References

OpenAI272

OpenAI. Entanglement with zero distillable secret key in local dimension ten. Edition dated 27 September 2026. External Family 272-01 manuscript. Consulted source sections include foundations, coherence, transcripts, secret key, introduction, applications, and the supplementary-checks scope. The dimension-ten geometric/arithmetic construction is not independently verified in the present manuscript. https://github.com/openai/math/tree/main/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026

Companion272

N. Cheung. Enter Calculus Companion: OpenAI 272-01 identification page. Consulted 14 October 2026. A source-identification and scope record, not proof verification. https://enter.grok.me/openai/272-01

LeanScope

OpenAI. Entanglement without distillable secret key: Lean scope note. Repository file lean/docs/272.md. Consulted 14 October 2026. The listed pair and channel targets exclude the zero-key theorem. https://github.com/openai/math/blob/main/lean/docs/272.md

LeanPair

OpenAI. DimensionTenPair.lean. Repository file lean/ComparatorChallenges/DimensionTenPair.lean. Consulted 14 October 2026. The inspected main_pair proof body contains sorry; no completed formal proof is asserted. https://github.com/openai/math/blob/main/lean/ComparatorChallenges/DimensionTenPair.lean

LeanChannel

OpenAI. DimensionTenChannel.lean. Repository file lean/ComparatorChallenges/DimensionTenChannel.lean. Consulted 14 October 2026. The inspected exists_channel_fin21 proof body contains sorry. https://github.com/openai/math/blob/main/lean/ComparatorChallenges/DimensionTenChannel.lean

Supplementary272

OpenAI. Supplementary exact checks. Appendix in the Family 272-01 source, build/source/appendices/supplementary-checks.tex. Scope of the two small integer-check programs for the separate combinatorial construction. No execution is reproduced here. https://github.com/openai/math/blob/main/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/build/source/appendices/supplementary-checks.tex

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Finite-dimensional channels, reference systems, Choi operators, and completely bounded trace norms. https://cs.uwaterloo.ca/ watrous/TQI/

Choi1975

M.-D. Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications 10 (1975), 285–290. doi:10.1016/0024-3795(75)90075-0.

HSR2003

M. Horodecki, P. W. Shor, and M. B. Ruskai. Entanglement breaking channels. Reviews in Mathematical Physics 15 (2003), 629–641. doi:10.1142/S0129055X03001709. Preprint title: General Entanglement Breaking Channels. https://arxiv.org/abs/quant-ph/0302031

HorodeckiRange

P. Horodecki. Separability criterion and inseparable mixed states with positive partial transposition. Physics Letters A 232 (1997), 333–339. doi:10.1016/S0375-9601(97)00416-7. https://arxiv.org/abs/quant-ph/9703004

HHHO2005

K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim. Secure key from bound entanglement. Physical Review Letters 94 (2005), 160502. doi:10.1103/PhysRevLett.94.160502. https://arxiv.org/abs/quant-ph/0309110

HHHO2009

K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim. General paradigm for distilling classical key from quantum states. IEEE Transactions on Information Theory 55 (2009), 1898–1929. doi:10.1109/TIT.2008.2009798. https://arxiv.org/abs/quant-ph/0506189

FuchsVanDeGraaf

C. A. Fuchs and J. van de Graaf. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Transactions on Information Theory 45 (1999), 1216–1227. https://arxiv.org/abs/quant-ph/9712042

CLMOW

E. Chitambar, D. Leung, L. Mančinska, M. Ozols, and A. Winter. Everything you always wanted to know about LOCC (but were afraid to ask). Communications in Mathematical Physics 328 (2014), 303–326. https://arxiv.org/abs/1210.4583

Kallenberg

O. Kallenberg. Foundations of Modern Probability. Second edition, Springer, 2002. Standard Borel kernels, conditional distributions, countable history extensions, and conditional expectation.

CMHW2019

M. Christandl, A. Müller-Hermes, and M. M. Wolf. When do composed maps become entanglement breaking? Annales Henri Poincaré 20 (2019). PPT-composition framework, Choi transfer, and block-switch comparisons. https://arxiv.org/abs/1807.01266

Filippov2021

S. N. Filippov. Capacity of trace decreasing quantum operations and superadditivity of coherent information for a generalized erasure channel. 2021. Trace-preserving flag completion is a comparison mechanism, not a key-distillation theorem. https://arxiv.org/abs/2101.05686

HansonRouzeFranca

E. P. Hanson, C. Rouzé, and D. Stilck França. Eventually entanglement breaking Markovian dynamics: structure and characteristic times. Annales Henri Poincaré 21 (2020), 1517–1571. doi:10.1007/s00023-020-00906-4.

