图像保真度不等于场保真度：神经网络层析成像中的联合热力学重建与错误定位

# Image Fidelity is Not Field Fidelity: Joint Thermodynamic Reconstruction and Error Localization in Neural Tomography
Source: [https://arxiv.org/html/2609.28868](https://arxiv.org/html/2609.28868)
Alan HsuJenna SamraAffiliation:Center for Astrophysics \| Harvard & SmithsonianAffiliation:Cambridge, MA, USAEmail:[jsamra@cfa\.harvard\.edu](mailto:)Alin Razvan ParaschivAffiliation:National Solar ObservatoryAffiliation:Boulder, CO, USAEmail:[arparaschiv@nso\.edu](mailto:)Liam ConnorAffiliation:Harvard UniversityAffiliation:Center for Astrophysics \| Harvard & SmithsonianAffiliation:Cambridge, MA, USAEmail:[liam\.connor@cfa\.harvard\.edu](mailto:)

###### Abstract

Neural fields for scientific tomography are optimized from 2D images, but the actual quantity of interest is often a latent 3D physical field\. Because the forward map is many\-to\-one, low 2D image error need not certify a correct 3D field\. Moreover, the latent field is not directly supervised during training, and its error cannot be evaluated against truth at deployment\. We develop CoroNeRF to jointly optimize 3D electron density and temperature fields directly from multiview, multiline intensities through a differentiable atomic\-emission renderer\. Using solar coronal tomography as a controlled testbed, we evaluate physical\-field recovery and test whether cross\-seed instability provides a ground\-truth\-free\-at\-inference indicator of local physical\-field error\. We underscore the following two observations\. \(i\) Image fidelity is not field fidelity: spectral ablations show that limited\-channel reconstructions can fit their available observations well while recovering substantially worse fields, whereas evaluation on a common richer probe exposes the discrepancy\. \(ii\) Cross\-seed instability ranks local physical\-field error across tested matched\-model conditions, supported by sparsification and physical signal\-strength controls\. Seed\-deviation projections provide complementary directional validation, but shared forward\-model mismatch can still produce incorrect cross\-seed consensus\. These results characterize joint thermodynamic recovery and the usefulness and limits of seed\-based error localization in a controlled, single\-scene solar tomography testbed\.

## 1Introduction

Scientific tomographic imaging problems aim to recover latent 3D physical fields where the data available for supervised learning are indirect, multiview 2D measurements, such as projections, line\-of\-sight \(LOS\) integrals, or other nonlinear measurements\. Neural fields\([Mildenhall et al\., 2020](https://arxiv.org/html/2609.28868#bib.bib5)\)and NeRF\-style frameworks\([Gao et al\., 2026](https://arxiv.org/html/2609.28868#bib.bib6)\)have been developed primarily for 3D scene representation and are a promising paradigm for scientific tomography\. These frameworks represent a spatial field as a continuous coordinate\-conditioned function and optimize it through a differentiable forward model\. Under these frameworks, neural rendering problems are often evaluated as novel view synthesis problems, where success is determined by how well the model reproduces the training images and novel views\. However, in scientific inverse problems, notably tomographic reconstruction, such a criterion is incomplete: a model can achieve low 2D image\-space error while recovering an incorrect latent 3D physical field, especially when the forward map is non\-injective and ill\-conditioned, allowing substantially different fields to produce similar measurements\. Moreover, in standard novel view synthesis pipelines, error in a physically interpreted latent field is typically not part of the task definition, let alone the training objective\.Thus, the latent field is not directly supervised during training and its error is unverifiable against truth during deployment,leaving image agreement as a common but unreliable validation signal\.

We investigate this limitation through joint reconstruction of the solar corona’s 3D electron density and temperature\. The solar corona is the outermost layer of the Sun, composed of hot plasma that is optically thin across a large wavelength range\. Multiline 2D intensity observations are LOS integrals of 3D plasma emissivities \(analogous to the RGBσ\\sigmaradiance proxies in standard NeRFs\), and these emissivities are functions of the latent 3D plasma density and temperature fields\. The goal is therefore not merely to reconstruct each line’s emitting structure, but to recover a shared density\-temperature state that explains all the observed channels\. This inverse problem is highly ill\-posed as distinct 3D density\-temperature fields can produce nearly indistinguishable measurements\. There is a standard geometric LOS degeneracy: since each pixel is an integral along a ray, a redistribution of the emissivities along that ray can still produce the same image\. In addition, without sufficient spectral coverage, there is a plasma thermodynamic degeneracy: many density\-temperature realizations can produce similar multiline emissivity values because the line\-response functions are not jointly injective\.

We thus introduce CoroNeRF, a differentiable multiline neural tomography framework for the direct joint recovery of 3D electron density and temperature from multiview spectral line\-intensity images\. Instead of learning the RGBσ\\sigmaradiance proxy of the scene, CoroNeRF parameterizes a neural plasma field that maps 3D position𝐱\\mathbf\{x\}to electron densitylog10⁡ne​\(𝐱\)\\log\_\{10\}n\_\{\\mathrm\{e\}\}\(\\mathbf\{x\}\)and temperaturelog10⁡Te​\(𝐱\)\\log\_\{10\}T\_\{\\mathrm\{e\}\}\(\\mathbf\{x\}\)\. A differentiable line\-emission renderer then maps these fields to multichannel spectral line intensities using precomputed emissivity tables on samples along each camera ray\. All channels are synthesized from the same density and temperature fields, which are optimized jointly using only image supervision\. Within this framework, we examine how observational information affects thermodynamic recovery and whether cross\-seed instability can localize the physical\-field error without ground truth at inference\.

Our contributions are as follows:

1. 1\.Direct joint thermodynamic neural tomography\.We develop CoroNeRF to optimize 3D electron density and temperature directly from multiview, forbidden\-line intensities through a differentiable atomic\-emission renderer\. At each spatial location, all modeled lines share the same inferred electron density and temperature, coupling the channel reconstructions through their atomic\-emission responses\.
2. 2\.Image fidelity is not field fidelity\.In a controlled synthetic scene, spectral ablations show that limited\-channel reconstructions can fit their available observations well while recovering substantially worse physical fields\. A common richer probe exposes these discrepancies, while noise\-view experiments characterize reconstruction performance under reduced observational information\.
3. 3\.Empirical field\-error localization and its limits\.Cross\-seed instability ranks local physical\-field error across tested matched\-model conditions, with emissivity, radial, sparsification, and seed\-stability controls\. Seed\-deviation projections provide complementary directional validation, while a specified abundance mismatch demonstrates that shared model error can produce incorrect consensus\.

## 2Related Work

In the past two decades, various tomography methods have been used for coronal reconstruction, such as recovering coronal density from LASCO\-C2 white\-light images via classical regularized\-least\-squares\([Frazin and Janzen, 2002](https://arxiv.org/html/2609.28868#bib.bib33)\)and 3D coronal density and temperature through EUV tomography\([Frazin et al\., 2009](https://arxiv.org/html/2609.28868#bib.bib34)\)\. Vector coronal tomographic inversion theory has also been developed for magnetic field inversion from spectropolarimetric observations, validated on both simulations\([Kramar et al\., 2006](https://arxiv.org/html/2609.28868#bib.bib35);[Kramar and Inhester, 2007](https://arxiv.org/html/2609.28868#bib.bib36);[Kramar et al\., 2013](https://arxiv.org/html/2609.28868#bib.bib37);[Kramar and Lin, 2026](https://arxiv.org/html/2609.28868#bib.bib39)\)and real observations\([Kramar et al\., 2016](https://arxiv.org/html/2609.28868#bib.bib38)\)\. These approaches use classical inverse\-problem formulations with explicit regularization or prior structure, rather than learned neural representations coupled with an end\-to\-end differentiable nonlinear emission model\.

Implicit neural representations and related learned inverse models have increasingly been used for scientific tomography\. In medical imaging\([Wang et al\., 2024](https://arxiv.org/html/2609.28868#bib.bib19);[Kabika et al\., 2026](https://arxiv.org/html/2609.28868#bib.bib21)\), uncertainty\-aware null\-space networks have been studied for MRI\([Angermann et al\., 2023](https://arxiv.org/html/2609.28868#bib.bib14)\)\. Neural reconstruction has also been used for heterogeneous macromolecular structures\([Zhong et al\., 2021](https://arxiv.org/html/2609.28868#bib.bib18)\)\. Related neural\-field approaches have been studied for physical flow field tomography\([Molnar and Grauer, 2022](https://arxiv.org/html/2609.28868#bib.bib15)\), seismic full\-waveform inversion\([Sun et al\., 2023](https://arxiv.org/html/2609.28868#bib.bib22)\), black hole dynamic tomography\([Levis et al\., 2022](https://arxiv.org/html/2609.28868#bib.bib24);[Feng et al\., 2026](https://arxiv.org/html/2609.28868#bib.bib25)\), and thermal tomography in inverse heat conduction problems\([Zhong et al\., 2026](https://arxiv.org/html/2609.28868#bib.bib23)\)\. NeRFs have also been used for solar coronal reconstruction:[Asensio Ramos \(2023\)](https://arxiv.org/html/2609.28868#bib.bib26)reconstructs density from polarized\-brightness observations, SuNeRF\([Jarolim et al\., 2024](https://arxiv.org/html/2609.28868#bib.bib27)\)models wavelength\-specific EUV emission and absorption, while SuNeRF\-CME\([Jarolim et al\., 2026](https://arxiv.org/html/2609.28868#bib.bib28)\)reconstructs time\-dependent electron density through Thomson\-scattering observations\. Related efforts have reported density\-temperature reconstruction from multithermal EUV imaging\([Southwest Research Institute, 2024](https://arxiv.org/html/2609.28868#bib.bib30);[Jarolim et al\., 2025](https://arxiv.org/html/2609.28868#bib.bib29)\)\. To our knowledge, CoroNeRF is the first neural\-field framework to directly and jointly reconstruct 3D coronal electron density and temperature from multiview forbidden infrared\-line intensities through a differentiable atomic\-emission model\.

Methods for uncertainty quantification in NeRFs have also been developed over the past five years\. Density\-aware NeRF ensembles employ deep ensembles to identify regions unobserved during training\([Sünderhauf et al\., 2023](https://arxiv.org/html/2609.28868#bib.bib11)\); full probabilistic frameworks such as Stochastic NeRFs\([Shen et al\., 2021](https://arxiv.org/html/2609.28868#bib.bib16)\)and Conditional\-Flow NeRFs\([Shen et al\., 2022](https://arxiv.org/html/2609.28868#bib.bib17)\)quantify uncertainty by learning the distribution over NeRFs\. Prior post\-hoc methods also study uncertainty in conventional radiance fields: Bayes’ Rays\([Goli et al\., 2024](https://arxiv.org/html/2609.28868#bib.bib12)\)evaluates geometric depth uncertainty, while FisherRF\([Jiang et al\., 2024](https://arxiv.org/html/2609.28868#bib.bib13)\)uses Fisher information\. Finally, concurrent work on sparse\-view CT shows that strong global uncertainty\-error association can deteriorate substantially within the reconstructed object, and that error shared across independently trained members can remain invisible to ensemble spread\([Zhao et al\., 2026](https://arxiv.org/html/2609.28868#bib.bib20)\)\. We study a complementary setting involving coupled density\-temperature recovery through nonlinear multiline atomic\-emission responses, characterizing when cross\-seed instability localizes physical\-field error and when shared forward\-model mismatch causes it to fail\.

## 3Methodology: Differentiable Multiline Neural Tomography

CoroNeRF parameterizes a neural plasma fieldfθ​\(𝐱\)↦\(log10⁡n^e,θ​\(𝐱\),log10⁡T^e,θ​\(𝐱\)\)f\_\{\\theta\}\(\\mathbf\{x\}\)\\mapsto\(\\log\_\{10\}\\hat\{n\}\_\{\\mathrm\{e\},\\theta\}\(\\mathbf\{x\}\),\\log\_\{10\}\\hat\{T\}\_\{\\mathrm\{e\},\\theta\}\(\\mathbf\{x\}\)\)\. Rather than first inverting observed line ratios, it jointly fits the absolute intensities of all lines through their emissivity responses\. Density and temperature diagnostic information therefore enters implicitly through the forward model, while multiview geometry constrains the 3D distribution\.

The observations are multiview, multichannel line\-intensity images\. Letv∈𝒱v\\in\\mathcal\{V\}index viewpoints,p∈Ωvp\\in\\Omega\_\{v\}index valid image pixels under the view mask, andc∈\{1,…,C\}c\\in\\\{1,\\ldots,C\\\}index the spectral line channels\. We writeq=\(v,p\)q=\(v,p\)for the ray associated with pixelppin viewvv\. Each ray is then parameterized as𝐱q​\(t\)=𝐨q\+t​𝐝q,t∈\[ts​\(q\),tf​\(q\)\]\\mathbf\{x\}\_\{q\}\(t\)=\\mathbf\{o\}\_\{q\}\+t\\mathbf\{d\}\_\{q\},\\;t\\in\[t\_\{s\}\(q\),t\_\{f\}\(q\)\], where𝐨q\\mathbf\{o\}\_\{q\}is the ray origin and𝐝q\\mathbf\{d\}\_\{q\}is the unit direction\. The observed multichannel intensity vector for this ray is𝐲q=\(yq,1,…,yq,C\)∈ℝC\\mathbf\{y\}\_\{q\}=\(y\_\{q,1\},\\ldots,y\_\{q,C\}\)\\in\\mathbb\{R\}^\{C\}\. The training ray dataset is then given by𝒟ray=\{\(𝐱q\(⋅\),𝐲q\):q=\(v,p\),v∈𝒱train,p∈Ωv\}\\mathcal\{D\}\_\{\\mathrm\{ray\}\}=\\left\\\{\\left\(\\mathbf\{x\}\_\{q\}\(\\cdot\),\\mathbf\{y\}\_\{q\}\\right\):q=\(v,p\),\\;v\\in\\mathcal\{V\}\_\{\\mathrm\{train\}\},\\;p\\in\\Omega\_\{v\}\\right\\\}\.

Given a neural plasma field, the differentiable line\-emission renderer maps physical variables to predicted line intensities\. For channelcc, the renderer first evaluates a channel\-dependent emissivityϵc​\(ne,Te,r\)\\epsilon\_\{c\}\(n\_\{\\mathrm\{e\}\},T\_\{\\mathrm\{e\}\},r\), whererris the heliocentric radius, using a precomputed atomic\-physics lookup table from the CHIANTI database \([Dere, K\. P\. et al\. \(1997\)](https://arxiv.org/html/2609.28868#bib.bib1);[Dere et al\. \(2023\)](https://arxiv.org/html/2609.28868#bib.bib2), Appendix[A\.1](https://arxiv.org/html/2609.28868#A1.SS1)\)\. It then integrates the emissivities along each ray to produce the predicted line intensity:

I^θ,q,c=∫ts​\(q\)tf​\(q\)ϵc​\(n^e,θ​\(𝐱q​\(t\)\),T^e,θ​\(𝐱q​\(t\)\),r⁡\(𝐱q​\(t\)\)\)​𝑑t\.\\hat\{I\}\_\{\\theta,q,c\}=\\int\_\{t\_\{s\}\(q\)\}^\{t\_\{f\}\(q\)\}\\epsilon\_\{c\}\\\!\\left\(\\hat\{n\}\_\{\\mathrm\{e\},\\theta\}\(\\mathbf\{x\}\_\{q\}\(t\)\),\\hat\{T\}\_\{\\mathrm\{e\},\\theta\}\(\\mathbf\{x\}\_\{q\}\(t\)\),r\(\\mathbf\{x\}\_\{q\}\(t\)\)\\right\)\\,dt\.\(1\)
In practice, we compute LOS integrals using numerical quadrature at the fixed sampling resolution described in Appendix[A\.2](https://arxiv.org/html/2609.28868#A1.SS2)\. The inverse problem is then latent physical\-field recovery ofne​\(𝐱\)n\_\{\\mathrm\{e\}\}\(\\mathbf\{x\}\)andTe​\(𝐱\)T\_\{\\mathrm\{e\}\}\(\\mathbf\{x\}\): we optimize the neural plasma field parametersθ\\thetausing image supervision only:

θ⋆=arg⁡minθ​1\|𝒟ray\|​∑\(xq​\(⋅\),yq\)∈𝒟ray1C​∑c=1Cρ⁡\(I^θ,q,c,yq,c\)\.\\theta^\{\\star\}=\\arg\\min\_\{\\theta\}\\frac\{1\}\{\|\\mathcal\{D\}\_\{\\mathrm\{ray\}\}\|\}\\sum\_\{\(x\_\{q\}\(\\cdot\),y\_\{q\}\)\\in\\mathcal\{D\}\_\{\\mathrm\{ray\}\}\}\\frac\{1\}\{C\}\\sum\_\{c=1\}^\{C\}\\rho\(\\hat\{I\}\_\{\\theta,q,c\},y\_\{q,c\}\)\.\(2\)
For most experiments, we use a fixed\-scale asinh image lossρ⁡\(I^,y\)=\|asinh⁡\(I^/sc\)−asinh⁡\(y/sc\)\|\\rho\(\\hat\{I\},y\)=\|\\operatorname\{asinh\}\(\\hat\{I\}/s\_\{c\}\)\-\\operatorname\{asinh\}\(y/s\_\{c\}\)\|, wherescs\_\{c\}is a channel\-dependent fixed characteristic intensity scale, chosen during development so as to not allow bright channels to dominate\. While our images have heteroscedastic Gaussian noise, image\-space fidelity is not the ultimate objective: we thus use the asinh\-L1 image loss to compress the intensity dynamic range\. Among the tested objectives, it yields lower field error than the inverse\-variance\-weighted squared\-residual objective \(Appendix[D\.3](https://arxiv.org/html/2609.28868#A4.SS3)\)\.

## 4Experimental Setup and Evaluation

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/fig_channel_strip.png)
![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/coronerf_diagram.png)

Figure 1:CoroNeRF overview\. The left2×22\\times 2grid displays multiline coronal observations of four spectral channels \(Fe XIII1075/10801075/1080nm, Si IX2585/39352585/3935nm\) forward\-rendered from a PSI cube\. The observations are used to fit a multiresolution hash\-grid neural plasma field through the same differentiable renderer by minimizing an image loss \(right panel\)\. The fitted neural field yields jointly recovered 3D density and temperature fields\.We construct synthetic observations \(left panel of Figure[1](https://arxiv.org/html/2609.28868#S4.F1)\) of four spectral channels \(Fe XIII1075/10801075/1080nm, Si IX2585/39352585/3935nm\) from one ground truth PSI MAS \(MHD\) coronal cube\([Mikić et al\., 2007](https://arxiv.org/html/2609.28868#bib.bib32);[Lionello et al\., 2008](https://arxiv.org/html/2609.28868#bib.bib31)\)based on Carrington Rotation 2283 \(April\-May 2024\)\. The GT field has \(299299,142142,154154\) longitude\-colatitude\-radius voxels spanning the full sphere and ranging from\[1,30\]\[1,30\]R⊙\. The master dataset consists of 3000 views evenly spaced in longitude at0∘0^\{\\circ\}latitude: we mimic observational coverage expected from real Earth measurements, but the viewing geometry is a static\-rotational tomography idealization\. Each view is 256 by 256 pixels with a FOV of\[−3,\+3\]\[\-3,\+3\]R⊙, and each pixel corresponds to a particular ray through the coronal field\. Because the corona is optically thin and faint \(≈10−6\\approx 10^\{\-6\}of the central photospheric disk brightness\), it is often observed by occulting the bright solar disk, as done in a coronagraph\. We mirror this procedure by masking pixels within11R⊙, which additionally confines the LOS integral to ther≥1r\\geq 1R⊙region where the emissivity model is well\-defined\. Consequently, all observation\-space training and evaluation images apply the occulting mask\. The canonical training set contains300300evenly\-spaced views \(19\.6619\.66M rays,78\.6478\.64M channel measurements, before the occulting mask\), excluding a30∘30^\{\\circ\}evaluation arc and a3∘3^\{\\circ\}guard region around it: evaluation uses the arc, while the guard region is excluded from both sets\. We chose this holdout protocol because it explicitly leaves a section of the corona views unseen by the model, which is a harder interpolation test rather than an interleaved\-view held\-out set\.

