Datasets:

HerrHruby
/

soda_fa0_luna_sample

problem_id stringclasses 8 values	sample_idx int64 0 7	source stringclasses 1 value	category stringclasses 2 values	problem stringclasses 8 values	thinking stringclasses 10 values	answer stringclasses 10 values	luna_score float64 0 8 ⌀	luna_max_score float64 7 8 ⌀	luna_reward float64 0 1	luna_concluded bool 2 classes	luna_auto_reason stringclasses 1 value	luna_rubric stringclasses 10 values	luna_explanation stringclasses 10 values	problem_luna_mean float64 0.13 0.77	problem_oss_mean float64 0.13 0.75	termination_reason stringclasses 1 value	wall_time_s float64 969 21.7k
train:1003	1	SODA	proof_writing	Let I = (A, B, E, ΣA, ΣB, {R_e}_{e\in E}) be a regular bipartite LabelCover instance of input size N, meaning that every vertex of the underlying bipartite graph has the same degree. For each edge e = (a,b), the constraint R_e is a subset of ΣA × ΣB. An ordinary labeling assigns to each a \in A a label in ΣA and to ea...	We need to prove two parts: 1. A statement about multilabelings: if a multilabeling φ has cost ≤ N^η (\|A\|+\|B\|) and covers at least N^{-η} fraction of edges, then for sufficiently large N there exists an ordinary labeling that satisfies at least N^{-8η} fraction of edges. 2. A hardness reduction: assuming a certain ha...	1. Multi‑labeling to ordinary labeling Let * $M=\|A\|+\|B\|$ – the total number of vertices, * $d$ the common degree of the regular bipartite graph, so $\|E\|=d\|A\|=dM/2$, * $\varphi$ a multilabelling with $\operatorname{cost}(\varphi)=\sum_{v}\| \varphi(v)\|\le N^{\eta}M$, * $C_0\subseteq E$ the set ...	7	7	1	true	null	1. Random-rounding setup — 1/1 point. The candidate defines independent uniform sampling from each $\varphi(v)$ and derives the correct per-edge survival probability $1/(\|\varphi(a)\|\,\|\varphi(b)\|)$. Strictly, it does not explicitly handle vertices with empty label sets, but such vertices can be assigned a...	1. Random-rounding setup — 1/1 point. The candidate defines independent uniform sampling from each $\varphi(v)$ and derives the correct per-edge survival probability $1/(\|\varphi(a)\|\,\|\varphi(b)\|)$. Strictly, it does not explicitly handle vertices with empty label sets, but such vertices can be assigned a...	0.5982	0.6607	max_layers	20,053.617152
train:1004	4	SODA	proof_writing	Let $G=(V,E)$ be a DAG on $n$ vertices, and let $E^T$ denote the transitive closure of $G$. Write $m:=\alpha_D d$, and assume $m\ge 18$ is divisible by $4$. Suppose there are directed paths $P_1,\dots,P_k$ in $G$ with \[ k\le \frac{9n}{m}, \] a map $a:V\to [k]$ such that $v\in P_{a(v)}$ for every $v\in V$, and a set $...	We need to understand the problem and provide a solution. The problem is about a DAG G, its transitive closure E^T, and certain parameters. We have m = α_D d, m >= 18 and divisible by 4. There are directed paths P_1,...,P_k in G with k <= 9n/m, and a map a: V -> [k] such that v ∈ P_{a(v)} for every v. There is a set H ...	We give the complete solution in two parts: the bound on $\|F_1\|$ and the high‑probability distance estimate. --- ### 1. Size of $F_1$ The set $H$ has size $O(n\log n)$. The random multisets are \[ \|R\| = r = \left\lceil C\frac{n\log n}{\beta}\right\rceil,\qquad \|\mathcal Q\| = q = \left\lceil C\frac{n\log n...	3	7	0.428571	true	null	1. Size bound — 1/1. The candidate correctly observes that $H$ contributes $O(n\log n)$ edges and that each sampled pair $(u,P)$ contributes at most two edges, giving $\|F_1\|\le \|H\|+2rq$. Substituting $m=\alpha_Dd$ yields the required asymptotic bound. 2. Intersection lemma — 0/2. The candidate do...	1. Size bound — 1/1. The candidate correctly observes that $H$ contributes $O(n\log n)$ edges and that each sampled pair $(u,P)$ contributes at most two edges, giving $\|F_1\|\le \|H\|+2rq$. Substituting $m=\alpha_Dd$ yields the required asymptotic bound. 2. Intersection lemma — 0/2. The candidate do...	0.4196	0.4107	max_layers	19,809.103438
