Paper Citation Record · LEDGER

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning

As of 7 August 2026, this Paper Citation Record lists 48 of 48 outbound references and 0 inbound Pith citation observations for arXiv:2607.25905.

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

pith.paper-citation-record.v1
2607.25905 v1

Coverage vector

measured 48 of 48 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-01T01:22:49.166462Z

measured 48 of 48 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-07T06:34:17.273281+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

48 of 48 outbound references displayed

verified exact0
verified fuzzy0
unresolved48
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 1124f8f9-fffa-4341-b669-d8ae0f532395 · outbound

This paper cites Mathematical analysis, 1958.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Mathematical analysis, 1958

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.927894Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 09efc6b0-1a48-4eb6-ad47-7d30838fd830 · outbound

This paper cites Functions of bounded variation and free discontinuity problems.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Functions of bounded variation and free discontinuity problems

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.941406Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2ee82292-8154-4f80-97d5-d24a302f472e · outbound

This paper cites Variational models for phase transitions, an approach via -convergence.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Variational models for phase transitions, an approach via -convergence

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.946985Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 0587230c-7d3b-42a2-bdaa-e1a8c4511475 · outbound

This paper cites Breaking the curse of dimensionality with convex neural networks.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Breaking the curse of dimensionality with convex neural networks

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.952168Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3a57f580-4e2c-482c-a4c2-c380e4465563 · outbound

This paper cites Discontinuous equilibrium solutions and cavitation in nonlinear elasticity.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Discontinuous equilibrium solutions and cavitation in nonlinear elasticity

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.957643Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8e59ce6d-faf1-44e0-8f88-53139ceadf1c · outbound

This paper cites Onsager's conjecture for admissible weak solutions.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Onsager's conjecture for admissible weak solutions

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.965992Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 225e6be9-bfab-4a6a-8820-ff1167a93cb3 · outbound

This paper cites New examples on L avrentiev gap using fractals.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning New examples on L avrentiev gap using fractals

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.973308Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 458a7b25-9ef4-4af6-b1fc-5af5600c1866 · outbound

This paper cites Penalising the biases in norm regularisation enforces sparsity.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Penalising the biases in norm regularisation enforces sparsity

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.978471Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 56fa74bb-d603-4ace-8b83-da6d302bc141 · outbound

This paper cites A global method for relaxation in w^ 1, p and in sbv^p.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning A global method for relaxation in w^ 1, p and in sbv^p

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.983574Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation fa62e9b1-c9df-4fce-9cfc-818cc5bf9d46 · outbound

This paper cites Finite Element Methods for the Stretching and Bending of Thin Structures with Folding.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Finite Element Methods for the Stretching and Bending of Thin Structures with Folding

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.988172Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6923c3e9-f127-4a3f-802b-7170876a0772 · outbound

This paper cites h-principle and rigidity for c^ 1, -isometric embeddings.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning h-principle and rigidity for c^ 1, -isometric embeddings

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.993183Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b541d1e3-502a-4d55-994a-01f3b21f3aff · outbound

This paper cites Deformation concentration for martensitic microstructures in the limit of low volume fraction.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Deformation concentration for martensitic microstructures in the limit of low volume fraction

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:48.998736Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 518a7e8d-3537-4a8b-bd4b-c7696425ec8e · outbound

This paper cites Onsager's conjecture on the energy conservation for solutions of E uler's equation.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Onsager's conjecture on the energy conservation for solutions of E uler's equation

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.003846Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b3373d55-39c7-4be0-bfaf-7246f2da3994 · outbound

This paper cites On turbulence and geometry: from Nash to Onsager.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning On turbulence and geometry: from Nash to Onsager

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.008076Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7d676a53-df96-44c2-abdd-d41732c74d9f · outbound

This paper cites Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-R \"a ume und elliptische Differentialgleichungen.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-R \"a ume und elliptische Differentialgleichungen

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.013124Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3c1086d8-9528-4931-8508-6069cc2c097c · outbound

This paper cites Measure theory and fine properties of functions.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Measure theory and fine properties of functions

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.019376Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 9ab5fd58-fd8a-4f61-b80b-d1cf16e362db · outbound

This paper cites The B arron space and the flow-induced function spaces for neural network models.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The B arron space and the flow-induced function spaces for neural network models

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.027560Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 87b5b00a-1c27-40e7-9612-e6c4b8351ded · outbound

This paper cites L’h \^o pital’s monotone rule, gromov’s theorem, and operations that preserve the monotonicity of quotients.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning L’h \^o pital’s monotone rule, gromov’s theorem, and operations that preserve the monotonicity of quotients

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.034241Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5616e724-09e5-426b-9251-91f7b3c1f735 · outbound

This paper cites The power of depth for feedforward neural networks.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The power of depth for feedforward neural networks

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.038305Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5e0027da-66fb-4f7e-a24d-ec49e104b187 · outbound

This paper cites Partial differential equations , volume 19.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Partial differential equations , volume 19

Reference 20

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.042589Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f4c45030-0b5e-4e89-b8cc-a101e9049788 · outbound

This paper cites On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics

Reference 21

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.046736Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c73ef5a8-2044-4885-b031-2c4c1b40dd7e · outbound

This paper cites Representation formulas and pointwise properties for B arron functions.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Representation formulas and pointwise properties for B arron functions

Reference 22

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.051495Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6b4f7af8-29ef-41f9-bce6-708aa729dc41 · outbound

This paper cites The L avrentiev gap phenomenon in nonlinear elasticity.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The L avrentiev gap phenomenon in nonlinear elasticity