We summarize each observational condition, or experiment family, by the tuple\(η,V,Λ,a\)\(\\eta,V,\\Lambda,a\)\. We add heteroscedastic Gaussian noise \(Equation[6](https://arxiv.org/html/2609.28868#A1.E6)\) to our images, with the shot\-noise variance multiplierη\\eta\(e\.g\.η=9\\eta=9is×3​σ\\times 3\\sigma\)\.VVis the number of views,Λ\\Lambdais the set of spectral channel indices, anda≠1a\\neq 1is a forward\-model elemental abundance mismatch scale \(Appendix[A\.4](https://arxiv.org/html/2609.28868#A1.SS4)\)\. We define the canonical noisy condition as\(η=9,V=300,Λ=all 4,a=1\)\(\\eta=9,V=300,\\Lambda=\\text\{all 4\},a=1\), used later in Section[5](https://arxiv.org/html/2609.28868#S5)\.

Our joint\-reconstruction model uses a multiresolution hash\-grid encoder \(Instant NGP\-style,[Müller et al\. \(2022\)](https://arxiv.org/html/2609.28868#bib.bib7)\), with separate decoder heads for density and temperature predictions \(Appendix[A\.5](https://arxiv.org/html/2609.28868#A1.SS5)\)\. We use the AdamW optimizer for6060k steps with batches of10241024rays, optimized using a fixed\-scale asinh loss \(Appendices[D\.3](https://arxiv.org/html/2609.28868#A4.SS3),[D\.4](https://arxiv.org/html/2609.28868#A4.SS4)\)\. We evaluate performance using field\-space mean absolute error on the inner coronal radial band\[1\.1,2\.0\]\[1\.1,2\.0\]R⊙, as well as held\-out image\-space loss \(Appendix[B](https://arxiv.org/html/2609.28868#A2)\)\. We show a visualization of the renderer and the training process in the right panel of Figure[1](https://arxiv.org/html/2609.28868#S4.F1)\. Training a single reconstruction takes≈3\\approx 3GPU\-hours on an NVIDIA A100100, and rendering a 300\-view dataset takes≈10\\approx 10\-100100minutes on an NVIDIA RTX 4090 depending on the resolution and integration step size \(Appendix[D\.4](https://arxiv.org/html/2609.28868#A4.SS4)\)\.

We stress that this is a controlled study using one static coronal scene\. Thus, we are evaluating joint thermodynamic reconstruction and within\-scene error localization, not a probabilistic calibration or generalization across multiple coronal states\. The objective family and architecture were selected based on physical\-field recovery on this scene; for noise\-level experiments, the asinh scalescs\_\{c\}is fixed at its baseline forη≤1\\eta\\leq 1and scaled byη\\sqrt\{\\eta\}forη\>1\\eta\>1\(Appendix[D\.6](https://arxiv.org/html/2609.28868#A4.SS6)\)\. All synthetic generation and inversion experiments also use the same discretized renderer \(Appendix[A\.2](https://arxiv.org/html/2609.28868#A1.SS2)\), so the study evaluates reconstruction under a matched forward model except for abundance perturbations described in Appendix[A\.4](https://arxiv.org/html/2609.28868#A1.SS4)\.

## 5Results

### 5\.1Joint Thermodynamic Reconstruction and Measurement Limitations

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_A2_density_shell_r1p5.png)Figure 2:Density\-field reconstruction \(given GT temperature\) at1\.511\.51R⊙\. Each panel is a shell unraveled on a latitude\-longitude plot\. The top row shows the ground truth and reconstructions using a classical grid with Tikhonov regularization, positional encoder \+ MLP, and hash grid \+ decoder MLP representations\. The bottom row shows corresponding residualsΔ​log10​ne=log10⁡ne,pred−log10⁡ne,GT\\Delta\\log\_\{10\}n\_\{e\}=\\log\_\{10\}n\_\{e,\\mathrm\{pred\}\}\-\\log\_\{10\}n\_\{e,\\mathrm\{GT\}\}\. The hash grid is able to reproduce sharper small\-scale features, substantially reducing inner\-bandMAE\\operatorname\{MAE\}relative to the MLP by more than2×2\\timesand classical grid methods by about4×4\\times\.We first isolate the spatial representation by reconstructing density while supplying GT temperature and then select the hash\-grid representation for the joint density\-temperature inversion\. Figure[2](https://arxiv.org/html/2609.28868#S5.F2)shows density\-field reconstruction results at1\.511\.51R⊙, trained using two matched\-model, noiseless intensity channels \(Fe XIII1075/10801075/1080\) while given the ground truth temperature field\. Latitude\-longitude plots \(flattened spherical shells\) of the reconstructed density are shown on the top, while the signed residuals are shown on the bottom\. We useMAEinner⁡\(log10⁡ne\)\\operatorname\{MAE\}\_\{\\text\{inner\}\}\(\\log\_\{10\}n\_\{\\rm e\}\), the mean absolute error of log\-density in the inner radial band\[1\.1,2\.0\]\[1\.1,2\.0\]R⊙, as our main metric\. Under the tested grid resolution, architectures, and optimization settings, the multiresolution hash grid \+ decoder head attains the lowest reconstruction error, performing over twice as well \(0\.0420\.042dexMAE\\operatorname\{MAE\}\) as a standard positional encoder \+ MLP \(0\.0970\.097dexMAE\\operatorname\{MAE\}\), and about four times as well as a classical voxelized representation \+ Tikhonov regularizer \(0\.1680\.168dexMAE\\operatorname\{MAE\}\)\. These comparisons are intended to isolate the effect of spatial representation under a common emission renderer, rather than to benchmark complete solar\-tomography systems, which are generally tailored to different observations and inversion settings\. See Appendix[D\.2](https://arxiv.org/html/2609.28868#A4.SS2)for detailed results\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_B2_los_filmstrip.png)Figure 3:LOS projections of the jointly\-reconstructed column density \(top\) and total\-emissivity\-weighted temperature \(bottom\), using a multiresolution hash\-grid representation trained on all four spectral channels \(Appendix[A\.3](https://arxiv.org/html/2609.28868#A1.SS3)\)\.Next, we jointly reconstruct density and temperature using all four spectral channels\. The resulting inner\-bandMAE\\operatorname\{MAE\}s are0\.0390\.039dex for density \(9\.2%9\.2\\%AbsRel\) and0\.0140\.014dex for temperature \(3\.2%3\.2\\%AbsRel\)\. Figure[3](https://arxiv.org/html/2609.28868#S5.F3)visualizes the recovered fields through column\-density and emissivity\-weighted\-temperature projections, and Figure[13](https://arxiv.org/html/2609.28868#A4.F13)in Appendix[D\.5](https://arxiv.org/html/2609.28868#A4.SS5)compares reconstructed fields with ground truth at three radii\. We find empirical evidence that increasing spectral coverage substantially reduces the density\-temperature degeneracies: single\-line models are underconstrained because the observed intensity depends jointly on density and temperature through a single emissivity response, allowing multiple density\-temperature states to produce similar measurements\. Because each line also has a finite temperature\-response range, additional lines with different thermodynamic responses provide complementary constraints that substantially reduce these degeneracies: a pair of lines drops thenen\_\{\\rm e\}MAE\\operatorname\{MAE\}by a factor of55\-1010, and using four lines further halves the error\. Finally, additional spectral lines also reduce the radialMAE\\operatorname\{MAE\}at larger radii \(Figure[14](https://arxiv.org/html/2609.28868#A4.F14)\)\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_B2_image_field_grid_main.png)Figure 4:Image fidelity is not field fidelity\. We show 3D density field\-spaceMAE\\operatorname\{MAE\}vs 2D held\-out image\-space loss\. Each marker is a trained model \(color==number of spectral lines, shape==the specific line set\)\. The left plot uses held\-out image error measured on each model’s own channels, which is a GT\-free image\-space validation quantity available on held\-out observed views\. Single\-channel models achieve lower observed\-channel image error than the four\-line model despite having substantially larger field error\. Evaluation on a common four\-line probe \(right panel\) exposes these failures and largely restores the density\-error ranking, showing that the low observed\-channel image error of these configurations reflects limited spectral coverage rather than correct field recovery\.We further show thatimage fidelity is not field fidelity: a low image\-space error need not certify a low field\-space error\. Figure[4](https://arxiv.org/html/2609.28868#S5.F4)shows a per\-model scatter of field\-spaceMAE\\operatorname\{MAE\}vs 2D held\-out validation image error \(color==number of spectral lines, shape==the specific line set\)\. The left panel uses the held\-out image error measured on each model’s own channels, which is a GT\-free image\-space validation quantity available on held\-out observed views\. We see that single\-channel experiments fit their observed images better than the four\-line models, yet they recover a far worse density field\. The right panel recomputes the image\-space evaluation on a common four\-channel set across all models with the same fixed observational setup, showing that own\-channel image error is not comparable as a certificate of field fidelity across spectral configurations\. In the density comparison, the common probe largely restores the field\-error ranking\. Temperature reconstructions likewise exhibit the same discrepancy, although the common\-probe image error does not perfectly order temperature error across line sets \(Figure[15](https://arxiv.org/html/2609.28868#A4.F15)\)\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_C2_noise_view_heatmaps_x1.png)Figure 5:Noise\-view sweep, fixed at 4 matched\-model spectral channels, ofnen\_\{\\rm e\}andTeT\_\{\\rm e\}MAE\\operatorname\{MAE\}\. With at least100100views, field recovery remains accurate across the tested noise levels under the noise\-matched loss\. Severe angular undersampling substantially increases error and makes the reconstruction much more vulnerable to noise\.Finally, we assess the robustness under sparse\-view and high\-noise regimes\. Figure[5](https://arxiv.org/html/2609.28868#S5.F5)crosses four noise levels,η∈\{0,1,9,25\}\\eta\\in\\\{0,1,9,25\\\}, with four view counts,V∈\{5,20,100,300\}V\\in\\\{5,20,100,300\\\}\. ForV≤20V\\leq 20, densityMAE\\operatorname\{MAE\}increases several\-fold relative toV≥100V\\geq 100, and severe angular undersampling substantially increases sensitivity to noise\. However, forV≥100V\\geq 100, increasing the noise scale from1→251\\rightarrow 25only modestly increases theMAE\\operatorname\{MAE\}\(for example,0\.052→0\.0810\.052\\rightarrow 0\.081at100100views\)\. We conclude that within this scene, reconstructions with at least100100views retain low field error across the tested noise levels\. We additionally note that becausescs\_\{c\}sets the scale of the asinh training loss, we scale it with the measurement\-noise standard deviation; Figure[5](https://arxiv.org/html/2609.28868#S5.F5)therefore compares reconstructions under a noise\-matched loss\.

### 5\.2Cross\-Seed Instability as a GT\-free\-at\-Inference Error\-Ranking Proxy

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/fig_ensemble_panels.png)Figure 6:Ensemble panels of the 3D density field at1\.511\.51R⊙using the canonical high\-noise condition\. We show the ensemble meanm¯​\(𝐱\)\\overline\{m\}\(\\mathbf\{x\}\), ensemble\-mean errorϵmean​\(𝐱\)\\epsilon\_\{\\rm mean\}\(\\mathbf\{x\}\), and cross\-seed instabilityσens​\(𝐱\)\\sigma\_\{\\rm ens\}\(\\mathbf\{x\}\)\. The maps exhibit similar spatial structure, which we quantify using correlations below\.We evaluate whether standard seed ensembles provide useful local error rankings for the jointly reconstructed thermodynamic fields\. We trainK=10K=10models for each condition, varying only the training seeds\([Lakshminarayanan et al\., 2017](https://arxiv.org/html/2609.28868#bib.bib10)\)while holding the model architecture, observational condition, and dataset fixed\. The per\-voxel standard deviation in the field predictions across the models is the cross\-seed instabilityσens\\sigma\_\{\\rm ens\}\(Appendix[C\.1](https://arxiv.org/html/2609.28868#A3.SS1)\)\. Figure[6](https://arxiv.org/html/2609.28868#S5.F6)shows reconstructions at1\.511\.51R⊙with the canonical high\-noise regime for ensemble meanm¯​\(𝐱\)\\overline\{m\}\(\\mathbf\{x\}\), absolute ensemble\-mean errorϵmean​\(𝐱\)\\epsilon\_\{\\rm mean\}\(\\mathbf\{x\}\), and cross\-seed instabilityσens​\(𝐱\)\\sigma\_\{\\rm ens\}\(\\mathbf\{x\}\)\.Largeσens\\sigma\_\{\\rm ens\}identifies regions where the recovered field is sensitive to training seeds under the fixed pipeline, while smallσens\\sigma\_\{\\rm ens\}identifies regions of cross\-seed consensus\.We note that consensus is not a correctness guarantee, because error shared by all ensemble members \(such as a common\-mode forward\-model discrepancy\) will go undetected\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/fig_baseline_summary.png)Figure 7:Left panel plots Spearman rank correlation between the true ensemble mean errorϵmean\\epsilon\_\{\\rm mean\}andσens\\sigma\_\{\\rm ens\}\(green circle\), inverse\-emissivitybℰb\_\{\\mathcal\{E\}\}\(orange\), and partial correlationρ⁡\(σens,ϵmean\|bℰ\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\)\(green triangle\), showing that the rank association persists after controlling for the oracle inverse\-emissivity proxy\. Error bars are95%95\\%within\-scene longitude\-slice bootstrap intervals\. We find thatσens\\sigma\_\{\\rm ens\}localizes error in three matched\-model regimes: noiseless, sparse\-view, and high\-noise, outperforming the emissivity proxybℰb\_\{\\mathcal\{E\}\}\. The partial correlation shows that the rank association remains after conditioning on an oracle inverse\-emissivity proxy\. Finally, we note that the localization ofσens\\sigma\_\{\\rm ens\}deteriorates substantially under positive Si over\-scaling and approaches chance for density ata=1\.4a=1\.4, but under\-scaling retains partial localization ability; whilebℰb\_\{\\mathcal\{E\}\}can still track the high\-error, low\-emissivity regions in this regime, it requires the GT field and is thus used only as an oracle baseline\. The right panel consists of sparsification curves for the canonical noisy condition, showing thatσens\\sigma\_\{\\rm ens\}tracks the oracle more closely than does the baselinebℰb\_\{\\mathcal\{E\}\}\(nAUSE of0\.270\.27compared to0\.520\.52\)\.We further examine whether rank association is explained by local signal strength\. The left panel in Figure[7](https://arxiv.org/html/2609.28868#S5.F7)shows the Spearman rank spatial correlation for the inner band density field betweenϵmean\\epsilon\_\{\\rm mean\}and \(i\)σens\\sigma\_\{\\rm ens\}and \(ii\) an oracle inverse\-emissivity signal\-strength baselinebℰb\_\{\\mathcal\{E\}\}\. Error bars are95%95\\%within\-scene longitude\-slice bootstrap intervals computed on the inner radial band using20002000resamples\. We find thatσens\\sigma\_\{\\rm ens\}localizes error across three matched\-model regimes: noiseless at300300and100100views \(0\.760\.76\-0\.770\.77\), sparse\-view at2020and55views \(0\.710\.71\-0\.780\.78\), and high\-noise \(0\.550\.55\-0\.570\.57\);bℰb\_\{\\mathcal\{E\}\}has weak rank correlation with error in the noiseless and sparse\-view regimes, whileσens\\sigma\_\{\\rm ens\}localizes strongly\. Moreover, the partial correlationρ⁡\(σens,ϵmean\|bℰ\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\)remains positive across matched\-model conditions, suggesting thatσens\\sigma\_\{\\rm ens\}captures error structure beyond what is explained by signal strength alone\. This association also remains positive within radial shells \(Tables[4](https://arxiv.org/html/2609.28868#A3.T4)and[5](https://arxiv.org/html/2609.28868#A3.T5)\), indicating that it is not driven solely by radial stratification, although its magnitude varies strongly under five views and high noise\. Finally, at the strongest tested Si over\-scaling \(a=1\.4a=1\.4\),σens\\sigma\_\{\\rm ens\}approaches chance for density, showing that error shared across ensemble members can produce incorrect cross\-seed consensus; althoughbℰb\_\{\\mathcal\{E\}\}can still localize low\-signal, high\-error regions in this regime, it requires the GT field and is used only as an oracle diagnostic baseline \(Tables[2](https://arxiv.org/html/2609.28868#A3.T2)and[3](https://arxiv.org/html/2609.28868#A3.T3)\)\.

Beyond correlation, we use a normalized Area Under the Sparsification Error curvenAUSE\\operatorname\{nAUSE\}\([Ilg et al\., 2018](https://arxiv.org/html/2609.28868#bib.bib8);[Poggi et al\., 2020](https://arxiv.org/html/2609.28868#bib.bib9)\)to quantify how well a ranking proxy approximates the oracleϵmean\\epsilon\_\{\\rm mean\}\(nAUSE\\operatorname\{nAUSE\}is00for oracle\-level sparsification and11for the expected random\-reference performance\)\. We find thatσens\\sigma\_\{\\rm ens\}\(nAUSE\\operatorname\{nAUSE\}0\.110\.11\-0\.300\.30\) is substantially closer to oracle sparsification than the tested baselines\. In particular, our inverse\-emissivity baselinebℰb\_\{\\mathcal\{E\}\}is consistently weaker thanσens\\sigma\_\{\\rm ens\}in the matched\-model conditions \(nAUSE\\operatorname\{nAUSE\}0\.470\.47\-1\.251\.25\) and becomes near\- or worse\-than\-random in sparse\-view cases\. Even in the high\-noise regime wherebℰb\_\{\\mathcal\{E\}\}can track the error,σens\\sigma\_\{\\rm ens\}performs better \(nAUSE\\operatorname\{nAUSE\}0\.270\.27vs0\.520\.52for×9\\times 9, right panel in Figure[7](https://arxiv.org/html/2609.28868#S5.F7)\)\. Finally, consistent with spatial correlations, under abundance mismatchσens\\sigma\_\{\\rm ens\}approaches chance \(nAUSE=0\.93\\operatorname\{nAUSE\}=0\.93for density ata=1\.4a=1\.4\)\. See Appendix[C\.2](https://arxiv.org/html/2609.28868#A3.SS2)\(Figure[10](https://arxiv.org/html/2609.28868#A3.F10)\) for more details on sparsification analysis and temperature field results\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/fig_errorcap_ladder.png)Figure 8:Per\-field error\-energy capture computed by projecting the error onto ensemble deviation subspaces\. Rank\-9 seed\-deviation spans capture2\.42\.4\-32\.1%32\.1\\%of the signed field error energy \(green\), which is above the0\.2%0\.2\\%isotropic reference \(dashed red\)\. Split\-seed evaluation, using span and error from disjoint seed halves \(orange\), shows that the directional overlap persists when the span and target error are constructed from disjoint seed subsets\. We additionally find that the original unrotated capture exceeds the longitude\-rotation\-null mean \(purple\) in all tested matched\-model regimes and lies at the top of the tested rotation distribution in 11 of 12 combinations\. Finally, we note that capture is largest under sparse\-view and high\-noise regimes\.Next, we provide a complementary field\-space consistency test by computing the directional per\-field energy overlapEc​a​pE\_\{cap\}\(Equation[27](https://arxiv.org/html/2609.28868#A3.E27)\), or fraction of the squaredL2L^\{2\}norm of the signed field erroree\(whose magnitude isϵmean\\epsilon\_\{\\rm mean\}\) that lies within the seed\-deviation subspace\. These percentages are computed on the full1\.051\.05\-33R⊙control grid, as opposed to the inner radial band used by localization metrics\. We also compute the overlap enrichment relative to the rank\-matched isotropic expectationEcaprand=r/ME\_\{\\rm cap\}^\{\\rm rand\}=r/M\. Across the three matched\-model regimes, shown in Figure[8](https://arxiv.org/html/2609.28868#S5.F8), the observed rank\-9 deviation span \(green\) captures4\.84\.8\-32\.1%32\.1\\%of the density signed\-error energy and2\.42\.4\-24\.0%24\.0\\%of the temperature signed\-error energy, strongest under severe view sparsity, and modest under full coverage \(all of which are substantially above the isotropic reference of0\.2%0\.2\\%in dashed red\)\. The corresponding enrichment ranges between2323\-154×154\\timesfor density and1111\-115×115\\timesfor temperature\. We additionally note that even under forward model mismatch,55\-58×58\\timesenrichment can persist above the rank\-matched isotropic expectation\.

