name string | instruction string | description string | keywords list | schema_version string | config unknown | metadata unknown | files list | location string |
|---|---|---|---|---|---|---|---|---|
arxiv-answer-verifiers/construct-rm-0704-0918-6-n18 | Construct a counterexample to commutation of two-fold joins and tree initial degeneration with 18 leaves, and report the least join multiplicity.
Use a connected, simple, undirected tree T on vertices 1,...,34. Vertices 1,...,18 must have degree one and all other vertices degree three. Each edge has length one. For i<... | construct/rm_0704_0918__6 | [
"arxiv-answer-verifiers",
"construct",
"rm_0704_0918__6"
] | 1.4 | {
"schema_version": "1.4",
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"name": "arxiv-answer-verifiers/construct-rm-0704-0918-6-n18",
"version": "0.1.0",
"description": "construct/rm_0704_0918__6",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0704_0918__6"
]
}... | {
"subset": "construct",
"family": "rm_0704_0918__6",
"problem_key": "gaussian-tree-grassmann-secant-commutation",
"domain": "algebra",
"tier": "research",
"level": "medium",
"source": "arXiv:0704.0918",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": ... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0704-0918-6-n18 |
arxiv-answer-verifiers/construct-rm-0704-1032-2-m3-p26 | Let x_1,...,x_9 be algebraically independent indeterminates over Q.
For a 3 by 3 array T containing each of the integers 1,...,9 exactly once,
with its rows increasing strictly, define
p_T = product over rows r of product over 1 <= i < j <= 3 of
(x_{T[r,i]} - x_{T[r,j]}). Array indices in this definition start at 1.
A ... | construct/rm_0704_1032__2 | [
"arxiv-answer-verifiers",
"construct",
"rm_0704_1032__2"
] | 1.4 | {
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"version": "0.1.0",
"description": "construct/rm_0704_1032__2",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0704_1032__2"
]
... | {
"subset": "construct",
"family": "rm_0704_1032__2",
"problem_key": "kopeliovich-standard-tableau-coefficients",
"domain": "algebra",
"tier": "research",
"level": "hard",
"source": "arXiv:0704.1032",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0704-1032-2-m3-p26 |
arxiv-answer-verifiers/construct-rm-0704-1174-2-d4 | Fix Q(x,y,z)=xz-y^2 over C and d=4. Construct two nonzero products L=prod(j=1..d) L_j and L'=prod(j=1..d) L'_j of linear forms with integer coefficients of absolute value at most 1000000. A triple [a,b,c] denotes ax+by+cz and cannot be [0,0,0]. Repeated proportional factors are allowed.
For any factor list F=(F_1,...,F... | construct/rm_0704_1174__2 | [
"arxiv-answer-verifiers",
"construct",
"rm_0704_1174__2"
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"description": "construct/rm_0704_1174__2",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0704_1174__2"
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},... | {
"subset": "construct",
"family": "rm_0704_1174__2",
"problem_key": "multipole-orbit-type-kernel-dimension",
"domain": "algebra",
"tier": "research",
"level": "medium",
"source": "arXiv:0704.1174",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0704-1174-2-d4 |
arxiv-answer-verifiers/construct-rm-0704-1540-0-k2 | Construct a negative-norm physical state in a twisted sector of an open free-bosonic permutation-type orbifold with K=2 copies of 26 bosons. The metric is g_0=-1, g_i=1 for i=1,...,25. Choose a permutation pi of the copies, an integer d in 1,...,26, and flip in {0,1}. The extra target-space automorphism multiplies coor... | construct/rm_0704_1540__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_0704_1540__0"
] | 1.4 | {
"schema_version": "1.4",
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"description": "construct/rm_0704_1540__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0704_1540__0"
]
},... | {
"subset": "construct",
"family": "rm_0704_1540__0",
"problem_key": "permutation-orbifold-extended-no-ghost",
"domain": "algebra",
"tier": "research",
"level": "easy",
"source": "arXiv:0704.1540",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"... | [
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0704-1540-0-k2 |
arxiv-answer-verifiers/construct-rm-0704-2007-0-n1 | Let k=F_2, the field with two elements. Let
R=(k[x,t,z_1,...,z_1]/J)_m, where m=(x,t,z_1,...,z_1) and
J=(xt, xz_i, tz_i, z_i z_j : 1<=i<=j<=1); xz_i and tz_i are included for every 1<=i<=1.
Let I=(x) in R and M=H_I^0(R)={u in R : x^s u=0 for some integer s>=1}.
The module M is generated by g_0=t and g_i=z_i for 1<=i<=1... | construct/rm_0704_2007__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_0704_2007__0"
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"description": "construct/rm_0704_2007__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0704_2007__0"
]
},... | {
"subset": "construct",
"family": "rm_0704_2007__0",
"problem_key": "local_cohomology_endomorphism_commutativity",
"domain": "algebra",
"tier": "research",
"level": "easy",
"source": "arXiv:0704.2007",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0704-2007-0-n1 |
arxiv-answer-verifiers/construct-rm-0704-3727-0-d2 | In R^2, construct two injective affine maps phi_i(x)=A_i x+b_i that are strictly contractive in the Euclidean norm, and a nonempty compact set K satisfying K=phi_1(K) union phi_2(K). Construct a nonempty bounded open convex set U with phi_i(U) contained in U and phi_1(U) intersection phi_2(U) empty, while the self-affi... | construct/rm_0704_3727__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_0704_3727__0"
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"description": "construct/rm_0704_3727__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0704_3727__0"
]
},... | {
"subset": "construct",
"family": "rm_0704_3727__0",
"problem_key": "self_affine_cosc_not_sosc",
"domain": "analysis",
"tier": "research",
"level": "easy",
"source": "arXiv:0704.3727",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qwen3_5_4b_r... | [
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"instruction.md",
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"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0704-3727-0-d2 |
arxiv-answer-verifiers/construct-rm-0705-3954-0-n2 | Construct a counterexample to the following equivalence in dimension n=2, over the rational field Q.
Choose exactly n nonempty subsets A_1,...,A_n of [n]={1,...,n}. Let U be the distinct exponent vectors of all monomials x_{i_1}...x_{i_n} with i_j in A_j, let R=Q[x^u : u in U], and let C={sum_{u in U} t_u u : t_u>=0}.
