Paper Citation Record · LEDGER

Non-Asymptotic Length Generalization

As of 7 August 2026, this Paper Citation Record lists 29 of 29 outbound references and 1 inbound Pith citation observation for arXiv:2506.03085.

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

pith.paper-citation-record.v1
2506.03085 v2

Coverage vector

measured 29 of 29 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T11:16:51.920045Z

measured 30 of 30 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-07T06:34:17.273281+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-07-14T03:23:31.727605Z

measured 0 of 1 external citation measurements

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

Source: cited_works

Reference resolution

29 of 29 outbound references displayed

verified exact0
verified fuzzy19
unresolved7
parse uncertain3
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 22e898fd-9973-45d2-9258-e9d26dfc2503 · outbound

This paper cites A outputs ˆp at least for one N ∈ N.

Non-Asymptotic Length Generalization A outputs ˆp at least for one N ∈ N

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:57.630509Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation abd59e88-eee0-4cc2-88e1-b2b503e6fe36 · outbound

This paper cites equivalent.

Non-Asymptotic Length Generalization equivalent

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:57.446627Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 39f83fef-a94b-48a0-bf45-aa11c8df0b89 · outbound

This paper cites , m− 2}, Y restricted to the interval [ Ti, Ti+1] can be rearranged into an equivalent monotone curve by Lemma D.13.

Non-Asymptotic Length Generalization , m− 2}, Y restricted to the interval [ Ti, Ti+1] can be rearranged into an equivalent monotone curve by Lemma D.13

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:54.786993Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 34e2a6ad-b77c-4c8d-b63b-f8c8bf78fa29 · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-07T11:16:57.232097Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation ed1cea82-f20b-48d3-8c2b-8f625d04c672 · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 5

Resolution: unresolved; raw_fallback, observed 2026-08-07T11:16:56.993421Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 71116b9d-dcca-493a-b0c5-ec06113151d1 · outbound

This paper cites functions.

Non-Asymptotic Length Generalization functions

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:56.800713Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 24be3c23-8f04-4869-a4a6-be27a41f331e · outbound

This paper cites S is 1-Lipschitz and monotone non-decreasing.

Non-Asymptotic Length Generalization S is 1-Lipschitz and monotone non-decreasing

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:56.609495Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 9d914eb2-38fa-4c7d-ad98-a021091fbcd0 · outbound

This paper cites minimal segment.

Non-Asymptotic Length Generalization minimal segment

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:56.388556Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation ff8c5179-b2fc-4d4a-bf9b-ddd569e6f7c5 · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-08-07T11:16:56.160012Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation e71a2aaa-7b75-49a1-85ea-0de7f104a6aa · outbound

This paper cites Corollary D.14.

Non-Asymptotic Length Generalization Corollary D.14

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:55.941248Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 52cf3ea0-4d8c-4ea7-a256-f7cd4107b13a · outbound

This paper cites The 1st monotone curve has start-point at the origin, (0, 0).

Non-Asymptotic Length Generalization The 1st monotone curve has start-point at the origin, (0, 0)

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:55.474938Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 6cdc2290-7a75-4b8b-9f91-ec7913d8b6c6 · outbound

This paper cites Finally, note that each monotone curve consists of multiple segments.

Non-Asymptotic Length Generalization Finally, note that each monotone curve consists of multiple segments

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:55.229252Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 7d529402-f1c9-45b8-9ab1-3746dd96bf3b · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 14

Resolution: parse uncertain; raw_fallback, observed 2026-08-07T11:16:55.084566Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 95a7fad2-24d3-4ea3-abbb-d4097dff3ec4 · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-07T11:16:54.973364Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 893848f4-3179-4e1c-b33d-edcdb5e5475e · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 17

Resolution: parse uncertain; raw_fallback, observed 2026-08-07T11:16:55.700521Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation b4220b5c-f009-4416-b5b2-238e3d4757bd · outbound

This paper cites The last issue about ensuring YPairs satisfies the property that ( y1 1, y1.

