Beyond Token Alignment: Event Completion for Cross-Tokenizer On-Policy Distillation
Paper • 2609.34738 • Published
Datasets:
id stringlengths 4 4 | title stringlengths 25 65 | problem_latex stringlengths 1.29k 3.16k | reference_solution_and_rubric_latex stringlengths 7.08k 15k | problem_sha256 stringlengths 64 64 | reference_sha256 stringlengths 64 64 |
|---|---|---|---|---|---|
A001 | Static Aristotelian Algebra Sector | \section*{Static Aristotelian Algebra Sector}
Consider a complex scalar field on flat \(d\)-dimensional space at fixed time. Treat \(\Phi\) and \(\Phi^{*}\) as independent canonical coordinates with conjugate momenta \(\Pi\) and \(\Pi^{*}\). The equal-time Poisson brackets are
\[
\{\Phi(\mathbf{x}),\Pi(\mathbf{y})\... | \section*{Solution}
\subsection*{1. Canonical momentum density and translation charge}
We work at fixed time with canonical brackets
\[
\{\Phi(\mathbf{x}),\Pi(\mathbf{y})\}=\delta^{(d)}(\mathbf{x}-\mathbf{y}),
\qquad
\{\Phi^{*}(\mathbf{x}),\Pi^{*}(\mathbf{y})\}=\delta^{(d)}(\mathbf{x}-\mathbf{y}),
\]
and all oth... | a37e46301cbed89aa5f59d5bb058fe30395aec8f389b15e15fdd2dfe8d622f41 | 49a9b817aa0b5ad9388a9ea3defc780070f53161a79b9cfc48cff6550bc76cf0 |
A008 | Quadrupole Algebra Sector | \section*{Quadrupole Algebra Sector}
Consider an equal-time charge density \(J^{(0)}(\mathbf{x})\) on flat \(d\)-dimensional Euclidean space. Spatial indices are raised and lowered with \(\delta_{ij}\), and
\[
x^2 := \delta_{ij}x^i x^j .
\]
The scalar second spatial moment of the \(U(1)\) charge density is introduced... | \section*{Solution}
\subsection*{1. Conventions and charge moments}
We work at fixed time on flat \(d\)-dimensional Euclidean space. The equal-time charge density is denoted by \(J^{(0)}(\mathbf{x})\). Its monopole, dipole, and scalar second moment are
\begin{center}
\begingroup
\setbox0=\hbox{\(\displaystyle
Q^{(0)}=... | 667c3c9076dff7067b848092c0c5d46f724dc12342b39d08da37607e4a9cb27e | f7ee0e3a557a968bdb019b7239b3ca54085eb0720a677041ee071a0ce33d7100 |
A015 | Ground-State Boundary CDD Contribution | \section*{Ground-State Boundary CDD Contribution}
Consider the massless Ising theory with a \(T\bar T\)-type bulk CDD deformation and a boundary reflection factor modified by higher odd-power boundary CDD phases. The one-particle momentum on the positive massless line is denoted by \(p>0\), the circumference by \(R\),... | \section*{Solution}
\subsection*{1. Reflection factor and logarithmic boundary kernel}
The higher boundary CDD deformation multiplies the minimal reflection factor by independent odd-power phase factors:
\begin{center}
\begingroup
\setbox0=\hbox{\(\displaystyle
\widetilde R_{\boldsymbol\delta}(p)
=R_0(p)\prod_{j\geq... | 991ae47212fd41583c1c86d44f46b04461bb746b1d6e08bb415aeb74cede701e | 2a7d70ad7aca98a6b353f2051a1a3557e00a6aa417dc8f43b0157c1180b7be32 |
A017 | Finite-Cutoff Gravity and the Large-\(c\) Partition Function | \section*{Finite-Cutoff Gravity and the Large-\(c\) Partition Function}
Consider Euclidean \(\mathrm{AdS}_3\) gravity with AdS radius set to one, Newton constant \(G\), and a radial cutoff surface at \(r=r_c\). The induced metric on the cutoff surface is denoted by \(\gamma_{\alpha\beta}\), while the two-dimensional ... | \section*{Solution}
\subsection*{1. Cutoff dictionary and central-charge normalization}
The finite-cutoff dictionary separates the two-dimensional source metric from the metric held fixed in the bulk Dirichlet problem. The QFT background is
\[
ds^2=g_{\alpha\beta}(x)\,dx^\alpha dx^\beta,
\]
whereas the induced met... | cff7af49574e45bdb1f4122d88c5630ff5a976d490ac57d1bf81d3a477282cb4 | 5d372b1444c1c1490999f7d402a5a07f4c98168e3fc5db661d8fe0ab1e545b7a |
A019 | Discrete Anomaly at the Self-Dual Radius | \section*{Discrete Anomaly at the Self-Dual Radius}
Consider the compact boson at the self-dual radius, with normalization
\[
R=1 .
