References

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.

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. arXiv:quant-ph/0605009.

BenyKempfKribs

C. Bény, A. Kempf, and D. W. Kribs. Quantum error correction of observables. Physical Review A 76 (2007), 042303. arXiv:0705.1574.

KlieschRoth

M. Kliesch and I. Roth. Theory of quantum system certification—a tutorial. PRX Quantum 2 (2021), 010201. arXiv:2010.05925.

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. doi:10.1109/18.761271. arXiv:quant-ph/9712042.

Uhlmann

A. Uhlmann. The “transition probability” in the state space of a ${}^{\ast }$-algebra. Reports on Mathematical Physics 9 (1976), 273–279. doi:10.1016/0034-4877(76)90060-4.

Watrous

J. Watrous. Simpler semidefinite programs for completely bounded norms. arXiv:1207.5726, 2012.

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.

Takesaki

M. Takesaki. Theory of Operator Algebras II. Encyclopaedia of Mathematical Sciences, vol. 125. Springer, 2003. doi:10.1007/978-3-662-10451-4.

HaagerupMusat

U. Haagerup and M. Musat. The Effros–Ruan conjecture for bilinear forms on $C^{\ast }$-algebras. Inventiones Mathematicae 174 (2008), 139–163. arXiv:0711.1851.

MarrakchiVaes

A. Marrakchi and S. Vaes. {Ergodic states on type ${\mathrm{III}}_1$ factors and ergodic actions}. arXiv:2305.14217. Cited for the separate ergodic-state context, not as a gravitational or finite-decoder theorem.

JLMS

D. L. Jafferis, A. Lewkowycz, J. Maldacena, and S. J. Suh. Relative entropy equals bulk relative entropy. Journal of High Energy Physics 2016, issue 6, article 004. arXiv:1512.06431.

DongHarlowWall

X. Dong, D. Harlow, and A. C. Wall. Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality. Physical Review Letters 117 (2016), 021601. arXiv:1601.05416.

Hormander

L. Hörmander. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Second edition, Springer, 1990. Especially the wavefront, product, and pullback calculus.

OpenAI29002

OpenAI, as attributed in the supplied document. Bounded recovery for modular spectral averages. 23 September 2026. User-supplied document -recovery-for-modular-spectral-averages-September-23-2026.pdf. The dossier identifies it as Family 290, preprint 290-02. The bounded-witness statement is Theorem 4.1. No independent verification of the complete paper, catalogue metadata, or formalization is asserted here.

Dossier

User-supplied development plan and comparative source dossier. Undated material supplied with this manuscript request. Includes corpus label 6f617598, retrieval label -3, Family 290 identifiers, other catalogue summaries, and selected Cheung excerpts. Retrieval and documentary status are not proof verification. The full source of 290-01 and formalization manifests were not supplied.

Oai265B

Oai265-B: Reference-Preserving Quantum-Memory Certification from Measured Correlations. Supplied companion source and version-2 revision, 12 October 2026. Finite-confidence and complete-record comparison. No new execution is imported into Oai290-A.

CheungI

N. Cheung. Landmark I, version 2. Undated manuscript; supplied .tex excerpts. Finite instruments, destructive access, records, and decoder boundaries. Enter Calculus Companion. Cited at the supplied-source level.

CheungHawk04A

N. Cheung. Hawk04-A, version 1. Undated manuscript; supplied statement summaries. Especially :global and :exact: global code preservation versus accessible recovery. Enter Calculus Companion. Full source not independently verified.

CheungHawk04B

N. Cheung. Hawk04-B: Black-Hole Information as an Access Problem: Enter, Quantum Recovery, and Complete Residuals. Version 2, undated manuscript. Supplied summaries and excerpts: :classical, :certificate, and decoder/error-normalization interfaces. Enter Calculus Companion.

CheungEn07A

N. Cheung. En07-A, version 1. Undated manuscript; supplied summaries of :exact, :dimension, and :release. Complete restoration and the separation from thermodynamic cost. Enter Calculus Companion.

CheungEn07B

N. Cheung. En07-B, version 1. Undated manuscript; supplied summary of :endpoint. Four same-model obligations are reported, but their complete verbatim clauses are not reconstructed here. Enter Calculus Companion.

CheungEn07C

N. Cheung. En07-C, version 1. Undated manuscript; supplied thermal-input and complete-channel scope records, including :release. Enter Calculus Companion.

CheungEn01C

N. Cheung. En01-C, version 1. Undated manuscript; supplied complete-record statistical excerpts and calibration-risk interface. Enter Calculus Companion.

CheungEn02B

N. Cheung. En02-B, version 1. Supplied 02-B-v1.tex excerpts. Especially :observable and the declared streaming and decoder-module interfaces. Enter Calculus Companion.

CheungEn02C

N. Cheung. En02-C, version 1. Supplied 02-C-v1.tex excerpts. Especially :C2, :C3, and :firstmeasurement. Exact information-theoretic recovery and a restricted one-live-qubit streaming obstruction. Enter Calculus Companion.

CheungEm05B

N. Cheung. Em05-B, version 1. Supplied 05-B-v1.tex excerpts. Especially :constantComplement and the flagged-instrument interface. Enter Calculus Companion.

CheungHawk03A

N. Cheung. Hawk03-A, version 1. Supplied 03-A.tex excerpts and summaries. Especially :exact, :partner, and the exact commuting-copy obstruction. Enter Calculus Companion.

CheungHawk01B

N. Cheung. Hawk01-B, version 1. Supplied 01-B.tex excerpts. Especially :exact and :rankcapacity. Enter Calculus Companion.

CheungVIIG

N. Cheung. VII-G, version 1. Supplied -G.tex excerpts. Especially :decouple and the constant-channel and informative-record distinctions. Enter Calculus Companion.

CheungEm03A

N. Cheung. Em03-A, version 1. Supplied fixed-background emission and documentary-audit excerpts. The physical source hypotheses are not imported here. Enter Calculus Companion.

CheungEm03B

N. Cheung. Em03-B, version 2. Supplied 03-B-v2.tex excerpts on finite packet marginals, apparatus access, complete failures, and non-transfer. Enter Calculus Companion.

CheungEm03C

N. Cheung. Em03-C, version 1. Supplied 03-C-v1.tex excerpts on joint packet interfaces, reference stability, and mechanism nonidentification. Enter Calculus Companion.

CheungEm03F

N. Cheung. Em03-F, version 1. Supplied 03-F-v1.tex excerpts. Source-dependent exterior field signals and an entanglement-breaking accessible channel are distinct from quantum recovery. Enter Calculus Companion.

CheungEm03G

N. Cheung. Em03-G, version 1. Supplied 03-G-v1.tex excerpts. Information-theoretic packet transmission and local coherent reception are separate release gates. Enter Calculus Companion.

CheungXVIIA

N. Cheung. XVII-A, version 1. Supplied -A.tex excerpts and summaries. One finite-matrix sine effect from a declared boundary region; source-alignment and non-premise ledger. Enter Calculus Companion.

CheungHawk02A

N. Cheung. Hawk02-A, version 1. Supplied 02-A.tex excerpts. Complete transfer interfaces, residual leakage, and the outstanding gravitational and analytic obligations. Enter Calculus Companion.

CheungHawk02D

N. Cheung. Hawk02-D, version 1. Supplied 02-D.tex excerpts. Especially the detailed :endpoint statement with $F_{\mathrm{out}}={\Omega }_{+}^{†}{J}_S$. The finite reservoir endpoint is not a gravitational endpoint. Enter Calculus Companion.

CheungHawk02E

N. Cheung. Hawk02-E, version 1. Supplied 02-E.tex excerpts. Gravitational correspondence, common validity intervals, and the auxiliary-mathematics non-transfer audit. Enter Calculus Companion.

CheungVIIIA

N. Cheung. VIII-A, version 2. Supplied -A-v2.tex excerpts. Accessible state functionals and independently specified conical geometric admissibility. Enter Calculus Companion.

CheungVIIIB

N. Cheung. VIII-B, version 2. Supplied -B-v2.tex excerpts. Especially :spectators and :source-map. A prescribed source dictionary is not derived geometry. Enter Calculus Companion.

CheungVIIIC

N. Cheung. VIII-C, version 2. Supplied -C-v2.tex excerpts. Especially the token-boundary and destructive-record distinctions. Enter Calculus Companion.

CheungEm06A

N. Cheung. Em06-A, version 1. Supplied 06-A-v1.tex excerpts. A scalar-only coherent limit on a prescribed background, with explicit non-emergence boundaries. Enter Calculus Companion.

CheungEm06B

N. Cheung. Em06-B, version 1. Supplied 06-B-v1.tex excerpts. Declared scalar-family concentration and residual fluctuations; the metric coefficient remains an input. Enter Calculus Companion.

CheungXIVH

N. Cheung. Landmark XIV-H: Internal kernel relations on the operator route. Version 1, supplied -H.tex excerpts. Especially :H2 and :K5exact. Historical internal-amplitude stop, not a current full-map release. Enter Calculus Companion.

CheungXIVI

N. Cheung. Landmark XIV-I. Version 1, supplied -I.tex excerpts. Especially :pole, :I2, :I3, and :release. Internal amplitudes are constructed at I2; the joint characteristic map remains unresolved at I3. Enter Calculus Companion.

CheungXIVJ

N. Cheung. Landmark XIV-J. Version 1, supplied -J.tex excerpts. Especially :descent, :null, :reference, :conditionalextension, :J2, and :noadmission. The supported-sector reference is not the full joint map. Enter Calculus Companion.

CheungXVIA

N. Cheung. XVI-A, version 1. Supplied -A.tex excerpts, especially the separate prepared-source obligations 4–6. Enter Calculus Companion.

CheungXVIN

N. Cheung. XVI-N, version r12. Supplied -N.tex package-boundary excerpt and documentary status summary. No coefficient admission or O.1 closure is inferred. Enter Calculus Companion.