Datasets:

HerrHruby
/

MR_rl_v4_buffer_warmstart

Tasks:

Text Generation

Tags:

meta-reasoning

reinforcement-learning

trajectory-buffer

License:

Dataset card Data Studio Files Files and versions

xet

Community

Dataset Preview

Duplicate

Split (1)

train

The full dataset viewer is not available (click to read why). Only showing a preview of the rows.

The dataset generation failed

Error code:   DatasetGenerationError
Exception:    CastError
Message:      Couldn't cast
key_algo: string
n_problems: int64
n_trajectories: int64
max_per_problem: int64
max_sampled_step: int64
note: string
termination: string
terminal_answer: string
problem: string
reward: null
key: string
layers: list<item: list<item: struct<direction_text: string, detailed: string, rationale: string, core_resul (... 34 chars omitted)
  child 0, item: list<item: struct<direction_text: string, detailed: string, rationale: string, core_result: string,  (... 22 chars omitted)
      child 0, item: struct<direction_text: string, detailed: string, rationale: string, core_result: string, direction_i (... 10 chars omitted)
          child 0, direction_text: string
          child 1, detailed: string
          child 2, rationale: string
          child 3, core_result: string
          child 4, direction_idx: int64
origin: string
version: int64
to
{'key': Value('string'), 'problem': Value('string'), 'layers': List(List({'direction_text': Value('string'), 'detailed': Value('string'), 'rationale': Value('string'), 'core_result': Value('string'), 'direction_idx': Value('int64')})), 'termination': Value('string'), 'terminal_answer': Value('string'), 'reward': Value('null'), 'origin': Value('string'), 'version': Value('int64')}
because column names don't match
Traceback:    Traceback (most recent call last):
                File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1827, in _prepare_split_single
                  for key, table in generator:
                                    ^^^^^^^^^
                File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 613, in wrapped
                  for item in generator(*args, **kwargs):
                              ~~~~~~~~~^^^^^^^^^^^^^^^^^
                File "/usr/local/lib/python3.14/site-packages/datasets/packaged_modules/json/json.py", line 343, in _generate_tables
                  self._cast_table(pa_table, json_field_paths=json_field_paths),
                  ~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                File "/usr/local/lib/python3.14/site-packages/datasets/packaged_modules/json/json.py", line 132, in _cast_table
                  pa_table = table_cast(pa_table, self.info.features.arrow_schema)
                File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2378, in table_cast
                  return cast_table_to_schema(table, schema)
                File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2306, in cast_table_to_schema
                  raise CastError(
                  ...<3 lines>...
                  )
              datasets.table.CastError: Couldn't cast
              key_algo: string
              n_problems: int64
              n_trajectories: int64
              max_per_problem: int64
              max_sampled_step: int64
              note: string
              termination: string
              terminal_answer: string
              problem: string
              reward: null
              key: string
              layers: list<item: list<item: struct<direction_text: string, detailed: string, rationale: string, core_resul (... 34 chars omitted)
                child 0, item: list<item: struct<direction_text: string, detailed: string, rationale: string, core_result: string,  (... 22 chars omitted)
                    child 0, item: struct<direction_text: string, detailed: string, rationale: string, core_result: string, direction_i (... 10 chars omitted)
                        child 0, direction_text: string
                        child 1, detailed: string
                        child 2, rationale: string
                        child 3, core_result: string
                        child 4, direction_idx: int64
              origin: string
              version: int64
              to
              {'key': Value('string'), 'problem': Value('string'), 'layers': List(List({'direction_text': Value('string'), 'detailed': Value('string'), 'rationale': Value('string'), 'core_result': Value('string'), 'direction_idx': Value('int64')})), 'termination': Value('string'), 'terminal_answer': Value('string'), 'reward': Value('null'), 'origin': Value('string'), 'version': Value('int64')}
              because column names don't match
              
              The above exception was the direct cause of the following exception:
              
