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 |
scideriv-O-06806 | [
{
"role": "user",
"content": "# From Definition of phi error term to Signed Fourier integral integration by parts\n\nThe function phi is written as explicit v,w-dependent terms plus a remaining signed integral\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n&\\phi_{\\k_2} (v,\\wedge; x,s) := \\frac{i ... | conclusion_assessment | From Definition of phi error term to Signed Fourier integral integration by parts | 06806 | O | 392e82d14aa33e2b4224ed2d56478e11cd7f71a61b099bbb526260cd1fdd67ca |
scideriv-R-01617 | [
{
"role": "user",
"content": "# From angular momenta to the rotation parameter\n\nThe BMPV black hole carries one independent combination of the two possible five-dimensional angular momenta. With \\(J_{\\pm}=(J_1\\pm J_2)/2\\), the source gives the following relation between the nonzero combination, the pa... | conclusion_reconstruction | From angular momenta to the rotation parameter | 01617 | R | c4795ee7cf20568cd27ddca129c5145aa0cbe510566d2e2e6a53431c91d37e81 |
scideriv-O-02038 | [
{
"role": "user",
"content": "# From constraints on analytic blocks to evenness of g n and h n\n\nUnitarity, crossing, Watson's equation, and form-factor crossing impose functional constraints on the block representation\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\begin{split}\n\\delta(\\theta) ... | conclusion_assessment | From constraints on analytic blocks to evenness of g n and h n | 02038 | O | caaf78837ab5295a734e68e199f64b0b812e43b498e505296bcf12a511d600eb |
scideriv-O-02941 | [
{
"role": "user",
"content": "# From collision invariants to weak moment equation\n\nThe Boltzmann-type collision operator annihilates mass, linear momentum, kinetic energy and generalized angular momentum invariants\n\n## Starting formula\n\n```latex\n\\int_{\\mathcal{M}} \\int_{\\mathbb{R}^3} \\int_{\\mat... | conclusion_assessment | From collision invariants to weak moment equation | 02941 | O | a4643956c6d43201d37771b1f45d6502cf27084734a4998d9cfd2493f8161341 |
scideriv-R-03909 | [
{
"role": "user",
"content": "# From the expanded \\(A_r\\) ODE to the evaluation of \\(Q_2\\)\n\nConsider the expanded scalar ODE used in the \\(A_r^{(1)}\\) WKB analysis:\n\n```latex\n\\left[(\\epsilon\\partial_{x})^{h}+\\sum_{i=2}^{h}\\epsilon^{i}\\frac{L_{i}}{x^{i}}(\\epsilon\\partial_{x})^{h-i}+(-1)^{r... | conclusion_reconstruction | From the expanded \(A_r\) ODE to the evaluation of \(Q_2\) | 03909 | R | d5f800bad9d422b34e859c6a20ddd677d112c378f7c1790347117cae7116f47d |
scideriv-O-08416 | [
{
"role": "user",
"content": "# Derive Shear-only correction truncation from Linearized correction amplitude\n\nParametrizes the nonequilibrium distribution correction by a small scalar amplitude multiplying equilibrium\n\n## Given starting formula\n\n```latex\n\\delta f_k^\\pm = f_{0k}^\\pm \\,\\delta y_k^... | conclusion_assessment | Derive Shear-only correction truncation from Linearized correction amplitude | 08416 | O | 2c68918a0a5fdf575ab4690911e4358c4d27e0585bccd857938bee935e4a9fb2 |
scideriv-O-04621 | [
{
"role": "user",
"content": "# From U(1)-twisted reference wavefunction to right eigenmode with U(1) phase\n\nA relative phase is inserted on the right half of the defect-free wavefunction basis\n\n## Starting formula\n\n```latex\n\\phi_k^0(m) \\to \\phi_k^{0,U(1)}(m) =\n\\begin{cases}\n\\phi_k^0(m),\n& 1 ... | conclusion_assessment | From U(1)-twisted reference wavefunction to right eigenmode with U(1) phase | 04621 | O | 81f8a7ae96b8e176190ae9818064a9c3fcc3ac233404c4d00351b88b135fb7fb |
scideriv-O-00316 | [
{
"role": "user",
