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REVIEW 4 major objections 5 minor 125 references

LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Under 1,000 offline solves now beat Laplace at Bayesian inversion.

desk verdict A genuinely useful combination of derivative-informed surrogates and lazy maps with convincing numerics, but the theory as written does not cover the implemented objective. read the letter →

arxiv 2411.12726 v1 pith:76AIRJZO submitted 2024-11-19 math.NA cs.LGcs.NAstat.COstat.ML

Lianghao Cao , Joshua Chen , Michael Brennan , Thomas O'Leary-Roseberry , Youssef Marzouk , Omar Ghattas This is my paper · ORCID
classification math.NAcs.LGcs.NAstat.COstat.ML MSC 65N2162F1565C6068T07
keywords Bayesianinverseproblemsvariationalinferencemeasuretransportlazymapsderivative-informedneuraloperatorsdimensionreductionamortizedPDE-constraineduncertaintyquantification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

LazyDINO attacks nonlinear Bayesian inverse problems whose parameter-to-observable (PtO) map is too expensive to evaluate inside a sampling loop. It builds one derivative-informed neural surrogate of the PtO map offline, then uses that surrogate to train a lazy map, a transport map that is nonlinear only in a low-dimensional latent subspace, for each new data set online. The paper proves that this two-step design is the right one for amortized inference: the reduced-basis architecture minimizes an upper bound on expected posterior error, and the derivative-informed training loss minimizes the expected optimality gap of the surrogate-driven transport optimization. In two PDE-governed examples, LazyDINO delivers posterior approximations that beat the Laplace approximation with fewer than 1,000 offline PDE solves, while conventional surrogate-driven and simulation-based amortized methods still struggle at 16,000 samples.

What carries the argument

The load-bearing object is the lazy map $T_\theta=(I-P)+D_r\,\mathcal{T}_\theta E_r$, which leaves the prior untouched in the complement of a $d_r$-dimensional subspace and transports the whitened latent coordinates by $\mathcal{T}_\theta$. It is driven by a DIPNet ridge-function surrogate $V g_w(E_r\cdot)$, whose parameter encoder comes from the eigenproblem $H_A\psi_j=\lambda_j\psi_j$ and whose weights are trained with the derivative-informed objective that includes the latent Jacobian $J_r^{(j)}=V^*D G(m^{(j)})D_r$. The machinery transfers the expensive likelihood evaluation into a cheap neural evaluation in $\mathbb{R}^{d_r}$, and the theory ties the resulting posterior error to the eigenvalue tail sum and the Sobolev error of the surrogate.

What would settle it

Run LazyDINO on an inverse problem with a deliberately slow eigenvalue decay so that $\sum_{j>d_r}\lambda_j$ is not small, and check whether the posterior mean/covariance errors and the ANIS effective sample size degrade as predicted by Theorem 3.1; a second check is to swap Remark 2's zero-mean objective for the true conditional-expectation ridge function and see whether the reported advantage reverses at small sample counts.

Watch

Extended reading notes

Core claim

The central claim is that surrogate-driven lazy-map variational inference becomes both accurate and amortizable when the surrogate is co-designed with the latent structure of the posterior update. Concretely, the PtO map is replaced by a ridge function $G(m)\approx V g_w(E_r m)$ with $E_r$ projecting onto the leading $d_r$ eigenfunctions of the prior-preconditioned Gauss-Newton Hessian $H_A=\mathbb{E}_\mu[D_H G^* D_H G]$, and $g_w$ is trained with joint samples of the map and its Jacobian. Theorem 3.1 bounds the expected forward KL error by the eigenvalue tail sum plus a latent-representation error, and Theorem 3.2 with Corollary 3.3 show that the derivative-informed $H^1_\mu$ loss controls the gradient error and optimality gap of the lazy-map objective. The numerical sections show these bounds are tight enough in practice: with $d_r=200$ and fewer than 1,000 offline samples, LazyDINO outperforms the Laplace posterior in moment, density, and sampling diagnostics on two infinite-dimensional PDE inverse problems.

