% Bibliographic metadata follows the cited manuscripts and their bibliographies.

@book{vershynin-2018,
  author    = {Vershynin, Roman},
  title     = {High-Dimensional Probability: An Introduction with Applications in Data Science},
  publisher = {Cambridge University Press},
  series    = {Cambridge Series in Statistical and Probabilistic Mathematics},
  volume    = {47},
  year      = {2018},
  doi       = {10.1017/9781108231596},
  url       = {https://webapps.math.uci.edu/~rvershyn/papers/HDP-book/HDP-1.pdf},
  note      = {First edition; numbered locators refer to this edition}
}

@article{brenier-1991,
  author  = {Brenier, Yann},
  title   = {Polar factorization and monotone rearrangement of vector-valued functions},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {44},
  number  = {4},
  pages   = {375--417},
  year    = {1991},
  doi     = {10.1002/cpa.3160440402},
  url     = {https://www.ceremade.dauphine.fr/~carlier/Brenier91.pdf}
}

@article{mccann-1995,
  author  = {McCann, Robert J.},
  title   = {Existence and uniqueness of monotone measure-preserving maps},
  journal = {Duke Mathematical Journal},
  volume  = {80},
  number  = {2},
  pages   = {309--323},
  year    = {1995},
  doi     = {10.1215/S0012-7094-95-08013-2},
  url     = {https://www.math.toronto.edu/~mccann/papers/monotone.pdf}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original equations (5)--(6): Gaussian pairing, including repeated variables.
@article{isserlis-1918,
  author  = {Isserlis, L.},
  title   = {On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables},
  journal = {Biometrika},
  volume  = {12},
  number  = {1--2},
  pages   = {134--139},
  year    = {1918},
  doi     = {10.1093/biomet/12.1-2.134},
  url     = {https://academic.oup.com/biomet/article-abstract/12/1-2/134/193428}
}

@misc{collins-woodfin-le-survey-2025,
  author        = {Collins-Woodfin, Elizabeth W. and Le, Han Gia},
  title         = {Free energy fluctuations in {SK} and related spin glass models: A literature survey},
  year          = {2025},
  eprint        = {2510.26960},
  archivePrefix = {arXiv},
  primaryClass  = {math.PR},
  doi           = {10.48550/arXiv.2510.26960},
  howpublished  = {\url{https://arxiv.org/abs/2510.26960v3}},
  url           = {https://arxiv.org/abs/2510.26960v3},
  note          = {Version 3, November 7, 2025}
}

@article{chatterjee-lower-2019,
  author        = {Chatterjee, Sourav},
  title         = {A general method for lower bounds on fluctuations of random variables},
  journal       = {Annals of Probability},
  volume        = {47},
  number        = {4},
  pages         = {2140--2171},
  year          = {2019},
  doi           = {10.1214/18-AOP1304},
  url           = {https://arxiv.org/abs/1706.04290v3},
  note          = {Statement locators use \url{https://arxiv.org/abs/1706.04290v3}}
}

@inproceedings{anari-koehler-vuong-2024,
  author        = {Anari, Nima and Koehler, Frederic and Vuong, Thuy-Duong},
  title         = {Trickle-Down in Localization Schemes and Applications},
  booktitle     = {Proceedings of the 56th Annual ACM Symposium on Theory of Computing},
  publisher     = {ACM},
  pages         = {1094--1105},
  year          = {2024},
  doi           = {10.1145/3618260.3649622},
  url           = {https://arxiv.org/abs/2407.16104v1},
  note          = {Statement locators use \url{https://arxiv.org/abs/2407.16104v1}}
}

@article{dey-wu-2023,
  author        = {Dey, Partha S. and Wu, Qiang},
  title         = {Mean Field Spin Glass Models Under Weak External Field},
  journal       = {Communications in Mathematical Physics},
  volume        = {402},
  number        = {2},
  pages         = {1205--1258},
  year          = {2023},
  doi           = {10.1007/s00220-023-04742-5},
  url           = {https://arxiv.org/abs/2112.11349v3},
  note          = {Statement locators use \url{https://arxiv.org/abs/2112.11349v3}}
}

@misc{oai-barriers,
  author       = {{OpenAI}},
  title        = {Stretched-exponential barriers for typical {SK} initial states},
  year         = {2026},
  month        = sep,
  howpublished = {OpenAI Math Release preprint, \url{https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf}},
  url          = {https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026/paper.pdf},
  note         = {September 24, 2026. Source at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}
}

@misc{oai-fluctuations,
  author       = {{OpenAI}},
  title        = {The low-temperature {Sherrington--Kirkpatrick} fluctuation scale},
  year         = {2026},
  month        = sep,
  howpublished = {OpenAI Math Release preprint, \url{https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf}},
  url          = {https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf},
  note         = {September 24, 2026. Source at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}
}

@article{parisi-formula,
  author       = {Talagrand, Michel},
  title        = {The {Parisi} formula},
  journal      = {Annals of Mathematics},
  volume       = {163},
  number       = {1},
  pages        = {221--263},
  year         = {2006},
  doi          = {10.4007/annals.2006.163.221},
  url          = {https://annals.math.princeton.edu/2006/163-1/p04},
  note         = {\url{https://doi.org/10.4007/annals.2006.163.221}}
}

@misc{full-rsb,
  author       = {Lopatto, Patrick},
  title        = {Full replica symmetry breaking in the {Sherrington--Kirkpatrick} model},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2607.11756v3}},
  eprint       = {2607.11756},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  url          = {https://arxiv.org/abs/2607.11756v3},
  note         = {Version 3, July 16, 2026}
}

@misc{quantitative-parisi,
  author       = {Aronow, P. M. and Lopatto, Patrick},
  title        = {Quantitative {Parisi} formulas and fluctuations in the {Sherrington--Kirkpatrick} model},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2609.09103v1}},
  eprint       = {2609.09103},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  url          = {https://arxiv.org/abs/2609.09103v1},
  note         = {Version 1, September 8, 2026}
}

@misc{hexagon-bias,
  author       = {Aronow, P. M. and Lopatto, Patrick},
  title        = {An {$N^{-7/10}$} Bound for the Finite-Size Bias of the {Sherrington--Kirkpatrick} Free Energy},
  year         = {2026},
  month        = oct,
  howpublished = {Hexagon preprint, \url{https://hexagonmath.org/pdf/2610.00177v1}},
  url          = {https://hexagonmath.org/pdf/2610.00177v1},
  note         = {Version 1; publication date October 12, 2026. hexagon:2610.00177v1}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. 1792--1796; model and scaling in (1)--(3).
@article{sk-original,
  author = {Sherrington, David and Kirkpatrick, Scott},
  title = {Solvable Model of a Spin-Glass},
  journal = {Physical Review Letters},
  volume = {35},
  number = {26},
  pages = {1792--1796},
  year = {1975},
  doi = {10.1103/PhysRevLett.35.1792},
  url = {https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.35.1792},
  note = {\url{https://doi.org/10.1103/PhysRevLett.35.1792}}
}

@article{parisi-1979,
  author = {Parisi, Giorgio},
  title = {Infinite Number of Order Parameters for Spin-Glasses},
  journal = {Physical Review Letters},
  volume = {43},
  number = {23},
  pages = {1754--1756},
  year = {1979},
  doi = {10.1103/PhysRevLett.43.1754},
  url = {https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.43.1754},
  note = {\url{https://doi.org/10.1103/PhysRevLett.43.1754}}
}

@article{guerra-2003,
  author = {Guerra, Francesco},
  title = {Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model},
  journal = {Communications in Mathematical Physics},
  volume = {233},
  pages = {1--12},
  year = {2003},
  doi = {10.1007/s00220-002-0773-5},
  eprint = {cond-mat/0205123},
  archivePrefix = {arXiv},
  url = {https://link.springer.com/article/10.1007/s00220-002-0773-5},
  note = {\url{https://doi.org/10.1007/s00220-002-0773-5}}
}

