bibkey: eftekhari2018embedology authors: Armin Eftekhari, Han Lun Yap, Michael B. Wakin, Christopher J. Rozell year: 2018 title: ‘Stabilizing Embedology: Geometry-Preserving Delay-Coordinate Maps’ doi: 10.1103/PhysRevE.97.022222 url: https://arxiv.org/abs/1609.06347v2 claim: Stable delay embeddings require quantitative geometry beyond injectivity; the conclusions pose a measurement-design question about increasing stable rank. strata_touched:

• D5/S3/ConceptDynamics/ObservationTopology/PartitionTopologyKernel license: citation-only triage: anchor

Stable observation and the measurement-design question

Eftekhari, Yap, Wakin, Rozell, Physical Review E 97, 022222 (2018). The DOI and author list agree with the publisher’s record.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevE.97.022222
• Author manuscript: https://arxiv.org/abs/1609.06347v2

Exact source locators

The publisher accepted manuscript, Section III.A, assumptions A1–A3 and Equations (14)–(16), defines a finite basis family H and the stable rank of trajectory difference matrices. A scalar sensor is subsequently selected as h_alpha = alpha^T H. The stable-rank quantity in Equation (16) depends on the basis family H, not on alpha within a fixed family.

Section V, the second open-problem bullet (accepted manuscript printed page 34), asks whether optimizing measurement functions can increase stable rank. The authors express a negative expectation in that paragraph. Their main stable-embedding theorem additionally controls geometric quantities such as bi-Lipschitz constants and reach. Increasing stable rank alone does not certify all of those conditions.

Current repository interface

Section 12 of SYMPLECTIC_PREDICTIVE_COMPLETION.md studies a circle rotation by pi/6 with two delays and designs the whole basis family. It already contains a four-dimensional comparison, a general separation-margin obstruction and a ten-dimensional approaching-extremal construction.

Sections 52–54 of COMPUTATIONAL_BEHAVIOR_REPRESENTATION_THEORY.md study the restricted class of paired sine/cosine harmonics. They prove an upper bound 4/3 for injective families with at most two harmonic pairs, a six-dimensional three-pair family approaching stable rank 2, and exact chord/noise tradeoffs. This is a restricted design result, not a solution for arbitrary attractors, arbitrary smooth sensors, or the original random scalar embedding theorem. The six-dimensional construction trades fewer basis functions for a worse separation-margin exponent than the earlier ten-dimensional construction.

The Scribe connection is a literature scope note on the existing observation kernel document. No new formal theorem is claimed, and the existing Lean result establishes none of these quantitative stable-rank claims.

Source scope and limitations

The source basis is arXiv:1609.06347v2 and the publisher accepted manuscript. The accepted manuscript explicitly identifies the basis-family/scalar-sensor distinction. The source comparison does not establish an exhaustive priority determination. This note contains an original summary and precise scope only; no full paper is reproduced.