Paper Citation Record · LEDGER

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks

As of 17 August 2026, this Paper Citation Record lists 44 of 44 outbound references and 2 inbound Pith citation observations for arXiv:2505.18266.

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

pith.paper-citation-record.v1
2505.18266 v1

Coverage vector

measured 44 of 44 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T14:40:57.219755Z

measured 46 of 46 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-17T06:30:58.91139+00:00

measured 2 of 2 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-07T18:32:14.733852Z

measured 0 of 1 external citation measurements

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

Source: arxiv_reference, observed 2026-05-20T12:28:16.606577Z

Reference resolution

44 of 44 outbound references displayed

verified exact2
verified fuzzy14
unresolved25
parse uncertain0
malformed identifier3
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External citation measurements

No source-named external measurement is stored.

Outbound references

Observation a40db060-55bb-4ecb-8bf1-c6b121d0e662 · outbound

This paper cites Zoom in: An introduction to circuits.Distill, 5(3):e00024–001, 2020.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Zoom in: An introduction to circuits.Distill, 5(3):e00024–001, 2020

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:53.166207Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8cb2f851-a49c-4ded-9d85-01088bc78e3f · outbound

This paper cites Convergent Learning: Do different neural networks learn the same representations?.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Convergent Learning: Do different neural networks learn the same representations?

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:53.254885Z

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Observation 7f50ce00-027f-48d1-8a17-5a3903f3833f · outbound

This paper cites The Platonic Representation Hypothesis.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks The Platonic Representation Hypothesis

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:53.360915Z

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Observation 652ef375-8c70-474d-88df-fe7283821d8e · outbound

This paper cites Progress mea- sures for grokking via mechanistic interpretability.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Progress mea- sures for grokking via mechanistic interpretability

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:53.491186Z

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Observation b99bb124-3bd4-4d54-92d7-c4b8cb3ee7ba · outbound

This paper cites The clock and the pizza: Two stories in mechanistic explanation of neural networks.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks The clock and the pizza: Two stories in mechanistic explanation of neural networks

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:53.563566Z

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Observation 326d5541-bcf5-4ddf-ad9c-0f012c39f40f · outbound

This paper cites Grokking modular arithmetic.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Grokking modular arithmetic

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:53.645862Z

Source-reported events for the cited work

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Observation e7bd76c8-14ee-4850-89e5-bccded7725a6 · outbound

This paper cites Edelman, Costin-Andrei Oncescu, Rosie Zhao, and Sham M.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Edelman, Costin-Andrei Oncescu, Rosie Zhao, and Sham M

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:41:02.014506Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation 75e51b3f-f411-4ab4-8ecf-6aba9f418d6d · outbound

This paper cites A toy model of universality: Reverse engineering how networks learn group operations.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks A toy model of universality: Reverse engineering how networks learn group operations

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:41:01.731397Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation 051f1374-a86b-4312-a005-f0fae742a0a3 · outbound

This paper cites Grokking group multiplication with cosets.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Grokking group multiplication with cosets

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:41:01.311382Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation ee1c2a1f-b733-45f9-95e8-18f7dfd094b5 · outbound

This paper cites Neural networks learn representation theory: Reverse engineering how networks perform group operations.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Neural networks learn representation theory: Reverse engineering how networks perform group operations

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:41:00.913368Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation a1dd26ae-20e4-4e1b-85c1-f2f8b4719421 · outbound

This paper cites Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:54.198192Z

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Observation 8d970e66-191a-4de5-8f5f-42613f4e0527 · outbound

This paper cites Grokking modular arithmetic can be explained by margin maximization.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Grokking modular arithmetic can be explained by margin maximization

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:41:00.566341Z

Source-reported events for the cited work

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Observation 30b1ff91-1129-4565-af17-7658331d90a5 · outbound

This paper cites Open Problems in Mechanistic Interpretability.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Open Problems in Mechanistic Interpretability

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:54.459356Z

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Observation ef968e82-c10e-4928-8742-bd8f35314717 · outbound

This paper cites Thread: Circuits.Distill, 2020.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Thread: Circuits.Distill, 2020

Reference 14

Resolution: malformed identifier; raw_fallback, observed 2026-08-07T14:41:00.247003Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation 06dd71ed-dc68-4526-b89e-c5722863c611 · outbound

This paper cites A mathematical framework for transformer circuits.Transformer Circuits Thread,.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks A mathematical framework for transformer circuits.Transformer Circuits Thread,

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:54.684451Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3fcf4fe0-b59d-4059-a787-e7d54097508e · outbound