              Traceback (most recent call last):
                File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1369, in compute_config_parquet_and_info_response
                  parquet_operations, partial, estimated_dataset_info = stream_convert_to_parquet(
                                                                        ~~~~~~~~~~~~~~~~~~~~~~~~~^
                      builder, max_dataset_size_bytes=max_dataset_size_bytes
                      ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                  )
                  ^
                File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 948, in stream_convert_to_parquet
                  builder._prepare_split(split_generator=splits_generators[split], file_format="parquet")
                  ~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1694, in _prepare_split
                  for job_id, done, content in self._prepare_split_single(
                                               ~~~~~~~~~~~~~~~~~~~~~~~~~~^
                      gen_kwargs=gen_kwargs, job_id=job_id, **_prepare_split_args
                      ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                  ):
                  ^
                File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1880, in _prepare_split_single
                  raise DatasetGenerationError("An error occurred while generating the dataset") from e
              datasets.exceptions.DatasetGenerationError: An error occurred while generating the dataset

Need help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.

key string	problem string	layers list	termination string	terminal_answer string	reward null	origin string	version int64
00be11639838633aa2f836a62dc69d83a29e8885	Fix parameters $0\le \varepsilon<1/9$, $0\le\delta<1$, $F\ge0$, and $a\ge100$, and set \[\nH:=\frac92,\qquad L:=\frac{100}{a}. \]\nAssume \[ \frac{F+16\delta}{1/9-\varepsilon}\le L, \qquad \frac{4-2000/a-100\delta a}{1/9+\varepsilon}\ge H. \tag{*} \] \nLet $u,v,w$ be goods with prices $0<p_u,p_v,p_w\le20$. ...	[ [ { "direction_text": "Develop the forced-allocation core. For a buyer in \$B_s\$, analyze the equality-\$p_s\$-bundle that obtains the maximum utility itself when \$p_s\\le L\$. Use the precise nontrivial utility guarantee and derive the resulting lower bound on its expenditure on \$w\$ required ...	terminated	Let \[ H=\frac92,\qquad L=\frac{100}{a}. \] We prove the two implications separately. --- ## 1. If $p_u,p_v\ge H$, then $p_w\le L$ Assume, toward a contradiction, that $p_u,p_v\ge H$ and $p_w>L>0$. Fix $s\in\{u,v\}$. Then $H\le p_s\le 20$. Since $p_w>L=100/a$, we have \[ a p_w>100. \] For any feasib...	null	offline	-1
3adee23046dc1d5efc82c23df8b546434b0b3a7b	Let $\Delta$ be sufficiently large and define $$ \tau:=\frac1{\log\log\log\Delta},\qquad\nL:=\log^*(1/\tau),\qquad \ell:=L^{1/100},\qquad \beta:=\frac{L}{\ell},\qquad\nq:=\tau^{1/10}. $$\nFix an integer $k$ and positive numbers $a_1,\ldots,a_{k+1}$ such that, for every $i\in[k]$, $$\na_1>\beta,\qquad\na_{i+1}=\frac{e^{...	[ [ { "direction_text": "Investigate the raw selection logic independently. For a fixed ordered edge {u,v}, decompose the event that v can belong to C by its first qualifying trial i, and compute an exact conditional probability expression after imposing r_u≥q. Focus on comparing the probabilities of v's po...	terminated	Let \[ A_i:=\{v\in C_i^+\},\qquad A_0:=C^-. \] We prove the required lower bound. All constants below can be taken absolute; no attempt is made to optimize $10^6$. Fix an edge $\{u,v\}\in E(G)$ with $d_v\le \ell$. Condition on $r_u\ge q$. --- ## 1. The vertex $u$ is never selected If $u\in C_i^+$, the...	null	offline	-1