While a highEcapE\_\{\\rm cap\}indicates directional alignment, it could be inflated by \(i\) reusing the same finite ensemble to constructeeand the deviation span, or \(ii\) generic spatial structure\. We provide two corresponding controls to test this\. First, we split theK=10K=10ensemble into two55\-seed groupsAAandBBand compute the error fromBBcaptured by the field deviation subspace ofAA\(EcapB→AE\_\{\\rm cap\}^\{B\\rightarrow A\}\), repeated over all252252splits\. Figure[8](https://arxiv.org/html/2609.28868#S5.F8)shows the mean and2\.52\.5th and97\.597\.5th percentile interval of the split\-sensitivity distribution in orange: across matched\-model conditions, a span estimated from one half of the seeds has a mean capture of the other half’s error up to18%18\\%versus0\.09%0\.09\\%for a random subspace of equal rank\. Thus, the observed alignment is not solely an artifact of using the same finite seed set to construct both quantities\. Second, we cyclically rotate the field deviations relative to the signed field error in longitude, preserving latitudinal/radial structure while disrupting their longitudinal alignment\. Shown in purple is the meanμ\\muof the error capture computed after the rotations, where the unrotated capture exceeds the mean and lies at the top of the tested distribution in11/1211/12field\-condition combinations\. The reduced capture indicates that the original longitudinal alignment contributes to the observed overlap beyond generic spatial structure alone\. We finally note that the captured energy from the seed\-deviation span is a minority of the total error energy but substantially greater than the rank\-matched reference expectations \(Tables[9](https://arxiv.org/html/2609.28868#A3.T9)and[10](https://arxiv.org/html/2609.28868#A3.T10)\)\.

To summarize, CoroNeRF reports the ensemble mean and cross\-seed instability without access to the true field; correlation, partial correlation, sparsification, and error\-capture diagnostics require ground truth and are used only to validate this inference\-time proxy\. In addition, errors shared by ensemble members are absent from cross\-seed instability, so consensus is not a correctness guarantee: we characterized this explicitly under a controlled Si abundance mismatch, where localization degrades under the tested range with a pronounced asymmetry between under and over\-scaling\. Notably, this relative abundance uncertainty can be reduced through co\-elemental line selection, although absolute\-intensity inversions remain sensitive to the common elemental scale\. Finally, we note that the present line set was selected for informative coronal density and temperature response, and conditioning under arbitrary line selections remains future work\.

## 6Conclusion

CoroNeRF jointly reconstructs 3D coronal electron density and temperature through a differentiable multiline atomic\-emission renderer\. In a controlled synthetic testbed, we show that image fidelity is not a sufficient certificate of thermodynamic\-field fidelity and evaluate cross\-seed instability as a GT\-free\-at\-inference error\-ranking proxy\. Additionally, increasing angular coverage substantially mitigates sparse\-view geometric degeneracies, while increasing spectral coverage mitigates plasma\-thermodynamic degeneracies\. Future work will assess generalization and calibration across coronal states and noise realizations, extend the framework to time\-dependent and spectropolarimetric observations, and evaluate it on DKIST\([Rimmele et al\., 2020](https://arxiv.org/html/2609.28868#bib.bib42);[Schad et al\., 2024](https://arxiv.org/html/2609.28868#bib.bib43)\)or UCoMP\([Landi et al\., 2016](https://arxiv.org/html/2609.28868#bib.bib40);[Tomczyk et al\., 2021](https://arxiv.org/html/2609.28868#bib.bib41)\)data\.

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#### AI use statement\.

The authors used large language models \(ChatGPT and Claude\) for finding related works, software development and debugging, checking mathematical rigor, feedback on experimental design and analysis of results, and for editing and proofreading the manuscript\. However, all methods, experiments, results, and claims were designed, executed, and verified by the authors, who take full responsibility for the content of this paper\.

#### Reproducibility statement\.

The authors conducted this study with reproducibility in mind\. Various parts of the forward model \(Appendix[A](https://arxiv.org/html/2609.28868#A1)\), hyperparameters \(Table[11](https://arxiv.org/html/2609.28868#A4.T11)\), and benchmark/ablation specifications are detailed in the Appendix of this manuscript\. All parameters associated with each experiment, as well as figures in the text, are documented explicitly in configuration code files\. Additionally, the full codebase and experiment configuration files are publicly available on the GitHub repository[https://github\.com/AlanHsu314/CoroNeRF\_public](https://github.com/AlanHsu314/CoroNeRF_public)\.

## Appendix AForward Model and Rendering Details

### A\.1Ground Truth Corona Cube and Emissivity Table Construction

Our ground truth coronal model is a MAS \(Magnetohydrodynamic Algorithm outside a Sphere\) cube from Predictive Science Inc\.\([Mikić et al\., 2007](https://arxiv.org/html/2609.28868#bib.bib32);[Lionello et al\., 2008](https://arxiv.org/html/2609.28868#bib.bib31)\), based on Carrington Rotation 2283 that occurred around April\-May 2024\. The corona cube is in \(longitude\[0,2​π\]\[0,2\\pi\], colatitude\[0,π\]\[0,\\pi\], radius\[1,30\]\[1,30\]R⊙\) coordinates with resolution\(299,142,154\)\(299,142,154\)\. Density and temperature have rangeslog10⁡\(ne/cm−3\)∈\[2\.16,12\.32\]\\log\_\{10\}\(n\_\{\\rm e\}/\\text\{cm\}^\{\-3\}\)\\in\[2\.16,12\.32\]andlog10⁡\(Te/K\)∈\[5\.75,6\.35\]\\log\_\{10\}\(T\_\{\\rm e\}/\\text\{K\}\)\\in\[5\.75,6\.35\], respectively\.

We use the CHIANTI 10\.1 atomic database\([Dere, K\. P\. et al\., 1997](https://arxiv.org/html/2609.28868#bib.bib1);[Dere et al\., 2023](https://arxiv.org/html/2609.28868#bib.bib2)\)and a Python packagepyCELP\([Schad and Dima, 2020](https://arxiv.org/html/2609.28868#bib.bib3)\)to construct the emissivity tables for our forward model\. Our spectral lines are two density\-sensitive pairs of forbidden coronal infrared lines: Fe XIII1075/10801075/1080nm and Si IX2585/39352585/3935nm\. While these are the nominal values used in scientific literature, the actual values used in the database are Fe XIII1074\.7/1079\.81074\.7/1079\.8nm and Si IX2584\.6/3929\.32584\.6/3929\.3nm, respectively\. When synthesizing these lines, we assume the elemental abundance described in the 2021 Asplund paper\([Asplund et al\., 2021](https://arxiv.org/html/2609.28868#bib.bib4)\)\.

The emissivity LUT is a function of densitynen\_\{\\rm e\}, temperatureTeT\_\{\\rm e\}, and heliocentric radiusrr, and its outputs are in units of erg s\-1cm\-3sr\-1\. We retain only the scalar Stokes\-I emissivity, neglecting the quadrupole atomic\-alignment/depolarization terms associated with the magnetic\-field geometry\. The LUT axes are chosen from the PSI cube statistics and atomic\-response ranges\. Queries outside the tabulated\(log10⁡ne,log10⁡Te,log10⁡r\)\(\\log\_\{10\}n\_\{\\rm e\},\\log\_\{10\}T\_\{\\rm e\},\\log\_\{10\}r\)domain are clamped to the nearest LUT boundary before interpolation\. In particular, the radial LUT spans approximately1\.0011\.001\-1010R⊙, and LOS samples farther than1010R⊙use the1010R⊙radial boundary response\. Our grid resolution is128×64×32128\\times 64\\times 32in\(log10⁡ne,log10⁡Te,log10⁡r\)\(\\log\_\{10\}n\_\{\\rm e\},\\log\_\{10\}T\_\{\\rm e\},\\log\_\{10\}r\)\. We implement the table in log\-space, and we usePyTorchto make it tensor\-compatible, notably a tensor\-product trilinear interpolator in LUT\-coordinate space that usesgrid\_samplefromtorch\.nn\.functional\. The trilinear interpolator additionally has a backstop that returns the nearest boundary even if it is given a value outside the bounds, rather than zeros or a linear extrapolation\.

This emissivity LUT is used both in \(i\) synthetic observation generation using the GT density and temperature values from the PSI cube, and \(ii\) CoroNeRF for rendering the outputs from the neural plasma field\. For \(i\), we trilinearly interpolate the PSI density and temperature values on a spherical grid using the samegrid\_sample, this time with the longitude dimension augmented to support periodic longitude wrapping\.

### A\.2Sampling Geometry and Numerical Quadrature

CoroNeRF contains objects that live in different coordinate systems\. We place the original\[−30,30\]3\[\-30,30\]^\{3\}R⊙PSI cube centered in a normalized axis\-aligned bounding box \(AABB\) that has bounds\[−1,1\]3\[\-1,1\]^\{3\}by normalizing all coordinates by3030R⊙\. Our ray sampler works in this coordinate system\.

A view is constructed as follows\. Given a selected longitude angle, we place a camera at a heliocentric distance of≈215\\approx 215R⊙\(Earth\-Sun distance\),0∘0^\{\\circ\}latitude, pointed towards the origin\. We use a perspective camera model: our rays start from the observer and span a\[−3,3\]\[\-3,3\]R⊙FOV, and each ray forms one pixel on the image plane\. At an observer distance of≈215\\approx 215R⊙, perspective rays are nearly parallel across the66R⊙FOV\. We then mask sightlines with impact parameter<1​R⊙<1R\_\{\\odot\}to mimic a coronagraph, leaving only a set of coronal pixels to render\. Given a pixel to render, we shoot a ray through the AABB, compute the near and far bounds as intersections with the AABB, and compute emissivities at uniform samples along the ray with step size1/2561/256\(the number of samples depends on the ray’s intersection length with the AABB\)\. To do so at a given sample \(in AABB coordinates\), we query the neural plasma fieldfθf\_\{\\theta\}to get density and temperature predictions \(log10⁡ne,log10⁡Te\\log\_\{10\}n\_\{\\rm e\},\\log\_\{10\}T\_\{\\rm e\}\) and use our LUT to compute the emissivity at that point\. Note that the LUT takes in the heliocentric radiusr=‖𝐱‖r=\|\|\\mathbf\{x\}\|\|, computed from physical Cartesian coordinates𝐱\\mathbf\{x\}\. Next, we compute the differential length segment corresponding to each sample point, by scaling our coordinates from AABB toR⊙R\_\{\\odot\}to cm\. We additionally convert the raw LUT emissivity from erg to photon counts, and from steradian\-1to arcsec\-2, so that the integrandϵc\\epsilon\_\{c\}has units photons s\-1cm\-3arcsec\-2, and our final intensity observations after integrating along the ray are in units of photons s\-1cm\-2arcsec\-2\. The LOS integration uses a midpoint Riemann sum\.

step sizeΔ​s\\Delta s\(R⊙\)rel\-L2 vs finestasinhErr\\operatorname\{asinhErr\}vs finest1/1281/1280\.2340\.23420\.6%20\.6\\%1\.0×10−21\.0\\times 10^\{\-2\}𝟏/𝟐𝟓𝟔\\mathbf\{1/256\}\(production\)0\.1170\.1177\.5%7\.5\\%1\.7×10−31\.7\\times 10^\{\-3\}1/5121/5120\.0590\.0592\.3%2\.3\\%5\.7×10−45\.7\\times 10^\{\-4\}1/10241/10240\.0290\.029— \(ref\)— \(ref\)Table 1:Convergence table for selected renderer step sizes\. For each step size, we render eight views using the GT cube and compute relative L2 andasinhErr\\operatorname\{asinhErr\}compared to the finest resolution\.We assess the renderer convergence by re\-rendering the GT forward model at integration stepsΔ​s∈\{1/128,1/256,1/512,1/1024\}\\Delta s\\in\\\{1/128,1/256,1/512,1/1024\\\}\(AABB coordinates\) on eight views and comparing to the finest resolution \(Table[1](https://arxiv.org/html/2609.28868#A1.T1)\)\. The images converge at roughly first\-to\-second order, and at production step \(1/256≈0\.1171/256\\approx 0\.117R⊙\) they differ from a4×4\\times\-finer render by7\.5%7\.5\\%in relative L2 but only1\.7×10−31\.7\\times 10^\{\-3\}in the asinh image metric, so production\-resolution renders are close to the finer reference under the optimization metric\. However, we do not claim discretization\-independent field recovery\. Finally, for synthetic dataset generation, the ray sampling and integral procedures are identical, except we use the oracle PSI density and temperature fields instead of the neural plasma field\. These fields live on a spherical coordinate system, so we transform AABB to spherical coordinates before sampling for density and temperature values using spherical trilinear field interpolators\.

### A\.3LOS visual projections

For visualizing the field, we use rendered field\-space projections for intuition, and reserve latitude\-longitude plots at fixed shell radii for metric evaluation\.

First, our LOS integration support for visualization is\[1\.1,2\.5\]\[1\.1,2\.5\]R⊙\. We reuse the ray geometry notationqqand define the column density and emissivity\-weighted temperature with LOS integrals analogous to Equation[1](https://arxiv.org/html/2609.28868#S3.E1):

Ne​\(q\)\\displaystyle N\_\{\\mathrm\{e\}\}\(q\)=∫𝒮⁡\(q\)ne​\(𝐱q​\(t\)\)​𝑑t,\\displaystyle=\\int\_\{\\mathcal\{S\}\(q\)\}n\_\{\\mathrm\{e\}\}\\big\(\\mathbf\{x\}\_\{q\}\(t\)\\big\)\\,dt,\(3\)⟨Te⟩ϵ​\(q\)\\displaystyle\\langle T\_\{\\mathrm\{e\}\}\\rangle\_\{\\epsilon\}\(q\)=∫𝒮⁡\(q\)Te​\(𝐱q​\(t\)\)​ℰ​\(𝐱q​\(t\)\)​𝑑t∫𝒮⁡\(q\)ℰ⁡\(𝐱q​\(t\)\)​𝑑t\.\\displaystyle=\\frac\{\\displaystyle\\int\_\{\\mathcal\{S\}\(q\)\}T\_\{\\mathrm\{e\}\}\\big\(\\mathbf\{x\}\_\{q\}\(t\)\\big\)\\,\\mathcal\{E\}\\big\(\\mathbf\{x\}\_\{q\}\(t\)\\big\)\\,dt\}\{\\displaystyle\\int\_\{\\mathcal\{S\}\(q\)\}\\mathcal\{E\}\\big\(\\mathbf\{x\}\_\{q\}\(t\)\\big\)\\,dt\}\.\(4\)
Here, we define a total summed emissivityℰ⁡\(𝐱\)\\mathcal\{E\}\(\\mathbf\{x\}\)that accounts for all the observed spectral channels

ℰ⁡\(𝐱\)=∑c=1Cϵc​\(ne​\(𝐱\),Te​\(𝐱\),r⁡\(𝐱\)\)\.\\mathcal\{E\}\(\\mathbf\{x\}\)=\\sum\_\{c=1\}^\{C\}\\epsilon\_\{c\}\\big\(n\_\{\\mathrm\{e\}\}\(\\mathbf\{x\}\),\\,T\_\{\\mathrm\{e\}\}\(\\mathbf\{x\}\),\\,r\(\\mathbf\{x\}\)\\big\)\.\(5\)
We choose the column density because it is a real physical quantity: the total electrons accumulated per cm2\. On the other hand, temperature is not additive, and thus we choose to compute a weighted representative temperature based on the total emission along the LOS\.

### A\.4Observational Noise and Forward Model Mismatch

We additionally add heteroscedastic Gaussian noise to our images: for a given channelccand pixelii, the observed intensity is drawn from

Ii,cobs∼𝒩⁡\(Ii,ctrue,η​αc​Ii,ctrue\+βc2\),I\_\{i,c\}^\{\\rm obs\}\\sim\\mathcal\{N\}\(I\_\{i,c\}^\{\\rm true\},\\eta\\alpha\_\{c\}I\_\{i,c\}^\{\\rm true\}\+\\beta\_\{c\}^\{2\}\),\(6\)whereαc\\alpha\_\{c\}is a channel\-dependent coefficient for the signal\-dependent shot\-like variance term, andβc\\beta\_\{c\}is the background/read\-noise floor\. We scale the noise by multiplyingαc\\alpha\_\{c\}byη\\eta, a variance multiplier \(η=9\\eta=9means a×3\\times 3increase in standard deviation\)\. Next, to introduce forward model mismatch, we define the truthIi,ctrueI\_\{i,c\}^\{\\rm true\}to be a scaled version of the raw forward model outputIi,crawI\_\{i,c\}^\{\\rm raw\}:

Ii,ctrue=\{a​Ii,craw,c∈\{Si IX channels\},Ii,craw,c∈\{Fe XIII channels\}\.I\_\{i,c\}^\{\\rm true\}=\\begin\{cases\}aI\_\{i,c\}^\{\\rm raw\},&c\\in\\\{\\text\{Si IX channels\}\\\},\\\\ I\_\{i,c\}^\{\\rm raw\},&c\\in\\\{\\text\{Fe XIII channels\}\\\}\.\\\\ \\end\{cases\}\(7\)
whereaais an abundance\-mismatch scale applied to the Si IX channels before noise is added\. This mismatch models a controlled family of multiplicative channel\-wise abundance discrepancies, in particular a relative Fe\-to\-Si abundance scaling, wherea=1a=1means no forward model error \(matched\-model\)\. Our implementation scales both Si IX channels by the same value while the Fe XIII channels remain at one, so we refer toaaspecifically as a Si scale\.

In practice, we selectαc\\alpha\_\{c\}such that radial bands across spectral channels achieve approximately the same SNR, reflecting slightly different per\-channel noise levels as a real instrument would have\. We set the background standard deviation floor toβc=0\\beta\_\{c\}=0\. Forη=0\\eta=0specifically, we bypass this noise generation process entirely\. Finally, we note that observation noise is applied to training views only, and held\-out image\-space evaluation uses corresponding noise\-free synthetic views\.

### A\.5Multiresolution Hash Grid Model

Our multiresolution hash\-grid representation is based on the encoding described by\([Müller et al\., 2022](https://arxiv.org/html/2609.28868#bib.bib7)\), where we encode spatial information as trainable grid features, and then separate decoder MLP heads combine the features into density and temperature estimates\. The grids themselves do not store features directly at each spatial point, as this does not scale with resolution\. The grid is indexed by hashing the spatial coordinate of the point, where we define the size as2G2^\{G\}byFF:GGis the hashmap table size inlog2\\log\_\{2\}, andFFis the number of features\. The full hash grid contains many such levels, each associated with a physical spatial resolution\. Given a sample position, the features at that point are computed by trilinearly interpolating the features at the eight corners of the cube surrounding that point\. The positions of those eight corners are computed relative to the fixed spatial resolution of that level\. The coordinates are then hashed and used in the trainable hashmap to retrieve the features\.

