References

Laflamme

R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek. Perfect quantum error correcting code. Physical Review Letters 77 (1996), 198–201. doi:10.1103/PhysRevLett.77.198. arXiv:quant-ph/9602019.

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.

HaPPY

F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill. Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. Journal of High Energy Physics 06 (2015), 149. doi:10.1007/JHEP06(2015)149. arXiv:1503.06237.

Gong

M. Gong et al. Experimental exploration of five-qubit quantum error correcting code with superconducting qubits. National Science Review 9 (2022), no. 1, nwab011. doi:10.1093/nsr/nwab011. arXiv:1907.04507.

PastawskiPreskill

F. Pastawski and J. Preskill. Code properties from holographic geometries. Physical Review X 7 (2017), 021022. doi:10.1103/PhysRevX.7.021022. arXiv:1612.00017.

OperatorQEC

D. W. Kribs, R. Laflamme, D. Poulin, and M. Lesosky. Operator quantum error correction. Quantum Information and Computation 6 (2006), 383–399. arXiv:quant-ph/0504189.

Harlow

D. Harlow. The Ryu–Takayanagi formula from quantum error correction. Communications in Mathematical Physics 354 (2017), 865–912. doi:10.1007/s00220-017-2904-z. arXiv:1607.03901.

JonesPollack

J. Jones and J. Pollack. Uniqueness of Complementary Recovery in Holographic Error-Correcting Codes. arXiv:2509.19299v1, 23 September 2025. Cited for the explicitly added correctability assumption; not used as a premise of a local recovery or uniqueness theorem.

Watrous

J. Watrous. The Theory of Quantum Information. Cambridge University Press, 2018. Finite-dimensional channels, Choi matrices, references, and completely bounded trace norms.

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. arXiv: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.

ClopperPearson

C. J. Clopper and E. S. Pearson. The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika 26 (1934), 404–413. doi:10.1093/biomet/26.4.404.

Oai280A

Oai280-A: Sector-Label Certificates and VOA–Net Separation: Locality, Fusion, Recovery, Extensions, and Energy. Version 1, 12 October 2026. User-supplied complete manuscript 280-A.tex; the supplied source file does not specify an author. Especially :exact, :main, :forty, :comparison, and the physical-identification scope boundaries.

CheungEn02B

N. Cheung. En02-B, version 1. Undated supplied record and excerpts identifying 02-B-v1.tex. Resource-limited holographic reconstruction; especially :privacy, :access, :perfect, :B1, :B5, and :observable. Enter Calculus Companion. The complete manuscript was not independently verified here.

CheungEn02C

N. Cheung. En02-C, version 1. Undated user-supplied excerpts of 02-C-v1.tex. Especially :C3 and its first-carrier separability argument for the declared ordered streaming interface. Enter Calculus Companion.

CheungEn02E

N. Cheung. En02-E, version 2. Undated supplied excerpts of 02-E-v2.tex. Complete protocols, off-support completion, and instruction accounting in the stated packet interface. Enter Calculus Companion. No packet model is imported as the boundary experiment here.

CheungOai265B

N. Cheung. Oai265-B, version 2. Undated supplied excerpts of 265-B-v2.tex. Especially :main and the conditional validation contract. Enter Calculus Companion. Reported numerical studies are not reused as validation evidence.

CheungOai290A

N. Cheung. Oai290-A, version 1. Undated supplied excerpts of 290-A.tex. Especially :assembly: implemented-decoder, all-decoder, effect-only, and inconclusive decisions. Enter Calculus Companion.

CheungOai265A

N. Cheung. Oai265-A, version 1. Undated supplied excerpts of 265-A.tex. Exact complete-complement recovery and separation of static records, recovery, entropy histories, and gravitational identification. Enter Calculus Companion.

CheungHawk04B

N. Cheung. Hawk04-B, version 2. Undated supplied excerpts of 04-B.tex. Especially :exact: a complete complement and a decoder defined on the whole received interface. Enter Calculus Companion.

CheungEn07A

N. Cheung. En07-A, version 1. Undated supplied excerpts of 07-A.tex. Especially :exact, the dimension-count distinction, and the holographic-identification refusal. Enter Calculus Companion.

CheungEn07B

N. Cheung. En07-B, version 1. Undated supplied excerpts of 07-B.tex. Exact complete-complement criterion and the separate same-model endpoint obligations. Enter Calculus Companion. No gravitational endpoint package is instantiated here.

CheungEn05C

N. Cheung. En05-C, version 1. Undated supplied excerpts of 05-C.tex. Especially :comparison: snapshot determination, UCP factorization, and adjoint containment. Enter Calculus Companion.

CheungEn06A

N. Cheung. En06-A, version 1. Undated supplied excerpts of 06-A.tex. Especially :algmicro and its product-preserving observable-limit hypotheses. Enter Calculus Companion.

CheungVIIg

N. Cheung. VII-G, version 1. Undated supplied excerpts of -G.tex. Finite hybrid channel structure, complete branch sums, and classical–quantum interfaces. Enter Calculus Companion.

CheungEm05B

N. Cheung. Em05-B, version 1. Undated supplied record and revision-audit excerpts of 05-B-v1.tex. Complete complements, flagged instruments, reference-preserving comparison, and the separation of verification instructions from execution reports. Enter Calculus Companion.

CheungXVIIA

N. Cheung. XVII-A, version 1. Undated supplied excerpts of -A.tex. Especially :main and :no-wedge: a named finite observable reconstruction is not full subsystem recovery or an entanglement wedge. Enter Calculus Companion.

CheungI

N. Cheung. Landmark I, version 2. Undated supplied excerpts of .tex. Born laws and destructive projective residuals within the specified finite strategy class. Enter Calculus Companion. No measurement-characterization theorem is a premise here.

CheungXIVI

N. Cheung. Landmark XIV-I, version 1. User-supplied excerpts of -I.tex. Internal amplitude construction at I2 and the unresolved complete joint characteristic map at I3. Cited only for the separate documentary status ledger.

CheungXIVJ

N. Cheung. Landmark XIV-J, version 1. User-supplied excerpts of -J.tex. Reconstruction descent, the supported-sector off-characteristic reference, the J2 boundary, and the unasserted quantum admission. Cited only for documentary separation.

CheungXVB

N. Cheung. XV-B, version 2. User-supplied excerpts of -B.tex. Relative lifting, normal-jet and extension interfaces; the inherited full characteristic map remains unconstructed at B1. Its technical existence is distinguished from opening the downstream admission exit.

CheungXVIN

N. Cheung. XVI-N, version r12. User-supplied excerpts of -N.tex. Conditional prepared massless-instance obstruction and the separately undischarged spatially compact Route B. Cited only for the unchanged documentary boundary.