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.055573Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 837a8dc3-40de-4191-b7b6-78411a29e314 · outbound

This paper cites A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence

Reference 24

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.059742Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c5b56031-3228-44e1-9f54-fb6fd25d715e · outbound

This paper cites An anisotropic P oincar \'e inequality in gsbv^p and the limit of strongly anisotropic Mumford--Shah functionals.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning An anisotropic P oincar \'e inequality in gsbv^p and the limit of strongly anisotropic Mumford--Shah functionals

Reference 25

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.064416Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f6e94dd0-2026-4403-88ce-85a15445ad99 · outbound

This paper cites Minimal surfaces and functions of bounded variation.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Minimal surfaces and functions of bounded variation

Reference 26

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.068688Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 75c1494c-8609-40de-812d-31aa9072d48a · outbound

This paper cites Elliptic partial differential equations of second order , volume 224.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Elliptic partial differential equations of second order , volume 224

Reference 27

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.073025Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 20012e66-d91d-42da-be69-377ebdca7e27 · outbound

This paper cites Sur quelques problemes du calcul des variations.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Sur quelques problemes du calcul des variations

Reference 28

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.077081Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bf23da10-0239-4f40-ac39-3de2ee940dd9 · outbound

This paper cites On the L avrentiev phenomenon.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning On the L avrentiev phenomenon

Reference 29

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.082051Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1bb79f4b-b9fb-4fb7-8897-bcd382cb08e6 · outbound

This paper cites Strong approximation of special functions of bounded variation functions with prescribed jump direction.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Strong approximation of special functions of bounded variation functions with prescribed jump direction

Reference 30

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.086397Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ca3a0ef8-9462-442c-b5c5-25ec90e54218 · outbound

This paper cites Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory

Reference 31

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.090309Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 36fecc8b-8b21-4ba5-9e8a-7e218b082923 · outbound

This paper cites an unresolved cited work.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Unresolved cited work

Reference 32

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.094471Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation cbe278d7-cffb-4321-a60c-150655652ef8 · outbound

This paper cites The L avrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The L avrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data

Reference 33

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.099576Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1e79cefb-4de3-474d-9179-cb4b979ca0f0 · outbound

This paper cites C ^1 -isometric imbeddings.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning C ^1 -isometric imbeddings

Reference 34

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.104931Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 90f2a156-ada0-4192-92d2-ed3702918066 · outbound

This paper cites The imbedding problem for R iemannian manifolds.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning The imbedding problem for R iemannian manifolds

Reference 35

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.109335Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation aaf9f8ac-bc47-4e64-9f04-9a0be85eb9ff · outbound

This paper cites A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case

Reference 36

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.114447Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f0294dd7-39f2-4106-af85-f8837e5840a9 · outbound

This paper cites Banach space representer theorems for neural networks and ridge splines.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Banach space representer theorems for neural networks and ridge splines

Reference 37

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.119593Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation cd80a9b4-86a6-4308-b918-bf19d595c769 · outbound

This paper cites Minimum norm interpolation by perceptra: Explicit regularization and implicit bias.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Minimum norm interpolation by perceptra: Explicit regularization and implicit bias

Reference 38

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.123582Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a68a6d9a-00d3-4771-a2ec-a70e1cce3c1f · outbound

This paper cites Real and complex analysis.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Real and complex analysis

Reference 39

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.127396Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6f69b81e-1194-4113-af1c-965553ad318d · outbound

This paper cites Benefits of depth in neural networks.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Benefits of depth in neural networks

Reference 40

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.131840Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 21144741-daba-4be8-a933-2561dd24560f · outbound

This paper cites Ridges, neural networks, and the radon transform.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Ridges, neural networks, and the radon transform

Reference 41

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.135793Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 840c6c9f-0283-4250-9883-dc7e3ce7462f · outbound

This paper cites Depth separation beyond radial functions.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Depth separation beyond radial functions

Reference 42

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.140342Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation bc115aac-ee50-49ef-bdc5-17ae04a7d49c · outbound

This paper cites Solving the Poisson Equation with Dirichlet data by shallow ReLU$^\alpha$-networks: A regularity and approximation perspective.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Solving the Poisson Equation with Dirichlet data by shallow ReLU$^\alpha$-networks: A regularity and approximation perspective

Reference 43

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.144572Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a6e0cb95-2625-41e1-b06c-63a100bb4933 · outbound

This paper cites Lipschitz algebras.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Lipschitz algebras

Reference 44

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.148894Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 57564083-5d7e-4f4b-b0ba-5eeea200b1f7 · outbound

This paper cites Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Optimal bump functions for shallow ReLU networks: Weight decay, depth separation and the curse of dimensionality

Reference 45

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.153491Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b9d73740-7ac6-4e0b-9b95-ae23514279ab · outbound

This paper cites A note on elliptic regularity theory in barron spaces.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning A note on elliptic regularity theory in barron spaces

Reference 46

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.157897Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 99504fbc-1174-4681-b456-4a5d6a0d9505 · outbound

This paper cites Averaging of functionals of the calculus of variations and elasticity theory.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Averaging of functionals of the calculus of variations and elasticity theory

Reference 47

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.162450Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 836941b6-3e19-4d84-b392-5526e7c2f33d · outbound

This paper cites Lavrentiev phenomenon and homogenization for some variational problems.

The Barron-Lipschitz Energy Gap and Depth Separation Phenomena in Scientific Machine Learning Lavrentiev phenomenon and homogenization for some variational problems

Reference 48

Resolution: unresolved; no resolver link, observed 2026-08-01T01:22:49.166462Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.