arxiv: 2604.24238 · v1 · submitted 2026-04-27 · 💻 cs.LG

Recognition: unknown

GeoEdit: Local Frames for Fast, Training-Free On-Manifold Editing in Diffusion Models

Yiming Zhang , Sitong Liu , Ke Li , Zhihong Wu , Alex Cloninger , Melvin Leok

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:24 UTC · model grok-4.3

classification 💻 cs.LG

keywords diffusion modelson-manifold editingtangent space estimationtraining-free editingmanifold approximationimage generationgenerative models

0 comments

The pith

Estimating local tangent spaces from perturbed samples enables fast on-manifold editing in diffusion models without training or full re-synthesis.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a local tangent space to the data manifold can be estimated directly from small perturbations of generated samples, along with a proof that this estimator approximates the true tangent closely enough for practical use. Building on the guarantee, they introduce an algorithm that builds a tangent frame from noise perturbations and alternates small moves along it with diffusion projections to stay on the manifold. If this holds, iterative edits become cheap because only local steps are needed instead of restarting the full denoising trajectory each time. A sympathetic reader cares because current training-free editing remains slow and inflexible for interactive refinement.

Core claim

We estimate a local manifold tangent space directly from perturbed samples and prove that this sample-based estimator closely approximates the true tangent. Building on this guarantee, we devise a Jacobian-free algorithm that constructs a tangent frame via small perturbations to the initial noise and alternates small tangent moves with diffusion-based projections. Updates within this frame follow principled on-manifold directions while suppressing off-manifold drift, enabling fine-grained edits without full re-diffusion or additional training, with edit strength controlled by the number of steps.

What carries the argument

The sample-based estimator of the local manifold tangent space, which builds a tangent frame from small perturbations to the initial noise for alternating tangent steps and diffusion projections.

If this is right

Edit strength is controlled by the number of steps for rapid continuous adjustments that preserve fidelity.
The method produces smooth semantic unsupervised traversals and supports effective CLIP-guided optimization.
It integrates directly into existing samplers without requiring model changes or retraining.
On-manifold directions suppress off-manifold drift during iterative refinement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same local-frame construction might apply to other score-based or flow-based generative models that share manifold structure.
Real-time user interfaces could let people drag along the estimated tangent directions for live editing sessions.
The approximation quality could be tested by measuring how perturbation size affects the accumulated error over many steps.

Load-bearing premise

That small local updates near the data manifold can replace repeated full re-synthesis and that the sample-based tangent estimator closely approximates the true tangent space without significant off-manifold drift.

What would settle it

Repeated tangent steps causing generated samples to accumulate visible off-manifold artifacts or sharply rising diffusion reconstruction loss beyond the paper's approximation bound.

Figures

Figures reproduced from arXiv: 2604.24238 by Alex Cloninger, Ke Li, Melvin Leok, Sitong Liu, Yiming Zhang, Zhihong Wu.

**Figure 1.** Figure 1: The illustration of proposed algorithm. Perturbed samples are first flowed onto the data manifold, yielding an initial tangent estimate. During editing, we alternate small tangent moves with diffusion-based projections, which refine the tangent space adaptively. so J(t) spreads sensitivity over many fine-grained modes, weakening any lowrank semantic structure as exhibited in [2]. To address these issues… view at source ↗

**Figure 2.** Figure 2: Comparison of tangent lines view at source ↗

**Figure 5.** Figure 5: Edits along different tangent bases. The DDPM is pretrained on CelebaA-HQ. (a) Editing along ub1. (b) Editing along ub2. (c) Editing along ub3 view at source ↗

**Figure 8.** Figure 8: Unsupervised edit results with Stable Diffusion model. our estimator returns an orthonormal basis Ubk(x∥) ∈ R n×k and the projector Pbx∥ = UbkUb⊤ k ≈ ProjTx∥M. We visualize unsupervised edits by selecting three basis directions ub1, ub2, ub3 from Ubk and traversing each independently from the same image x view at source ↗