For a multiresolution hash grid withLLlevels, we define the coarsest \(base\) resolution asN0N\_\{0\}and the finest resolution asNL−1N\_\{L\-1\}\. We then spread the resolution geometrically, where the resolution at levelℓ\\ellis given by

Nℓ=N0​bℓ,b=exp⁡\(ln⁡\(NL−1/N0\)L−1\)\.N\_\{\\ell\}=N\_\{0\}b^\{\\ell\}\\,,\\quad b=\\exp\\left\(\\frac\{\\ln\(N\_\{L\-1\}/N\_\{0\}\)\}\{L\-1\}\\right\)\.\(8\)
Our hash function is a bitwiseXOR\\operatorname\{XOR\}hash\. While hashing integer coordinates may introduce many collisions, we use multiple levels, each with a different physical spatial resolution\. Thus, the multiresolution nature of our grid not only encodes information at different spatial levels, but also mitigates the effect of collisions once we concatenate the features at all levels\.

In our implementation, each level has a hash table with2202^\{20\}entries andF=2F=2features per entry, with a total ofL=20L=20levels that span a spatial resolution going fromN0=16N\_\{0\}=16\(coarsest\) toNL−1=512N\_\{L\-1\}=512\(finest\)\. Our decoder heads are each two hidden layers of width6464\.

During training \(per minibatch of rays\), because our forward model is differentiable with respect to the neural plasma field \(and thus the model parameters\), the gradients first backpropagate through the differentiable line\-emission renderer, and then through the emissivity lookup table to the predictedlog10⁡ne\\log\_\{10\}n\_\{\\rm\{e\}\}andlog10⁡Te\\log\_\{10\}T\_\{\\rm\{e\}\}fields, and finally to the multiresolution hash grid\. The lookup table itself and ray geometry and sampling are fixed during optimization\.

## Appendix BMetrics

Our evaluations are conducted in two spaces\.*Image\-space metrics*compare rendered line\-intensity images to held\-out observations\.*Field\-space metrics*compare the reconstructed density and temperature fields to their ground\-truth counterparts on spherical shells\. Field\-space metrics are the primary metrics in this work as the goal is a latent joint physical\-field recovery rather than image synthesis\.

#### Notation\.

For any finite index set𝒜\\mathcal\{A\}, we write

⟨f⁡\(a\)⟩a∈𝒜=1\|𝒜\|​∑a∈𝒜f⁡\(a\)\\langle f\(a\)\\rangle\_\{a\\in\\mathcal\{A\}\}=\\frac\{1\}\{\|\\mathcal\{A\}\|\}\\sum\_\{a\\in\\mathcal\{A\}\}f\(a\)\(9\)
for the empirical average\. We also use the same indexing as in Section[3](https://arxiv.org/html/2609.28868#S3)of the main text:v∈𝒱=\{1,…,V\}v\\in\\mathcal\{V\}=\\\{1,\\ldots,V\\\}indexes viewpoints,p∈Ωvp\\in\\Omega\_\{v\}indexes valid pixels in viewvv, andc∈𝒞=\{1,…,C\}c\\in\\mathcal\{C\}=\\\{1,\\ldots,C\\\}indexes spectral line channels\.

### B\.1Image\-Space Metrics

LetIv,p,cI\_\{v,p,c\}andI^v,p,c\\hat\{I\}\_\{v,p,c\}denote the observed and predicted line intensities for viewvv, pixelpp, and channelcc\. The per\-view, per\-channel mean\-squared error is

MSEv,c​\(I\)=⟨\(I^v,p,c−Iv,p,c\)2⟩p∈Ωv\.\\text\{MSE\}\_\{v,c\}\(I\)=\\left\\langle\(\\hat\{I\}\_\{v,p,c\}\-I\_\{v,p,c\}\)^\{2\}\\right\\rangle\_\{p\\in\\Omega\_\{v\}\}\.\(10\)
The per\-view, per\-channel PSNR is

PSNRv,c​\(I\)=10​log10⁡\(Rv,c2MSEv,c​\(I\)\),\\text\{PSNR\}\_\{v,c\}\(I\)=10\\log\_\{10\}\\left\(\\frac\{R\_\{v,c\}^\{2\}\}\{\\text\{MSE\}\_\{v,c\}\(I\)\}\\right\),\(11\)
whereRv,cR\_\{v,c\}is the11st\-to\-9999th\-percentile target\-intensity range for that view and channel, with max\-minus\-min as a fallback if the percentile range is degenerate\. Dataset\-level image\-space metrics are obtained by averaging over views and channels:

PSNR​\(I\)=⟨PSNRv,c​\(I\)⟩v∈𝒱,c∈𝒞\.\\text\{PSNR\}\(I\)=\\left\\langle\\text\{PSNR\}\_\{v,c\}\(I\)\\right\\rangle\_\{v\\in\\mathcal\{V\},c\\in\\mathcal\{C\}\}\.\(12\)
Next, our fixed\-scale asinh image error is defined as follows:

asinhErr​\(I\)=⟨\|asinh⁡\(I^v,p,csc\)−asinh⁡\(Iv,p,csc\)\|⟩p∈Ωv,v∈𝒱,c∈𝒞,\\text\{asinhErr\}\(I\)=\\left\\langle\\left\|\\operatorname\{asinh\}\\\!\\left\(\\frac\{\\hat\{I\}\_\{v,p,c\}\}\{s\_\{c\}\}\\right\)\-\\operatorname\{asinh\}\\\!\\left\(\\frac\{I\_\{v,p,c\}\}\{s\_\{c\}\}\\right\)\\right\|\\right\\rangle\_\{p\\in\\Omega\_\{v\},\\;v\\in\\mathcal\{V\},\\;c\\in\\mathcal\{C\}\},\(13\)
wherescs\_\{c\}is the channel\-dependent intensity scale specified by the evaluation protocol \(in the noise\-view study, held\-outasinhErruses the common base per\-channel scales\)\. This metric is well\-defined for signed intensities and behaves logarithmically for large intensity magnitudes\.

### B\.2Field\-Space Metrics

Field metrics are computed on spherical shells\. Letr∈ℛr\\in\\mathcal\{R\}index evaluation shell radii, and lets∈𝒮rs\\in\\mathcal\{S\}\_\{r\}index valid samples on the shell at radiusrr\. We usemmto denote a density or temperature log\-field

m∈\{log10⁡ne,log10⁡Te\}\.m\\in\\\{\\log\_\{10\}n\_\{\\rm e\},\\;\\log\_\{10\}T\_\{\\rm e\}\\\}\.\(14\)
Letmr,sm\_\{r,s\}andm^θ,r,s\\hat\{m\}\_\{\\theta,r,s\}be the ground\-truth and predicted log\-field values\. For a radial bandℛband⊆ℛ\\mathcal\{R\}\_\{\\text\{band\}\}\\subseteq\\mathcal\{R\}, we define the mean absolute log error

MAEband​\(m\)=⟨\|m^θ,r,s−mr,s\|⟩s∈𝒮r,r∈ℛband\.\\text\{MAE\}\_\{\\text\{band\}\}\(m\)=\\left\\langle\|\\hat\{m\}\_\{\\theta,r,s\}\-m\_\{r,s\}\|\\right\\rangle\_\{s\\in\\mathcal\{S\}\_\{r\},\\;r\\in\\mathcal\{R\}\_\{\\text\{band\}\}\}\.\(15\)
We also report the absolute relative error

AbsRelband​\(m\)=⟨\|10m^θ,r,s−10mr,s\|max⁡\(\|10mr,s\|,ϵ0\)⟩s∈𝒮r,r∈ℛband,\\text\{AbsRel\}\_\{\\text\{band\}\}\(m\)=\\left\\langle\\frac\{\|10^\{\\hat\{m\}\_\{\\theta,r,s\}\}\-10^\{m\_\{r,s\}\}\|\}\{\\operatorname\{max\}\(\|10^\{m\_\{r,s\}\}\|,\\epsilon\_\{0\}\)\}\\right\\rangle\_\{s\\in\\mathcal\{S\}\_\{r\},\\;r\\in\\mathcal\{R\}\_\{\\text\{band\}\}\},\(16\)
whereϵ0=10−30\\epsilon\_\{0\}=10^\{\-30\}is a small numerical floor\.

Finally, for model\-mismatch studies, the sign of the error is also important, so we additionally report signed shell mean error:

MEband​\(m\)=⟨m^θ,r,s−mr,s⟩s∈𝒮r,r∈ℛband\.\\text\{ME\}\_\{\\text\{band\}\}\(m\)=\\left\\langle\\hat\{m\}\_\{\\theta,r,s\}\-m\_\{r,s\}\\right\\rangle\_\{s\\in\\mathcal\{S\}\_\{r\},\\;r\\in\\mathcal\{R\}\_\{\\text\{band\}\}\}\.\(17\)
Unlike absolute error metrics, this signed metric indicates whether a reconstructed field systematically overestimates or underestimates the ground truth field\.

## Appendix CEnsemble Uncertainty Quantification

### C\.1Cross\-Seed Instability and Correlation Analysis

We denote a plasma quantity asm∈\{log10⁡ne,log10⁡Te\}m\\in\\\{\\log\_\{10\}n\_\{\\rm e\},\\log\_\{10\}T\_\{\\rm e\}\\\}\. Every definition below is per\-mmfor readability \(fields are in dex\), and voxels are indexed by position𝐱\\mathbf\{x\}\. Let us reuse the shell and radial notation from Appendix[B](https://arxiv.org/html/2609.28868#A2):𝒮r=\{𝐱:r⁡\(𝐱\)=r\}\\mathcal\{S\}\_\{r\}=\\\{\\mathbf\{x\}:r\(\\mathbf\{x\}\)=r\\\}as a radial shell, andℛ\[r1,r2\]=\{𝐱:r1≤r⁡\(𝐱\)≤r2\}\\mathcal\{R\}\_\{\[r\_\{1\},r\_\{2\}\]\}=\\\{\\mathbf\{x\}:r\_\{1\}\\leq r\(\\mathbf\{x\}\)\\leq r\_\{2\}\\\}as a radial band fromr1r\_\{1\}tor2r\_\{2\}\. Finally, letm∗​\(𝐱\)m^\{\*\}\(\\mathbf\{x\}\)be the ground truth discretized field\.

We trainKKmodels, with fixed model architecture, observational protocol\(η,V,Λ,a\)\(\\eta,V,\\Lambda,a\), and training dataset, only varying the seedsk=1​…​Kk=1\\dots K\([Lakshminarayanan et al\., 2017](https://arxiv.org/html/2609.28868#bib.bib10)\)\. In particular, seedkkgives a predicted fieldm^k​\(𝐱\)\\hat\{m\}\_\{k\}\(\\mathbf\{x\}\)\. Define the ensemble mean and cross\-seed instability across the seeds, respectively, as follows:

m¯​\(𝐱\)\\displaystyle\\overline\{m\}\(\\mathbf\{x\}\)=1K​∑km^k​\(𝐱\),\\displaystyle=\\frac\{1\}\{K\}\\sum\_\{k\}\\hat\{m\}\_\{k\}\(\\mathbf\{x\}\),\(18\)σens​\(𝐱\)\\displaystyle\\sigma\_\{\\rm ens\}\(\\mathbf\{x\}\)=\[1K−1​∑k\(m^k​\(𝐱\)−m¯​\(𝐱\)\)2\]1/2\.\\displaystyle=\\left\[\\frac\{1\}\{K\-1\}\\sum\_\{k\}\(\\hat\{m\}\_\{k\}\(\\mathbf\{x\}\)\-\\overline\{m\}\(\\mathbf\{x\}\)\)^\{2\}\\right\]^\{1/2\}\.\(19\)
σens\\sigma\_\{\\rm ens\}is a GT\-free\-at\-inference empirical metric that is a field\-error localization proxy\. Largeσens\\sigma\_\{\\rm ens\}identifies regions where the recovered field is sensitive to training seeds under the fixed pipeline, while smallσens\\sigma\_\{\\rm ens\}identifies regions of cross\-seed consensus\. On the other hand, the absolute ensemble mean error, which can never be known without access to the GT fieldm∗m^\{\*\}, quantifies the error of the ensemble\-mean field:

ϵmean​\(𝐱\)=\|m¯​\(𝐱\)−m∗​\(𝐱\)\|\.\\epsilon\_\{\\rm mean\}\(\\mathbf\{x\}\)=\|\\overline\{m\}\(\\mathbf\{x\}\)\-m^\{\*\}\(\\mathbf\{x\}\)\|\.\(20\)
Note thatϵmean\\epsilon\_\{\\rm mean\}is not a statistical bias, which would require an expectation over multiple noise realizations\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/fig_ensemble_panels_appendix.png)Figure 9:Ensemble reconstruction panels of the 3D density \(top\) and temperature \(bottom\) fields at1\.511\.51R⊙in the high\-noise regime \(η=9,V=300,all four lines,a=1\\eta=9,V=300,\\text\{all four lines\},a=1\)\. We show the ensemble meanm¯​\(𝐱\)\\overline\{m\}\(\\mathbf\{x\}\)\(Equation[18](https://arxiv.org/html/2609.28868#A3.E18)\), ensemble\-mean errorϵmean​\(𝐱\)\\epsilon\_\{\\rm mean\}\(\\mathbf\{x\}\)\(unobservable, Equation[20](https://arxiv.org/html/2609.28868#A3.E20)\), and cross\-seed instabilityσens​\(𝐱\)\\sigma\_\{\\rm ens\}\(\\mathbf\{x\}\)\(Equation[19](https://arxiv.org/html/2609.28868#A3.E19)\)\.Finally, to quantitatively express localization correlation, let us use error targetϵmean\\epsilon\_\{\\rm mean\}and spatial domainΩ∈\{𝒮r,ℛ\[r1,r2\]\}\\Omega\\in\\\{\\mathcal\{S\}\_\{r\},\\mathcal\{R\}\_\{\[r\_\{1\},r\_\{2\}\]\}\\\}\. Define the Spearman rank correlation coefficient between the cross\-seed instability and the error overΩ\\Omegaas

ρϵ=Spearmanx∈Ω⁡\(σens,ϵmean\)=corr⁡\(rank⁡σens,rank⁡ϵmean\)\.\\rho\_\{\\epsilon\}=\\operatorname\{Spearman\}\_\{x\\in\\Omega\}\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)=\\operatorname\{corr\}\(\\operatorname\{rank\}\\sigma\_\{\\rm ens\},\\operatorname\{rank\}\\epsilon\_\{\\rm mean\}\)\.\(21\)
We note that our ensemble\-localization study uses300300evenly\-spaced training views with a disjoint3030\-view interleaved evenly\-spaced holdout\. All members within each condition share the same split, and for noisy conditions they share the same fixed noise realization\. Figure[9](https://arxiv.org/html/2609.28868#A3.F9)shows sample ensemble meanm¯​\(𝐱\)\\overline\{m\}\(\\mathbf\{x\}\), ensemble\-mean errorϵmean​\(𝐱\)\\epsilon\_\{\\rm mean\}\(\\mathbf\{x\}\), and cross\-seed instabilityσens​\(𝐱\)\\sigma\_\{\\rm ens\}\(\\mathbf\{x\}\)panels for our canonical noisy condition\.

Next, our baseline signal\-strength proxy is the inverse\-emissivity proxy, given bybℰ=1/ℰb\_\{\\mathcal\{E\}\}=1/\\mathcal\{E\}, whereℰ\>0\\mathcal\{E\}\>0is the total local emissivity summed over all spectral channels \(Equation[5](https://arxiv.org/html/2609.28868#A1.E5)\)\. We use this as a baseline signal\-strength confound: low\-emissivity regions are a plausible confound because they are more weakly constrained and may exhibit larger reconstruction error\. We note that in mismatch conditions we intentionally retain the nominal, unscaled GT emissivity, sobℰb\_\{\\mathcal\{E\}\}remains an oracle probe of the underlying physical signal rather than an observation\-consistent proxy\.

We then compute per\-condition pairwise correlations between the cross\-seed instabilityσens\\sigma\_\{\\rm ens\}, the ensemble mean errorϵmean\\epsilon\_\{\\rm mean\}, and our emissivity proxybℰb\_\{\\mathcal\{E\}\}, to answer the following question: does the rank association betweenσens\\sigma\_\{\\rm ens\}andϵmean\\epsilon\_\{\\rm mean\}remain after controlling for the oracle inverse\-emissivity proxy? To do so, we control forbℰb\_\{\\mathcal\{E\}\}by rank\-transforming each variable and computing the standard partial Pearson correlation on those ranks:

ρ⁡\(σens,ϵmean\|bℰ\)=ρ⁡\(σens,ϵmean\)−ρ⁡\(σens,bℰ\)​ρ​\(bℰ,ϵmean\)\(1−ρ2​\(σens,bℰ\)\)​\(1−ρ2​\(bℰ,ϵmean\)\)\.\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\)=\\frac\{\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)\-\\rho\(\\sigma\_\{\\rm ens\},b\_\{\\mathcal\{E\}\}\)\\rho\(b\_\{\\mathcal\{E\}\},\\epsilon\_\{\\rm mean\}\)\}\{\\sqrt\{\(1\-\\rho^\{2\}\(\\sigma\_\{\\rm ens\},b\_\{\\mathcal\{E\}\}\)\)\(1\-\\rho^\{2\}\(b\_\{\\mathcal\{E\}\},\\epsilon\_\{\\rm mean\}\)\)\}\}\.\(22\)
A strong positive partial correlation indicates a nontrivial linear association between the rank\-transformedσens\\sigma\_\{\\rm ens\}andϵmean\\epsilon\_\{\\rm mean\}, after linear adjustment for the rank\-transformedbℰb\_\{\\mathcal\{E\}\}proxy \(in other words, suggesting thatσens\\sigma\_\{\\rm ens\}captures error structure beyond what is explained by signal strength alone\)\.

