Paper Citation Record · LEDGER

On the Supremum of Singleton Ratios in Submodular Functions

As of 15 August 2026, this Paper Citation Record lists 12 of 12 outbound references and 0 inbound Pith citation observations for arXiv:2604.23634.

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

pith.paper-citation-record.v1
2604.23634 v1

Coverage vector

measured 12 of 12 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-05-08T06:05:55.260140Z

measured 12 of 12 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-15T06:32:42.880941+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

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

Source: cited_works

Reference resolution

12 of 12 outbound references displayed

verified exact2
verified fuzzy6
unresolved1
parse uncertain0
malformed identifier3
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 7e1860d1-7cb1-4213-afea-b7c9f111f899 · outbound

This paper cites Submodular functions: Learnability, structure, and optimization.SIAM J.

On the Supremum of Singleton Ratios in Submodular Functions Submodular functions: Learnability, structure, and optimization.SIAM J

Reference 1

Resolution: verified exact; doi, observed 2026-05-08T23:24:26.023131Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation c8b6dd48-3e1a-4dca-b6de-7f4ffb3f3764 · outbound

This paper cites Learning with Submodular Functions: A Convex Optimization Perspective.

On the Supremum of Singleton Ratios in Submodular Functions Learning with Submodular Functions: A Convex Optimization Perspective

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-05-26T19:12:55.617150Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 88cc2a6d-ad60-43b7-af62-3faeebc7158d · outbound

This paper cites Secret-sharing schemes: a survey.

On the Supremum of Singleton Ratios in Submodular Functions Secret-sharing schemes: a survey

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-05-26T19:12:55.621204Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation ec878cf5-cadb-4b5b-8172-822ae06518e1 · outbound

This paper cites Decomposition Polyhedra of Piecewise Linear Functions.

On the Supremum of Singleton Ratios in Submodular Functions Decomposition Polyhedra of Piecewise Linear Functions

Reference 4

Resolution: malformed identifier; arxiv_id, observed 2026-06-05T02:16:16.697629Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 42d8846c-7952-46f1-a0a2-6830ef315187 · outbound

This paper cites The size of a share must be large.

On the Supremum of Singleton Ratios in Submodular Functions The size of a share must be large

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-05-26T19:12:55.637968Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation a50d0f91-ed50-492d-872a-dacd6d36bbea · outbound

This paper cites Enumerating Extremal Submodular Func- tions for n = 6.Mathematics2024,13, 97.https://doi.org/10.3390/ math13010097.

On the Supremum of Singleton Ratios in Submodular Functions Enumerating Extremal Submodular Func- tions for n = 6.Mathematics2024,13, 97.https://doi.org/10.3390/ math13010097

Reference 6

Resolution: malformed identifier; raw_fallback, observed 2026-05-26T19:12:55.613143Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation e1c55a1a-d640-42f1-897e-83f2549eb5f1 · outbound

This paper cites Submodular Functions, Matroids, and Certain Polyhedra.

On the Supremum of Singleton Ratios in Submodular Functions Submodular Functions, Matroids, and Certain Polyhedra

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-05-26T19:12:55.625459Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 9eef9557-0894-4b56-9c41-88fc0e330570 · outbound

This paper cites Polymatroidal dependence structure of a set of random vari- ables.Information and Control,197839(1), 55–72.

On the Supremum of Singleton Ratios in Submodular Functions Polymatroidal dependence structure of a set of random vari- ables.Information and Control,197839(1), 55–72

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-05-26T19:12:55.633932Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 2e310850-2491-4b84-b770-3212e9081e11 · outbound

This paper cites Many rays of the submodular cone.

On the Supremum of Singleton Ratios in Submodular Functions Many rays of the submodular cone

Reference 9

Resolution: verified exact; arxiv_id, observed 2026-05-11T21:16:26.213805Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 5799f644-7838-4331-a06b-2b6ab564240b · outbound

This paper cites Supermodularity and Supermodular Games.The New Pal- grave Dictionary of Economics,2008, pp 1–9.https://doi.org/10.1057/ 978-1-349-95121-5_2443-1.

On the Supremum of Singleton Ratios in Submodular Functions Supermodularity and Supermodular Games.The New Pal- grave Dictionary of Economics,2008, pp 1–9.https://doi.org/10.1057/ 978-1-349-95121-5_2443-1

Reference 10

Resolution: malformed identifier; raw_fallback, observed 2026-05-26T19:12:55.608740Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 8e5f2f6c-539c-4a43-aebe-192f8c7a2849 · outbound

This paper cites W.A First Course in Information Theory.

On the Supremum of Singleton Ratios in Submodular Functions W.A First Course in Information Theory

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-05-26T19:12:55.629755Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Observation 2100b0f4-faa1-4c12-b3ad-a8e2399f3b8a · outbound

This paper cites an unresolved cited work.

On the Supremum of Singleton Ratios in Submodular Functions Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-05-26T19:12:55.642352Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-15T06:32:42.880941+00:00.

Pith citing papers

No inbound Pith citation observations are available.