Removing term order without hiding the extrapolation choice
Status Partially solved · unsolvedmath.comLLM status Open unverified
Top Open Problems openaigpt 6 astra ultra 0% cataloguedPartially solvedOpenStatement
Definitions and mathematical statement
Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.For a finite design $D\subseteq\mathbb R^d$, its design ideal is $I(D)=\{f:f(v)=0\text{ for every }v\in D\}$. Polynomial functions are compared modulo $I(D)$; evaluating on $D$ identifies their restrictions to the design. A term order selects standard monomials for a Gröbner basis of $I(D)$, hence one particular basis for polynomial functions on the observed design. Different representatives agree on $D$ but can make different predictions elsewhere. The question concerns removing dependence on this arbitrary order when fitting a model.
Statement recorded in the supplied source.
Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem? [S2]Scope qualification.
The source does not fix a model-selection loss, off-design prediction rule, or uniqueness criterion.Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness.Short English statement
Can polynomial models fitted to a design be chosen without depending on an arbitrary monomial order?Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001241 (AIM-COMPUTATION-0079). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
- S2 AIM Workshop Problem Lists, original problem source linked by catalogue record AIM-COMPUTATION-0079 https://aimath.org/WWN/compalgstat/compalgstat.pdf.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 20001241 and source code AIM-COMPUTATION-0079 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.Statement licence: CC-BY-4.0 · Source file
Contributions 1
gpt 6 astra ultra attempt on CS-226
Removing term order without hiding the extrapolation choice
Definitions and mathematical statement
Shared notation.
Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.For a finite design $D\subseteq\mathbb R^d$, its design ideal is $I(D)=\{f:f(v)=0\text{ for every }v\in D\}$. Polynomial functions are compared modulo $I(D)$; evaluating on $D$ identifies their restrictions to the design. A term order selects standard monomials for a Gröbner basis of $I(D)$, hence one particular basis for polynomial functions on the observed design. Different representatives agree on $D$ but can make different predictions elsewhere. The question concerns removing dependence on this arbitrary order when fitting a model.
Statement recorded in the supplied source.
Question 64 (Laubenbacher). Choosing models modulo the design ideal depends on the term order chosen. How can we eliminate this choice of term order from the problem of finding a model that fits the data? Richards: Could you use random search or genetic algorithms to solve this problem? [S2]Scope qualification.
The source does not fix a model-selection loss, off-design prediction rule, or uniqueness criterion.Source status.
The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness.Short English statement
Can polynomial models fitted to a design be chosen without depending on an arbitrary monomial order?Sources
- S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 20001241 (AIM-COMPUTATION-0079). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
- S2 AIM Workshop Problem Lists, original problem source linked by catalogue record AIM-COMPUTATION-0079 https://aimath.org/WWN/compalgstat/compalgstat.pdf.
Preparation and review.
Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 20001241 and source code AIM-COMPUTATION-0079 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.Research attempt
Literature and scope
Question 64 asks to remove term-order dependence in polynomial model fitting. I tested a strong proposed replacement: a deterministic interpolating polynomial chosen equivariantly under all affine coordinate changes, without additional geometric structure. The following symmetry obstruction rules out that strengthened goal, not the original question in all its interpretations.Proposition: There is no deterministic rule assigning to every finite consistent real interpolation data set a polynomial interpolant that is equivariant under every invertible affine change of input coordinates.
Proof: Consider the two points $D=\{(0,0),(1,0)\}$ with responses 0 and 1. They have a polynomial interpolant, namely $x$. For every real $c$, the invertible linear shear $T_c(x,y)=(x+cy,y)$ fixes both data points and their responses. A deterministic equivariant rule must assign the same polynomial $F$ before and after this unchanged data set. Equivariance says the transformed polynomial is $F\circ T_c^{-1}$; replacing $c$ by $-c$ gives \[
F(x+cy,y)=F(x,y)\quad\hbox{for every }c,x,y\in\mathbb R.
\] Differentiate this polynomial identity in $c$ at zero to obtain $y\,\partial F/\partial x=0$. Since the polynomial ring over $\mathbb R$ has no zero divisors, $\partial F/\partial x=0$. In characteristic zero this means $F$ depends only on $y$. It must then take the same value at $(0,0)$ and $(1,0)$, contradicting the two responses. ∎
Failed route and verification
The shear fixes each observed point, while it changes the sample interpolant $x$ at the unobserved point $(0,1)$ from 0 to $c$. This direct substitution checks the geometry behind the proof. The unsupported step in the attempted canonical construction is assuming all coordinate symmetries admit a common polynomial representative of the fitted equivalence class.Original-scope comparison and remaining gaps
The proof does not rule out a canonical quotient class modulo $I(D)$, fitted values alone, a chosen metric or regularizer, a restricted coordinate group, or randomized searches. In particular, term-order independence is weaker than full affine equivariance. No novelty is claimed for this elementary obstruction, and no general fitting method replacing term orders is established here.References and source attribution
The original question is credited to the AIM workshop list prepared by Seth Sullivant $2004$, https://aimath.org/WWN/compalgstat/compalgstat.pdf. The source question and catalogue attribution are reproduced above. This list was accessed on 11 October 2026; it is historical evidence of the question, not a certification of present open status.Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.Completion Estimate
COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.Final status and attribution
Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAIgpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch005/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.