Datasets:
qid stringlengths 26 26 | did stringlengths 26 26 | dense_rank int8 1 100 ⌀ | stage stringclasses 8
values | label int8 0 2 ⌀ | judge stringclasses 2
values | rm8b_score float32 -1.06 3.13 ⌀ | dense_score float32 0.42 0.98 ⌀ | reranker_score float32 -16.89 11.2 ⌀ | answer_validity stringclasses 5
values | reason_codes listlengths 1 5 ⌀ | confidence stringclasses 3
values | rationale stringlengths 22 730 ⌀ | rubric_sha256 stringclasses 2
values | query_evaluable bool 2
classes | is_source_page bool 2
classes | bm25_rank int8 1 100 ⌀ | ngram5_rank int8 1 100 ⌀ | fused_rank int8 1 20 ⌀ | rrf_score float64 0.01 0.07 ⌀ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
p_00021f40cdc830fe40aee5e8 | d_15935b322320fcd1255b4744 | 10 | short_doc | 0 | rm8b | 0.1514 | 0.6938 | -3.1875 | null | null | null | null | null | null | false | null | null | 18 | 0.028571 |
p_00021f40cdc830fe40aee5e8 | d_19c91960529e2a01e258d746 | 30 | fused_short | 0 | rm8b | 0.127 | 0.6758 | null | null | null | null | null | null | null | false | 31 | null | 8 | 0.033211 |
p_00021f40cdc830fe40aee5e8 | d_229a5b650c79be148003e263 | 8 | short_doc | 1 | rm8b | 0.832 | 0.6953 | -4.1875 | null | null | null | null | null | null | false | null | null | 15 | 0.029412 |
p_00021f40cdc830fe40aee5e8 | d_28e84410927a5e02e795dce4 | 65 | fused_short | 0 | rm8b | 0.0549 | 0.6675 | null | null | null | null | null | null | null | false | 15 | null | 16 | 0.029333 |
p_00021f40cdc830fe40aee5e8 | d_33a37b8682cfaabc8fe454a2 | 3 | dense_top5 | 0 | llm | null | 0.7017 | null | major_error | [
"major_math_error",
"wrong_problem"
] | high | The document addresses a different income problem and does not explain when to use Z rather than t. It also incorrectly gives the standard error of a sample mean using sqrt(n-1) instead of sqrt(n). | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 12 | 0.031746 |
p_00021f40cdc830fe40aee5e8 | d_3e005cb38f9a9c256c5bdd7e | null | fused_short | 0 | rm8b | 0.2734 | null | null | null | null | null | null | null | null | false | 2 | 3 | 10 | 0.032002 |
p_00021f40cdc830fe40aee5e8 | d_423ce158a879881ae25a9ece | 19 | rerank_top5 | 0 | llm | 0.0835 | 0.6836 | 0.25 | not_applicable | [
"contaminated_or_spliced",
"unresolved_attempt"
] | high | This is a large splice of unrelated statistics questions. Although one multiple-choice prompt mentions an unknown population standard deviation, no answer is selected or explanation of the query's z-versus-t distinction is supplied. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 7 | null | 6 | 0.040242 |
p_00021f40cdc830fe40aee5e8 | d_524bf915e309b6fb159811be | 16 | short_doc | 1 | rm8b | 0.7969 | 0.687 | -2.4375 | null | null | null | null | null | null | false | null | null | null | 0.026316 |
p_00021f40cdc830fe40aee5e8 | d_525c81484d42f2827661d9f7 | 18 | rerank_top5 | 0 | llm | 1.5391 | 0.6836 | 1.8125 | major_error | [
"major_math_error"
] | high | The document incorrectly treats a small sample by itself as grounds for using t and even uses t in an example where a population standard deviation is stated. This reinforces the query's misconception instead of explaining that known population sigma gives a z statistic even for n=25. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.025641 |
p_00021f40cdc830fe40aee5e8 | d_6a72b7a20b8b722f5c46e100 | 15 | rerank_top5 | 1 | llm | 1.1797 | 0.6875 | 0.5 | correct | [
"correct_partial_method"
] | medium | The document correctly states that t is used for a small normal sample when the population standard deviation is unknown, which explains part (b). It does not directly state or apply the complementary fact that known population sigma makes z appropriate in part (a). | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.026667 |
p_00021f40cdc830fe40aee5e8 | d_6d53564bcc4a56b44bc6bed6 | 14 | short_doc | 0 | rm8b | 0.3145 | 0.6875 | -2.75 | null | null | null | null | null | null | false | null | null | null | 0.027027 |
p_00021f40cdc830fe40aee5e8 | d_72db36a3beac90ff2e9592ea | 2 | dense_top5 | 0 | llm | null | 0.7046 | null | not_applicable | [
"topic_only"
] | high | This is a collection of statistics multiple-choice questions. Although it mentions small-sample tests and known standard deviations, it does not explain the known-versus-estimated standard-deviation distinction asked about. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 9 | 0.032258 |