Non-Asymptotic Length Generalization The last issue about ensuring YPairs satisfies the property that ( y1 1, y1

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:54.584557Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation d2a6660c-3cf1-4e95-ac4f-a464f1184428 · outbound

This paper cites The issue arises when Span(Y) ⊊ [k], so that Y does not span all k lines.

Non-Asymptotic Length Generalization The issue arises when Span(Y) ⊊ [k], so that Y does not span all k lines

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:54.238691Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation fe953422-4e47-4a1d-a4ee-1ae1cef2639c · outbound

This paper cites positive.

Non-Asymptotic Length Generalization positive

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:54.082823Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation b26814ef-eee7-4cc0-98e6-3280d1c90efc · outbound

This paper cites , O(T d)}M −1.

Non-Asymptotic Length Generalization , O(T d)}M −1

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:53.891518Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation fc03f7cb-c861-47ef-9773-528312f48164 · outbound

This paper cites crossing points.

Non-Asymptotic Length Generalization crossing points

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:53.700867Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation ab6bfda1-1b4d-4dd7-878e-7cb2101d2b97 · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 24

Resolution: parse uncertain; raw_fallback, observed 2026-08-07T11:16:54.435708Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 07db6190-43b6-4d33-8fcf-5240d8f0c713 · outbound

This paper cites adjacent.

Non-Asymptotic Length Generalization adjacent

Reference 25

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:53.513583Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 5402cb01-18b3-4144-b8c7-19c79e274741 · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 26

Resolution: unresolved; raw_fallback, observed 2026-08-07T11:16:53.325993Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation d648edcb-c058-40df-a28d-001f757d2929 · outbound

This paper cites an unresolved cited work.

Non-Asymptotic Length Generalization Unresolved cited work

Reference 27

Resolution: unresolved; raw_fallback, observed 2026-08-07T11:16:53.135140Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 5498339e-9ed4-4c02-ad8a-8ec5457ba134 · outbound

This paper cites Lemma D.34 implies that either all Aj are type II or all Aj are type III, which can be seen as follows.

Non-Asymptotic Length Generalization Lemma D.34 implies that either all Aj are type II or all Aj are type III, which can be seen as follows

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:52.936995Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation e2bf0045-844e-462c-bdd1-2c6566cfa8a5 · outbound

This paper cites Thus, O is disjoint from S j∈[Nk] int(Aj).

Non-Asymptotic Length Generalization Thus, O is disjoint from S j∈[Nk] int(Aj)

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:52.712088Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation f7405b54-b945-434d-8d19-761a43e51663 · outbound

This paper cites x + y >2, x+ y >1.

Non-Asymptotic Length Generalization x + y >2, x+ y >1

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:52.393386Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 16521e21-b34d-403b-adbe-232462d0bbfd · outbound

This paper cites Once again, {Aj}j∈[Nk] are either all type II or all type III, contradicting that z, z′ > 0, and PK i=1 λi > z andPK′ i=1 λ′ i > z′ respectively.

Non-Asymptotic Length Generalization Once again, {Aj}j∈[Nk] are either all type II or all type III, contradicting that z, z′ > 0, and PK i=1 λi > z andPK′ i=1 λ′ i > z′ respectively

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T11:16:52.114933Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-07T06:34:17.273281+00:00.

Observation 65efd611-76cd-4376-9fbd-0125e9f514c9 · outbound

This paper cites Investigating the Limitations of Transformers with Simple Arithmetic Tasks.

Non-Asymptotic Length Generalization Investigating the Limitations of Transformers with Simple Arithmetic Tasks

Reference 2021

Resolution: unresolved; no resolver link, observed 2026-08-07T11:16:49.682246Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

Observation 9af35948-d31e-4e0c-81b3-44c16424fe67 · inbound

From Expressivity to Sample Complexity: Narrow Teachers for Transformers via C-RASP cites this paper.

From Expressivity to Sample Complexity: Narrow Teachers for Transformers via C-RASP Non-Asymptotic Length Generalization

Reference 2

Resolution: unresolved; no resolver link, observed 2026-07-14T03:23:31.727605Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.