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The dataset generation failed because of a cast error
Error code:   DatasetGenerationCastError
Exception:    DatasetGenerationCastError
Message:      An error occurred while generating the dataset

All the data files must have the same columns, but at some point there are 1 new columns ({'prompt'}) and 4 missing columns ({'pred_len', 'out_len', 'true_len', 'text'}).

This happened while the json dataset builder was generating data using

hf://datasets/dungnv/qwen36-27b-length-traces/benchmark_prompts.json (at revision a198ac9438475bd736dd76b898942b72789add61), ['hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_olympiad.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_short.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/benchmark_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/collect_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36_uncapped.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/longcot600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/manifest_L32.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/olympiad600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_aug.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_base.json'], ['hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_olympiad.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_short.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/benchmark_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/collect_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36_uncapped.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/longcot600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/manifest_L32.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/olympiad600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_aug.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_base.json']

Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)
Traceback:    Traceback (most recent call last):
                File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1848, in _prepare_split_single
                  writer.write_table(table)
                  ~~~~~~~~~~~~~~~~~~^^^^^^^
                File "/usr/local/lib/python3.14/site-packages/datasets/arrow_writer.py", line 765, in write_table
                  self._write_table(pa_table, writer_batch_size=writer_batch_size)
                  ~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
                File "/usr/local/lib/python3.14/site-packages/datasets/arrow_writer.py", line 773, in _write_table
                  pa_table = table_cast(pa_table, self._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
              prompt: string
              dataset: string
              to
              {'text': Value('string'), 'true_len': Value('int64'), 'out_len': Value('int64'), 'dataset': Value('string'), 'pred_len': Value('int64')}
              because column names don't match
              
              During handling of the above exception, another exception occurred:
              
              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 1850, in _prepare_split_single
                  raise DatasetGenerationCastError.from_cast_error(
                  ...<4 lines>...
                  )
              datasets.exceptions.DatasetGenerationCastError: An error occurred while generating the dataset
              
              All the data files must have the same columns, but at some point there are 1 new columns ({'prompt'}) and 4 missing columns ({'pred_len', 'out_len', 'true_len', 'text'}).
              
              This happened while the json dataset builder was generating data using
              
              hf://datasets/dungnv/qwen36-27b-length-traces/benchmark_prompts.json (at revision a198ac9438475bd736dd76b898942b72789add61), ['hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_olympiad.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_short.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/benchmark_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/collect_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36_uncapped.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/longcot600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/manifest_L32.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/olympiad600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_aug.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_base.json'], ['hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_olympiad.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/bench_short.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/benchmark_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/collect_prompts.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/layer_sweep_qwen36_uncapped.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/longcot600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/manifest_L32.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/olympiad600_prompts.jsonl', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_aug.json', 'hf://datasets/dungnv/qwen36-27b-length-traces@a198ac9438475bd736dd76b898942b72789add61/split_base.json']
              
              Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)