p_00021f40cdc830fe40aee5e8 | d_734333e4dc6cb1a1b37dde1d | 12 | short_doc | 0 | rm8b | 0.1895 | 0.6909 | -1.125 | null | null | null | null | null | null | false | 5 | null | 5 | 0.043162 |
p_00021f40cdc830fe40aee5e8 | d_7eedcd06d8592d9b60ab7e21 | 13 | short_doc | 1 | rm8b | 0.6875 | 0.6895 | -2.125 | null | null | null | null | null | null | false | null | null | 20 | 0.027397 |
p_00021f40cdc830fe40aee5e8 | d_80de20b0ab0c046c40c7139f | 6 | short_doc | 0 | rm8b | 0.4414 | 0.6978 | -5 | null | null | null | null | null | null | false | null | null | 14 | 0.030303 |
p_00021f40cdc830fe40aee5e8 | d_92697cb8b1dd3a5468f8c58d | 17 | rerank_top5 | 0 | llm | 1.1953 | 0.6865 | 1.5 | major_error | [
"major_math_error"
] | high | Its claim that t should be used whenever n<30, regardless of whether population sigma is known, is the central misconception the query asks to resolve. It does not justify the correct z choice in part (a). | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.025974 |
p_00021f40cdc830fe40aee5e8 | d_9c5ab899601bbbd8086e4d57 | null | fused_short | 1 | rm8b | 0.6172 | null | null | null | null | null | null | null | null | false | 3 | 2 | 11 | 0.032002 |
p_00021f40cdc830fe40aee5e8 | d_a2b0aaeef753c170a7e74389 | 71 | fused_long | 1 | rm8b | 1.2734 | 0.6665 | null | null | null | null | null | null | null | false | 12 | null | 17 | 0.029156 |
p_00021f40cdc830fe40aee5e8 | d_ae07b24f9ac862b20c91baf5 | 7 | rerank_top5 | 0 | llm | 0.125 | 0.6973 | -0.875 | not_applicable | [
"contaminated_or_spliced",
"unresolved_attempt"
] | high | The document is an incoherent compilation of many unrelated exercises. Its uncompleted multiple-choice item about t distributions does not explain why known sigma leads to z in part (a) and estimated sigma leads to t in part (b). | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 8 | null | 3 | 0.044557 |
p_00021f40cdc830fe40aee5e8 | d_b8db37356cfe1930448d8e5e | 4 | dense_top5 | 2 | llm | null | 0.6992 | null | correct | [
"full_solution"
] | high | The relevant portions give the exact distinction: with known sigma, the standardized sample mean is normal; with unknown sigma replaced by S, it has a t distribution with n-1 degrees of freedom under normal sampling. This directly resolves why the two cases use different statistics regardless of both sample sizes being... | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 13 | 0.03125 |
p_00021f40cdc830fe40aee5e8 | d_bcd007b73eda43299fbd14d6 | 5 | dense_top5 | 1 | llm | null | 0.6978 | null | correct | [
"correct_partial_subquestion"
] | medium | The document contains the correct key criterion that t is used when the population standard deviation is unknown and the normal-model conditions apply. It does not coherently contrast this with the known-sigma Z case or explain why 30 is not the deciding cutoff. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 9 | null | 2 | 0.045262 |
p_00021f40cdc830fe40aee5e8 | d_bd5c8c73e603184ec80fba24 | 20 | short_doc | 0 | rm8b | 0.2812 | 0.6816 | -5 | null | null | null | null | null | null | false | null | null | null | 0.025 |
p_00021f40cdc830fe40aee5e8 | d_d89a0808db08bf4bafa852bd | 9 | short_doc | 0 | rm8b | 0.127 | 0.6948 | -1.0625 | null | null | null | null | null | null | false | 6 | null | 4 | 0.044137 |
p_00021f40cdc830fe40aee5e8 | d_e256974ca72e587a766c71ff | 11 | long_doc | 2 | llm | null | 0.6924 | null | correct | [
"full_solution"
] | high | The document gives the decisive distinction: with a normal population and known population standard deviation, standardizing the sample mean uses the normal Z distribution; when the standard deviation is estimated by the sample SD, the statistic has a t distribution with n−1 degrees of freedom. Thus sample size 30 is o... | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 19 | 0.028169 |
p_00021f40cdc830fe40aee5e8 | d_e87081db946a164fc262ee41 | 31 | fused_short | 0 | rm8b | 0.0234 | 0.6758 | null | null | null | null | null | null | null | false | 16 | null | 7 | 0.035136 |
p_00043540acc32a47a5b1d2bf | d_00555ef95da9445585ead3b8 | 25 | fused_short | 1 | rm8b | 0.8828 | 0.7222 | null | null | null | null | null | null | null | false | 2 | 25 | 10 | 0.051423 |
p_00043540acc32a47a5b1d2bf | d_1551c73be8b6e3ba25e34c98 | 11 | rerank_top5 | 0 | llm | 0.2383 | 0.7661 | -0.5 | not_applicable | [
"target_mismatch",
"topic_only"
] | high | The document discusses geometric reducedness and a general topological irreducibility lemma but neither states nor proves the transitivity of geometric irreducibility in the query. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.028169 |
