Paper Citation Record · LEDGER

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers

As of 10 August 2026, this Paper Citation Record lists 44 of 44 outbound references and 4 inbound Pith citation observations for arXiv:2502.02393.

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

pith.paper-citation-record.v1
2502.02393 v3

Coverage vector

measured 44 of 44 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-09T12:23:57.989726Z

measured 48 of 48 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-10T06:31:04.303077+00:00

measured 4 of 4 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-03T20:57:09.438395Z

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

44 of 44 outbound references displayed

verified exact0
verified fuzzy20
unresolved17
parse uncertain4
malformed identifier3
metadata mismatch0

External citation measurements

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

Outbound references

Observation 2ed6408d-2aa5-4927-a47f-17693ea09256 · outbound

This paper cites The situation is different for parallel computation as performed by transformers: Many problems do have direct parallel solutions.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers The situation is different for parallel computation as performed by transformers: Many problems do have direct parallel solutions

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.667527Z

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No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation 7e24c091-8eae-44d9-aa3f-9e05bf61ecb7 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.652970Z

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Observation 2fff638b-a1a9-4bbf-af5d-92983cb89be9 · outbound

This paper cites Theoretical limitations of multi-layer Transformer.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Theoretical limitations of multi-layer Transformer

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-09T12:23:57.804636Z

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Observation 41e8c9a9-6673-437b-9213-253eef0872b1 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 4

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.623062Z

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Observation a20aa124-19bc-45f0-874c-eda0e7b4521b · outbound

This paper cites How Numerical Precision Affects Arithmetical Reasoning Capabilities of LLMs.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers How Numerical Precision Affects Arithmetical Reasoning Capabilities of LLMs

Reference 5

Resolution: malformed identifier; no resolver link, observed 2026-08-09T12:23:57.814261Z

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Observation e0ae232d-ede8-4e86-b337-a761520ba6f4 · outbound

This paper cites Lower bounds on transformers with infinite precision.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Lower bounds on transformers with infinite precision

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-09T12:23:57.818820Z

Source-reported events for the cited work

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Observation f41969af-0b35-486d-b939-d9d04147137e · outbound

This paper cites Ask, and it shall be given: On the Turing completeness of prompting.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Ask, and it shall be given: On the Turing completeness of prompting

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-09T12:23:57.824032Z

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Observation 5e8362be-6f0a-4a8c-9a01-03e2a3e326b5 · outbound

This paper cites Average-Hard Attention Transformers are Constant-Depth Uniform Threshold Circuits.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Average-Hard Attention Transformers are Constant-Depth Uniform Threshold Circuits

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-09T12:23:57.829205Z

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Observation d68a1469-0d65-4685-b3b6-2b12c737b463 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.547854Z

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Observation 160a6dfd-a02d-4545-95f5-a77718590eb3 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 11

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.638548Z

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Observation e7bc93ff-3c56-4480-8101-d3fd7ec7f248 · outbound

This paper cites For instance, PARITY is a simple example of state tracking, a family of reasoning problems that have been of substantial interests and that still pose challenges for LLMs (e.g.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers For instance, PARITY is a simple example of state tracking, a family of reasoning problems that have been of substantial interests and that still pose challenges for LLMs (e.g

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.607322Z

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Observation 3289191e-3553-4a60-8361-1998690b9f9b · outbound

This paper cites Why? The input alphabet needs to be finite for Theorem 3.3 to go through, because the proof of Lemma B.3 by Hahn (2020) involves a union bound over the alphabet.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Why? The input alphabet needs to be finite for Theorem 3.3 to go through, because the proof of Lemma B.3 by Hahn (2020) involves a union bound over the alphabet

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.592084Z

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Observation 7c68ef69-46b5-46f5-bab8-be4058460db9 · outbound

This paper cites Isn’t this unrealistic? What about the role of length generalization? It is true that, in practice, one will expect transformers to perform the tasks across input lengths.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Isn’t this unrealistic? What about the role of length generalization? It is true that, in practice, one will expect transformers to perform the tasks across input lengths

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.577022Z

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Observation 88fcc03d-6a11-4ca2-81fb-b5e14441d394 · outbound

This paper cites In particular, UHAT is bounded by AC 0, whereas softmax transformers are bounded by T C0.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers In particular, UHAT is bounded by AC 0, whereas softmax transformers are bounded by T C0