**Figure 10.** Figure 10: Compatibility with edit prompts in Stable Diffusion. The leftmost image is generated without the edit prompt. The following images exhibit tangent–space edits under the prompt “cartoon style.” Our method is directly applicable to latent diffusion models. We instantiate it with Stable Diffusion and report results of GeoEdit in the latent space view at source ↗

**Figure 12.** Figure 12: Ablation study. Both results were produced with CLIP guidance using the identical prompt “an old male.” GeoEdit preserves the on-manifold component and produces edits that are consistent with the given prompt. (a) Effect of edit step size η in one-step editing. (b) Effect of local subspace dimension k view at source ↗

**Figure 14.** Figure 14: CLIP score (smoothed) view at source ↗

read the original abstract

Diffusion models are a leading paradigm for data generation, but training-free editing typically re-runs the full denoising trajectory for every edit strength, making iterative refinement expensive. To address this issue, we instead edit near the data manifold, where small local updates can replace repeated re-synthesis. To enable this, we estimate a local manifold tangent space directly from perturbed samples and prove that this sample-based estimator closely approximates the true tangent. Building on this guarantee, we devise a Jacobian-free algorithm that constructs a tangent frame via small perturbations to the initial noise and alternates small tangent moves with diffusion-based projections. Updates within this frame follow principled on-manifold directions while suppressing off-manifold drift, enabling fine-grained edits without full re-diffusion or additional training. Edit strength is controlled by the number of steps for rapid, continuous adjustments that preserve fidelity and plug into existing samplers. Empirically, the resulting tangent directions yield smooth, semantic unsupervised traversals and effective CLIP-guided optimization, demonstrating practical interactive continuous editing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

GeoEdit's sample-based tangent frames let you edit diffusion outputs locally without full re-runs, but the multi-step error control is not yet tightly bounded.

read the letter

The paper's core move is to estimate a local tangent frame straight from small perturbations around a sample, prove that estimator is close to the true tangent, then use it for Jacobian-free edits that alternate short tangent steps with standard diffusion projections. This replaces the usual full denoising trajectory for each edit strength with cheaper local updates, and edit size is just the number of steps. That is the practical payoff they are after, and the abstract plus the described experiments on unsupervised traversals and CLIP-guided optimization show it can produce smooth semantic changes while staying roughly on-manifold. The approach is training-free and plugs into existing samplers, which is the part that would actually matter to people doing iterative refinement in tools. The single-step approximation claim is stated clearly and the circularity burden is low because the tangent comes from external perturbations rather than from fitted parameters inside the method itself. The experiments appear to support the usability claim at least for modest edits. The soft spot is exactly the one the stress-test flags. The proof covers the single-step estimator, but stronger edits require chaining multiple tangent moves plus projections, and nothing in the description supplies a composition bound or drift analysis showing that local approximation errors plus imperfect projections stay controlled rather than accumulate off-manifold. Without that, the claim that small local updates can safely replace full re-synthesis is only partially supported for non-trivial distances. The paper would be stronger with either a multi-step guarantee or explicit empirical checks of fidelity versus step count against full-trajectory baselines. This is aimed at researchers working on efficient sampling and editing inside diffusion models. Anyone already running iterative edits in generative pipelines would find the method and the reported speed-ups worth looking at. It is coherent enough on its own terms to deserve a serious referee, even though the multi-step analysis needs work.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces GeoEdit, a training-free method for on-manifold editing of diffusion models. It estimates a local tangent space directly from perturbed samples (via small perturbations to initial noise), states a proof that this sample-based estimator approximates the true tangent, and presents a Jacobian-free algorithm that constructs a tangent frame and alternates small tangent-space moves with diffusion-based projections. Edit strength is controlled by the number of steps, enabling continuous semantic traversals and CLIP-guided optimization without repeated full denoising trajectories.

