Paper Citation Record · LEDGER

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies

As of 21 August 2026, this Paper Citation Record lists 42 of 42 outbound references and 4 inbound Pith citation observations for arXiv:2502.05627.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2502.05627 v1

Coverage vector

measured 42 of 42 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-08T18:55:48.977887Z

measured 46 of 46 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-21T06:32:19.484+00:00

measured 4 of 4 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-08T15:46:28.636096Z

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: arxiv_reference, observed 2026-05-23T04:32:34.007990Z

Reference resolution

42 of 42 outbound references displayed

verified exact0
verified fuzzy37
unresolved5
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 5e48d347-f793-4483-b684-94a258d01f76 · outbound

This paper cites Nesterov and A.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Nesterov and A

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.625222Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation ba0fd928-c009-48bf-b050-2145676434d2 · outbound

This paper cites Matrix monotonicity and self-c oncordance: how to handle quantum entropy in optimization problems,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Matrix monotonicity and self-c oncordance: how to handle quantum entropy in optimization problems,

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.610908Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 84fa2da8-9f3a-43f7-b966-dd706a1acd73 · outbound

This paper cites Conic optimization with sp ectral functions on Euclidean Jordan algebras,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Conic optimization with sp ectral functions on Euclidean Jordan algebras,

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.596552Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 2212754e-fc00-4589-a0b4-6622e0d7ef76 · outbound

This paper cites Optimal self-concordant barrie rs for quantum relative en- tropies,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Optimal self-concordant barrie rs for quantum relative en- tropies,

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.582371Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 6cb1d41e-3354-4498-8a67-afc00367af22 · outbound

This paper cites On quantum R´ enyi entropies: A new generalization and some properties,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies On quantum R´ enyi entropies: A new generalization and some properties,

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.568071Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 08cc0787-aaed-4d4f-afb1-9caa16911194 · outbound

This paper cites Strong converse for the cla ssical capacity of entanglement-breaking and Hadamard channels via a sandwiched R´ enyi relative entropy,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Strong converse for the cla ssical capacity of entanglement-breaking and Hadamard channels via a sandwiched R´ enyi relative entropy,

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.553029Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation de11e514-60bc-4be9-ae57-43b24a04c1bf · outbound

This paper cites Quantum hypothesis testing and the operational interpretation of the quantum R´ enyi relative entropies,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Quantum hypothesis testing and the operational interpretation of the quantum R´ enyi relative entropies,

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.538008Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 40e9a923-6e91-4324-9ee7-260e84dc7289 · outbound

This paper cites Privacy ampliﬁcation and decoupling without smoothing ,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Privacy ampliﬁcation and decoupling without smoothing ,

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.523093Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 3038655d-39f2-49e4-b96b-fe01b6e4fc5d · outbound

This paper cites Monotonicity of a relative R´ enyi ent ropy,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Monotonicity of a relative R´ enyi ent ropy,

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.507658Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 3cd0f369-f353-4f21-94cd-5bdcce09855c · outbound

This paper cites Remarks on two theorems of E. Lieb,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Remarks on two theorems of E. Lieb,

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.491410Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 5373a3ef-bcb8-48f8-8474-655221879e53 · outbound

This paper cites Concavity of certain matrix trace functions,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Concavity of certain matrix trace functions,

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.476454Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 76fd98ad-4081-4023-8339-f967e81e15ca · outbound

This paper cites Concavity of certain matrix trace and norm functions,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Concavity of certain matrix trace and norm functions,

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.461387Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 2fedd667-b55a-47eb-9f94-5b6d3d232bc6 · outbound

This paper cites Concavity of certain matrix trace and norm functions. II ,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Concavity of certain matrix trace and norm functions. II ,

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.446137Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 993e31ff-e250-4b60-a77a-79c40a0b2f64 · outbound

This paper cites From Wigner-Yanase-Dyson conjecture to Carlen -Frank-Lieb conjecture,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies From Wigner-Yanase-Dyson conjecture to Carlen -Frank-Lieb conjecture,

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.430435Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 7133bda6-f93a-408f-91d4-ec2e72c0bfc2 · outbound

