Paper Citation Record · LEDGER

Thermodynamic Irreversibility of Training Algorithms

As of 31 July 2026, this Paper Citation Record lists 43 of 43 outbound references and 0 inbound Pith citation observations for arXiv:2605.21933.

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

pith.paper-citation-record.v1
2605.21933 v1

Coverage vector

measured 43 of 43 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-05-22T04:38:51.773377Z

measured 43 of 43 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-07-31T06:34:12.847434+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

43 of 43 outbound references displayed

verified exact10
verified fuzzy24
unresolved8
parse uncertain0
malformed identifier0
metadata mismatch1

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation f60164b5-8377-466a-b445-7e4e45a9c936 · outbound

This paper cites Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit.

Thermodynamic Irreversibility of Training Algorithms Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit

Reference 1

Resolution: verified exact; local_arxiv, observed 2026-05-22T04:41:04.241477Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation f1c76b03-535f-4cbe-b6fa-aabcc8b14ef0 · outbound

This paper cites Halverson, A.

Thermodynamic Irreversibility of Training Algorithms Halverson, A

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.695336Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 7d65ebdf-4267-4f81-8d8e-38858bf0b995 · outbound

This paper cites Rotskoff and E.

Thermodynamic Irreversibility of Training Algorithms Rotskoff and E

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.683904Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation c9babf64-7776-47a8-9cc1-f5a82a91f1b2 · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 4

Resolution: verified exact; arxiv_id, observed 2026-05-22T04:41:04.274914Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 16e5f64e-e377-4fe1-9fde-d22be7d914d5 · outbound

This paper cites A Diffusion Theory For Deep Learning Dynamics: Stochastic Gradient Descent Exponentially Favors Flat Minima.

Thermodynamic Irreversibility of Training Algorithms A Diffusion Theory For Deep Learning Dynamics: Stochastic Gradient Descent Exponentially Favors Flat Minima

Reference 5

Resolution: verified exact; arxiv_id, observed 2026-05-22T04:41:04.270581Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation d615f377-7eb8-468e-8886-837221cd03c0 · outbound

This paper cites Prigogine and R.

Thermodynamic Irreversibility of Training Algorithms Prigogine and R

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.764921Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation c0541980-2bd8-4e59-9091-72839d8cae24 · outbound

This paper cites Seifert, The European Physical Journal B64, 423 (2008).

Thermodynamic Irreversibility of Training Algorithms Seifert, The European Physical Journal B64, 423 (2008)

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.758215Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation c5b6a752-e98d-4984-b861-575b4addd5a9 · outbound

This paper cites O’Byrne, Y.

Thermodynamic Irreversibility of Training Algorithms O’Byrne, Y

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.751875Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 3f6da75b-828a-4876-bbee-c4ae31967ffe · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 9

Resolution: unresolved; raw_fallback, observed 2026-05-22T04:41:04.749060Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation d7fbb36c-9575-468e-b9de-6f2090fe635e · outbound

This paper cites Adam: A Method for Stochastic Optimization.

Thermodynamic Irreversibility of Training Algorithms Adam: A Method for Stochastic Optimization

Reference 10

Resolution: verified exact; local_arxiv, observed 2026-05-22T04:41:04.255779Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 76d1af30-087b-456c-9ede-2e813c64c710 · outbound

This paper cites Tieleman and G.

Thermodynamic Irreversibility of Training Algorithms Tieleman and G

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.746064Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 85e60875-b36c-4936-b336-428e6bdd9a45 · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-05-22T04:41:04.743130Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 0f48fb44-ebb0-4173-ae34-17abc7ee215e · outbound

This paper cites Hairer, M.

Thermodynamic Irreversibility of Training Algorithms Hairer, M

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.674103Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation b66cfb7d-6808-4888-a784-2e0447680fd7 · outbound

This paper cites On the Origin of Implicit Regularization in Stochastic Gradient Descent.

Thermodynamic Irreversibility of Training Algorithms On the Origin of Implicit Regularization in Stochastic Gradient Descent

Reference 14

Resolution: verified exact; arxiv_id, observed 2026-05-22T04:41:04.236302Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 3152b507-0779-4e32-82ad-fa339ef32253 · outbound

This paper cites Implicit Gradient Regularization.

Thermodynamic Irreversibility of Training Algorithms Implicit Gradient Regularization

Reference 15

Resolution: verified exact; arxiv_id, observed 2026-05-22T04:41:04.218335Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation bd6d1732-4dea-4ce3-9b02-e851238d21dd · outbound

This paper cites Ziyin, H.