train:10	1	SODA	proof_writing	Rectangle Covering Problem (RCP). An instance consists of a finite set $\mathcal R$ of axis-parallel rectangles with integer $x$-coordinates; each rectangle $Q\in\mathcal R$ has $\operatorname{left}(Q)<\operatorname{right}(Q)$, cost $c(Q)>0$, and weight $p(Q)>0$. The rectangles are partitioned into horizont...	We need to prove that for every ε>0, arbitrary RCP instances admit a (1+ε)α-approximation algorithm with running time 2^{((K/ε) log n log p_max)^{O(1)}}. The given black-box guarantee is that for any instance where rectangle costs assume at most M distinct values, there is an α-approximation algorithm with running time...	We describe a reduction that uses the assumed black‑box to obtain the claimed algorithm. 1. Compressing the coordinates. For every distinct $x$-coordinate that appears as a left or right endpoint, replace it by its rank (the position of that coordinate in the increasing list of all distinct coordinates). This...	0	7	0	true	null	1. Coordinate compression — 0/1. The candidate compresses only rectangle endpoint coordinates and omits the $x$-coordinates of rays. Thus ray–rectangle incidences need not be preserved, contrary to the claim that “it preserves every ray-intersection relation.” The bound $X\le 2n$ is therefore not justified...	1. Coordinate compression — 0/1. The candidate compresses only rectangle endpoint coordinates and omits the $x$-coordinates of rays. Thus ray–rectangle incidences need not be preserved, contrary to the claim that “it preserves every ray-intersection relation.” The bound $X\le 2n$ is therefore not justified...	0.125	0.125	max_layers	969.275255
train:1028	1	SODA	proof_strategy	Let A = {A_1, ..., A_r} \subseteq \mathbb{R}^{n \times n} be PSD matrices. For \varepsilon \in [0,1), an \varepsilon-sparsifier of a subset T \subseteq [r] is a weight function \mu : T \to \mathbb{R}_{\ge 0} such that \[ (1-\varepsilon)\sum_{i\in T} A_i \preccurlyeq \sum_{i\in T} \mu_i A_i \preccurlyeq (1+\varepsilon)\...	We need to produce a high-level but mathematically concrete proof strategy for the claimed lower bound: existence of a subset T of size N^*_ε(A) - 1 that admits no non-trivial ε-sparsifier. Essentially, we need to show that if T has that size, it cannot have a non-trivial sparsifier. This is reminiscent of a threshold ...	We prove the claim by selecting a subset $T\subseteq [r]$ of size $N^_\varepsilon(A)-1$ that lacks the property forced by the definition of $N^_\varepsilon(A)$, converting that property into spectral witnesses, and then using any hypothetical non‑trivial $\varepsilon$-sparsifier to derive incompatible bounds ...	8	8	1	true	null	1. Main bottleneck identified — 2/2. The candidate explicitly identifies the need to convert the failure of the threshold condition into “spectral witnesses” and then derive “incompatible bounds on its weights,” rather than treating the threshold definition as directly implying non-sparsifiability. 2. **Corre...	1. Main bottleneck identified — 2/2. The candidate explicitly identifies the need to convert the failure of the threshold condition into “spectral witnesses” and then derive “incompatible bounds on its weights,” rather than treating the threshold definition as directly implying non-sparsifiability. 2. **Corre...	0.5469	0.6719	max_layers	14,262.371313
train:1028	0	SODA	proof_strategy	Let A = {A_1, ..., A_r} \subseteq \mathbb{R}^{n \times n} be PSD matrices. For \varepsilon \in [0,1), an \varepsilon-sparsifier of a subset T \subseteq [r] is a weight function \mu : T \to \mathbb{R}_{\ge 0} such that \[ (1-\varepsilon)\sum_{i\in T} A_i \preccurlyeq \sum_{i\in T} \mu_i A_i \preccurlyeq (1+\varepsilon)\...	We are given a problem. We need to produce a high-level but mathematically concrete proof strategy for a lower bound claim. The claim: There exists a subset T ⊆ [r] of size N*_ε(A)-1 that admits no non-trivial ε-sparsifier. We are asked to provide a proof strategy, not a complete formal proof. The strategy should speci...	We are to prove the lower bound: Claim. There exists a subset $T\subseteq[r]$ of size $N^_\varepsilon(A)-1$ that admits no non‑trivial $\varepsilon$-sparsifier. Proof strategy.