"content": "# From Infinitesimal deformation of action to Flat-space mass-ratio limit\n\nA single relevant bulk scalar deformation changes the AdS action by insertion of its integrated operator\n\n## Starting formula\n\n```latex\nS(\\lambda+\\delta\\lambda)=S(\\lambda)+\\delta\\lambda\\int... | conclusion_assessment | From Infinitesimal deformation of action to Flat-space mass-ratio limit | 00316 | O | 047a4f965d628738067ef1c2bebf4e625390d1e382a75bd51fc07f7a95aae95d |
scideriv-O-06819 | [
{
"role": "user",
"content": "# From Ising fusion rules to Ising periodic topological algebra\n\nThe simple objects +, -, and nu obey the Ising fusion category rules\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n& [+] \\otimes [+] = [+] \\,,\n&\\quad& [+] \\otimes [-] = [-] \\otimes [+] = [-] \\,,\n... | conclusion_assessment | From Ising fusion rules to Ising periodic topological algebra | 06819 | O | 34041870f85a0dbb6d1db3692e7c93a9e076493253966e09245b893a242e262a |
scideriv-R-02208 | [
{
"role": "user",
"content": "# From a one-instanton sector at a Borel singularity to its regular tail\n\nFor each singular point \\(\\omega\\) on an active ray \\(d\\), the source introduces an exponentially weighted formal sector with possible polar powers.\n\n## Starting formula\n\n```latex\nF^{(\\omega)... | conclusion_reconstruction | From a one-instanton sector at a Borel singularity to its regular tail | 02208 | R | 963a8ecf1783f58ecd0689f9f1e03d93816c3e44122642fb71cecfc19e35f200 |
scideriv-R-03310 | [
{
"role": "user",
"content": "# From projected effective evolution to a pure-state distance bound\n\nWithin the constrained computational subspace, consider the projected pulse evolution.\n\n## Starting formula\n\n```latex\n\\begin{aligned}\nU_{\\Gamma} &= \\mathcal{T} \\exp \\left( -i\\int_0^\\tau P H_t P ... | conclusion_reconstruction | From projected effective evolution to a pure-state distance bound | 03310 | R | d4067e2f45a2e758bde4eb646d2ebcde6f207953234622d06cd4a7f008c181c3 |
scideriv-O-07267 | [
{
"role": "user",
"content": "# From rank and block constraints on A to exact cut-charge weights\n\nThe identity upper block and rank condition constrain the positive-sector operator A\n\n## Starting formula\n\n```latex\nA_{\\mathbf1\\mathbf1}=I_{\\mathcal H_{\\mathbf1}},\\qquad\n\\mathrm{rank}(A)=\\dim\\ma... | conclusion_assessment | From rank and block constraints on A to exact cut-charge weights | 07267 | O | 753c2594518cf031824d9e2291d9caca1088912dd44c2e914ed4bdf0f01a9633 |
scideriv-R-01847 | [
{
"role": "user",
"content": "# From an IR-completed initial droplet function to initial quenched phase-space data\n\nWork in the Thomas–Fermi approximation at the quench time \\(t=-T\\). For this problem, restrict to\n\n\\[\ng>0, \\qquad 0<r<2,\n\\]\n\nso that the real power \\(|x-p|^r\\) is pointwise defi... | conclusion_reconstruction | From an IR-completed initial droplet function to initial quenched phase-space data | 01847 | R | 5495e844dc419c8a11a402b29329a7852cdc8951e191cf079b012ac1eb92f517 |
scideriv-O-08897 | [
{
"role": "user",
"content": "# Reconstruct differential cross-section measure from transition rate factorization\n\nThe scattering rate is written with an invariant kernel and four-momentum conservation\n\n## Given starting formula\n\n```latex\nW_{kk'\\to pp'}\n=\n\\chi(s,\\Theta_s)\\,\n\\delta^{(4)}(k+k'-... | conclusion_assessment | Reconstruct differential cross-section measure from transition rate factorization | 08897 | O | 1453dba8d9850a9b9836af26b657370043071e99e4045a7b2aa6dace83b71e96 |
scideriv-O-01669 | [
{
"role": "user",
"content": "# From ordered double soft factor notation to assembled collinear double soft expansion\n\nA product of individual positive-helicity soft factors is denoted by a double-soft symbol\n\n## Starting formula\n\n```latex\nS^{(m)}(1^+)S^{(n)}(2^+) \\equiv \\mathbb{S}_{12}^{(m,n)}.\n`... | conclusion_assessment | From ordered double soft factor notation to assembled collinear double soft expansion | 01669 | O | b438b9158e3dcd15116ab50d2c97ed63f67f5cb51f38ed800a88fec1e9344d67 |
