Datasets:

mytcl
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SciDeriv

id stringlengths 16 16	prompt listlengths 1 1	task_type stringclasses 2 values	topic stringlengths 16 133	source_id stringlengths 5 5	source_variant stringclasses 2 values	prompt_sha256 stringlengths 64 64
scideriv-R-00090	[ { "role": "user", "content": "# From a curvature-modified Higgs decay rate to a reheating equality condition\n\nA nonminimal Higgs-curvature coupling modifies the inflaton partial decay rate into Higgs scalars.\n\n## Starting formula\n\n```latex\n\\Gamma(\\phi \\to HH) = \\frac{N_H(1-6\\xi)^2}{192\\pi}\\fra...	conclusion_reconstruction	From a curvature-modified Higgs decay rate to a reheating equality condition	00090	R	12874e5219e8f9042b8075566245f8e254bf8da7f32bfedcded6571fd343a389
scideriv-O-05476	[ { "role": "user", "content": "# From rational Stokes constant to alien derivative on partition function\n\nThe rational Stokes constant is a divisor sum of primitive Stokes constants\n\n## Starting formula\n\n```latex\n\\overline{S}_\\mu = \\sum_{n\|\\mu} \\frac{S_{\\mu/n}}{n^2}\n```\n\n### Paper-local LaTeX...	conclusion_assessment	From rational Stokes constant to alien derivative on partition function	05476	O	facc4286e1541018fa6803a868bd9311a3fe534f51005f8307d982eb30d71588
scideriv-R-02175	[ { "role": "user", "content": "# From linear sector fits to the PBH tail probability\n\nThe fixed-defect-number conditional distributions are modeled phenomenologically by Gamma distributions. Their sector-dependent means and widths are calibrated from lattice data, fitted in the statistically reliable range...	conclusion_reconstruction	From linear sector fits to the PBH tail probability	02175	R	7070dc6a2676b51df035955d3d4d276d11e01a689da6e100e0023e6806fa75b4
scideriv-O-02032	[ { "role": "user", "content": "# Relation analysis: omitted-rapidity notation and the S-factor expansion\n\nThe two FormulaForest endpoints are displayed below exactly as recorded. Analyze their relation without silently correcting either endpoint.\n\n## Endpoint 1: deleted rapidity tuple\n\n```latex\n\\wh{\...	conclusion_assessment	Relation analysis: omitted-rapidity notation and the S-factor expansion	02032	O	d6475387bd00f0d2ba8390f07e9f565739319a4e1a3a44ee8ac2a6a42d80fab4
scideriv-O-08933	[ { "role": "user", "content": "# Reconstruct integrated BBb inner term from BBb block from Bbb block\n\nA bulk-bulk-boundary block is constructed by acting with descendant Laplacians on a bulk-boundary-boundary block\n\n## Given starting formula\n\n```latex\n\\begin{aligned}\nG_{\\text{BBb}}^{\\Delta_l,\\Del...	conclusion_assessment	Reconstruct integrated BBb inner term from BBb block from Bbb block	08933	O	7774d3ce47bfc6e03a79098afd6da8a28fc387793e5f8757deee02e04a059b05
scideriv-O-08923	[ { "role": "user", "content": "# Analyze the relation between unit-modulus ticks and the three-party impossibility condition\n\nThe source first verifies that the numerator and denominator in the tick quotient have equal modulus.\n\n## Endpoint 1: unit-modulus verification\n\n```latex\n\\left\|\\sin\\tfrac{\\...	conclusion_assessment	Analyze the relation between unit-modulus ticks and the three-party impossibility condition	08923	O	b0568fffe7017610c174201a3eb7cb705a5271c2c3bdd4f412f6d65b05297f2a
scideriv-O-07447	[ { "role": "user", "content": "# From twisted x boundary state to flux quantization condition\n\nBasis states translated by one x period acquire the gauge-transition phase\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\ket{x+N_x,\\,y}\n&= \\exp(-2\\pi i\\,\\alpha\\,N_x\\,y)\\,\\ket{x,y}\n= \\exp\\!\...	conclusion_assessment	From twisted x boundary state to flux quantization condition	07447	O	9da7dfe4454d19e5cb2409d9d8b6ec348777dcc850dda653f08adffbfa725a6a