... | construct/rm_0705_3954__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_0705_3954__0"
] | 1.4 | {
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"name": "arxiv-answer-verifiers/construct-rm-0705-3954-0-n2",
"version": "0.1.0",
"description": "construct/rm_0705_3954__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0705_3954__0"
]
},... | {
"subset": "construct",
"family": "rm_0705_3954__0",
"problem_key": "stefan-conjecture-4-5",
"domain": "algebra",
"tier": "research",
"level": "easy",
"source": "arXiv:0705.3954",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qwen3_5_4b_reward... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0705-3954-0-n2 |
arxiv-answer-verifiers/construct-rm-0706-1379-0-n10 | For the fixed integer radix n=10, let [n]={0,...,n-1}. Construct
a nonempty period (p_1,...,p_L), 1 <= L <= 16, of permutations of [n]. Each p_j must
have all pairs (r-s,(p_j(r)-p_j(s)) mod (n+1)) distinct as 0 <= s < r < n varies.
Set f_i=p_{1+((i-1) mod L)} for every integer i>=1. For t in [0,1), use its
base-n expan... | construct/rm_0706_1379__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_0706_1379__0"
] | 1.4 | {
"schema_version": "1.4",
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"name": "arxiv-answer-verifiers/construct-rm-0706-1379-0-n10",
"version": "0.1.0",
"description": "construct/rm_0706_1379__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0706_1379__0"
]
}... | {
"subset": "construct",
"family": "rm_0706_1379__0",
"problem_key": "digitwise-modulo-costas-real-displacement",
"domain": "combinatorics",
"tier": "research",
"level": "hard",
"source": "arXiv:0706.1379",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward... | [
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"environment/instance.json",
"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0706-1379-0-n10 |
arxiv-answer-verifiers/construct-rm-0706-3425-0-r5-k3 | Let F_5 be the free group on generators x_1,...,x_5. Set Gamma_1(F)=F and Gamma_{i+1}(F)=[Gamma_i(F),F], where [H,F] is generated by hgh^(-1)g^(-1) for h in H and g in F. Let G=F_5/Gamma_4(F_5), the free nilpotent group of rank 5 and class 3. Construct an integer 5 by 5 matrix A such that the endomorphism of G defined ... | construct/rm_0706_3425__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_0706_3425__0"
] | 1.4 | {
"schema_version": "1.4",
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"version": "0.1.0",
"description": "construct/rm_0706_3425__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0706_3425__0"
]
... | {
"subset": "construct",
"family": "rm_0706_3425__0",
"problem_key": "free_nilpotent_finite_reidemeister",
"domain": "algebra",
"tier": "research",
"level": "hard",
"source": "arXiv:0706.3425",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qwen... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0706-3425-0-r5-k3 |
arxiv-answer-verifiers/construct-rm-0706-4417-2-k17 | For the fixed parameter k=17, let RR(k) be the least positive integer N such that every 2-coloring of {1,...,N} has positive integers x,y,z,w in that set, all of one color, satisfying x+y+kz=3w. Repeated variable values are allowed. Let P(k)=floor((k+4)/3)^2+C(k). In the following cases all congruences are modulo 9: C(... | construct/rm_0706_4417__2 | [
"arxiv-answer-verifiers",
"construct",
"rm_0706_4417__2"
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"schema_version": "1.4",
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"description": "construct/rm_0706_4417__2",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0706_4417__2"
]
}... | {
"subset": "construct",
"family": "rm_0706_4417__2",
"problem_key": "discrepancy_x_plus_y_plus_kz_eq_3w",
"domain": "combinatorics",
"tier": "research",
"level": "easy",
"source": "arXiv:0706.4417",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0706-4417-2-k17 |
arxiv-answer-verifiers/construct-rm-0707-1459-8-n34 | For the fixed level N=34, exhibit a finite certificate of d(2,N)<D(2,N).
Let zeta_N=exp(2*pi*i/N). For positive integers s_1,...,s_k and residues i_1,...,i_k modulo N, with (s_1,i_1)!=(1,0), define
L_N(s_1,...,s_k | i_1,...,i_k)=sum_(n_1>...>n_k>0) zeta_N^(i_1*n_1+...+i_k*n_k)/(n_1^s_1...n_k^s_k).
The weight is s_1+...... | construct/rm_0707_1459__8 | [
"arxiv-answer-verifiers",
"construct",
"rm_0707_1459__8"
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0707_1459__8"
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}... | {
"subset": "construct",
"family": "rm_0707_1459__8",
"problem_key": "weight_two_cyclotomic_mpv_dimension_defect",
"domain": "number_theory",
"tier": "research",
"level": "easy",
"source": "arXiv:0707.1459",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_rewar... | [
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0707-1459-8-n34 |
arxiv-answer-verifiers/construct-rm-0707-3010-2-d25 | Fix d=25. For every pair of integers 0 <= j < k <= d, let
f_(j,k)(z) = (1/d) product_(t=j)^(k-1) (z-t)(conjugate(z)+d-t), and
D_(j,k)(x,y) = (-1)^(k-j-1) (f_(j,k)(z)-f_(j,k)(conjugate(z)))/(z-conjugate(z)),
where z=x+i*y and f_(j,k)(conjugate(z)) swaps z with conjugate(z) in both factors.
For k-j >= 2 the real zero set... | construct/rm_0707_3010__2 | [
"arxiv-answer-verifiers",
"construct",
"rm_0707_3010__2"
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0707_3010__2"
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}... | {
"subset": "construct",
"family": "rm_0707_3010__2",
"problem_key": "pfeifle-binomial-oval-nonindexed-intersection",
"domain": "geometry",
"tier": "research",
"level": "research",
"source": "arXiv:0707.3010",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_rew... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0707-3010-2-d25 |
arxiv-answer-verifiers/construct-rm-0708-1695-2-n40 | Construct a complemented semidistributive lattice L with exactly 40 elements whose atomic-cover components are not all isomorphic.
Label the elements 0,...,39; labels need not be a topological ordering. Submit an 40 by 40 matrix order with integer entries 0 or 1, where order[i][j]=1 exactly when i<=j in L. This relatio... | construct/rm_0708_1695__2 | [
"arxiv-answer-verifiers",
"construct",
"rm_0708_1695__2"
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
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}... | {
"subset": "construct",
"family": "rm_0708_1695__2",
"problem_key": "complemented-semidistributive-regularity",
"domain": "algebra",
"tier": "research",
"level": "research",
"source": "arXiv:0708.1695",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": ... | [
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"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0708-1695-2-n40 |
arxiv-answer-verifiers/construct-rm-0709-0743-1-n7 | Let X = {1,...,7}. A partial bijection of X is an injective map from a subset of X into X. Encode it by a length-7 array a: a[i-1] is the image of i, or 0 if i is outside the domain. Composition applies the first map and then the second; an undefined intermediate image remains undefined.