@misc{chatterjee-2009,
  author = {Chatterjee, Sourav},
  title = {Disorder chaos and multiple valleys in spin glasses},
  year = {2009},
  howpublished = {\url{https://arxiv.org/abs/0907.3381v4}},
  eprint = {0907.3381},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  url = {https://arxiv.org/abs/0907.3381v4},
  note = {Version 4, 23 November 2009}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original Theorem 4.1, pp. 375--376: covariance-inverse variance bound.
@article{brascamp-lieb-1976,
  author = {Brascamp, Herm Jan and Lieb, Elliott H.},
  title = {On extensions of the {Brunn--Minkowski} and {Pr\'{e}kopa--Leindler} theorems, including inequalities for log concave functions, and with an application to the diffusion equation},
  journal = {Journal of Functional Analysis},
  volume = {22},
  number = {4},
  pages = {366--389},
  year = {1976},
  doi = {10.1016/0022-1236(76)90004-5},
  url = {https://www.sciencedirect.com/science/article/pii/0022123676900045},
  note = {\url{https://doi.org/10.1016/0022-1236(76)90004-5}}
}

@article{prekopa-1973,
  author = {Pr\'{e}kopa, Andr\'{a}s},
  title = {On logarithmic concave measures and functions},
  journal = {Acta Scientiarum Mathematicarum (Szeged)},
  volume = {34},
  pages = {335--343},
  year = {1973},
  url = {https://acta.bibl.u-szeged.hu/14411/},
  note = {\url{https://acta.bibl.u-szeged.hu/14411/1/math_034_335-343.pdf}}
}

@article{parisi-rizzo-2008,
  author = {Parisi, Giorgio and Rizzo, Tommaso},
  title = {Large Deviations in the Free Energy of Mean-Field Spin Glasses},
  journal = {Physical Review Letters},
  volume = {101},
  pages = {117205},
  year = {2008},
  doi = {10.1103/PhysRevLett.101.117205},
  eprint = {0706.1180},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/0706.1180v2},
  note = {Statement locators use \url{https://arxiv.org/abs/0706.1180v2}}
}

@article{parisi-rizzo-2009,
  author = {Parisi, Giorgio and Rizzo, Tommaso},
  title = {Phase diagram and large deviations in the free energy of mean-field spin glasses},
  journal = {Physical Review B},
  volume = {79},
  pages = {134205},
  year = {2009},
  doi = {10.1103/PhysRevB.79.134205},
  eprint = {0811.1524},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/0811.1524v2},
  note = {Statement locators use \url{https://arxiv.org/abs/0811.1524v2}}
}

@article{auffinger-chen-uniqueness-2015,
  author = {Auffinger, Antonio and Chen, Wei-Kuo},
  title = {The {Parisi} formula has a unique minimizer},
  journal = {Communications in Mathematical Physics},
  volume = {335},
  pages = {1429--1444},
  year = {2015},
  doi = {10.1007/s00220-014-2254-z},
  eprint = {1402.5132},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1402.5132v3},
  note = {Statement locators use \url{https://arxiv.org/abs/1402.5132v3}}
}

@article{auffinger-chen-2015,
  author = {Auffinger, Antonio and Chen, Wei-Kuo},
  title = {On properties of {Parisi} measures},
  journal = {Probability Theory and Related Fields},
  volume = {161},
  pages = {817--850},
  year = {2015},
  doi = {10.1007/s00440-014-0563-y},
  eprint = {1303.3573},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1303.3573v1},
  note = {Statement locators use \url{https://arxiv.org/abs/1303.3573v1}}
}

@article{jagannath-tobasco-2017,
  author = {Jagannath, Aukosh and Tobasco, Ian},
  title = {Some properties of the phase diagram for mixed {$p$}-spin glasses},
  journal = {Probability Theory and Related Fields},
  volume = {167},
  pages = {615--672},
  year = {2017},
  doi = {10.1007/s00440-015-0691-z},
  eprint = {1504.02731},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/1504.02731v2},
  note = {Statement locators use \url{https://arxiv.org/abs/1504.02731v2}}
}

@article{guerra-toninelli-2002,
  author = {Guerra, Francesco and Toninelli, Fabio L.},
  title = {The Thermodynamic Limit in Mean Field Spin Glass Models},
  journal = {Communications in Mathematical Physics},
  volume = {230},
  number = {1},
  pages = {71--79},
  year = {2002},
  doi = {10.1007/s00220-002-0699-y},
  eprint = {cond-mat/0204280},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/cond-mat/0204280v1},
  note = {Statement locators use \url{https://arxiv.org/abs/cond-mat/0204280v1}}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. L129--L130, (9)--(11): fifth-order replica dependence in the truncated model.
@article{kondor-1983,
  author = {Kondor, Imre},
  title = {Parisi's mean-field solution for spin glasses as an analytic continuation in the replica number},
  journal = {Journal of Physics A: Mathematical and General},
  volume = {16},
  number = {4},
  pages = {L127--L131},
  year = {1983},
  doi = {10.1088/0305-4470/16/4/006},
  note = {\url{https://doi.org/10.1088/0305-4470/16/4/006}}
}

@article{alr-1987,
  author = {Aizenman, Michael and Lebowitz, Joel L. and Ruelle, David},
  title = {Some rigorous results on the {Sher\-rington--Kirkpatrick} spin glass model},
  journal = {Communications in Mathematical Physics},
  volume = {112},
  number = {1},
  pages = {3--20},
  year = {1987},
  doi = {10.1007/BF01217677},
  note = {\url{https://doi.org/10.1007/BF01217677}}
}

@article{chen-lam-2019,
  author = {Chen, Wei-Kuo and Lam, Wai-Kit},
  title = {Order of fluctuations of the free energy in the {SK} model at critical temperature},
  journal = {ALEA. Latin American Journal of Probability and Mathematical Statistics},
  volume = {16},
  number = {1},
  pages = {809--816},
  year = {2019},
  doi = {10.30757/ALEA.v16-29},
  url = {https://alea.impa.br/articles/v16/16-29.pdf},
  note = {\url{https://doi.org/10.30757/ALEA.v16-29}}
}

@misc{du-huang-2026,
  author = {Du, Hang and Huang, Brice},
  title = {Fluctuations of the {Sherrington--Kirkpatrick} free energy at critical temperature},
  howpublished = {\url{https://arxiv.org/abs/2607.02172v2}},
  year = {2026},
  eprint = {2607.02172},
  archivePrefix = {arXiv},
  primaryClass = {math.PR},
  note = {Version 2, 8 July 2026},
  url = {https://arxiv.org/abs/2607.02172v2}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. 295--296, (8)--(17): master equation and single-spin rates.
@article{glauber-1963,
  author       = {Glauber, Roy J.},
  title        = {Time-Dependent Statistics of the {Ising} Model},
  journal      = {Journal of Mathematical Physics},
  volume       = {4},
  number       = {2},
  pages        = {294--307},
  year         = {1963},
  doi          = {10.1063/1.1703954},
  url          = {https://doi.org/10.1063/1.1703954},
  note         = {\url{https://doi.org/10.1063/1.1703954}}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original Propositions 3.4 and 4.4, (3.11), (4.6): constrained comparisons and mismatch costs.
@incollection{talagrand-2007,
  author       = {Talagrand, Michel},
  title        = {Mean Field Models for Spin Glasses: Some Obnoxious Problems},
  booktitle    = {Spin Glasses},
  editor       = {Bolthausen, Erwin and Bovier, Anton},
  series       = {Lecture Notes in Mathematics},
  volume       = {1900},
  publisher    = {Springer},
  address      = {Berlin},
  pages        = {63--80},
  year         = {2007},
  doi          = {10.1007/978-3-540-40908-3_3},
  url          = {https://doi.org/10.1007/978-3-540-40908-3_3},
  note         = {\url{https://doi.org/10.1007/978-3-540-40908-3_3}}
}

@article{panchenko-2018,
  author       = {Panchenko, Dmitry},
  title        = {Free energy in the mixed {$p$}-spin models with vector spins},
  journal      = {The Annals of Probability},
  volume       = {46},
  number       = {2},
  pages        = {865--896},
  year         = {2018},
  doi          = {10.1214/17-AOP1194},
  url          = {https://doi.org/10.1214/17-AOP1194},
  note         = {\url{https://doi.org/10.1214/17-AOP1194}}
}

@article{chen-2017,
  author       = {Chen, Wei-Kuo},
  title        = {Variational representations for the {Parisi} functional and the two-dimensional {Guerra--Talagrand} bound},
  journal      = {The Annals of Probability},
  volume       = {45},
  number       = {6A},
  pages        = {3929--3966},
  year         = {2017},
  doi          = {10.1214/16-AOP1154},
  url          = {https://arxiv.org/abs/1501.06635v2},
  note         = {Statement locators use \url{https://arxiv.org/abs/1501.06635v2}}
}

@article{jagannath-tobasco-2016,
  author       = {Jagannath, Aukosh and Tobasco, Ian},
  title        = {A dynamic programming approach to the {Parisi} functional},
  journal      = {Proceedings of the American Mathematical Society},
  volume       = {144},
  number       = {7},
  pages        = {3135--3150},
  year         = {2016},
  doi          = {10.1090/proc/12968},
  url          = {https://arxiv.org/abs/1502.04398v4},
  note         = {Statement locators use \url{https://arxiv.org/abs/1502.04398v4}}
}

@article{comets-neveu-1995,
  author       = {Comets, Francis and Neveu, Jacques},
  title        = {The {Sherrington--Kirkpatrick} model of spin glasses and stochastic calculus: the high temperature case},
  journal      = {Communications in Mathematical Physics},
  volume       = {166},
  number       = {3},
  pages        = {549--564},
  year         = {1995},
  doi          = {10.1007/BF02099887},
  url          = {https://doi.org/10.1007/BF02099887},
  note         = {\url{https://doi.org/10.1007/BF02099887}}
}

@article{chen-dey-panchenko-2017,
  author       = {Chen, Wei-Kuo and Dey, Partha and Panchenko, Dmitry},
  title        = {Fluctuations of the free energy in the mixed {$p$}-spin models with external field},
  journal      = {Probability Theory and Related Fields},
  volume       = {168},
  number       = {1--2},
  pages        = {41--53},
  year         = {2017},
  doi          = {10.1007/s00440-016-0705-5},
  url          = {https://arxiv.org/abs/1509.07071v2},
  note         = {Statement locators use \url{https://arxiv.org/abs/1509.07071v2}}
}

@article{franz-parisi-ricci-2026,
  author       = {Franz, Silvio and Parisi, Giorgio and Ricci-Tersenghi, Federico},
  title        = {Overlap locking and nonperturbative effects in spin glasses},
  journal      = {Proceedings of the National Academy of Sciences},
  volume       = {123},
  number       = {25},
  pages        = {e2606171123},
  year         = {2026},
  doi          = {10.1073/pnas.2606171123},
  url          = {https://arxiv.org/abs/2602.18399v2},
  note         = {Statement and equation locators use \url{https://arxiv.org/abs/2602.18399v2}}
}

@article{eldan-koehler-zeitouni-2022,
  author       = {Eldan, Ronen and Koehler, Frederic and Zeitouni, Ofer},
  title        = {A spectral condition for spectral gap: fast mixing in high-temperature {Ising} models},
  journal      = {Probability Theory and Related Fields},
  volume       = {182},
  number       = {3--4},
  pages        = {1035--1051},
  year         = {2022},
  doi          = {10.1007/s00440-021-01085-x},
  url          = {https://arxiv.org/abs/2007.08200v2},
  note         = {Statement locators use \url{https://arxiv.org/abs/2007.08200v2}}
}

@inproceedings{lean4,
  author       = {de Moura, Leonardo and Ullrich, Sebastian},
  title        = {The {Lean 4} Theorem Prover and Programming Language},
  booktitle    = {Automated Deduction -- CADE 28},
  series       = {Lecture Notes in Computer Science},
  volume       = {12699},
  publisher    = {Springer},
  pages        = {625--635},
  year         = {2021},
  doi          = {10.1007/978-3-030-79876-5_37},
  url          = {https://doi.org/10.1007/978-3-030-79876-5_37},
  note         = {\url{https://doi.org/10.1007/978-3-030-79876-5_37}}
}

@inproceedings{mathlib,
  author       = {{The mathlib Community}},
  title        = {The {Lean} Mathematical Library},
  booktitle    = {Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs},
  publisher    = {ACM},
  pages        = {367--381},
  year         = {2020},
  doi          = {10.1145/3372885.3373824},
  url          = {https://doi.org/10.1145/3372885.3373824},
  note         = {\url{https://doi.org/10.1145/3372885.3373824}}
}

@misc{al-overlap-structure,
  author       = {Aronow, P. M. and Lopatto, Patrick},
  title        = {Overlap structure of mixed even {$p$}-spin models},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2609.17442v1}},
  eprint       = {2609.17442},
  archivePrefix= {arXiv},
  url          = {https://arxiv.org/abs/2609.17442v1},
  note         = {Version 1, September 15, 2026}
}

@article{ben-arous-jagannath-2018,
  author       = {Ben Arous, G\'{e}rard and Jagannath, Aukosh},
  title        = {Spectral Gap Estimates in Mean Field Spin Glasses},
  journal      = {Communications in Mathematical Physics},
  volume       = {361},
  number       = {1},
  pages        = {1--52},
  year         = {2018},
  doi          = {10.1007/s00220-018-3152-6},
  url          = {https://arxiv.org/abs/1705.04243v3},
  note         = {Statement locators use \url{https://arxiv.org/abs/1705.04243v3}, version 3, March 2, 2018}
}

@misc{sellke-2025,
  author       = {Sellke, Mark},
  title        = {Exponentially Slow Mixing of the Low Temperature {SK} Model},
  year         = {2025},
  howpublished = {\url{https://arxiv.org/abs/2511.22621v2}},
  eprint       = {2511.22621},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2511.22621v2},
  note         = {Version 2, December 9, 2025}
}

@misc{pei-fifth-order-2026,
  author       = {Pei, Yan Ru},
  title        = {Upper deviations of the {Sherrington--Kirkpatrick} free energy: the fifth-order law and the onset of constrained {Parisi} measures},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2610.06480v2}},
  eprint       = {2610.06480},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2610.06480v2},
  note         = {Version 2, October 7, 2026}
}

@inproceedings{el-alaoui-montanari-sellke-sampling,
  author       = {El Alaoui, Ahmed and Montanari, Andrea and Sellke, Mark},
  title        = {Sampling from the {Sherrington--Kirkpatrick} Gibbs measure via algorithmic stochastic localization},
  booktitle    = {2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)},
  publisher    = {IEEE},
  pages        = {323--334},
  year         = {2022},
  doi          = {10.1109/FOCS54457.2022.00038},
  url          = {https://arxiv.org/abs/2203.05093v2},
  note         = {Numbered statements here use the extended preprint \url{https://arxiv.org/abs/2203.05093v2}, version 2, February 15, 2024}
}

% ---------------------------------------------------------------- OpenAI preprints
% CHECK: title, author, date and source URL verified against the release README and
% catalogue at commit fd4aeeb; Theorems 1.1--1.2 read in
% the local LaTeX source (sections/01-introduction.tex).
% CHECK: OpenAI's limiting-law preprint and release catalogue at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb:
% author, title, date and source PDF URL; Theorems 1.1--1.2 and the appendix of inputs in the LaTeX source
% (sections/01-introduction.tex, lines 89--135; sections/11-companion-interface.tex, compiled as Appendix A).
@misc{oai-limiting-law,
  author       = {{OpenAI}},
  title        = {The low-temperature {Sherrington--Kirkpatrick} free-energy limiting law},
  year         = {2026},
  month        = sep,
  howpublished = {OpenAI Math Release preprint, \url{https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf}},
  url          = {https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf},
  note         = {September 24, 2026. Source at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}
}

% CHECK: Theorem 1.1 read in the supplied PDF text (main.txt); metadata from the release README.
@misc{oai-spectral-gap,
  author       = {{OpenAI}},
  title        = {A spectral gap throughout the high-temperature {Sherrington--Kirkpatrick} phase},
  year         = {2026},
  month        = sep,
  howpublished = {OpenAI Math Release preprint, \url{https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf}},
  url          = {https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/main.pdf},
  note         = {September 24, 2026. Source at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}
}

% CHECK: Theorem 1.1 read in the supplied PDF text (paper.txt); metadata from the release README.
@misc{oai-cutoff,
  author       = {{OpenAI}},
  title        = {Cutoff throughout the high-temperature {Sherrington--Kirkpatrick} phase},
  year         = {2026},
  month        = sep,
  howpublished = {OpenAI Math Release preprint, \url{https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf}},
  url          = {https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026/paper.pdf},
  note         = {September 24, 2026. Source at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}
}

% CHECK: Theorems 1.1--1.3 read in the supplied PDF text (paper.txt); metadata from the release README.
@misc{oai-critical-slowing,
  author       = {{OpenAI}},
  title        = {Critical slowing down in the {Sherrington--Kirkpatrick} model},
  year         = {2026},
  month        = sep,
  howpublished = {OpenAI Math Release preprint, \url{https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf}},
  url          = {https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026/paper.pdf},
  note         = {September 24, 2026. Source at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}
}

% CHECK: Theorem 1.1 read in the supplied PDF text (paper.txt); metadata from the release README.
@misc{oai-critical-mixing,
  author       = {{OpenAI}},
  title        = {Critical mixing in the {Sherrington--Kirkpatrick} model},
  year         = {2026},
  month        = sep,
  howpublished = {OpenAI Math Release preprint, \url{https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf}},
  url          = {https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Critical-mixing-in-the-Sherrington-Kirkpatrick-model-September-25-2026/paper.pdf},
  note         = {September 25, 2026. Source at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original (11)--(16) and (32)--(35): disorder tilt, physical-density width and log-Z cumulants.
@article{crisanti-paladin-sommers-vulpiani-1992,
  author  = {Crisanti, Andrea and Paladin, Giovanni and Sommers, Hans-J{\"u}rgen and Vulpiani, Angelo},
  title   = {Replica trick and fluctuations in disordered systems},
  journal = {Journal de Physique I},
  volume  = {2},
  number  = {7},
  pages   = {1325--1332},
  year    = {1992},
  doi     = {10.1051/jp1:1992213}
}

% CHECK: primary record checked (Section 2, (7)--(10) of the arXiv preprint); journal metadata verified.
@article{aspelmeier-billoire-marinari-moore-2008,
  author  = {Aspelmeier, Timo and Billoire, Alain and Marinari, Enzo and Moore, Michael A.},
  title   = {Finite size corrections in the {Sherrington--Kirkpatrick} model},
  journal = {Journal of Physics A: Mathematical and Theoretical},
  volume  = {41},
  number  = {32},
  pages   = {324008},
  year    = {2008},
  doi     = {10.1088/1751-8113/41/32/324008},
  note    = {Section locators refer to arXiv:0711.3445v1}
}

% CHECK: primary record checked (exact identities (14) and (31) of arXiv:0802.3007).
@article{aspelmeier-2008,
  author  = {Aspelmeier, Timo},
  title   = {An exact relation between free energy fluctuations and bond chaos in the {Sherrington--Kirkpatrick} model},
  journal = {Journal of Statistical Mechanics: Theory and Experiment},
  volume  = {2008},
  number  = {4},
  pages   = {P04018},
  year    = {2008},
  doi     = {10.1088/1742-5468/2008/04/P04018},
  note    = {Equation locators refer to arXiv:0802.3007v1}
}

% CHECK: primary record checked (abstract; numerical evaluation along the Guerra--Toninelli interpolation).
@article{billoire-2006,
  author  = {Billoire, Alain},
  title   = {Numerical estimate of the finite-size corrections to the free energy of the {Sherrington--Kirkpatrick} model using {Guerra--Toninelli} interpolation},
  journal = {Physical Review B},
  volume  = {73},
  pages   = {132201},
  year    = {2006},
  doi     = {10.1103/PhysRevB.73.132201}
}

% CHECK: primary record checked (SK discussion and Figure 2 of cond-mat/0212070).
@article{bouchaud-krzakala-martin-2003,
  author  = {Bouchaud, Jean-Philippe and Krzakala, Florent and Martin, Olivier C.},
  title   = {Energy exponents and corrections to scaling in {Ising} spin glasses},
  journal = {Physical Review B},
  volume  = {68},
  pages   = {224404},
  year    = {2003},
  doi     = {10.1103/PhysRevB.68.224404}
}

% CHECK: primary record checked (author PDF; zero temperature, binary couplings).
@article{boettcher-2005,
  author  = {Boettcher, Stefan},
  title   = {Extremal optimization for {Sherrington--Kirkpatrick} spin glasses},
  journal = {The European Physical Journal B},
  volume  = {46},
  number  = {4},
  pages   = {501--505},
  year    = {2005},
  doi     = {10.1140/epjb/e2005-00280-6}
}

% CHECK: primary record checked (Section IV and Figures 7--10 of arXiv:0906.1292v2).
@article{boettcher-2010,
  author  = {Boettcher, Stefan},
  title   = {Simulations of ground state fluctuations in mean-field {Ising} spin glasses},
  journal = {Journal of Statistical Mechanics: Theory and Experiment},
  volume  = {2010},
  number  = {7},
  pages   = {P07002},
  year    = {2010},
  doi     = {10.1088/1742-5468/2010/07/P07002},
  note    = {Section locators refer to arXiv:0906.1292v2}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Published 2008 original, Section 4 and Conclusions; distinct from the 2003 preprint.
@article{palassini-2008,
  author  = {Palassini, Matteo},
  title   = {Ground-state energy fluctuations in the {Sherrington--Kirkpatrick} model},
  journal = {Journal of Statistical Mechanics: Theory and Experiment},
  volume  = {2008},
  number  = {10},
  pages   = {P10005},
  year    = {2008},
  doi     = {10.1088/1742-5468/2008/10/P10005}
}

% CHECK: primary record checked ((12)--(15) of arXiv:0901.1100: asymmetric tails 6/5 and 12/5).
@article{parisi-rizzo-2010,
  author  = {Parisi, Giorgio and Rizzo, Tommaso},
  title   = {Universality and deviations in disordered systems},
  journal = {Physical Review B},
  volume  = {81},
  pages   = {094201},
  year    = {2010},
  doi     = {10.1103/PhysRevB.81.094201},
  note    = {Equation locators refer to arXiv:0901.1100v1}
}

% ---------------------------------------------------------------- rigorous fluctuations
% CHECK: primary record checked (Theorems 1--2).
@misc{schertzer-2026,
  author       = {Schertzer, Adrien},
  title        = {The order of free energy fluctuations in the critical {Sherrington--Kirkpatrick} model revisited},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2606.21360v1}},
  eprint       = {2606.21360},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2606.21360v1},
  note         = {Version 1, June 19, 2026}
}

% CHECK: primary record checked (Theorem 1.1).
@misc{dey-kang-2026,
  author       = {Dey, Partha S. and Kang, Taegu},
  title        = {Fluctuations for the {Sherrington--Kirkpatrick} spin glass model near the critical temperature},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2603.05636v1}},
  eprint       = {2603.05636},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2603.05636v1},
  note         = {Version 1, March 5, 2026}
}

% CHECK: primary record checked (Theorems 2.10--2.11 of arXiv:1505.07349v5).
@article{baik-lee-2016,
  author  = {Baik, Jinho and Lee, Ji Oon},
  title   = {Fluctuations of the free energy of the spherical {Sherrington--Kirkpatrick} model},
  journal = {Journal of Statistical Physics},
  volume  = {165},
  number  = {2},
  pages   = {185--224},
  year    = {2016},
  doi     = {10.1007/s10955-016-1610-0},
  note    = {Statement locators refer to arXiv:1505.07349v5}
}

% CHECK: primary record checked (Theorem 2.1, equation (34) of the author preprint).
% CHECK: Theorem 2.1, equation (34), author preprint p.6 (multiscale Parisi formula); its one-level
% case gives the fixed-order fractional-moment limit for zero-field SK; also checked against
% the introduction of arXiv:2609.09103v1.
@article{contucci-mingione-2019,
  author  = {Contucci, Pierluigi and Mingione, Emanuele},
  title   = {A multi-scale spin-glass mean-field model},
  journal = {Communications in Mathematical Physics},
  volume  = {368},
  number  = {3},
  pages   = {1323--1344},
  year    = {2019},
  doi     = {10.1007/s00220-019-03308-8}
}

% CHECK: primary record checked (Theorem 2.1, equations (2.3)--(2.4)).
% CHECK: arXiv:2603.06368v1 (6 March 2026, only listed version); Theorem 2.1, (2.3)-(2.4): for fixed
% 0<s<1 the fractional-moment limit is a Parisi minimum over measures with mass at least s at zero.
@misc{chen-guionnet-ko-lacroix-mourrat-2026,
  author       = {Chen, Hong-Bin and Guionnet, Alice and Ko, Justin and Lacroix-A-Chez-Toine, Bertrand and Mourrat, Jean-Christophe},
  title        = {One-sided large deviations for the ground-state energy of spin glasses},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2603.06368v1}},
  eprint       = {2603.06368},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2603.06368v1},
  note         = {Version 1, March 6, 2026}
}

% CHECK: primary record checked (Theorem 1; journal publication verified).
@article{zhou-2026,
  author  = {Zhou, Yuxin},
  title   = {Existence of full replica symmetry breaking for the {Sherrington--Kirkpatrick} model at low temperature},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {79},
  number  = {7},
  pages   = {1746--1770},
  year    = {2026},
  doi     = {10.1002/cpa.70038}
}

% CHECK: primary record checked (strict inequality (11) under condition (12)).
@article{toninelli-2002,
  author  = {Toninelli, Fabio Lucio},
  title   = {About the {Almeida--Thouless} transition line in the {Sher\-rington--Kirkpatrick} mean-field spin glass model},
  journal = {Europhysics Letters},
  volume  = {60},
  number  = {5},
  pages   = {764--767},
  year    = {2002},
  doi     = {10.1209/epl/i2002-00667-5}
}

% CHECK: primary record checked (Section III, (26)--(29) of the v5 preprint).
@article{crisanti-rizzo-2002,
  author  = {Crisanti, Andrea and Rizzo, Tommaso},
  title   = {Analysis of the {$\infty$}-replica symmetry breaking solution of the {Sherrington--Kirkpatrick} model},
  journal = {Physical Review E},
  volume  = {65},
  pages   = {046137},
  year    = {2002},
  doi     = {10.1103/PhysRevE.65.046137},
  note    = {Equation locators refer to arXiv:cond-mat/0111037v5}
}

% CHECK: primary record checked (Section 5 and Theorem 6.2: multidimensional interpolation and the matrix Parisi PDE).
% CHECK: Section 6, Proposition 6.1 and (6.5), journal p. 205: the matrix Parisi-type PDE,
% verified in the journal PDF (EJP 14, paper 8).
@article{bovier-klimovsky-2009,
  author  = {Bovier, Anton and Klimovsky, Anton},
  title   = {The {Aizenman--Sims--Starr} and {Guerra}'s schemes for the {SK} model with multidimensional spins},
  journal = {Electronic Journal of Probability},
  volume  = {14},
  number  = {8},
  pages   = {161--241},
  year    = {2009},
  doi     = {10.1214/EJP.v14-611}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Publisher original Sections 3--4: finite and continuous probability cascades.
@article{ruelle-1987,
  author  = {Ruelle, David},
  title   = {A mathematical reformulation of {Derrida}'s {REM} and {GREM}},
  journal = {Communications in Mathematical Physics},
  volume  = {108},
  number  = {2},
  pages   = {225--239},
  year    = {1987},
  doi     = {10.1007/BF01210613}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Published book, Section 2.3, Theorem 2.9; Chapter 3, (3.10)--(3.15): recursion/cascade correspondence.
@book{panchenko-2013,
  author    = {Panchenko, Dmitry},
  title     = {The {Sherrington--Kirkpatrick} Model},
  series    = {Springer Monographs in Mathematics},
  publisher = {Springer},
  address   = {New York},
  year      = {2013},
  doi       = {10.1007/978-1-4614-6289-7}
}

% CHECK: primary record checked (Theorem 1, equations (11)--(13) of cond-mat/0306386v1).
@article{aizenman-sims-starr-2003,
  author  = {Aizenman, Michael and Sims, Robert and Starr, Shannon L.},
  title   = {Extended variational principle for the {Sherrington--Kirkpatrick} spin-glass model},
  journal = {Physical Review B},
  volume  = {68},
  pages   = {214403},
  year    = {2003},
  doi     = {10.1103/PhysRevB.68.214403}
}

% ---------------------------------------------------------------- rigorous dynamics
% CHECK: primary record checked (Theorem 1, preprint p. 3, arXiv:1608.06609v3).
@article{gheissari-jagannath-2019,
  author  = {Gheissari, Reza and Jagannath, Aukosh},
  title   = {On the spectral gap of spherical spin glass dynamics},
  journal = {Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
  volume  = {55},
  number  = {2},
  pages   = {756--776},
  year    = {2019},
  doi     = {10.1214/18-AIHP897},
  note    = {Statement locators refer to arXiv:1608.06609v3}
}

% CHECK: primary record checked (Theorem 1 and Corollary 2 of arXiv:1712.03676v2).
@article{bauerschmidt-bodineau-2019,
  author  = {Bauerschmidt, Roland and Bodineau, Thierry},
  title   = {A very simple proof of the {LSI} for high temperature spin systems},
  journal = {Journal of Functional Analysis},
  volume  = {276},
  number  = {8},
  pages   = {2582--2588},
  year    = {2019},
  doi     = {10.1016/j.jfa.2019.01.007},
  note    = {Statement locators refer to arXiv:1712.03676v2}
}

% CHECK: primary record checked (Theorem 12, p. 10 of arXiv:2106.04105v2; STOC metadata verified).
@incollection{anari-jain-koehler-pham-vuong-2022,
  author    = {Anari, Nima and Jain, Vishesh and Koehler, Frederic and Pham, Huy Tuan and Vuong, Thuy-Duong},
  title     = {Entropic independence: optimal mixing of down-up random walks},
  booktitle = {Proceedings of the 54th Annual {ACM} {SIGACT} Symposium on Theory of Computing},
  publisher = {ACM},
  pages     = {1418--1430},
  year      = {2022},
  doi       = {10.1145/3519935.3520048},
  note      = {Theorem locators refer to the preprint arXiv:2106.04105v2}
}

% CHECK: primary record checked (Theorem 1.1 and abstract).
@misc{wang-2026,
  author       = {Wang, Sihan},
  title        = {Optimal mixing of {Glauber} dynamics for the {Sherrington--Kirkpatrick} model at {$\beta<1/2$}},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2608.22159v2}},
  eprint       = {2608.22159},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2608.22159v2},
  note         = {Version 2}
}

% CHECK: primary record checked (Theorem 1.5 and Corollary 1.6, pp. 4--5).
@misc{boban-li-oveis-gharan-2026,
  author       = {Boban, Mathews and Li, Anqi and Oveis Gharan, Shayan},
  title        = {Rank-1-perturbed trickledown theorems: mixing time of {Glauber} dynamics for the {Sherrington--Kirkpatrick} model up to {$\beta\leq\frac12+\varepsilon$}},
  year         = {2026},
  howpublished = {\url{https://arxiv.org/abs/2609.13138v1}},
  eprint       = {2609.13138},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2609.13138v1},
  note         = {Version 1}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Published second edition: Props. 1.20, 4.2; Ex. 4.2; Thm. 7.4; Section 20.1, (20.1)--(20.6); Lem. 20.5.
@book{levin-peres-2017,
  author    = {Levin, David A. and Peres, Yuval},
  title     = {Markov Chains and Mixing Times},
  edition   = {Second},
  publisher = {American Mathematical Society},
  address   = {Providence, RI},
  year      = {2017},
  doi       = {10.1090/mbk/107},
  note      = {With contributions by Elizabeth L. Wilmer},
  url       = {https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf}
}

% CHECK: primary record checked (arXiv preprint and DOI verified; journal pagination corroborated).
@article{franz-parisi-1995,
  author  = {Franz, Silvio and Parisi, Giorgio},
  title   = {Recipes for metastable states in spin glasses},
  journal = {Journal de Physique I},
  volume  = {5},
  number  = {11},
  pages   = {1401--1415},
  year    = {1995},
  doi     = {10.1051/jp1:1995201}
}

% ---------------------------------------------------------------- physics of dynamics
% CHECK: primary record checked (APS publisher abstract, p. 301).
@article{mackenzie-young-1982,
  author  = {Mackenzie, N. D. and Young, A. P.},
  title   = {Lack of ergodicity in the infinite-range {Ising} spin-glass},
  journal = {Physical Review Letters},
  volume  = {49},
  pages   = {301--304},
  year    = {1982},
  doi     = {10.1103/PhysRevLett.49.301}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. 5321--5337, Sections 3--4, (30), (40)--(41); full page range confirmed.
@article{mackenzie-young-1983,
  author  = {Mackenzie, N. D. and Young, A. P.},
  title   = {Statics and dynamics of the infinite-range {Ising} spin glass model},
  journal = {Journal of Physics C: Solid State Physics},
  volume  = {16},
  number  = {27},
  pages   = {5321--5337},
  year    = {1983},
  doi     = {10.1088/0022-3719/16/27/015}
}