p_00043540acc32a47a5b1d2bf | d_1ac47d6a594abd7556e44252 | 7 | rerank_top5 | 0 | llm | 0.1943 | 0.8096 | -0.5625 | not_applicable | [
"target_mismatch"
] | high | It proves a sufficient condition for a field extension K/k to be geometrically irreducible, whereas the query already assumes that fact and asks for the resulting geometric irreducibility of S over k. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | 19 | 16 | 0.042509 |
p_00043540acc32a47a5b1d2bf | d_278cd83d89764b1720f67450 | 17 | short_doc | 0 | rm8b | 0.0152 | 0.7344 | -6.9062 | null | null | null | null | null | null | false | null | null | null | 0.025974 |
p_00043540acc32a47a5b1d2bf | d_3086d4754d62f49a83a02dd9 | 22 | fused_short | 0 | rm8b | 0.1011 | 0.7256 | null | null | null | null | null | null | null | false | 24 | 6 | 9 | 0.051447 |
p_00043540acc32a47a5b1d2bf | d_4276c2f8821877d0f4afe2d5 | 5 | dense_top5 | 1 | llm | null | 0.8354 | null | correct | [
"correct_partial_subquestion"
] | medium | The document correctly proves that tensoring a reduced algebra with a separable field extension preserves reducedness. This helps prove K⊗_k k' is a field after irreducibility is supplied, but it neither supplies that step nor concludes that S is geometrically irreducible over k. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 8 | 18 | 5 | 0.058296 |
p_00043540acc32a47a5b1d2bf | d_5eb647907b9182cd2bd0d821 | 9 | short_doc | 0 | rm8b | 0.3477 | 0.7759 | -3.4375 | null | null | null | null | null | null | false | null | null | null | 0.028986 |
p_00043540acc32a47a5b1d2bf | d_66492d6b51f5f2a41a2fa6ba | 2 | dense_top5 | 1 | llm | null | 0.8398 | null | correct | [
"correct_partial_method"
] | high | The document supplies the definition and finite-separable testing criterion for geometric irreducibility, which gives the appropriate initial reduction. It does not perform the crucial argument that K⊗_k k' is a field or apply the hypothesis on S to finish the proof. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 1 | 7 | 2 | 0.063577 |
p_00043540acc32a47a5b1d2bf | d_66f9562aac7ccdb731dc43b1 | 20 | short_doc | 0 | rm8b | 0.4043 | 0.7275 | -5.5938 | null | null | null | null | null | null | false | null | null | null | 0.025 |
p_00043540acc32a47a5b1d2bf | d_7007526eed5ddcbad3a8ec71 | 14 | short_doc | 0 | rm8b | 0.0172 | 0.7529 | -6.4375 | null | null | null | null | null | null | false | null | null | null | 0.027027 |
p_00043540acc32a47a5b1d2bf | d_96150241cf528ae0b42f0be8 | 3 | dense_top5 | 1 | llm | null | 0.8364 | null | correct | [
"correct_partial_subquestion"
] | medium | Lemma 10.42.6 gives a useful required substep: for finite separable k'/k, the tensor product K⊗_k k' is reduced. Combined with its irreducibility this helps show it is a field, but the document does not establish that irreducibility or complete the argument for S. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 13 | 17 | 4 | 0.058432 |
p_00043540acc32a47a5b1d2bf | d_997e1e96d009e11434689503 | 8 | short_doc | 0 | rm8b | 0.3906 | 0.7773 | -3.5 | null | null | null | null | null | null | false | null | 100 | 20 | 0.035662 |
p_00043540acc32a47a5b1d2bf | d_9acbcadd24e4b955f641ebe0 | 4 | dense_top5 | 1 | llm | null | 0.8364 | null | correct | [
"correct_partial_subquestion"
] | medium | The result that separable scalar extension preserves reducedness can be applied to show K⊗_k k' is reduced for finite separable k'/k, a genuine intermediate step in the desired proof. The document treats geometric reducedness rather than completing the geometric-irreducibility argument. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 7 | 16 | 3 | 0.059333 |
p_00043540acc32a47a5b1d2bf | d_af2eed69291ab140bcf6d79a | 26 | fused_short | 0 | rm8b | 0.1206 | 0.7217 | null | null | null | null | null | null | null | false | 22 | 3 | 11 | 0.051324 |
p_00043540acc32a47a5b1d2bf | d_c3824fe13114ee61ac0c7e19 | 39 | fused_short | 0 | rm8b | 0.2012 | 0.7085 | null | null | null | null | null | null | null | false | 20 | 20 | 15 | 0.045202 |
p_00043540acc32a47a5b1d2bf | d_c432dcf6e64ccb4eb1edf12b | 32 | fused_short | 0 | rm8b | 0.3184 | 0.7148 | null | null | null | null | null | null | null | false | 3 | 28 | 12 | 0.048976 |
p_00043540acc32a47a5b1d2bf | d_c46612c0437279aade8f6ad2 | 21 | fused_short | 0 | rm8b | 0.1348 | 0.7266 | null | null | null | null | null | null | null | false | 23 | 5 | 8 | 0.052124 |
p_00043540acc32a47a5b1d2bf | d_c61cfd67beceb0fa6827a95b | 16 | rerank_top5 | 0 | llm | 0.8438 | 0.7378 | -2.875 | major_error | [