Construct a nonempty set S of a... | construct/rm_0709_0743__1 | [
"arxiv-answer-verifiers",
"construct",
"rm_0709_0743__1"
] | 1.4 | {
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"description": "construct/rm_0709_0743__1",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_0709_0743__1"
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},... | {
"subset": "construct",
"family": "rm_0709_0743__1",
"problem_key": "inclusion_minimal_semitransitive_unequal_r_classes",
"domain": "algebra",
"tier": "research",
"level": "medium",
"source": "arXiv:0709.0743",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_r... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0709-0743-1-n7 |
arxiv-answer-verifiers/construct-rm-0709-1570-0-p43-q53-r2963 | For the distinct odd primes p=43, q=53, r=2963, let Phi_{pqr}(x)=sum_{j=0}^{6469008} a_j x^j be the pqr-th cyclotomic polynomial, whose roots are precisely the primitive pqr-th roots of unity. Exhibit a zero-based exponent k with 0<=k<=6469008 and |a_k|>(p+1)/2. The coefficients a_j are integers. Return the instance pr... | construct/rm_0709_1570__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_0709_1570__0"
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"problem_key": "sister_beiter_coefficient_counterexamples",
"domain": "number_theory",
"tier": "research",
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"source": "arXiv:0709.1570",
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"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-0709-1570-0-p43-q53-r2963 |
arxiv-answer-verifiers/construct-rm-0709-3259-5-n6 | Construct a finite simple graph G on the labelled vertex set {1,...,6} with the following two properties.
Define A_G in R^6 to consist of x_i-x_j=0 for every edge {i,j}. Let r_0 be the region containing (1,...,6), and let R_G(q)=sum_r q^d(r,r_0), where d counts the hyperplanes separating the two regions. First, R_G(q) ... | construct/rm_0709_3259__5 | [
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},... | {
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"problem_key": "opy-nice-elimination-q-factorization",
"domain": "graph_theory",
"tier": "research",
"level": "easy",
"source": "arXiv:0709.3259",
"source_license": "CC-BY-4.0",
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"glm_5_3_flash_reward": 1,
... | [
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"tests/instance.json",
"tests/run.py",
"tests/test.sh"
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arxiv-answer-verifiers/construct-rm-1104-4534-2-n1 | On T^1=(R/Z)^1, construct real smooth vector fields b_1,b_2 and real smooth initial functions u_01,u_02 for which the viscosity solution of
(u_1)_t+|Du_1-b_1(x)|^2-|b_1(x)|^2+u_1-u_2=0,
(u_2)_t+|Du_2-b_2(x)|^2-|b_2(x)|^2+u_2-u_1=0
does not converge uniformly to a shared real constant as t tends to infinity. No divergen... | construct/rm_1104_4534__2 | [
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"problem_key": "weakly_coupled_hj_constant_limit_without_divergence",
"domain": "analysis",
"tier": "research",
"level": "easy",
"source": "arXiv:1104.4534",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
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"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1104-4534-2-n1 |
arxiv-answer-verifiers/construct-rm-1309-0643-0-n18 | Construct a finite simple connected undirected graph H on exactly 18 vertices labeled 0,...,17, and a vertex v with the following properties.
The distance d(a,b) is the length of a shortest path. A vertex x is maximally distant from y if d(y,z)<=d(y,x) for every neighbor z of x. Distinct vertices x,y are mutually maxim... | construct/rm_1309_0643__0 | [
"arxiv-answer-verifiers",
"construct",
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"arxiv-answer-verifiers",
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}... | {
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"family": "rm_1309_0643__0",
"problem_key": "rooted-product-matching-boundary-conjecture",
"domain": "graph_theory",
"tier": "research",
"level": "hard",
"source": "arXiv:1309.0643",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_rewar... | [
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1309-0643-0-n18 |
arxiv-answer-verifiers/construct-rm-1506-01091-5-s9 | Construct two different probability laws on isomorphism types of simple edge-weighted trees with 3 leaves. Each tree must be a star: one central vertex joined to three distinct leaves. Its three edge lengths must be positive integers summing to 9. Two such stars are isomorphic exactly when their unordered triples of ed... | construct/rm_1506_01091__5 | [
"arxiv-answer-verifiers",
"construct",
"rm_1506_01091__5"
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... | {
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"family": "rm_1506_01091__5",
"problem_key": "random-tree-length-law-identifiability",
"domain": "probability",
"tier": "research",
"level": "easy",
"source": "arXiv:1506.01091",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": ... | [
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"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1506-01091-5-s9 |
arxiv-answer-verifiers/construct-rm-1610-03873-4-a10-l20 | Fix a=10, l=20, r=2, and n=l. The vertices are 1,...,l in cyclic order, with indices reduced modulo l to 1,...,l. Let H have exactly the unordered pairs contained in at least one block of a consecutive vertices. An a-clique is a set Q of a distinct vertices for which every pair is an edge of H. An a-clique-free edge su... | construct/rm_1610_03873__4 | [
"arxiv-answer-verifiers",
"construct",
"rm_1610_03873__4"
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... | {
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"family": "rm_1610_03873__4",
"problem_key": "hyperweb-tightness",
"domain": "combinatorics",
"tier": "research",
"level": "hard",
"source": "arXiv:1610.03873",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qwen3_5_4b_r... | [
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"tests/test.sh"
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arxiv-answer-verifiers/construct-rm-1703-03800-0-n8 | Let K_8 be the complete simple undirected graph with vertices 1 through 8. Partition all its 28 edges into the minimum possible number of nonempty layers. Each layer must be planar and have girth at least 4, meaning that it contains no triangle; a forest has girth infinity. A layer may have isolated vertices. Every uno... | construct/rm_1703_03800__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_1703_03800__0"
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
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... | {
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"family": "rm_1703_03800__0",
"problem_key": "4_girth_thickness_complete_graph",
"domain": "graph_theory",
"tier": "research",
"level": "easy",
"source": "arXiv:1703.03800",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
... | [
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"tests/instance.json",
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"tests/test.sh"
] | tasks/construct-rm-1703-03800-0-n8 |
arxiv-answer-verifiers/construct-rm-1703-05929-1-k5 | Construct a counterexample to complete transitivity for the following code, with k=5, q=4, c=3 and n=341.