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text
string
true_len
int64
out_len
int64
dataset
string
pred_len
int64
<|im_start|>user $$ \begin{array}{ll} g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m...
24,313
24,313
olympiad
10,011
<|im_start|>user ## Real Expansion Consider a "real" monatomic gas consisting of $N$ atoms of negligible volume and mass $m$ in equilibrium inside a closed cubical container of volume $V$. In this "real" gas, the attractive forces between atoms is small but not negligible. Because these atoms have negligible volume,...
20,205
20,205
olympiad
22,377
<|im_start|>user Suppose that $n>5$ and that the numbers $t_{1}, t_{2}, t_{3}, \ldots, t_{n-2}, t_{n-1}, t_{n}$ form an arithmetic sequence with $n$ terms. If $t_{3}=5, t_{n-2}=95$, and the sum of all $n$ terms is 1000 , what is the value of $n$ ? (An arithmetic sequence is a sequence in which each term after the firs...
8,885
8,885
olympiad
5,889
<|im_start|>user $$ \begin{array}{ll} g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m...
7,676
7,676
olympiad
8,384
<|im_start|>user Points $A, R, M$, and $L$ are consecutively the midpoints of the sides of a square whose area is 650. The coordinates of point $A$ are $(11,5)$. If points $R, M$, and $L$ are all lattice points, and $R$ is in Quadrant I, compute the number of possible ordered pairs $(x, y)$ of coordinates for point $R$...
30,089
30,089
olympiad
25,261
<|im_start|>user Consider the sequence $t_{1}=1, t_{2}=-1$ and $t_{n}=\left(\frac{n-3}{n-1}\right) t_{n-2}$ where $n \geq 3$. What is the value of $t_{1998}$ ? Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
7,964
7,964
olympiad
14,590
<|im_start|>user ## Real Expansion Consider a "real" monatomic gas consisting of $N$ atoms of negligible volume and mass $m$ in equilibrium inside a closed cubical container of volume $V$. In this "real" gas, the attractive forces between atoms is small but not negligible. Because these atoms have negligible volume,...
18,148
18,148
olympiad
22,960
<|im_start|>user Compute the least positive integer $n$ such that the set of angles $$ \left\{123^{\circ}, 246^{\circ}, \ldots, n \cdot 123^{\circ}\right\} $$ contains at least one angle in each of the four quadrants. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assista...
23,930
23,930
olympiad
22,687
<|im_start|>user What is the value of $x$ such that $\log _{2}\left(\log _{2}(2 x-2)\right)=2$ ? Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
1,018
1,018
olympiad
3,016
<|im_start|>user Define $\log ^{*}(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to 1 . For example, $\log ^{*}(1000)=2$ since $\log 1000=3$ and $\log (\log 1000)=\log 3=0.477 \ldots \leq 1$. Let $a$ be the smallest integer such that $\log ...
25,883
25,883
olympiad
14,207
<|im_start|>user Define a reverse prime to be a positive integer $N$ such that when the digits of $N$ are read in reverse order, the resulting number is a prime. For example, the numbers 5, 16, and 110 are all reverse primes. Compute the largest two-digit integer $N$ such that the numbers $N, 4 \cdot N$, and $5 \cdot N...
25,163
25,163
olympiad
26,838
<|im_start|>user In non-right triangle $A B C$, distinct points $P, Q, R$, and $S$ lie on $\overline{B C}$ in that order such that $\angle B A P \cong \angle P A Q \cong \angle Q A R \cong \angle R A S \cong \angle S A C$. Given that the angles of $\triangle A B C$ are congruent to the angles of $\triangle A P Q$ in so...
24,259
24,259
olympiad
39,803
<|im_start|>user Determine all real values of $x$ for which $\log _{2}\left(2^{x-1}+3^{x+1}\right)=2 x-\log _{2}\left(3^{x}\right)$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
13,302
13,302
olympiad
10,553
<|im_start|>user The equation $2^{x+2} 5^{6-x}=10^{x^{2}}$ has two real solutions. Determine these two solutions. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
15,078
15,078
olympiad
6,582
<|im_start|>user In equilateral hexagon $A B C D E F, \mathrm{~m} \angle A=2 \mathrm{~m} \angle C=2 \mathrm{~m} \angle E=5 \mathrm{~m} \angle D=10 \mathrm{~m} \angle B=10 \mathrm{~m} \angle F$, and diagonal $B E=3$. Compute $[A B C D E F]$, that is, the area of $A B C D E F$. Please reason step by step, and put your fi...
32,448
32,448
olympiad
29,510
<|im_start|>user For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
14,555
14,555
olympiad
10,111
<|im_start|>user Let $T=16$. An isosceles trapezoid has an area of $T+1$, a height of 2 , and the shorter base is 3 units shorter than the longer base. Compute the sum of the length of the shorter base and the length of one of the congruent sides. Please reason step by step, and put your final answer within \boxed{}.<|...
2,624
2,624
olympiad
4,529
<|im_start|>user Let $T=688$. Let $a$ be the least nonzero digit in $T$, and let $b$ be the greatest digit in $T$. In square $N O R M, N O=b$, and points $P_{1}$ and $P_{2}$ lie on $\overline{N O}$ and $\overline{O R}$, respectively, so that $O P_{1}=O P_{2}=a$. A circle centered at $O$ has radius $a$, and quarter-circ...
19,931
19,931
olympiad
20,476
<|im_start|>user Let $10^{y}$ be the product of all real numbers $x$ such that $\log x=\frac{3+\left\lfloor(\log x)^{2}\right\rfloor}{4}$. Compute $y$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
19,509
19,509
olympiad
18,829
<|im_start|>user Particles from the Sun ${ }^{1}$ Photons from the surface of the Sun and neutrinos from its core can tell us about solar temperatures and also confirm that the Sun shines because of nuclear reactions. Throughout this problem, take the mass of the Sun to be $M_{\odot}=2.00 \times 10^{30} \mathrm{~kg}...
10,259
10,259
olympiad
12,701
<|im_start|>user All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual ...
