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

proved→proved

Scope and record

Occurred: Oct 29, 2026

Delta type: REGISTRY REVISION

Assumptions: VibeMathed verification: site-confirmed. Publication: preprint. AI contribution: ai-co-developed. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their 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

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Attribution

VibeMathed
registry · event recorded by

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

Qihang Wang
human · human collaborator

Dongming Zhang
human · human collaborator

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

Corrects Fröberg’s conjecture for quintics and septics in four variables →

Supersedes Fröberg’s conjecture for quintics and septics in four variables →

This event attributed to Dongming Zhang

This event attributed to Qihang Wang

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 parent of this event

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

This event attributed to Grok 4.6

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

This event attributed to GPT-5.6 Sol

This event attributed to Claude Fable 5

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