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.562746Z

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Observation 47cca165-cb83-40e6-a3f9-4ada7330bf89 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 18

Resolution: parse uncertain; raw_fallback, observed 2026-08-09T12:23:58.533204Z

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Observation 42643bd6-9925-4fc4-9560-f1cbd1d9b4ec · outbound

This paper cites By applying Lemma B.3 iteratively, L times.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers By applying Lemma B.3 iteratively, L times

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.519420Z

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Observation 054c180d-0d4d-4765-8f39-b5cc456cfa41 · outbound

This paper cites Here, “ρ(3)” corresponds to the restriction ρ′ from the statement of the lemma.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Here, “ρ(3)” corresponds to the restriction ρ′ from the statement of the lemma

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.505231Z

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Observation 790de37a-4d11-4b3b-944d-1e5f35be3cb4 · outbound

This paper cites First, we obtain a restriction ρ(0), and integers c, k from Lemma 3.4, applying T only on the input x ∈ Σ∗ itself.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers First, we obtain a restriction ρ(0), and integers c, k from Lemma 3.4, applying T only on the input x ∈ Σ∗ itself

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.491288Z

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Observation 13642b46-bee7-4ff2-8aa0-272493bd69ef · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 22

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.478548Z

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Observation 08664bc3-d590-4078-949c-1d84e1a8de71 · outbound

This paper cites dot-by-dot CoT.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers dot-by-dot CoT

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.464323Z

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Observation 157215f7-71d2-406a-af1f-a882e090f143 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 24

Resolution: parse uncertain; raw_fallback, observed 2026-08-09T12:23:58.450359Z

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Observation f1e57c3a-99c7-4b0a-9693-54c9964c5094 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 25

Resolution: parse uncertain; raw_fallback, observed 2026-08-09T12:23:58.436030Z

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Observation 65cecb05-ccae-44f3-83c6-aaebec03afa6 · outbound

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Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 26

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.421806Z

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Observation 54acb11d-eeab-4eef-be72-c0bc16b5b5ae · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 27

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.406030Z

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Observation 4899fcfa-a50e-4ef0-a77b-72c5578f60e1 · outbound

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Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 28

Resolution: parse uncertain; raw_fallback, observed 2026-08-09T12:23:58.390936Z

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Observation 1e5201c1-c51b-4ff7-955b-58cb8b844950 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 29

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.376170Z

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Observation 1fbb4e47-c178-4452-b2c6-e2e43406cb73 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 30

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.361906Z

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Observation ba150e93-b063-4aa8-94f4-3767c6da8f63 · outbound

This paper cites All remaining tape positions hold a blank symbol BLANK ∈ Σ.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers All remaining tape positions hold a blank symbol BLANK ∈ Σ

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.347133Z

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Observation 231408a6-7c7f-469b-8fcb-43f2c04743ea · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 32

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.332312Z

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Observation d93dc880-48ae-4efa-88f2-0c6d7b894b6a · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 33

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.317110Z

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Observation 66396782-3b8a-482b-8ed3-2229c4503236 · outbound

This paper cites We first encode the input word as a string over the input alphabet Σ, followed by a separator symbol.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers We first encode the input word as a string over the input alphabet Σ, followed by a separator symbol

Reference 34

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.302892Z

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Observation 624657ac-bbaf-4f47-bfac-da67bbce207e · outbound

This paper cites 19 Lower Bounds for CoT Reasoning.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers 19 Lower Bounds for CoT Reasoning

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.288385Z

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Observation 4fdc6f53-e880-4772-ae9d-898d2da33c02 · outbound

This paper cites Turing programs.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Turing programs

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.274603Z

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Observation 21395cfc-2124-44c7-b7da-3d1181b7e691 · outbound

This paper cites an unresolved cited work.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Unresolved cited work

Reference 37

Resolution: unresolved; raw_fallback, observed 2026-08-09T12:23:58.260035Z

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Observation ea12ce18-32fd-4b26-b059-4e16490af1a1 · outbound

This paper cites We show this using the characterization of regular languages in AC 0 by Barrington et al.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers We show this using the characterization of regular languages in AC 0 by Barrington et al

Reference 38

Resolution: malformed identifier; raw_fallback, observed 2026-08-09T12:23:58.244879Z

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Observation 951fa4a9-3778-4454-a0c6-52528dc742bd · outbound

This paper cites , M2N (and analogously for addition) in parallel.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers , M2N (and analogously for addition) in parallel

Reference 39

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.229227Z

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Observation 59588778-f524-46d8-9af1-93f369e2cef7 · outbound

This paper cites M2N (and analogously for addition).