Conditionρ⁡\(σens,ϵmean\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)↑\\uparrowρ⁡\(bℰ,ϵmean\)\\rho\(b\_\{\\mathcal\{E\}\},\\epsilon\_\{\\rm mean\}\)partialρ⁡\(σens,ϵmean\|bℰ\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\)↑\\uparrowρ⁡\(σens,bℰ\)\\rho\(\\sigma\_\{\\rm ens\},b\_\{\\mathcal\{E\}\}\)noiseless \(300300v\)0\.761−0\.029\+0\.0220\.761\_\{\-0\.029\}^\{\+0\.022\}0\.072−0\.058\+0\.0520\.072\_\{\-0\.058\}^\{\+0\.052\}0\.759−0\.027\+0\.0220\.759\_\{\-0\.027\}^\{\+0\.022\}0\.0810\.081100100views0\.773−0\.027\+0\.0210\.773\_\{\-0\.027\}^\{\+0\.021\}0\.085−0\.056\+0\.0500\.085\_\{\-0\.056\}^\{\+0\.050\}0\.772−0\.025\+0\.0200\.772\_\{\-0\.025\}^\{\+0\.020\}0\.0860\.0862020views0\.778−0\.034\+0\.0290\.778\_\{\-0\.034\}^\{\+0\.029\}−0\.019−0\.056\+0\.045\-0\.019\_\{\-0\.056\}^\{\+0\.045\}0\.780−0\.035\+0\.0290\.780\_\{\-0\.035\}^\{\+0\.029\}0\.0380\.03855views0\.712−0\.050\+0\.0420\.712\_\{\-0\.050\}^\{\+0\.042\}−0\.103−0\.094\+0\.088\-0\.103\_\{\-0\.094\}^\{\+0\.088\}0\.708−0\.051\+0\.0430\.708\_\{\-0\.051\}^\{\+0\.043\}−0\.161\-0\.161Noise×9\\times 90\.572−0\.052\+0\.0410\.572\_\{\-0\.052\}^\{\+0\.041\}0\.405−0\.042\+0\.0380\.405\_\{\-0\.042\}^\{\+0\.038\}0\.447−0\.059\+0\.0520\.447\_\{\-0\.059\}^\{\+0\.052\}0\.6210\.621Noise×25\\times 250\.555−0\.055\+0\.0390\.555\_\{\-0\.055\}^\{\+0\.039\}0\.438−0\.044\+0\.0360\.438\_\{\-0\.044\}^\{\+0\.036\}0\.393−0\.059\+0\.0520\.393\_\{\-0\.059\}^\{\+0\.052\}0\.6630\.663Si×0\.6\\times 0\.60\.438−0\.082\+0\.0730\.438\_\{\-0\.082\}^\{\+0\.073\}0\.167−0\.063\+0\.0600\.167\_\{\-0\.063\}^\{\+0\.060\}0\.417−0\.087\+0\.0790\.417\_\{\-0\.087\}^\{\+0\.079\}0\.2290\.229Si×0\.8\\times 0\.80\.387−0\.071\+0\.0670\.387\_\{\-0\.071\}^\{\+0\.067\}0\.209−0\.068\+0\.0560\.209\_\{\-0\.068\}^\{\+0\.056\}0\.359−0\.070\+0\.0690\.359\_\{\-0\.070\}^\{\+0\.069\}0\.2070\.207Si×1\.2\\times 1\.20\.166−0\.071\+0\.0670\.166\_\{\-0\.071\}^\{\+0\.067\}0\.435−0\.061\+0\.0550\.435\_\{\-0\.061\}^\{\+0\.055\}0\.070−0\.061\+0\.0610\.070\_\{\-0\.061\}^\{\+0\.061\}0\.2430\.243Si×1\.4\\times 1\.4−0\.017−0\.087\+0\.081\-0\.017\_\{\-0\.087\}^\{\+0\.081\}0\.476−0\.055\+0\.0490\.476\_\{\-0\.055\}^\{\+0\.049\}−0\.082−0\.079\+0\.070\-0\.082\_\{\-0\.079\}^\{\+0\.070\}0\.1150\.115Table 2:\(Density\) Pairwise Spearman correlations between cross\-seed instabilityσens\\sigma\_\{\\rm ens\}, ensemble mean errorϵmean\\epsilon\_\{\\rm mean\}, and emissivity proxybℰb\_\{\\mathcal\{E\}\}\.Conditionρ⁡\(σens,ϵmean\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)↑\\uparrowρ⁡\(bℰ,ϵmean\)\\rho\(b\_\{\\mathcal\{E\}\},\\epsilon\_\{\\rm mean\}\)partialρ⁡\(σens,ϵmean\|bℰ\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\)↑\\uparrowρ⁡\(σens,bℰ\)\\rho\(\\sigma\_\{\\rm ens\},b\_\{\\mathcal\{E\}\}\)noiseless \(300300v\)0\.726−0\.027\+0\.0250\.726\_\{\-0\.027\}^\{\+0\.025\}0\.001−0\.066\+0\.0670\.001\_\{\-0\.066\}^\{\+0\.067\}0\.732−0\.027\+0\.0240\.732\_\{\-0\.027\}^\{\+0\.024\}0\.1220\.122100100views0\.732−0\.023\+0\.0190\.732\_\{\-0\.023\}^\{\+0\.019\}0\.014−0\.068\+0\.0630\.014\_\{\-0\.068\}^\{\+0\.063\}0\.735−0\.022\+0\.0190\.735\_\{\-0\.022\}^\{\+0\.019\}0\.1150\.1152020views0\.717−0\.034\+0\.0310\.717\_\{\-0\.034\}^\{\+0\.031\}0\.092−0\.055\+0\.0530\.092\_\{\-0\.055\}^\{\+0\.053\}0\.716−0\.032\+0\.0290\.716\_\{\-0\.032\}^\{\+0\.029\}0\.0610\.06155views0\.611−0\.041\+0\.0340\.611\_\{\-0\.041\}^\{\+0\.034\}0\.015−0\.057\+0\.0490\.015\_\{\-0\.057\}^\{\+0\.049\}0\.626−0\.040\+0\.0320\.626\_\{\-0\.040\}^\{\+0\.032\}−0\.189\-0\.189Noise×9\\times 90\.590−0\.046\+0\.0440\.590\_\{\-0\.046\}^\{\+0\.044\}0\.325−0\.052\+0\.0460\.325\_\{\-0\.052\}^\{\+0\.046\}0\.521−0\.049\+0\.0460\.521\_\{\-0\.049\}^\{\+0\.046\}0\.5790\.579Noise×25\\times 250\.601−0\.049\+0\.0420\.601\_\{\-0\.049\}^\{\+0\.042\}0\.382−0\.051\+0\.0450\.382\_\{\-0\.051\}^\{\+0\.045\}0\.503−0\.058\+0\.0470\.503\_\{\-0\.058\}^\{\+0\.047\}0\.6240\.624Si×0\.6\\times 0\.60\.424−0\.073\+0\.0680\.424\_\{\-0\.073\}^\{\+0\.068\}0\.317−0\.067\+0\.0580\.317\_\{\-0\.067\}^\{\+0\.058\}0\.361−0\.072\+0\.0750\.361\_\{\-0\.072\}^\{\+0\.075\}0\.3120\.312Si×0\.8\\times 0\.80\.448−0\.065\+0\.0620\.448\_\{\-0\.065\}^\{\+0\.062\}0\.368−0\.052\+0\.0460\.368\_\{\-0\.052\}^\{\+0\.046\}0\.365−0\.064\+0\.0650\.365\_\{\-0\.064\}^\{\+0\.065\}0\.3560\.356Si×1\.2\\times 1\.20\.250−0\.071\+0\.0680\.250\_\{\-0\.071\}^\{\+0\.068\}0\.228−0\.069\+0\.0690\.228\_\{\-0\.069\}^\{\+0\.069\}0\.226−0\.065\+0\.0670\.226\_\{\-0\.065\}^\{\+0\.067\}0\.1410\.141Si×1\.4\\times 1\.40\.096−0\.063\+0\.0690\.096\_\{\-0\.063\}^\{\+0\.069\}0\.138−0\.066\+0\.0660\.138\_\{\-0\.066\}^\{\+0\.066\}0\.101−0\.057\+0\.0630\.101\_\{\-0\.057\}^\{\+0\.063\}−0\.027\-0\.027Table 3:\(Temperature\) Pairwise Spearman correlations between cross\-seed instabilityσens\\sigma\_\{\\rm ens\}, ensemble mean errorϵmean\\epsilon\_\{\\rm mean\}, and emissivity proxybℰb\_\{\\mathcal\{E\}\}\.Tables[2](https://arxiv.org/html/2609.28868#A3.T2)\(density\) and[3](https://arxiv.org/html/2609.28868#A3.T3)\(temperature\) show pairwise correlations betweenσens\\sigma\_\{\\rm ens\},ϵmean\\epsilon\_\{\\rm mean\}, andbℰb\_\{\\mathcal\{E\}\}, as well as the partial correlationρ⁡\(σens,ϵmean\|bℰ\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\), in four different regimes: noiseless \(300v, 100v\), sparse\-view \(20v, 5v\), high\-noise \(×9\\times 9,×25\\times 25\), and forward model abundance mismatch \(Si×0\.6\\times 0\.6,×0\.8\\times 0\.8,×1\.2\\times 1\.2,×1\.4\\times 1\.4\)\.

We evaluate the diagnostics on a30×18×830\\times 18\\times 8longitude\-latitude\-radius grid, with the radial coordinates spanning1\.051\.05\-33R⊙\. To assess within\-scene resampling variability, we treat each longitude slice, containing its associated latitude\-radius samples, as one cluster\. Resampling entire slices preserves their internal spatial dependence but does not explicitly model dependence between neighboring longitude slices\. For each of20002000bootstrap replicates, we draw3030longitude indices independently with replacement from the3030available indices and concatenate the corresponding slices\. The same sampled indices are applied toσens\\sigma\_\{\\rm ens\},ϵmean\\epsilon\_\{\\rm mean\}, andbℰb\_\{\\mathcal\{E\}\}\. We recompute the correlation, partial correlation, andnAUSE\\operatorname\{nAUSE\}\(Appendix[C\.2](https://arxiv.org/html/2609.28868#A3.SS2)\) on each resample and report the2\.52\.5th and97\.597\.5th percentiles\. Finally, correlation and sparsification calculations retain only samples in the inner radial band, and all statistics use unweighted samples rather than spherical area or volume weighting\.

Conditionpooledr=1\.33r=1\.33R⊙r=1\.61r=1\.61R⊙r=1\.89r=1\.89R⊙noiseless \(300300v\)0\.760\.76\[0\.760\.76\]0\.740\.74\[0\.740\.74\]0\.770\.77\[0\.750\.75\]0\.760\.76\[0\.750\.75\]100100views0\.770\.77\[0\.770\.77\]0\.750\.75\[0\.750\.75\]0\.780\.78\[0\.770\.77\]0\.770\.77\[0\.750\.75\]2020views0\.780\.78\[0\.780\.78\]0\.780\.78\[0\.780\.78\]0\.780\.78\[0\.780\.78\]0\.760\.76\[0\.730\.73\]55views0\.710\.71\[0\.710\.71\]0\.680\.68\[0\.680\.68\]0\.750\.75\[0\.750\.75\]0\.610\.61\[0\.600\.60\]Noise×9\\times 90\.570\.57\[0\.450\.45\]0\.590\.59\[0\.580\.58\]0\.540\.54\[0\.440\.44\]0\.410\.41\[0\.200\.20\]Noise×25\\times 250\.560\.56\[0\.390\.39\]0\.530\.53\[0\.510\.51\]0\.540\.54\[0\.410\.41\]0\.380\.38\[0\.150\.15\]Si×0\.6\\times 0\.60\.440\.44\[0\.420\.42\]0\.470\.47\[0\.500\.50\]0\.460\.46\[0\.440\.44\]0\.370\.37\[0\.330\.33\]Si×0\.8\\times 0\.80\.390\.39\[0\.360\.36\]0\.390\.39\[0\.430\.43\]0\.400\.40\[0\.370\.37\]0\.370\.37\[0\.280\.28\]Si×1\.2\\times 1\.20\.170\.17\[0\.070\.07\]0\.120\.12\[0\.140\.14\]0\.130\.13\[0\.090\.09\]0\.170\.17\[0\.080\.08\]Si×1\.4\\times 1\.4−0\.02\-0\.02\[−0\.08\-0\.08\]−0\.09\-0\.09\[−0\.02\-0\.02\]−0\.04\-0\.04\[−0\.01\-0\.01\]−0\.04\-0\.04\[−0\.05\-0\.05\]Table 4:\(Density\) Per\-shell correlationρ⁡\(σens,ϵmean\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)\(partialρ⁡\(σens,ϵmean\|bℰ\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\)in brackets\), for selected conditions\.Conditionpooledr=1\.33r=1\.33R⊙r=1\.61r=1\.61R⊙r=1\.89r=1\.89R⊙noiseless \(300300v\)0\.730\.73\[0\.730\.73\]0\.700\.70\[0\.700\.70\]0\.720\.72\[0\.700\.70\]0\.770\.77\[0\.750\.75\]100100views0\.730\.73\[0\.740\.74\]0\.730\.73\[0\.730\.73\]0\.700\.70\[0\.680\.68\]0\.760\.76\[0\.750\.75\]2020views0\.720\.72\[0\.720\.72\]0\.720\.72\[0\.730\.73\]0\.670\.67\[0\.650\.65\]0\.750\.75\[0\.720\.72\]55views0\.610\.61\[0\.630\.63\]0\.700\.70\[0\.720\.72\]0\.500\.50\[0\.500\.50\]0\.530\.53\[0\.510\.51\]Noise×9\\times 90\.590\.59\[0\.520\.52\]0\.660\.66\[0\.660\.66\]0\.540\.54\[0\.450\.45\]0\.490\.49\[0\.300\.30\]Noise×25\\times 250\.600\.60\[0\.500\.50\]0\.610\.61\[0\.610\.61\]0\.550\.55\[0\.440\.44\]0\.500\.50\[0\.290\.29\]Si×0\.6\\times 0\.60\.420\.42\[0\.360\.36\]0\.300\.30\[0\.310\.31\]0\.380\.38\[0\.300\.30\]0\.500\.50\[0\.430\.43\]Si×0\.8\\times 0\.80\.450\.45\[0\.360\.36\]0\.350\.35\[0\.350\.35\]0\.430\.43\[0\.300\.30\]0\.520\.52\[0\.380\.38\]Si×1\.2\\times 1\.20\.250\.25\[0\.230\.23\]0\.300\.30\[0\.330\.33\]0\.180\.18\[0\.190\.19\]0\.240\.24\[0\.210\.21\]Si×1\.4\\times 1\.40\.100\.10\[0\.100\.10\]0\.140\.14\[0\.200\.20\]0\.080\.08\[0\.120\.12\]0\.060\.06\[0\.070\.07\]Table 5:\(Temperature\) Per\-shell correlationρ⁡\(σens,ϵmean\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)\(partialρ⁡\(σens,ϵmean\|bℰ\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\)in brackets\), for selected conditions\.In Tables[4](https://arxiv.org/html/2609.28868#A3.T4)\(density\) and[5](https://arxiv.org/html/2609.28868#A3.T5)\(temperature\), we show within\-shell Spearman correlationρ⁡\(σens,ϵmean\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)between cross\-seed instability and absolute ensemble mean error, with the partial correlationρ⁡\(σens,ϵmean\|bℰ\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\\,\|\\,b\_\{\\mathcal\{E\}\}\)in brackets\. We compute these correlations for 3 shells \(columns\) at R=\[1\.33,1\.61,1\.89\]=\[1\.33,1\.61,1\.89\]R⊙, each with 540 voxels on a controlled grid\. Within\-shell values track the pooled correlation for the selected conditions, showing that the correlation is not solely explained by a radial trend\.

### C\.2Sparsification Analysis

A useful error\-ranking proxy should allow the average error to improve by systematically discarding voxels it flags as least trustworthy \(sparsification process\)\. A metric we use is the Area Under the Sparsification Error curve \(AUSE\\operatorname\{AUSE\},[Ilg et al\. \(2018\)](https://arxiv.org/html/2609.28868#bib.bib8);[Poggi et al\. \(2020\)](https://arxiv.org/html/2609.28868#bib.bib9)\), a standard protocol used to assess neural\-field uncertainty quantification\([Shen et al\., 2022](https://arxiv.org/html/2609.28868#bib.bib17);[Sünderhauf et al\., 2023](https://arxiv.org/html/2609.28868#bib.bib11);[Goli et al\., 2024](https://arxiv.org/html/2609.28868#bib.bib12)\)\. To construct this, we pick a ranking scalarzz\(such asσens\\sigma\_\{\\rm ens\},bℰb\_\{\\mathcal\{E\}\},ϵmean\\epsilon\_\{\\rm mean\}\(oracle\), random\), and sort all voxels by decreasingzz\. We note that the random baseline is the exact expected uniform\-retention curve \(the global mean error\), not a finite permutation average\. For a total ofNNvoxels, we then parameterize the process of removing voxels viaKz​\(t\)K\_\{z\}\(t\), the subset retaining approximately the lowest\-ranked\(1−t\)\(1\-t\)fraction of voxels\. We then define the risk as the mean error over the kept set

Riskz⁡\(t\)=1\|Kz​\(t\)\|​∑i∈Kz​\(t\)ϵmean,i\.\\operatorname\{Risk\}\_\{z\}\(t\)=\\frac\{1\}\{\|K\_\{z\}\(t\)\|\}\\sum\_\{i\\in K\_\{z\}\(t\)\}\\epsilon\_\{\\rm\{mean\},i\}\.\(23\)
PlottingRiskz⁡\(t\)\\operatorname\{Risk\}\_\{z\}\(t\)againstttgives the sparsification curve\. Att=0t=0, all curves retain all voxels\. The oraclez=ϵmeanz=\\epsilon\_\{\\rm mean\}decreases the fastest as it removes the largest errors first, while a random ranking keeps the MAE relatively flat\. A useful ranking proxy should approach the oracle, while arbitrary proxies may perform similar to or worse than the random baseline\.

To summarize each curve with a single metric, we first compute the sparsification error

SEz⁡\(t\)=Riskz⁡\(t\)−Riskoracle⁡\(t\)≥0,\\operatorname\{SE\}\_\{z\}\(t\)=\\operatorname\{Risk\}\_\{z\}\(t\)\-\\operatorname\{Risk\}\_\{\\rm oracle\}\(t\)\\geq 0,\(24\)
which is the gap to the oracle at each level, and then compute the area under the curve

AUSEz=∫0tmaxSEz⁡\(t\)​𝑑t,\\operatorname\{AUSE\}\_\{z\}=\\int\_\{0\}^\{t\_\{\\rm max\}\}\\operatorname\{SE\}\_\{z\}\(t\)dt,\(25\)
where we integrate up totmax=0\.95t\_\{\\rm max\}=0\.95to avoid a noisy terminal spike when few voxels remain\. In practice, this is done using a discretizedtt\-grid\. A lowerAUSE\\operatorname\{AUSE\}indicates a ranking closer to that of the oracle \(AUSE=0\\operatorname\{AUSE\}=0\)\. Finally, to compare across conditions, we normalize theAUSE\\operatorname\{AUSE\}by a random ranking, such that00gives perfect, while11gives no\-better\-than\-random, and\>1\>1gives worse\-than\-random:

nAUSEz=AUSEzAUSErandom\.\\operatorname\{nAUSE\}\_\{z\}=\\frac\{\\operatorname\{AUSE\_\{z\}\}\}\{\\operatorname\{AUSE\}\_\{\\rm random\}\}\.\(26\)
![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/fig_sparsification.png)Figure 10:Density\-error sparsification curves for the noiseless, high\-noise, sparse\-view, and Si\-mismatch regimes\. Lower remaining MAE is better, and the random reference is the exact expected uniform\-retention curve\.Figure[10](https://arxiv.org/html/2609.28868#A3.F10)shows the sparsification curves for four main regimes: noiseless, high\-noise, sparse\-view, and forward\-model mismatch\. In each plot, we show curves for various ranking scalarszz: the cross\-seed instability \(green\), emissivity proxy \(orange\), oracle \(black\), and random \(gray\)\. We find that across the matched\-model noiseless, high\-noise, and sparse\-view regimes, the cross\-seed instability tracks the oracle closely and much better than our physical emissivity proxy, and degrades towards random ranking under the strongest tested forward\-model mismatch\. Tables[6](https://arxiv.org/html/2609.28868#A3.T6)\(density\) and[7](https://arxiv.org/html/2609.28868#A3.T7)\(temperature\) tabulate the values in the curves, along with the rawAUSE\\operatorname\{AUSE\}values\. ThenAUSE\\operatorname\{nAUSE\}intervals are95%95\\%within\-scene longitude\-slice bootstrap intervals using the same20002000\-resample procedure\. Note that rawAUSE\\operatorname\{AUSE\}combines the scale of reconstruction error with ranking quality, so we therefore usenAUSE\\operatorname\{nAUSE\}for comparisons of relative ranking performance across conditions and report rawAUSE\\operatorname\{AUSE\}for reference\.