Load-bearing premise

The whole construction assumes the data move the posterior almost entirely within a 200-dimensional derivative-informed subspace, so the discarded eigenvalue tail of the prior-preconditioned Gauss-Newton Hessian is negligible; if informative directions fall outside it, the ridge surrogate cannot see them and the posterior error bound degrades.

Editorial extensions

If this is right

  • The offline surrogate cost is amortized across every future data set sharing the same PtO map and prior, since the online phase only optimizes the cheap latent transport map.
  • Posterior sampling and density evaluation become as fast as evaluating the trained lazy map, which enables real-time uncertainty quantification for digital twins and experimental design.
  • Controlling the surrogate Jacobian, not just the map values, is what makes surrogate-driven transport optimization reliable; standard $L^2_\mu$ training can fail at 16,000 samples where derivative-informed training succeeds at 1,000.
  • Because the parameter dimension is reduced to $d_r=200$ before any neural network or transport map is trained, the method's offline and online costs are independent of the discretization dimension of the PDE parameter field.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same derivative-informed ridge-function architecture should accelerate other query-intensive algorithms that differentiate through the PtO map, such as Bayesian optimal experimental design and PDE-constrained optimization under uncertainty, where the paper's optimality-gap bound would carry over.
  • If the eigenvalue tail decays fast, the theory suggests an adaptive strategy: increase $d_r$ until the tail sum in (33) falls below the target posterior error, making LazyDINO's guarantee quantitative rather than heuristic.
  • Remark 2's use of the zero prior mean instead of the conditional expectation leaves a testable gap: at very small sample budgets the practical objective differs from the theoretically optimal ridge function, and the paper's empirical choice suggests a bias-variance trade-off worth isolating in controlled experiments.
  • A natural stress test is to push the method to multiple independent observations per parameter, where the posterior concentrates and the derivative-informed subspace may need to grow; the eigenvalue-tail criterion predicts exactly when the lazy-map ansatz breaks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces LazyDINO, a two-phase amortized Bayesian inversion method. In an offline phase, a derivative-informed reduced-basis neural operator (RB-DINO) surrogate of the parameter-to-observable map is trained using joint samples of the map and its Jacobian. In an online phase, this surrogate drives the training of a lazy map, a transport map whose nonlinearity acts only on a low-dimensional derivative-informed latent subspace. The authors claim two main theoretical results: (i) the DIPNet architecture and derivative-informed training minimize upper bounds on the expected surrogate posterior approximation error and on the expected optimality gap of surrogate-driven lazy-map optimization (Theorems 3.1, 3.2 and Corollary 3.3), and (ii) numerically, LazyDINO achieves high posterior accuracy at substantially lower offline cost than LazyNO, SBAI, LazyMap, and the Laplace approximation. The numerical study covers two nonlinear PDE-constrained inverse problems with four observation instances each and a wide range of posterior diagnostics.

Significance. The paper addresses an important practical problem: amortized Bayesian inversion for expensive PDE-governed models, where posterior approximation must be cheap after the model evaluations are performed offline. The co-design of the reduced basis, the neural surrogate, and the lazy-map transport is conceptually appealing, and the numerical comparison is unusually thorough: it includes ground-truth MCMC, moment discrepancies, density-based diagnostics, marginal visualizations, and careful computational-cost accounting. The algorithmic pseudocode and appendices are also detailed and would allow replication. If the theoretical characterization were fully established, the paper would make a solid contribution to surrogate-driven variational inference. However, as it stands, a load-bearing gap in the equivalence between the full-space and latent rKL objectives, together with an acknowledged deviation between the implemented and analyzed objectives, means that the central theoretical claims are not established for the algorithm as implemented.

