References

OpenAI265A

OpenAI, A two-dimensional area law from a global spectral gap, 24 September 2026. External catalogue manuscript 265-01. Documentary statement and supplied excerpts; not independently verified here. https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf

OpenAI265B

OpenAI, Polynomial PEPS approximation of gapped square-grid ground states, 24 September 2026. External catalogue manuscript 265-02. Existence record, not an algorithmic premise. https://github.com/openai/math/blob/main/preprints/Polynomial-PEPS-approximation-of-gapped-square-grid-ground-states-September-24-2026/paper.pdf

OpenAIOverview

OpenAI, Mathematics release overview, repository file overview.tex. Catalogue descriptions of Family 265 and the named comparison families. Consulted 14 October 2026. Catalogue statements are not proof verification. https://github.com/openai/math/blob/main/overview.tex

Companion265

N. Cheung, Enter Calculus Companion: OpenAI 265-01 identification page. Attributes the external manuscript to OpenAI and states the non-verification and non-premise boundary. Consulted 14 October 2026. https://enter.grok.me/openai/265-01

LiebRuskai

E. H. Lieb and M. B. Ruskai, “Proof of the strong subadditivity of quantum-mechanical entropy,” Journal of Mathematical Physics 14 (1973), 1938–1941. doi:10.1063/1.1666274.

Watrous

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

Audenaert

K. M. R. Audenaert, “A sharp continuity estimate for the von Neumann entropy,” Journal of Physics A: Mathematical and Theoretical 40 (2007), 8127–8136. Preprint title: A Sharp Fannes-type Inequality for the von Neumann Entropy. https://arxiv.org/abs/quant-ph/0610146

VerstraeteCirac

F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, 2004. PEPS framework; no generic efficient exact contraction conclusion is imported. https://arxiv.org/abs/cond-mat/0407066

SchuchEtAl

N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, “Computational complexity of projected entangled pair states,” Physical Review Letters 98 (2007), 140506. Generic PEPS complexity, not a promise-preserving hardness theorem for the frozen class. https://arxiv.org/abs/quant-ph/0611050

Hastings

M. B. Hastings, “An area law for one-dimensional quantum systems,” Journal of Statistical Mechanics: Theory and Experiment (2007), P08024. One-dimensional theorem, not verification of the external two-dimensional claim. https://arxiv.org/abs/0705.2024

KimLinRanardShi

I. H. Kim, T.-C. Lin, D. Ranard, and B. Shi, Strict area law implies commuting parent Hamiltonian, 2024. Exact entropic assumptions and their own region geometry are retained. https://arxiv.org/abs/2404.05867

KnillLaflamme

E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Physical Review A 55 (1997), 900–911. doi:10.1103/PhysRevA.55.900.

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

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.

RyuTakayanagi

S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Physical Review Letters 96 (2006), 181602. A geometric holographic proposal, not a lattice-boundary identification in this manuscript. https://arxiv.org/abs/hep-th/0603001

EngelhardtWall

N. Engelhardt and A. C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, 2014. Generalized-entropy extremization is a separate interface. https://arxiv.org/abs/1408.3203

AlmheiriEtAl

A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, 2019. Gravitational replica constructions are not supplied by finite replica tensor identities. https://arxiv.org/abs/1911.12333

CheungOai265A

N. Cheung, Oai265-A, version 1, supplied manuscript Oai265-A.tex. Relevant labels: def:determination, prop:entropy, prop:gappedpath, thm:separation. Static area-law records, recovery, and temporal histories are different interfaces. Enter Calculus Companion; supplied passages. This citation identifies the source version, not the present unregistered development.

CheungOai265B

N. Cheung, Oai265-B, version 2, supplied manuscript Oai265-B-v2.tex. Relevant material: the complete-record statistical contract, thm:main, coverage and false-acceptance statements, small-sample nonacceptance, and release/provenance declarations. A supplied simulation report is not independently reproduced here. Enter Calculus Companion.

CheungEn02A

N. Cheung, En02-A, version 1, supplied manuscript En02-A-v1.tex. Relevant supplied labels: thm:Dprime, thm:E, thm:F. One-shot environment accounting, adaptive-storage distinctions, cut monotones, and reachability. No resource constant is identified with an entropy coefficient. Enter Calculus Companion.

CheungEn02B

N. Cheung, En02-B, version 1, supplied manuscript En02-B-v1.tex. Relevant labels: thm:B1, thm:B5, prop:observable, thm:tree, def:moduleclass. Complete finite programmes and model-specific decoder resources. Enter Calculus Companion.

CheungEn02C

N. Cheung, En02-C, version 1, supplied manuscript En02-C-v1.tex. Relevant label: thm:C1. Reference-preserving sequential soundness and separate laboratory and external-storage ledgers. Enter Calculus Companion.

CheungEn07B

N. Cheung, En07-B, version 1, supplied manuscript En07-B.tex. Relevant labels: thm:global, thm:exact, and the conditional endpoint interface. Complete access is not radiation-only access. Enter Calculus Companion.

CheungEn07C

N. Cheung, En07-C, version 1, supplied manuscript En07-C.tex. Relevant labels: prop:entropy, thm:exact, and the thermal-input and endpoint scope declarations. Selected thermal spectra and entropy do not determine a channel. Enter Calculus Companion.

CheungI

N. Cheung, Landmark I, version 2, supplied manuscript I.tex. Finite strategy characterization and destructive-readout semantics within the declared structural class. No many-body preparation or gravitational identification is imported. Enter Calculus Companion.

CheungXXII

N. Cheung, XXII-A, XXII-B, XXII-D, and XXII-F, supplied version-1 manuscripts. Relevant material: three-relation separation; static-state and operational-realization separation; conditional transport certificates; conditional assembly and the nonempty operational-instantiation target. No affine selector, transport realization, or cone-selection hypothesis is supplied by an entropy record. Enter Calculus Companion.

Oai283Draft

Oai283-A: Conditional Operational Certification of Constant-Error Oracle Unitary Synthesis. Earlier conversation development manuscript. Cited for explicit norm normalization and the supplied-resource distinction only. No independent registration, compilation, or verification of OpenAI 283-01 is asserted.

CheungEmission

N. Cheung, Em03-A–H and Hawk01–04, as identified in the supplied comparative dossier. Model-specific emission, apparatus, complete-channel, recovery, and endpoint scope records. No joint premise transfer and no identification with a gapped lattice ground state are made. Only the supplied passages are being attributed. Enter Calculus Companion.

CheungComparisons

N. Cheung, Oai290-A, Oai211-A, and Oai280-A/B/C, as identified in the supplied comparative dossier. Typed finite recovery, conditioned-tree marks, sector and logical-algebra certificates. Comparison interfaces only; no modular/tree/VOA/lattice or gravitational identification is assumed. Enter Calculus Companion.

CheungAnalytic

N. Cheung, IX-Free-2, version 1; IX-D, version 3; XI-E, version 1; XI-F, version 2; XI-G, version 1; XVI-B/D, version 1; and the supplied XIV-I/J, XV-B version 2, and XVI-N boundary passages. These are separate analytic source records, not premises of the finite lattice or channel theorems. No characteristic quantum-coefficient admission is inferred. Enter Calculus Companion.

CheungPortal

N. Cheung, Enter Calculus Companion. Searchable manuscript registry and scope descriptions. Consulted 14 October 2026. A source companion, not a proof checker. https://enter.grok.me/