Source authenticated

Fröberg’s conjecture for quintics and septics in four variables

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: site-confirmed. Publication: preprint. AI contribution: ai-co-developed. Editorial metadata: CC BY 4.0; linked content retains its own rights.

Canonical aliases: Fröberg’s conjecture for quintics and septics in four variables · Fröberg for 5D and 7D in 4 variables

Confidence: Not scored

Registry verification: site confirmed · preprint · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Qihang Wang
human · human collaborator

Dongming Zhang
human · human collaborator

GPT-5.6 Sol
model · ai model contributor · OpenAI

Claude Fable 5
model · ai model contributor · Anthropic

Grok 4.6
model · ai model contributor · xAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Fröberg’s conjecture for quintics and septics in four variables

specialist registry · passed

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.

Source ↗

Fröberg’s conjecture for quintics and septics in four variables

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Ancillary exact verifiers and septic certificate (arXiv anc/)

code · pending

code link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Ancillary exact verifiers and septic certificate (arXiv anc/)

code · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Qihang Wang

This event attributed to Dongming Zhang

This event attributed to GPT-5.6 Sol

This event attributed to Claude Fable 5

This event attributed to Grok 4.6

Fröberg’s conjecture for quintics and septics in four variables parent of this event

Ancillary exact verifiers and septic certificate (arXiv anc/) evidence for this event

VibeMathed record: Fröberg’s conjecture for quintics and septics in four variables evidence for this event

Fröberg’s conjecture for quintics and septics in four variables evidence for this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph