References

BroutinMarckert

N. Broutin and J.-F. Marckert. Asymptotics of trees with a prescribed degree sequence and applications. Random Structures & Algorithms 44 (2014), 290–316. doi:10.1002/rsa.20463. arXiv:1110.5203. Theorem 3, Lemma 11, and Section 6 are the identified coding input.

Janson

S. Janson. Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probability Surveys 9 (2012), 103–252. doi:10.1214/11-PS188. arXiv:1112.0510.

LyonsPemantlePeres

R. Lyons, R. Pemantle, and Y. Peres. Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. The Annals of Probability 23 (1995), 1125–1138. doi:10.1214/aop/1176988176. arXiv:math/0404083. Section 2 supplies the size-biased-tree context.

Dwass

M. Dwass. The total progeny in a branching process and a related random walk. Journal of Applied Probability 6 (1969), 682–686. doi:10.2307/3212112.

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.

AddarioBerry

L. Addario-Berry. A probabilistic approach to block sizes in random maps. ALEA, Latin American Journal of Probability and Mathematical Statistics 16 (2019). arXiv:1503.08159. Cited for the rooted corner-indexed block-tree organization.

DuquesneLeGall

T. Duquesne and J.-F. Le Gall. The Hausdorff measure of stable trees. arXiv:math/0509690. Cited for the Brownian-CRT Hausdorff-measure and dimension facts, not for a gravitational interpretation.

Sheffield

S. Sheffield. Quantum gravity and inventory accumulation. The Annals of Probability 44 (2016), 3804–3848. arXiv:1108.2241. Inventory encoding and the $p=1/2,q=4$ transition context.

Watrous

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

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.

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.

OpenAI211

OpenAI, as attributed in the supplied document. Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four. 24 September 2026. User-supplied .pdf. The supplied dossier places it in Family 211. No independent verification of the complete proof or catalogue provenance is asserted.

OpenAI290

OpenAI, as attributed in the supplied document. Bounded recovery for modular spectral averages. 23 September 2026. User-supplied manuscript, identified in the dossier as Family 290, preprint 290-02. Its modular statement is documentary context, not a tree-theorem premise.

Dossier

User-supplied Oai211-A development plan and comparative source dossier. Includes Family 211 companion titles, Families 265 and 290 records, supplied manuscript excerpts, and retrieval label 6f2e9611. Retrieval, catalogue identifiers, and reported formalization metadata are not mathematical verification.

Oai265A

Oai265-A, version 1. Supplied companion manuscript excerpts, especially :determination, :entropy, and :separation. A record-only area-law/recovery comparison. No independent verification or physical identification imported.

Oai265B

Oai265-B: Reference-Preserving Quantum-Memory Certification from Measured Correlations, version 2. Supplied 265-B-v2.tex excerpts and reported version-1 execution summary. The report is not independently rerun; no laboratory calibration is supplied by its documentary use.

Oai290A

Oai290-A: Typed Boundary Recovery, Complete Complements, and Modular-Band Witnesses, version 1. Supplied companion draft, especially :assembly and :endpoint. Its operational endpoint inputs are independently defined there and are not En07-B's verbatim four clauses.

CheungI

N. Cheung. Landmark I, version 2. Undated supplied manuscript excerpts. Finite instruments, destructive readout, and quantum-return boundaries. Enter Calculus Companion. Cited at the supplied-source level.

CheungHawk04A

N. Cheung. Hawk04-A, version 1. Undated supplied 04-A.tex excerpts. Especially :exact: complete-complement recovery and its explicit normal form. Enter Calculus Companion.

CheungHawk04B

N. Cheung. Hawk04-B. Undated supplied manuscript summaries and excerpts. Classical readout, deterministic quantum recovery, and reference-sensitive certification. Enter Calculus Companion. No physical model transferred.

CheungEn07A

N. Cheung. En07-A, version 1. Undated supplied 07-A.tex excerpts. Especially :exact and the complete-channel, restoration, and thermodynamic distinctions. Enter Calculus Companion.

CheungEn07B

N. Cheung. En07-B, version 1. Undated supplied 07-B.tex excerpts. Especially :four and :endpoint. Complete preparation, evolution, access-preserving endpoint, and complete complementary-channel obligations. Enter Calculus Companion.

CheungEn07C

N. Cheung. En07-C, version 1. Undated supplied 07-C.tex excerpts. Thermal-spectrum input and the exact complete-complement criterion; a residual dimension count is not an area law. Enter Calculus Companion.

CheungEn02B

N. Cheung. En02-B, version 1. Supplied 02-B-v1.tex excerpts. Especially :B1 and :observable. Complete programme soundness, resource ledgers, and the distinction between logical readout and subsystem recovery. Enter Calculus Companion.

CheungEn02C

N. Cheung. En02-C, version 1. Supplied 02-C-v1.tex excerpts. Especially :C1 and the stated restricted streaming interfaces. No streaming-error number is transferred to a different channel. Enter Calculus Companion.

CheungEn01B

N. Cheung. En01-B, version 2. Supplied 01-B-v2.tex excerpts. Especially :bounded. Independent attempted runs, complete records, and robust bounded statistics. Enter Calculus Companion.

CheungEm03ABC

N. Cheung. Em03-A, version 1; Em03-B, version 2; Em03-C, version 1. Supplied excerpts. Fixed-background source, finite packet reductions, apparatus-model access, and physical-correspondence boundaries. Their field states and receiver models are not inputs here. Enter Calculus Companion.

CheungVIII

N. Cheung. VIII-A, VIII-B, and VIII-C, version 2. Supplied excerpts on matching contracts, the prescribed conical dictionary, and certified fixed-packet kinematics. These conical data are not the Brownian CRT. Enter Calculus Companion.

CheungXVII

N. Cheung. XVII-A, version 1. Supplied manuscript description and excerpts. One finite-matrix sine effect from one declared boundary region. Not a gravitational identification of the present channel marks. Enter Calculus Companion.

CheungXIVHIJ

N. Cheung. Landmark XIV-H, Landmark XIV-I, and Landmark XIV-J, version 1. Supplied excerpts. Historical H2 stop; internal amplitude at I2; full-map stops at I3/J2; reconstruction descent and partial supported-sector reference. No inherited quantum admission is imported. Enter Calculus Companion.

CheungXVB

N. Cheung. XV-B, version 2. Supplied -B.tex excerpts. Relative lifting, complete normal-jet null tests, and conditional extension interfaces. The inherited full six-branch map remains unconstructed at B1 in the supplied version. Enter Calculus Companion.

CheungXVIN

N. Cheung. XVI-N, version r12. Supplied excerpts. Conditional massless prepared-insertion obstruction and the separate unresolved compact-data/universal obligations. No discharge or enlargement of its obstruction class is claimed. Enter Calculus Companion.