19,759
19,759
olympiad
13,599
<|im_start|>user Diagonal $W Y$ of square $W X Y Z$ has slope 2. Determine the sum of the slopes of $W X$ and $X Y$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
19,806
19,806
olympiad
11,297
<|im_start|>user If $2 \leq x \leq 5$ and $10 \leq y \leq 20$, what is the maximum value of $15-\frac{y}{x}$ ? Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
4,383
4,383
olympiad
3,036
<|im_start|>user Let $T\neq 0$. Suppose that $a, b, c$, and $d$ are real numbers so that $\log _{a} c=\log _{b} d=T$. Compute $$ \frac{\log _{\sqrt{a b}}(c d)^{3}}{\log _{a} c+\log _{b} d} $$ Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
10,484
10,484
olympiad
14,565
<|im_start|>user Let $T=3$. In $\triangle A B C, A C=T^{2}, \mathrm{~m} \angle A B C=45^{\circ}$, and $\sin \angle A C B=\frac{8}{9}$. Compute $A B$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
10,493
10,493
olympiad
9,579
<|im_start|>user A list $a_{1}, a_{2}, a_{3}, a_{4}$ of rational numbers is defined so that if one term is equal to $r$, then the next term is equal to $1+\frac{1}{1+r}$. For example, if $a_{3}=\frac{41}{29}$, then $a_{4}=1+\frac{1}{1+(41 / 29)}=\frac{99}{70}$. If $a_{3}=\frac{41}{29}$, what is the value of $a_{1} ?$ P...
9,730
9,730
olympiad
11,377
<|im_start|>user 4. A complex dance In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number $i$, strange...
24,669
24,669
olympiad
10,544
<|im_start|>user ## Toffee Pudding A box of mass $m$ is at rest on a horizontal floor. The coefficients of static and kinetic friction between the box and the floor are $\mu_{0}$ and $\mu$ (less than $\mu_{0}$ ), respectively. One end of a spring with spring constant $k$ is attached to the right side of the box, and...
24,704
24,704
olympiad
18,738
<|im_start|>user David builds a circular table; he then carves one or more positive integers into the table at points equally spaced around its circumference. He considers two tables to be the same if one can be rotated so that it has the same numbers in the same positions as the other. For example, a table with the nu...
23,355
23,355
olympiad
13,469
<|im_start|>user Let $T=5$. Compute the smallest positive integer $n$ such that there are at least $T$ positive integers in the domain of $f(x)=\sqrt{-x^{2}-2 x+n}$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
10,643
10,643
olympiad
14,243
<|im_start|>user An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are...
5,348
5,348
olympiad
18,791
<|im_start|>user Compute the number of ordered pairs of integers $(a, b)$ such that $1<a \leq 50,1<b \leq 50$, and $\log _{b} a$ is rational. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
32,420
32,420
olympiad
13,444
<|im_start|>user Suppose that the functions $f(x)$ and $g(x)$ satisfy the system of equations $$ \begin{aligned} f(x)+3 g(x) & =x^{2}+x+6 \\ 2 f(x)+4 g(x) & =2 x^{2}+4 \end{aligned} $$ for all $x$. Determine the values of $x$ for which $f(x)=g(x)$. Please reason step by step, and put your final answer within \boxed{}...
5,200
5,200
olympiad
5,182
<|im_start|>user A computer is programmed to choose an integer between 1 and 99, inclusive, so that the probability that it selects the integer $x$ is equal to $\log _{100}\left(1+\frac{1}{x}\right)$. Suppose that the probability that $81 \leq x \leq 99$ is equal to 2 times the probability that $x=n$ for some integer $...
12,355
12,355
olympiad
14,840
<|im_start|>user Given a positive integer $n \geq 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_{0}, a_{1}, \ldots, a_{N}$ such that (1) $a_{0}+a_{1}=-\frac{1}{n}$, and (2) $\left(a_{k}+a_{k-1}\right)\left(a_{k}+a_{k+1}\right)=a_{k-1}-a_{k+1}$ for $1 \leq k \leq N-1$. Plea...
25,691
25,691
olympiad
26,775
<|im_start|>user Each positive integer less than or equal to 2019 is written on a blank sheet of paper, and each of the digits 0 and 5 is erased. Compute the remainder when the product of the remaining digits on the sheet of paper is divided by 1000 . Please reason step by step, and put your final answer within \boxed{...
32,140
32,140
olympiad
20,495
<|im_start|>user Particles from the Sun ${ }^{1}$ Photons from the surface of the Sun and neutrinos from its core can tell us about solar temperatures and also confirm that the Sun shines because of nuclear reactions. Throughout this problem, take the mass of the Sun to be $M_{\odot}=2.00 \times 10^{30} \mathrm{~kg}...
11,439
11,439
olympiad
12,318
<|im_start|>user Suppose that neither of the three-digit numbers $M=\underline{4} \underline{A} \underline{6}$ and $N=\underline{1} \underline{B} \underline{7}$ is divisible by 9 , but the product $M \cdot N$ is divisible by 9 . Compute the largest possible value of $A+B$. Please reason step by step, and put your final...
20,028
20,028
olympiad
13,079
<|im_start|>user If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
2,611
2,611
olympiad
5,698
<|im_start|>user All matters in the universe have fundamental properties called spin, besides their mass and charge. Spin is an intrinsic form of angular momentum carried by particles. Despite the fact that quantum mechanics is needed for a full treatment of spin, we can still study the physics of spin using the usual ...
19,731
19,731
olympiad
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<|im_start|>user Ross starts with an angle of measure $8^{\circ}$ and doubles it 10 times until he obtains $8192^{\circ}$. He then adds up the reciprocals of the sines of these 11 angles. That is, he calculates $$ S=\frac{1}{\sin 8^{\circ}}+\frac{1}{\sin 16^{\circ}}+\frac{1}{\sin 32^{\circ}}+\cdots+\frac{1}{\sin 4096^...
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20,962
olympiad