Use F_4=F_2[a]/(a^2+a+1), encoding 0,1,a,a+1 by the integers 0,1,2,3. Supply a k-by-n matrix H whose columns represent all one-dimensional subspaces of F_4^k exactly once. Its row space must be invariant under righ... | construct/rm_1703_05929__1 | [
"arxiv-answer-verifiers",
"construct",
"rm_1703_05929__1"
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"description": "construct/rm_1703_05929__1",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1703_05929__1"
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... | {
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"family": "rm_1703_05929__1",
"problem_key": "cyclic_hamming_concatenation_complete_transitivity",
"domain": "coding_theory",
"tier": "research",
"level": "hard",
"source": "arXiv:1703.05929",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_f... | [
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"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1703-05929-1-k5 |
arxiv-answer-verifiers/construct-rm-1703-06262-0-n2 | Work in R^2, with B_1 = {x: |x|<1}. Construct a counterexample to preservation of the upper contact boundary by quadratic blowup for the following hybrid double obstacle problem. Use rational polynomials u, psi of total degree at most 4 and f of total degree at most 2, and rational numbers c>0, M>0.
Define Omega(w)=B_... | construct/rm_1703_06262__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_1703_06262__0"
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1703_06262__0"
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... | {
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"family": "rm_1703_06262__0",
"problem_key": "double_obstacle_upper_contact_blowup",
"domain": "analysis",
"tier": "research",
"level": "research",
"source": "arXiv:1703.06262",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1703-06262-0-n2 |
arxiv-answer-verifiers/construct-rm-1703-06347-4-q25 | Construct a finite projective plane of order q=25 and a polarity giving a counterexample to this implication: if S or E induces a triangle-free graph, the plane is PG(2,q) and its polarity has exactly q+1 absolute points.
Use points and lines indexed 0,...,650. There are 651 of each. Every line contains q+1 distinct po... | construct/rm_1703_06347__4 | [
"arxiv-answer-verifiers",
"construct",
"rm_1703_06347__4"
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
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... | {
"subset": "construct",
"family": "rm_1703_06347__4",
"problem_key": "polarity_triangle_free_partition_characterization",
"domain": "discrete_geometry",
"tier": "research",
"level": "hard",
"source": "arXiv:1703.06347",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_... | [
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"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1703-06347-4-q25 |
arxiv-answer-verifiers/construct-rm-1703-06690-2-n1 | Construct a counterexample in dimension n=1 to the following avoidance claim: when an Armijo/weak-Wolfe BFGS method minimizes a semi-algebraic strongly convex function with bounded sublevel sets from an absolutely continuous random initial point, every accepted iterate lies in the interior V of the set of twice differe... | construct/rm_1703_06690__2 | [
"arxiv-answer-verifiers",
"construct",
"rm_1703_06690__2"
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"version": "0.1.0",
"description": "construct/rm_1703_06690__2",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1703_06690__2"
]
... | {
"subset": "construct",
"family": "rm_1703_06690__2",
"problem_key": "bfgs-semialgebraic-C2-avoidance",
"domain": "analysis",
"tier": "research",
"level": "research",
"source": "arXiv:1703.06690",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1703-06690-2-n1 |
arxiv-answer-verifiers/construct-rm-1703-06800-5-d1 | Let H=C^1, with standard basis indexed 0,...,0. Construct a POVM
E_1,...,E_n on H for which the equivalence below fails. You may choose 1<=n<=16.
Each E_alpha must be Hermitian, 0<=E_alpha<=I, and sum_alpha E_alpha=I.
For every unit Psi in H tensor H define p_(alpha,beta)(Psi)=
<Psi|E_alpha tensor E_beta|Psi> and t(Ps... | construct/rm_1703_06800__5 | [
"arxiv-answer-verifiers",
"construct",
"rm_1703_06800__5"
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"keywords": [
"arxiv-answer-verifiers",
"construct",
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... | {
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"family": "rm_1703_06800__5",
"problem_key": "conical-design-nonregular-monotone-equivalence",
"domain": "algebra",
"tier": "research",
"level": "easy",
"source": "arXiv:1703.06800",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_rewar... | [
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"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1703-06800-5-d1 |
arxiv-answer-verifiers/construct-rm-1703-07439-5-order04 | Work over the rational polynomial ring R=Q[a0,a1,b0,b1,b2,c0,c1,c2,c3].