Significance. If the single-step approximation holds and off-manifold drift remains controlled under iteration, the approach would enable substantially faster iterative editing in diffusion pipelines, supporting interactive refinement while preserving fidelity and integrating with existing samplers.

major comments (2)

[Abstract / algorithm description] Abstract and algorithm description: the stated proof applies only to the single-step sample-based tangent estimator; the central claim that small local updates can replace full re-synthesis for non-trivial edit distances requires a composition bound showing that local approximation errors plus imperfect projections do not accumulate off-manifold drift across multiple tangent-move + projection steps. No such bound or error-propagation analysis is supplied.
[Tangent estimation and projection loop] Tangent estimation and projection loop: the guarantee that the estimator 'closely approximates the true tangent' and 'suppresses off-manifold drift' is load-bearing for the multi-step editing procedure, yet the provided description supplies neither explicit error bounds on the estimator nor empirical controls (e.g., manifold-distance metrics) that quantify accumulation as edit strength (step count) increases.

minor comments (2)

[Notation and algorithm pseudocode] Clarify notation for the tangent-frame construction (how perturbations to initial noise are mapped to the local frame basis) and the precise projection operator used after each tangent step.
[Experiments] The empirical section would benefit from explicit reporting of manifold-distance or reconstruction-error metrics across varying step counts to support the claim of controlled drift.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed and constructive feedback. The comments correctly identify that our theoretical analysis focuses on the single-step tangent estimator and that additional justification is needed for the iterated editing procedure. We respond point-by-point below and indicate planned revisions.

read point-by-point responses

Referee: [Abstract / algorithm description] Abstract and algorithm description: the stated proof applies only to the single-step sample-based tangent estimator; the central claim that small local updates can replace full re-synthesis for non-trivial edit distances requires a composition bound showing that local approximation errors plus imperfect projections do not accumulate off-manifold drift across multiple tangent-move + projection steps. No such bound or error-propagation analysis is supplied.

Authors: We agree that the proof in the manuscript establishes the approximation property only for the single-step estimator. For the multi-step case, the algorithm relies on repeated small tangent moves followed by diffusion projections that are intended to return samples to the manifold. While we do not supply a formal composition bound, the design keeps steps small and the projection operator is the same denoising process used in standard sampling. In the revision we will add a dedicated paragraph in the method section discussing the absence of a closed-form error bound and will include new experiments that track a manifold-distance proxy (reconstruction error after projection) as a function of edit-step count. These additions will make the empirical control of drift explicit. revision: partial
Referee: [Tangent estimation and projection loop] Tangent estimation and projection loop: the guarantee that the estimator 'closely approximates the true tangent' and 'suppresses off-manifold drift' is load-bearing for the multi-step editing procedure, yet the provided description supplies neither explicit error bounds on the estimator nor empirical controls (e.g., manifold-distance metrics) that quantify accumulation as edit strength (step count) increases.

Authors: We acknowledge that the current manuscript does not include explicit error bounds beyond the single-step case nor quantitative plots of drift versus step count. The existing experiments demonstrate stable semantic edits and CLIP optimization for moderate step counts, but they do not systematically report manifold-distance accumulation. In the revised version we will add (i) a short theoretical remark clarifying the scope of the existing guarantee and (ii) empirical controls consisting of curves that plot a simple manifold-deviation metric against increasing numbers of tangent-projection steps on the same datasets used in the paper. This will directly address the request for quantification of accumulation. revision: yes

standing simulated objections not resolved

A rigorous composition bound on accumulated approximation error for the iterated tangent-move plus projection process is not derived in the manuscript and would require substantial additional theoretical analysis beyond the current scope.