scideriv-O-01043	[ { "role": "user", "content": "# From Plane gravitational wave metric to Potential quadratures\n\nWeak plane gravitational wave propagating along x1 in TT form\n\n## Starting formula\n\n```latex\ng_{\\mu\\nu}=\\left(\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 & -1 & 0 & 0 \\\\\n0 & 0 & h_{22}-1 & h_{23} \\...	conclusion_assessment	From Plane gravitational wave metric to Potential quadratures	01043	O	654b8dcd7e1b9f454fb9c7bc282ec8927387fbd18f2d1d764368b2e792987493
scideriv-O-06317	[ { "role": "user", "content": "# From Kerr base metric in vartheta to Kerr beta-four EFT term\n\nWeyl-rescaled two-dimensional base metric and measure after the angular coordinate change\n\n## Starting formula\n\n```latex\n\\rmd \\hat{s}^2 = \\frac{\\rmd \\vartheta^2}{\\beta ^2 \\left(1+\\Omega ^2 \\cos^2 \\...	conclusion_assessment	From Kerr base metric in vartheta to Kerr beta-four EFT term	06317	O	c71d5ffe1a6633c0000fce25f848eed33d472fafb44cdce249a60b39d44d061c
scideriv-O-04587	[ { "role": "user", "content": "# From Charlie clock tuple to layered Alice-Bob measurements\n\nA clock datum is a tuple containing period, tick increments, and a real offset parameter\n\n## Starting formula\n\n```latex\n\\Gamma=(N,t,s_0,s_1,\\mu)\n```\n\n### Paper-local LaTeX definitions used by the displaye...	conclusion_assessment	From Charlie clock tuple to layered Alice-Bob measurements	04587	O	b8c1133a9fdbf3a5d3ee68570b42ec33b9e3e2f213873b2cd2ec0a29f3fd66f5
scideriv-O-02760	[ { "role": "user", "content": "# From first beta subset to ordered B1 tuple\n\nA subset B1 of beta rapidities is selected by indices i_a\n\n## Starting formula\n\n```latex\nB_1 \\, = \\, \\{ \\, \\beta_{i_a} \\}_{a=1}^{p} \\quad \\text{and} \\quad\nB_2 \\, = \\, \\{ \\, \\beta_{j_a} \\}_{a=1}^{n-p}\n```\n\n...	conclusion_assessment	From first beta subset to ordered B1 tuple	02760	O	a6874c5646799c6dcbcf30f8ee233ce86ab14d4c5ad241dd4fc47521d42925e7
scideriv-O-08713	[ { "role": "user", "content": "# Reconstruct localized inflow variation from bulk 4d CS gauge variation\n\nGauge variation of the 4d Chern-Simons action is supported by poles through d omega\n\n## Given starting formula\n\n```latex\n\\delta S_{\\rm CS}[A]=\\frac{1}{2\\pi \\hbar}\\int_{\\cM\\times \\mathbb{CP...	conclusion_assessment	Reconstruct localized inflow variation from bulk 4d CS gauge variation	08713	O	0dfcad35472b201139407f9436a90cbcda34cedf8d6b5360f28a4e764398f33f
scideriv-O-01156	[ { "role": "user", "content": "# From instantaneous jet-velocity bundle to instantaneous Legendre transform\n\nThe instantaneous Lagrangian density is defined on spatial first jets together with vertical velocity data\n\n## Starting formula\n\n```latex\nJ^1(Y_t)\\times_{Y_t}VY_t\n```\n\n## Intermediate relat...	conclusion_assessment	From instantaneous jet-velocity bundle to instantaneous Legendre transform	01156	O	88a0b946e571ebae253e087ccb59c3aeb92473682bc306d99d742d99c7c29137
scideriv-R-04227	[ { "role": "user", "content": "# From the SCI Hessian block matrix to the explicit Cardy saddle SCI\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\Big(\\mathbb{H}[U;\\Delta]\\Big)_{I,J}=\\begin{pmatrix}\n\\mathbb{J}[U;\\Delta] & 0 \\\\\n0 & -\\overline{\\mathbb{J}[U;\\Delta]}\n\\end{pmatrix}\\,,\\qq...	conclusion_reconstruction	From the SCI Hessian block matrix to the explicit Cardy saddle SCI	04227	R	010e6556e6847044771eebb2b7fb09bffd8d1c9d7a9b6e4cbd92283df57f8293
scideriv-O-09983	[ { "role": "user", "content": "# Research problem: determine the homogeneous shear-stress relaxation equation\n\n## Given starting formula\n\n```latex\nS^\\mu\n\\equiv\n-\\delta h_\\nu \\omega^{\\mu\\nu}\n+\\dot u_\\nu \\delta\\pi^{\\mu\\nu}\n-\\Delta^\\mu_{\\ \\alpha}\\nabla_\\beta \\delta\\pi^{\\alpha\\bet...	conclusion_assessment	Research problem: determine the homogeneous shear-stress relaxation equation	09983	O	91c0c8206b070d52819a14ae92a9527bcedd56b6cf6c0e293004de2177b4b511
scideriv-O-03175	[ { "role": "user", "content": "# From flat and AdS radial constraints to E-classification\n\nThe uniform constraint specializes to four-dimensional flat and five-dimensional AdS radial equations\n\n## Starting formula\n\n```latex\n\\begin{aligned}\nn=4:\\qquad\n\\dot a^{\\,2}&=1-{a_0^4\\over a^4},\n&a_0^4&={...	conclusion_assessment	From flat and AdS radial constraints to E-classification	03175	O	12cc55602a2417b77b76277dba4642975dc2042bae691bdb603e1097c87aead5
