Paper Citation Record · LEDGER

Learning words in groups: fusion algebras, tensor ranks and grokking

As of 20 August 2026, this Paper Citation Record lists 34 of 34 outbound references and 0 inbound Pith citation observations for arXiv:2509.06931.

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

pith.paper-citation-record.v1
2509.06931 v1

Coverage vector

measured 34 of 34 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-04T22:57:12.270615Z

measured 34 of 34 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-20T06:33:59.587034+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

34 of 34 outbound references displayed

verified exact2
verified fuzzy12
unresolved20
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 1e7465f0-2bc2-4954-ab9a-2278cfde047a · outbound

This paper cites an unresolved cited work.

Learning words in groups: fusion algebras, tensor ranks and grokking Unresolved cited work

Reference 1

Resolution: unresolved; raw_fallback, observed 2026-08-04T22:57:13.377311Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 1c5ff3e5-7f41-4f97-8e5b-7e021b3d8d2c · outbound

This paper cites Barak, B.

Learning words in groups: fusion algebras, tensor ranks and grokking Barak, B

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.354406Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 410022db-c0a1-4546-9c75-b276b14736f8 · outbound

This paper cites Bl ¨aser.

Learning words in groups: fusion algebras, tensor ranks and grokking Bl ¨aser

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.324421Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 59fb7154-22df-4b65-b1c5-964a4016313d · outbound

This paper cites Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges.

Learning words in groups: fusion algebras, tensor ranks and grokking Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.064758Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6417496d-dcd7-4aed-a9d0-f9fd2338c55f · outbound

This paper cites Chughtai, L.

Learning words in groups: fusion algebras, tensor ranks and grokking Chughtai, L

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.277589Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation a2cfc99a-6c63-4e7d-b442-b9c1d18587e8 · outbound

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

Learning words in groups: fusion algebras, tensor ranks and grokking To grok or not to grok: Disentangling generalization and memorization on corrupted algorithmic datasets

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.078755Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9204c1b0-623d-4178-9221-ff3448285cf5 · outbound

This paper cites Grokking Modular Polynomials.

Learning words in groups: fusion algebras, tensor ranks and grokking Grokking Modular Polynomials

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.084835Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8f98a357-1f12-4e32-9685-f53fee496fca · outbound

This paper cites Deep Grokking: Would Deep Neural Networks Generalize Better?.

Learning words in groups: fusion algebras, tensor ranks and grokking Deep Grokking: Would Deep Neural Networks Generalize Better?

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.089881Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bc36a3e1-3a50-40a2-8da7-b271c7388d92 · outbound

This paper cites Fawzi, M.

Learning words in groups: fusion algebras, tensor ranks and grokking Fawzi, M

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.231051Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation a198c29e-d4ee-4910-a238-dbbe433c2154 · outbound

This paper cites Fukushima.

Learning words in groups: fusion algebras, tensor ranks and grokking Fukushima

Reference 10

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.186595Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation cbb4cfad-b6c5-484e-b7ff-2d3341f73830 · outbound

This paper cites Fulton and J.

Learning words in groups: fusion algebras, tensor ranks and grokking Fulton and J

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.157123Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation e806ea79-a2bc-4f1a-91c0-195a10f5fee4 · outbound

This paper cites Goller and A.

Learning words in groups: fusion algebras, tensor ranks and grokking Goller and A

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.130848Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 0c6465a0-a671-4931-99c8-528f59ed8fa8 · outbound

This paper cites Grokking modular arithmetic.

Learning words in groups: fusion algebras, tensor ranks and grokking Grokking modular arithmetic

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.116867Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9a1ec924-3bb1-42c1-b0e8-a6b7e8446d9f · outbound

This paper cites an unresolved cited work.

Learning words in groups: fusion algebras, tensor ranks and grokking Unresolved cited work

Reference 14

Resolution: unresolved; raw_fallback, observed 2026-08-04T22:57:13.107868Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 71ce2773-ead6-4970-8cb6-70bc99ab70d5 · outbound

This paper cites an unresolved cited work.

Learning words in groups: fusion algebras, tensor ranks and grokking Unresolved cited work

Reference 15

Resolution: unresolved; raw_fallback, observed 2026-08-04T22:57:13.071922Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 2ba03143-8a1e-4c0b-89e6-ac072f76aff2 · outbound

This paper cites Deep Networks Always Grok and Here is Why.

Learning words in groups: fusion algebras, tensor ranks and grokking Deep Networks Always Grok and Here is Why

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.131495Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9db89af4-c0f7-452e-975d-9ca7e7014767 · outbound

This paper cites Kanatani.Group-theoretical methods in image understanding, volume 20.

