranges are $6\le r\le11$ for quintics and $6\le r\le21$ for septics. These are reduced to finitely many exact rank tests of Macaulay multiplication matrices; explicit maximal minors are nonzero mod $2$, so the required ranks hold in characteristic zero. Thus the conjecture is completely settled for the equal-degree four-variable slices $d=5$ and $d=7$, but not in general.
Known method families
computation (source-reported)
Source-reported tools: computation.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.