ConditionAUSEσ\\operatorname\{AUSE\}\_\{\\sigma\}AUSEb\\operatorname\{AUSE\}\_\{b\}nAUSEσ\\operatorname\{nAUSE\}\_\{\\sigma\}nAUSEb\\operatorname\{nAUSE\}\_\{b\}noiseless \(300300v\)0\.00210\.00210\.01450\.01450\.115−0\.012\+0\.0190\.115\_\{\-0\.012\}^\{\+0\.019\}0\.809−0\.096\+0\.1200\.809\_\{\-0\.096\}^\{\+0\.120\}100100views0\.00210\.00210\.01470\.01470\.113−0\.012\+0\.0160\.113\_\{\-0\.012\}^\{\+0\.016\}0\.794−0\.094\+0\.1150\.794\_\{\-0\.094\}^\{\+0\.115\}2020views0\.00620\.00620\.06570\.06570\.117−0\.020\+0\.0260\.117\_\{\-0\.020\}^\{\+0\.026\}1\.239−0\.102\+0\.1011\.239\_\{\-0\.102\}^\{\+0\.101\}55views0\.03270\.03270\.18340\.18340\.222−0\.035\+0\.0430\.222\_\{\-0\.035\}^\{\+0\.043\}1\.245−0\.135\+0\.1391\.245\_\{\-0\.135\}^\{\+0\.139\}Noise×9\\times 90\.00850\.00850\.01610\.01610\.274−0\.036\+0\.0470\.274\_\{\-0\.036\}^\{\+0\.047\}0\.520−0\.063\+0\.0750\.520\_\{\-0\.063\}^\{\+0\.075\}Noise×25\\times 250\.01080\.01080\.01680\.01680\.304−0\.044\+0\.0560\.304\_\{\-0\.044\}^\{\+0\.056\}0\.472−0\.055\+0\.0670\.472\_\{\-0\.055\}^\{\+0\.067\}Si×0\.6\\times 0\.60\.06760\.06760\.13830\.13830\.408−0\.062\+0\.0740\.408\_\{\-0\.062\}^\{\+0\.074\}0\.834−0\.097\+0\.1020\.834\_\{\-0\.097\}^\{\+0\.102\}Si×0\.8\\times 0\.80\.03120\.03120\.05790\.05790\.416−0\.050\+0\.0600\.416\_\{\-0\.050\}^\{\+0\.060\}0\.772−0\.096\+0\.1070\.772\_\{\-0\.096\}^\{\+0\.107\}Si×1\.2\\times 1\.20\.03020\.03020\.02270\.02270\.610−0\.060\+0\.0720\.610\_\{\-0\.060\}^\{\+0\.072\}0\.459−0\.047\+0\.0630\.459\_\{\-0\.047\}^\{\+0\.063\}Si×1\.4\\times 1\.40\.06760\.06760\.03350\.03350\.932−0\.084\+0\.0980\.932\_\{\-0\.084\}^\{\+0\.098\}0\.462−0\.046\+0\.0600\.462\_\{\-0\.046\}^\{\+0\.060\}Table 6:\(Density\) Sparsification summary for selected ranking metrics across selected conditions\.ConditionAUSEσ\\operatorname\{AUSE\}\_\{\\sigma\}AUSEb\\operatorname\{AUSE\}\_\{b\}nAUSEσ\\operatorname\{nAUSE\}\_\{\\sigma\}nAUSEb\\operatorname\{nAUSE\}\_\{b\}noiseless \(300300v\)0\.00080\.00080\.00520\.00520\.140−0\.025\+0\.0280\.140\_\{\-0\.025\}^\{\+0\.028\}0\.855−0\.156\+0\.1900\.855\_\{\-0\.156\}^\{\+0\.190\}100100views0\.00090\.00090\.00510\.00510\.142−0\.026\+0\.0290\.142\_\{\-0\.026\}^\{\+0\.029\}0\.838−0\.152\+0\.1840\.838\_\{\-0\.152\}^\{\+0\.184\}2020views0\.00170\.00170\.00980\.00980\.161−0\.026\+0\.0330\.161\_\{\-0\.026\}^\{\+0\.033\}0\.910−0\.123\+0\.1440\.910\_\{\-0\.123\}^\{\+0\.144\}55views0\.00650\.00650\.02400\.02400\.281−0\.031\+0\.0370\.281\_\{\-0\.031\}^\{\+0\.037\}1\.037−0\.085\+0\.0921\.037\_\{\-0\.085\}^\{\+0\.092\}Noise×9\\times 90\.00250\.00250\.00600\.00600\.251−0\.045\+0\.0520\.251\_\{\-0\.045\}^\{\+0\.052\}0\.599−0\.107\+0\.1300\.599\_\{\-0\.107\}^\{\+0\.130\}Noise×25\\times 250\.00300\.00300\.00610\.00610\.266−0\.042\+0\.0560\.266\_\{\-0\.042\}^\{\+0\.056\}0\.550−0\.095\+0\.1210\.550\_\{\-0\.095\}^\{\+0\.121\}Si×0\.6\\times 0\.60\.01550\.01550\.02870\.02870\.349−0\.076\+0\.0900\.349\_\{\-0\.076\}^\{\+0\.090\}0\.644−0\.090\+0\.0960\.644\_\{\-0\.090\}^\{\+0\.096\}Si×0\.8\\times 0\.80\.00670\.00670\.00980\.00980\.344−0\.060\+0\.0690\.344\_\{\-0\.060\}^\{\+0\.069\}0\.507−0\.051\+0\.0570\.507\_\{\-0\.051\}^\{\+0\.057\}Si×1\.2\\times 1\.20\.00640\.00640\.00760\.00760\.587−0\.062\+0\.0670\.587\_\{\-0\.062\}^\{\+0\.067\}0\.706−0\.089\+0\.1020\.706\_\{\-0\.089\}^\{\+0\.102\}Si×1\.4\\times 1\.40\.01140\.01140\.01120\.01120\.827−0\.078\+0\.0770\.827\_\{\-0\.078\}^\{\+0\.077\}0\.811−0\.086\+0\.0930\.811\_\{\-0\.086\}^\{\+0\.093\}Table 7:\(Temperature\) Sparsification summary for selected ranking metrics across selected conditions\.Finally, to confirm that no single training run drives the localization signal, we recompute the full correlationρ⁡\(σens,ϵmean\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)andnAUSEσ\\operatorname\{nAUSE\}\_\{\\sigma\}on each of the ten leave\-one\-seed\-out sub\-ensembles \(Table[8](https://arxiv.org/html/2609.28868#A3.T8)\)\. We find that the LOO values stay within a comparably tight band, so the reported correlation values are not artifacts of any individual seed\.

Conditionnen\_\{\\rm e\}fullρ\\rhonen\_\{\\rm e\}LOOρ\\rhonen\_\{\\rm e\}LOOnAUSEσ\\operatorname\{nAUSE\}\_\{\\sigma\}TeT\_\{\\rm e\}fullρ\\rhoTeT\_\{\\rm e\}LOOρ\\rhoTeT\_\{\\rm e\}LOOnAUSEσ\\operatorname\{nAUSE\}\_\{\\sigma\}300300views0\.7610\.761\[0\.755,0\.767\]\[0\.755,0\.767\]\[0\.115,0\.120\]\[0\.115,0\.120\]0\.7260\.726\[0\.717,0\.735\]\[0\.717,0\.735\]\[0\.138,0\.144\]\[0\.138,0\.144\]100100views0\.7730\.773\[0\.765,0\.775\]\[0\.765,0\.775\]\[0\.112,0\.128\]\[0\.112,0\.128\]0\.7320\.732\[0\.726,0\.735\]\[0\.726,0\.735\]\[0\.141,0\.145\]\[0\.141,0\.145\]2020views0\.7780\.778\[0\.773,0\.783\]\[0\.773,0\.783\]\[0\.114,0\.124\]\[0\.114,0\.124\]0\.7170\.717\[0\.708,0\.722\]\[0\.708,0\.722\]\[0\.160,0\.166\]\[0\.160,0\.166\]55views0\.7120\.712\[0\.695,0\.731\]\[0\.695,0\.731\]\[0\.205,0\.240\]\[0\.205,0\.240\]0\.6110\.611\[0\.599,0\.619\]\[0\.599,0\.619\]\[0\.276,0\.293\]\[0\.276,0\.293\]Noise×9\\times 90\.5720\.572\[0\.559,0\.568\]\[0\.559,0\.568\]\[0\.280,0\.287\]\[0\.280,0\.287\]0\.5900\.590\[0\.582,0\.590\]\[0\.582,0\.590\]\[0\.251,0\.262\]\[0\.251,0\.262\]Noise×25\\times 250\.5550\.555\[0\.540,0\.552\]\[0\.540,0\.552\]\[0\.307,0\.324\]\[0\.307,0\.324\]0\.6010\.601\[0\.585,0\.603\]\[0\.585,0\.603\]\[0\.263,0\.286\]\[0\.263,0\.286\]Si×0\.6\\times 0\.60\.4380\.438\[0\.436,0\.442\]\[0\.436,0\.442\]\[0\.405,0\.414\]\[0\.405,0\.414\]0\.4240\.424\[0\.408,0\.440\]\[0\.408,0\.440\]\[0\.337,0\.356\]\[0\.337,0\.356\]Si×0\.8\\times 0\.80\.3870\.387\[0\.380,0\.403\]\[0\.380,0\.403\]\[0\.409,0\.422\]\[0\.409,0\.422\]0\.4480\.448\[0\.437,0\.450\]\[0\.437,0\.450\]\[0\.339,0\.352\]\[0\.339,0\.352\]Si×1\.2\\times 1\.20\.1660\.166\[0\.158,0\.174\]\[0\.158,0\.174\]\[0\.605,0\.625\]\[0\.605,0\.625\]0\.2500\.250\[0\.241,0\.256\]\[0\.241,0\.256\]\[0\.585,0\.604\]\[0\.585,0\.604\]Si×1\.4\\times 1\.4−0\.017\-0\.017\[−0\.023,−0\.012\]\[\-0\.023,\-0\.012\]\[0\.926,0\.951\]\[0\.926,0\.951\]0\.0960\.096\[0\.088,0\.107\]\[0\.088,0\.107\]\[0\.812,0\.835\]\[0\.812,0\.835\]Table 8:Leave\-one\-out \(LOO\) ranges ofρ⁡\(σens,ϵmean\)\\rho\(\\sigma\_\{\\rm ens\},\\epsilon\_\{\\rm mean\}\)andnAUSEσ\\operatorname\{nAUSE\}\_\{\\sigma\}for both density and temperature\.
### C\.3Seed\-Deviation Error Capture

As a complementary field\-space consistency test, we project the signed ensemble error onto the vectors spanned by the ensemble\-deviations\. Define the ensemble\-deviation subspace asSens=span⁡\{δ1,…,δK\}S\_\{\\rm ens\}=\\operatorname\{span\}\\\{\\delta\_\{1\},\\dots,\\delta\_\{K\}\\\}, whereδk=m^k−m¯\\delta\_\{k\}=\\hat\{m\}\_\{k\}\-\\overline\{m\}are the per\-seed field deviations\. Define the rank ofSensS\_\{\\rm ens\}asr≤K−1r\\leq K\-1, since∑kδk=0\\sum\_\{k\}\\delta\_\{k\}=0removes one degree of freedom\. Then, we can define the orthogonal projection on this subspace by lettingQ∈ℝM×rQ\\in\\mathbb\{R\}^\{M\\times r\}be an orthonormal basis ofSensS\_\{\\rm ens\}, andPens=Q​QTP\_\{\\rm ens\}=QQ^\{T\}as the orthogonal projection operator\. This allows us to define an error\-capture energy fraction

Ecap=‖Pens​e‖2‖e‖2=‖QT​e‖2‖e‖2∈\[0,1\],E\_\{\\rm cap\}=\\frac\{\|\|P\_\{\\rm ens\}e\|\|^\{2\}\}\{\|\|e\|\|^\{2\}\}=\\frac\{\|\|Q^\{T\}e\|\|^\{2\}\}\{\|\|e\|\|^\{2\}\}\\quad\\in\[0,1\],\(27\)
which is the fraction of squaredL2L^\{2\}norm of the signed reconstruction error that lies within the seed\-deviation subspace\. We additionally report enrichment relative to the rank\-matched isotropic expectationEcaprand=r/ME\_\{\\rm cap\}^\{\\rm rand\}=r/Mfor both density and temperature\. Note that in our experiment, we useK=10K=10seeds and all reported full\-ensemble deviation matrices have numerical rankr=K−1=9r=K\-1=9under a relative tolerance of10−810^\{\-8\}on the singular values\. This is computed using the same control grid as above \(M=4320M=4320per\-field\), so our isotropic baseline sits at aroundEcaprand≈0\.002E\_\{\\rm cap\}^\{\\rm rand\}\\approx 0\.002\. Additionally,EcapE\_\{\\rm cap\}depends on the chosen control\-grid discretization and Euclidean inner product\.

A highEcapE\_\{\\rm cap\}indicates directional alignment, but such alignment could be inflated by \(i\) reusing the same finite ensemble to construct both the signed errore=m¯−m∗e=\\overline\{m\}\-m^\{\*\}and the deviation span, or \(ii\) generic spatial structure, where the signed erroreeand the seed\-deviation fields share smooth, low\-frequency modes even without scene\-specific alignment\.

We test \(i\) by computing a split\-seed validation test, where we split theK=10K=10seeds into two groupsAAandBB\. We construct the span from the per\-seed field deviations inAA,SA=span⁡\{m^k−m¯A:k∈A\}S\_\{A\}=\\operatorname\{span\}\\\{\\hat\{m\}\_\{k\}\-\\overline\{m\}\_\{A\}\\,:\\,k\\in A\\\}, and separately construct the signed ensemble mean erroreB=m¯B−m∗e\_\{B\}=\\overline\{m\}\_\{B\}\-m^\{\*\}fromBB\. We then compute the error from ensembleBBcaptured by the span from ensembleAA:

EcapB→A=‖PA​eB‖2‖eB‖2,E\_\{\\rm cap\}^\{B\\rightarrow A\}=\\frac\{\|\|P\_\{A\}e\_\{B\}\|\|^\{2\}\}\{\|\|e\_\{B\}\|\|^\{2\}\},\(28\)
repeated over all balanced splits, and report the mean and the2\.52\.5th and97\.597\.5th percentile interval of the finite split\-sensitivity distribution\. This explicitly tests whether a subspace learned from one subset of seeds captures the error of a disjoint seed subset, rather than evaluating the subspace and error from exactly the same finite ensemble\. For ourK=10K=10ensemble, we enumerate all\(105\)=252\\binom\{10\}\{5\}=252ordered choices ofAA\(126126complementary partitions evaluated in both directions\), withBBas its complement, and report the exact finite split\-sensitivity distribution, rather than a Monte Carlo approximation\. We find that across matched\-model conditions, ourEcapB→AE\_\{\\rm cap\}^\{B\\rightarrow A\}values are substantially larger than the rank\-matched isotropic expectation\. We note that our five\-member group has a numerical rank of four under the same tolerance, so its rank\-matched isotropic baseline is4/4320≈0\.09%4/4320\\approx 0\.09\\%, as opposed to the0\.2%0\.2\\%rank\-9 baseline for the originalEcapE\_\{\\rm cap\}\.

We test \(ii\) by using a stronger structured null: we cyclically rotate the ensemble\-deviation fieldsδk\\delta\_\{k\}relative to the signed error fieldeethrough all2929nonzero longitude shifts, preserving spatial structure radially and latitudinally while disrupting the longitudinal alignment\. More specifically, we choose a cyclic longitude shift, apply that same shift to all deviationsδk\\delta\_\{k\}, keepingeefixed\. We then computeEc​a​pE\_\{cap\}for the shifted deviation span, and repeat for all2929nonzero cyclic longitude shifts\. In implementation, we equivalently keep the deviation span fixed and cyclically shift the signed field error\. Finally, we compute the meanμ\\muand compare with the originalEc​a​pE\_\{cap\}with no longitudinal rotations\. Again across matched\-model conditions, we find that the non\-rotatedEc​a​pE\_\{cap\}exceeds the rotation\-null mean and is at the top of the distribution in all but the 100\-view temperature reconstruction\. The reduced capture indicates that the original longitudinal alignment contributes to the observed overlap beyond generic spatial structure alone\.

ConditionEcapE\_\{\\rm cap\}%\\%Ecap/EcaprandE\_\{\\rm cap\}/E\_\{\\rm cap\}^\{\\rm rand\}\(enrichment\)EcapB→AE\_\{\\rm cap\}^\{B\\rightarrow A\}%\\%rot\-nullμ\\mu%\\%EcapE\_\{\\rm cap\}rot%\\%noiseless \(300300v\)7\.57\.536×36\\times3\.6−3\.3\+2\.73\.6^\{\+2\.7\}\_\{\-3\.3\}2\.52\.5100100100100views4\.84\.823×23\\times2\.5−2\.2\+2\.72\.5^\{\+2\.7\}\_\{\-2\.2\}2\.12\.11001002020views27\.227\.2130×130\\times14\.4−8\.0\+8\.614\.4^\{\+8\.6\}\_\{\-8\.0\}5\.55\.510010055views32\.132\.1154×154\\times17\.9−13\.3\+11\.617\.9^\{\+11\.6\}\_\{\-13\.3\}20\.920\.9100100Noise×9\\times 96\.06\.029×29\\times2\.7−2\.0\+2\.32\.7^\{\+2\.3\}\_\{\-2\.0\}0\.90\.9100100Noise×25\\times 2511\.011\.053×53\\times5\.9−4\.4\+4\.75\.9^\{\+4\.7\}\_\{\-4\.4\}2\.32\.3100100Si×0\.6\\times 0\.63\.43\.416×16\\times1\.6−1\.1\+1\.71\.6^\{\+1\.7\}\_\{\-1\.1\}2\.12\.19797Si×0\.8\\times 0\.88\.08\.038×38\\times4\.3−3\.6\+3\.84\.3^\{\+3\.8\}\_\{\-3\.6\}2\.02\.0100100Si×1\.2\\times 1\.24\.84\.823×23\\times2\.1−1\.5\+1\.52\.1^\{\+1\.5\}\_\{\-1\.5\}1\.71\.7100100Si×1\.4\\times 1\.41\.01\.05×5\\times0\.6−0\.4\+0\.60\.6^\{\+0\.6\}\_\{\-0\.4\}0\.80\.87272Table 9:\(Density\) Error\-capture diagnostics as an additional field\-space consistency metric\.ConditionEcapE\_\{\\rm cap\}%\\%Ecap/EcaprandE\_\{\\rm cap\}/E\_\{\\rm cap\}^\{\\rm rand\}\(enrichment\)EcapB→AE\_\{\\rm cap\}^\{B\\rightarrow A\}%\\%rot\-nullμ\\mu%\\%EcapE\_\{\\rm cap\}rot%\\%noiseless \(300300v\)5\.05\.024×24\\times2\.6−1\.9\+2\.02\.6^\{\+2\.0\}\_\{\-1\.9\}1\.51\.5100100100100views2\.42\.411×11\\times1\.2−0\.9\+1\.41\.2^\{\+1\.4\}\_\{\-0\.9\}1\.41\.479792020views20\.520\.598×98\\times11\.0−7\.3\+7\.711\.0^\{\+7\.7\}\_\{\-7\.3\}4\.04\.010010055views24\.024\.0115×115\\times12\.8−8\.3\+8\.312\.8^\{\+8\.3\}\_\{\-8\.3\}18\.718\.7100100Noise×9\\times 99\.39\.345×45\\times4\.6−3\.6\+3\.14\.6^\{\+3\.1\}\_\{\-3\.6\}2\.12\.1100100Noise×25\\times 2520\.620\.699×99\\times12\.8−7\.0\+7\.012\.8^\{\+7\.0\}\_\{\-7\.0\}3\.43\.4100100Si×0\.6\\times 0\.612\.112\.158×58\\times6\.3−3\.7\+4\.46\.3^\{\+4\.4\}\_\{\-3\.7\}1\.01\.0100100Si×0\.8\\times 0\.86\.06\.029×29\\times3\.5−2\.8\+3\.73\.5^\{\+3\.7\}\_\{\-2\.8\}0\.80\.8100100Si×1\.2\\times 1\.22\.52\.512×12\\times1\.4−1\.3\+1\.11\.4^\{\+1\.1\}\_\{\-1\.3\}1\.11\.19393Si×1\.4\\times 1\.41\.91\.99×9\\times1\.2−1\.0\+1\.21\.2^\{\+1\.2\}\_\{\-1\.0\}0\.50\.59797Table 10:\(Temperature\) Error\-capture diagnostics as an additional field\-space consistency metric\.Tables[9](https://arxiv.org/html/2609.28868#A3.T9)and[10](https://arxiv.org/html/2609.28868#A3.T10)summarize our error capture diagnostics\. For selected conditions, we report theEcapE\_\{\\rm cap\}%\\%as the fraction of the squared signed\-error energy lying in the rank\-9 span, and the enrichment relative to an equal\-rank isotropic subspace\. We also report the split\-seed capture percentageEcapB→AE\_\{\\rm cap\}^\{B\\rightarrow A\}%\\%with its2\.52\.5th\-97\.597\.5th percentile interval, the mean capture percentageμ\\muof the longitude\-rolled deviations, and the percentile rot%\\%of the unrotatedEcapE\_\{\\rm cap\}relative to the rotation\-null distribution\.

