References

OpenAI069Catalogue

OpenAI, as attributed in the catalogue. Global quantum geometric Langlands at irrational level. Family 069-01, 4 October 2026. Enter Calculus Companion: https://enter.grok.me/openai/069-01. Documentary source record; not independently verified here.

OpenAI069Introduction

OpenAI, as attributed in the public preprint source. Global quantum geometric Langlands at irrational level. Introduction, including the shifted-level convention and :main, 4 October 2026. Public source: https://github.com/openai/math/blob/main/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/build/sections/01-introduction.tex. Source consultation is not proof verification.

KapustinWitten

A. Kapustin and E. Witten. Electric-magnetic duality and the geometric Langlands program. Communications in Number Theory and Physics 1 (2007), 1–236. arXiv:hep-th/0604151. https://arxiv.org/abs/hep-th/0604151. Gauge-theoretic context, not a gravitational premise of this paper.

GaitsgoryQGL

D. Gaitsgory. Quantum Langlands Correspondence. Informal notes written in 2007, posted in 2016. arXiv:1601.05279. https://arxiv.org/abs/1601.05279.

GaitsgoryWhit

D. Gaitsgory. The local and global versions of the Whittaker category. arXiv:1811.02468. https://arxiv.org/abs/1811.02468. The independently stated local/global Whittaker framework is not replaced by the finite cyclic example.

ArakawaFrenkel

T. Arakawa and E. Frenkel. Quantum Langlands duality of representations of $W$-algebras. Compositio Mathematica 155 (2019), 2235–2262. doi:10.1112/S0010437X19007553. arXiv:1807.01536. https://arxiv.org/abs/1807.01536.

PolishchukRothstein

A. Polishchuk and M. Rothstein. {Fourier transform for $D$-algebras}. arXiv:math/9901009. https://arxiv.org/abs/math/9901009. Abelian differential-operator transforms have their own geometric hypotheses and are not the finite matrix transform proved here.

Maldacena

J. M. Maldacena. The large $N$ limit of superconformal field theories and supergravity. Advances in Theoretical and Mathematical Physics 2 (1998), 231–252. arXiv:hep-th/9711200. https://arxiv.org/abs/hep-th/9711200. External AdS/CFT context only.

Choi

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.

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Author's book materials: https://cs.uwaterloo.ca/ watrous/TQI/. Channel, Choi, trace-norm, and reference conventions.

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. arXiv:quant-ph/9604034. https://arxiv.org/abs/quant-ph/9604034.

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.

CCJ

C. G. Callan, Jr., S. Coleman, and R. Jackiw. A new improved energy-momentum tensor. Annals of Physics 59 (1970), 42–73. doi:10.1016/0003-4916(70)90394-5.

HollandsWald

S. Hollands and R. M. Wald. Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes. arXiv:gr-qc/0404074. https://arxiv.org/abs/gr-qc/0404074. Derivative, product, and conservation normalization distinctions.

Dossier

User-supplied Oai069-A development plan and comparative manuscript dossier. Includes source statements, retrieved passages, and status records. Retrieval label 0d99297f. No exhaustive corpus scan or independent manuscript-proof verification is asserted.

Cheung280C

N. Cheung. Oai280-C, version 1. Supplied 280-C.tex excerpts. Especially :simultaneous and the operational-map contract. Logical-code, subsystem, and finite-algebra certification. Enter Calculus Companion, https://enter.grok.me.

Cheung280A

N. Cheung. Oai280-A, version 1. Supplied 280-A.tex excerpts. Sector-label certificates and VOA–net separation. The strong-rationality, positivity, and locality boundaries are retained. Enter Calculus Companion.

Cheung280B

N. Cheung. Oai280-B, version 1. Supplied 280-B.tex record. Finite perfect-tensor and five-qubit decoder benchmarks. No five-qubit theorem is imported into the different encoder of this paper. Enter Calculus Companion.

Cheung290A

N. Cheung. Oai290-A, version 1. Supplied 290-A.tex excerpts. Implemented-decoder versus optimal-decoder decisions and modular/operational separation. Enter Calculus Companion.

Cheung265B

N. Cheung. Oai265-B, version 2. Supplied 265-B-v2.tex excerpts. Especially :main. Complete finite-confidence recovery accounting. Enter Calculus Companion.

CheungEn02B

N. Cheung. En02-B, version 1. Supplied 02-B-v1.tex excerpts. Especially :B1. Complete decoder-programme soundness and resource ledgers. Enter Calculus Companion.

CheungEn02C

N. Cheung. En02-C, version 1. Supplied 02-C-v1.tex excerpts. Especially :C1. Reference-preserving sequential execution and actual-storage accounting. Enter Calculus Companion.

CheungEn05C

N. Cheung. En05-C, version 1. Supplied 05-C.tex excerpts. Especially :comparison. Tensor-free snapshot–channel–adjoint equivalence. Only its finite-algebra interface is reproved and used here. Enter Calculus Companion.

CheungVIIG

N. Cheung. VII-G, version 1. Supplied -G.tex excerpts. Especially :structure. Finite direct-sum hybrid channels and complete branch maps. Enter Calculus Companion.

Cheung282A

N. Cheung, as attributed in the supplied dossier. Oai282-A, version 1. Supplied predecessor text and excerpts. Especially :ward. Declared improvement data, finite Ward-defect certificates, and local/global separation. Enter Calculus Companion.

Cheung282B

N. Cheung. Oai282-B, version 1. Supplied 282-B.tex excerpts and description. Especially :coverage. Free-scalar homogeneous stress-class coverage and its stated physical-field boundaries. The full manuscript is not an unchecked proof premise here. Enter Calculus Companion.

CheungIII

N. Cheung. Landmark III, version 2. Supplied .tex excerpts. Classical equation-space quotient and jet distinctions. Enter Calculus Companion.

CheungVI

N. Cheung. Landmark VI, version 1. Supplied .tex excerpts. Especially :descent. Finite-time second-order classical relational descent. Enter Calculus Companion.

CheungEm02E

N. Cheung. Em02-E, version 1. Supplied 02-E-v1.tex excerpts. Especially :lapse. Relational lapse on a declared Einstein–scalar background; not the improvement scalar or a local-system coefficient. Enter Calculus Companion.

CheungXIVH

N. Cheung. Landmark XIV-H: Internal kernel relations on the operator route, version 1. Supplied -H.tex excerpts. Historical H2 boundary and exact internal kernel maps. Enter Calculus Companion.

CheungXIVI

N. Cheung. Landmark XIV-I, version 1. Supplied -I.tex excerpts. Especially :I2, :I3, and :release. Internal amplitude construction and the missing full characteristic map. Enter Calculus Companion.

CheungXIVJ

N. Cheung. Landmark XIV-J, version 1. Supplied -J.tex excerpts. Especially :descent, :J2, and :noadmission. Supported-sector references do not admit the complete coefficient. Enter Calculus Companion.

CheungXVB

N. Cheung. XV-B, version 2. Supplied -B.tex excerpts. Relative lifting and normal-jet extension interfaces. The technical manuscript is distinct from the downstream quantum-admission exit. Enter Calculus Companion.

CheungXVIN

N. Cheung. XVI-N, version r12. Supplied -N.tex excerpts. Especially :massless-obstruction. Conditional prepared-insertion obstruction and the retained Route B limitation. Enter Calculus Companion.

CheungXIG

N. Cheung. XI-G, version 1. Supplied -G.tex excerpts. Frozen-class microlocal separation on its stated quantifiers; not a universal obstruction to every bimodule. Enter Calculus Companion.