"major_math_error",
"condition_mismatch",
"unresolved_attempt"
] | high | The discussion concerns tensor products of domains and relies on finite-type, algebraically closed, or separability assumptions absent from the query. Its broad opening claim that tensor products of domains over an arbitrary field are domains is false, and it never proves the requested irreducibility result. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 11 | null | 17 | 0.0404 |
p_00043540acc32a47a5b1d2bf | d_c7c28052d97a4ee7fe95be23 | 18 | short_doc | 1 | rm8b | 0.6211 | 0.7324 | -3.3125 | null | null | null | null | null | null | false | null | null | null | 0.025641 |
p_00043540acc32a47a5b1d2bf | d_c7c8a925d25b8c3f3869376f | 10 | short_doc | 0 | rm8b | 0.4004 | 0.7754 | -3.5 | null | null | null | null | null | null | false | null | 79 | 19 | 0.035766 |
p_00043540acc32a47a5b1d2bf | d_c96f5916c58878f44a3d5716 | 6 | rerank_top5 | 1 | llm | 0.3867 | 0.8169 | 0.375 | correct | [
"correct_partial_method"
] | medium | The valid tensor-product lemma for geometrically integral algebras supplies the analogous argument under stronger domain and reducedness hypotheses. It does not handle merely geometrically irreducible, possibly nonreduced algebras or complete the requested transitivity proof. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 9 | 23 | 6 | 0.056844 |
p_00043540acc32a47a5b1d2bf | d_d762e1221983ae53a7209746 | 12 | rerank_top5 | 0 | llm | 0.1836 | 0.7646 | -2 | not_applicable | [
"target_mismatch",
"topic_only"
] | high | The document proves that the generic-point residue field of a geometrically irreducible scheme is geometrically irreducible. It does not address composition of base fields or the K-algebra S. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 95 | 26 | 14 | 0.045857 |
p_00043540acc32a47a5b1d2bf | d_db0cf5e8cd5e42a9fc94f06d | 33 | fused_short | 0 | rm8b | 0.0908 | 0.7148 | null | null | null | null | null | null | null | false | 26 | 4 | 13 | 0.048758 |
p_00043540acc32a47a5b1d2bf | d_dcd855d5eecbf72d8173291c | 15 | short_doc | 0 | rm8b | 0.2832 | 0.7388 | -3.625 | null | null | null | null | null | null | false | null | 59 | null | 0.03507 |
p_00043540acc32a47a5b1d2bf | d_de644d78813c7e239d257126 | 19 | short_doc | 0 | rm8b | 0.2266 | 0.7295 | -3.875 | null | null | null | null | null | null | false | null | 57 | null | 0.033863 |
p_00043540acc32a47a5b1d2bf | d_f4475917c98f036b68290568 | 13 | short_doc | 0 | rm8b | 0.0513 | 0.7588 | -6.6875 | null | null | null | null | null | null | false | null | 43 | 18 | 0.037106 |
p_00043540acc32a47a5b1d2bf | d_f554a48db24bc1f8ac60b781 | 24 | fused_short | 0 | rm8b | 0.1357 | 0.7227 | null | null | null | null | null | null | null | false | 21 | 2 | 7 | 0.052284 |
p_00043540acc32a47a5b1d2bf | d_fda5fb51aa3abbde127d7a1c | 1 | dense_top5 | 2 | llm | null | 0.8657 | null | correct | [
"full_solution"
] | high | The document proves the exact claim by reducing to finite separable extensions k'/k, showing K⊗_k k' is a finite separable field extension of K, and applying the geometric irreducibility of S over K. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | true | 5 | 1 | 1 | 0.064565 |
p_00092ac075db954489913e2b | d_113c98023ac3e528b9236c59 | null | source_page | 2 | llm | null | null | null | null | null | null | null | null | null | true | null | 3 | null | 0.015873 |
p_00092ac075db954489913e2b | d_16ee47a9c6d42a470377fdc5 | 8 | rerank_top5 | 0 | llm | 1.9766 | 0.7451 | 2.1875 | not_applicable | [
"unresolved_attempt"
] | high | The document merely recalls that the units of F[x] are F* and uses that fact later; it supplies no proof of the requested assertion. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 10 | 0.029412 |
p_00092ac075db954489913e2b | d_515b9ef18beae1f5f9e5562d | 10 | short_doc | 1 | rm8b | 1.3281 | 0.7378 | 1.25 | null | null | null | null | null | null | false | null | null | 12 | 0.028571 |
p_00092ac075db954489913e2b | d_581aa36bfec30ee56cadd54c | 63 | fused_short | 0 | rm8b | 0.0845 | 0.7109 | null | null | null | null | null | null | null | false | 6 | null | 4 | 0.031412 |
p_00092ac075db954489913e2b | d_5b0ac9622108bc7b38259e6e | 65 | fused_short | 0 | rm8b | -0.0452 | 0.7095 | null | null | null | null | null | null | null | false | null | 9 | 7 | 0.030493 |
p_00092ac075db954489913e2b | d_72d836e281e3336d06e76d86 | 3 | dense_top5 | 2 | llm | null | 0.7622 | null | correct | [
"full_solution"