% CHECK: primary record checked (author-uploaded original, abstract p. 2325: alpha = 0.34 +- 0.08).
@article{vertechi-virasoro-1989,
  author  = {Vertechi, D. and Virasoro, M. A.},
  title   = {Energy barriers in {SK} spin-glass model},
  journal = {Journal de Physique},
  volume  = {50},
  number  = {17},
  pages   = {2325--2332},
  year    = {1989},
  doi     = {10.1051/jphys:0198900500170232500}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original Section 4, pp. 4017--4019, (4.3)--(4.4): mean ergodic barrier exponent.
@article{colborne-1990,
  author  = {Colborne, S. G. W.},
  title   = {A {Monte Carlo} study of the dynamics of the {Ising} {SK} model},
  journal = {Journal of Physics A: Mathematical and General},
  volume  = {23},
  number  = {17},
  pages   = {4013--4023},
  year    = {1990},
  doi     = {10.1088/0305-4470/23/17/030}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original Section VIII, (8.4), p. 112: exponential N^(1/3) relaxation scales.
@article{kinzelbach-horner-1991,
  author  = {Kinzelbach, Harald and Horner, Heinz},
  title   = {Dynamics of the finite {SK} spin-glass},
  journal = {Zeitschrift f{\"u}r Physik B Condensed Matter},
  volume  = {84},
  pages   = {95--114},
  year    = {1991},
  doi     = {10.1007/BF01453763}
}

% CHECK: primary record checked (arXiv:cond-mat/0101177v1, abstract and p. 4).
@article{billoire-marinari-2001,
  author  = {Billoire, Alain and Marinari, Enzo},
  title   = {Correlation time scales in the {Sherrington--Kirkpatrick} model},
  journal = {Journal of Physics A: Mathematical and General},
  volume  = {34},
  pages   = {L727--L734},
  year    = {2001},
  doi     = {10.1088/0305-4470/34/50/101}
}

% CHECK: primary record checked (published reprint, equations (7)--(11), p. 197).
@article{bittner-janke-2006,
  author  = {Bittner, Elmar and Janke, Wolfhard},
  title   = {Free-energy barriers in the {Sherrington--Kirkpatrick} model},
  journal = {Europhysics Letters},
  volume  = {74},
  number  = {2},
  pages   = {195--201},
  year    = {2006},
  doi     = {10.1209/epl/i2006-10007-y}
}

% CHECK: primary record checked (arXiv:0910.4833v2 abstract; journal metadata).
@article{monthus-garel-2009,
  author  = {Monthus, C{\'e}cile and Garel, Thomas},
  title   = {An eigenvalue method for computing the largest relaxation time of disordered systems},
  journal = {Journal of Statistical Mechanics: Theory and Experiment},
  volume  = {2009},
  number  = {12},
  pages   = {P12017},
  year    = {2009},
  doi     = {10.1088/1742-5468/2009/12/P12017}
}

% CHECK: primary record checked (arXiv:1003.6086v3: abstract, p. 3, equation (6), Figure 3 and
% the following discussion of the centered variable and its temperature scaling).
@article{billoire-2010,
  author  = {Billoire, Alain},
  title   = {Distribution of timescales in the {Sherrington--Kirkpatrick} model},
  journal = {Journal of Statistical Mechanics: Theory and Experiment},
  volume  = {2010},
  number  = {11},
  pages   = {P11034},
  year    = {2010},
  doi     = {10.1088/1742-5468/2010/11/P11034},
  note    = {Equation and figure locators refer to arXiv:1003.6086v3}
}

% CHECK: primary record checked (APS abstract and arXiv:2111.06753v1).
@article{aspelmeier-moore-2022,
  author  = {Aspelmeier, T. and Moore, M. A.},
  title   = {Free-energy barriers in the {Sherrington--Kirkpatrick} model},
  journal = {Physical Review E},
  volume  = {105},
  pages   = {034138},
  year    = {2022},
  doi     = {10.1103/PhysRevE.105.034138}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. 359--362: dynamical mean-field theory and diverging relaxation scales.
@article{sompolinsky-zippelius-1981,
  author  = {Sompolinsky, Haim and Zippelius, Annette},
  title   = {Dynamic theory of the spin-glass phase},
  journal = {Physical Review Letters},
  volume  = {47},
  pages   = {359--362},
  year    = {1981},
  doi     = {10.1103/PhysRevLett.47.359}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. 6860--6875, especially p. 6862 and Sections II and VI.
@article{sompolinsky-zippelius-1982,
  author  = {Sompolinsky, Haim and Zippelius, Annette},
  title   = {Relaxational dynamics of the {Edwards--Anderson} model and the mean-field theory of spin-glasses},
  journal = {Physical Review B},
  volume  = {25},
  pages   = {6860--6875},
  year    = {1982},
  doi     = {10.1103/PhysRevB.25.6860}
}

% CHECK: primary record checked (arXiv:cond-mat/9311016v1, abstract; the SK paper, not the 1993 spherical one).
@article{cugliandolo-kurchan-1994,
  author  = {Cugliandolo, Leticia F. and Kurchan, Jorge},
  title   = {On the out-of-equilibrium relaxation of the {Sherrington--Kirkpatrick} model},
  journal = {Journal of Physics A: Mathematical and General},
  volume  = {27},
  pages   = {5749--5772},
  year    = {1994},
  doi     = {10.1088/0305-4470/27/17/011}
}

% CHECK: primary record checked (verified preprint, Section 3.7 beginning p. 36).
@misc{bouchaud-cugliandolo-kurchan-mezard-1997,
  author       = {Bouchaud, Jean-Philippe and Cugliandolo, Leticia F. and Kurchan, Jorge and M{\'e}zard, Marc},
  title        = {Out of equilibrium dynamics in spin-glasses and other glassy systems},
  year         = {1997},
  howpublished = {\url{https://arxiv.org/abs/cond-mat/9702070v1}},
  eprint       = {cond-mat/9702070},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/cond-mat/9702070v1},
  note         = {Version 1}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original Section 3, pp. 595--596, (11)--(13): TAP equations and single-site response.
@article{thouless-anderson-palmer-1977,
  author  = {Thouless, D. J. and Anderson, P. W. and Palmer, R. G.},
  title   = {Solution of `{Solvable} model of a spin glass'},
  journal = {Philosophical Magazine},
  volume  = {35},
  number  = {3},
  pages   = {593--601},
  year    = {1977},
  doi     = {10.1080/14786437708235992}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. L377--L378, (1)--(3): TAP states and metastability.
@article{bray-moore-1981,
  author  = {Bray, A. J. and Moore, M. A.},
  title   = {Metastable states in the solvable spin glass model},
  journal = {Journal of Physics A: Mathematical and General},
  volume  = {14},
  number  = {9},
  pages   = {L377--L383},
  year    = {1981},
  doi     = {10.1088/0305-4470/14/9/012}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original Proposition 3.2, p. 285, (3.35): support contact identity.
@article{talagrand-2006-parisi-measures,
  author  = {Talagrand, Michel},
  title   = {{Parisi} measures},
  journal = {Journal of Functional Analysis},
  volume  = {231},
  number  = {2},
  pages   = {269--286},
  year    = {2006},
  doi     = {10.1016/j.jfa.2005.03.001}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. 606--608, (13)--(15): near-critical truncated-model marginal stability.
@article{de-dominicis-kondor-1983,
  author  = {De Dominicis, Cirano and Kondor, Imre},
  title   = {Eigenvalues of the stability matrix for {Parisi} solution of the long-range spin-glass},
  journal = {Physical Review B},
  volume  = {27},
  number  = {1},
  pages   = {606--608},
  year    = {1983},
  doi     = {10.1103/PhysRevB.27.606}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% All six original pages visually inspected: N-to-N+1 cavity construction and RSB assumptions.
@article{mezard-parisi-virasoro-1986,
  author  = {M{\'e}zard, Marc and Parisi, Giorgio and Virasoro, Miguel A.},
  title   = {{SK} Model: The Replica Solution without Replicas},
  journal = {Europhysics Letters},
  volume  = {1},
  number  = {2},
  pages   = {77--82},
  year    = {1986},
  doi     = {10.1209/0295-5075/1/2/006}
}