scideriv-R-04746	[ { "role": "user", "content": "# From discretized defect path integrals to a hidden quantum-integrability identity\n\nThe 4d Chern–Simons path integral is coupled to chiral and anti-chiral one-dimensional defect variables at discretized sites.\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n&\\int \\ma...	conclusion_reconstruction	From discretized defect path integrals to a hidden quantum-integrability identity	04746	R	842f7484e02b38a59cdab9b5138fffae8aea03928279c62f71a7c405f5fadc49
scideriv-O-00241	[ { "role": "user", "content": "# From doubling operator to symmetric and antisymmetric eigenvectors\n\nThe recursive block operation is named as the operator D_L acting on a square matrix A\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\mathcal D_L(A)=\n\\begin{pmatrix}\nA&-L \\ \\mathrm{Id}\\\\\n-L...	conclusion_assessment	From doubling operator to symmetric and antisymmetric eigenvectors	00241	O	d28cdd745ce8050f3014f9375bdb5c98ed4dc4c2206221c4aaec1b62d7efdb15
scideriv-O-03755	[ { "role": "user", "content": "# Relation between the mixed Hull-curvature formulas\n\nThe extracted source-node text is a continuation row and is not standalone mathematics. Preserve it verbatim as node evidence:\n\n```formula-fragment\n&=\\nabla_\\rho\\D_{\\alpha\\mb}{}^\\nu\n```\n\nFor this task, the auth...	conclusion_assessment	Relation between the mixed Hull-curvature formulas	03755	O	afe214ab1cc740e64acb4e40c77d361069ca51d921f4bd47e556467556a280c9
scideriv-R-01346	[ { "role": "user", "content": "# From the time-evolved two-qubit angular state to the Newtonian action increment\n\nIn the large-angular-momentum regime, the evolved source–probe state contains a relative frame-dragging phase between its same-sign and opposite-sign angular-momentum branches.\n\n## Starting f...	conclusion_reconstruction	From the time-evolved two-qubit angular state to the Newtonian action increment	01346	R	41c7cbc10adefce859a3cff63a4c3f70607e386b2cf9913f400277fca843c5e3
scideriv-O-08252	[ { "role": "user", "content": "# Derive bulk T y symmetry from T y after bulk Hamiltonian\n\nFor a non-boundary state, T_y shifts the y coordinate in each hopping term\n\n## Given starting formula\n\n```latex\n\\begin{aligned}\nT_yh\\ket{x,y}\n=\n-t\\Big[\n&\\ket{x-1,y+1}\n+\\ket{x+1,y+1}\n\\nonumber\\\\\n&+...	conclusion_assessment	Derive bulk T y symmetry from T y after bulk Hamiltonian	08252	O	28de3b191ab0acccfd4a05c6d0f387b007f2fc3456d955809e907420c4a233b0
scideriv-O-04353	[ { "role": "user", "content": "# From three D and four B partition sums to final two D and three B partition sums\n\nThis version uses a three-part partition of A2 and a constrained four-part partition of B\n\n## Starting formula\n\n```latex\n\\sul{ A_2 = \\cup\\, _{a=1}^{3} D_a }{} \\sul{ B = \\underse...	conclusion_assessment	From three D and four B partition sums to final two D and three B partition sums	04353	O	397ccef455c2365fcf4b9c06ef76a56884ac953f476fd94dc585a0d2937d5073
scideriv-O-01825	[ { "role": "user", "content": "# From Parameter specialization A and B to Central product P0\n\nA proof step specializes shifts to A=0 and B=beta epsilon_1\n\n## Starting formula\n\n```latex\nA=0 , \\qquad B=\\beta\\eps_1\n```\n\n### Paper-local LaTeX definitions used by the displayed formulas\n\n```latex-de...	conclusion_assessment	From Parameter specialization A and B to Central product P0	01825	O	930fbd2b2e94c4d7a6e46c94aa75cee954c6574e352d28faa2ee7439734de0b0
scideriv-O-03381	[ { "role": "user", "content": "# From classical connection matrix A(z) to semiclassical quantization diagram\n\nThe coefficients of the classical equation are collected into a Laurent polynomial connection matrix\n\n## Starting formula\n\n```latex\nA(z)=\\left(\\frac{u}{z^{p+1}}+\\frac{A_p}{z^p}+\\cdots+\\fr...	conclusion_assessment	From classical connection matrix A(z) to semiclassical quantization diagram	03381	O	6014d589b52ccea90ea80db156f5ad155f2787118d4426d84391b679c058aaf9