Park

S.-J. Park. Every PPT channel has finite entanglement-breaking index. Preprint, 2026, arXiv:2608.13551. First submitted 13 August 2026. Cited for its stated eventual-breaking theorem, not independently verified here. https://arxiv.org/abs/2608.13551

Hoeffding

W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58 (1963), 13–30. doi:10.1080/01621459.1963.10500830.

CheungI

N. Cheung. Landmark I, version 2. Supplied manuscript I.tex. Relevant items: prop:classical-recovery and the record/continuation, destructive-readout, and retained-memory interfaces. Enter Calculus Companion. The structural measurement class is not replaced by a secret-key or gravity model.

CheungVIIIB

N. Cheung. VIII-B, version 3. Supplied manuscript VIII-B-v3.tex. Relevant supplied label: prop:classical-recovery. Sharp record-only recovery obstruction and its retained-quantum-memory exception. Enter Calculus Companion.

CheungHawk04B

N. Cheung. Hawk04-B. Supplied source descriptions and comparative excerpts. Typed readout, operational discard, complete preservation, and record-only quantum-recovery limitations. No evaporation endpoint or secret-key theorem is imported. Enter Calculus Companion.

CheungEn02A

N. Cheung. En02-A, version 1. Supplied manuscript En02-A-v1.tex. Relevant label: thm:Dprime. Exact one-shot environment ledger and its distinction from adaptive live-memory occupancy. Enter Calculus Companion.

CheungEn02B

N. Cheung. En02-B, version 1. Supplied manuscript En02-B-v1.tex. Relevant labels: thm:batch, thm:B5, prop:observable, and thm:communication. Specified subsystem decoders, streaming access, and communication costs; not Family 272 inputs. Enter Calculus Companion.

CheungEn02C

N. Cheung. En02-C, version 1. Supplied manuscript En02-C-v1.tex. Relevant labels: thm:C1, thm:C2, thm:C3, and cor:robustgap. Complete sequential soundness and model-specific memory/recovery obstructions. Enter Calculus Companion.

CheungEn01B

N. Cheung. En01-B, version 2. Supplied manuscript En01-B-v2.tex. Relevant label: thm:bounded. Predeclared bounded complete-record statistics, uniform family gaps, and implementation allowances. Enter Calculus Companion.

CheungEn01C

N. Cheung. En01-C, version 1. Supplied manuscript En01-C.tex. Relevant label: thm:certificate. Clipping, complete attempted-run records, implementation allowances, and finite-sample coverage for its declared statistic. No diffusion parameter or apparatus is transferred here. Enter Calculus Companion.

CheungOai265A

N. Cheung. Oai265-A, supplied version 1. Relevant labels: def:determination and thm:separation. Static entropy records, selected spectra, complete channels, and radiation histories are distinct interfaces. The earlier conversation development under that identifier is not independently registered or machine-verified here. Enter Calculus Companion.

CheungEn07C

N. Cheung. En07-C, version 1, as supplied in the comparative material. Relevant comparison: selected entropy and thermal spectra do not determine a complete channel or its referenced recovery behavior. No thermal input is identified with the Family 272 state. Enter Calculus Companion.

CheungEm03F

N. Cheung. Em03-F, version 1. Supplied manuscript Em03-F-v1.tex. Relevant labels: thm:EB, thm:recovery, thm:witness, and thm:counts. A declared packet channel, a classical signal, and a referenced quantum-recovery obstruction. No Family 272 or gravitational identification is imported. Enter Calculus Companion.

CheungEm03G

N. Cheung. Em03-G, version 1. Supplied manuscript Em03-G-v1.tex. Relevant labels: prop:Fobstruction, thm:compacttransmission, and thm:lower. Receiver postprocessing, changed source dynamics, resource-dependent recovery, and finite-coupling obstructions are distinguished. Enter Calculus Companion.

ComparisonDossier

User-supplied Oai272-A development plan and comparative source dossier. Includes Family 272 and comparison-family records, supplied Cheung excerpts, and nominations including En07-A, Oai265-B, Oai290-A, Oai280-C, En10-B, and Oai283-A. A nomination is not a complete theorem statement, an inherited premise, or independent verification. Retrieval label supplied by the user: 178f233f (retrieval-3).

AnalyticDossier

N. Cheung. Supplied version-specific excerpts from IX-Free-2, IX-D, XI-E/F/G, XIV-A/B, and XVI-B/D, together with the stated XIV–XV–XVI programme boundaries. Separate analytic source records, not premises of this finite-dimensional certificate. No quantum-admission or standing-problem conclusion is transferred. Enter Calculus Companion.