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers M2N (and analogously for addition)

Reference 40

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.212842Z

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Observation e59177fa-2132-437b-865d-e0afb470c97e · outbound

This paper cites We expect that this setup will make the task feasible for transformers.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers We expect that this setup will make the task feasible for transformers

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.197213Z

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Observation 63b5920b-757d-43c4-8014-479ad69567c4 · outbound

This paper cites increment-by-1.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers increment-by-1

Reference 42

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.181087Z

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Observation a3f84ea5-440e-45b7-85c9-69f409b1830e · outbound

This paper cites Conjecture E.2 thus provides a more optimistic prediction here than our results.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Conjecture E.2 thus provides a more optimistic prediction here than our results

Reference 43

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.164516Z

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Observation 1083912c-4c02-44b4-8733-ddba00e93559 · outbound

This paper cites (2024) requires a linear-length CoT by Theorem 3.3, and can also be shown to not be representable at sub-linear average sensitivity under a reasonable input encoding.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers (2024) requires a linear-length CoT by Theorem 3.3, and can also be shown to not be representable at sub-linear average sensitivity under a reasonable input encoding

Reference 44

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.148411Z

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Observation ff1f9449-eced-40e6-aaca-1e4c845ba365 · outbound

This paper cites Abb´e, E., Bengio, S., Lotfi, A., Sandon, C., and Saremi, O.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Abb´e, E., Bengio, S., Lotfi, A., Sandon, C., and Saremi, O

Reference 2023

Resolution: verified fuzzy; raw_fallback, observed 2026-08-09T12:23:58.682081Z

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No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation 5ef45449-9471-4a80-b362-2a02d7e0eaa6 · outbound

This paper cites Learning High-Degree Parities: The Crucial Role of the Initialization.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Learning High-Degree Parities: The Crucial Role of the Initialization

Reference 2024

Resolution: malformed identifier; local_arxiv, observed 2026-08-09T12:23:58.131874Z

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Observation 62b6c806-972d-42b7-bc7e-9af5a335e79c · outbound

This paper cites Arithmetic Transformers Can Length-Generalize in Both Operand Length and Count.

Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers Arithmetic Transformers Can Length-Generalize in Both Operand Length and Count

Reference 8856

Resolution: unresolved; no resolver link, observed 2026-08-09T12:23:57.809581Z

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Unavailable: canonical work link unavailable.

Pith citing papers

Observation f0ea446a-36d9-4569-9757-835e1ba637ff · inbound

Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models cites this paper.

Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers

Reference 12

Resolution: verified exact; arxiv_id, observed 2026-05-12T08:40:41.311770Z

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No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation 4030c640-1a10-41e9-9fc0-0e6efc0c3cd3 · inbound

Transformers with RL or SFT Provably Learn Sparse Boolean Functions, But Differently cites this paper.

Transformers with RL or SFT Provably Learn Sparse Boolean Functions, But Differently Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-03T20:57:09.438395Z

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Unavailable: canonical work link unavailable.

Observation d57bf956-5b30-4e9a-80dc-8d8181c248be · inbound

On the Spatiotemporal Dynamics of Generalization in Neural Networks cites this paper.

On the Spatiotemporal Dynamics of Generalization in Neural Networks Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers

Reference 26

Resolution: verified exact; arxiv_id, observed 2026-05-16T09:00:46.785226Z

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No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.

Observation d415cefe-afb7-4efa-87d2-7840d60e20f9 · inbound

Agentic Transformers Provably Learn to Search via Reinforcement Learning cites this paper.

Agentic Transformers Provably Learn to Search via Reinforcement Learning Lower Bounds for Chain-of-Thought Reasoning in Hard-Attention Transformers

Reference 23

Resolution: metadata mismatch; arxiv_id, observed 2026-06-28T23:42:49.970053Z

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No event found in the named queried sources as of 2026-08-10T06:31:04.303077+00:00.