scideriv-R-03391	[ { "role": "user", "content": "# From the kite maximal-cut representation to \\(P_0\\)-localization residue points\n\nConsider the five-propagator kite sector with inverse propagators \\(\\{\\sigma_2,\\sigma_3,\\sigma_4,\\sigma_6,\\sigma_7\\}\\) and the two ISPs \\(z_1=\\sigma_5\\) and \\(z_2=\\sigma_9\\).\n...	conclusion_reconstruction	From the kite maximal-cut representation to \(P_0\)-localization residue points	03391	R	9e4d57766cb598f21821e7a6f1df3029cb18f9a38ef0953a6c6c9be265aa89ce
scideriv-O-03577	[ { "role": "user", "content": "# From covariant derivative of deformation tensor to corrected mixed Hull curvature equals covariant variation\n\nThe mixed Hull-curvature expression is written as a covariant derivative of the deformation tensor\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\begin{spl...	conclusion_assessment	From covariant derivative of deformation tensor to corrected mixed Hull curvature equals covariant variation	03577	O	c50bf2f9dc5f89b680bebd754cd40258ec947e094d71621eecf15c580460ee76
scideriv-O-03557	[ { "role": "user", "content": "# From polar-coordinate integral form to smeared-field arcsine law\n\nA polar-coordinate representation is used in the derivation of the sinh integral identity\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\int_0^{\\infty}\\int_0^{\\infty}\\frac{du\\,dv}{uv}e^{-\\frac1...	conclusion_assessment	From polar-coordinate integral form to smeared-field arcsine law	03557	O	3e36b797003fd2fb90179e58e42565c928e2ce876da19d0d44d08768f58b8fb0
scideriv-O-04863	[ { "role": "user", "content": "# From A r Riccati equation to A r WKB periods Q 2 through Q 5\n\nRiccati equation obtained by substituting the WKB ansatz into the expanded A_r ODE\n\n## Starting formula\n\n```latex\n(\\epsilon\\partial_{x}+S)^{r}S+\\sum_{i=2}^{r+1}\\frac{L_{i}}{x^{i}}(\\epsilon\\partial_{x}+...	conclusion_assessment	From A r Riccati equation to A r WKB periods Q 2 through Q 5	04863	O	084dc6ed47941c88d0e9e58c0752285b60980290fba622352d4dbdf15e6b871d
scideriv-R-05534	[ { "role": "user", "content": "# From derivative-invariant definitions to the thermal effective action\n\nConsider the thermal EFT on the two-dimensional base \\(\\mathcal M_2\\) of a circle bundle in the regime where the circle is small. The local geometric data are built from the Weyl-rescaled metric \\(\\...	conclusion_reconstruction	From derivative-invariant definitions to the thermal effective action	05534	R	a3e85a67ab36a126aba88156d3e29adb2d455a0461eb78aa2e5ad59359f465a8
scideriv-R-02286	[ { "role": "user", "content": "# From the abstract CCR product to sections of the pure-state bundle\n\nLet \\((\\mathcal A,+)\\) be an abelian group and let \\(\\varpi:\\mathcal A\\times\\mathcal A\\to S^1\\) be the normalized bicharacter defining the CCR \\(*\\)-algebra \\(\\Delta(\\mathcal A,\\varpi)\\). I...	conclusion_reconstruction	From the abstract CCR product to sections of the pure-state bundle	02286	R	a9d8c54651abd77090354cfd4f96e63199b6fa9779422d4769acce50309d9433
scideriv-O-01612	[ { "role": "user", "content": "# From instantaneous Legendre transform to vertical derivative formula\n\nThe instantaneous Legendre transform is defined as the vertical fiber derivative of the instantaneous Lagrangian\n\n## Starting formula\n\n```latex\n\\mathbb{F}\\mathcal{L}_{t,\\zeta}(\\varphi_t,\\dot{\\v...	conclusion_assessment	From instantaneous Legendre transform to vertical derivative formula	01612	O	203cb945900b8e6a6884373397b128b2f93417e71c58214168f1f89c1e139085
scideriv-R-06579	[ { "role": "user", "content": "# From Initial-branch sign data to small-\\(\\xi\\) angular asymptotics\n\nOn each monotonic branch \\(0<\\xi<1\\), introduce\n\n```latex\n\\xi=\\sin^2\\varphi,\\qquad \\eta=\\tan\\frac{\\theta}{2}.\n```\n\nThe initial value \\(\\varphi_0\\) fixes the signs by the exact startin...	conclusion_reconstruction	From Initial-branch sign data to small-\(\xi\) angular asymptotics	06579	R	6ecfa458a24c35048df9a7333e2ab3127942153b7d62d132a6039d601db39572