] | high | The response correctly uses the nonvanishing leading coefficient of a product over a field to rule out a positive-degree unit, then observes that every nonzero constant has an inverse in F. Both inclusions in F[x]^*=F^* are established. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 3 | 0.031746 |
p_00092ac075db954489913e2b | d_7ffaecb692288bce763b3b93 | 18 | rerank_top5 | 2 | llm | 1.7578 | 0.7334 | 1.25 | correct | [
"full_solution"
] | high | The Herstein 3.11.4 solution takes uv=1, applies degree additivity in the integral domain to force deg(u)=deg(v)=0, and concludes u is a unit of the coefficient ring. Together with the evident converse, this proves the equality for a field. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 20 | 0.025641 |
p_00092ac075db954489913e2b | d_84529ecb9054a52429171658 | 15 | short_doc | 1 | rm8b | 0.7539 | 0.7344 | 0.25 | null | null | null | null | null | null | false | null | null | 17 | 0.026667 |
p_00092ac075db954489913e2b | d_8b670c4ebf819a1d60325606 | 1 | dense_top5 | 2 | llm | null | 0.7671 | null | correct | [
"full_solution"
] | high | The document states that a polynomial is a unit exactly when it divides 1, which in F[x] occurs exactly for nonzero constant polynomials, and notes that these are precisely the invertible elements of F. This proves F[x]^*=F^*. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 1 | 0.032787 |
p_00092ac075db954489913e2b | d_9dd862d6ed24b9e5a36c30e2 | 19 | short_doc | 1 | rm8b | 0.543 | 0.7329 | -2.1875 | null | null | null | null | null | null | false | null | null | null | 0.025316 |
p_00092ac075db954489913e2b | d_a593258cfad155bc09cc5fe6 | 16 | rerank_top5 | 0 | llm | 1.4844 | 0.7334 | 1.3125 | not_applicable | [
"target_mismatch"
] | high | The argument only proves that the indeterminate x is not a unit and hence that R[x] is not a field. It does not establish that every nonconstant polynomial is noninvertible or characterize all units. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 18 | 0.026316 |
p_00092ac075db954489913e2b | d_b5e84535f83185e90d3f0af3 | 12 | rerank_top5 | 2 | llm | 1.8516 | 0.7368 | 2.375 | correct | [
"equivalent_solution"
] | high | The stated lemma proves the stronger integral-domain result: a unit f in R[x] and its inverse g satisfy 0=deg(fg)=deg(f)+deg(g), so both are constant units of R; the converse inclusion is immediate. Applying this to a field gives the requested equality. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 14 | 0.027778 |
p_00092ac075db954489913e2b | d_b6556c11432f4fc2ea321d46 | 4 | dense_top5 | 0 | llm | null | 0.7524 | null | not_applicable | [
"question_restatement"
] | high | The document merely presents the more general domain result as an exercise and supplies no proof or substantive derivation. An explicitly requested proof is therefore unsupported. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 5 | 0.03125 |
p_00092ac075db954489913e2b | d_c13e9c16b0372b76bc2a1d62 | 5 | dense_top5 | 2 | llm | null | 0.751 | null | correct | [
"full_solution"
] | high | It correctly shows that a product involving a positive-degree polynomial cannot equal 1 because the leading coefficient remains nonzero over a field. Hence every unit is a nonzero constant, and conversely every nonzero constant is invertible in the field. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 6 | 0.030769 |
p_00092ac075db954489913e2b | d_c2f7b833620a1e2316b1a8d3 | 9 | rerank_top5 | 2 | llm | 1.4453 | 0.7451 | 2.4375 | correct | [
"full_solution"
] | high | The document proves that if A has polynomial inverse B, then deg(AB)=deg(A)+deg(B)=0, forcing both polynomials to be constants. Thus every unit is a nonzero field element, while every nonzero constant is plainly invertible. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 11 | 0.028986 |
p_00092ac075db954489913e2b | d_c9c768bc0aee8cc0d6d03265 | 14 | short_doc | 1 | rm8b | 1.0859 | 0.7349 | -1.375 | null | null | null | null | null | null | false | null | null | 16 | 0.027027 |
p_00092ac075db954489913e2b | d_dd0e8172303fa9d1ab4df5de | 6 | short_doc | 1 | rm8b | 0.7227 | 0.749 | 0.75 | null | null | null | null | null | null | false | null | null | 8 | 0.030303 |
p_00092ac075db954489913e2b | d_e192b48c40125b82f02bdf9e | 17 | short_doc | 0 | rm8b | 0.3613 | 0.7334 | -1.6875 | null | null | null | null | null | null | false | null | null | 19 | 0.025974 |
p_00092ac075db954489913e2b | d_e422dfdd80433cf609e24d48 | 20 | long_doc | 2 | llm | null | 0.7329 | null | correct | [