This paper cites In-context learning and induction heads.Transformer Circuits Thread, 2022.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks In-context learning and induction heads.Transformer Circuits Thread, 2022

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:54.885770Z

Source-reported events for the cited work

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Observation 5f417759-0ca5-49cd-a230-a409296389e7 · outbound

This paper cites Toy Models of Superposition.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Toy Models of Superposition

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:55.005013Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 27a22bb0-6496-4730-9314-aed84e835155 · outbound

This paper cites From understanding computation to understanding neural circuitry.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks From understanding computation to understanding neural circuitry

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:59.976611Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation d9915aaa-6aa8-4b09-b845-c1be79042c9a · outbound

This paper cites Levels of Analysis for Machine Learning.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Levels of Analysis for Machine Learning

Reference 19

Resolution: verified exact; local_arxiv, observed 2026-08-07T14:40:57.796916Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation 48b7ea23-a93e-41a9-b6a6-562e195a25f0 · outbound

This paper cites Multilevel Interpretability Of Artificial Neural Networks: Leveraging Framework And Methods From Neuroscience.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Multilevel Interpretability Of Artificial Neural Networks: Leveraging Framework And Methods From Neuroscience

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:55.305178Z

Source-reported events for the cited work

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Observation b5fe882a-6337-4b89-a591-23fa2d4d14d8 · outbound

This paper cites Vilas, Federico Adolfi, David Poeppel, and Gemma Roig.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Vilas, Federico Adolfi, David Poeppel, and Gemma Roig

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:59.719652Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation 9cd3a607-a758-4574-88bb-dca4b36257c9 · outbound

This paper cites SALSA: Attacking Lattice Cryptography with Transformers.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks SALSA: Attacking Lattice Cryptography with Transformers

Reference 22

Resolution: verified exact; local_arxiv, observed 2026-08-07T14:40:57.618997Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation af69a3a9-3a5f-480a-b960-e48003467287 · outbound

This paper cites Towards understanding grokking: An effective theory of representation learning.Advances in Neural Information Processing Systems, 35:34651–34663, 2022.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Towards understanding grokking: An effective theory of representation learning.Advances in Neural Information Processing Systems, 35:34651–34663, 2022

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:55.572073Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 68c6dd4c-1615-43c9-b540-98d286f74253 · outbound

This paper cites To grok or not to grok: Disentangling generalization and memorization on corrupted algorithmic datasets.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks To grok or not to grok: Disentangling generalization and memorization on corrupted algorithmic datasets

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:55.660165Z

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Observation cf772343-b122-42ed-944c-57473aa521d7 · outbound

This paper cites Emergence in non-neural models: grokking modular arithmetic via average gradient outer product.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Emergence in non-neural models: grokking modular arithmetic via average gradient outer product

Reference 25

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:55.768975Z

Source-reported events for the cited work

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Observation 1393798c-22c0-462c-be71-f9a6aa34aaab · outbound

This paper cites Towards empirical interpretation of internal circuits and properties in grokked transformers on modular polynomials,.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Towards empirical interpretation of internal circuits and properties in grokked transformers on modular polynomials,

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:59.487152Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation ec22ac2e-d3c4-4590-b745-6bb9bb1a5297 · outbound

This paper cites Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking

Reference 27

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:56.021018Z

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Observation 3b4aea52-0d78-4cb3-aa47-0b6b6ecbfafc · outbound

This paper cites Gershman, and Cengiz Pehlevan.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Gershman, and Cengiz Pehlevan

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:59.201326Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation 1a5f812d-08d5-4a78-83e0-206dd977c582 · outbound

This paper cites Grokking Modular Polynomials.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Grokking Modular Polynomials

Reference 29

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:56.187966Z

Source-reported events for the cited work

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Observation 93d996df-6eb8-4d27-8cdd-2035bcb69607 · outbound

This paper cites Learning to grok: Emergence of in-context learning and skill composition in modular arithmetic tasks.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Learning to grok: Emergence of in-context learning and skill composition in modular arithmetic tasks

Reference 30

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:56.283173Z

Source-reported events for the cited work

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Observation 47c3e015-b686-47de-a8a8-e8f7ed09e69e · outbound

This paper cites The evolution of statistical induction heads: In-context learning markov chains.Advances in Neural Information Processing Systems, 37:64273–64311, 2024.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks The evolution of statistical induction heads: In-context learning markov chains.Advances in Neural Information Processing Systems, 37:64273–64311, 2024