major comments (4)
  1. [Section 2.5, Proposition 2.1] The equivalence in Eq. (22) is not valid under the stated assumptions. The full-space objective contains E_{z∼π,m⊥∼µ⊥}[1/2||V^*G(D_r Tθ(z)+m⊥)-V^*y||^2], while the latent objective contains E_{z∼π}[1/2||gopt(Tθ(z))-V^*y||^2]. The difference is E_{z∼π}[1/2 Var_{m⊥}(V^*G(D_r Tθ(z)+m⊥))], which depends on θ unless the conditional covariance of V^*G given the reduced coordinate is independent of that coordinate. No such condition is stated or verified. Since Theorem 3.2 and Corollary 3.3 operate in the latent formulation, this gap is load-bearing for the paper's main theoretical claims.
  2. [Remark 2 and Section 3.5, Eq. (43)] The implemented LazyDINO objective sets the complementary-space sample m⊥ to E[µ⊥]=0 rather than integrating over µ⊥. The surrogate-driven objective in Eq. (43a) is therefore E_z[1/2||g_w(Tθ(z))-V^*y||^2 + regularizer], not the expectation of the potential appearing in Eq. (25). This is a different variational problem from the one analyzed in Corollary 3.3. The text acknowledges the deviation but supplies neither a condition under which the two objectives coincide nor numerical evidence that the resulting bias is small. Consequently, claim (C1), that derivative-informed training minimizes the expected optimality gap of surrogate-driven lazy-map optimization, is not established for the algorithm as implemented.
  3. [Corollary 3.3] The optimality-gap result depends on assumptions (i) and (ii): the surrogate minimizer eθ^{y,†} must lie in a ball around the true minimizer θ^{y,†}, and the true objective must be locally strongly convex on that ball, γ-a.e. These are assumptions about problem- and surrogate-dependent quantities, and no verification is provided. The surrounding text states that derivative-informed learning minimizes the expected optimality gap without repeating these caveats, so the corollary should be presented as a conditional statement rather than an unconditional justification of the method.
  4. [Section 6.2, Figures 8 and 10] The numerical comparison is extensive, but the role of the chosen reduced basis dimension dr=200 and the MC approximation of the derivative-informed subspace with only 1000 samples is not investigated. The theoretical bounds in Theorem 3.1 and Eq. (33) assume the exact eigenbasis of the expected Gauss-Newton Hessian, while the experiments use a fixed sample-based approximation. This is not fatal, but a sensitivity study or at least a discussion of the approximation gap would be needed to connect the theory quantitatively to the reported results.
minor comments (5)
  1. [Abstract and Section 7] The abstract claims LazyDINO "consistently outperforms Laplace approximation" with fewer than 1000 samples, but Figure 8 shows that for Example I the Laplace baseline achieves lower covariance error than all methods, and Section 7 itself notes this exception. The abstract should be qualified.
  2. [Section 4, contribution list (C2)] There is a typo: "Scabalility" should be "Scalability".
  3. [Table 3] The note in Table 3 says "LazyDINO (1k) achieves smaller relative mean error than LazyDINO (128k)", but the comparison is with LazyMap (128k); the label should be corrected.
  4. [Section 6.1, Figures 6 and 7] The figures report a "statistical anomaly" in RB-DINO training at 500 samples. Since this anomaly propagates to the posterior comparisons, the paper should state whether the anomaly reflects a single seed or a systematic effect, and ideally report uncertainty over training seeds.
  5. [Section 5.4] The transport-map training schedule is reported as a list of (iterations, batch size, learning rate) tuples, but the notation is dense; a short table or clearer formatting would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theoretical bounds are genuine inequalities and the training losses coincide with the bound terms by design, which is a consistency result rather than a circular derivation.

