References

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.

Polchinski

J. Polchinski. Scale and conformal invariance in quantum field theory. Nuclear Physics B 303 (1988), 226–236. doi:10.1016/0550-3213(88)90179-4.

KomargodskiSchwimmer

Z. Komargodski and A. Schwimmer. On Renormalization Group Flows in Four Dimensions. arXiv:1107.3987. Cited for the independently stated dilaton and anomaly framework, not as a premise of Family 282.

LPR

M. A. Luty, J. Polchinski, and R. Rattazzi. {The $a$-theorem and the Asymptotics of 4D Quantum Field Theory}. arXiv:1204.5221. The hypotheses and scope of that approach are retained.

DKST

A. Dymarsky, Z. Komargodski, A. Schwimmer, and S. Theisen. On Scale and Conformal Invariance in Four Dimensions. arXiv:1309.2921.

DFKLP

A. Dymarsky, K. Farnsworth, Z. Komargodski, M. A. Luty, and V. Prilepina. Scale Invariance, Conformality, and Generalized Free Fields. arXiv:1402.6322. Additional assumptions and improvement questions are not erased.

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. Wick products, time-ordered products, and their conservation and derivative-normalization conditions.

Nelson

E. Nelson. Analytic vectors. Annals of Mathematics 70 (1959), 572–615. doi:10.2307/1970331. Cited for sufficient integration and operator-domain tools, not applied without their hypotheses.

HislopLongo

P. D. Hislop and R. Longo. Modular structure of the local algebras associated with the free massless scalar field theory. Communications in Mathematical Physics 84 (1982), 71–85. doi:10.1007/BF01208372.

BGL

R. Brunetti, D. Guido, and R. Longo. Modular structure and duality in conformal quantum field theory. Communications in Mathematical Physics 156 (1993), 201–219. arXiv:funct-an/9302008.

BisognanoWichmann

J. J. Bisognano and E. H. Wichmann. On the duality condition for a Hermitian scalar field. Journal of Mathematical Physics 16 (1975), 985–1007. doi:10.1063/1.522605.

Wiesbrock

H.-W. Wiesbrock. Half-sided modular inclusions of von-Neumann-algebras. Communications in Mathematical Physics 157 (1993), 83–92. doi:10.1007/BF02098019.

ArakiZsido

H. Araki and L. Zsidó. Extension of the structure theorem of Borchers and its application to half-sided modular inclusions. Reviews in Mathematical Physics 17 (2005), 491–543. arXiv:math/0412061.

Pedersen

G. K. Pedersen. {$C^{\ast }$-Algebras and Their Automorphism Groups}. Academic Press, 1979.

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.

Watrous

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

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.

OpenAI282

OpenAI, as attributed in the supplied source and catalogue. Scale and conformal symmetry in four-dimensional operational quantum field theory. 26 September 2026. Family 282-01. Catalogue: https://enter.grok.me/openai/282-01. The catalogue and supplied :main account are documentary records, not independently verified premises. The catalogue reports no Lean scope note for this family.

Dossier

User-supplied Oai282-A development plan and comparative source dossier. Includes Family 282 framework and theorem statements, source excerpts, standing status records, and comparative family descriptions. Retrieval label 6f2e9611. No comprehensive corpus scan or independent source-proof verification is asserted.

CheungPortal

N. Cheung. Enter Calculus Companion. https://enter.grok.me. Searchable companion, not a proof checker. Individual manuscript identifiers and supplied versions govern the attributions below.

CheungII

N. Cheung. Landmark II, version 2. Supplied manuscript .tex. Finite contact-membership certificate and its stated coverage boundaries. A finite fit is not a global functional identity. Enter Calculus Companion.

CheungIII

N. Cheung. Descent of first-order relational scalars and the free-product obstruction: Classical equation spaces, retarded sections, and compact Ward observables. Landmark III, version 2. Supplied .tex excerpts. Classical quotient and coupled-jet results on their declared equation spaces; companion comparisons are not premises. Enter Calculus Companion.

CheungVI

N. Cheung. Relational Einstein–scalar observables at second order: Classical descent and a compact-switch obstruction. Landmark VI, version 1. Supplied .tex excerpts. Especially :descent. Second-order classical relational descent; the supplied quantum certificate remains RG-open. Enter Calculus Companion.

CheungEn05C

N. Cheung. En05-C, version 1. Supplied 05-C.tex excerpts. Especially :comparison. Tensor-free snapshot–channel–adjoint equivalence for declared unital local algebras. Enter Calculus Companion.

CheungEn05A

N. Cheung. En05-A, version 1. Supplied 05-A.tex excerpts. Especially :cone and :comparison. Schedule-past support and a prescribed metric envelope. Enter Calculus Companion.

CheungEn03A

N. Cheung. En03-A, version 1. Supplied 03-A.tex excerpts. Especially :locality. Complete finite schedule support, not a derived field-theoretic causal structure. Enter Calculus Companion.