Learning words in groups: fusion algebras, tensor ranks and grokking Kanatani.Group-theoretical methods in image understanding, volume 20

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.042743Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 2692c4de-ed6f-4363-a4f0-7f070d64067e · outbound

This paper cites Grokking as the Transition from Lazy to Rich Training Dynamics.

Learning words in groups: fusion algebras, tensor ranks and grokking Grokking as the Transition from Lazy to Rich Training Dynamics

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.143845Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 740c4e13-4765-404a-a5df-94351027bce1 · outbound

This paper cites Lenz.Group theoretical methods in image processing, volume 413.

Learning words in groups: fusion algebras, tensor ranks and grokking Lenz.Group theoretical methods in image processing, volume 413

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:13.011911Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation d9ad088c-cdc5-4ffd-b687-a335bc346de4 · outbound

This paper cites an unresolved cited work.

Learning words in groups: fusion algebras, tensor ranks and grokking Unresolved cited work

Reference 20

Resolution: unresolved; raw_fallback, observed 2026-08-04T22:57:12.964103Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 56573034-664b-4182-8a78-c7a71df93708 · outbound

This paper cites an unresolved cited work.

Learning words in groups: fusion algebras, tensor ranks and grokking Unresolved cited work

Reference 21

Resolution: unresolved; raw_fallback, observed 2026-08-04T22:57:12.936562Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation f6bd2900-ac4b-4d19-a26d-4b27d95f54d0 · outbound

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

Learning words in groups: fusion algebras, tensor ranks and grokking Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.175287Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation af1cf6d3-46a1-4638-8586-3e7fbc6d3f89 · outbound

This paper cites Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition.

Learning words in groups: fusion algebras, tensor ranks and grokking Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.180002Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a25e524b-c2d7-4151-a496-2574be67c10e · outbound

This paper cites Progress measures for grokking via mechanistic interpretability.

Learning words in groups: fusion algebras, tensor ranks and grokking Progress measures for grokking via mechanistic interpretability

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.185267Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 14af041e-336d-46ed-83b7-f76edc573423 · outbound

This paper cites Tensor Rank and Complexity.

Learning words in groups: fusion algebras, tensor ranks and grokking Tensor Rank and Complexity

Reference 25

Resolution: verified exact; local_arxiv, observed 2026-08-04T22:57:12.522531Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 6d92e549-674e-4867-9d2d-ddcb3ff14135 · outbound

This paper cites Is there a faster method to compute the composition of 2 3d rotations than directly using the formula for quaternion product? Mathematics Stack Exchange.

Learning words in groups: fusion algebras, tensor ranks and grokking Is there a faster method to compute the composition of 2 3d rotations than directly using the formula for quaternion product? Mathematics Stack Exchange

Reference 26

Resolution: verified exact; arxiv_id, observed 2026-08-04T22:57:12.494045Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 92650b4c-99cf-4ed3-ac41-648b24e507ad · outbound

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

Learning words in groups: fusion algebras, tensor ranks and grokking Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets

Reference 27

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.200452Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9dfa9ee7-7e88-4ca3-9c1e-22ad9c26e7f4 · outbound

This paper cites Grokking as a First Order Phase Transition in Two Layer Networks.

Learning words in groups: fusion algebras, tensor ranks and grokking Grokking as a First Order Phase Transition in Two Layer Networks

Reference 28

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.241241Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6f2c958a-4ec3-443a-a1b2-35d50042661d · outbound

This paper cites Sperduti.

Learning words in groups: fusion algebras, tensor ranks and grokking Sperduti

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:12.907176Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation d0a5e7a2-7aef-4202-90e2-823052a69a1c · outbound

This paper cites Strassen.

Learning words in groups: fusion algebras, tensor ranks and grokking Strassen

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:12.886800Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation 710e4e68-8017-4f3e-aa4f-134079e650a9 · outbound

This paper cites The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon.

Learning words in groups: fusion algebras, tensor ranks and grokking The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon

Reference 31

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.255903Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation cd5cc67e-24c3-4e72-b914-83d3ac30757f · outbound

This paper cites Explaining grokking through circuit efficiency.

Learning words in groups: fusion algebras, tensor ranks and grokking Explaining grokking through circuit efficiency

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.260487Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d864f75e-0030-4b3b-9622-ff9faed94ab5 · outbound

This paper cites Wood and J.

Learning words in groups: fusion algebras, tensor ranks and grokking Wood and J

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-08-04T22:57:12.862209Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-20T06:33:59.587034+00:00.

Observation c11111f0-5303-4cdf-ab69-bc36fe8db78f · outbound

This paper cites Benign Overfitting and Grokking in ReLU Networks for XOR Cluster Data.

Learning words in groups: fusion algebras, tensor ranks and grokking Benign Overfitting and Grokking in ReLU Networks for XOR Cluster Data

Reference 34

Resolution: unresolved; no resolver link, observed 2026-08-04T22:57:12.270615Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.