Thermodynamic Irreversibility of Training Algorithms Ziyin, H

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.701184Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 7147e5a4-0978-4823-89e8-3caea8e312c3 · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 17

Resolution: unresolved; raw_fallback, observed 2026-05-22T04:41:04.686447Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 629d5f67-c9f6-4aa7-8ded-a65550d522d6 · outbound

This paper cites Ziyin and M.

Thermodynamic Irreversibility of Training Algorithms Ziyin and M

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.739681Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 653a38f6-7960-4a47-a6da-8c82d32c3fb7 · outbound

This paper cites Thermodynamic Uncertainty Relation for Arbitrary Initial States.

Thermodynamic Irreversibility of Training Algorithms Thermodynamic Uncertainty Relation for Arbitrary Initial States

Reference 19

Resolution: verified exact; arxiv_id, observed 2026-05-22T04:41:04.246630Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 7ba0233d-0ddf-4cdd-a2bd-cfd6bb4ed877 · outbound

This paper cites Goldt and U.

Thermodynamic Irreversibility of Training Algorithms Goldt and U

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.773976Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 29d4c86a-5da5-46d1-b28e-8b2957e1b33f · outbound

This paper cites Murashita, K.

Thermodynamic Irreversibility of Training Algorithms Murashita, K

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.777122Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 10c88cfd-b70c-4b3e-b802-c1d85bdcfa87 · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 22

Resolution: unresolved; raw_fallback, observed 2026-05-22T04:41:04.768138Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation e2acecba-54d4-4490-b820-4d4e50c88b57 · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 23

Resolution: unresolved; raw_fallback, observed 2026-05-22T04:41:04.771012Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation fb451b88-15e2-4900-b577-343e509179c3 · outbound

This paper cites Parameter Symmetry Potentially Unifies Deep Learning Theory.

Thermodynamic Irreversibility of Training Algorithms Parameter Symmetry Potentially Unifies Deep Learning Theory

Reference 24

Resolution: verified exact; arxiv_id, observed 2026-05-22T04:41:04.261521Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 64503291-11bf-43a8-aae2-0ba0d53bdb47 · outbound

This paper cites Fluctuation-dissipation relations for stochastic gradient descent.

Thermodynamic Irreversibility of Training Algorithms Fluctuation-dissipation relations for stochastic gradient descent

Reference 25

Resolution: metadata mismatch; local_arxiv, observed 2026-05-22T04:41:04.251165Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 7d2b96b4-2b2b-4aa7-946a-b69aae0ea03d · outbound

This paper cites Ziyin, Y.

Thermodynamic Irreversibility of Training Algorithms Ziyin, Y

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.761649Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 4aa122e1-216d-4e62-b755-bd5200da4a36 · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 27

Resolution: verified exact; arxiv_id, observed 2026-05-22T04:41:04.266323Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation e883348a-4936-437c-9519-a75cbf7a69bf · outbound

This paper cites Poggio, R.

Thermodynamic Irreversibility of Training Algorithms Poggio, R

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.754845Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 397c6b61-4d87-47cb-95e8-88a2e12d5a22 · outbound

This paper cites Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations.

Thermodynamic Irreversibility of Training Algorithms Stochastic Modified Equations and Dynamics of Stochastic Gradient Algorithms I: Mathematical Foundations

Reference 29

Resolution: verified exact; local_arxiv, observed 2026-05-22T04:41:04.213211Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation a83c4d37-d675-49d9-ae22-2005861f09ca · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 30

Resolution: unresolved; raw_fallback, observed 2026-05-22T04:41:04.670939Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation eabbc429-9939-4532-a041-cd7d9d45276c · outbound

This paper cites symmetry.

Thermodynamic Irreversibility of Training Algorithms symmetry

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.736700Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation e3d1d1c3-6a07-43cb-884b-28e1825982a6 · outbound

This paper cites Theorem 6(Continuous Symmetry Breaking).Let K(θ, λ) =θ+λQ(θ) +O(λ 2)be a continuous symmetry generated byQ(θ).

Thermodynamic Irreversibility of Training Algorithms Theorem 6(Continuous Symmetry Breaking).Let K(θ, λ) =θ+λQ(θ) +O(λ 2)be a continuous symmetry generated byQ(θ)

Reference 32

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.724667Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 71860016-78f6-4143-b1d9-6fa08b2cbf25 · outbound

This paper cites Theorem 7(Discrete Symmetry Preservation).Let the transformation beK(θ) =Oθ, whereOis an orthogonal matrix (O T O=I).