This paper cites an unresolved cited work.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Unresolved cited work

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-08T18:55:48.855611Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 23114f5e-4d09-466a-8e1c-c1a343bfd3e1 · outbound

This paper cites Minimizing quantum R´ enyi d ivergences via mirror descent with Polyak step size,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Minimizing quantum R´ enyi d ivergences via mirror descent with Polyak step size,

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.405654Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation e4702ad7-3177-44b6-a8f0-2523152e261a · outbound

This paper cites Simpler semideﬁnite programs for completely boun ded norms,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Simpler semideﬁnite programs for completely boun ded norms,

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.391005Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 7bbe2770-f3ca-4fb3-b87e-d0a209f275fc · outbound

This paper cites A matrix convexity approach to some celebrated quantum inequalities,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies A matrix convexity approach to some celebrated quantum inequalities,

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.377010Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation a68ea3f3-cf45-4ed4-bcc3-a02b7835b99e · outbound

This paper cites Quasi-entropies for ﬁnite quantum systems,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Quasi-entropies for ﬁnite quantum systems,

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.362849Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation c0d87633-30a7-4701-8d32-4c42fb03f14a · outbound

This paper cites Convex trace functions and the Wigner-Yanase-D yson conjecture,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Convex trace functions and the Wigner-Yanase-D yson conjecture,

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.347875Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 0b169dae-2e28-4460-ae45-84a0fc79878f · outbound

This paper cites Concavity of certain maps on positive deﬁnite matrice s and applications to Hadamard products,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Concavity of certain maps on positive deﬁnite matrice s and applications to Hadamard products,

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.333411Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 920a6bcd-7ee3-4477-a803-e82a7abd1bf3 · outbound

This paper cites Interior Point Methods for Structured Quantum Relative Entropy Optimization Problems.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Interior Point Methods for Structured Quantum Relative Entropy Optimization Problems

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-08T18:55:48.887176Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 54750a89-3204-487e-aee4-83b62dcfcb94 · outbound

This paper cites QICS: Quantum Information Conic Solver.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies QICS: Quantum Information Conic Solver

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-08T18:55:48.892037Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a4653324-9b95-4fa9-bde7-9fe001d8c1e5 · outbound

This paper cites Eﬃcient implementation of interior-po int methods for quantum relative entropy,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Eﬃcient implementation of interior-po int methods for quantum relative entropy,

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.318652Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 1f2b9097-ded6-41a5-bfa3-085b02cfc5a0 · outbound

This paper cites Performance enhanc ements for a generic conic interior point algorithm,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Performance enhanc ements for a generic conic interior point algorithm,

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.304230Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation f82d7d02-d6d3-42ce-bf53-c01f8ca00c77 · outbound

This paper cites Semideﬁnite appro ximations of the matrix logarithm,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Semideﬁnite appro ximations of the matrix logarithm,

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.289798Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 45bfff4f-d236-43e1-b68e-0c96fc238e94 · outbound

This paper cites Lieb’s concavity theorem, matr ix geometric means, and semideﬁnite optimization,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Lieb’s concavity theorem, matr ix geometric means, and semideﬁnite optimization,

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.274076Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation b2ea5c85-34b8-489c-9b1f-9b83991f692a · outbound

This paper cites Inequalities for quan tum divergences and the Audenaert–Datta conjecture,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Inequalities for quan tum divergences and the Audenaert–Datta conjecture,

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.259484Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 7a969437-1850-4a1a-a733-426b180c1eaf · outbound

This paper cites Matrix analysis: matrix monotone functions, matrix mea ns, and majorization,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Matrix analysis: matrix monotone functions, matrix mea ns, and majorization,

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.244671Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 7009339d-3083-4701-a6ec-1c8edcfda822 · outbound

This paper cites Simon, Loewner’s theorem on monotone matrix functions.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Simon, Loewner’s theorem on monotone matrix functions

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.229284Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation d93183e7-1e13-4920-82de-cd7754137e7e · outbound

This paper cites Nesterov, Lectures on Convex Optimization.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Nesterov, Lectures on Convex Optimization