"full_solution"
] | high | The document proves that deg(fg)=deg(f)+deg(g). Thus fg=1 forces both factors to have degree zero, while every nonzero constant has an inverse in F, establishing F[x]^*=F^*. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.025 |
p_00092ac075db954489913e2b | d_e635fb7cfb66b50d73830f13 | 11 | short_doc | 2 | rm8b | 1.8438 | 0.7378 | -0.25 | null | null | null | null | null | null | false | null | null | 13 | 0.028169 |
p_00092ac075db954489913e2b | d_fd1f5a0b61bf8f2b2decc9c1 | 7 | short_doc | 1 | rm8b | 1.0625 | 0.748 | -1.0625 | null | null | null | null | null | null | false | null | null | 9 | 0.029851 |
p_000bc940bd868650bebfdfcb | d_09357d567272db888e6a33b4 | 4 | dense_top5 | 1 | llm | null | 0.7974 | null | correct | [
"correct_partial_method"
] | high | It correctly counts the six sum-7 outcomes and states P(sum 7)=1/6, while also giving the general complement rule, but does not explicitly calculate the requested 1-1/6=5/6. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.03125 |
p_000bc940bd868650bebfdfcb | d_1051b252bea001aa0c709000 | 1 | dense_top5 | 2 | llm | null | 0.8271 | null | correct | [
"correct_short_answer"
] | high | The document explicitly states that the probability of not rolling a 7 on one roll of two fair dice is 5/6, which is the requested result. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 1 | 2 | 1 | 0.065309 |
p_000bc940bd868650bebfdfcb | d_2b6939c179bf461b6c595459 | 18 | rerank_top5 | 1 | llm | 1.0312 | 0.7637 | 3.125 | correct | [
"correct_partial_method"
] | high | It correctly lists the 36 outcomes and the six outcomes summing to 7, yielding 1/6, but stops without computing the requested complementary probability 5/6. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.025641 |
p_000bc940bd868650bebfdfcb | d_2d481f3201ec5cf8ce2ac7c7 | null | fused_long | 1 | rm8b | 1.1172 | null | null | null | null | null | null | null | null | false | 6 | 1 | 20 | 0.031545 |
p_000bc940bd868650bebfdfcb | d_32d5366d51ac9d964e3dbfc9 | 13 | short_doc | 1 | rm8b | 1.3516 | 0.7715 | 1.625 | null | null | null | null | null | null | false | null | 41 | 11 | 0.037298 |
p_000bc940bd868650bebfdfcb | d_3d5db9257e9d601f98dfcefa | 24 | fused_short | 1 | rm8b | 1.3203 | 0.7559 | null | null | null | null | null | null | null | false | 56 | null | 16 | 0.03243 |
p_000bc940bd868650bebfdfcb | d_3d7a59aac72b7135f360e7c7 | 9 | short_doc | 1 | rm8b | 1.2891 | 0.7769 | 1.875 | null | null | null | null | null | null | false | null | null | null | 0.028986 |
p_000bc940bd868650bebfdfcb | d_443b67e9c8651584d64b5ecb | 49 | fused_long | 1 | rm8b | 0.8477 | 0.7407 | null | null | null | null | null | null | null | false | 32 | 89 | 12 | 0.03593 |
p_000bc940bd868650bebfdfcb | d_4ba7c461bbc9f9d3cf924579 | 12 | short_doc | 1 | rm8b | 1.0547 | 0.7725 | 0.125 | null | null | null | null | null | null | false | null | null | null | 0.027778 |
p_000bc940bd868650bebfdfcb | d_4f7e46a2841329de03669eb0 | 45 | fused_long | 1 | rm8b | 0.75 | 0.7432 | null | null | null | null | null | null | null | false | 9 | 39 | 7 | 0.043641 |
p_000bc940bd868650bebfdfcb | d_59a4f4ecac7cfb62243ec5b8 | 58 | fused_short | 1 | rm8b | 0.6289 | 0.7378 | null | null | null | null | null | null | null | false | 8 | null | 19 | 0.031655 |
p_000bc940bd868650bebfdfcb | d_67668d803724c9617e6c958c | 39 | fused_short | 1 | rm8b | 1.2734 | 0.7456 | null | null | null | null | null | null | null | false | 15 | 3 | 4 | 0.049408 |
p_000bc940bd868650bebfdfcb | d_76c90ba1da009b43d8d4a125 | 14 | rerank_top5 | 2 | llm | 1.625 | 0.7705 | 3.625 | correct | [
"full_solution"
] | high | It correctly enumerates all six sum-7 pairs, obtains probability 1/6, and takes the complement to give 5/6. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.027027 |
p_000bc940bd868650bebfdfcb | d_76ccba9f2fff15558ba8e6ca | 6 | rerank_top5 | 1 | llm | 1.7422 | 0.7871 | 5.125 | minor_error | [
"correct_partial_method",
"minor_math_error"
] | high | The complement calculation and final value 5/6 are correct, but the displayed list contains only five sum-7 outcomes because it omits (4,3) while nevertheless claiming six. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.030303 |
p_000bc940bd868650bebfdfcb | d_7b96e0b4cfedb1557b64f503 | 7 | short_doc | 1 | rm8b | 1.0703 | 0.7822 | 2.6875 | null | null | null | null | null | null | false | null | 9 | 5 | 0.044343 |
