Paper Citation Record · LEDGER

Fairness Through Matching

As of 19 August 2026, this Paper Citation Record lists 13 of 13 outbound references and 0 inbound Pith citation observations for arXiv:2501.02793.

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

pith.paper-citation-record.v1
2501.02793 v1

Coverage vector

measured 13 of 13 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-10T22:13:01.484905Z

measured 13 of 13 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-19T06:32:44.657259+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

13 of 13 outbound references displayed

verified exact0
verified fuzzy6
unresolved6
parse uncertain0
malformed identifier1
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 6e58109f-3e2c-40b0-88b1-86dcbdb9f9b7 · outbound

This paper cites We partition the interval[0,z] into subintervals of length at mostδ.

Fairness Through Matching We partition the interval[0,z] into subintervals of length at mostδ

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T22:13:02.815989Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation b79a0bee-1706-4576-bff4-5af38cdcf9d6 · outbound

This paper cites By Brenier’s Theorem (Villani, 2008; Hütter & Rigollet, 2021), there exists a transport mapT(1) 0,k from P0,k(⋅) to P1,k(⋅), under (C).

Fairness Through Matching By Brenier’s Theorem (Villani, 2008; Hütter & Rigollet, 2021), there exists a transport mapT(1) 0,k from P0,k(⋅) to P1,k(⋅), under (C)

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T22:13:02.744077Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 8dfc5d66-67b2-46a9-950d-c018627b8a75 · outbound

This paper cites Fix d ∈D0,1.

Fairness Through Matching Fix d ∈D0,1

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T22:13:02.625057Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation fb8d98ec-1c95-49ba-922e-9fb41ebe9de9 · outbound

This paper cites Because P0(⋃m k=1 X0,k)= P1(⋃m k=1 X1,k) and P0(⋃d∈D0,1 ˜X0,d)= P1(⋃d∈D0,1 ˜X1,d), we haveP0(X′ 0)= 1− P0(⋃k∈{1,...,m} X0,k)− P0(⋃d∈D0,1 ˜X0,d)= P1(X′ 1)≤δ.

Fairness Through Matching Because P0(⋃m k=1 X0,k)= P1(⋃m k=1 X1,k) and P0(⋃d∈D0,1 ˜X0,d)= P1(⋃d∈D0,1 ˜X1,d), we haveP0(X′ 0)= 1− P0(⋃k∈{1,...,m} X0,k)− P0(⋃d∈D0,1 ˜X0,d)= P1(X′ 1)≤δ

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T22:13:02.562063Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 6c9497ff-3eb5-4878-8c69-cf683e2cec33 · outbound

This paper cites (11) We note that{{X0,k}m k=1,{ ˜X0,d}d∈D0,1, X′ 0} and {{X1,k}m k=1,{ ˜X1,d}d∈D0,1, X′ 1} are partitions ofX0 and X1, respectively.

Fairness Through Matching (11) We note that{{X0,k}m k=1,{ ˜X0,d}d∈D0,1, X′ 0} and {{X1,k}m k=1,{ ˜X1,d}d∈D0,1, X′ 1} are partitions ofX0 and X1, respectively

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T22:13:02.397411Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation abd99d4e-8edb-43b4-9c56-f5d57746a938 · outbound

This paper cites an unresolved cited work.

Fairness Through Matching Unresolved cited work

Reference 6

Resolution: unresolved; raw_fallback, observed 2026-08-10T22:13:02.301338Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation cf6ee7aa-b64c-4f9e-b420-774bbf3816cf · outbound

This paper cites Furthermore, sincef(⋅)∈ [0, 1], we have that E0(∣f(X, 0)−f(T(3) 0 (X), 1)∣⋅ 1 (X∈ X′ 0))=∫ ∣f(X, 0)−f(T(3) 0 (X), 1)∣⋅ 1 (X∈ X′ 0)dP0(X) ≤∫ 1 (X∈ X′ 0)dP0(X)= P0(X′ 0)≤δ.

Fairness Through Matching Furthermore, sincef(⋅)∈ [0, 1], we have that E0(∣f(X, 0)−f(T(3) 0 (X), 1)∣⋅ 1 (X∈ X′ 0))=∫ ∣f(X, 0)−f(T(3) 0 (X), 1)∣⋅ 1 (X∈ X′ 0)dP0(X) ≤∫ 1 (X∈ X′ 0)dP0(X)= P0(X′ 0)≤δ

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-10T22:13:02.460313Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 95e840ac-4110-43bd-859f-a64162e2de74 · outbound

This paper cites an unresolved cited work.

Fairness Through Matching Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-10T22:13:02.204755Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation b91daef1-f0a0-482b-b977-599b5b93846c · outbound

This paper cites an unresolved cited work.

Fairness Through Matching Unresolved cited work

Reference 11

Resolution: unresolved; raw_fallback, observed 2026-08-10T22:13:02.093518Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 7aa81de9-9875-44a8-9c82-2ff5ea963bf5 · outbound

This paper cites an unresolved cited work.

Fairness Through Matching Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-08-10T22:13:02.038101Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation 3693b8b3-e326-4206-b903-137cc5d8f8ba · outbound

This paper cites Last, note that theΓ is a coupling matrix (i.e., satisfying the constraints∑n0 i=1γi,j = 1/n0,∀j ∈ [n1] and ∑n1 j=1γi,j = 1/n1,∀i∈ [n0]).

Fairness Through Matching Last, note that theΓ is a coupling matrix (i.e., satisfying the constraints∑n0 i=1γi,j = 1/n0,∀j ∈ [n1] and ∑n1 j=1γi,j = 1/n1,∀i∈ [n0])

Reference 13

Resolution: malformed identifier; raw_fallback, observed 2026-08-10T22:13:01.956662Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-19T06:32:44.657259+00:00.

Observation b9eeba0f-6f6a-438f-828f-d7c9ee9699a0 · outbound

This paper cites URL http://proceedings.mlr.press/v108/wei20a.html.

Fairness Through Matching URL http://proceedings.mlr.press/v108/wei20a.html

Reference 2020

Resolution: unresolved; no resolver link, observed 2026-08-10T22:13:01.164756Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d8fd0cb2-2811-47c8-b079-d0e483a74aa9 · outbound

This paper cites URL https://proceedings.neurips.cc/paper/2020/file/ 51cdbd2611e844ece5d80878eb770436-Paper.pdf.

Fairness Through Matching URL https://proceedings.neurips.cc/paper/2020/file/ 51cdbd2611e844ece5d80878eb770436-Paper.pdf

Reference 7331

Resolution: unresolved; no resolver link, observed 2026-08-10T22:13:01.129760Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.