% CHECK: Paouris--Valettas, "A Gaussian small deviation inequality for convex functions".
% Journal metadata (Ann. Probab. 46 (2018), no. 3, 1441--1454, DOI 10.1214/17-AOP1206) verified
% through the Crossref record of the DOI on 2026-10-18. The statement used is Theorem 2.1 and
% Remark 2.4(3), equation (2.4), in the numbering of arXiv:1611.01723v2 (15 June 2017), read
% P(f(Z) - E f(Z) < -t sqrt(Var f(Z))) <= (1/2) exp(-(pi/1024)(t-1)^2).
@article{paouris-valettas-2018,
  author       = {Paouris, Grigoris and Valettas, Petros},
  title        = {A {G}aussian small deviation inequality for convex functions},
  journal      = {The Annals of Probability},
  volume       = {46},
  number       = {3},
  pages        = {1441--1454},
  year         = {2018},
  doi          = {10.1214/17-AOP1206},
  eprint       = {1611.01723},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/1611.01723v2},
  note         = {Theorem and equation locators refer to arXiv:1611.01723v2, June 15, 2017}
}

% CHECK: OpenAI mathematics repository, commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb:
% Spin.lean line 453 (PositiveLaplaceWindow), line 531 (MainFluctuationStatement),
% EventuallyNatPos.lean line 294 (main_of_positiveLaplaceWindow), lean/docs/227.md (barrier scope),
% and OAI.SK.main in lean/OAI/Probability/SKBarriers/Main.lean.
@misc{oai-math-lean,
  author       = {{OpenAI}},
  title        = {Lean formalizations in the {OpenAI} mathematics repository},
  year         = {2026},
  howpublished = {\url{https://github.com/openai/math/tree/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean}},
  url          = {https://github.com/openai/math/tree/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean},
  note         = {Commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb; directories lean/OAI/Probability/SKBarriers and lean/OAI/Probability/SKFluctuations, and lean/docs/227.md}
}

% CHECK: Chen, Theorem 1, arXiv:2311.08351v2; journal metadata AIHP 62(1) (2026),
% 195--206, DOI 10.1214/24-AIHP1527. Also cited in arXiv:2609.09103v1, Lemma 5.2.
@article{chen-gaussian-convexity,
  author       = {Chen, Wei-Kuo},
  title        = {A {Gaussian} convexity for logarithmic moment generating functions with applications in spin glasses},
  journal      = {Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
  volume       = {62},
  number       = {1},
  pages        = {195--206},
  year         = {2026},
  doi          = {10.1214/24-AIHP1527},
  url          = {https://doi.org/10.1214/24-AIHP1527},
  note         = {Preprint version arXiv:2311.08351v2}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Published Volume I: Lemma 1.3.1, Appendix A (A.6), (A.7), (A.11), (A.17), Section A.13.
@book{talagrand-2011,
  author    = {Talagrand, Michel},
  title     = {Mean Field Models for Spin Glasses. {Volume I}: Basic Examples},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge},
  volume    = {54},
  publisher = {Springer},
  address   = {Berlin},
  year      = {2011},
  doi       = {10.1007/978-3-642-15202-3}
}

% CHECK: Theorem 3.3.6 (Section 3.3.A "The Ito rule", p. 149 ff.): X = X_0 + M + B with M a vector of
% continuous local martingales and B adapted of bounded variation, f(t,x) on [0,oo) x R^d of class
% C^{1,2}; formula (3.5). Theorem 5.7.6 (Section 5.7.B "The Cauchy problem and a Feynman--Kac
% representation", p. 366): v continuous, of class C^{1,2}([0,T) x R^d), solving the Cauchy problem
% with polynomial growth, "admits the stochastic representation". Both statements read in the
% full-text search of the 1988 first edition (archive.org item brownianmotionst0000kara, via
% openlibrary.org/search/inside); the 1991 second edition is an almost unchanged softcover reprint
% with the same section pages (p. 149, p. 366) in the publisher's front matter.
@book{karatzas-shreve-1991,
  author    = {Karatzas, Ioannis and Shreve, Steven E.},
  title     = {Brownian Motion and Stochastic Calculus},
  edition   = {Second},
  series    = {Graduate Texts in Mathematics},
  volume    = {113},
  publisher = {Springer},
  address   = {New York},
  year      = {1991},
  doi       = {10.1007/978-1-4612-0949-2}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original pp. 837--840 and Section 7, pp. 870--872: positive fractional-moment limits.
@article{talagrand-2007-large-deviations,
  author       = {Talagrand, Michel},
  title        = {Large Deviations, {Guerra}'s and {A.S.S.} Schemes, and the {Parisi} Hypothesis},
  journal      = {Journal of Statistical Physics},
  volume       = {126},
  number       = {4-5},
  pages        = {837--894},
  year         = {2007},
  doi          = {10.1007/s10955-006-9108-9}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original Theorem 1A, pp. 25--26; bounded-stopping Gaussian representation.
@incollection{tsirelson-ibragimov-sudakov-1976,
  author    = {Tsirel'son, B. S. and Ibragimov, I. A. and Sudakov, V. N.},
  title     = {Norms of {Gaussian} sample functions},
  booktitle = {Proceedings of the {Third Japan--USSR} Symposium on Probability Theory ({Tashkent}, 1975)},
  series    = {Lecture Notes in Mathematics},
  volume    = {550},
  publisher = {Springer},
  address   = {Berlin},
  pages     = {20--41},
  year      = {1976}
}

% CHECK: README at https://github.com/pmaronow/sk-mean-bias-palomar/blob/7601a9d5ade08066632bc68b9a615e163f2dd02c/README.md (read 2026-10-19): title "Finite-size mean bias in the Sherrington--Kirkpatrick model", P. M. Aronow and Patrick Lopatto, formalizes hexagon:2610.00177v1, Lean 4.35.0-rc2, Mathlib 065356127b1dc0016f66b7283ce0ce2c4055aa55, MIT license; QuantitativePaper/ShortVarianceMeanProposition.lean (theorem short_variance_mean) and HexagonPaper/Main.lean (theorem main_mean_bias) present at this commit.
@misc{hexagon-bias-lean,
  author       = {Aronow, P. M. and Lopatto, Patrick},
  title        = {Finite-size mean bias in the {Sherrington--Kirkpatrick} model},
  year         = {2026},
  howpublished = {Lean development, \url{https://github.com/pmaronow/sk-mean-bias-palomar/tree/7601a9d5ade08066632bc68b9a615e163f2dd02c}},
  url          = {https://github.com/pmaronow/sk-mean-bias-palomar/tree/7601a9d5ade08066632bc68b9a615e163f2dd02c},
  note         = {Commit 7601a9d5ade08066632bc68b9a615e163f2dd02c}
}

% CHECK: Theorem 1 (typical-start bound
% P{T(X_0) <= exp(n^{1-c_beta})} -> 1, c_beta = 1/40000000 permitted, rate-one-per-site continuous time) and
% Corollary 2 (two-sided bound with OB's lower bound) read in the local LaTeX source introduction.tex, lines 49-98,
% of the release at commit fd4aeeb; theorems are numbered consecutively, not by section.
@misc{oai-typical-upper,
  author       = {{OpenAI}},
  title        = {A typical-start upper bound for low-temperature {SK} {Glauber} dynamics},
  year         = {2026},
  month        = sep,
  howpublished = {OpenAI Math Release preprint, \url{https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf}},
  url          = {https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-typical-start-upper-bound-for-low-temperature-SK-Glauber-dynamics-September-24-2026/paper.pdf},
  note         = {September 24, 2026. Source at commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb}
}

% CHECK: arXiv abstract page and HTML of 2610.23777v1 (read 2026-10-19): Vasyl Paliy and Youngtak Sohn,
% submitted 16 October 2026. Corollary 1.5: SK coupling beta W/2 with W ~ GOE(n), every fixed beta >= 0,
% min_i h_i >= C(1 + beta sqrt(log(1+beta))) and average field <= H; with probability 1-o(1) over W, the
% modified log-Sobolev constant is at least c_0(beta,H)/n, and discrete Glauber dynamics mixes in
% O_{beta,H}(n log n) steps; proved in Section 7.
@misc{paliy-sohn-2026,
  author        = {Paliy, Vasyl and Sohn, Youngtak},
  title         = {Rapid mixing within a phase for {Ising} models},
  year          = {2026},
  eprint        = {2610.23777},
  archivePrefix = {arXiv},
  primaryClass  = {math.PR},
  howpublished  = {\url{https://arxiv.org/abs/2610.23777v1}},
  url           = {https://arxiv.org/abs/2610.23777v1},
  note          = {Version 1, October 16, 2026}
}