9e71a84799c82d7545e4151a4a24c46ed80bdb46	Consider a repeated single-item allocation process over T rounds. There are n agents with shares α_1,\dots,α_n \in (0,1) satisfying $\sum_{j=1}^n α_j = 1$. Agent j may bid in at most \[\nB_j=(1+\delta_j^T)α_jT, \qquad \delta_j^T=\sqrt{\frac{6\ln T}{α_jT}}. \]\nIn round t, an arbitrary set $S_t$ of agents with remai...	[ [ { "direction_text": "Develop a dyadic-access lemma specifically tailored to agent i's fixed Bernoulli-α_i activity. Condition on the full public history available just before each eligible round, while retaining i's own anticipated station as the only randomness. Design stopping times and truncations th...	terminated	The claimed inequality is false as stated. Here is a complete counterexample. ## 1. Parameters and allocation rule Let \[ n=2,\qquad \alpha_i=a=\frac1{100},\qquad \alpha_j=1-a=\frac{99}{100},\qquad \bar p=\frac12. \] Use the following admissible allocation rule: - if only $i$ bids, $i$ wins with probabilit...	null	offline	-1
1067e1ce6c13b87c56761f2e05e4f22bdcf7c50f	Let $X$ be a random real symmetric matrix. Assume that there exist constants $\tau>0$ and $\delta\in(0,1)$ such that \[ \mathbb P\big(\operatorname{spec}(X)\cap[-\tau,\tau]=\varnothing\big)\ge 1-\delta, \]\nand that for every Lipschitz function $f:\mathbb R\to\mathbb R$ with Lipschitz constant $L$, \[ \mathbb...	[ [ { "direction_text": "Develop a focused proof of the implication using a spectral-gap decomposition and Hölder/moment comparison. Fix a spectral-gap event A and explore consequences of the event that all eigenvalues lie outside [-τ,τ], including monotonicity of the event under enlarging the excluded inte...	terminated	Let $d$ be the dimension of $X$, and write its eigenvalues as \[ \lambda_1,\dots,\lambda_d. \] Define \[ g(x):=\log\max(\|x\|,\tau),\qquad x\in\mathbb R. \] Then $g$ is $1/\tau$-Lipschitz. Indeed: - if $\|x\|,\|y\|>\tau$, then \[ \|g(x)-g(y)\|=\left\|\log\frac{\|x\|}{\|y\|}\right\| \le \frac{\|\|x\|-\|y\|\|}{\tau} \le ...	null	offline	-1
6d0720d55792a5ceec95ba2cde767fb69fdb9e02	Let c,r∈ℕ, let 0<l≤1/2, and let ≤ be a linear order on [0,1]^2. Let d denote Euclidean distance. For each integer k with 0≤k≤⌊r/c⌋, let Bad_{ck} be a finite set of dyadic squares of side length 2^{-ck}. For every Q∈Bad_{ck} choose a point p_Q∈Q, a strip σ_Q, and two disjoint rectangles R_1(Q),R_2(Q)⊂Q∩σ_Q, each of leng...	[ [ { "direction_text": "Develop a local interval-stability lemma under the stated hypotheses. Fix a dyadic square Q=Bad_{ck} and p_Q, and analyze a strictly linearly ordered finite point configuration whose only ordered-separation violation of R(Q) is the fourth defined condition. Use parallel rays through...	terminated	Assume the intended convention that the marked points are distinct: \[ Q\mapsto p_Q \text{ is injective}. \] This is necessary: otherwise equal representatives cannot be distinguished by the single ordered list $S$. Under this convention, the hypotheses actually imply the slightly stronger bound \[ 2\operatorname{cos...	null	offline	-1