## Appendix DBenchmark Studies and Ablations

### D\.1Reproducibility

Table[11](https://arxiv.org/html/2609.28868#A4.T11)provides a list of the main canonical hyperparameters used in our experiments\. Note that control configurations are rerun independently within each ablation study\. Because GPU hash\-grid optimization uses nondeterministic operations, fixing the nominal RNG seed does not guarantee bitwise\-identical training trajectories\. Additionally, different studies may use different RNG streams, so numerically identical configurations can differ slightly across tables\. We therefore interpret within\-study mean±\\pmstandard deviation comparisons, and do not treat small cross\-table differences between nominally identical controls as scientific effects\.

Unless otherwise stated, we fix the model architecture to be a multiresolution hash grid with a hashmap size of2202^\{20\}×2\\times 2features, and2020levels spanning resolutions from1616\(coarsest\) to512512\(finest\)\. Separate decoder heads, each containing two hidden layers of width6464, are used to predict log\-density and log\-temperature, and their biases are initialized to the midpoints of their corresponding clamp ranges, which lie within the respective LUT domains\. We train with300300evenly\-spaced views outside a30∘30^\{\\circ\}held\-out arc with a3∘3^\{\\circ\}buffer\. We use the AdamW optimizer with our asinh loss, and joint reconstructions optimize using batches of10241024rays for6060k steps, while our density\-only representations \(Appendix[D\.2](https://arxiv.org/html/2609.28868#A4.SS2)\) use3030k steps\. No observation noise or abundance mismatch is applied\. Our evaluation metrics are field\-space errors and held\-out image\-space errors\.

GroupNameDescriptionRenderingAABBxAABB∈\[−1,1\]3x\_\{\\rm AABB\}\\in\[\-1,1\]^\{3\}sample step sized​xAABB=1/256dx\_\{\\rm AABB\}=1/256LOS integrationmidpoint Riemann sumArchitecturehash grid2202^\{20\}hash map,×2\\times 2featureslevels2020levels,16→51216\\rightarrow 512coarse\-to\-fine resolutionsMLP headper\-field,22hidden layers, width6464TrainingOptimizerAdamWlearning rate5×10−45\\times 10^\{\-4\}weight decay10−510^\{\-5\}ray batch sizeb=1024b=1024stepss=60s=60kasinh scalesc=\[1\.3×10−2,4\.5×10−4,10−2,5\.8×10−2\]s\_\{c\}=\[1\.3\\times 10^\{\-2\},4\.5\\times 10^\{\-4\},10^\{\-2\},5\.8\\times 10^\{\-2\}\]ensemble seeds0xC0FFEE, 0xDECAF, 0xBADD1E, 0xD1CE0x8888, 0xB0BA, 0xFACADE, 0xCAFE0xF00D, 0xBADA55Forward Modelshot coefficientsαc=\[10−2,5×10−4,10−2,5×10−2\]\\alpha\_\{c\}=\[10^\{\-2\},5\\times 10^\{\-4\},10^\{\-2\},5\\times 10^\{\-2\}\]noise multiplierη∈\{0,1,9,25\}\\eta\\in\\\{0,1,9,25\\\}mismatch scalea∈\{0\.6,0\.8,1,1\.2,1\.4\}a\\in\\\{0\.6,0\.8,1,1\.2,1\.4\\\}Table 11:Table of canonical hyperparameters used in CoroNeRF\. Note that the listedscs\_\{c\}values are the base scales used forη≤1\\eta\\leq 1\. Forη\>1\\eta\>1,scs\_\{c\}is multiplied byη\\sqrt\{\\eta\}as described in Appendix[D\.6](https://arxiv.org/html/2609.28868#A4.SS6)\. The ordering ofαc\\alpha\_\{c\}andscs\_\{c\}values corresponds to Fe XIII 1075, Fe XIII 1080, Si IX 2585, and Si IX 3935, respectively\.
### D\.2Density Representation Baselines

Our density representational baselines below share the following properties\. We only reconstruct density with the GT temperature field supplied\. The observational dataset has300300evenly\-spaced views with a30∘30^\{\\circ\}arc holdout and a3∘3^\{\\circ\}buffer on both ends \(the arc holdout is used as validation\)\. Each view consists of a\[−3,3\]\[\-3,3\]R⊙FOV of two Fe XIII channels \(1075/10801075/1080nm\), with no observational noise or abundance scale mismatch\. All models take in as inputs a 3D Cartesian position mapped to the AABB\. We use the AdamW optimizer, with learning rate5×10−45\\times 10^\{\-4\}, weight decay10−510^\{\-5\}, and10241024ray batches for3030k steps\. Our loss is the fixed\-scale asinh image loss\. Each model is trained with33seeds \(0xC0FFEE, 0xDECAF, 0xBADD1E\)\.

The first baseline we test is a classical voxel grid, which we call theGridmodel\. The optimized variable is a dense trainable log\-density field with no encoder or MLP\. The grid dimensions match those of the native PSI cube, with colatitude converted to latitude:299×142×154299\\times 142\\times 154longitude\-latitude\-radius, where the longitude∈\[0,2​π\]\\in\[0,2\\pi\]is periodic \(wrap\-padded\), latitude∈\[−π/2,π/2\]\\in\[\-\\pi/2,\\pi/2\], and radius∈\[1,30\]\\in\[1,30\]R⊙\. Retrieving values from this trainable grid is done using trilinear interpolation via PyTorch’sgrid\_samplefunction, with a longitude wrap for periodicity\. All grid nodes are initialized as the midpoint of thelog⁡ne\\log n\_\{\\rm e\}clamp range\. The number of trainable parameters in this model is the total number of voxels6,538,5326,538,532\(=299⋅142⋅154=299\\cdot 142\\cdot 154\)\.

We additionally test the grid representation with a first\-order \(Tikhonov/gradient\) smoothness regularizer on the learned density grid, which we will call theGrid\-Regmodel\. For a gridgg, the regularizer has the formℒR=λlon​⟨\(Δlon​g\)2⟩\+λlat​⟨\(Δlat​g\)2⟩\+λr​⟨\(Δr​g\)2⟩\\mathcal\{L\}\_\{R\}=\\lambda\_\{\\rm lon\}\\langle\(\\Delta\_\{\\rm lon\}\\,g\)^\{2\}\\rangle\+\\lambda\_\{\\rm lat\}\\langle\(\\Delta\_\{\\rm lat\}\\,g\)^\{2\}\\rangle\+\\lambda\_\{\\rm r\}\\langle\(\\Delta\_\{\\rm r\}\\,g\)^\{2\}\\rangle, whereΔ\\Deltais the first difference along each axis and⟨⋅⟩\\langle\\cdot\\rangleis the mean over nodes\. We set the directional weightsλlon=λlat=λr=1\\lambda\_\{\\rm lon\}=\\lambda\_\{\\rm lat\}=\\lambda\_\{\\rm r\}=1\. The coefficient we use for the overall total loss isλsmooth=10\.0\\lambda\_\{\\rm smooth\}=10\.0, where the training objective isasinhErr\+10\.0×ℒR\\operatorname\{asinhErr\}\+10\.0\\times\\mathcal\{L\}\_\{R\}\. We chose10\.010\.0as it gave us the best preliminary results compared to other regularization values\. Finally, the regularizer does not change the amount of trainable parameters \(6\.56\.5M\)\.

The second baseline we use is a positional encoder \+ MLP, which we will call theMLPmodel\. The encoding frequencies are axis\-wise Fourier positional encoding withL=6L=6bands with corresponding frequencies\{20,…,25\}\\\{2^\{0\},\\ldots,2^\{5\}\\\}\(features aresin⁡\(2k​π​x\)\\sin\(2^\{k\}\\pi\\,x\)andcos⁡\(2k​π​x\)\\cos\(2^\{k\}\\pi\\,x\)per axis\)\. This gives a total of22features×\\times33axes×\\times66bands=36=36dims\. We additionally have a spherical\-harmonic encoding of the unit position direction up to degree22\(\(2\+1\)2=9\(2\+1\)^\{2\}=9dims\)\. We concatenate the raw position with the positional encoding and spherical\-harmonic encoding for a total of4848dims to the MLP input\. Our MLP has44hidden layers each with width256256\. SiLU activations connect the hidden layers, and the final layer is a linear output layer\. The output is a single scalarlog10⁡ne\\log\_\{10\}n\_\{\\rm e\}, and the final\-layer bias is initialized to the midpoint of thelog⁡ne\\log n\_\{\\rm e\}clamp range\. The total number of trainable parameters is210,177210,177\(=48⋅256\+256\+3⋅\(2562\+256\)\+\(256\+1\)\)\(=48\\cdot 256\+256\+3\\cdot\(256^\{2\}\+256\)\+\(256\+1\)\)\.

Our final and main representation that we eventually select for the main experiments is a multiresolution hash grid as described in Appendix[A\.5](https://arxiv.org/html/2609.28868#A1.SS5)\. The encoding hash grid has2020levels,22features per level, with the resolutions from1616\(coarsest\) to512512\(finest\)\. Each level has a2G2^\{G\}hashmap \(we testG∈\{17,20\}G\\in\\\{17,20\\\}\)\. Given a position, the hash grid produces a4040\-dimensional feature vector by concatenating the22features across all2020levels\. This is then fed into a decoder head with22hidden layers each with width6464\. Activations are SiLU, and the final layer is a linear output layer\. The decoder head has68496849trainable parameters\. OurHash20model has2020levels×220\\times 2^\{20\}hashmap×2\\times 2features \+6849=41,949,889≈41\.96849=41,949,889\\approx 41\.9M parameters\. OurHash17model has2020levels×217\\times 2^\{17\}hashmap×2\\times 2features \+6849=5,249,729≈5\.26849=5,249,729\\approx 5\.2M parameters\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_A2_density_slice_lon0.png)Figure 11:Density\-field reconstruction \(with GT temperature supplied\) at0∘0^\{\\circ\}longitude, spanning\[1\.1,3\.0\]\[1\.1,3\.0\]R⊙\. Each panel is a meridional slice \(latitude\-radius plot\)\. The top row shows the ground truth and reconstructions using classical spherical grid with Tikhonov regularization, positional\-encoder \+ MLP, and hash grid \+ decoder head representations\. The bottom row shows corresponding signed residualsΔ​log10​ne=log10⁡ne,pred−log10⁡ne,GT\\Delta\\log\_\{10\}n\_\{e\}=\\log\_\{10\}n\_\{e,\\mathrm\{pred\}\}\-\\log\_\{10\}n\_\{e,\\mathrm\{GT\}\}\. The multiresolution hash grid is able to reconstruct consistent radial structures much farther out than the classical method, and achieves lower errors on the inner radial boundaries near the photosphere\.ModelMAEinner​\(log10⁡ne\)\\text\{MAE\}\_\{\\text\{inner\}\}\(\\log\_\{10\}n\_\{\\rm e\}\)↓\\downarrowAbsRelinner​\(ne\)\\text\{AbsRel\}\_\{\\text\{inner\}\}\(n\_\{\\rm e\}\)↓\\downarrowasinhErr​\(I\)\\text\{asinhErr\}\(I\)↓\\downarrowPSNR​\(I\)\\text\{PSNR\}\(I\)↑\\uparrowHash200\.042±<0\.001\\mathbf\{0\.042\\pm<0\.001\}0\.094±<0\.001\\mathbf\{0\.094\\pm<0\.001\}0\.003±<0\.001\\mathbf\{0\.003\\pm<0\.001\}39\.59±0\.67\\mathbf\{39\.59\\pm 0\.67\}Hash170\.042±<0\.001\\mathbf\{0\.042\\pm<0\.001\}0\.095±<0\.0010\.095\\pm<0\.0010\.004±0\.0010\.004\\pm 0\.00138\.77±0\.6538\.77\\pm 0\.65MLP0\.097±0\.0010\.097\\pm 0\.0010\.254±0\.0040\.254\\pm 0\.0040\.024±0\.0030\.024\\pm 0\.00324\.65±0\.6224\.65\\pm 0\.62Grid\-Reg0\.168±<0\.0010\.168\\pm<0\.0010\.449±<0\.0010\.449\\pm<0\.0010\.053±<0\.0010\.053\\pm<0\.00111\.57±<0\.0111\.57\\pm<0\.01Grid0\.217±<0\.0010\.217\\pm<0\.0010\.571±<0\.0010\.571\\pm<0\.0010\.080±<0\.0010\.080\\pm<0\.00111\.30±<0\.0111\.30\\pm<0\.01Table 12:Density representation benchmark\. Each row indicates a model configuration, and metrics are reported as mean±\\pmstandard deviation over 3 seeds, with deviations below5×10−45\\times 10^\{\-4\}reported as<0\.001<0\.001\. The first column indicates the experiment family, and the last four columns are the relevant metrics\. Image metrics are averaged over views and channels, and density metrics are averaged over inner coronal radial shells1\.11\.1\-22R⊙\. The multiresolution hash grid outperforms the MLP by more than2×2\\times, and the classical grid by about4×4\\times\.Figure[11](https://arxiv.org/html/2609.28868#A4.F11)shows a meridional \(latitude\-radius\) slice of the reconstructed density at0∘0^\{\\circ\}longitude for various representational baselines, providing a complementary view of the reconstructed corona compared to a latitude\-longitude shell shown in Figure[2](https://arxiv.org/html/2609.28868#S5.F2)in the main text\. Table[12](https://arxiv.org/html/2609.28868#A4.T12)provides the field\-space and image\-space metrics for our density representations\. We find that the two tested hash\-table sizes,G∈\{17,20\}G\\in\\\{17,20\\\}, achieve nearly identical field error\.

### D\.3Loss Ablation Study

In our loss ablation study, we keep the same model architecture and observational setup as those described in Appendix[D\.1](https://arxiv.org/html/2609.28868#A4.SS1), except we introduce observational noise atη=1\\eta=1and alter the objective function\. We use the following objective family:

ℒtrain=λG​ℒG\+λA​ℒA,\\mathcal\{L\}\_\{\\mathrm\{train\}\}=\\lambda\_\{G\}\\mathcal\{L\}\_\{G\}\+\\lambda\_\{A\}\\mathcal\{L\}\_\{A\},\(29\)
whereℒG\\mathcal\{L\}\_\{G\}is the heteroscedastic Gaussian image loss, andℒA\\mathcal\{L\}\_\{A\}is the fixed\-scale asinh image loss\. More specifically, given rayqqand channelcc, the heteroscedastic Gaussian term is the inverse\-variance\-weighted squared image residual\(\(I^θ,q,c−yq,c\)/σq,c\)2\(\(\\hat\{I\}\_\{\\theta,q,c\}\-y\_\{q,c\}\)/\\sigma\_\{q,c\}\)^\{2\}, which is equivalent for optimization to the Gaussian NLL up to a positive constant factor and additive terms independent ofθ\\theta\. The whitening\-scaleσq,c\\sigma\_\{q,c\}is the target\-σ\\sigma, or the known per\-pixel, per\-channel observation noise standard deviation of our generative model \(Equation[6](https://arxiv.org/html/2609.28868#A1.E6)\): this is the exact heteroscedastic standard deviation used to add noise to the training images\. By contrast, the asinh term uses\|asinh⁡\(I^q,c/sc\)−asinh⁡\(yq,c/sc\)\|\|\\operatorname\{asinh\}\(\\hat\{I\}\_\{q,c\}/s\_\{c\}\)\-\\operatorname\{asinh\}\(y\_\{q,c\}/s\_\{c\}\)\|, wherescs\_\{c\}is a channel\-dependent fixed characteristic intensity scale\. Our experiment families varyλG\\lambda\_\{G\}andλA\\lambda\_\{A\}so as to test objectives that are pure Gaussian, mixed Gaussian and asinh, and pure asinh\. Each experiment family is trained with 3 seeds, and replicate runs use different observation\-noise realizations coupled to the run seed\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_E2_loss_transition_shell_residuals_r1p5.png)Figure 12:Density and temperature reconstructed residuals for different loss metric settings\. We find that the Gaussian\-only loss performs worst while the asinh\-only loss performs best, with hybrid losses landing in between the two regimes\.ObjectiveMAEinner​\(log10⁡ne\)\\text\{MAE\}\_\{\\text\{inner\}\}\(\\log\_\{10\}n\_\{\\rm e\}\)↓\\downarrowMAEinner​\(log10⁡Te\)\\text\{MAE\}\_\{\\text\{inner\}\}\(\\log\_\{10\}T\_\{\\rm e\}\)↓\\downarrowasinhErr​\(I\)\\text\{asinhErr\}\(I\)↓\\downarrowGaussian\-onlyλG=1\\lambda\_\{G\}=10\.145±0\.0100\.145\\pm 0\.0100\.075±0\.0200\.075\\pm 0\.0200\.060±0\.0060\.060\\pm 0\.006Fixed hybridλG=10−1\\lambda\_\{G\}=10^\{\-1\}0\.134±0\.0090\.134\\pm 0\.0090\.064±0\.0140\.064\\pm 0\.0140\.047±0\.0050\.047\\pm 0\.005Fixed hybridλG=10−3\\lambda\_\{G\}=10^\{\-3\}0\.087±0\.0010\.087\\pm 0\.0010\.036±0\.0010\.036\\pm 0\.0010\.032±0\.0020\.032\\pm 0\.002Fixed hybridλG=10−5\\lambda\_\{G\}=10^\{\-5\}0\.061±0\.0020\.061\\pm 0\.0020\.022±0\.0020\.022\\pm 0\.0020\.021±<0\.001\\mathbf\{0\.021\\pm<0\.001\}Fixed asinh onlyλG=0\\lambda\_\{G\}=00\.047±0\.001\\mathbf\{0\.047\\pm 0\.001\}0\.017±<0\.0010\.017\\pm<0\.0010\.022±<0\.0010\.022\\pm<0\.001Target\-σ\\sigmaasinhλG=0\\lambda\_\{G\}=00\.047±<0\.0010\.047\\pm<0\.0010\.016±<0\.001\\mathbf\{0\.016\\pm<0\.001\}0\.022±0\.0010\.022\\pm 0\.001Table 13:Loss ablation benchmark\. For the Gaussian\-only experiment,λA=0\\lambda\_\{A\}=0, otherwise it is fixed atλA=1\\lambda\_\{A\}=1\. We find that objectives that favor the asinh termℒA\\mathcal\{L\}\_\{A\}generally perform better than those that favor the heteroscedastic Gaussian loss term\.Under the tested training protocol, the pure asinh objective yields lower log\-fieldMAE\\operatorname\{MAE\}than the Gaussian\-only objective \(Figure[12](https://arxiv.org/html/2609.28868#A4.F12)and Table[13](https://arxiv.org/html/2609.28868#A4.T13)\)\. We therefore use it as the task\-aware objective for the subsequent reconstruction experiments\. As an auxiliary oracle variant, the final row, our target\-σ\\sigmacase, replacesscs\_\{c\}inℒA\\mathcal\{L\}\_\{A\}with the synthetic generator’s per\-pixel targetσ\\sigma\. The corresponding loss term becomes\|asinh⁡\(I^q,c/σq,c\)−asinh⁡\(yq,c/σq,c\)\|\|\\operatorname\{asinh\}\(\\hat\{I\}\_\{q,c\}/\\sigma\_\{q,c\}\)\-\\operatorname\{asinh\}\(y\_\{q,c\}/\\sigma\_\{q,c\}\)\|, and is thus included only as an oracle diagnostic ablation\. Finally, we note that this ablation compares complete objective choices rather than separately isolating the effects of intensity transformation, residual norm, and weighting\.