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<|im_start|>user Let $P(x)=x^{2}+T x+800$, and let $r_{1}$ and $r_{2}$ be the roots of $P(x)$. The polynomial $Q(x)$ is quadratic, it has leading coefficient 1, and it has roots $r_{1}+1$ and $r_{2}+1$. Find the sum of the coefficients of $Q(x)$. Please reason step by step, and put your final answer within \boxed{}.<|i...
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9,769
olympiad
9,347
<|im_start|>user In rectangle $P A U L$, point $D$ is the midpoint of $\overline{U L}$ and points $E$ and $F$ lie on $\overline{P L}$ and $\overline{P A}$, respectively such that $\frac{P E}{E L}=\frac{3}{2}$ and $\frac{P F}{F A}=2$. Given that $P A=36$ and $P L=25$, compute the area of pentagon $A U D E F$. Please rea...
28,397
28,397
olympiad
18,008
<|im_start|>user Let $r=1$ and $R=5$. A circle with radius $r$ is centered at $A$, and a circle with radius $R$ is centered at $B$. The two circles are internally tangent. Point $P$ lies on the smaller circle so that $\overline{B P}$ is tangent to the smaller circle. Compute $B P$. Please reason step by step, and put y...
8,535
8,535
olympiad
13,975
<|im_start|>user Let $T=5$. If $|T|-1+3 i=\frac{1}{z}$, compute the sum of the real and imaginary parts of $z$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
3,324
3,324
olympiad
8,430
<|im_start|>user For some positive integers $k$, the parabola with equation $y=\frac{x^{2}}{k}-5$ intersects the circle with equation $x^{2}+y^{2}=25$ at exactly three distinct points $A, B$ and $C$. Determine all such positive integers $k$ for which the area of $\triangle A B C$ is an integer. Please reason step by st...
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20,867
olympiad
19,817
<|im_start|>user Problem T3. Protostar formation Let us model the formation of a star as follows. A spherical cloud of sparse interstellar gas, initially at rest, starts to collapse due to its own gravity. The initial radius of the ball is $r_{0}$ and the mass is $m$. The temperature of the surroundings (much sparser ...
15,789
15,789
olympiad
21,218
<|im_start|>user A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$. Determ...
16,475
16,475
olympiad
11,997
<|im_start|>user Chinara starts with the point $(3,5)$, and applies the following three-step process, which we call $\mathcal{P}$ : Step 1: Reflect the point in the $x$-axis. Step 2: Translate the resulting point 2 units upwards. Step 3: Reflect the resulting point in the $y$-axis. As she does this, the point $(3,5...
14,609
14,609
olympiad
8,294
<|im_start|>user An Electrified Soap Bubble A spherical soap bubble with internal air density $\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\gamma$, density $\rho_{s}$ and thickne...
7,480
7,480
olympiad
15,098
<|im_start|>user An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$, and the other focus is at $(d, 7)$. Compute $d$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
16,231
16,231
olympiad
22,839
<|im_start|>user A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$...
10,246
10,246
olympiad
20,843
<|im_start|>user Compute the sum of all positive two-digit factors of $2^{32}-1$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
25,467
25,467
olympiad
9,070
<|im_start|>user Triangle $A B C$ is inscribed in circle $\omega$. The tangents to $\omega$ at $B$ and $C$ meet at point $T$. The tangent to $\omega$ at $A$ intersects the perpendicular bisector of $\overline{A T}$ at point $P$. Given that $A B=14, A C=30$, and $B C=40$, compute $[P B C]$. Please reason step by step, a...
31,097
31,097
olympiad
34,764
<|im_start|>user Let $T=49$. Compute the last digit, in base 10, of the sum $$ T^{2}+(2 T)^{2}+(3 T)^{2}+\ldots+\left(T^{2}\right)^{2} $$ Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
27,791
27,791
olympiad
9,329
<|im_start|>user Digital images consist of a very large number of equally spaced dots called pixels The resolution of an image is the number of pixels/cm in each of the horizontal and vertical directions. Thus, an image with dimensions $10 \mathrm{~cm}$ by $15 \mathrm{~cm}$ and a resolution of 75 pixels/cm has a total...
7,547
7,547
olympiad
12,112
<|im_start|>user Find all triples $(a, b, c)$ of real numbers such that $a b+b c+$ $c a=1$ and $$ a^{2} b+c=b^{2} c+a=c^{2} a+b \text {. } $$ Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
20,022
20,022
olympiad
14,085
<|im_start|>user A cylinder with radius $r$ and height $h$ has volume 1 and total surface area 12. Compute $\frac{1}{r}+\frac{1}{h}$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
7,859
7,859
olympiad
6,917
<|im_start|>user ## String Cheese a. When a faucet is turned on, a stream of water flows down with initial speed $v_{0}$ at the spout. For this problem, we define $y$ to be the vertical coordinate with its positive direction pointing up. Assuming the water speed is only affected by gravity as the water falls, find th...
14,134
14,134
olympiad
10,331
<|im_start|>user Particles and Waves Wave-particle duality, which states that each particle can be described as a wave and vice versa, is one of the central concepts of quantum mechanics. In this problem, we will rely on this notion and just a few other basic assumptions to explore a selection of quantum phenomena cov...
4,231
4,231
olympiad
9,579
<|im_start|>user What are all values of $x$ such that $$ \log _{5}(x+3)+\log _{5}(x-1)=1 ? $$ Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
2,969
2,969
olympiad
3,216
<|im_start|>user $\quad$ Suppose that $p$ and $q$ are two-digit prime numbers such that $p^{2}-q^{2}=2 p+6 q+8$. Compute the largest possible value of $p+q$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
11,382
11,382
olympiad
13,258
