Paper Citation Record · LEDGER

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks

As of 11 August 2026, this Paper Citation Record lists 17 of 17 outbound references and 1 inbound Pith citation observation for arXiv:2607.21366.

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

pith.paper-citation-record.v1
2607.21366 v1

Coverage vector

measured 17 of 17 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-01T07:45:50.679709Z

measured 18 of 18 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-11T06:34:44.6726+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-08-07T01:03:26.262556Z

measured 0 of 1 external citation measurements

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

Source: pith, observed 2026-08-07T01:03:31.697365Z

Reference resolution

17 of 17 outbound references displayed

verified exact0
verified fuzzy0
unresolved16
parse uncertain0
malformed identifier1
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 98aca707-ff14-4fd6-8eaf-7db39dad3540 · outbound

This paper cites an unresolved cited work.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Unresolved cited work

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:48.504351Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 67b4b0bd-c7e2-4112-86ab-69ddb2db0cc6 · outbound

This paper cites closeness.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks closeness

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:48.663065Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 8062e3a9-0d90-41dd-8f66-430bd7d26d2f · outbound

This paper cites We embed this function into a scalar Hilbert spaceHin ≜𝐿 2(X,𝑃X;ℝ).

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks We embed this function into a scalar Hilbert spaceHin ≜𝐿 2(X,𝑃X;ℝ)

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:48.926656Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation dbe60e1d-25d6-4c9b-973d-b8af4b3c66ff · outbound

This paper cites We defined the inner product for any functions𝑓,𝑔 . But in HOPE, we only ever calculate it for single ReLU neurons. Is that enough to define the whole space?.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks We defined the inner product for any functions𝑓,𝑔 . But in HOPE, we only ever calculate it for single ReLU neurons. Is that enough to define the whole space?

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:49.054243Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 86a06bee-3aa3-4463-814e-24bfd73a95f8 · outbound

This paper cites 3.Partition Invariance:∀𝑓∈H,∀𝑁∈ℤ ≥1, 𝐸((𝑓))=𝐸 (𝑓/𝑁,..., 𝑓/𝑁 | {z } 𝑁times ).

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks 3.Partition Invariance:∀𝑓∈H,∀𝑁∈ℤ ≥1, 𝐸((𝑓))=𝐸 (𝑓/𝑁,..., 𝑓/𝑁 | {z } 𝑁times )

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:49.124250Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a2243eec-43b6-4a28-80ec-af9921c13fc5 · outbound

This paper cites By Lemma C.1, capacity scales linearly𝐸(𝑘Φ)=𝑘𝐸(Φ) , so¤𝐸(𝑘Φ(𝑡))=𝑘 ¤𝐸(𝑡).

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks By Lemma C.1, capacity scales linearly𝐸(𝑘Φ)=𝑘𝐸(Φ) , so¤𝐸(𝑘Φ(𝑡))=𝑘 ¤𝐸(𝑡)

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:49.229666Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2b65aa20-b411-4135-8ea2-f0f6c113d93f · outbound

This paper cites Scaling the state by𝑘yields: 𝑐(𝑘Φ)=𝜉(𝑘Φ)𝐸(𝑘Φ)=(𝑘 −1𝜉(Φ))(𝑘𝐸(Φ))=𝑐(Φ) This shows𝑐(Φ) is scale-invariant.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Scaling the state by𝑘yields: 𝑐(𝑘Φ)=𝜉(𝑘Φ)𝐸(𝑘Φ)=(𝑘 −1𝜉(Φ))(𝑘𝐸(Φ))=𝑐(Φ) This shows𝑐(Φ) is scale-invariant

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:49.412975Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e8866f39-f57f-4843-bac7-c406e57de6d0 · outbound

This paper cites an unresolved cited work.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Unresolved cited work

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:49.512243Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 4a421c77-4f3d-451a-b1c5-d56d24f6527b · outbound

This paper cites Thus, 𝑘(1)=𝑘 ′(1).

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Thus, 𝑘(1)=𝑘 ′(1)

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:49.648835Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 5963bf7f-cf8e-44b5-85ac-efa966718d49 · outbound

This paper cites 2.Cauchy-Schwarz Compliance:The magnitude is bounded:|𝐾(𝑖, 𝑗)|≤ √︁ 𝐾(𝑖,𝑖)𝐾(𝑗, 𝑗).

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks 2.Cauchy-Schwarz Compliance:The magnitude is bounded:|𝐾(𝑖, 𝑗)|≤ √︁ 𝐾(𝑖,𝑖)𝐾(𝑗, 𝑗)

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:49.858093Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 2fe1bb1e-4a63-4325-8e7d-5b03cb993c14 · outbound

This paper cites an unresolved cited work.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Unresolved cited work

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:49.986546Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 19ab4fe4-6335-4cb1-bf08-9d59eb6136dc · outbound

This paper cites Note that BN layers are positionedbetweenthe affine transformations (Conv2D/Dense) and the ReLU non-linearities.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Note that BN layers are positionedbetweenthe affine transformations (Conv2D/Dense) and the ReLU non-linearities

Reference 12

Resolution: malformed identifier; no resolver link, observed 2026-08-01T07:45:50.095866Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3a450b40-8657-46d0-995f-8503deadbe82 · outbound

This paper cites Thus,∥𝒘core,𝑗∥2≤∥𝒘 out,𝑗∥2.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Thus,∥𝒘core,𝑗∥2≤∥𝒘 out,𝑗∥2

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:50.165608Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1f112bfe-befb-43da-acff-4e51d90effdb · outbound

This paper cites an unresolved cited work.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Unresolved cited work

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:50.262677Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 67a9604d-3aa5-4c6b-b1e3-e241341be838 · outbound

This paper cites Deep networks often fragment a feature across𝑀 correlated neurons.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Deep networks often fragment a feature across𝑀 correlated neurons

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:50.383710Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 7e5c8435-bfe7-4780-90af-dfa1f006d5c1 · outbound

This paper cites However, because Theorem H.2 guarantees zero interference during training, the network’s total error does not compound exponentially.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks However, because Theorem H.2 guarantees zero interference during training, the network’s total error does not compound exponentially

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:50.513738Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 41ca6d0c-7c9f-4d64-bc38-40b460da1fc7 · outbound

This paper cites Therefore, the functional distortion is strictly bounded by: Δ𝒔init core H(𝑙) ≤ ∑︁ 𝑗∈N(𝑙) slack 𝜏(𝑙) =𝜏(𝑙)|N(𝑙) slack|(112) □ H.2.1.

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks Therefore, the functional distortion is strictly bounded by: Δ𝒔init core H(𝑙) ≤ ∑︁ 𝑗∈N(𝑙) slack 𝜏(𝑙) =𝜏(𝑙)|N(𝑙) slack|(112) □ H.2.1

Reference 17

Resolution: unresolved; no resolver link, observed 2026-08-01T07:45:50.679709Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

Observation 042bcce7-0f05-4bb1-8c1a-a9daafb8fb33 · inbound

When Compression Scores Cannot Decide: Information Boundaries for Group-Robust LLM Pruning cites this paper.

When Compression Scores Cannot Decide: Information Boundaries for Group-Robust LLM Pruning Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks

Reference 13

Resolution: verified exact; local_arxiv, observed 2026-08-07T01:03:31.782750Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-11T06:34:44.6726+00:00.