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:58.941980Z

Source-reported events for the cited work

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Observation 2e41c6ef-512c-41e0-aca8-aefd4b56ab63 · outbound

This paper cites Emergent properties with repeated examples.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Emergent properties with repeated examples

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:56.425685Z

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Observation 8e1dc65c-15ff-4d36-9200-7a3579cdd3a0 · outbound

This paper cites Self-Improving Transformers Overcome Easy-to-Hard and Length Generalization Challenges.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Self-Improving Transformers Overcome Easy-to-Hard and Length Generalization Challenges

Reference 33

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:56.496449Z

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Observation d09cb9bb-7f37-4852-a02f-15dfcdb5a0c0 · outbound

This paper cites Length Generalization in Arithmetic Transformers.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Length Generalization in Arithmetic Transformers

Reference 34

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:56.590936Z

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Observation 2cd08920-8ff8-4e2e-9c43-a193cb95eed6 · outbound

This paper cites Learning the greatest common divisor: explaining transformer predictions,.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Learning the greatest common divisor: explaining transformer predictions,

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:58.621238Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation 469a7c4a-97c5-48ed-8b4f-e6884400c660 · outbound

This paper cites Ruiz, Julian Schrittwieser, Grzegorz Swirszcz, et al.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Ruiz, Julian Schrittwieser, Grzegorz Swirszcz, et al

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:58.468800Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation ede7620b-16eb-4b72-9470-173565196f76 · outbound

This paper cites Faster sorting algorithms discovered using deep reinforcement learning.Nature, 618(7964):257–263, 2023.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Faster sorting algorithms discovered using deep reinforcement learning.Nature, 618(7964):257–263, 2023

Reference 37

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:56.934379Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bc7cda93-b047-4ab4-9a59-d72a9f11b882 · outbound

This paper cites Learning the greatest common divisor: explaining transformer predictions.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Learning the greatest common divisor: explaining transformer predictions

Reference 38

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:56.780389Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8438e0db-b2a7-48de-a67e-3e2e84dd19e1 · outbound

This paper cites Can deep reinforcement learning solve erdos-selfridge-spencer games? InInternational Conference on Machine Learning, pages 4238–4246.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Can deep reinforcement learning solve erdos-selfridge-spencer games? InInternational Conference on Machine Learning, pages 4238–4246

Reference 39

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:58.185207Z

Source-reported events for the cited work

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Observation eeac6ae6-0ded-4138-8f32-0f1739e8e36c · outbound

This paper cites Adam: A Method for Stochastic Optimization.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Adam: A Method for Stochastic Optimization

Reference 40

Resolution: malformed identifier; no resolver link, observed 2026-08-07T14:40:57.145750Z

Source-reported events for the cited work

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Observation 7682a127-9ffe-4f25-836f-89375ee2696e · outbound

This paper cites McGill University (Canada), 2021.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks McGill University (Canada), 2021

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T14:40:58.326536Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation af24f564-67b3-4031-b7e8-57e226eea6e8 · outbound

This paper cites error correct.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks error correct

Reference 44

Resolution: malformed identifier; raw_fallback, observed 2026-08-07T14:40:58.027784Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.

Observation 2857b8d2-ca1e-4936-a9fb-94ebca8084f3 · outbound

This paper cites an unresolved cited work.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Unresolved cited work

Reference 2021

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:54.785452Z

Source-reported events for the cited work

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Observation 9cabe313-7d16-422a-b59d-950c20328dd4 · outbound

This paper cites Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomials.

Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomials

Reference 2024

Resolution: unresolved; no resolver link, observed 2026-08-07T14:40:55.955596Z

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Unavailable: canonical work link unavailable.

Pith citing papers

Observation f3e92fce-8555-4c51-b983-84e7284a6016 · inbound

(How) Can Transformers Predict Pseudo-Random Numbers? cites this paper.

(How) Can Transformers Predict Pseudo-Random Numbers? Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-07T18:32:14.733852Z

Source-reported events for the cited work

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Observation bab6ce43-d1d8-43bd-aa96-d6b34fe1ddd8 · inbound

Unveiling Memorization-Generalization Coexistence: A Case Study on Arithmetic Tasks with Label Noise cites this paper.

Unveiling Memorization-Generalization Coexistence: A Case Study on Arithmetic Tasks with Label Noise Uncovering a Universal Abstract Algorithm for Modular Addition in Neural Networks

Reference 39

Resolution: verified exact; arxiv_id, observed 2026-05-20T12:28:16.608417Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-17T06:30:58.91139+00:00.