* --- ### 1. Choice of the set $T$ Let $N^*_\varepsilon(A)=N(\alpha_\varepsilon;A)$ be the connectivity threshold. By def...	3	8	0.375	true	null	1. Main bottleneck identified — 2/2. The response recognizes that the key issue is converting the threshold condition into a spectral obstruction to a proper-support two-sided approximation, and it attempts to use both Loewner inequalities rather than relying on counting alone. 2. **Correct extremal subset ch...	1. Main bottleneck identified — 2/2. The response recognizes that the key issue is converting the threshold condition into a spectral obstruction to a proper-support two-sided approximation, and it attempts to use both Loewner inequalities rather than relying on counting alone. 2. **Correct extremal subset ch...	0.5469	0.6719	max_layers	14,378.23326
train:1077	0	SODA	proof_strategy	Suppose you are given the following static tree-embedding scheme for a point set P in R^d. For each point x in P, choose an independent random priority π(x) in [0,1]. For each level i, there is a scale w_i, a random radius r_i in [w_i/(4Γ), w_i/(2Γ)], and a hash function φ_i whose buckets have diameter at most w_i/2. ...	We need to design a dynamic data structure maintaining an implicit tree whose restriction to the current active set has distortion O(Γ log Γ log n). The given static scheme defines an HST (hierarchical tree of nets) based on random priorities and approximate balls. The labels ℓ_p^(i) = min priority among points in the ...	Dynamic representation. For each level $i$ we maintain a dictionary $\operatorname{min}_i$ that stores the point with the smallest priority among the points currently in each bucket $b$ (the bucket IDs are computed from $\phi_i$). Within each bucket we keep a data structure (e.g. a min-heap or a doubly‑l...	6	7	0.857143	true	null	1. Main obstacle identified — 2/2. The answer recognizes that changing a bucket minimum can change labels of existing points, and that deletions can invalidate minima: it explicitly discusses labels becoming stale and labels “that change because the minimum of one of their intersecting buckets changed.” 2. **...	1. Main obstacle identified — 2/2. The answer recognizes that changing a bucket minimum can change labels of existing points, and that deletions can invalidate minima: it explicitly discusses labels becoming stale and labels “that change because the minimum of one of their intersecting buckets changed.” 2. **...	0.7679	0.75	max_layers	20,950.957999
train:100	0	SODA	proof_strategy	Consider the infinite balls-into-bins process with insertions and random deletions. For a configuration $\mathbf{x}$, a base height $h$, and a level $\ell$, let $m_\ell^{(h)}(\mathbf{x})$ denote the number of balls at height at least $h+\ell$. There are thresholds $\alpha_0,\alpha_1,\dots,\alpha_{\ell^*+1}$...	We need to produce a proof blueprint for showing that the probability that over the first T=n^4 steps, the configuration remains hvalid (i.e., all levels valid) is at least 1 - n^{-1}. The problem statement gives some definitions and facts. We need to outline a plan converting the global failure event into tractable pa...	Proof blueprint Let $T=n^{4}$ and denote the configuration at time $t$ by $\mathbf{x}(t)$. For a level $\ell$ and base height $h$, write \[ m_{\ell}^{(h)}(\mathbf{x})=\|\{\text{balls whose height}\ge h+\ell\}\|. \] The thresholds $\alpha_{0},\dots,\alpha_{\ell^{}+1}$ satisfy \(\alpha_{\ell^{}}=\Th...	2.5	7	0.357143	true	null	1. Identifies the main obstacle — 1/2 points. The candidate recognizes that the argument must control the process over the entire $T=n^4$ horizon and account for the moving base height. However, it does not clearly identify the key pathwise issue of rare safe-to-invalid excursions or explain why fixed-time c...	1. Identifies the main obstacle — 1/2 points. The candidate recognizes that the argument must control the process over the entire $T=n^4$ horizon and account for the moving base height. However, it does not clearly identify the key pathwise issue of rare safe-to-invalid excursions or explain why fixed-time c...	0.3393	0.625	max_layers	1,205.384694