Circularity Check

0 steps flagged

No circularity: tangent estimator and algorithm derive from independent perturbations and diffusion projections

full rationale

The paper's core derivation estimates a local tangent space from perturbed samples and proves its approximation to the true manifold tangent using standard diffusion properties. The subsequent Jacobian-free editing algorithm alternates small tangent steps with diffusion-based projections. Neither step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the proof and construction rely on external sample perturbations and existing sampler mechanics rather than re-expressing the target result as input. This matches the default expectation of a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unverified approximation guarantee for the tangent estimator and the assumption that manifold-local steps suffice for editing. No explicit free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The sample-based estimator from perturbed samples closely approximates the true manifold tangent space
Stated as proven in the abstract but without the proof details or assumptions listed.

pith-pipeline@v0.9.0 · 5489 in / 1207 out tokens · 54468 ms · 2026-05-08T04:24:22.356363+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 11 canonical work pages · 4 internal anchors

[1]

In: Proceedings of the IEEE/CVF international conference on computer vision

Cao, M., Wang, X., Qi, Z., Shan, Y., Qie, X., Zheng, Y.: Masactrl: Tuning- free mutual self-attention control for consistent image synthesis and editing. In: Proceedings of the IEEE/CVF international conference on computer vision. pp. 22560–22570 (2023)

2023
[2]

Advances in neural information processing systems37, 27340–27371 (2024)

Chen, S., Zhang, H., Guo, M., Lu, Y., Wang, P., Qu, Q.: Exploring low- dimensional subspace in diffusion models for controllable image editing. Advances in neural information processing systems37, 27340–27371 (2024)

2024
[3]

arXiv preprint arXiv:2406.08070(2024)

Chung, H., Kim, J., Park, G.Y., Nam, H., Ye, J.C.: Cfg++: Manifold- constrained classifier free guidance for diffusion models. arXiv preprint arXiv:2406.08070 (2024)

work page arXiv 2024
[4]

Advances in Neural Information Processing Systems35, 25683–25696 (2022)

Chung, H., Sim, B., Ryu, D., Ye, J.C.: Improving diffusion models for in- verse problems using manifold constraints. Advances in Neural Information Processing Systems35, 25683–25696 (2022)

2022
[5]

DiffEdit: Diffusion-based seman- tic image editing with mask guidance

Couairon, G., Verbeek, J., Schwenk, H., Cord, M.: Diffedit: Diffusion- based semantic image editing with mask guidance. arXiv preprint arXiv:2210.11427 (2022)

work page arXiv 2022
[6]

Advances in neural information processing systems34, 8780–8794 (2021)

Dhariwal, P., Nichol, A.: Diffusion models beat gans on image synthesis. Advances in neural information processing systems34, 8780–8794 (2021)

2021
[7]

Journal of the American Mathematical Society29(4), 983–1049 (2016)

Fefferman, C., Mitter, S., Narayanan, H.: Testing the manifold hypothesis. Journal of the American Mathematical Society29(4), 983–1049 (2016)

2016
[8]

Manifold preserv- ing guided diffusion.arXiv preprint arXiv:2311.16424, 2023

He, Y., Murata, N., Lai, C.H., Takida, Y., Uesaka, T., Kim, D., Liao, W.H., Mitsufuji, Y., Kolter, J.Z., Salakhutdinov, R., et al.: Manifold preserving guided diffusion. arXiv preprint arXiv:2311.16424 (2023)

work page arXiv 2023
[9]

Prompt-to-Prompt Image Editing with Cross Attention Control

Hertz, A., Mokady, R., Tenenbaum, J., Aberman, K., Pritch, Y., Cohen-Or, D.: Prompt-to-prompt image editing with cross attention control.(2022). URL https://arxiv. org/abs/2208.016263(2022)

work page internal anchor Pith review arXiv 2022
[10]

In: Advances in Neural Information Processing Systems

Ho, J., Jain, A., Abbeel, P.: Denoising diffusion probabilistic mod- els. In: Advances in Neural Information Processing Systems. vol. 33 (2020),https : / / proceedings . neurips . cc / paper / 2020 / hash / 4c5bcfec8584af0d967f1ab10179ca4b-Abstract.html

2020
[11]

Advances in Neural Information Processing Systems37, 38307–38354 (2024)

Kamkari, H., Ross, B., Hosseinzadeh, R., Cresswell, J., Loaiza-Ganem, G.: A geometric view of data complexity: Efficient local intrinsic dimension es- timation with diffusion models. Advances in Neural Information Processing Systems37, 38307–38354 (2024)