full rationale

The derivation chain is not circular. Theorem 3.1 (Section 3.1) proves a genuine inequality, E_y DKL(mu^y || emu^y) <= Tr((I-P) H_A (I-P)) + E_z ||gopt - g||^2. Choosing the derivative-informed eigenbasis to minimize the first term and training the neural network to minimize the second term matches the bound's terms, but the bound itself does not presuppose DIPNet or the training loss; it is derived from H^1_mu regularity and the conditional-expectation optimality of eGopt. Similarly, Theorem 3.2 and Corollary 3.3 prove that the expected rKL gradient error and the expected optimality gap are controlled by the pi-weighted Sobolev norm of (gopt - g, grad gopt - grad g); the derivative-informed objective (40) is an unbiased single-sample estimator of exactly that norm, so minimizing it targets the bound. This is a consistency result, not circularity: the theorems provide the inequalities, and the losses are chosen to match the right-hand sides. The numerical superiority claim is benchmarked against MCMC ground truth and the Laplace baseline, so it is not forced by construction. The paper does cite prior work by overlapping authors ([24], [29], [30]), but those are used as proven mathematical results and algorithmic building blocks, and the core inequalities are re-proved in Appendices D.1-D.2; no load-bearing argument reduces to an unverified self-citation. Two correctness risks should be noted separately: Proposition 2.1's equality of full-space and latent rKL objectives requires the conditional covariance of G given the reduced coordinates to be independent of the reduced coordinate, a condition not stated in the paper; and Remark 2 explicitly replaces the m_perp expectation by E[mu_perp] = 0, so the implemented objective differs from the analyzed one. Neither is a circular step; both are gaps between assumptions and implementation. Verdict: no significant circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim depends on the choice of the dimension-reducing subspace (dr=200) and on the assumption that the posterior update is confined to that subspace. The bounds are proven under Gaussian prior/noise and Sobolev differentiability assumptions. No invented entities are introduced.

free parameters (4)
  • reduced basis dimension dr = 200
    Selected by hand for both examples and fixed across all compared methods. The eigenvalue tail at dr=200 is implicitly assumed small, directly controlling the expressiveness of the ridge function surrogate.
  • neural surrogate width and depth = 7 layers, 400 units
    Capacity of the MLP gw affects the representation error and the Sobolev norm in the bounds. Chosen by common practice, not by a principled criterion.
  • lazy map IAF size = 30 layers, 400 hidden units
    Flexibility of the transport map; affects how well the latent posterior can be approximated. Set to the same architecture for all compared methods.
  • stochastic optimization schedule = Adamax batches 200-7500, learning rates 5e-3 to 5e-4; Adam with 1500 epochs
    The training schedule for the transport map and surrogate influences the achieved optimality gap and generalization. Chosen empirically and reported in Section 5.
assumptions (6)
  • standard math Gaussian prior distribution (Assumption 2.1)
    The method assumes a Gaussian prior on the parameter. Non-Gaussian priors require a change of variables as noted in Remark 1.
  • standard math Additive Gaussian noise (Assumption 2.2)
    The likelihood is Gaussian, leading to the quadratic potential in Eq. (5).
  • domain assumption H1_mu-differentiable PtO map (Assumption 2.3)
    The Malliavin derivative and the Hessian-based bounds require the PtO map to be differentiable in the Gaussian Sobolev sense. This is a technical assumption that holds for typical PDE maps under sufficient regularity.
  • domain assumption Subspace concentration: data uninformative in Ker(P)
    The ridge function approximation (11) and the use of the derivative-informed subspace assume the posterior-to-prior update is confined to a dr-dimensional subspace; the eigenvalue tail in (33) must be small. This is the main structural assumption of the method.
  • domain assumption Local strong convexity and boundedness assumptions in Corollary 3.3
    The optimality gap bound assumes the rKL objective is locally strongly convex near the true optimum and the surrogate minimizer lands within that region. These conditions are not verified for the numerical examples.
  • ad hoc to paper Implementation uses E[mu_perp]=0 instead of the optimal conditional expectation (Remark 2)
    The numerical method replaces the theoretical objective based on the optimal ridge function with a single-sample unconditional prior mean, changing the optimization problem that the theorems bound. This is an acknowledged deviation, not an assumption in the published theory.