<|im_start|>user Let $T=184$. A group of $n$ friends goes camping; two of them are selected to set up the campsite when they arrive and two others are selected to take down the campsite the next day. Compute the smallest possible value of $n$ such that there are at least $T$ ways of selecting the four helpers. Please r...
15,919
15,919
olympiad
7,682
<|im_start|>user For each positive integer $n$, let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$. For example, $T(6)=1$ since the only such triangle with a perimeter of 6 has side lengths 2,2 and 2 . Determine the values of $T(10), T(11)$ and $T(12)$. Please reason step ...
21,039
21,039
olympiad
9,607
<|im_start|>user Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles at the same constant speed. On days when he paddles with the current, it takes him 18 minutes to get from $A$ to $B$. When he ...
3,498
3,498
olympiad
5,167
<|im_start|>user Compute all ordered pairs of real numbers $(x, y)$ that satisfy both of the equations: $$ x^{2}+y^{2}=6 y-4 x+12 \quad \text { and } \quad 4 y=x^{2}+4 x+12 $$ Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
6,011
6,011
olympiad
5,471
<|im_start|>user Let $n$ be a positive integer such that $\frac{3+4+\cdots+3 n}{5+6+\cdots+5 n}=\frac{4}{11}$. Compute $\frac{2+3+\cdots+2 n}{4+5+\cdots+4 n}$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
20,549
20,549
olympiad
18,854
<|im_start|>user Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation holds for all rational numbers $x$ and $y$. $$ f(x f(x)+y)=f(y)+x^{2} $$ Here, $\mathbb{Q}$ denotes the set of rational numbers. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_star...
27,413
27,413
olympiad
15,660
<|im_start|>user Particles from the Sun ${ }^{1}$ Photons from the surface of the Sun and neutrinos from its core can tell us about solar temperatures and also confirm that the Sun shines because of nuclear reactions. Throughout this problem, take the mass of the Sun to be $M_{\odot}=2.00 \times 10^{30} \mathrm{~kg}...
10,377
10,377
olympiad
12,944
<|im_start|>user Hexagon $A B C D E F$ has vertices $A(0,0), B(4,0), C(7,2), D(7,5), E(3,5)$, $F(0,3)$. What is the area of hexagon $A B C D E F$ ? Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
17,104
17,104
olympiad
10,264
<|im_start|>user Let $T=125$. Cube $\mathcal{C}_{1}$ has volume $T$ and sphere $\mathcal{S}_{1}$ is circumscribed about $\mathcal{C}_{1}$. For $n \geq 1$, the sphere $\mathcal{S}_{n}$ is circumscribed about the cube $\mathcal{C}_{n}$ and is inscribed in the cube $\mathcal{C}_{n+1}$. Let $k$ be the least integer such th...
18,556
18,556
olympiad
14,817
<|im_start|>user ## Hot Pocket a. It's winter and you want to keep warm. The temperature is $T_{0}=263 \mathrm{~K}$ outside and $T_{1}=290 \mathrm{~K}$ in your room. You have started a fire, which acts as a hot reservoir at temperature $T_{2}=1800 \mathrm{~K}$. You want to add a small amount of heat $d Q_{1}$ to you...
19,610
19,610
olympiad
18,403
<|im_start|>user Linh is driving at $60 \mathrm{~km} / \mathrm{h}$ on a long straight highway parallel to a train track. Every 10 minutes, she is passed by a train travelling in the same direction as she is. These trains depart from the station behind her every 3 minutes and all travel at the same constant speed. What ...
8,590
8,590
olympiad
6,587
<|im_start|>user Paul was planning to buy 20 items from the ARML shop. He wanted some mugs, which cost $\$ 10$ each, and some shirts, which cost $\$ 6$ each. After checking his wallet he decided to put $40 \%$ of the mugs back. Compute the number of dollars he spent on the remaining items. Please reason step by step, a...
3,498
3,498
olympiad
2,983
<|im_start|>user 2. Trajectory of a point mass A point mass on the ground is thrown with initial velocity $\vec{v}_{0}$ that makes an angle $\theta$ with the horizontal. Assuming that air friction is negligible, Context question: (a) What value of $\theta$ maximizes the range? Context answer: \boxed{$\frac{\pi}{4}$} ...
23,580
23,580
olympiad
14,954
<|im_start|>user 5. Polarization and Oscillation In this problem, we will understand the polarization of metallic bodies and the method of images that simplifies the math in certain geometrical configurations. Throughout the problem, suppose that metals are excellent conductors and they polarize significantly faster...
12,391
12,391
olympiad
12,456
<|im_start|>user The four zeros of the polynomial $x^{4}+j x^{2}+k x+225$ are distinct real numbers in arithmetic progression. Compute the value of $j$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
14,984
14,984
olympiad
18,254
<|im_start|>user Point $P$ is on the hypotenuse $\overline{E N}$ of right triangle $B E N$ such that $\overline{B P}$ bisects $\angle E B N$. Perpendiculars $\overline{P R}$ and $\overline{P S}$ are drawn to sides $\overline{B E}$ and $\overline{B N}$, respectively. If $E N=221$ and $P R=60$, compute $\frac{1}{B E}+\fr...
14,744
14,744
olympiad
21,312
<|im_start|>user Particles and Waves Wave-particle duality, which states that each particle can be described as a wave and vice versa, is one of the central concepts of quantum mechanics. In this problem, we will rely on this notion and just a few other basic assumptions to explore a selection of quantum phenomena cov...
6,178
6,178
olympiad
11,173
<|im_start|>user $$ \begin{array}{ll} g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m...
18,275
18,275
olympiad
9,654
<|im_start|>user $$ \begin{array}{ll} g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m...
5,440
5,440
olympiad
8,380
<|im_start|>user Let $T=1$. Circles $L$ and $O$ are internally tangent and have radii $T$ and $4 T$, respectively. Point $E$ lies on circle $L$ such that $\overline{O E}$ is tangent to circle $L$. Compute $O E$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <thin...