Let I be generated by all five 2-by-2 minors of the following three matrices,
together with the polynomial h below:
[[a0*(b0-b2), a1*(b1-b2)], [b0,b1]],
[[a0*(c0-c2), a1*(c1-c2)], [c0,c1]],
[[b0*(c0-c3), b1*(c1-c3), b2*(c2-c3)], [c0,c1,c2]],
h=a0*b... | construct/rm_1703_07439__5 | [
"arxiv-answer-verifiers",
"construct",
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"description": "construct/rm_1703_07439__5",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1703_07439__5"
... | {
"subset": "construct",
"family": "rm_1703_07439__5",
"problem_key": "monin-rana-groebner-degree-count",
"domain": "algebra",
"tier": "research",
"level": "medium",
"source": "arXiv:1703.07439",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qw... | [
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"instruction.md",
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"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1703-07439-5-order04 |
arxiv-answer-verifiers/construct-rm-1703-07461-0-n3 | Construct a counterexample to an ordinary local error bound for a reversible mass-action reaction network with two species A,B and exactly n=3 distinct reversible reactions. Each complex has between 1 and 3 molecules, and each reaction conserves A+B. Reaction j has reactant vector F[:,j] and product vector R[:,j], for ... | construct/rm_1703_07461__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_1703_07461__0"
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"description": "construct/rm_1703_07461__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1703_07461__0"
]
... | {
"subset": "construct",
"family": "rm_1703_07461__0",
"problem_key": "mass_conserved_kinetics_metric_subregularity",
"domain": "analysis",
"tier": "research",
"level": "easy",
"source": "arXiv:1703.07461",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1703-07461-0-n3 |
arxiv-answer-verifiers/construct-rm-1703-08291-4-d5 | Work over the field F_2, with q=2 and r=1. Construct two nonempty sets K_1 and K_2 of distinct nonzero column vectors, spanning F_2^2 and F_2^5, respectively. Each set generates a binary linear code: for a row vector a, its codeword has coordinates a dot x (modulo 2), one for each column x. Every such codeword must hav... | construct/rm_1703_08291__4 | [
"arxiv-answer-verifiers",
"construct",
"rm_1703_08291__4"
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"description": "construct/rm_1703_08291__4",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1703_08291__4"
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... | {
"subset": "construct",
"family": "rm_1703_08291__4",
"problem_key": "projective-divisible-code-embedding-interval",
"domain": "coding_theory",
"tier": "research",
"level": "medium",
"source": "arXiv:1703.08291",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1703-08291-4-d5 |
arxiv-answer-verifiers/construct-rm-1703-08327-2-d128 | For dimension d=128, exhibit finite rational exponents
p,q>d/(d-1) and a certified sequence witnessing failure of a uniform bound
for the spherical maximal operator on L^p(R^d;ell^q). Here
M_S f(x)=sup_{r>0} |integral_{S^{d-1}} f(x-r*theta) d sigma(theta)|,
sigma is normalized surface measure, and ||g||_p=(integral |g|... | construct/rm_1703_08327__2 | [
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arxiv-answer-verifiers/construct-rm-1703-10741-4-n10 | Construct a finite simple undirected graph G on the labelled vertex set {0,...,9} and an integer r with 2 <= r < 10, such that the minimum degree is exactly floor(10/2) + min{1,r-3} and m(G,r) > r. Here an initially infected set A evolves in simultaneous rounds: every uninfected vertex with at least r infected neighbou... | construct/rm_1703_10741__4 | [
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... | {
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"domain": "graph_theory",
"tier": "research",
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"source": "arXiv:1703.10741",
"source_license": "CC-BY-4.0",
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arxiv-answer-verifiers/construct-rm-1704-01562-4-n8 | For n=8, give an exact counterexample to the following rational-function identity for some integer 1<=i<=n:
G(1^(n-i) 0 1^(i-1);q,t,a)=(-qt)^(1-i)(a^(i-1)+a^i).
Here juxtaposition concatenates binary words and 1^j means j copies of 1. G is defined on the empty word and binary words containing a zero by the following re... | construct/rm_1704_01562__4 | [
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... | {
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"problem_key": "hogancamp-hook-shadow-literal-normalization",
"domain": "algebra",
"tier": "research",
"level": "hard",
"source": "arXiv:1704.01562",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward":... | [
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"tests/test.sh"
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arxiv-answer-verifiers/construct-rm-1704-01892-2-m10-chi3 | Construct a family of exactly 10 distinct polygonal paths in the plane. Each path consists of one or two nonzero closed straight line segments joined consecutively; specify it by two or three distinct points in traversal order. Intersections at endpoints count as intersections. Its disjointness graph has one vertex per... | construct/rm_1704_01892__2 | [
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... | {
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"domain": "discrete_geometry",
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"source": "arXiv:1704.01892",
"source_license": "CC-BY-4.0",
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arxiv-answer-verifiers/construct-rm-1704-04385-5-e1 | Let e=1, N=2^e=2, and let A_N=F_2[x,y]/(x^N,y^N),
where F_2 is the field with two elements. A polynomial is represented uniquely
as a sum of c_ij x^i y^j with 0<=i,j<N and c_ij in {0,1}.
All polynomial and matrix operations are performed in this quotient ring.
Let H_N be the group of 2 by 2 matrices of determinant one ... | construct/rm_1704_04385__5 | [
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... | {
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"domain": "algebra",
"tier": "research",
"level": "easy",
"source": "arXiv:1704.04385",
"source_license": "CC-BY-4.0",
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"glm_5_3_flash_reward": 1... | [
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arxiv-answer-verifiers/construct-rm-1704-05115-2-n5 | Construct a symmetric 5-by-5 integer matrix A with zero diagonal and off-diagonal entries between 0 and 9, inclusive, together with a perfect elimination ordering (PEO). An ordering p of vertices [n] is a PEO if A[y,z] >= min(A[x,y], A[x,z]) for every triple occurring in the order x before y before z.
Require that no ... | construct/rm_1704_05115__2 | [
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... | {
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"tier": "research",
"level": "easy",
"source": "arXiv:1704.05115",
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... | [
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arxiv-answer-verifiers/construct-rm-1704-05335-2-d2 | Construct a certificate that a log-covariance likelihood is not convex.
Let D=2, L=D and n=1. For a real symmetric positive definite D by D
matrix C define f_C(X)=L*tr(X+C exp(-X)) on all complex Hermitian matrices X.
Here exp denotes the matrix exponential, tr the trace. Such C is an admissible
observation because C=e... | construct/rm_1704_05335__2 | [
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... | {
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"domain": "analysis",
"tier": "research",
"level": "research",
"source": "arXiv:1704.05335",
"source_license": "CC-BY-4.0",
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"glm_5_3_flash_reward": 1,
... | [
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arxiv-answer-verifiers/construct-rm-1704-07735-1-n102 | Construct a real unit equiangular tight frame of n=102 vectors in R^51, together with a nonzero vector in its integer span whose norm is strictly less than 1.
Encode the frame by its Seidel matrix S: an n by n symmetric integer matrix with zero diagonal, off-diagonal entries in {-1,1}, and S^2=(n-1)I. Its Gram matrix i... | construct/rm_1704_07735__1 | [
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"keywords": [
"arxiv-answer-verifiers",
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"rm_1704_07735__1"
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"problem_key": "integer_spans_of_real_equiangular_tight_frames",
"domain": "geometry",
"tier": "research",
"level": "hard",
"source": "arXiv:1704.07735",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_rewa... | [
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arxiv-answer-verifiers/construct-rm-1704-07752-7-n4 | Construct two distinct 4 x 4 x 4 alternating sign hypermatrices A and B with equal weighted projections. An alternating sign hypermatrix has entries in {-1,0,1}. On every line obtained by fixing any two of its three indices, the nonzero entries must form a nonempty alternating sequence beginning and ending with +1. Ind... | construct/rm_1704_07752__7 | [
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... | {
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"family": "rm_1704_07752__7",
"problem_key": "ashm_projection_noninjectivity",
"domain": "combinatorics",
"tier": "research",
"level": "easy",
"source": "arXiv:1704.07752",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"... | [
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arxiv-answer-verifiers/construct-rm-1704-07930-0-k4 | Let k=4, p=2, and s=k+1/2. Construct a one-dimensional counterexample to bounded multiplication on the intrinsic Sobolev-Slobodeckij space W^{s,2}(Omega), in the following finite polynomial representation.