train:1003	0	SODA	proof_writing	Let I = (A, B, E, ΣA, ΣB, {R_e}_{e\in E}) be a regular bipartite LabelCover instance of input size N, meaning that every vertex of the underlying bipartite graph has the same degree. For each edge e = (a,b), the constraint R_e is a subset of ΣA × ΣB. An ordinary labeling assigns to each a \in A a label in ΣA and to ea...		We are given a problem. We need to prove two statements. Let's restate: We have a regular bipartite LabelCover instance I = (A, B, E, Σ_A, Σ_B, {R_e}_{e∈E}) of input size N, meaning that every vertex of the underlying bipartite graph has the same degree. The input size N presumably is something like total description ...	0	null	0	false	no_think_close			0.5982	0.6607	max_layers	20,279.364867
train:1131	7	SODA	proof_writing	Consider the weighted k-server problem on a uniform metric, where moving server i to a different point costs w_i, and assume w_i \ge 2w_{i-1} \qquad (i=2,\dots,k). Fix l \in \{2,\dots,k\}, and set t:=d_l-1. Assume the following recursive data are available. - For every H with \|H\|\le k-l and every ((l-1),H)-multiphas...	We need to prove that we can define a randomized online algorithm A_l with the stated properties, using the given recursive data and the existence of strategy B. The problem is about weighted k-server on a uniform metric. The definitions are quite complex, involving phases, multiplicase, etc. We need to construct A_l b...	Solution. We are given the recursive data for levels $l$ and $l-1$ and a $((l-1))$-multiphase strategy $B$ with the stated properties. We construct a randomized online algorithm $A_l$ for level $l$ as follows. Let $H$ be a set with $\|H\|\le k-l$ and assume the initial configuration in which ever...	null	null	0	false	null	1. Explore-stage setup — 2/2. The candidate correctly runs $B$ with parameter $H$, identifies the multiphase $Q_0$ served when $B$ terminates, and extracts $\mathrm{Crit}(Q_0)=\{p_1,\ldots,p_t\}$. 2. Exploit-stage construction — 0/2. The required construction needs one copy $B^p$ for every \(p\...	1. Explore-stage setup — 2/2. The candidate correctly runs $B$ with parameter $H$, identifies the multiphase $Q_0$ served when $B$ terminates, and extracts $\mathrm{Crit}(Q_0)=\{p_1,\ldots,p_t\}$. 2. Exploit-stage construction — 0/2. The required construction needs one copy $B^p$ for every \(p\...	0.3469	0.3929	max_layers	14,872.967177
train:3527	0	SODA	proof_writing	Let T be the recursive tree of subproblems for a load-1 open-addressed hash table. For a subproblem s: - s is in a weakly common-case configuration if 1. for each child c of s, every element hashing to c or a descendant of c is stored in the buffers of c and its descendants; and 2. the only elements stored in desc...	We need to solve the problem. The description: T is the recursive tree of subproblems for a load-1 open-addressed hash table. Definitions given. Then we are given facts: (1) whether s is strongly feasible can be determined in O(\|T_s\| + log log n). (2) If s is weakly feasible and all elements hashing to s or descendants...	We prove that such an in‑place procedure exists. The construction is recursive and works on the whole subtree $T'$ rooted at $r$. Notation and observations. For a node $s$ let $T_s$ be the subtree rooted at $s$ (the set of buffers of all its descendants including $s$ itself). \$ Fact 1: For any s...	3	8	0.375	true	null	1. Recursive case split at root — 1/2 points. The candidate proposes a recursive repair and distinguishes strongly feasible nodes from nodes whose bits are not marked feasible, but it does not give the required explicit root-level non-feasible case with a clear free-slot move and subtree-wide bit update. 2. *...	1. Recursive case split at root — 1/2 points. The candidate proposes a recursive repair and distinguishes strongly feasible nodes from nodes whose bits are not marked feasible, but it does not give the required explicit root-level non-feasible case with a clear free-slot move and subtree-wide bit update. 2. *...	0.2969	0.4219	max_layers	21,704.283045