p_000bc940bd868650bebfdfcb | d_7e466147ef175bd127a86145 | 54 | fused_short | 1 | rm8b | 0.9336 | 0.7393 | null | null | null | null | null | null | null | false | 2 | null | 14 | 0.033673 |
p_000bc940bd868650bebfdfcb | d_966543f917fcaf6b9c7053e1 | 5 | dense_top5 | 1 | llm | null | 0.7886 | null | minor_error | [
"correct_partial_method",
"minor_math_error"
] | high | It gives the requested result 5/6 using the complement of 1/6, but its supporting list omits the outcome (4,3) while nonetheless claiming six favorable outcomes. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | null | 0.030769 |
p_000bc940bd868650bebfdfcb | d_9788cc9ba179fe6919206dd4 | 11 | long_doc | 1 | llm | null | 0.7734 | null | minor_error | [
"correct_partial_method",
"minor_math_error"
] | high | It correctly establishes P(sum 7)=6/36=1/6, from which the requested complement 5/6 follows, but it never states that answer and incorrectly calls mutually exclusive sum events independent. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | 4 | 10 | 2 | 0.05808 |
p_000bc940bd868650bebfdfcb | d_a1183f37eaaa4cf4648a883f | 3 | dense_top5 | 1 | llm | null | 0.7988 | null | correct | [
"correct_partial_method"
] | high | The document correctly establishes P(sum 7)=1/6 and explains complementary events elsewhere, but it does not apply the complement to state the requested probability 5/6. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | null | 18 | 0.031746 |
p_000bc940bd868650bebfdfcb | d_a31c43fc44d30b6191aa4f83 | 10 | rerank_top5 | 2 | llm | 1.7891 | 0.7739 | 5.8125 | correct | [
"full_solution"
] | high | It correctly counts six sum-7 outcomes among 36 and explicitly gives the complementary probability 1-1/6=5/6. | 031ac72b649ae9793c04c9cbb8a7acc64fc9de65e871d6e20589f11522c5eaa6 | true | false | null | 4 | 6 | 0.044196 |
p_000bc940bd868650bebfdfcb | d_ac6748520388813e9d3afc24 | 22 | fused_short | 1 | rm8b | 1.2344 | 0.7588 | null | null | null | null | null | null | null | false | 3 | 6 | 3 | 0.055415 |
p_000bc940bd868650bebfdfcb | d_b31647a1170840d8ed695dfe | 20 | short_doc | 1 | rm8b | 0.6445 | 0.7617 | 1.625 | null | null | null | null | null | null | false | null | null | null | 0.025 |
p_000bc940bd868650bebfdfcb | d_b96bf8667c75a58d7cde52e5 | 19 | short_doc | 1 | rm8b | 1.3672 | 0.7632 | 2.75 | null | null | null | null | null | null | false | null | null | null | 0.025316 |
p_000bc940bd868650bebfdfcb | d_c3acbd412ca7aab05a0b6b9d | 15 | short_doc | 1 | rm8b | 0.8398 | 0.7695 | 2.5625 | null | null | null | null | null | null | false | 7 | null | 10 | 0.041592 |
p_000bc940bd868650bebfdfcb | d_c48cb96b7048f754bd5f2b54 | 32 | fused_short | 1 | rm8b | 0.9922 | 0.75 | null | null | null | null | null | null | null | false | 21 | 58 | 9 | 0.042559 |
p_000bc940bd868650bebfdfcb | d_c82a1fbc9d644bc8cd656666 | 16 | short_doc | 1 | rm8b | 0.832 | 0.7646 | 2.3125 | null | null | null | null | null | null | false | null | null | null | 0.026316 |
End of preview. Expand in Data Studio
MathPinpoint
Status: v2. Every count below comes from the released tables. For what a retriever fine-tuned on it gains, see Training a retriever on MathPinpoint.
MathPinpoint is training data for problem-level math retrieval. Given a math question, the task is to find the web page that solves that problem, and solves it correctly. Each (query, page) pair carries a relevance label of 0, 1 or 2 under one written rubric, prompts/judge_prompt.md. The rubric is strict in two ways:
- Same problem, not same topic. A page gets 0 when it addresses a different problem (
wrong_problem), changes the conditions (condition_mismatch), answers a different target (target_mismatch), or only shares the topic or keywords (topic_only). - Correctness-aware. A page whose central mathematics is wrong gets 0, even when it addresses the right problem. A correct final answer reached through invalid reasoning is also 0.
What the release contains
| config | split | rows | content |
|---|---|---|---|
corpus |
train | 3,525,546 | deduplicated mathematical documents, shared by both splits |
queries |
train | 1,853,389 | query text, the page it was extracted from, extraction metadata |
judgments |
train | 43,506,169 | one row per judged (query, document) pair: its rank in each retrieval route, label, which judge produced it, and raw score where available |
queries |
test | 6,173 | held-out test queries |
qrels |
test | 83,833 | the relevant documents of each test query |
Column-level schemas are in docs/schema.md. The test split is described in docs/test_split.md.