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.213365Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 9c5b6c3f-8938-428f-845a-86d897924aa2 · outbound

This paper cites an unresolved cited work.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Unresolved cited work

Reference 32

Resolution: unresolved; raw_fallback, observed 2026-08-08T18:55:49.196659Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 8e783255-7c33-4c27-ba1b-8d1fd00225ab · outbound

This paper cites Proper analytic free m aps,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Proper analytic free m aps,

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.181898Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 96041537-b8a2-4318-a1c1-39f62ea2aa91 · outbound

This paper cites Perspective s of matrix convex functions,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Perspective s of matrix convex functions,

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.167129Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation a47835eb-be78-4370-802e-c18860c4a854 · outbound

This paper cites Diﬀerent quantum f-divergences and t he reversibility of quantum operations,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Diﬀerent quantum f-divergences and t he reversibility of quantum operations,

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.152315Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation e32b627e-b19e-4621-ada6-a03c1d9153bf · outbound

This paper cites PICOS: A Python interface to co nic optimization solvers,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies PICOS: A Python interface to co nic optimization solvers,

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.137633Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 8901d239-c5bf-4759-94ce-14710c3206ff · outbound

This paper cites an unresolved cited work.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Unresolved cited work

Reference 37

Resolution: unresolved; no resolver link, observed 2026-08-08T18:55:48.955244Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 98e24b06-a837-450e-b44a-423c8642366b · outbound

This paper cites Hiai and D.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Hiai and D

Reference 38

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.110728Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation b3408795-a134-408a-af12-25bbaabfce4c · outbound

This paper cites Bhatia, Matrix analysis.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Bhatia, Matrix analysis

Reference 39

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.095421Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation a191e0a8-7bae-494c-9c2e-b3518e47fdaa · outbound

This paper cites Correlation detection and an op erational interpretation of the R´ enyi mutual information,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Correlation detection and an op erational interpretation of the R´ enyi mutual information,

Reference 40

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.080077Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation f887f796-fb6a-46e8-aac2-6b92db4fc72d · outbound

This paper cites Eﬃcient computation of t he quantum rate-distortion function,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Eﬃcient computation of t he quantum rate-distortion function,

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.064190Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation b3a75f6d-ace1-4a44-ac9b-f62320b1ecc9 · outbound

This paper cites Lifting for sim- plicity: Concise descriptions of convex sets,.

Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies Lifting for sim- plicity: Concise descriptions of convex sets,

Reference 42

Resolution: verified fuzzy; raw_fallback, observed 2026-08-08T18:55:49.048122Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Pith citing papers

Observation 52937825-1de5-4db4-97bf-7e8107f64059 · inbound

Generalized Numerical Framework for Improved Finite-Sized Key Rates with R\'enyi Entropy cites this paper.

Generalized Numerical Framework for Improved Finite-Sized Key Rates with R\'enyi Entropy Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies

Reference 46

Resolution: verified exact; arxiv_id, observed 2026-05-23T04:32:34.011451Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 4003baa2-67d3-4214-9554-e1fcb5c6e98b · inbound

A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean cites this paper.

A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies

Reference 34

Resolution: unresolved; no resolver link, observed 2026-08-08T15:46:28.636096Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5898bb92-729e-4c21-90ff-d697127951ec · inbound

Finite-size quantum key distribution rates from R\'enyi entropies using conic optimization cites this paper.

Finite-size quantum key distribution rates from R\'enyi entropies using conic optimization Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies

Reference 18

Resolution: verified exact; arxiv_id, observed 2026-05-17T22:05:21.569325Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation ee449b2e-26cc-43a2-8650-dd1eeb2bbb43 · inbound

Finite-size general security for relativistic phase shift keying via variable-length quantum key distribution cites this paper.

Finite-size general security for relativistic phase shift keying via variable-length quantum key distribution Operator convexity along lines, self-concordance, and sandwiched R\'enyi entropies

Reference 104

Resolution: verified exact; arxiv_id, observed 2026-05-11T19:36:14.948904Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.