scideriv-R-00795 | [
{
"role": "user",
"content": "# From a relevant deformation of a CFT in AdS to flat-space spectral data\n\nConsider a QFT in AdS defined by a single relevant bulk scalar deformation of a solvable CFT. Work in units in which the AdS radius is \\(R=1\\).\n\n## Starting formula\n\n```latex\nS_{\\text{QFT}}(\\l... | conclusion_reconstruction | From a relevant deformation of a CFT in AdS to flat-space spectral data | 00795 | R | 91ac8f796ee78f3a2948de581c9441571fc527cf9914b5a48a292e06bf1a5cf7 |
scideriv-R-01018 | [
{
"role": "user",
"content": "# Relation and derivation: field-action representation and iterated crossing recursion\n\nThe source represents the \\(n\\)-particle output component of the field action as a sum over incoming particle number.\n\n## Starting formula: field-action decomposition\n\n```latex\n\\Bi... | conclusion_reconstruction | Relation and derivation: field-action representation and iterated crossing recursion | 01018 | R | 3c054b6a392606d6c41286c09857d7830cb5c497eba3b243ef7007f322e3aab6 |
scideriv-O-03781 | [
{
"role": "user",
"content": "# From One-gate approximation bound to Per-gate physical error\n\nA product of Z waits and driven pulses approximates an intended indexed operation with errors from phase resolution and weak drive\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n&\\left\\| \\left(U_Z(\\tau... | conclusion_assessment | From One-gate approximation bound to Per-gate physical error | 03781 | O | a476bc7b6482fb8f713290024c5a41157ff6b8b5af4e63469a7df4034dd7abac |
scideriv-R-03001 | [
{
"role": "user",
"content": "# From odd-spin analytic blocks to the deformed phase and minimal-factor logarithm\n\nFor a pure deformation, take \\(g^{(0)}=0\\) and \\(\\epsilon=1\\). Before selecting the analytic blocks, the phase and logarithm of the deformation contribution to the minimal form factor are... | conclusion_reconstruction | From odd-spin analytic blocks to the deformed phase and minimal-factor logarithm | 03001 | R | 5be01b3514b7dad3c8c966a785b3be6c00213048c8908f838fdc3edc3e96e8fe |
scideriv-R-01195 | [
{
"role": "user",
"content": "# From the phase convention for \\(S\\) to a Fourier transform of the phase\n\nConsider an interacting factorized scattering theory with diagonal \\(S\\)-matrix. The scattering phase \\(\\delta(\\theta)\\) is introduced through the following convention.\n\n## Starting formula\n... | conclusion_reconstruction | From the phase convention for \(S\) to a Fourier transform of the phase | 01195 | R | 16a354510274a8492b9befc4dae70437598c68205de7cc4ccf51a16fa3286671 |
scideriv-R-04992 | [
{
"role": "user",
"content": "# Serre product and a hidden composed-current relation\n\nConsider distinct Dynkin-diagram nodes \\(a\\neq b\\), with \\(\\mathfrak m_{a,b}=1-\\mathfrak a_{a,b}\\). Work in the case\n\\[\n\\mathfrak b_{a,b}\\neq0,\\qquad \\mathfrak m_{a,b}=2,\\qquad a>b.\n\\]\n\n## Starting end... | conclusion_reconstruction | Serre product and a hidden composed-current relation | 04992 | R | 638358e6d4fe76e48cc7c3a175cb03ab42d3f6b18b669de0b5ec2f52cf8f95cd |
scideriv-O-08689 | [
{
"role": "user",
"content": "# Reconstruct A r Q3-I3 matching from A r WKB period Q3\n\nThe third A_r WKB period is proportional to s_3 and J_{2,3}\n\n## Given starting formula\n\n```latex\n\\begin{split}\nQ_{3} = \\frac{1}{h}J_{2,3}s_{3}.\n\\end{split}\n```\n\n## Local source context\n\n\\subsubsection*{T... | conclusion_assessment | Reconstruct A r Q3-I3 matching from A r WKB period Q3 | 08689 | O | 9908dbc0a15336853e2d2846db57da273ab3f730577e07abbabba2c07b6ce3f2 |