Thermodynamic Irreversibility of Training Algorithms Theorem 7(Discrete Symmetry Preservation).Let the transformation beK(θ) =Oθ, whereOis an orthogonal matrix (O T O=I)

Reference 33

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.721383Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 3a20a6df-7a52-4ca3-9151-9892159ec15c · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 34

Resolution: unresolved; raw_fallback, observed 2026-05-22T04:41:04.677635Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation af932534-2fb2-4dc5-918b-29883eed6af0 · outbound

This paper cites Define Θcoarse(θ;η) =θ−ηU(θ),(G10) and Θfine(θ;η) = Θ η/2 ◦Θ η/2(θ),Θ η/2(θ) =θ− η 2 U(θ).

Thermodynamic Irreversibility of Training Algorithms Define Θcoarse(θ;η) =θ−ηU(θ),(G10) and Θfine(θ;η) = Θ η/2 ◦Θ η/2(θ),Θ η/2(θ) =θ− η 2 U(θ)

Reference 35

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.718528Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation e723b048-c0da-4972-ace6-a149224c4910 · outbound

This paper cites Starting at θt, take one forward step and then one backward step with the sign of the step size reversed: θt+1 =θ t −ηU(θ t), ˜θt =θ t −ηU(θ t) +ηU(θ t+1).

Thermodynamic Irreversibility of Training Algorithms Starting at θt, take one forward step and then one backward step with the sign of the step size reversed: θt+1 =θ t −ηU(θ t), ˜θt =θ t −ηU(θ t) +ηU(θ t+1)

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.715225Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation ee432604-30d8-43b5-ab5b-db2daa616ccb · outbound

This paper cites We in- troduce a virtual Gaussian transition kernel pσ(θ′|θ)∝exp − ∥θ′ −θ+ηU(θ)∥ 2 2σ2 ,(G27) whereσ 2 is a small virtual noise variance.

Thermodynamic Irreversibility of Training Algorithms We in- troduce a virtual Gaussian transition kernel pσ(θ′|θ)∝exp − ∥θ′ −θ+ηU(θ)∥ 2 2σ2 ,(G27) whereσ 2 is a small virtual noise variance

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.711791Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 7673364d-e4ac-48e6-90d7-bd41b6ed3c61 · outbound

This paper cites (G33) Here bϕTA denotes the normalized quantity defined in Eq.

Thermodynamic Irreversibility of Training Algorithms (G33) Here bϕTA denotes the normalized quantity defined in Eq

Reference 38

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.708470Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation af75dcea-bf8a-45a1-8805-610f94e1d060 · outbound

This paper cites We consider a quadratic potential E(θ) = 1 2 θ⊤Aθ,(G34) whereA∈R d×d is positive definite.

Thermodynamic Irreversibility of Training Algorithms We consider a quadratic potential E(θ) = 1 2 θ⊤Aθ,(G34) whereA∈R d×d is positive definite

Reference 39

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.705007Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 52cf208a-3aad-4e51-9331-221d65f366a3 · outbound

This paper cites The model is a 2- layer causal Transformer (GPT-style) withd model = 128, nhead = 4 attention heads, and a feedforward dimension of 512.

Thermodynamic Irreversibility of Training Algorithms The model is a 2- layer causal Transformer (GPT-style) withd model = 128, nhead = 4 attention heads, and a feedforward dimension of 512

Reference 40

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.692394Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 2e998348-4f58-4a15-9b63-bd62fbed87e2 · outbound

This paper cites Our model is a gated recurrent unit (GRU) lan- guage model withL= 2 recurrent layers and hidden size h= 256.

Thermodynamic Irreversibility of Training Algorithms Our model is a gated recurrent unit (GRU) lan- guage model withL= 2 recurrent layers and hidden size h= 256

Reference 41

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.689401Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation b6d02fdc-0ab4-4a6e-8ac1-ec32faf908cb · outbound

This paper cites linear re- gression).

Thermodynamic Irreversibility of Training Algorithms linear re- gression)

Reference 42

Resolution: verified fuzzy; raw_fallback, observed 2026-05-22T04:41:04.697936Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Observation 59c88fe2-2661-4da6-bfd5-ea001825d176 · outbound

This paper cites an unresolved cited work.

Thermodynamic Irreversibility of Training Algorithms Unresolved cited work

Reference 43

Resolution: unresolved; raw_fallback, observed 2026-05-22T04:41:04.681056Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-07-31T06:34:12.847434+00:00.

Pith citing papers

No inbound Pith citation observations are available.