Paper Citation Record · LEDGER

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason

As of 19 August 2026, this Paper Citation Record lists 8 of 8 outbound references and 1 inbound Pith citation observation for arXiv:2604.15350.

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

pith.paper-citation-record.v1
2604.15350 v1

Coverage vector

measured 8 of 8 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-05-13T20:48:51.783586Z

measured 9 of 9 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-18T06:34:40.430872+00:00

measured 1 of 1 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links, observed 2026-07-13T05:02:55.608400Z

measured 0 of 1 external citation measurements

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

Source: cited_works

Reference resolution

8 of 8 outbound references displayed

verified exact0
verified fuzzy5
unresolved3
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 2a33b218-8d16-4b9e-aee6-3c8ef38b7a68 · outbound

This paper cites an unresolved cited work.

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason Unresolved cited work

Reference 1

Resolution: unresolved; raw_fallback, observed 2026-05-14T01:03:31.705211Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation a6a2ca4e-1d06-40a3-a6af-7cd7c118656a · outbound

This paper cites an unresolved cited work.

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason Unresolved cited work

Reference 2

Resolution: unresolved; raw_fallback, observed 2026-05-14T01:03:31.713165Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 998a3b28-84c0-4bb8-ad88-a1d19059b300 · outbound

This paper cites an unresolved cited work.

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason Unresolved cited work

Reference 3

Resolution: unresolved; raw_fallback, observed 2026-05-14T01:03:31.707797Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation c975f2a4-9fd7-4ee3-9987-d9caa7ffe614 · outbound

This paper cites For token-level dynamics (Finding 5–6), we use a sliding window of w= 10 tokens, computing SVD on H(ℓ) window ∈ Rw×d at each generation step.

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason For token-level dynamics (Finding 5–6), we use a sliding window of w= 10 tokens, computing SVD on H(ℓ) window ∈ Rw×d at each generation step

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-05-14T01:03:31.715664Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 022d5260-e205-4261-a321-23009d896462 · outbound

This paper cites The spectral trajectory shows clear α gradient spikes at calculation boundaries (e.g., after “=”), with Layer 9 showing the most pronounced punctuation effect.

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason The spectral trajectory shows clear α gradient spikes at calculation boundaries (e.g., after “=”), with Layer 9 showing the most pronounced punctuation effect

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-05-14T01:03:31.710809Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 12ce60fa-2fb1-4796-a431-4f7ad663af2a · outbound

This paper cites Similar punctuation pattern to Task 0, with additional spikes at variable introduction (“Letx=”).

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason Similar punctuation pattern to Task 0, with additional spikes at variable introduction (“Letx=”)

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-05-14T01:03:31.702574Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation 25f08585-038b-400c-be9b-17b4181a0ede · outbound

This paper cites therefore.

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason therefore

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-05-14T01:03:31.717859Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Observation db7519c9-2ad5-4536-bf9c-c1deb8718727 · outbound

This paper cites Step 1:”, “therefore.

The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason Step 1:”, “therefore

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-05-14T01:03:31.720027Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.

Pith citing papers

Observation cae8744d-8b3d-488b-ad1f-b8cd9e7e5eca · inbound

KV-PRM: Efficient Process Reward Modeling via KV-Cache Transfer for Multi-Agent Test-Time Scaling cites this paper.

KV-PRM: Efficient Process Reward Modeling via KV-Cache Transfer for Multi-Agent Test-Time Scaling The Spectral Geometry of Thought: Phase Transitions, Instruction Reversal, Token-Level Dynamics, and Perfect Correctness Prediction in How Transformers Reason

Reference 18

Resolution: unresolved; no resolver link, observed 2026-07-13T05:02:55.608400Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.