16,062
16,062
olympiad
19,907
<|im_start|>user Let $T=\frac{9}{17}$. When $T$ is expressed as a reduced fraction, let $m$ and $n$ be the numerator and denominator, respectively. A square pyramid has base $A B C D$, the distance from vertex $P$ to the base is $n-m$, and $P A=P B=P C=P D=n$. Compute the area of square $A B C D$. Please reason step by...
11,195
11,195
olympiad
8,347
<|im_start|>user Compute the value of $$ \sin \left(6^{\circ}\right) \cdot \sin \left(12^{\circ}\right) \cdot \sin \left(24^{\circ}\right) \cdot \sin \left(42^{\circ}\right)+\sin \left(12^{\circ}\right) \cdot \sin \left(24^{\circ}\right) \cdot \sin \left(42^{\circ}\right) \text {. } $$ Please reason step by step, and ...
19,655
19,655
olympiad
17,096
<|im_start|>user Let $T=19$. Let $\mathcal{S}_{1}$ denote the arithmetic sequence $0, \frac{1}{4}, \frac{1}{2}, \ldots$, and let $\mathcal{S}_{2}$ denote the arithmetic sequence $0, \frac{1}{6}, \frac{1}{3}, \ldots$ Compute the $T^{\text {th }}$ smallest number that occurs in both sequences $\mathcal{S}_{1}$ and $\math...
15,415
15,415
olympiad
14,285
<|im_start|>user Regular hexagon $A B C D E F$ and regular hexagon $G H I J K L$ both have side length 24 . The hexagons overlap, so that $G$ is on $\overline{A B}, B$ is on $\overline{G H}, K$ is on $\overline{D E}$, and $D$ is on $\overline{J K}$. If $[G B C D K L]=\frac{1}{2}[A B C D E F]$, compute $L F$. Please rea...
31,036
31,036
olympiad
40,000
<|im_start|>user An Electrified Soap Bubble A spherical soap bubble with internal air density $\rho_{i}$, temperature $T_{i}$ and radius $R_{0}$ is surrounded by air with density $\rho_{a}$, atmospheric pressure $P_{a}$ and temperature $T_{a}$. The soap film has surface tension $\gamma$, density $\rho_{s}$ and thickne...
18,536
18,536
olympiad
16,061
<|im_start|>user Let $T=5$. Hannah flips two fair coins, while Otto flips $T$ fair coins. Let $p$ be the probability that the number of heads showing on Hannah's coins is greater than the number of heads showing on Otto's coins. If $p=q / r$, where $q$ and $r$ are relatively prime positive integers, compute $q+r$. Plea...
17,222
17,222
olympiad
13,074
<|im_start|>user Compute the least integer greater than 2023 , the sum of whose digits is 17 . Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
13,040
13,040
olympiad
8,915
<|im_start|>user In ARMLopolis, every house number is a positive integer, and City Hall's address is 0. However, due to the curved nature of the cowpaths that eventually became the streets of ARMLopolis, the distance $d(n)$ between house $n$ and City Hall is not simply the value of $n$. Instead, if $n=3^{k} n^{\prime}$...
19,692
19,692
olympiad
15,404
<|im_start|>user Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8}$ be a regular octagon. Let $\mathbf{u}$ be the vector from $A_{1}$ to $A_{2}$ and let $\mathbf{v}$ be the vector from $A_{1}$ to $A_{8}$. The vector from $A_{1}$ to $A_{4}$ can be written as $a \mathbf{u}+b \mathbf{v}$ for a unique ordered pair of re...
27,433
27,433
olympiad
20,802
<|im_start|>user Let $T=15$. For complex $z$, define the function $f_{1}(z)=z$, and for $n>1, f_{n}(z)=$ $f_{n-1}(\bar{z})$. If $f_{1}(z)+2 f_{2}(z)+3 f_{3}(z)+4 f_{4}(z)+5 f_{5}(z)=T+T i$, compute $|z|$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
30,487
30,487
olympiad
21,681
<|im_start|>user Suppose that $a(x+b(x+3))=2(x+6)$ for all values of $x$. Determine $a$ and $b$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
3,121
3,121
olympiad
5,214
<|im_start|>user Large Hadron Collider Please read the general instructions in the separate envelope before you start this problem. In this task, the physics of the particle accelerator LHC (Large Hadron Collider) at CERN is discussed. CERN is the world's largest particle physics laboratory. Its main goal is to get i...
8,197
8,197
olympiad
9,918
<|im_start|>user Let $T=7$. Compute the number of positive perfect cubes that are divisors of $(T+10) !$. Please reason step by step, and put your final answer within \boxed{}.<|im_end|> <|im_start|>assistant <think>
15,749
15,749
olympiad
9,982
<|im_start|>user $$ \begin{array}{ll} g=9.8 \mathrm{~N} / \mathrm{kg} & G=6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2} \\ k=1 / 4 \pi \epsilon_{0}=8.99 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} & k_{\mathrm{m}}=\mu_{0} / 4 \pi=10^{-7} \mathrm{~T} \cdot \mathrm{m} / \m...
32,368
32,368
olympiad
10,667
<|im_start|>user A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$. Determ...
11,151
11,151
olympiad
8,923
<|im_start|>user LIGO-GW150914 In 2015, the gravitational-wave observatory LIGO detected, for the first time, the passing of gravitational waves (GW) through Earth. This event, named GW150914, was triggered by waves produced by two black holes that were orbiting on quasi-circular orbits. This problem will make you est...
6,735
6,735
olympiad
11,925
<|im_start|>user The taxicab distance between points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$ is defined as $d(A, B)=$ $\left|x_{A}-x_{B}\right|+\left|y_{A}-y_{B}\right|$. Given some $s>0$ and points $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right)$, define the taxicab ellipse with ...
29,479
29,479
olympiad
15,971
<|im_start|>user Let $T=19$. The first two terms of a sequence are $a_{1}=3 / 5$ and $a_{2}=4 / 5$. For $n>2$, if $n$ is odd, then $a_{n}=a_{n-1}^{2}-a_{n-2}^{2}$, while if $n$ is even, then $a_{n}=2 a_{n-2} a_{n-3}$. Compute the sum of the squares of the first $T-3$ terms of the sequence. Please reason step by step, a...
29,640
29,640
olympiad
22,949
End of preview.