Choose rational numbers a<c<b and set Omega=(a,c) union (c,b), an open subset of R. Choose a polynomial phi(x)=su... | construct/rm_1704_07930__0 | [
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... | {
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"problem_key": "smooth_multipliers_intrinsic_slobodeckij_arbitrary_open_sets",
"domain": "analysis",
"tier": "research",
"level": "medium",
"source": "arXiv:1704.07930",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"gl... | [
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arxiv-answer-verifiers/construct-rm-1704-08184-0-n3-p99of100-l1of10 | Let D={z in C: |z|<1}, n=3, p=99/100, and lambda=1/10. Exhibit a normalized meromorphic f on D with f(0)=0, f'(0)=1, no pole in |z|<p, and |z/f(z)-z(z/f(z))'-1|<lambda for every z in D, whose expansion f(z)=z+sum_{k>=2} a_k*z^k satisfies
|a_n| > p^(1-n)*sum_{j=0}^{n-1}(lambda*p^2)^j.
Use this finite encoding: choose re... | construct/rm_1704_08184__0 | [
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"description": "construct/rm_1704_08184__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
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"rm_1704_0... | {
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"problem_key": "meromorphic_U_lambda_coefficient_conjecture",
"domain": "analysis",
"tier": "research",
"level": "easy",
"source": "arXiv:1704.08184",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
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arxiv-answer-verifiers/construct-rm-1705-01758-0-n3 | Construct an integer 3 by 3 matrix A and an integer z for which z belongs to Phi(A) but not Theta(A). Each entry of A and z must lie between -1000000 and 1000000, inclusive. Integer matrices are regarded as complex matrices.
Indices range from 1 to 3. Define r_i = sum_{k != i} |a_ik| and r_i^j = r_i - |a_ij| for j != i... | construct/rm_1705_01758__0 | [
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
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... | {
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"family": "rm_1705_01758__0",
"problem_key": "li-li-li-phi-theta-inclusion",
"domain": "analysis",
"tier": "research",
"level": "research",
"source": "arXiv:1705.01758",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qwe... | [
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arxiv-answer-verifiers/construct-rm-1705-02658-2-g32 | Let N_0 = {0,1,2,...}. A numerical semigroup is a subset S of N_0 containing 0, closed under addition, with finite complement. Its genus is the number of missing positive integers. In this instance g = 32.
Call S 1-hyperelliptic if exactly one of 2 and 4 belongs to S and 6 belongs to S. Let c be its conductor (the leas... | construct/rm_1705_02658__2 | [
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... | {
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"problem_key": "hyperelliptic_canonical_weight_unique_maximizer",
"domain": "number_theory",
"tier": "research",
"level": "research",
"source": "arXiv:1705.02658",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_... | [
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arxiv-answer-verifiers/construct-rm-1705-05686-6-r1 | Construct a certificate that finite local inverse-system recovery fails at socle degree r=1.
Work over Q, in R=Q[[x,y]], with contraction module Gamma=Q_DP[X,Y]. The monomial x^a y^b sends X^[i]Y^[j] to X^[i-a]Y^[j-b] if i>=a and j>=b, and to zero otherwise. Brackets denote divided powers, so contraction has no factori... | construct/rm_1705_05686__6 | [
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1705_05686__6"
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... | {
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"family": "rm_1705_05686__6",
"problem_key": "local_gorenstein_finite_recovery",
"domain": "algebra",
"tier": "research",
"level": "research",
"source": "arXiv:1705.05686",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"... | [
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arxiv-answer-verifiers/construct-rm-1705-09100-0-p5d1 | Fix N=1, s=3/4, mu2=1, and p=5/1. With these N and s the fractional Sobolev critical exponent is infinity, so p satisfies the required subcritical range. Construct rational mu1>1 and beta>p-1 such that
f(tau)=(1+tau^2)/(mu1+2*beta*tau^p+tau^(2*p))^(1/p), tau>=0,
has local minima at 0 and tau1>0 and local maxima at tau2... | construct/rm_1705_09100__0 | [
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"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1705_09100__0"
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"family": "rm_1705_09100__0",
"problem_key": "fractional-system-interior-minimum-comparison",
"domain": "analysis",
"tier": "research",
"level": "research",
"source": "arXiv:1705.09100",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_r... | [
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"tests/test.sh"
] | tasks/construct-rm-1705-09100-0-p5d1 |
arxiv-answer-verifiers/construct-rm-1705-10353-10-n6 | For n=6, construct a certificate of a negative elementary-symmetric coefficient as follows.
Use [n]={1,...,n} and interpret vertex indices modulo n. Choose integers a_1,...,a_n
with 0<=a_i<=n-1 and a_i-1<=a_(i+1). Let Gamma have exactly the directed edges
i->i+s for 1<=s<=a_i. A corner is an ordered pair (i,j) absent f... | construct/rm_1705_10353__10 | [
"arxiv-answer-verifiers",
"construct",
"rm_1705_10353__10"
] | 1.4 | {
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"version": "0.1.0",
"description": "construct/rm_1705_10353__10",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
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... | {
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"family": "rm_1705_10353__10",
"problem_key": "alexandersson_panova_conjecture_46",
"domain": "combinatorics",
"tier": "research",
"level": "research",
"source": "arXiv:1705.10353",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward... | [
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"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1705-10353-10-n6 |
arxiv-answer-verifiers/construct-rm-1706-00986-15-q3-p2-m1 | Construct an exact certificate of a nontrivial continuous deformation
of the following partial complex Hadamard matrix. Such a matrix has entries
of absolute value one and pairwise orthogonal rows.