How the labels were made
- Documents. Mathematical web pages go through a content extractor, are normalized, and are deduplicated with MinHash. Documents whose extraction degenerated, and documents that bundle five or more problems, are removed. See
docs/corpus.md. - Queries. One query per page, produced with a minimal-edit prompt: if the page contains a question somebody actually asked, that question is the query, copied with as few changes as possible. The prompt is
prompts/query_gen_prompt.txt. Queries are then deduplicated and filtered, leakage against evaluation sets is removed, and queries that two LLM judges both find not self-contained are dropped. Seedocs/queries.md. - Candidates.
- Three retrieval routes each contribute their top 100: dense (Qwen3-Embedding-4B), BM25, and word 5-gram. They are combined with weighted reciprocal rank fusion (dense weight 2, k = 60).
- A query's candidates are its dense top 20 and its fused top 20. On average, 4.55 candidates per query come from the fused top 20 alone.
- See
docs/judging.md.
- Judging.
- The LLM judge (GPT-5.6-Sol) labels, under
prompts/judge_prompt.md: dense ranks 1–5; within dense ranks 6–20, the top 5 as ordered by a zero-shot reranker and every document longer than 8,144 tokens; and, among the candidates from the fused top 20 alone, documents longer than 8,144 tokens at fused positions 1–5. - An 8B relevance model, trained on the LLM labels, judges everything else. It also scores the reranker's picks, so those pairs carry both labels.
- Every row records which judge produced its label, and the 8B's raw score is kept wherever it exists. See
docs/judging.md.
- The LLM judge (GPT-5.6-Sol) labels, under
Training a retriever on MathPinpoint
Qwen3-Embedding-0.6B was fine-tuned on training rows built from this release and compared with the same model before fine-tuning, on the test split (6,173 queries).
| metric | before | after | difference [95% CI] |
|---|---|---|---|
| nDCG@10 | 0.441 | 0.477 | +0.036 [+0.028, +0.045] |
| R@100 | 0.561 | 0.568 | +0.007 [−0.001, +0.016] |
Both metrics count unjudged documents as not relevant, and the test pool covers the two models unequally: 66% of the top 10 of the model before fine-tuning is judged, but only 35% of the fine-tuned model's. Ranking only the judged documents instead (condensed nDCG@10) gives 0.486 before and 0.617 after, a difference of +0.131 [+0.123, +0.139]. On a 500-query sample where both models' top 10 was judged in full, the nDCG@10 difference was +0.114, close to the condensed value.
Training rows. Each query gives one row:
- the positive is drawn at random from the query's score-2 candidates;
- the three score-0 candidates with the best dense rank are the hard negatives; candidates outside the dense top 100 come last;
- queries with fewer than three score-0 candidates are skipped.
Labels from both judges are used:
- Where both judges labeled a pair, the LLM label is used.
- A pair labeled only by the 8B model counts as score 2 if its raw score is at least 1.5, and as score 0 if it is below 0.5.
- Rows from the source-page check are not used, and pairs marked not evaluable are dropped. A query's own source page that retrieval also returned is a candidate like any other; it is the positive in 46.8% of rows.
Documents are cut to 4,000 characters and queries to 2,000. This gives 1,225,057 rows.
Training.
- Loss: multiple-negatives ranking loss, using in-batch negatives plus the three hard negatives, and a Matryoshka loss over 768, 512, 256 and 128 dimensions.
- Batch: effective batch 512 (8 GPUs × 64, GradCache with mini-batch 32).
- Schedule: learning rate 2e-5 with 10% warmup, sequence length 512, 1,000 steps. A run therefore sees 512,000 of the 1,225,057 rows.
- Runs: three, with seeds 42, 43 and 44. Per-query scores are averaged over the three.
Evaluation.
Search runs over all 3,525,546 corpus documents, with last-token pooling at 512 tokens. Each query is prefixed with the instruction below; documents get no prefix.
Instruct: Given a web search query, retrieve relevant passages that answer the query Query:Relevant documents are the test split's
qrels: documents judged score 2, which answer the same problem correctly.The test queries are held out: no training query matches one verbatim, at word 5-gram Jaccard ≥ 0.50 (checked exhaustively), or as a semantic duplicate judged to be the same problem.
Intervals come from a paired bootstrap over queries, with 10,000 resamples.
Documentation
| page | covers |
|---|---|
docs/corpus.md |
source pages, extraction, question removal, dedup, degenerate and multi-problem documents, document ids |
docs/queries.md |
query extraction, dedup, leakage removal, source-page check, self-containedness check |
docs/judging.md |
candidates, who judged what, the rubric, the LLM judge, the 8B model |
docs/test_split.md |
test queries and how their relevant documents were judged |
docs/schema.md |
release layout |
docs/limitations.md |
what the labels do and do not tell you |
Prompts and schemas
prompts/ holds the exact texts that produced the data. prompts/SHA256SUMS pins them. The judge prompt's guideline_version string is fixed in the output schema, so it does not identify which prompt text produced a label; the SHA-256 does.
| file | sha256 |
|---|---|
judge_prompt.md |
031ac72b649ae979… |
query_gen_prompt.txt |
d75d30a5b8cb0b03… |
query_check_prompt.md |
5de21b985641bc6a… |
Provenance and license
TBD before any public release. The source pages come from a gated upstream dataset that declares no license, and the documents here are derived from that content.
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