Qwen3.6-27B generation-length prediction: heads, calibrations and workloads

Artifacts for conformal length-aware LLM scheduling on Qwen/Qwen3.6-27B — predicting a request's remaining generation length from a hidden layer during decoding, wrapping it in a split-conformal interval, and scheduling with SRPT inside vLLM. Extends TRAIL (Don't Stop Me Now, ICLR'25) to a hybrid-attention reasoning model.

This repo contains the derived artifacts, not the raw activations. The 3250 decode-step traces are 322 GB and are not uploaded. Everything needed to use the predictors is here; everything needed to retrain them is reproducible with the commands below (20 GPU-hours).

Setup

model Qwen/Qwen3.6-27B (Qwen3_5ForConditionalGeneration)
probe layer 32 of 64
sampling temperature 0.6, max_tokens 32768, max_model_len 40960
engine vLLM 0.17.0, enforce_eager=True
feature last-token hidden state at layer 32 per decode step, fp16, 5120-dim

Qwen3.6-27B is hybrid-attention: 64 layers with full_attention_interval=4, so only 16 carry a token-proportional KV cache and 48 hold fixed-size recurrent state. Consequences: KV is 64 KiB/token (vs 192 for a comparable dense 14B), but each request commits ~144 MiB of recurrent state at admission — equal to ~2300 tokens of KV. Per-request memory therefore barely grows with age until well past the median generation.

Files

Heads — all trained on the same 1000 base traces, so they differ only in target/readout. Held-out metrics on 300 base traces (mean T = 4625):

file target readout MAE Spearman
abs.pt sqrt(remaining), SmoothL1 clamp(z,0,zcap)² 1431 0.737
prog.pt p=(g+1)/T via sigmoid t(1−p)/p, capped 1539 0.638
trail_mlp.pt bin index, CE (TRAIL) softmax(z)·bin_mids 1439 0.720
trail.pt as above, 20 bins over 8000
psi_L32.pt prompt-only → total T 2090 0.70

trail_mlp.pt is the faithful TRAIL baseline: the paper's Linear(h→512)→ReLU→Linear(512→bins) with 80 bins over 32768. trail.pt keeps the original 20-bins-over-8000 ceiling, which saturates on 7.5% of the short workload (34% of MATH) and understates the baseline — prefer trail_mlp.pt.