scideriv-O-02269 | [
{
"role": "user",
"content": "# From effective 4D gravitational action to explicit corrected Planck relation\n\nIntegrating over the extra dimension identifies the coefficient of the 4D Ricci scalar as M_Pl squared\n\n## Starting formula\n\n```latex\n\\mathcal S_{eff} \\supset \\underbrace{M_*^3\\int_0^b dy... | conclusion_assessment | From effective 4D gravitational action to explicit corrected Planck relation | 02269 | O | da2ffeb40ca1d8289586229225b69096154f75f2afd9f2a38932d1428a59e8a5 |
scideriv-O-08503 | [
{
"role": "user",
"content": "# Analyze the recursive free-field relation for the \\(WD_r\\) generator\n\nThe paper describes the \\(WD_r\\) theory as the result of adjoining one free boson to \\(WD_{r-1}\\), with\n\n```latex\nWD_r=\\{\\tilde{\\phi}_r \\} \\oplus WD_{r-1}\n```\n\nwhere\n\n```latex\n\\tilde{... | conclusion_assessment | Analyze the recursive free-field relation for the \(WD_r\) generator | 08503 | O | 639be44483bcb05cdc3ad1064ef700ca00e714a9824630b13d306f20f078f177 |
scideriv-O-08417 | [
{
"role": "user",
"content": "# Analyze the localization relation for the Wess–Zumino inflow term\n\nThe paper relates the following two FormulaForest endpoints.\n\n## Endpoint A: effective meromorphic one-form shift\n\n```latex\n\\omega \\mapsto \\omega_{\\text{eff}}=\\omega-\\frac{\\hbar}{4\\pi i}\\left(\... | conclusion_assessment | Analyze the localization relation for the Wess–Zumino inflow term | 08417 | O | aafbe8b435afdb43021b80cd3d768e218a9f32373c424b6573d08087d0ae9e9a |
scideriv-R-01319 | [
{
"role": "user",
"content": "# From non-holomorphic to perturbative holomorphic free energy\n\nThe full non-holomorphic topological-string free energy is expanded by genus in the string coupling.\n\n## Starting formula\n\n```latex\nF(z,\\bar{z};g_s) = \\sum_{g=0}^\\infty g_s^{2g-2} F_g(z,\\bar{z}).\n```\n\... | conclusion_reconstruction | From non-holomorphic to perturbative holomorphic free energy | 01319 | R | d64e39136943543b175a8f7a737d66c3dd3106f1e872e986060e44bb735f8f51 |
scideriv-O-03009 | [
{
"role": "user",
"content": "# From T-modulus kinetic term to fluctuation parametrization of T\n\nThe alpha no-scale geometry gives the kinetic term for the complex modulus T\n\n## Starting formula\n\n```latex\n{\\cal L}_{\\rm kin}\n\\;=\\;\n-\\,3\\alpha M_P^2\\,\n\\frac{\\partial_\\mu T\\,\\partial^{\\mu}... | conclusion_assessment | From T-modulus kinetic term to fluctuation parametrization of T | 03009 | O | 7f14b9d34282eed3982df66bbbca056dfa512653f2ff59a5784e9b43e5bd51d1 |
scideriv-O-08101 | [
{
"role": "user",
"content": "# Derive the closed near-degenerate gap equation\n\nConsider two boundary operators with scaling dimensions \\(\\Delta_1\\) and \\(\\Delta_2\\). Define\n\n\\[\n\\Delta_{21}\\equiv\\Delta_2-\\Delta_1,\n\\]\n\nand work in the near-degenerate regime\n\n\\[\n0<\\Delta_{21}\\ll |\\D... | conclusion_assessment | Derive the closed near-degenerate gap equation | 08101 | O | 52f16ff39b79eb6ce6d3e7524f67b0940c4980829264b117112dd7df316441ff |
scideriv-R-06948 | [
{
"role": "user",
"content": "# From zeroth-order longitudinal action pieces to a compact covariant hydrodynamic action\n\nExplicit dissipative and non-dissipative zeroth-order longitudinal action functionals are given below.\n\n## Starting formula\n\n```latex\n\\begin{aligned}\nS_{\\mathrm{diss}}[ \\xi_{\\... | conclusion_reconstruction | From zeroth-order longitudinal action pieces to a compact covariant hydrodynamic action | 06948 | R | 8a5e0cd4a15c5a92f6a2636181c34771e9d9ab2f12c30568342f9402d6587d80 |
scideriv-O-04747 | [
{
"role": "user",
