References

CheungHawk01B

N. Cheung, Hawk01-B, version 1, unpublished manuscript, n.d. Supplied excerpts: “Controlled island comparison without a reconstruction overclaim” and “Conclusions and mathematical release,” including tab:islandaudit, eq:Petz, and thm:release. Citation identifies the supplied version, not independent verification of the complete manuscript.

CheungHawk02A

N. Cheung, Hawk02-A, version 1, unpublished manuscript, n.d. Supplied excerpt: “Decision ledger and exact ending.” The quoted scope retains the missing microscopic transfer, gravitational, and endpoint obligations.

CheungHawk02C

N. Cheung, Hawk02-C, version 2, unpublished manuscript, n.d. Supplied abstract and “Outstanding gravitational obligations.” Autonomous source–boson comparison; no gravitational identification is imported.

CheungHawk03A

N. Cheung, Hawk03-A, version 1, unpublished manuscript, n.d. Supplied abstract and “Question, contribution, and scope.” Finite encoded-partner interface, not an independent gravitational interior construction.

CheungEn02B

N. Cheung, En02-B, version 1, unpublished manuscript, n.d. Supplied excerpt: “Programme soundness and elementary monotonicities: Finite certificates,” including thm:B1.

CheungEn02C

N. Cheung, En02-C, version 1, unpublished manuscript, n.d. Supplied excerpt: “Information availability versus executable recovery.”

CheungEn02E

N. Cheung, En02-E, version 2, unpublished manuscript, n.d. Supplied abstract and “Scope, dependencies, and evidence.” Selected Gaussian packet states and declared cutoff interfaces; no evaporation-channel identification is imported.

CheungEm03C

N. Cheung, Em03-C, version 1, unpublished manuscript, n.d. Supplied excerpts: “Source repairs and logical checks” and “One controlled correction and microscopic dynamics.”

Watrous

J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.

KSW

D. Kretschmann, D. Schlingemann, and R. F. Werner, “The information-disturbance tradeoff and the continuity of Stinespring's representation,” IEEE Transactions on Information Theory 54 (2008), 1708–1717. doi: 10.1109/TIT.2008.917696. arXiv:quant-ph/0605009.

KSWContinuity

D. Kretschmann, D. Schlingemann, and R. F. Werner, “A continuity theorem for Stinespring's dilation,” arXiv:0710.2495.

Kato

T. Kato, “Perturbation of continuous spectra by trace class operators,” Proceedings of the Japan Academy 33 (1957), 260–264. doi: 10.3792/pja/1195525063.

NachtergaeleSimsYoung

B. Nachtergaele, R. Sims, and A. Young, “Lieb–Robinson bounds, the spectral flow, and stability of the spectral gap for lattice fermion systems,” Contemporary Mathematics 717 (2018), 93–115. doi: 10.1090/conm/717/14443. arXiv:1705.08553.

ChenQiZhang

Y. Chen, X.-L. Qi, and P. Zhang, “Replica wormhole and information retrieval in the SYK model coupled to Majorana chains,” Journal of High Energy Physics 06 (2020), 121. doi: 10.1007/JHEP06(2020)121. arXiv:2003.13147.

KourkoulouMaldacena

I. Kourkoulou and J. Maldacena, “Pure states in the SYK model and nearly-${\mathrm{AdS}}_2$ gravity,” arXiv:1707.02325.

MaldacenaStanford

J. Maldacena and D. Stanford, “Comments on the Sachdev–Ye–Kitaev model,” Physical Review D 94 (2016), 106002. doi: 10.1103/PhysRevD.94.106002. arXiv:1604.07818.

NakayamaMiyataUgajin

Y. Nakayama, A. Miyata, and T. Ugajin, “The Petz (lite) recovery map for scrambling channel,” arXiv:2310.18991.

NakataTezuka

Y. Nakata and M. Tezuka, “Hayden–Preskill recovery in Hamiltonian systems,” Physical Review Research 6 (2024), L022021. doi: 10.1103/PhysRevResearch.6.L022021. arXiv:2303.02010.

GaikwadEtAl

A. Gaikwad, A. Kaushal, G. Mandal, and S. R. Wadia, “A microscopic model of black hole evaporation in two dimensions,” Journal of High Energy Physics 08 (2023), 171. doi: 10.1007/JHEP08(2023)171. arXiv:2210.15579.

AEMM

A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, “The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole,” Journal of High Energy Physics 12 (2019), 063. doi: 10.1007/JHEP12(2019)063. arXiv:1905.08762.

PSSY

G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, “Replica wormholes and the black hole interior,” arXiv:1911.11977.

Mathur

S. D. Mathur, “The information paradox: A pedagogical introduction,” arXiv:0909.1038.

AMPS

A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, “Black holes: Complementarity or firewalls?” arXiv:1207.3123.

SuppliedCatalogue

OpenAI, External mathematics catalogue, supplied manuscript records and incomplete excerpts, including families 227, 229, 260, 262, 264, 265, 272, and 273. The supplied record identifies a catalogue dated 7 October 2026. This citation is documentary only: the catalogue's claimed proofs are not independently verified here and none is a theorem premise of this manuscript.