\]
At this point the continuous symmetry is enhanced from the generic torus symmetry to the quotient of the two chiral level-one current-algebra symmetries by their common center. Use ... | \section*{Solution}
\subsection*{1. Enhanced symmetry and the quotient at \(R=1\)}
At the self-dual radius the compact boson has enhanced left- and right-moving current algebras. With the stated normalization,
\[
R=1,
\]
the chiral symmetry is generated by two level-one affine \(SU(2)\) algebras. The simply conne... | 2b3755484f2e730ed4868e4973b77373c68f284bcdda5161f39e893d1fbe5118 | 28821c5c3a79f7259e81d5ecdfbabdd9c0d6e9b5c379563eaac385f24d11f3f1 |
A020 | Momentum Shift in an \(O(3)\) Twisted Sector | \section*{Momentum Shift in an \(O(3)\) Twisted Sector}
Consider the \(O(3)=SO(3)\rtimes \mathbb Z_{2}^{C}\) twist family of a two-chiral-sector theory. The twist may be described either by momentum--winding variables \((\eta_m,\eta_w)\) or by chiral variables \((\eta,\bar\eta)\), related by
\[
\eta=\eta_m+\eta_w,
... | \section*{Solution}
\subsection*{1. Twist variables and the chiral endpoint}
The two coordinate systems on the twist torus are related by
\[
\eta=\eta_m+\eta_w,
\qquad
\bar\eta=\eta_m-\eta_w,
\]
or equivalently
\[
\eta_m={\eta+\bar\eta\over 2},
\qquad
\eta_w={\eta-\bar\eta\over 2}.
\]
Thus the elementary ... | 7afe8c418708acc85b56dfc0eabd0624437d3f73196e7be8306c5ccf05ec8fc3 | 657640c6557db2b71014bc22661d3a87c20fabb056f51fa16ab26cc835982136 |
A021 | Charge Conjugation at the Self-Dual Radius | \section*{Charge Conjugation at the Self-Dual Radius}
Consider the compact boson at the self-dual radius in the normalization
\[
R=1 .
\]
At this radius the theory has the enhanced chiral symmetry generated by the level-one affine algebras associated with the left- and right-moving \(SU(2)\) factors. The faithful co... | \section*{Solution}
\subsection*{1. Enhanced symmetry and the quotient at \(R=1\)}
The starting point is the self-dual-radius specialization
\[
R=1 .
\]
In the compact-boson normalization used here, the left- and right-moving momenta may be written as
\[
p_L=\frac{n}{R}+wR,
\qquad
p_R=\frac{n}{R}-wR,
\qquad ... | 417a84670d17427892a47b4645f44a74810702ba3b4a520cca3d833acd45d3ae | 770ad740f6c54007904d89cb2e120b810b7a6d79f1270384f7fb55118757f051 |
A030 | Finite-Cutoff Sound Speed in BTZ Thermodynamics | \section*{Finite-Cutoff Sound Speed in BTZ Thermodynamics}
Consider non-rotating BTZ thermodynamics with a finite Dirichlet wall at radius \(r=r_c\). Use Lorentzian signature \((-+)\), set the AdS radius to one, and denote the three-dimensional Newton constant by \(G\). The physical spatial circumference of the wall ... | \section*{Solution}
\subsection*{1. Wall energy density and pressure}
Introduce the redshift factor
\[
\alpha
:=\sqrt{1-\frac{r_+^2}{r_c^2}},
\qquad
0<\alpha\leq 1 .