"content": "# From integral representation of I(alpha) to resummed imaginary G0\n\nRepresents I(alpha) by a Schwinger-type integral and decomposes it into perturbative and nonperturbative parts\n\n## Starting formula\n\n```latex\n\\mathcal{I}(\\alpha)\\, =\\, \\frac{\\alpha^2}{4} \\sum_{n ... | conclusion_assessment | From integral representation of I(alpha) to resummed imaginary G0 | 04747 | O | ad9386f884c5eb224a12554260424a553e2f92ba0c4cefee9e1bb0c020b4c8e0 |
scideriv-O-01813 | [
{
"role": "user",
"content": "# From Static spherical line element to Exact circular energy\n\nThe regular black-hole geometry is written with lapse f and areal radius rho\n\n## Starting formula\n\n```latex\nds^2=-f(r)dt^2+\\frac{dr^2}{f(r)}+\\rho(r)^2\\left(d\\theta^2+\\sin^2\\theta\\,d\\varphi^2\\right)\n... | conclusion_assessment | From Static spherical line element to Exact circular energy | 01813 | O | 92efc0385ded9d4eec55d146443b00fed50c58d1380d65fb00b0f698bac50da5 |
scideriv-O-09083 | [
{
"role": "user",
"content": "# Reconstruct N=4 corrected sign sum from N=4 anomaly sign condition\n\nThe N=4 polarization signs must sum to one\n\n## Given starting formula\n\n```latex\n\\sigma_1+\\sigma_2+\\sigma_3=1.\n```\n\n## Local source context\n\nwhereas cancellation of the gauge-dependent \\(\\omeg... | conclusion_assessment | Reconstruct N=4 corrected sign sum from N=4 anomaly sign condition | 09083 | O | 21048e7588651969f84c57136f489a74ceed9c8566467bf4e24ea4e91d99a3e3 |
scideriv-R-03948 | [
{
"role": "user",
"content": "# From a fixed-boundary TL decomposition to the Fibonacci operator-algebra dimension\n\n## Starting data\n\nFor fixed boundary labels \\(b,c\\), an orthonormal basis of seed states gives\n\n```latex\n\\V_{bc}(N) = \\bigoplus_{k=0,\\frac 12,\\dots,\\frac N2}\n\\bigoplus_{i=1}^{(... | conclusion_reconstruction | From a fixed-boundary TL decomposition to the Fibonacci operator-algebra dimension | 03948 | R | 4c83a94e9c0f12811f78a62c8b218e9ac0418c644454c97d81727c17bb836486 |
scideriv-R-06067 | [
{
"role": "user",
"content": "# From sections with an infinity condition to a K-theoretic interpretation\n\nSections \\(F=(\\cdot,f(\\cdot))\\) are considered under the condition that \\(\\lim_{|p|\\to\\infty}f(p)\\) exists. The source states that the underlying section–function correspondence is homotopy-p... | conclusion_reconstruction | From sections with an infinity condition to a K-theoretic interpretation | 06067 | R | 9a0713df1f124c39cb13299f6d4542c72cb094f0d638fd1242588b6748011e17 |
scideriv-O-00029 | [
{
"role": "user",
"content": "# From bicharacter identities to abstract CCR product\n\nA bicharacter on an abelian group is specified by skew-conjugacy and multiplicativity in the second variable\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\varpi (z_1,z_2) \\;&=\\; \\overline{\\varpi(z_2,z_1)} \\... | conclusion_assessment | From bicharacter identities to abstract CCR product | 00029 | O | 9e62ef133cd03cdbef5b997ba70d631b811d442e2a1ed706653c23e5b7fe03d6 |
scideriv-R-03438 | [
{
"role": "user",
"content": "# Euclidean BTZ saddle and complex brane profile\n\nConsider the connected configuration for the $B\\bar B$ cylinder.\n\n## Starting geometry\n\nThe Euclidean BTZ metric is\n\n```latex\nds^2 = d\\chi^2 + \\left(\\frac{2\\pi}{L}\\right)^2\\sinh^2 \\chi \\,d\\tau^2 + \\left(\\fra... | conclusion_reconstruction | Euclidean BTZ saddle and complex brane profile | 03438 | R | fd3d1bcd20f55771a15d2f2fe41be180e083453ef216ca0cb532e0f220f38e3d |
scideriv-O-09049 | [
{
"role": "user",
"content": "# Reconstruct spin-resolved cumulative OPE asymptotic from two-dimensional crossed singularity\n\nThe four-point function has a product power-law singularity as both cross ratios approach the crossed limit\n\n## Given starting formula\n\n```latex\n\\lim_{|1-\\eta|\\rightarrow 0... | conclusion_assessment | Reconstruct spin-resolved cumulative OPE asymptotic from two-dimensional crossed singularity | 09049 | O | f8c68b1b9d17716c3925b9ccfebb31f2188e84e1f0e2f46fafa4043ae5165c00 |