SODA FA-0 + Luna — 10-row format sample

A tiny, public 10-row sample to illustrate the data format of the (private) full dataset HerrHruby/soda_fa0_luna. Use it to understand the schema before requesting/loading the full set — it is not meant for training or evaluation.

Each row is one FA-0 rollout (a model's direct final answer, no exploration) on a SODA-2026 proof problem, graded by GPT-5.6-Luna against a per-problem rubric.

⚠️ Long text fields are truncated in this sample. thinking, answer, luna_rubric, luna_explanation, and problem are cut (with a … [truncated for sample — full text (N chars) …] marker) so the file stays small and readable. The full dataset has them untruncated — real thinking/answer can be tens of thousands of characters (and when a rollout never closes </think>, the whole runaway generation lands in answer).

These 10 rows cover

both categories — 6 proof_writing, 4 proof_strategy
both rubric maxes — luna_max_score 7 and 8
8 graded (luna_concluded=True) + 2 not — one no_think_close auto-zero and one score-parse failure (rubric text present, numeric score null)
a reward spread: luna_reward ∈ {0.0 (incl. a graded zero), 0.357, 0.375, 0.429, 0.857, 1.0}

Columns

column	type	description
`problem_id`	str	`train:<i>` — stable id (8 samples share it in the full set)
`sample_idx`	int	FA-0 sample index
`source`	str	`SODA`
`category`	str	`proof_writing` or `proof_strategy`
`problem`	str	the problem statement
`thinking`	str	model reasoning (the `<think>` content)
`answer`	str	the final answer / written proof (after `</think>`)
`luna_score`	int?	rubric points earned (null on parse-failure)
`luna_max_score`	int?	rubric max, 7 or 8 (null if auto-zeroed)
`luna_reward`	float	normalized `[0,1]` = `luna_score / luna_max_score`, else `0.0`
`luna_concluded`	bool	True if the answer closed `</think>` and got a real rubric grade
`luna_auto_reason`	str?	`no_think_close` when auto-zeroed (else null)
`luna_rubric`	str	Luna's rubric-by-rubric grading (`raw_judge`)
`luna_explanation`	str	Luna's free-form explanation (distinct from the rubric)
`problem_luna_mean`	float	mean `luna_reward` over the problem's 8 samples (full set)
`problem_oss_mean`	float	per-problem gpt-oss-120b mean (gated selection into the Luna pool)
`n_layers_completed`	int	0 (FA-0)
`termination_reason`	str	e.g. `max_layers`
`wall_time_s`	float	rollout wall-clock

Load

from datasets import load_dataset
ds = load_dataset("HerrHruby/soda_fa0_luna_sample", split="train")   # public, 10 rows
print(ds[0])

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