\]
The Brown--York energy may then be written as
\[
E(r_c,r_+)=\frac{r_c}{4G}(1-\alpha).
\]
Dividing by the physical circumference of the cutof... | 69f45b51fde6795911c632c0ffdddf8459938414e291457c3a867eab8c1d5e67 | f74610a042d9313cecaf13ac8210395b27825a91bba34154a083929589f5a84d |
A031 | Parallel IR-Resummed Power Spectrum | \section*{Parallel IR-Resummed Power Spectrum}
Consider a Lagrangian-space density kernel \(K\) depending on an external momentum, a Lagrangian separation, and two time labels. We use three-dimensional vectors \(\mathbf k,\mathbf q\in\mathbb R^{3}\), the Fourier phase convention \(e^{-i\mathbf k\cdot\mathbf q}\), and ... | \section*{Solution}
\subsection*{1. Perturbative bookkeeping and the compensating factor}
The density kernel is expanded at fixed \((\mathbf k,\mathbf q;t_{1},t_{2})\) as
\begin{center}
\begingroup
\setbox0=\hbox{\(\displaystyle
K(\mathbf k,\mathbf q;t_{1},t_{2})
=\sum_{j=0}^{\infty}K(\mathbf k,\mathbf q;t_{1},t_{2}... | eb6d2d121638709edc5b7c1449f79e6f374b1ac47b09f6d768d3cf5f877fc038 | 86e350f0a14d7149c592dccd4c1f78ae46540714d3de79268909c46fa97be01d |
A033 | Linear Displacement Kernel | \section*{Linear Displacement Kernel}
Consider a statistically homogeneous and isotropic linear density field in three spatial dimensions. Use the Fourier convention
\[
f(\mathbf{x},t)=\int_{\mathbf{p}} e^{i\mathbf{p}\cdot\mathbf{x}}f(\mathbf{p},t),
\qquad
\int_{\mathbf{p}}\equiv \int\frac{d^3p}{(2\pi)^3}.
\]
Th... | \section*{Solution}
\subsection*{1. Linear displacement and endpoint difference}
The inverse Fourier representation of the linear displacement at a Lagrangian position \(\mathbf{x}\) is
\[
s^{(1)}_i(\mathbf{x},t)
=\int_{\mathbf{p}}e^{i\mathbf{p}\cdot\mathbf{x}}s^{(1)}_i(\mathbf{p},t)
=\int_{\mathbf{p}}e^{i\mathb... | 72e657b130b60ee25bbb316bca48cd621b423c3894cff05db61d251e900f8a34 | b72075d06318e6b4fb13b3250ef5ee35b71345dc3e50a3d50906cdfeac0b13d0 |
A036 | IR-Resummed Density Kernel | \section*{IR-Resummed Density Kernel}
Consider a density-spectrum kernel written in Lagrangian separation variables. We use the Fourier convention
\begin{center}
\begingroup
\setbox0=\hbox{\(\displaystyle
f(\mathbf{k})=\int d^3x\,e^{-i\mathbf{k}\cdot\mathbf{x}}f(\mathbf{x}),
\qquad
f(\mathbf{x})=\int_{\mathbf{k}}e... | \section*{Solution}
\subsection*{1. Kernel conventions and cumulant factorization}
The displacement kernel is
\begin{center}
\begingroup
\setbox0=\hbox{\(\displaystyle
K(\mathbf k,\mathbf q;t_1,t_2)
=
\left\langle
\exp\!\left[-i k_iX^i(\mathbf q;t_1,t_2)\right]
\right\rangle,
\qquad
X^i(\mathbf q;t_1,t_2... | ad56548a8050f337ed6422a78ffd0383d9a6658931724944cb80f8397a394fd0 | 0867f30ae00fa6078718e6bb005790b14827687a406701ad1210c11cae7caf2f |
A040 | Resummed Momentum Spectrum | \section*{Resummed Momentum Spectrum}
Consider an IR-resummed large-scale-structure calculation in which scalar two-point spectra are normalized by
\begin{center}
\begingroup
\setbox0=\hbox{\(\displaystyle
\bigl\langle X(\mathbf{k},t_1)Y(\mathbf{k}',t_2)\bigr\rangle
=(2\pi)^3\delta_{\rm D}^{(3)}(\mathbf{k}+\mathbf{k}... | \section*{Solution}
\subsection*{1. Fourier normalization and kernel action}
The two-point normalization fixes the meaning of every power spectrum that appears below:
\begin{center}
\begingroup
\setbox0=\hbox{\(\displaystyle
\bigl\langle X(\mathbf{k},t_1)Y(\mathbf{k}',t_2)\bigr\rangle
=(2\pi)^3\delta_{\rm D}^{(3)}(\... | f841d88ed6cf3d3dc000b93d8b1be6fdd7c5c9fd9e8784bd8861bce6e7c1eb28 | 94d34f130da7814d4c706f243177a9f6d90ee628c82b242d7cb646441aed8ad4 |
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PHYRD-40
40 physics reasoning problems with reference solutions and problem-specific grading rubrics.
