Paper Citation Record · LEDGER

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks

As of 8 August 2026, this Paper Citation Record lists 19 of 19 outbound references and 0 inbound Pith citation observations for arXiv:2607.08561.

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

pith.paper-citation-record.v1
2607.08561 v2

Coverage vector

measured 19 of 19 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-03T02:04:47.731641Z

measured 19 of 19 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-08T06:32:00.761636+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

19 of 19 outbound references displayed

verified exact0
verified fuzzy0
unresolved18
parse uncertain0
malformed identifier1
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 98d33e68-a930-481a-99e9-55080ef8e74e · outbound

This paper cites an unresolved cited work.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Unresolved cited work

Reference 1

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:46.653230Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 001f37ac-ae4d-473a-9e30-d469fccefca9 · outbound

This paper cites an unresolved cited work.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Unresolved cited work

Reference 2

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:46.693690Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation ab4a5a82-f1cd-49c1-8ac5-f046bfdf9a29 · outbound

This paper cites Layers at Similar Depths Generate Similar Activations Across LLM Architectures.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Layers at Similar Depths Generate Similar Activations Across LLM Architectures

Reference 3

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:46.623828Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 85d57070-008f-4ab7-8d5f-89563025280f · outbound

This paper cites Sincegj is affine, the residual−vigj can be absorbed into the affine parta.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Sincegj is affine, the residual−vigj can be absorbed into the affine parta

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:46.962107Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c813549c-8323-454b-a6bd-418d339a9142 · outbound

This paper cites generically.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks generically

Reference 5

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.024086Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 33264d7f-68c7-4512-a46a-e8c4f8b704af · outbound

This paper cites Ifv i +vj = 0, both nonlinear terms disappear.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Ifv i +vj = 0, both nonlinear terms disappear

Reference 6

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:46.798652Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 96d4b96d-6ab0-4884-917e-cba2ec925513 · outbound

This paper cites Minimality is an intrinsic condition on each expansion: every counted unit leaves a “private” task-visible nonlinear signature.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Minimality is an intrinsic condition on each expansion: every counted unit leaves a “private” task-visible nonlinear signature

Reference 9

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.090574Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation a32204df-69c2-4aab-a94d-1d04e91c1504 · outbound

This paper cites an unresolved cited work.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Unresolved cited work

Reference 10

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.149974Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 25cbb2b0-f51f-499d-9dc6-340d3057b7b2 · outbound

This paper cites obvious self-collisions.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks obvious self-collisions

Reference 11

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.194748Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3d67689a-2647-441f-92ac-99fbb0fbf1c0 · outbound

This paper cites an unresolved cited work.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Unresolved cited work

Reference 12

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.262156Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 26f30ee1-5163-43da-84e7-85fbdc6a7282 · outbound

This paper cites an unresolved cited work.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Unresolved cited work

Reference 13

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.322622Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 1b34fa9a-791d-445c-9f71-6a1c3be4c068 · outbound

This paper cites verification tool.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks verification tool

Reference 14

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.332423Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation b4e5db6b-9be2-4812-88d6-a5d72b13db54 · outbound

This paper cites an unresolved cited work.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Unresolved cited work

Reference 15

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.438141Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation d8e05da4-a820-47fd-844a-f0d1b96788c7 · outbound

This paper cites an unresolved cited work.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Unresolved cited work

Reference 16

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.545084Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 6fa037e7-071d-4dcf-a8a5-20c061a3da7c · outbound

This paper cites all ends well.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks all ends well

Reference 17

Resolution: malformed identifier; no resolver link, observed 2026-08-03T02:04:47.607758Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 34b35ab1-9f63-416b-93d2-d52893fe3e42 · outbound

This paper cites Combining with the previous bound yields ∥P[u]−P[v]∥ F≤ 2 √ 2δ√ 1−δ 2 = 2 √ 2(1−θ)√ 1−(1−θ) 2.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Combining with the previous bound yields ∥P[u]−P[v]∥ F≤ 2 √ 2δ√ 1−δ 2 = 2 √ 2(1−θ)√ 1−(1−θ) 2

Reference 18

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.671628Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 3cd16482-d81b-4ec4-8de8-2dba57c6cae5 · outbound

This paper cites Orthogonal projection is nonexpansive, so the same bound holds after applyingΠ H.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Orthogonal projection is nonexpansive, so the same bound holds after applyingΠ H

Reference 19

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:47.731641Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation e6d9657a-09ee-4826-b650-5414529094f8 · outbound

This paper cites Amil Merchant, Elahe Rahimtoroghi, Ellie Pavlick, and Ian Tenney.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks Amil Merchant, Elahe Rahimtoroghi, Ellie Pavlick, and Ian Tenney

Reference 2020

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:46.609362Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 73e0fb4d-5ed9-4b5d-943a-3624321b383b · outbound

This paper cites 21 Ansh Soni, Sudhanshu Srivastava, Marvin Maechler, Konrad Kording, and Meenakshi Khosla.

Contravariance Theory: Strong Alignment for Minimal Solutions to Hard Tasks 21 Ansh Soni, Sudhanshu Srivastava, Marvin Maechler, Konrad Kording, and Meenakshi Khosla

Reference 2024

Resolution: unresolved; no resolver link, observed 2026-08-03T02:04:46.595375Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Pith citing papers

No inbound Pith citation observations are available.