Put q=3, p=2, w=exp(2*pi*i/3), and t_0=0, t_1=2. The matrix H has three
rows indexed by a=0,1,2 and six columns indexed b... | construct/rm_1706_00986__15 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_00986__15"
] | 1.4 | {
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"name": "arxiv-answer-verifiers/construct-rm-1706-00986-15-q3-p2-m1",
"version": "0.1.0",
"description": "construct/rm_1706_00986__15",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_00986__15... | {
"subset": "construct",
"family": "rm_1706_00986__15",
"problem_key": "arithmetic_isolation_mcnulty_weigert",
"domain": "algebra",
"tier": "research",
"level": "research",
"source": "arXiv:1706.00986",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-00986-15-q3-p2-m1 |
arxiv-answer-verifiers/construct-rm-1706-01212-6-n4-q9 | For n=4, let [n]={1,...,n} and let P have vertices 0,...,8. Its complete list of strict order relations is 0<1, 0<2, 1<2; all other pairs are incomparable. A family consists of distinct subsets of [n]. A copy of P is an injective map from its vertices to sets such that each strict order relation maps to proper inclusio... | construct/rm_1706_01212__6 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_01212__6"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-01212-6-n4-q9",
"version": "0.1.0",
"description": "construct/rm_1706_01212__6",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_01212__6"
... | {
"subset": "construct",
"family": "rm_1706_01212__6",
"problem_key": "trace_closed_poset_arrow",
"domain": "combinatorics",
"tier": "research",
"level": "medium",
"source": "arXiv:1706.01212",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qwen... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-01212-6-n4-q9 |
arxiv-answer-verifiers/construct-rm-1706-02516-2-m15 | Let m=15. The graph M_(2m) has vertices 1,...,2m, ring edges {i,i+1} with labels modulo 2m, and rung edges {i,i+m} for 1<=i<=m. Its cut polytope is the convex hull in R^(3m) of the vectors delta_S, whose coordinate on an edge is 1 exactly when one endpoint belongs to S.
For an odd divisor t>1 of m, C(m,t) is the cycle... | construct/rm_1706_02516__2 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_02516__2"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-02516-2-m15",
"version": "0.1.0",
"description": "construct/rm_1706_02516__2",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_02516__2"
]
... | {
"subset": "construct",
"family": "rm_1706_02516__2",
"problem_key": "mobius_ladder_cycle_facet_orbits",
"domain": "geometry",
"tier": "research",
"level": "medium",
"source": "arXiv:1706.02516",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"q... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-02516-2-m15 |
arxiv-answer-verifiers/construct-rm-1706-03879-0-n12 | Construct a finite simple undirected connected graph G on vertices 0,...,11, other than the star K_1,11, with tmc(G) <= vmc(G), together with the following exact certificates.
A total coloring assigns colors to all vertices and edges. It is TMC if every pair of vertices has a path whose edges and internal vertices shar... | construct/rm_1706_03879__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_03879__0"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-03879-0-n12",
"version": "0.1.0",
"description": "construct/rm_1706_03879__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_03879__0"
]
... | {
"subset": "construct",
"family": "rm_1706_03879__0",
"problem_key": "tmc-versus-vmc-nonstar",
"domain": "graph_theory",
"tier": "research",
"level": "research",
"source": "arXiv:1706.03879",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qwen3... | [
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"environment/instance.json",
"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-03879-0-n12 |
arxiv-answer-verifiers/construct-rm-1706-04476-0-m23 | Construct a connected loopless multigraph G on vertices 0,1,2,3,4 with exactly 23 edges, counting parallel edges separately, and a permitted vertex order for which Conditional_Greedy stops with an uncolored edge. Require chi'(G) > Delta(G)+1, where Delta is maximum degree and chi' is the minimum number of colors in a p... | construct/rm_1706_04476__0 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_04476__0"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-04476-0-m23",
"version": "0.1.0",
"description": "construct/rm_1706_04476__0",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_04476__0"
]
... | {
"subset": "construct",
"family": "rm_1706_04476__0",
"problem_key": "goldberg-conditional-greedy",
"domain": "graph_theory",
"tier": "research",
"level": "easy",
"source": "arXiv:1706.04476",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"qwen... | [
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"environment/instance.json",
"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-04476-0-m23 |
arxiv-answer-verifiers/construct-rm-1706-04476-2-m31 | Construct a connected loopless undirected multigraph G on n=5 vertices, labelled 0,...,4, with exactly m=31 edges counted with multiplicity, and a permitted vertex order for which the following algorithm fails. Give G by a symmetric n by n matrix A of nonnegative integers with zero diagonal; A[u][v] counts parallel edg... | construct/rm_1706_04476__2 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_04476__2"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-04476-2-m31",
"version": "0.1.0",
"description": "construct/rm_1706_04476__2",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_04476__2"
]
... | {
"subset": "construct",
"family": "rm_1706_04476__2",
"problem_key": "goldberg_conditional_greedy_failure",
"domain": "graph_theory",
"tier": "research",
"level": "research",
"source": "arXiv:1706.04476",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward"... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-04476-2-m31 |
arxiv-answer-verifiers/construct-rm-1706-04476-3-w14 | Construct a connected loopless multigraph on vertices 0,1,2,3,4, with at most 6 parallel edges between each pair, together with a vertex order for which the following procedure halts with an uncolored edge. Its density must equal 14 and exceed its maximum degree plus one.
For S a vertex subset of size at least two, le... | construct/rm_1706_04476__3 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_04476__3"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-04476-3-w14",
"version": "0.1.0",
"description": "construct/rm_1706_04476__3",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_04476__3"
]
... | {
"subset": "construct",
"family": "rm_1706_04476__3",
"problem_key": "goldberg-conditional-greedy",
"domain": "graph_theory",
"tier": "research",
"level": "research",
"source": "arXiv:1706.04476",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
"... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-04476-3-w14 |
arxiv-answer-verifiers/construct-rm-1706-05187-5-g4 | Fix g=4. Let Gamma_g be g oriented circles with their
base points identified to one vertex. The group G=(Z_2)^g semidirect S_g
acts by independently reversing circle orientations and by permuting circles.