The arXiv paper is an earlier version. Please refer to the latest PDF for the updated technical report.
Training code · SciDeriv · PHYRD-40
Dataset description
PHYRD-40 is the physics evaluation set used in ESCD. The collection covers advanced physics derivations, including field theory, gravity, string theory, scattering, cosmology and tensor-network methods. The paper describes the problems as developed by scientists invited by the authors.
This release contains the exact 40 selected problems from the evaluation
platform's matrix_lab_40 data file, not the larger 103-problem source collection.
All 40 problem IDs and the complete LaTeX problem and solution/rubric strings
are preserved. Internal filesystem paths are removed.
Load
from datasets import load_dataset
data = load_dataset("mytcl/PHYRD-40", split="test")
problem = data[0]["problem_latex"]
# For grading only; do not include this in the model's generation prompt.
reference = data[0]["reference_solution_and_rubric_latex"]
There is one test split with 40 rows. The upstream storage field
original_split="train" describes an internal collection convention; it is not
the evaluation split used by ESCD. This release uses test to make its intended
benchmark role explicit. It should not be mixed into ESCD training data.
Parquet powers the viewer; data/test.jsonl is an equivalent line-oriented copy.
Fields
| Field | Type | Description |
|---|---|---|
id |
string | Original stable problem ID. |
title |
string | Problem title. |
problem_latex |
string | Complete question; feed this to the model. |
reference_solution_and_rubric_latex |
string | Complete combined reference solution and rubric; use only for grading. |
problem_sha256 |
string | SHA-256 of the released UTF-8 problem text. |
reference_sha256 |
string | SHA-256 of the released UTF-8 combined reference/rubric text. |
The source stores the solution and rubric together. They are intentionally not split using heuristic headings that could discard equations or scoring criteria.
Evaluation protocol
The ESCD report samples one response per problem, grades it against the reference and rubric on a 0–100 scale, and reports the arithmetic mean across all 40 items. The report names DeepSeek-V4 Pro Preview as the judge. This repository releases the data only; it does not bundle the judge service, generated responses or the authors' internal evaluation platform. Exact prompt and generation settings are documented in the accompanying PDF. Judge-based results can depend on the judge version and prompt; the presence of a reference is not a guarantee of evaluator reproducibility across implementations.
Limitations and intended use
Use for evaluating advanced physics reasoning. Forty problems constitute a small, curated benchmark; scores should not be interpreted as comprehensive coverage of physics. Answers and rubrics are public, so users must manage contamination in training and evaluation. Release validation checks exact data preservation, not an independent scientific re-evaluation of each solution.
License
The authors release this dataset under CC BY 4.0. Please attribute PHYRD-40 and ESCD when using or redistributing it.
Citation
@misc{liu2026escd,
title={Beyond Token Alignment: Event Completion for Cross-Tokenizer On-Policy Distillation},
author={Jiacheng Liu and Jingwei Song and Qituan Zhang and Siheng Chen and Linfeng Zhang},
year={2026},
eprint={2609.34738},
archivePrefix={arXiv},
primaryClass={cs.CL},
url={https://arxiv.org/abs/2609.34738}
}
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