Calibrations (split-conformal, α=0.5, q_t pooled per decode step):

file calib set M0 median q_t
conf_abs_base_a5.npz 495 base 17185 2436
conf_prog_base_a5.npz 495 base 17185 3289
conf_abs_aug_a5.npz + 236 Olympiad 28750 2935
conf_prog_aug_a5.npz + 236 Olympiad 28750 4539

⚠️ q_by_step[0] is NaN in every file. The collector records only true decode steps (nst == 1), so the first token — produced by the prefill forward — is never captured and step 0 has no residuals to take a quantile over. Every waiting request has g = 0, so a scheduler that indexes q_by_step[g] without a guard gets NaN, and NaN compares False against everything: sorted() silently leaves the admission queue in arbitrary order. Note beta * q does not save you — 0.0 * nan = nan. Fall back to the nearest finite step.

⚠️ The augmented calibrations are not exchangeable with base traffic. Olympiad's median true_len is 13722 vs base's 1585, so its residuals inflate q_t by +20% (abs) / +38% (prog), and the extra support between step 17185 and 28750 comes almost entirely from Olympiad traces. Use *_base_* when serving base-distribution traffic; the augmented files are provided for the M0-vs-exchangeability comparison, not as a drop-in improvement.

Splitssplit_base.json (train 1000 / calib 495 / eval 300, base only), split_aug.json (same train, calib 731 = 495 base + 236 Olympiad, eval 300 Olympiad). manifest_L32.json maps every trace path to (T, dataset).

Workloads — served request files with text, true_len, pred_len, dataset:

file n median true_len Spearman(pred_len, true_len)
bench_short.json 845 1573 0.695
bench_olympiad.json 228 13722 0.690

pred_len comes from ψ (prompt-only prefill predictor) and is never a copy of true_len — 0.0% identical in both. It is packed into the vLLM request id as the static admission priority, so setting it to true_len would hand the oracle key to every predictor policy.

Promptscollect_prompts.json (1800: gsm8k 500, mmlu_pro 500, bbh 400, math 400), benchmark_prompts.json (850, verified zero overlap with the collection pool), olympiad600_prompts.jsonl (OlympiadBench OE_TO_*_en_COMP, text-only English), longcot600_prompts.jsonl (LongHorizonReasoning/longcot, 5 domains × 3 difficulties).

Prefill featuresprefill_{collect,bench,olymp}.pt, mean-pooled layer-32 prompt states, one row per prompt, the input to ψ.

Layer sweeplayer_sweep_qwen36.json (all 100 traces) and ..._uncapped.json (59 uncapped).

Choosing the probe layer

8 layers swept, 4 full_attention / 4 linear_attention, depth 0.17–0.75, ranked by held-out cross-request Spearman over 5 resampled splits:

set best ρ vs runner-up
all 100 traces L32 0.623 ± 0.044 +0.032 ± 0.012 (separable)
59 uncapped only L32 0.700 ± 0.033 +0.009 ± 0.022 (indistinguishable)

On clean data L32 sits on a plateau with {32, 35, 43} — the choice within it is not load-bearing. The depth-ratio heuristic (0.5625 → L36) would have been meaningfully worse; its nearest swept neighbour L40 ranks 6th of 8. Layer type mattered little (linear 0.663 vs full 0.644).

Trace schema (for reproduction)

Each .pt is {"L32": float16 (n_steps, 5120), "T": int, "g": int32 (n_steps,), "dataset": str, "prompt": str}.

  • g is the 0-based index of the token an activation produced. Remaining is T − (g+1). Recording starts at g=1 because the first token comes from the prefill forward, which is skipped.
  • A trace is not a contiguous step sequence — always index through g, never arange(len(H)).
  • Capped traces are mislabelled: a generation that never emits </think> stores T = cap. Base capped 1% (5/500); Olympiad ~22%. Drop them from calibration — the residual |R_true − R̂| uses R_true = T − t, so a capped trace injects a too-small residual and makes q_t overconfident exactly in the tail.

Reproduction

export SCHED_MODEL=Qwen/Qwen3.6-27B SCHED_LAYER=32
python collect_activations_vllm.py --layers 32 --prompt_file collected/collect_prompts.json \
    --max_tokens 32768 --max_model_len 40960 --gpu_mem 0.85 --chunk 100 --out <traces>
python build_split.py --init --base-glob '<traces>/*.pt' --sizes 1000,500,300 --out split_base.json
python train_head.py --head absolute --arch linear --split split_base.json --out abs.pt
python calibrate.py --ckpt abs.pt --split split_base.json --alphas 0.5 --out conf_abs_base_a5.npz

Collection is ~20 GPU-hours on one 80–96 GB card. enforce_eager=True is mandatory — CUDA graphs bypass forward hooks, so the collector silently records nothing without it.

Limitations

  • Raw activations are not included (~322 GB).
  • Metrics are single-seed; head training variance was not measured.
  • The layer sweep used max_tokens 2500, where 41% of traces capped; the full collection used 32768, where only 1% of base traces cap.
  • Scheduling results are not included in this repo and remain under evaluation. One measured property worth stating: with max_num_seqs=64 this workload never saturates the concurrency cap (~41 of 64 concurrent), so no queue forms and scheduling cannot help; a queue requires max_num_seqs=24. Preemption was exactly zero across ~100k schedule calls, so TRAIL's limited-preemption parameter c is inert here.
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