scideriv-R-06522	[ { "role": "user", "content": "# From effective curvature parameters to the visible-brane graviton expansion\n\n## Starting formula\n\nThe effective curvature and back-reaction parameters are\n\n```latex\nK = k + \\frac{10}{3}\\alpha k^3 \\;\\;\\;\\;;\\;\\;\\;\\; \\varepsilon = \\alpha b_0\n```\n\nThe correc...	conclusion_reconstruction	From effective curvature parameters to the visible-brane graviton expansion	06522	R	94ce84b1df8f08678ba99582af47e3ea7cb45b7f578b5a518c770e1adf26f2e7
scideriv-O-06131	[ { "role": "user", "content": "# From commutator form of RLL to lattice Maillet-type Poisson bracket\n\nThe RLL relation is rearranged as a commutator of two local L-operators involving 1 minus R\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\begin{split}\n[L_{n,1}(z),L_{n,2}(w)]&=\\big(1-R_{12}(z-w...	conclusion_assessment	From commutator form of RLL to lattice Maillet-type Poisson bracket	06131	O	f4b912dca39f307ed13a63df8cffbf304ff232773763b2308f7c58842c7f6b6d
scideriv-O-07561	[ { "role": "user", "content": "# From First A r WKB coefficients to Evaluation of A r Q 2\n\nSolutions for S_0, S_1, and S_2 in the chosen WKB branch\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\begin{split}\nS_{0}(x)&=-p(x)^{\\frac{1}{h}},\\\\\nS_{1}(x)&=\\frac{h-1 }{2 h}\\partial_{x}\\log(p),\\\...	conclusion_assessment	From First A r WKB coefficients to Evaluation of A r Q 2	07561	O	f5dcdf2e4d997ddddc6d2d2cd1b3ed8bb63b0d83899aa763d1aa58c6d0a79d2b
scideriv-O-04527	[ { "role": "user", "content": "# From SCI Hessian block matrix to Absolute determinant identification\n\nThe Gaussian fluctuation Hessian is expressed through second derivatives of the leading potential\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\Big(\\mathbb{H}[U;\\Delta]\\Big)_{I,J}=\\begin{pma...	conclusion_assessment	From SCI Hessian block matrix to Absolute determinant identification	04527	O	7179c1e9d120929908f5fb56e43e42b81cc545eb55632412c9de15764a4ef87e
scideriv-O-05110	[ { "role": "user", "content": "# From D0 Lagrangian in proper coordinate to D0 crossover time\n\nUsing y rewrites the reduced D0 Lagrangian as a massive-particle form\n\n## Starting formula\n\n```latex\n{\\cal L}= \\frac{k}{2} \\sqrt{r^2-\\frac{\\dot{y}^2}{L_{\\text{ABJM}}^2}}.\n```\n\n## Local source contex...	conclusion_assessment	From D0 Lagrangian in proper coordinate to D0 crossover time	05110	O	d40cffd24fc0a3dcdf584ccaec0155abf5b034c53cfa94b8e32bcce2ce71fb79
scideriv-R-04397	[ { "role": "user", "content": "# From the general second-family corank to the odd-step linking-number-one asymptotic\n\n## Starting formula\n\n```latex\n\\Cork\\bigl(C_n^{(L)}\\bigr)=\n\\binom{n-1}{\\frac{n-1}{2}+\\frac{1}{L}}\\nn\n```\n\n### Paper-local LaTeX definitions\n\n```latex-definitions\n\\newcomman...	conclusion_reconstruction	From the general second-family corank to the odd-step linking-number-one asymptotic	04397	R	1e7d821869a013a0fe90a7ce86803edd10cfc1b2696585a94f6421e0d7495be5
scideriv-O-07329	[ { "role": "user", "content": "# From Full Kähler potential with matter to Matter-zero reduced Kähler potential\n\nThe Kähler potential is considered as a function of moduli-like fields Φ and charged matter fields Ψ\n\n## Starting formula\n\n```latex\nK=K(\\Phi,\\bar\\Phi,\\Psi,\\bar\\Psi) \\, .\n```\n\n## L...	conclusion_assessment	From Full Kähler potential with matter to Matter-zero reduced Kähler potential	07329	O	b731c6ba35a2f32bdaa32c626653799739ee94419b1654b994565586784bcd31
scideriv-O-02403	[ { "role": "user", "content": "# From factorization of H1 to first partner potential from superpotential\n\nThe first SUSY Hamiltonian is factorized into adjoint first-order operators\n\n## Starting formula\n\n```latex\nH_1=\\mathcal{A}^\\dagger(a_1)\\,\\mathcal{A}(a_1)\n```\n\n## Local source context\n\nThi...	conclusion_assessment	From factorization of H1 to first partner potential from superpotential	02403	O	af98569a12034698559eae5dbc2857311d5be8e188d8dbd2ca27904c89f2f47d