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Cite this review

Pith. "Pith review of LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport." pith.science (2026). https://pith.science/paper/76AIRJZO

@misc{pith2026241112726,
  author       = {Pith},
  title        = {Pith review of: LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/76AIRJZO}},
  note         = {Machine review of arXiv:2411.12726}
}
read the original abstract

We present LazyDINO, a transport map variational inference method for fast, scalable, and efficiently amortized solutions of high-dimensional nonlinear Bayesian inverse problems with expensive parameter-to-observable (PtO) maps. Our method consists of an offline phase in which we construct a derivative-informed neural surrogate of the PtO map using joint samples of the PtO map and its Jacobian. During the online phase, when given observational data, we seek rapid posterior approximation using surrogate-driven training of a lazy map [Brennan et al., NeurIPS, (2020)], i.e., a structure-exploiting transport map with low-dimensional nonlinearity. The trained lazy map then produces approximate posterior samples or density evaluations. Our surrogate construction is optimized for amortized Bayesian inversion using lazy map variational inference. We show that (i) the derivative-based reduced basis architecture [O'Leary-Roseberry et al., Comput. Methods Appl. Mech. Eng., 388 (2022)] minimizes the upper bound on the expected error in surrogate posterior approximation, and (ii) the derivative-informed training formulation [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] minimizes the expected error due to surrogate-driven transport map optimization. Our numerical results demonstrate that LazyDINO is highly efficient in cost amortization for Bayesian inversion. We observe one to two orders of magnitude reduction of offline cost for accurate posterior approximation, compared to simulation-based amortized inference via conditional transport and conventional surrogate-driven transport. In particular, LazyDINO outperforms Laplace approximation consistently using fewer than 1000 offline samples, while other amortized inference methods struggle and sometimes fail at 16,000 offline samples.

Figures

Figures reproduced from arXiv: 2411.12726 by the authors.