86d54a26c768c0c8ee845dd9d484da354cfa25d6	Let $G$ be a digraph and $T\subseteq V(G)$. A separation of $G$ is a pair $(A,B)$ of vertex sets with $A\cup B=V(G)$ such that either there is no edge from $B\setminus A$ to $A\setminus B$, or there is no edge from $A\setminus B$ to $B\setminus A$. In the first case call $(A,B)$ $A$-positive, and in the second case $B$...	[ [ { "direction_text": "Work through the alternative assumption X1⊆A2∖B2 directly. Let C=A1∩A2 and D=A1∪A2; use positivity to classify internal disjoint orders and orders at most 2 up to the deletion of entrance/tail regions. Focus especially on the role of B2∖A2∖X2 and T∖(C∩D), the possible J=A1∩B2 and I=...	terminated	Let \[ Y_i=B_i\setminus A_i \qquad (i=1,2) \] be the “outside” parts of the two separations. Since $S_i$ has order $2$, \[ \|A_i\cap B_i\|=2. \] Also $X_i\subseteq Y_i$ and $X_i\neq\varnothing$. We only prove the case \[ X_2\subseteq A_1\setminus B_1. \] The other case is exactly symmetric. Define \[ Z=(...	null	offline	-1
69d991c30f9cffa436e4d43bb334fc1d2822f14a	Let \[ \Lambda := \frac{\log\log(1/\epsilon)}{\Delta}, \qquad 0<\epsilon<\tfrac12,\ \Delta>0. \]\nConsider a recursive randomized procedure on weighted directed graphs with positive edge weights. Fix an original edge $e=(u,v)$ of weight $w(e)$, and let $m$ be the number of edges in the original graph. \nAssume th...	[ [ { "direction_text": "Develop a self-contained deterministic chance-compressed lemma tailored to the hypotheses. Model a surviving copy of e by pairs consisting of a recursive call, a phase/round, and the child inherited by e (if any). Use only the stated preprocessing cut rate, per-round conditional rat...	max_layers	The statement is false as written. The listed hypotheses do not imply the claimed bound. One can make a single top-level call run many rounds before producing any inherited child, so the per-round rates $2^i\Lambda w(e)$ can accumulate to a positive constant even though the claimed bound can be made $o(1)$. He...	null	offline	-1
f8c13c6ba599f9202412903ecd2f15cbc0f90a85	"Consider a compact computable manifold M (possibly with boundary). A computational dynamical system(...TRUNCATED)	[[{"direction_text":"Investigate a realization problem for a Turing universal computation by a gener(...TRUNCATED)	max_layers	"# Research-level plan\n\nFix a universal Turing machine \$U\$. Let \$A\$ be the finite alphabet(...TRUNCATED)	null	offline	-1
f33de9fd0f51bdb2356a08edb30a67fe49ecd44d	"Let $G=(V,E)$ be a directed graph with $n=\|V\|$ vertices, $m=\|E\|$ arcs, a designated root $r\\in V$,(...TRUNCATED)	[[{"direction_text":"Develop an explicit generic prereduction theorem candidate. Starting from only (...TRUNCATED)	max_layers	"The requested theorem is not derivable from the stated primitive. The primitive is terminal-sen(...TRUNCATED)	null	offline	-1
7aae385c5e1f660e880ad78a8b357d77dbcfce4e	"Let $n\\ge 4$ and $[n-1]=\\{1,\\ldots,n-1\\}$. For $i\\in[n-1]$, set\n$$a_i(x)=\\binom{n-1}{i-1}x^{(...TRUNCATED)	[[{"direction_text":"Analyze the kernel through directed-shift total positivity of Bernstein basis f(...TRUNCATED)	max_layers	"Let \$m=n-1\$ and write the standard Bernstein basis of degree \$m\$ as\n\n\\[\nB_j(x)=\\binom (...TRUNCATED)	null	offline	-1

