Paper Citation Record · LEDGER

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law

As of 19 August 2026, this Paper Citation Record lists 10 of 10 outbound references and 5 inbound Pith citation observations for arXiv:2505.19227.

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

pith.paper-citation-record.v1
2505.19227 v1

Coverage vector

measured 10 of 10 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T14:31:48.359506Z

measured 15 of 15 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 5 of 5 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-06T20:20:11.937785Z

measured 1 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-08-05T02:28:24.338817Z

Reference resolution

10 of 10 outbound references displayed

verified exact0
verified fuzzy5
unresolved5
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

0
arxiv_reference, observed 2026-08-05T02:28:24.338817Z

Outbound references

Observation 11090121-99b6-402d-80f1-2f4acc753923 · outbound

This paper cites Explaining Neural Scaling Laws.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law Explaining Neural Scaling Laws

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-07T14:31:47.667736Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 22a93578-5904-49cb-ac10-1bb1a5115209 · outbound

This paper cites A.2 Additional details about the figures Fig.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law A.2 Additional details about the figures Fig

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:31:49.443302Z

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 0287c687-2287-46b9-87be-bc9f9638f49b · outbound

This paper cites For a twice-differentiable function, this is equivalent to assuming that the eigenvalues of the Hessian are bounded by µ ≤ λij ≤ L for all i, j∈ [d] at every possible input.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law For a twice-differentiable function, this is equivalent to assuming that the eigenvalues of the Hessian are bounded by µ ≤ λij ≤ L for all i, j∈ [d] at every possible input

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:31:49.044341Z

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 485486e8-1179-4dae-80a8-dfd73bdfc5c1 · outbound

This paper cites an unresolved cited work.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-07T14:31:48.819103Z

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 1f76ed2f-06ac-4850-89fd-01b4abb05698 · outbound

This paper cites Normalizing the loss and using the same approach as in Proposition B.1 gives Ld(t) − L∗ d Ld(0) − L∗ d ≤ ϵ2 2 , after t ≥ ˜O(d).

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law Normalizing the loss and using the same approach as in Proposition B.1 gives Ld(t) − L∗ d Ld(0) − L∗ d ≤ ϵ2 2 , after t ≥ ˜O(d)

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:31:48.704560Z

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 28473bcc-4f22-4e04-a0db-48d8eededc6d · outbound

This paper cites Super Con- sistency of Neural Network Landscapes and Learning Rate Transfer.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law Super Con- sistency of Neural Network Landscapes and Learning Rate Transfer

Reference 87

Resolution: unresolved; no resolver link, observed 2026-08-07T14:31:47.717991Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 99978a26-86dc-4794-a1c4-4fb6d6741582 · outbound

This paper cites an unresolved cited work.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law Unresolved cited work

Reference 768

Resolution: unresolved; raw_fallback, observed 2026-08-07T14:31:49.162693Z

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 951ac39f-1b3a-453e-906d-1034c75889e8 · outbound

This paper cites (2024) and Liu et al.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law (2024) and Liu et al

Reference 2011

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:31:48.913684Z

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 8a7d5653-8c35-4a41-9db9-f94bccf66040 · outbound

This paper cites an unresolved cited work.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law Unresolved cited work

Reference 2018

Resolution: unresolved; raw_fallback, observed 2026-08-07T14:31:49.294581Z

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 1adf5540-8c1c-4e6b-b760-0f7b0def2f11 · outbound

This paper cites effectively.

Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law effectively

Reference 2025

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:31:48.568247Z

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.

Pith citing papers

Observation 9ca8a9d8-710a-46dd-8f5c-1669876f500a · inbound

On the Effectiveness of the z-Transform Method in Quadratic Optimization cites this paper.

On the Effectiveness of the z-Transform Method in Quadratic Optimization Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law

Reference 42

Resolution: unresolved; no resolver link, observed 2026-08-06T20:20:11.937785Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7d446638-3dd0-4a1c-8dcf-86f8670ec610 · inbound

Scaling Laws and Spectra of Shallow Neural Networks in the Feature Learning Regime cites this paper.

Scaling Laws and Spectra of Shallow Neural Networks in the Feature Learning Regime Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law

Reference 34

Resolution: unresolved; no resolver link, observed 2026-08-04T13:54:25.146793Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6518e1ba-3ef6-4021-853a-fe44b36fe820 · inbound

Muon in Associative Memory Learning: Training Dynamics and Scaling Laws cites this paper.

Muon in Associative Memory Learning: Training Dynamics and Scaling Laws Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-03T04:14:15.171584Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 56b85639-3a29-4ce3-a1b2-e06552f43fb4 · inbound

Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory cites this paper.

Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law

Reference 28

Resolution: verified exact; arxiv_id, observed 2026-05-14T23:38:16.479805Z

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 3ced0ac8-7004-4bea-a098-feca1ac064fd · inbound

Unlearning with Asymmetric Sources: Improved Unlearning-Utility Trade-off with Public Data cites this paper.

Unlearning with Asymmetric Sources: Improved Unlearning-Utility Trade-off with Public Data Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law

Reference 35

Resolution: verified exact; arxiv_id, observed 2026-05-13T05:57:21.544554Z

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.