scideriv-R-04649	[ { "role": "user", "content": "# From the supersymmetric potential identity to the BPS on-shell action\n\n## Starting formula\n\nAlong the complexified supersymmetric trajectory, the thermodynamic potentials obey\n\n```latex\n\\beta \\left(1+ \\Omega_1 +\\Omega_2 -2 \\Phi \\right) = 2\\pi i\\ .\n```\n\nIntr...	conclusion_reconstruction	From the supersymmetric potential identity to the BPS on-shell action	04649	R	1ba3a9c03fc5c0b673bfa579969ff2078cccc7ba5e9e2de042786ef01319f0bf
scideriv-O-09643	[ { "role": "user", "content": "# Research problem: determine the selected \\(A\\)-\\(B\\) difference denominator\n\nLet \\(A=\\{a_1,\\ldots,a_M\\}\\) and \\(B=\\{b_1,\\ldots,b_N\\}\\) be collections of mutually pairwise distinct complex numbers. Use the paper convention \\(x_{ab}=x_a-x_b\\).\n\nFor an ordere...	conclusion_assessment	Research problem: determine the selected \(A\)-\(B\) difference denominator	09643	O	b9422d9d18264825f0c40acf36e9e583d096a5104ac12b1c4a95014f76238c22
scideriv-O-04770	[ { "role": "user", "content": "# From Cartesian product adjacency to product zero-mode condition\n\nAdjacency matrix of a graph Cartesian product decomposes as a Kronecker sum\n\n## Starting formula\n\n```latex\n\\Adj(G\\square H)=\\Adj(G)\\otimes \\mathrm{Id}+\\mathrm{Id}\\otimes\\Adj(H)\n```\n\n### Paper-l...	conclusion_assessment	From Cartesian product adjacency to product zero-mode condition	04770	O	d12cc57352a5b33ef88fefadc056417d14156fdd5bdcb9ef27698110f9a2b125
scideriv-O-09586	[ { "role": "user", "content": "# From odd-spin Ising rho before summation to odd-spin Ising rho after summation\n\nThe odd-spin Ising log MFF is written with an infinite hyperbolic sum still present\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\rho_{\\alpha_{2n-1}}^{{\\rm Ising}}(\\theta) &= \\log\...	conclusion_assessment	From odd-spin Ising rho before summation to odd-spin Ising rho after summation	09586	O	63aec88e07853c37b57552810722c41255ca15dde611d0008584290944a925ca
scideriv-O-03251	[ { "role": "user", "content": "# From the full inverse-propagator family to merged localized master integrands\n\nThe source begins with the following printed definition of the nine inverse propagators.\n\n## Starting endpoint: full-family inverse propagators\n\n```latex\n\\begin{aligned}\n\\sigma_1&=(k1_p1)...	conclusion_assessment	From the full inverse-propagator family to merged localized master integrands	03251	O	0a8bec74a32cab2150f03b741e8da35721c0b89a98b50c8969a14ed2643cf372
scideriv-R-03529	[ { "role": "user", "content": "# Relation between a forbidden-pair clause and a hidden reverse implication\n\nOutcome labels lie in \\(\\mathbb Z_2\\). The complement of a literal is defined by toggling its outcome bit:\n\\[\n\\neg(A,a)=(A,a\\oplus 1),\n\\qquad\n(x\\oplus 1)\\oplus 1=x.\n\\]\n\nFor every for...	conclusion_reconstruction	Relation between a forbidden-pair clause and a hidden reverse implication	03529	R	94c3f5415e627d98c4349d397d1265467a8e077466c7525c757d716f2594734d
scideriv-R-07202	[ { "role": "user", "content": "# Boundary magnetic-translation comparison\n\nConsider a rectangular magnetic torus with flux per plaquette\n\n```latex\n\\alpha=\\frac{k}{N_xN_y}.\n```\n\nAt the \\(x\\)-boundary, the magnetic translation uses the transition function\n\n```latex\nT_x\\ket{N_x-1,y}\n=\ne^{-2\\p...	conclusion_reconstruction	Boundary magnetic-translation comparison	07202	R	5e4ca268b20669c5301975d852222dbfd18a5cd899246233d40573bd32fd76ad
scideriv-R-02661	[ { "role": "user", "content": "# From one collinear double-soft factor to the assembled \\(\\epsilon\\)-expansion\n\n## Starting formula\n\nThe collinear limit of the leading-leading ordered double-soft factor is\n\n```latex\n\\lim_{1\|\|2}\\mathbb{S}_{12}^{(0,0)} = \\frac{1}{\\epsilon^2}\\frac{1}{t(1-t)}S_P...	conclusion_reconstruction	From one collinear double-soft factor to the assembled \(\epsilon\)-expansion	02661	R	99041e4f61bc8fab91f211ff266b1414f81cfc6411e51a0e769a1f9351a42d9e