2024
[12]

arXiv preprint arXiv:2302.09301 (2023)

Kvinge, H., Brown, D., Godfrey, C.: Exploring the representation manifolds of stable diffusion through the lens of intrinsic dimension. arXiv preprint arXiv:2302.09301 (2023)

work page arXiv 2023
[13]

In: Proceedings of the Computer Vision and Pattern Recognition Conference

Kwon, M., Jeong, J., Hsiao, Y.T., Uh, Y., et al.: Tcfg: Tangential damping classifier-free guidance. In: Proceedings of the Computer Vision and Pattern Recognition Conference. pp. 2620–2629 (2025)

2025
[14]

arXiv preprint arXiv:2402.13929 (2024) 5

Lin, S., Wang, A., Yang, X.: Sdxl-lightning: Progressive adversarial diffusion distillation. arXiv preprint arXiv:2402.13929 (2024) GeoEdit: Local Frames for Fast, Training-Free On-Manifold Editing 17

work page arXiv 2024
[15]

In: Proceedings of the IEEE international conference on computer vision

Liu, Z., Luo, P., Wang, X., Tang, X.: Deep learning face attributes in the wild. In: Proceedings of the IEEE international conference on computer vision. pp. 3730–3738 (2015)

2015
[16]

arXiv preprint arXiv:2210.12100 (2022)

Luzi, L., Mayer, P.M., Casco-Rodriguez, J., Siahkoohi, A., Baraniuk, R.G.: Boomerang: Local sampling on image manifolds using diffusion models. arXiv preprint arXiv:2210.12100 (2022)

work page arXiv 2022
[17]

SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations

Meng, C., He, Y., Song, Y., Song, J., Wu, J., Zhu, J.Y., Ermon, S.: Sdedit: Guided image synthesis and editing with stochastic differential equations. arXiv preprint arXiv:2108.01073 (2021)

work page internal anchor Pith review arXiv 2021
[18]

Mokady, R., Hertz, A., Aberman, K., Pritch, Y., Cohen-Or, D.: Null-text in- versionforeditingrealimagesusingguideddiffusionmodels.In:Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 6038–6047 (2023)

2023
[19]

Park, Y.H., Kwon, M., Choi, J., Jo, J., Uh, Y.: Understanding the latent spaceofdiffusionmodelsthroughthelensofriemanniangeometry.Advances in Neural Information Processing Systems36, 24129–24142 (2023)

2023
[20]

In: ACM SIGGRAPH 2023 conference proceed- ings

Parmar, G., Kumar Singh, K., Zhang, R., Li, Y., Lu, J., Zhu, J.Y.: Zero-shot image-to-image translation. In: ACM SIGGRAPH 2023 conference proceed- ings. pp. 1–11 (2023)

2023
[21]

SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis

Podell, D., English, Z., Lacey, K., Blattmann, A., Dockhorn, T., Müller, J., Penna, J., Rombach, R.: Sdxl: Improving latent diffusion models for high-resolution image synthesis. arXiv preprint arXiv:2307.01952 (2023)

work page internal anchor Pith review arXiv 2023
[22]

In: International conference on machine learning

Radford, A., Kim, J.W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sas- try, G., Askell, A., Mishkin, P., Clark, J., et al.: Learning transferable visual models from natural language supervision. In: International conference on machine learning. pp. 8748–8763. PmLR (2021)

2021
[23]

In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition

Rombach, R., Blattmann, A., Lorenz, D., Esser, P., Ommer, B.: High- resolution image synthesis with latent diffusion models. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 10684–10695 (2022)

2022
[24]

In: Proceedings of the Computer Vision and Pattern Recognition Conference

Rotstein, N., Yona, G., Silver, D., Velich, R., Bensaid, D., Kimmel, R.: Pathways on the image manifold: Image editing via video generation. In: Proceedings of the Computer Vision and Pattern Recognition Conference. pp. 7857–7866 (2025)

2025
[25]