Let d_g be the smallest d admitting a faithful orthogonal G-action on the
unit sphere S^d and a G-equivariant embed... | construct/rm_1706_05187__5 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_05187__5"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-05187-5-g4",
"version": "0.1.0",
"description": "construct/rm_1706_05187__5",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_05187__5"
]
... | {
"subset": "construct",
"family": "rm_1706_05187__5",
"problem_key": "bouquet_equivariant_sphere_dimension",
"domain": "geometry",
"tier": "research",
"level": "easy",
"source": "arXiv:1706.05187",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,
... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-05187-5-g4 |
arxiv-answer-verifiers/construct-rm-1706-05444-3-ell4 | For ell = 4, construct a finite set A of nonnegative integers. It must contain 0, have exactly 32 elements, and have maximum either 162 or 324. Put N = 243. For all x,y,z in A, including repeated elements and with no ordering restriction, the congruence x = 2*y-z (mod N) must imply x=y=z. Also, for every integer r, the... | construct/rm_1706_05444__3 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_05444__3"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-05444-3-ell4",
"version": "0.1.0",
"description": "construct/rm_1706_05444__3",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_05444__3"
]... | {
"subset": "construct",
"family": "rm_1706_05444__3",
"problem_key": "near_modular_power_of_three_maximum",
"domain": "combinatorics",
"tier": "research",
"level": "research",
"source": "arXiv:1706.05444",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-05444-3-ell4 |
arxiv-answer-verifiers/construct-rm-1706-06994-8-n32 | For n=32, let [n]={1,...,n}. Exhibit a nonempty set A of distinct
three-element subsets of [n]. Its lower shadow is the set of all two-element
subsets contained in at least one member of A. The upper shadow of that lower
shadow is the set of all three-element subsets of [n] containing at least one
pair in the lower sha... | construct/rm_1706_06994__8 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_06994__8"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
"task": {
"name": "arxiv-answer-verifiers/construct-rm-1706-06994-8-n32",
"version": "0.1.0",
"description": "construct/rm_1706_06994__8",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_06994__8"
]
... | {
"subset": "construct",
"family": "rm_1706_06994__8",
"problem_key": "triple_system_lower_upper_shadow",
"domain": "combinatorics",
"tier": "research",
"level": "medium",
"source": "arXiv:1706.06994",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash_reward": 1,... | [
"environment/Dockerfile",
"environment/instance.json",
"instruction.md",
"solution/solve.sh",
"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-06994-8-n32 |
arxiv-answer-verifiers/construct-rm-1706-07085-1-n10 | Construct a finite simple connected undirected graph G on the vertex set {0,...,9} that is not an even cycle and whose Laplacian simplex is 2-reflexive.
The Laplacian is L[i,i]=degree(i), L[i,j]=-1 when {i,j} is an edge, and L[i,j]=0 otherwise. Define n=10, and let V[i,j]=sum(L[i,k] for k=0,...,j), for 0<=i<n and 0<=j<... | construct/rm_1706_07085__1 | [
"arxiv-answer-verifiers",
"construct",
"rm_1706_07085__1"
] | 1.4 | {
"schema_version": "1.4",
"artifacts": [],
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"version": "0.1.0",
"description": "construct/rm_1706_07085__1",
"authors": [],
"keywords": [
"arxiv-answer-verifiers",
"construct",
"rm_1706_07085__1"
]
... | {
"subset": "construct",
"family": "rm_1706_07085__1",
"problem_key": "noncycle-2-reflexive-laplacian-simplex",
"domain": "discrete_geometry",
"tier": "research",
"level": "research",
"source": "arXiv:1706.07085",
"source_license": "CC-BY-4.0",
"first_solved_by": "z-ai/glm-5.3-flash",
"glm_5_3_flash... | [
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"task.toml",
"tests/check.py",
"tests/instance.json",
"tests/run.py",
"tests/test.sh"
] | tasks/construct-rm-1706-07085-1-n10 |
End of preview. Expand in Data Studio
arxiv-answer-verifiers — hard sample (500 tasks, Harbor)
A 500-task sample of problems that are beyond Qwen3.5-35B-A3B but within reach of a stronger model. It is built from the final snapshot of
amphora/arxiv-answer-verifiers-Envs-harbor (tag snapshot-2026-10-20).
| subset | tasks | how they were chosen | selection |
|---|---|---|---|
| construct | 305 | Qwen3.5-4B and Qwen3.5-35B-A3B each failed 4 attempts; GLM-5.3-Flash solved it (reward 1). One task per family, seeded pick (20261022). |
35b_failed_glm_solved |
| optimize | 64 | same profile, all of them | 35b_failed_glm_solved |
| short_form | 61 | same profile, all of them | 35b_failed_glm_solved |
| proof | 20 | from the 100-task proof pilot: every Qwen3.5 solver (4B, 27B, 122B, 397B) fell short of a full proof, and Qwen3.8-2.4T wrote one (GPT-6.1 Sol grade 1.0). A degenerate task is excluded. | proof_qwen35_failed_qwen38_2p4t_full |
| proof | 50 | no solve labels: random proof tasks to fill to 500, one per paper, following the proof split's domain mix, none from the pilot | proof_random_unlabeled |
tasks/<task_id>/: Harbor task directories as in the snapshot.data/<subset>.parquet: the matching rows ofamphora/arxiv-answer-verifiers-Envsatsnapshot-2026-10-20, plusselection.proof.parquetalso carriespilot_qwen35_best_gpt_scoreandpilot_qwen38_2p4t_gpt_scorefor the 20 pilot proofs.- 500 tasks from 499 families (two optimize tasks share a family) and 472 source papers. Top domains: combinatorics 118, graph theory 100, analysis 88, algebra 57, number theory 43, geometry 33.
Evidence and caveats
- Construct / optimize / short_form (430): the evidence is the Qwen solve ladder (Harbor 0.23.0, terminus-2, up to 4 attempts per model, infrastructure failures not counted) plus GLM-5.3-Flash's single successful attempt from the 2026-10-04 run.
- That GLM result is carried only for tasks unchanged since then.
- A GLM solve means a model actually satisfied the deterministic checker, so these tasks are solvable and their checkers accept a real solution.
- Proofs (70): proof tasks have no ladder labels.
- Only the 20 pilot proofs carry solve evidence: one attempt per model, graded by GPT-6.1 Sol with the task's own
tests/judge.pyrubric. - The other 50 are unlabeled and may be much easier or much harder.
- Proof tasks are graded by an LLM judge: set
OPENROUTER_API_KEYandJUDGE_MODELSin the verifier environment (seetests/judge.py).
- Only the 20 pilot proofs carry solve evidence: one attempt per model, graded by GPT-6.1 Sol with the task's own
- Agent timeouts are wall-clock. Some Qwen attempts ran on A40s, where timeouts are more frequent, so "35B failed" partly reflects serving speed.
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