scideriv-O-05293	[ { "role": "user", "content": "# From Potential split to Polarization-resolved intensity\n\nElectromagnetic potential is decomposed into Coulomb background and gravitational-wave perturbation\n\n## Starting formula\n\n```latex\nA^\\alpha={\\bar A^\\alpha}+\\tilde A^\\alpha\n```\n\n### Paper-local LaTeX defin...	conclusion_assessment	From Potential split to Polarization-resolved intensity	05293	O	09b3e530b8cdf66e1e04988ab012983620f0f3a4203a4e29c5be83e9ea91ea43
scideriv-O-00406	[ { "role": "user", "content": "# From Universal gauge connection to Mixed Bianchi component\n\nExtends the fibre gauge connection by a horizontal component along moduli space\n\n## Starting formula\n\n```latex\n\\IA=A+A_a^{\\sharp}\\dd y^a~,\n\\qquad\n\\mathbb{F}=\\mathbb{d}\\IA+\\IA^2~.\n```\n\n### Paper-lo...	conclusion_assessment	From Universal gauge connection to Mixed Bianchi component	00406	O	12af09a0cddb7ce16e0144451ddd8e85c1bcac2e7dab1325c5414e1af628d88b
scideriv-O-02563	[ { "role": "user", "content": "# From Large-radius normalized F to q0 attractor equation\n\nThe normalized generalized prepotential is specialized to cubic, genus-one, and G-dependent perturbative terms\n\n## Starting formula\n\n```latex\nF(Y, \\Upsilon) = \\frac{D_{a b c} Y^a Y^b Y^c}{Y^0} + d_a\\, \\frac{Y...	conclusion_assessment	From Large-radius normalized F to q0 attractor equation	02563	O	305bab263e326694db6b9fcd6118f1a39c1f1987bf82d577e093b8ee5800e27c
scideriv-O-06924	[ { "role": "user", "content": "# From Photon-orbit angular frequency to Damping time\n\nThe photon angular frequency is set by the square root of f/rho^2 at the orbit\n\n## Starting formula\n\n```latex\n\\Omega_{\\rm ph}=\\sqrt{\\frac{f(3M)}{\\rho_{\\rm ph}^2}}=\\frac{\\sqrt{\\Xi(u)}}{3M}.\n```\n\n## Local s...	conclusion_assessment	From Photon-orbit angular frequency to Damping time	06924	O	9c47e5657ca8ecdfc57f56fe9a3c2c945720fa34d3d1452560e152c0f67cd1b1
scideriv-R-07158	[ { "role": "user", "content": "# Relate a local quadratic congruence to a two-adic shifted count\n\nThe source treats an odd-prime local congruence and then states that the case \\(p=2\\) is similar. Use only the formulas and conditions displayed below to analyze the relation. The supplied material does not ...	conclusion_reconstruction	Relate a local quadratic congruence to a two-adic shifted count	07158	R	e5f93e739c7e47df7b8c7781af80525a45b56f7620b6e9660491f0d6ea3eb655
scideriv-R-06581	[ { "role": "user", "content": "# Consistency audit of the \\(A_r^{(1)}\\) WKB recursion\n\nFor the \\(A_r^{(1)}\\) scalar ODE, let\n\\[\nh=r+1,\\qquad p(x)=x^{hM}-1.\n\\]\nThe source displays the expanded equation\n\\[\n\\left[(\\epsilon\\partial_x)^h+\\sum_{i=2}^{h}\\epsilon^i\\frac{L_i}{x^i}(\\epsilon\\par...	conclusion_reconstruction	Consistency audit of the \(A_r^{(1)}\) WKB recursion	06581	R	2b944148d8a15ec7ed1ebd3674b9e819c420f92b2e85c94c2580bfc2bff4c47b
scideriv-O-03163	[ { "role": "user", "content": "# From pairing verification for cotangent map to cotangent-density orbit embedding\n\nPairing computation showing that the Madelung transform yields dθ⊗ρ in the dual variables\n\n## Starting formula\n\n```latex\n\\langle\\Mom\\circ \\mathbf{S}^{-1}(\\varrho,\\theta), (v,f) \\ra...	conclusion_assessment	From pairing verification for cotangent map to cotangent-density orbit embedding	03163	O	8baa5df36567daab94a9531f6f7a16e4eec2e3ad30ade271a1a975a0d29f15b1
scideriv-R-00156	[ { "role": "user", "content": "# From superpotential charge constraints to a localized SCI\n\nConsider a three-dimensional \\(\\mathcal N=2\\) theory with two chiral multiplets and superpotential\n\n```latex\nW=\\Psi_1(\\Psi_2)^2\\,.\n```\n\nEach \\(\\Psi_i\\) has U(1)\\(_R\\) and U(1)\\(_F\\) charges \\((r_...	conclusion_reconstruction	From superpotential charge constraints to a localized SCI	00156	R	1db6fb594f28145d63d6ed9fd0bbdfc08eead493006dc90e6fdf49dcab2a0abd