In: International Conference on Learning Representations (2021),https: //openreview.net/forum?id=PxTIG12RRHS

Song,Y.,Sohl-Dickstein,J.,Kingma,D.P.,Kumar,A.,Ermon,S.,Poole,B.: Score-based generative modeling through stochastic differential equations. In: International Conference on Learning Representations (2021),https: //openreview.net/forum?id=PxTIG12RRHS

2021
[26]

In: Forty-first Interna- tional Conference on Machine Learning (2024)

Stanczuk, J.P., Batzolis, G., Deveney, T., Schönlieb, C.B.: Diffusion models encode the intrinsic dimension of data manifolds. In: Forty-first Interna- tional Conference on Machine Learning (2024)

2024
[27]

In: 2024 IEEE Interna- tional Conference on Multimedia and Expo (ICME)

Su, X., Jia, D., Wu, F., Zhao, J., Zheng, C., Qiang, W.: Unbiased image synthesis via manifold guidance in diffusion models. In: 2024 IEEE Interna- tional Conference on Multimedia and Expo (ICME). pp. 1–6. IEEE (2024)

2024
[28]

In: 18 Y

Sun, S., Wei, L., Xing, J., Jia, J., Tian, Q.: Sddm: score-decomposed dif- fusion models on manifolds for unpaired image-to-image translation. In: 18 Y. Zhang et al. International Conference on Machine Learning. pp. 33115–33134. PMLR (2023)

2023
[29]

In: International Conference on Artificial Intelligence and Statistics

Tang, R., Yang, Y.: Adaptivity of diffusion models to manifold structures. In: International Conference on Artificial Intelligence and Statistics. pp. 1648–1656. PMLR (2024)

2024
[30]

In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition

Tumanyan, N., Geyer, M., Bagon, S., Dekel, T.: Plug-and-play diffusion features for text-driven image-to-image translation. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. pp. 1921–1930 (2023)

1921
[31]

arXiv preprint arXiv:2311.06792 (2023)

Yang, Z., Yu, Z., Xu, Z., Singh, J., Zhang, J., Campbell, D., Tu, P., Hart- ley, R.: Impus: Image morphing with perceptually-uniform sampling using diffusion models. arXiv preprint arXiv:2311.06792 (2023)

work page arXiv 2023
[32]

LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop

Yu, F., Seff, A., Zhang, Y., Song, S., Funkhouser, T., Xiao, J.: Lsun: Con- struction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365 (2015)

work page internal anchor Pith review arXiv 2015
[33]

In: Proceedings of the IEEE/CVF international confer- ence on computer vision

Zhai, X., Mustafa, B., Kolesnikov, A., Beyer, L.: Sigmoid loss for language image pre-training. In: Proceedings of the IEEE/CVF international confer- ence on computer vision. pp. 11975–11986 (2023)

2023
[34]

In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition

Zhang, K., Zhou, Y., Xu, X., Dai, B., Pan, X.: Diffmorpher: Unleashing the capability of diffusion models for image morphing. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 7912–7921 (2024) GeoEdit: Local Frames for Fast, Training-Free On-Manifold Editing 19 7 Appendix 7.1 Related Works Manifold-aware Generative...

2024
[35]

, Rk] withR i := σ2 2 D2Φ(z)[ξi, ξi]

Assume in addition that∥P⊥ T DΦ(z)∥ 2 ≤L ⊥ ρ, and the normal second–order deviation is curvature–controlled:∥P⊥ T R∥2 ≤C curvκ σ2∥Ξ∥ 2 2, whereR:= [R 1, . . . , Rk] withR i := σ2 2 D2Φ(z)[ξi, ξi]. Then, for sufficiently smallσ, ρ, ∥P ⊥ T PSk ∥2 ≤ L⊥ ρ∥Ξ∥ 2 smin + Ccurv κ σ∥Ξ∥ 2 2 smin + C3 σ2 ∥Ξ∥ 3 2 smin . In particular, the first two terms vanish asρ, κ...