scideriv-O-07138 | [
{
"role": "user",
"content": "# From Cubic polynomial coefficients to Orbifold leading entropy\n\nThe coefficients p2, p1, and p0 encode charges, angular momenta, the leading anomaly tensor, and the leading central charge\n\n## Starting formula\n\n```latex\n\\begin{aligned}\np_2\\,&=\\, -12 k^{(0)}{}^{IJK}r... | conclusion_assessment | From Cubic polynomial coefficients to Orbifold leading entropy | 07138 | O | 3c62bfc336ce181c8098609476226cb8480194c0e030acd7822f6ada6b827503 |
scideriv-R-00269 | [
{
"role": "user",
"content": "# From the \\(f(T)\\) action to effective torsion density and pressure\n\n## Starting formula\n\n```latex\n\\mathcal{S}=\\frac{1}{2\\kappa^2}\\int d^{4}x\\,|e|f(T)+\\mathcal{S}_{M}\n```\n\nHere \\(T\\) is the torsion scalar, \\(|e|=\\sqrt{-g}=\\det(e_\\mu{}^a)\\), and \\(\\kapp... | conclusion_reconstruction | From the \(f(T)\) action to effective torsion density and pressure | 00269 | R | 7cd2476a4fb644158031a98ce1a786e268360928d000d09fe653e5ab8bfa2850 |
scideriv-R-03437 | [
{
"role": "user",
"content": "# From the graviton mode-separation ansatz to a Euclidean radial propagator\n\nWork in the conformal extra-dimensional coordinate \\(\\xi\\) and in RS transverse-traceless gauge. Begin with the KK separation ansatz\n\n```latex\n\\tilde h_{\\mu\\nu}(x,\\xi) = \\hat h_{\\mu\\nu}(... | conclusion_reconstruction | From the graviton mode-separation ansatz to a Euclidean radial propagator | 03437 | R | da737d57299be798c3c1b756f277a42731adf435f325a709f9292b35bd04d758 |
scideriv-O-00006 | [
{
"role": "user",
"content": "# From phase convention for S to minimal form factor equations\n\nThe undeformed scattering matrix is written as a sign times an exponential phase\n\n## Starting formula\n\n```latex\nS(\\theta) = \\epsilon\\, e^{i \\delta(\\theta)}\\;.\n```\n\n## Intermediate relations and tran... | conclusion_assessment | From phase convention for S to minimal form factor equations | 00006 | O | d3e3093a36eea06876e4b6e5a3a98fb566b604478c2631f27b73ae6690bf7f07 |
scideriv-O-07476 | [
{
"role": "user",
"content": "# From D2-D6 attractor ansatz to D0 charge attractor equation\n\nFor the no-D4 configuration the rescaled special coordinates obey a real/imaginary split fixed by p0\n\n## Starting formula\n\n```latex\nCX^0= \\text{Re}\\; CX^0 + i\\, \\frac{p^0}{2}\\, , \\qquad CX^a= \\bar{C} \... | conclusion_assessment | From D2-D6 attractor ansatz to D0 charge attractor equation | 07476 | O | 82dee768792d56b84706ed72ea518578e7d09611c7545170b27dca9d1ddf8f73 |
scideriv-O-08120 | [
{
"role": "user",
"content": "# Derive Product U times X from X formal solution\n\nThe omega-dependent factor X has a regular omega-series, an irregular exponential, and a logarithmic power\n\n## Given starting formula\n\n```latex\n\\widehat{X}(\\omega)=(1+\\sum_{m=1} C_m \\omega^m) \\cdot e^{\\frac{-u^{(2)... | conclusion_assessment | Derive Product U times X from X formal solution | 08120 | O | 3b317d6a0c739c0327f13c6a9b5d26e9899c50a9c04a4b1f53e43fa31dd748ae |
scideriv-O-02518 | [
{
"role": "user",
"content": "# From variation of momentum component to component-to-orbit momentum map\n\nVariational computation showing that the component Hamiltonian generates the infinitesimal action\n\n## Starting formula\n\n```latex\n\\begin{aligned}\n\\frac{d}{d\\epsilon}H_{(v,f)}(\\psi_\\epsilon) &... | conclusion_assessment | From variation of momentum component to component-to-orbit momentum map | 02518 | O | 3ce7c9a2bbdfa50637ce7e8d5bef5cf6e5ab9afde22ad686609fe3fcae9a480c |
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