% CHECK: arXiv abstract page and HTML of 2610.00011v2 (read 2026-10-19): Miguel Tierz; v1 9 July 2026,
% v2 16 October 2026. Theorem 1.2, (1.5): at beta = 1, h = 0, for every N >= 1,
% 2N^2 Var F_N(1) <= 1 + log chi_SG(N) and 2N^2 Var F_N^off(1) <= log chi_SG(N), where F_N is the free
% energy per spin (counting measure) and chi_SG(N) = N E<R_{13}^2> is the spin-glass susceptibility.
@misc{tierz-critical-2026,
  author        = {Tierz, Miguel},
  title         = {Critical Free-Energy Variance in the {Sherrington--Kirkpatrick} Model},
  year          = {2026},
  eprint        = {2610.00011},
  archivePrefix = {arXiv},
  primaryClass  = {math.PR},
  howpublished  = {\url{https://arxiv.org/abs/2610.00011v2}},
  url           = {https://arxiv.org/abs/2610.00011v2},
  note          = {Version 2, October 16, 2026}
}

% CHECK: metadata copied from OpenAI's references.tex (typical-start upper-bound release, fd4aeeb), entry Bolthausen2014:
% CMP 325 (2014) 333--366, doi 10.1007/s00220-013-1862-3, arXiv:1201.2891.
% Independently confirmed against the Springer publisher record; first online December 27, 2013.
@article{bolthausen-2014,
  author  = {Bolthausen, Erwin},
  title   = {An iterative construction of solutions of the {TAP} equations for the {Sherrington}--{Kirkpatrick} model},
  journal = {Communications in Mathematical Physics},
  volume  = {325},
  year    = {2014},
  pages   = {333--366},
  doi     = {10.1007/s00220-013-1862-3},
  eprint  = {1201.2891},
  archivePrefix = {arXiv}
}

% CHECK: metadata copied from OpenAI's references.tex, entry Montanari2021: SIAM J. Comput. 54 (2025), no. 4, FOCS19-1--FOCS19-38
% (first published online 2021), doi 10.1137/20M132016X, arXiv:1812.10897v2 (OpenAI: numbered sections refer to this preprint version).
% Independently confirmed against the SIAM publisher record: volume 54, issue 4, August 2025,
% FOCS19-1--FOCS19-38; first published online January 7, 2021.
% CHECK: arXiv:1812.10897v2 (read 2026-10-19) has Section 2, "A general message passing algorithm,"
% and Section 3, "Proof of the main theorem." The supplied upper-bound release's references.tex
% explicitly assigns numbered section locators to this version; the printed note retains that convention.
@article{montanari-2021,
  author  = {Montanari, Andrea},
  title   = {Optimization of the {Sherrington}--{Kirkpatrick} {Hamiltonian}},
  journal = {SIAM Journal on Computing},
  volume  = {54},
  number  = {4},
  year    = {2025},
  pages   = {FOCS19-1--FOCS19-38},
  doi     = {10.1137/20M132016X},
  eprint  = {1812.10897v2},
  archivePrefix = {arXiv},
  note    = {First published online 2021. Numbered section locators refer to
             \url{https://arxiv.org/abs/1812.10897v2}}
}

% CHECK: publisher metadata at Math-Net.Ru confirms Sb. Math. 196 (2005), no. 3,
% 309--335, doi 10.1070/SM2005v196n03ABEH000882. Corollary 3.10, (3.12),
% independently checked in the author preprint https://bibos.math.uni-bielefeld.de/preprints/04-10-159.pdf:
% uniform convexity with constant C gives quadratic transport cost at most 2/C times relative entropy.
@article{bogachev-kolesnikov-medvedev-2005,
  author  = {Bogachev, V. I. and Kolesnikov, A. V. and Medvedev, K. V.},
  title   = {Triangular transformations of measures},
  journal = {Sbornik: Mathematics},
  volume  = {196},
  number  = {3},
  year    = {2005},
  pages   = {309--335},
  doi     = {10.1070/SM2005v196n03ABEH000882}
}

% CHECK: the publisher's record and BibTeX at https://proceedings.mlr.press/v178/lugosi22a.html
% independently confirm the conference title, editors, series, volume, publisher, year, and pages.
% Lemma 11, independently checked in the publisher PDF, bounds the relative entropy of
% Gaussian convolutions by the mean squared coupling cost divided by twice the noise variance.
@inproceedings{lugosi-neu-2022,
  author    = {Lugosi, G\'abor and Neu, Gergely},
  title     = {Generalization bounds via convex analysis},
  booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory},
  editor    = {Loh, Po-Ling and Raginsky, Maxim},
  series    = {Proceedings of Machine Learning Research},
  volume    = {178},
  publisher = {PMLR},
  year      = {2022},
  pages     = {3524--3546},
  url       = {https://proceedings.mlr.press/v178/lugosi22a.html}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Original Theorem 3.5, p. 570; coarea proof in Theorem 3.1, p. 566.
@article{lawler-sokal-1988,
  author  = {Lawler, Gregory F. and Sokal, Alan D.},
  title   = {Bounds on the {$L^2$} spectrum for {Markov} chains and {Markov} processes: a generalization of {Cheeger}'s inequality},
  journal = {Transactions of the American Mathematical Society},
  volume  = {309},
  number  = {2},
  year    = {1988},
  pages   = {557--580},
  doi     = {10.1090/S0002-9947-1988-0930082-9}
}

% CHECK: author-hosted published PDF independently confirms Ann. Appl. Probab.
% 6 (1996), no. 3, 695--750, doi 10.1214/aoap/1034968224, and the entropy
% dissipation tools in Lemmas 2.5 and 2.7.
@article{diaconis-saloff-coste-1996,
  author  = {Diaconis, Persi and Saloff-Coste, Laurent},
  title   = {Logarithmic {Sobolev} inequalities for finite {Markov} chains},
  journal = {Annals of Applied Probability},
  volume  = {6},
  number  = {3},
  year    = {1996},
  pages   = {695--750},
  doi     = {10.1214/aoap/1034968224}
}

% ORIGINAL CHECK 2026-10-20: all in-text uses checked against the supplied original.
% Published 2013 edition: cited numbers agree with the 2012 draft; Chapter 4, Ex. 4.10, Cors. 2.11, 4.14, Thms. 2.10, 3.20, 4.20, 5.5--5.6, Lem. 2.2.
@book{boucheron-lugosi-massart-2013,
  author    = {Boucheron, St\'ephane and Lugosi, G\'abor and Massart, Pascal},
  title     = {Concentration Inequalities: A Nonasymptotic Theory of Independence},
  publisher = {Oxford University Press},
  year      = {2013},
  doi       = {10.1093/acprof:oso/9780199535255.001.0001},
  url       = {https://academic.oup.com/book/26549}
}

% CHECK: The existing oai-typical-upper entry identifies the September 24, 2026 release at commit
% fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. The supplied release source was
% compiled locally to paper.pdf, and its printed result numbers were checked:
% Theorem 1 and Corollary 2 on p. 2; Lemma 3 on p. 6; Lemma 4 on p. 8;
% Proposition 5 on p. 9; Proposition 6 on p. 14; Proposition 7 on p. 15;
% Lemma 8 on p. 19. Their sections are 1, 1, 3, 3, 4, 5, 6, and 7.
% The exact remote PDF was subsequently retrieved and checked independently.
% CHECK: The entropy-energy formula in oai-typical-upper, Lemma 4, is
% log 2 - j(V) + beta e(V) = P_beta - beta^2 p(V)^2/4. Its potential has
% terminal value log(2 cosh y), so its pressure is our counting-measure P_beta.
% CHECK: The post-support kernel used in Section 13 is the mass-one
% specialization of the step kernel in the proof of Lemma 3, Section 3.2,
% pp. 6-7 of that locally compiled PDF; the prefactor exp(-t/2) follows by
% inserting U(v,y)=log(2 cosh y)+(beta^2-v)/2 in that kernel.
% CHECK: The existing karatzas-shreve-1991 entry records Theorem 3.3.6 as
% the Ito product rule, which is the stochastic-calculus locator used here.
% The existing tsirelson-ibragimov-sudakov-1976 entry supplies the standard
% Gaussian concentration reference used with the explicit Lipschitz constant.