End of preview.

MR_rl_v4_buffer_warmstart

Offline warm-start trajectory buffer for the MR_rl_v4 meta-reasoning RL run (verl fully-async GRPO/CISPO). Each row is a BufferedTrajectory (JSONL) providing a frozen MR/E exploration prefix that the rollouter can sample as a starting frontier, amortising E-generation cost during training.

mr_buffer_warmstart.jsonl — one BufferedTrajectory per problem: key, problem, layers, termination, terminal_answer, reward, origin, version. layers holds MR/E exploration only (FA is never a prefix).
mr_buffer_warmstart.jsonl.meta.json — sidecar (key_algo, counts, max_sampled_step, max_per_problem).

Composition

1804 problems, full coverage of the HerrHruby/MR_rl_v4 train set (keyed by sha1-normws-v1 normalized-problem hash, so _buffer_filter_train keeps every problem).
1784 real prefixes (98.9%) + 20 zero-layer seeds (uncovered problems roll fresh, sampled_step=0).
Prefix depth histogram: {1: 34, 2: 83, 3: 436, 4: 1231} (mean 3.48).

Provenance

Prefixes rolled offline with the frozen MR_midtrain_9B_v4_condgen model via inference.mrv4.run_entropy (single-model DP gather), then converted with meta_reason_rl.buffer.build_trajectory_buffer --prompt-set v4 --min-layers 1 --max-sampled-step 4.

Use

export MR_BUFFER_WARMSTART=<path>/mr_buffer_warmstart.jsonl
# meta_reasoning.buffer.enable=true, meta_reasoning.buffer.warmstart_path=$MR_BUFFER_WARMSTART

Downloads last month: 13

Total file size:

99.8 MB

key string	problem string	layers list	termination string	terminal_answer string	reward null	origin string	version int64
00be11639838633aa2f836a62dc69d83a29e8885	Fix parameters \(0\le \varepsilon<1/9\), \(0\le\delta<1\), \(F\ge0\), and \(a\ge100\), and set \[\nH:=\frac92,\qquad L:=\frac{100}{a}. \]\nAssume \[ \frac{F+16\delta}{1/9-\varepsilon}\le L, \qquad \frac{4-2000/a-100\delta a}{1/9+\varepsilon}\ge H. \tag{*} \] \nLet \(u,v,w\) be goods with prices \(0<p_u,p_v,p_w\le20\). ...	[ [ { "direction_text": "Develop the forced-allocation core. For a buyer in \\(B_s\\), analyze the equality-\\(p_s\\)-bundle that obtains the maximum utility itself when \\(p_s\\le L\\). Use the precise nontrivial utility guarantee and derive the resulting lower bound on its expenditure on \\(w\\) required ...	terminated	Let \[ H=\frac92,\qquad L=\frac{100}{a}. \] We prove the two implications separately. --- ## 1. If \(p_u,p_v\ge H\), then \(p_w\le L\) Assume, toward a contradiction, that \(p_u,p_v\ge H\) and \(p_w>L>0\). Fix \(s\in\{u,v\}\). Then \(H\le p_s\le 20\). Since \(p_w>L=100/a\), we have \[ a p_w>100. \] For any feasib...	null	offline	-1
3adee23046dc1d5efc82c23df8b546434b0b3a7b	Let $\Delta$ be sufficiently large and define $$ \tau:=\frac1{\log\log\log\Delta},\qquad\nL:=\log^*(1/\tau),\qquad \ell:=L^{1/100},\qquad \beta:=\frac{L}{\ell},\qquad\nq:=\tau^{1/10}. $$\nFix an integer $k$ and positive numbers $a_1,\ldots,a_{k+1}$ such that, for every $i\in[k]$, $$\na_1>\beta,\qquad\na_{i+1}=\frac{e^{...	[ [ { "direction_text": "Investigate the raw selection logic independently. For a fixed ordered edge {u,v}, decompose the event that v can belong to C by its first qualifying trial i, and compute an exact conditional probability expression after imposing r_u≥q. Focus on comparing the probabilities of v's po...	terminated	Let \[ A_i:=\{v\in C_i^+\},\qquad A_0:=C^-. \] We prove the required lower bound. All constants below can be taken absolute; no attempt is made to optimize \(10^6\). Fix an edge \(\{u,v\}\in E(G)\) with \(d_v\le \ell\). Condition on \(r_u\ge q\). --- ## 1. The vertex \(u\) is never selected If \(u\in C_i^+\), the...	null	offline	-1
9e71a84799c82d7545e4151a4a24c46ed80bdb46	Consider a repeated single-item allocation process over T rounds. There are n agents with shares α_1,\dots,α_n \in (0,1) satisfying \(\sum_{j=1}^n α_j = 1\). Agent j may bid in at most \[\nB_j=(1+\delta_j^T)α_jT, \qquad \delta_j^T=\sqrt{\frac{6\ln T}{α_jT}}. \]\nIn round t, an arbitrary set \(S_t\) of agents with remai...	[ [ { "direction_text": "Develop a dyadic-access lemma specifically tailored to agent i's fixed Bernoulli-α_i activity. Condition on the full public history available just before each eligible round, while retaining i's own anticipated station as the only randomness. Design stopping times and truncations th...	terminated	The claimed inequality is false as stated. Here is a complete counterexample. ## 1. Parameters and allocation rule Let \[ n=2,\qquad \alpha_i=a=\frac1{100},\qquad \alpha_j=1-a=\frac{99}{100},\qquad \bar p=\frac12. \] Use the following admissible allocation rule: - if only \(i\) bids, \(i\) wins with probabilit...	null	offline	-1
1067e1ce6c13b87c56761f2e05e4f22bdcf7c50f	Let \(X\) be a random real symmetric matrix. Assume that there exist constants \(\tau>0\) and \(\delta\in(0,1)\) such that \[ \mathbb P\big(\operatorname{spec}(X)\cap[-\tau,\tau]=\varnothing\big)\ge 1-\delta, \]\nand that for every Lipschitz function \(f:\mathbb R\to\mathbb R\) with Lipschitz constant \(L\), \[ \mathbb...	[ [ { "direction_text": "Develop a focused proof of the implication using a spectral-gap decomposition and Hölder/moment comparison. Fix a spectral-gap event A and explore consequences of the event that all eigenvalues lie outside [-τ,τ], including monotonicity of the event under enlarging the excluded inte...	terminated	Let \(d\) be the dimension of \(X\), and write its eigenvalues as \[ \lambda_1,\dots,\lambda_d. \] Define \[ g(x):=\log\max(\|x\|,\tau),\qquad x\in\mathbb R. \] Then \(g\) is \(1/\tau\)-Lipschitz. Indeed: - if \(\|x\|,\|y\|>\tau\), then \[ \|g(x)-g(y)\|=\left\|\log\frac{\|x\|}{\|y\|}\right\| \le \frac{\|\|x\|-\|y\|\|}{\tau} \le ...	null	offline	-1
