Paper Citation Record · LEDGER

Towards Fair Representation: Clustering and Consensus

As of 8 August 2026, this Paper Citation Record lists 22 of 22 outbound references and 0 inbound Pith citation observations for arXiv:2506.08673.

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

pith.paper-citation-record.v1
2506.08673 v3

Coverage vector

measured 22 of 22 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T05:14:36.398965Z

measured 22 of 22 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-08T06:32:00.761636+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

22 of 22 outbound references displayed

verified exact0
verified fuzzy12
unresolved9
parse uncertain1
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation b0c89e6b-de96-4e13-8124-fe4e1129910f · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 1

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.681570Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation e00aa2d6-5f3f-4916-a4fa-3894ac3b2f74 · outbound

This paper cites 35 CHAKRABORTYCHATTERJEEDASNGUYENNOBAHARI For proving Theorem 24, we need the help of the following proposition.

Towards Fair Representation: Clustering and Consensus 35 CHAKRABORTYCHATTERJEEDASNGUYENNOBAHARI For proving Theorem 24, we need the help of the following proposition

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.669431Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation b213135f-e0d4-4ecc-b9ce-ae9246e4031b · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 3

Resolution: parse uncertain; raw_fallback, observed 2026-08-07T05:14:36.658604Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation d3853a9e-38aa-42f5-b483-aa91f71543fd · outbound

This paper cites Proof:Without loss of generality, assume thatx 1 ≤x 2 ≤ · · · ≤xt.

Towards Fair Representation: Clustering and Consensus Proof:Without loss of generality, assume thatx 1 ≤x 2 ≤ · · · ≤xt

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.646525Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 85936efe-0993-4696-95dd-5dfe12fe2d31 · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 5

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.634414Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 56afbfce-5fc3-481b-bd52-cb86f261a730 · outbound

This paper cites Now, to prove the previous claim, we need to prove the other claims below.

Towards Fair Representation: Clustering and Consensus Now, to prove the previous claim, we need to prove the other claims below

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.624415Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 492ed86d-ee92-4c03-bed3-4381c111ae87 · outbound

This paper cites , Yj−1, Yj+1,.

Towards Fair Representation: Clustering and Consensus , Yj−1, Yj+1,

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.614057Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 4c770895-18b7-4338-ae52-c3c02e7c9a35 · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.603148Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation aa729efa-1e44-44f9-ab6a-391848b1bcb2 · outbound

This paper cites We move the red points ofX i,ℓ for allℓ̸=jto the setT ∗ rj.

Towards Fair Representation: Clustering and Consensus We move the red points ofX i,ℓ for allℓ̸=jto the setT ∗ rj

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.591158Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation f288f52a-a251-4e22-97b4-3cf58bc63a56 · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.580657Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation d53fee42-d2f3-4cac-a9fe-788459cd6372 · outbound

This paper cites This implies [ ℓ̸=j Xi,ℓ modp=s i Hence, [ ℓ̸=j Xi,ℓ =ap+s i for some constanta.

Towards Fair Representation: Clustering and Consensus This implies [ ℓ̸=j Xi,ℓ modp=s i Hence, [ ℓ̸=j Xi,ℓ =ap+s i for some constanta

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.570118Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 7b55aba0-f9f2-4aaf-8a5d-7d178835dc49 · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.558479Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 346eff4a-345a-48f5-a843-ffec33c4402b · outbound

This paper cites Recall inT ∗,D i gets split intoX i,1,.

Towards Fair Representation: Clustering and Consensus Recall inT ∗,D i gets split intoX i,1,

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.546034Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 470119e8-7b0c-449c-ad6c-cbacb1458bb0 · outbound

This paper cites For eachD i ∈MERGE,s i ≥ p.

Towards Fair Representation: Clustering and Consensus For eachD i ∈MERGE,s i ≥ p

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.533971Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 3e48a31e-4b61-415c-96fc-51ebc97db825 · outbound

This paper cites These observations imply that X Di∈CUT\CUT ′ 1 2 s2 i + X Di∈MERGE 1 2 d2 i + X Di∈CUT ′ si(p−s i) 2 ≤ X Di∈D si(p−s i) 2 ≤OPT, where the last inequality is from Claim 12.

Towards Fair Representation: Clustering and Consensus These observations imply that X Di∈CUT\CUT ′ 1 2 s2 i + X Di∈MERGE 1 2 d2 i + X Di∈CUT ′ si(p−s i) 2 ≤ X Di∈D si(p−s i) 2 ≤OPT, where the last inequality is from Claim 12

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.519104Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 1749cf44-137e-4df0-9bfc-dadb7dae8365 · outbound

This paper cites Now we are ready to prove Theorem 23.

Towards Fair Representation: Clustering and Consensus Now we are ready to prove Theorem 23

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.507618Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation f79dde84-84e3-4f9a-9ba2-df06f5852eab · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 17

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.496363Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation baa8fcbd-ab30-43f9-8259-407c6a20fdd8 · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 18

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.482049Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 52b23c34-1150-4ee8-ade7-30bf84493705 · outbound

This paper cites Let us assumeW i,z,r denotes thezth red subset of the clusterD i.y i,z,r takes the value1if we cutzth red subset fromD i.

Towards Fair Representation: Clustering and Consensus Let us assumeW i,z,r denotes thezth red subset of the clusterD i.y i,z,r takes the value1if we cutzth red subset fromD i

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.470626Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 129d835e-0abb-4e46-8e7d-d087da610c65 · outbound

This paper cites Claim 25cost 1(Q)blue +cost 1(Q)blue +cost 3(Q)blue +cost 4(Q)blue ≤3dist(D,Q ∗).

Towards Fair Representation: Clustering and Consensus Claim 25cost 1(Q)blue +cost 1(Q)blue +cost 3(Q)blue +cost 4(Q)blue ≤3dist(D,Q ∗)

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T05:14:36.457711Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation daf688e8-57bd-4850-8545-6cc267e73b9c · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 21

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.442663Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 869114c0-40ab-406c-a2e0-151e8e00c4af · outbound

This paper cites an unresolved cited work.

Towards Fair Representation: Clustering and Consensus Unresolved cited work

Reference 22

Resolution: unresolved; raw_fallback, observed 2026-08-07T05:14:36.429186Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Pith citing papers

No inbound Pith citation observations are available.