### D\.4Compute and Runtime

Our multiresolution hash grid has around4040M parameters \(2​M2Mper hash table,2020levels\), and is trained in a cluster GPU setting: training a single reconstruction takes≈3\\approx 3GPU\-hours on an8080GB NVIDIA A100100, and so aK=10K=10ensemble takes≈30\\approx 30aggregate GPU\-hours\. The ten localization conditions reported in Tables[2](https://arxiv.org/html/2609.28868#A3.T2)\-[10](https://arxiv.org/html/2609.28868#A3.T10)require approximately300300aggregate GPU\-hours\. Rendering a300300\-view set, as a dataset or post\-training analysis, can take≈10\\approx 10\-100100minutes on a local NVIDIA RTX 4090 depending on the resolution and integration step size\. Finally, diagnostics such asσens\\sigma\_\{\\rm ens\}are cheap as they only requireKKforward evaluations\.

We performed pilot tests to pick batch sizebband step countssfrom a compute\-quality trade\-off ablation study \(Table[14](https://arxiv.org/html/2609.28868#A4.T14)\)\. Note that these pilot runs use the earlier density\-only configuration with no observational noise, and are not directly comparable to the noise\-view sweep of the joint thermodynamic reconstruction in Figure[5](https://arxiv.org/html/2609.28868#S5.F5)\. We find that field error varies modestly across the tested settings, and note that because these tests were done only for hyperparameter selection at an earlier stage of the pipeline, they should not be used for scientific conclusions\.

ExpbbssVVDDMAE\(log10⁡ne\\log\_\{10\}n\_\{e\}\)↓\\downarrowPSNR\(I\)↑\\uparrowMSE\(log10⁡I\\log\_\{10\}I\)↓\\downarrowbig409640966060k303064640\.0960\.09630\.5630\.561\.75×10−41\.75\\times 10^\{\-4\}big409640966060k100010002562560\.0970\.09730\.7430\.741\.78×10−41\.78\\times 10^\{\-4\}med409640963030k100010002562560\.1060\.10626\.6026\.605\.46×10−45\.46\\times 10^\{\-4\}med409640963030k303064640\.1070\.10727\.4727\.474\.81×10−44\.81\\times 10^\{\-4\}base102410243030k100010002562560\.1090\.10924\.8824\.886\.40×10−46\.40\\times 10^\{\-4\}base102410243030k303064640\.1090\.10927\.8227\.825\.01×10−45\.01\\times 10^\{\-4\}Table 14:Compute\-quality trade\-off ablation study\. We vary the batch sizebb, stepsss, viewsVV, and image dimensionDD\. We find that the log\-densityMAE\\operatorname\{MAE\}varies modestly across the tested settings, with slightly better performance atb=4096b=4096,s=60s=60k\.
### D\.5Spectral Channel Ablation Study

This section provides details on our spectral channel ablation study\. We ablate the number of spectral channels for a total of 5 experiment families: Fe XIII10751075, Si IX39353935, Fe XIII pair, Si IX pair, and All four\. Each experiment family is trained with 3 seeds\.

Channel SetMAEinner​\(log10⁡ne\)\\text\{MAE\}\_\{\\text\{inner\}\}\(\\log\_\{10\}n\_\{\\rm e\}\)↓\\downarrowMAEinner​\(log10⁡Te\)\\text\{MAE\}\_\{\\text\{inner\}\}\(\\log\_\{10\}T\_\{\\rm e\}\)↓\\downarrowasinhErr\(observed channels\)↓\\downarrowall40\.039±<0\.001\\mathbf\{0\.039\\pm<0\.001\}0\.014±<0\.001\\mathbf\{0\.014\\pm<0\.001\}0\.008±0\.0020\.008\\pm 0\.002Fe XIII pair0\.097±0\.0040\.097\\pm 0\.0040\.023±0\.0010\.023\\pm 0\.0010\.009±0\.0030\.009\\pm 0\.003Si IX pair0\.068±0\.0010\.068\\pm 0\.0010\.058±0\.0010\.058\\pm 0\.0010\.004±0\.001\\mathbf\{0\.004\\pm 0\.001\}Fe107510750\.605±0\.0040\.605\\pm 0\.0040\.072±<0\.0010\.072\\pm<0\.0010\.006±0\.0010\.006\\pm 0\.001Si393539350\.530±0\.0280\.530\\pm 0\.0280\.195±0\.0060\.195\\pm 0\.0060\.005±0\.0020\.005\\pm 0\.002Table 15:Reconstruction metrics for our spectral ablation study\. The four\-line configuration achieves the lowest field errors and low across\-seed variability in this three\-seed ablation\. We can also observe that a good image\-space reconstruction does not imply a good physical\-field reconstruction: while the Si IX pair has the lowest image\-space validation loss, it still has higher physical\-field errors\.![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_B2_all4_multi_radius_shells.png)Figure 13:Field Reconstruction at three selected radii:1\.20,1\.51,1\.901\.20,1\.51,1\.90R⊙\(columns\)\. We show the field\-space GT, prediction, and signed residualsΔ​m=m^−m∗\\Delta m=\\hat\{m\}\-m^\{\*\}form∈\{log10⁡ne,log10⁡Te\}m\\in\\\{\\log\_\{10\}n\_\{\\rm e\},\\log\_\{10\}T\_\{\\rm e\}\\\}\.![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_B2_spectral_radial_curves.png)Figure 14:Field\-spaceMAE\\operatorname\{MAE\}radial dependence for the spectral ablation study\. Color denotes the number of spectral lines, and marker shape denotes the line set\. We find that having at least a pair of lines substantially reduces radial errors across1\.11\.1\-33R⊙\.![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_B2_image_field_grid_appendix.png)Figure 15:3D field\-spaceMAE\\operatorname\{MAE\}vs 2D held\-out image\-space validation loss, showing that a good image\-space reconstruction need not imply a good field\-space reconstruction\. Each point is one seed from the spectral ablation benchmark trained with different sets of spectral channels\. We find that fewer\-channel models can achieve lower observed\-channel image error than the four\-line model despite worse field recovery\.Table[15](https://arxiv.org/html/2609.28868#A4.T15)summarizes the field\-space and image\-space performance for all five experiment families, and Figure[13](https://arxiv.org/html/2609.28868#A4.F13)shows a sample joint reconstruction at three selected radii and corresponding residual panels\. We also provide radial profiles of the field\-spaceMAE\\operatorname\{MAE\}s \(Figure[14](https://arxiv.org/html/2609.28868#A4.F14)\)\. Finally, Figure[15](https://arxiv.org/html/2609.28868#A4.F15)shows image\-space versus field\-space error for all five experiment families, for both density and temperature \(as opposed to the density\-only Figure[4](https://arxiv.org/html/2609.28868#S5.F4)in the main text\)\. We note that for visualizations of radial shells, we select the median\-performing model \(out of three seeds\) usingMAE\\operatorname\{MAE\}of the inner band temperature field\.

### D\.6Noise/View Ablation Study

This section details the noise\-view stress ablation study described in Figure[5](https://arxiv.org/html/2609.28868#S5.F5)in the main text\. We keep the same model architecture and observational setup as described in Appendix[D\.1](https://arxiv.org/html/2609.28868#A4.SS1)\. However, we ablate the number of viewsVVby selecting evenly sampled views from the original300300views\. Likewise, we also ablate the noise scaleη\\eta\. Additionally, we calibrate the per\-channel asinh scale to the noise level:scs\_\{c\}is fixed at the baseline forη∈\{0,1\}\\eta\\in\\\{0,1\\\}, and scaled byη\\sqrt\{\\eta\}for higher noise, so it tracks the noise standard deviation \(variance scales asη\\eta, standard deviation asη\\sqrt\{\\eta\}\)\. The scale is constant within any single run\.

Our ablation study has four noise levels \(η∈\{0,1,9,25\}\\eta\\in\\\{0,1,9,25\\\}\) crossed with four view counts \(V∈\{5,20,100,300\}V\\in\\\{5,20,100,300\\\}\), for a total of 16 distinct combinations trained with three different seeds for 48 total models\. For noisy conditions, the three replicate seeds also use distinct observation\-noise realizations derived from the run seed\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_C2_field_robustness_shell_r1p5.png)Figure 16:Robustness in physical field reconstruction\. The columns show ground truth and the reconstructed density \(top row\) and temperature \(bottom row\) for different combinations of views and noise multipliers\. We see that with 300 views, there is moderate degradation between a clean reconstruction with no noise and one with a noise multiplier of 25, while retaining low absolute error\. However, once we reduce the number of views significantly to 20 views \(last column\), the combined effect of a large noise multiplier and sparse views is a failure mode for physical field reconstruction\.Noise SettingMAEinner​\(log10⁡ne\)\\text\{MAE\}\_\{\\text\{inner\}\}\(\\log\_\{10\}n\_\{\\rm e\}\)↓\\downarrowMAEinner​\(log10⁡Te\)\\text\{MAE\}\_\{\\text\{inner\}\}\(\\log\_\{10\}T\_\{\\rm e\}\)↓\\downarrowasinhErr​\(I\)\\text\{asinhErr\}\(I\)↓\\downarrowη=0\\eta=0, 5 views0\.410±0\.0200\.410\\pm 0\.0200\.070±0\.0050\.070\\pm 0\.0051\.454±0\.0411\.454\\pm 0\.041η=0\\eta=0, 20 views0\.122±0\.0230\.122\\pm 0\.0230\.026±0\.0030\.026\\pm 0\.0030\.458±0\.0600\.458\\pm 0\.060η=0\\eta=0, 100 views0\.039±<0\.0010\.039\\pm\{<\}0\.0010\.014±<0\.0010\.014\\pm\{<\}0\.0010\.007±0\.0010\.007\\pm 0\.001η=0\\eta=0, 300 views0\.039±<0\.0010\.039\\pm\{<\}0\.0010\.014±<0\.0010\.014\\pm\{<\}0\.0010\.007±0\.0010\.007\\pm 0\.001η=1\\eta=1, 5 views0\.516±0\.0260\.516\\pm 0\.0260\.126±0\.0100\.126\\pm 0\.0101\.513±0\.0461\.513\\pm 0\.046η=1\\eta=1, 20 views0\.209±0\.0180\.209\\pm 0\.0180\.044±0\.0030\.044\\pm 0\.0030\.823±0\.0500\.823\\pm 0\.050η=1\\eta=1, 100 views0\.052±0\.0020\.052\\pm 0\.0020\.018±<0\.0010\.018\\pm\{<\}0\.0010\.039±<0\.0010\.039\\pm\{<\}0\.001η=1\\eta=1, 300 views0\.047±0\.0010\.047\\pm 0\.0010\.017±<0\.0010\.017\\pm\{<\}0\.0010\.022±<0\.0010\.022\\pm\{<\}0\.001η=9\\eta=9, 5 views0\.519±0\.0210\.519\\pm 0\.0210\.970±0\.3700\.970\\pm 0\.3701\.565±0\.0501\.565\\pm 0\.050η=9\\eta=9, 20 views0\.275±0\.0370\.275\\pm 0\.0370\.115±0\.0300\.115\\pm 0\.0301\.239±0\.0841\.239\\pm 0\.084η=9\\eta=9, 100 views0\.069±0\.0030\.069\\pm 0\.0030\.022±<0\.0010\.022\\pm\{<\}0\.0010\.092±0\.0040\.092\\pm 0\.004η=9\\eta=9, 300 views0\.057±0\.0020\.057\\pm 0\.0020\.019±<0\.0010\.019\\pm\{<\}0\.0010\.041±0\.0010\.041\\pm 0\.001η=25\\eta=25, 5 views0\.760±0\.4580\.760\\pm 0\.4582\.404±1\.1542\.404\\pm 1\.1541\.529±0\.0591\.529\\pm 0\.059η=25\\eta=25, 20 views0\.336±0\.0770\.336\\pm 0\.0770\.380±0\.1380\.380\\pm 0\.1381\.418±0\.1071\.418\\pm 0\.107η=25\\eta=25, 100 views0\.081±0\.0050\.081\\pm 0\.0050\.026±0\.0010\.026\\pm 0\.0010\.136±0\.0080\.136\\pm 0\.008η=25\\eta=25, 300 views0\.063±0\.0020\.063\\pm 0\.0020\.021±<0\.0010\.021\\pm\{<\}0\.0010\.054±0\.0010\.054\\pm 0\.001Table 16:Noise\-view stress benchmark results \(also shown in Figure[5](https://arxiv.org/html/2609.28868#S5.F5)in the main text\)\. With sufficient angular coverage, the reconstruction remains robust even under strong heteroscedastic observation noise: increasing the noise multiplier to×25\\times 25at 300 training views only moderately increases the density and temperature field errors\. In contrast, sparse\-view regimes are substantially less stable, and the combination of high noise and sparse views produces a clear failure mode in latent physical\-field recovery\. The training objective uses the noise\-matchedscs\_\{c\}scaling described above, while held\-outasinhErris evaluated on noise\-free held\-out views using the same base per\-channel scales across allη\\eta\.![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_C2_noisy_obs_denoising.png)Figure 17:Demonstration of the accompanying denoising effect in the framework\. We show the clean ground truth target, and sample noisy training observations\. The final column shows a reconstructed render using a model trained on noise×25\\times 25images, where multiview consistency suppresses pixel\-scale noise in the reconstructed render\.Figure[16](https://arxiv.org/html/2609.28868#A4.F16)shows a sample reconstruction at1\.511\.51R⊙, and Table[16](https://arxiv.org/html/2609.28868#A4.T16)shows detailed results of our ablation study\. Additionally, we show an apparent denoising effect of training on noisy observations with sufficient views in Figure[17](https://arxiv.org/html/2609.28868#A4.F17)\.

### D\.7Abundance Mismatch Ablation Study

This section details an abundance scale ablation study that probes a controlled common\-mode forward\-model mismatch\. We keep the same model architecture and observational setup as described in Appendix[D\.1](https://arxiv.org/html/2609.28868#A4.SS1)\. However, we ablate the abundance scaleaaof the Si IX channels to produce a forward model mismatch between the model used to generate synthetic observations and that used to train CoroNeRF\. Because this ablation has no added observation noise,Icobs=Ictrue=a​IcrawI\_\{c\}^\{\\rm obs\}=I\_\{c\}^\{\\rm true\}=aI\_\{c\}^\{\\rm raw\}for the Si IX channels with varyingaawhile the Fe XIII channels remain unscaled; the inversion renderer continues to use the nominala=1a=1for all channels\. Our study has 5 experiment families that varya∈\{0\.6,0\.8,1\.0,1\.2,1\.4\}a\\in\\\{0\.6,0\.8,1\.0,1\.2,1\.4\\\}, each trained with 3 seeds for a total of 15 runs\. Figure[18](https://arxiv.org/html/2609.28868#A4.F18)shows sample reconstructions at1\.511\.51R⊙, and Table[17](https://arxiv.org/html/2609.28868#A4.T17)details the field\-space and image\-space errors\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_D2_abundance_shell_r1p5.png)Figure 18:Errors in physical field reconstruction due to forward model mismatch\. The columns show ground truth and the reconstructed density \(top row\) and temperature \(bottom row\) for different abundance mismatch scales\. We see a clear failure mode when the Si IX intensities are scaled by±40%\\pm 40\\%: the reconstructed field has spurious sharp features that do not exist in the ground truth fields\.aaasinhErr​\(I\)\\text\{asinhErr\}\(I\)↓\\downarrownen\_\{\\rm e\}MAE\\operatorname\{MAE\}↓\\downarrownen\_\{\\rm e\}ME\\operatorname\{ME\}TeT\_\{\\rm e\}MAE\\operatorname\{MAE\}↓\\downarrowTeT\_\{\\rm e\}ME\\operatorname\{ME\}0\.60\.60\.030±<0\.0010\.030\\pm<0\.0010\.301±0\.0060\.301\\pm 0\.006−0\.211±0\.004\-0\.211\\pm 0\.0040\.076±0\.0010\.076\\pm 0\.001−0\.030±0\.002\-0\.030\\pm 0\.0020\.80\.80\.017±<0\.0010\.017\\pm<0\.0010\.146±0\.0050\.146\\pm 0\.005−0\.076±0\.004\-0\.076\\pm 0\.0040\.036±0\.0010\.036\\pm 0\.001−0\.010±0\.001\-0\.010\\pm 0\.0011\.01\.00\.006±<0\.001\\mathbf\{0\.006\\pm<0\.001\}0\.039±<0\.001\\mathbf\{0\.039\\pm<0\.001\}−0\.001±0\.001\-0\.001\\pm 0\.0010\.014±<0\.001\\mathbf\{0\.014\\pm<0\.001\}0\.000±<0\.001\\mathbf\{0\.000\\pm<0\.001\}1\.21\.20\.016±<0\.0010\.016\\pm<0\.0010\.092±0\.0010\.092\\pm 0\.0010\.062±0\.0010\.062\\pm 0\.0010\.023±<0\.0010\.023\\pm<0\.001−0\.002±<0\.001\-0\.002\\pm<0\.0011\.41\.40\.027±<0\.0010\.027\\pm<0\.0010\.142±0\.0010\.142\\pm 0\.0010\.107±0\.0010\.107\\pm 0\.0010\.031±<0\.0010\.031\\pm<0\.001−0\.006±<0\.001\-0\.006\\pm<0\.001Table 17:Abundance\-mismatch benchmark\. We scale the Si IX intensity channels in the synthetic observations byaawhile evaluating against the clean ground\-truth physical fields\.MAE\\operatorname\{MAE\}reports inner\-band absolute log\-field error, while ME reports signed inner\-band errors\. The clean setting, corresponding to scale1\.01\.0, gives the lowest image and field errors\. Scaling the Si IX channels below or above the correct value produces systematic signed density errors, indicating that forward\-model mismatch can be absorbed into incorrect latent physical fields\.
### D\.8Rendered Views

This section provides renders of the column density in Figure[19](https://arxiv.org/html/2609.28868#A4.F19), total\-emissivity weighted temperature in Figure[20](https://arxiv.org/html/2609.28868#A4.F20), and total emissivity in Figure[21](https://arxiv.org/html/2609.28868#A4.F21), for selected conditions and longitude view angles\.

![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_F_los_filmstrip.png)Figure 19:LOS projections of recovered column density for selected conditions and view angles\.![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_F_los_filmstrip_temp.png)Figure 20:LOS projections of recovered emissivity\-weighted temperature for selected conditions and view angles\.![Refer to caption](https://arxiv.org/html/2609.28868v1/Figures/paper_F_los_filmstrip_emissivity.png)Figure 21:LOS projections of total\-emissivity log\-intensities for selected conditions and view angles\.