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SciDeriv

17,577 scientific derivation prompts for on-policy distillation.

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

SciDeriv is the large-scale training corpus used in ESCD. It contains prompts about relationships between mathematical and scientific formulas, including the assumptions and conventions needed to justify a conclusion. The source pipeline constructs reasoning questions from mathematical and scientific documents. The release preserves every training prompt, including its chat role and text.

Task	Rows	Meaning
`conclusion_assessment`	10,000	Assess a supplied candidate conclusion against the displayed relations.
`conclusion_reconstruction`	7,577	Reconstruct the strongest conclusion supported by the information given.
Total	17,577	One `train` split; no supplied validation or test split.

This is prompt-only data. There are no reference completions or scalar rewards. The old training file used empty-string labels; those placeholders are omitted here. An empty label must not be interpreted as an incorrect answer or reward zero. Some problems intentionally ask the model to identify underdetermination.

Load

from datasets import load_dataset
data = load_dataset("mytcl/SciDeriv", split="train")
messages = data[0]["prompt"]

For JSONL consumers, download data/train.jsonl. Parquet is the canonical file used by the dataset viewer; both files contain the same records in the same order.

Fields

Field	Type	Description
`id`	string	Unique release ID: `scideriv-O-<source_id>` or `scideriv-R-<source_id>`.
`prompt`	list of role/content records	Original chat messages, unchanged.
`task_type`	string	One of the two task types above.
`topic`	string	Original topic/title.
`source_id`	string	Original example identifier, which may recur across task types.
`source_variant`	string	Original `O` (assessment) or `R` (reconstruction).
`prompt_sha256`	string	SHA-256 of compact UTF-8 JSON messages (`ensure_ascii=False`).

If constructing held-out splits, group the two variants by source_id at minimum; randomly splitting rows can separate related questions. This does not establish document-level independence: source-document identifiers are not provided in the processed input. Do not claim a document-disjoint evaluation from these IDs alone.

Scope and limitations

SciDeriv is not the DAPO-Math + TACO mixture used for the small-model experiments in Table 1. That mixture is available from the original BPM release. This release supplies no model outputs and no automatic correctness guarantee for the generated prompts. Dataset preparation checks structure, counts and exact preservation, not the scientific validity of every problem. Original document PDFs are not bundled.

License

The authors release this dataset under CC BY 4.0. Attribute SciDeriv and ESCD when redistributing or adapting it. Third-party source material, where applicable, remains subject to its original rights.

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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Paper for mytcl/SciDeriv

Beyond Token Alignment: Event Completion for Cross-Tokenizer On-Policy Distillation

Paper • 2609.34738 • Published 15 days ago