CheungEn06A

N. Cheung. En06-A, version 1. Supplied 06-A.tex excerpts. Especially :algmicro and its localized observable-limit hypotheses. Cross-product preservation is not replaced by channel commutation. Enter Calculus Companion.

CheungEn06B

N. Cheung. En06-B, version 1. Supplied 06-B.tex excerpts. Especially :identity. Complete finite instrument transport with prescribed routes, ports, and speed. Enter Calculus Companion.

CheungEn01B

N. Cheung. En01-B, version 2. Supplied 01-B-v2.tex excerpts. Especially :bounded. Bounded complete-record validation under a declared statistical contract. Enter Calculus Companion.

CheungOai265B

N. Cheung. Oai265-B, version 2. Supplied 265-B-v2.tex record. Finite-confidence certification for one implemented qubit decoder; coverage is not a posterior or a field-instrument construction. Enter Calculus Companion.

CheungEn07A

N. Cheung. En07-A, version 1. Supplied 07-A.tex excerpts. Especially :exact. Complete-complement recovery and the separation of operational discard, quantum return, and holographic terminology. Enter Calculus Companion.

CheungEn07B

N. Cheung. En07-B, version 1. Supplied 07-B.tex excerpts. Especially :exact and :endpoint. Finite endpoint statements retain their four same-model obligations. Enter Calculus Companion.

CheungHawk04

N. Cheung. Hawk04-A, version 1, and Hawk04-B, version 2. Supplied manuscript excerpts. Exact complete-complement recovery, complete preservation, and classical-record limitations. No gravitational evaporation interface is imported. Enter Calculus Companion.

CheungOai290A

N. Cheung. Oai290-A, version 1. Supplied 290-A.tex excerpts. Especially :exact, :bounded, and :assembly. Finite channel decisions and a modular-band witness are separately typed. No application to the Family 282 null cuts is supplied. Enter Calculus Companion.

CheungOai265A

N. Cheung. Oai265-A, version 1. Supplied 265-A.tex excerpts. Especially :determination and :separation. Static area-law records, selected entropy, recovery, and histories remain separate. Enter Calculus Companion.

CheungEm02E

N. Cheung. Em02-E, version 1. Supplied 02-E-v1.tex excerpts. Especially :lapse and :release. Relational lapse on a declared Einstein–scalar background. Not the scalar improvement field $L$. Enter Calculus Companion.

CheungEm03A

N. Cheung. Em03-A, version 1. Supplied 03-A-v1.tex excerpts. Especially :main and :release. Fixed-background scalar emission and its stated exterior, state, scattering, and normalization inputs. Enter Calculus Companion.

CheungXVIPrepared

N. Cheung. XVI-A, version 1; XVI-H, version 1; and XVI-I, version 1. Supplied manuscript excerpts. Prepared-source declarations, a conditional compact polarized kernel, and a reduced two-edge extension family. Their normalization and admission boundaries are retained. Enter Calculus Companion.

CheungXVIN

N. Cheung. XVI-N, version r12. Supplied -N.tex excerpts. Especially :massless-obstruction. Conditional prepared-insertion obstruction; the spatially compact Route B is not discharged. Enter Calculus Companion.

CheungXVIP

N. Cheung. XVI-P, version p23. Supplied -P.tex excerpt and status record. Technical subsector and build-report provenance are documentary. .1 remains _CLOSED. No execution is independently reproduced here. Enter Calculus Companion.

CheungXIVH

N. Cheung. Landmark XIV-H: Internal kernel relations on the operator route, version 1. Supplied -H.tex excerpts. Historical H2 amplitude 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 reconstruction is not the full characteristic composition. Enter Calculus Companion.

CheungXIVJ

N. Cheung. Landmark XIV-J, version 1. Supplied -J.tex excerpts. Especially :descent, :reference, :J2, and :noadmission. The full joint map remains unconstructed in the supplied release. Enter Calculus Companion.

CheungXVB

N. Cheung. XV-B, version 2. Supplied -B.tex excerpts. Relative lifting, finite normal-jet null tests, and conditional extension interfaces; the inherited full graph remains unresolved at B1 in the supplied account. The technical manuscript is distinct from the closed downstream quantum-admission exit. Enter Calculus Companion.

CheungI

N. Cheung. Landmark I, version 2. Supplied .tex excerpts. Especially :main. Born strategy and destructive projective residuals inside a stated finite strategy class. No field instrument or stress-tensor Ward theorem is imported. Enter Calculus Companion.

Oai280A

Oai280-A: Sector-Label Certificates and VOA–Net Separation. Earlier conversation draft. Cited only for its scoped source-record and operational comparison architecture. No independent registration, build, or proof verification is asserted.

Oai211A

Oai211-A. Earlier comparison draft as described in the supplied dossier. Conditional path-sum and metric-measure comparison only. No derivation of four-dimensional conformal symmetry is imported.