6d0720d55792a5ceec95ba2cde767fb69fdb9e02	Let c,r∈ℕ, let 0<l≤1/2, and let ≤ be a linear order on [0,1]^2. Let d denote Euclidean distance. For each integer k with 0≤k≤⌊r/c⌋, let Bad_{ck} be a finite set of dyadic squares of side length 2^{-ck}. For every Q∈Bad_{ck} choose a point p_Q∈Q, a strip σ_Q, and two disjoint rectangles R_1(Q),R_2(Q)⊂Q∩σ_Q, each of leng...	[ [ { "direction_text": "Develop a local interval-stability lemma under the stated hypotheses. Fix a dyadic square Q=Bad_{ck} and p_Q, and analyze a strictly linearly ordered finite point configuration whose only ordered-separation violation of R(Q) is the fourth defined condition. Use parallel rays through...	terminated	Assume the intended convention that the marked points are distinct: \[ Q\mapsto p_Q \text{ is injective}. \] This is necessary: otherwise equal representatives cannot be distinguished by the single ordered list \(S\). Under this convention, the hypotheses actually imply the slightly stronger bound \[ 2\operatorname{cos...	null	offline	-1
86d54a26c768c0c8ee845dd9d484da354cfa25d6	Let $G$ be a digraph and $T\subseteq V(G)$. A separation of $G$ is a pair $(A,B)$ of vertex sets with $A\cup B=V(G)$ such that either there is no edge from $B\setminus A$ to $A\setminus B$, or there is no edge from $A\setminus B$ to $B\setminus A$. In the first case call $(A,B)$ $A$-positive, and in the second case $B$...	[ [ { "direction_text": "Work through the alternative assumption X1⊆A2∖B2 directly. Let C=A1∩A2 and D=A1∪A2; use positivity to classify internal disjoint orders and orders at most 2 up to the deletion of entrance/tail regions. Focus especially on the role of B2∖A2∖X2 and T∖(C∩D), the possible J=A1∩B2 and I=...	terminated	Let \[ Y_i=B_i\setminus A_i \qquad (i=1,2) \] be the “outside” parts of the two separations. Since \(S_i\) has order \(2\), \[ \|A_i\cap B_i\|=2. \] Also \(X_i\subseteq Y_i\) and \(X_i\neq\varnothing\). We only prove the case \[ X_2\subseteq A_1\setminus B_1. \] The other case is exactly symmetric. Define \[ Z=(...	null	offline	-1
69d991c30f9cffa436e4d43bb334fc1d2822f14a	Let \[ \Lambda := \frac{\log\log(1/\epsilon)}{\Delta}, \qquad 0<\epsilon<\tfrac12,\ \Delta>0. \]\nConsider a recursive randomized procedure on weighted directed graphs with positive edge weights. Fix an original edge \(e=(u,v)\) of weight \(w(e)\), and let \(m\) be the number of edges in the original graph. \nAssume th...	[ [ { "direction_text": "Develop a self-contained deterministic chance-compressed lemma tailored to the hypotheses. Model a surviving copy of e by pairs consisting of a recursive call, a phase/round, and the child inherited by e (if any). Use only the stated preprocessing cut rate, per-round conditional rat...	max_layers	The statement is false as written. The listed hypotheses do not imply the claimed bound. One can make a single top-level call run many rounds before producing any inherited child, so the per-round rates \(2^i\Lambda w(e)\) can accumulate to a positive constant even though the claimed bound can be made \(o(1)\). He...	null	offline	-1
f8c13c6ba599f9202412903ecd2f15cbc0f90a85	"Consider a compact computable manifold M (possibly with boundary). A computational dynamical system(...TRUNCATED)	[[{"direction_text":"Investigate a realization problem for a Turing universal computation by a gener(...TRUNCATED)	max_layers	"# Research-level plan\n\nFix a universal Turing machine \\(U\\). Let \\(A\\) be the finite alphabet(...TRUNCATED)	null	offline	-1
f33de9fd0f51bdb2356a08edb30a67fe49ecd44d	"Let $G=(V,E)$ be a directed graph with $n=\|V\|$ vertices, $m=\|E\|$ arcs, a designated root $r\\in V$,(...TRUNCATED)	[[{"direction_text":"Develop an explicit generic prereduction theorem candidate. Starting from only (...TRUNCATED)	max_layers	"The requested theorem is not derivable from the stated primitive. The primitive is terminal-sen(...TRUNCATED)	null	offline	-1
7aae385c5e1f660e880ad78a8b357d77dbcfce4e	"Let $n\\ge 4$ and $[n-1]=\\{1,\\ldots,n-1\\}$. For $i\\in[n-1]$, set\n$$a_i(x)=\\binom{n-1}{i-1}x^{(...TRUNCATED)	[[{"direction_text":"Analyze the kernel through directed-shift total positivity of Bernstein basis f(...TRUNCATED)	max_layers	"Let \\(m=n-1\\) and write the standard Bernstein basis of degree \\(m\\) as\n\n\\[\nB_j(x)=\\binom (...TRUNCATED)	null	offline	-1

MR_rl_v4_buffer_warmstart

Contents

Composition

Provenance

Use