Figure 1
Figure 1. Overview of the RB-DINO construction. Step 2: RB-DINO Surrogate-Driven Lazy Map Optimization Given Observational Data y 0.0 0.2 0.4 0.6 0.8 1.0 Observed data y ∈ R dy gw : R dr → R dy V ∗ 1 2 ∥(gw ◦ Tθ)(z) − V ∗y∥ 2 Trained Surrogate Latent Representation Tθ : R dr → R dr Latent Space Transport Map Whitened latent prior on R dr Sample z ∼ π = N (0, IdRdr ) Latent representation transport map training min θ Ez∼π h 1 … view at source ↗
Figure 2
Figure 2. Overview of the latent representation lazy map construction. [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Overview of LazyDINO amortization procedure. Comparison of LazyDINO to other SBAI methods. Given joint samples of the latent prior and simulated observational data, the SBVI methods optimize for a conditional transport map that matches the pullback distributions, y (j) 7→ Tθ(y (j) , ·) ♯π, to posteriors at the simulated data samples using an fKL objective. The sample generation and the transport map construction are… view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Example I. Setup for inferring the diffusivity field in a nonlinear reaction–diffusion PDE detailed in Section 5.1. For each BIP (#1–4), we show the data-generating synthetic parameter m drawn from the prior, the synthetic data y placed on top of the PDE solution at m,…
Figure 5
Figure 5. Figure 5: Example II. Setup for inferring a heterogeneous hyperelastic material property detailed in Section 5.2. For each BIP (#1–4), we visualize the synthetic parameter m drawn from the prior, the corresponding deformed configuration, the synthetic displacement data y, and th…
Figure 6
Figure 6. Figure 6: Example I neural ridge function surrogate testing. Percentage accuracy, 100% × (1 − error), is also overlaid. Overall, these results demonstrate a significant cost reduction for achieving any given generalization accuracy in both the PtO map and the latent Jacobian via…
Figure 7
Figure 7. Figure 7: Example II neural operator ridge function generalization. In a similar pattern as [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: Example I moment discrepancies. In all cases, a lower error indicates a better posterior approximation. (LazyDINO vs. LazyNO): Apart from the statistical anomaly at 500 samples attributed to the stochasticity in surrogate training seen in [PITH_FULL_IMAGE:figures/full…
Figure 9
Figure 9. Figure 9: Example II moment discrepancies. We observe similar trends as in Example I ( [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]
Figure 10
Figure 10. Figure 10: Example I density-based diagnostics. Higher values of ESS100K% and lower values of all other diagnostics imply better posterior approximation. We observe similar trends to those observed in the moment discrepancy comparisons in [PITH_FULL_IMAGE:figures/full_fig_p030_…
Figure 11
Figure 11. Figure 11: Example II density-based diagnostics. We observe similar trends as in Example I. Notably, LazyDINO achieves nearly 50% ANIS effective sample percentage as a 100,000-sample independent sampler with 16,000 training samples, 50 to 50,000 times the percentage achieved for…
Figure 12
Figure 12. Figure 12: Example I: LazyDINO vs Laplace approximation marginals. At 250 training samples, we see clear deviation in the marginals for both approaches, though LA-baseline has contours that match the posterior marginals more closely. By 2k training samples, LazyDINO seems to mat…
Figure 13
Figure 13. Figure 13: Example I: LazyDINO vs LazyNO marginals. Consistent with [PITH_FULL_IMAGE:figures/full_fig_p035_13.png]
Figure 14
Figure 14. Figure 14: Example I: LazyDINO vs SBAI marginals. The marginals produced via SBAI are much further from the true posterior marginals than the other approaches. Notably, SBAI consistently overestimates the uncertainty in posterior reconstruction and still yields a poor reconstruc…
Figure 15
Figure 15. Figure 15: Example I: LazyDINO vs LazyMap marginals. The LazyMap marginals are quite poor for 1k training samples and only get more concentrated as the number of training samples increases. The equivalent sample-cost LazyDINO posterior marginals match the ground truth substantia…
Figure 16
Figure 16. Figure 16: Example II: LazyDINO vs Laplace approximation marginals. Due to the non-Gaussianity of the posterior, Laplace approximation struggles to capture the overall behavior of the marginals depicted, whereas the LazyDINO marginals are close to the posterior marginals at 2k t…
Figure 17
Figure 17. Figure 17: Example II: LazyDINO vs LazyNO marginals. With 250 training samples, LazyNO produces far underconcentrated samples in these marginals. With 16k training samples, both methods produce similar marginal distributions. The notable takeaway is that LazyNO requires multiple…
Figure 18
Figure 18. Figure 18: Example II: LazyDINO vs SBAI marginals. SBAI produces samples that are highly under-concentrated. SBAI is simultaneously off in capturing the peak locations of the marginals while overestimating the uncertainty in the parameter. 1 x 2 2 −0.25 0.00 0.25 x 3 0 2 x 4 0.2…
Figure 19
Figure 19. Figure 19: Example II: LazyDINO vs. LazyMap marginals. LazyMap fails to capture the contours of the posterior marginals faithfully. LazyMap is off by a constant error in capturing the peak of the marginals. As more data is available, it tends to over-concentrate, leading to na¨ı…
Figure 20
Figure 20. Figure 20: Example I: progression of mean estimators. LazyDINO already visually captures the mean well with 250 training samples and leads to significantly smaller point-wise errors at 16, 000 samples. The next best performing method is LazyNO, which struggles to resolve the ess…
Figure 21
Figure 21. Figure 21: Example I: progression of MAP estimators. In a similar story to [PITH_FULL_IMAGE:figures/full_fig_p040_21.png]
Figure 22
Figure 22. Figure 22: Example I: progression of point-wise marginal variance estimators. Again, the LazyDINO estimate of point￾wise marginal variance is superior to the other methods as in the previous studies of mean and MAP estimation. Notably, SBAI overestimates the point-wise marginal …
Figure 23
Figure 23. Figure 23: Example II: progression of mean estimators with N. As with [PITH_FULL_IMAGE:figures/full_fig_p042_23.png]
Figure 24
Figure 24. Figure 24: Example II: progression of MAP estimators. As with [PITH_FULL_IMAGE:figures/full_fig_p043_24.png]
Figure 25
Figure 25. Figure 25: Example II: progression of point-wise marginal variance estimators. We see, yet again, the same trend of the previous studies: LazyDINO yields a superior approximation of the point-wise marginal variance and, notably, can deliver a faithful approximation for as little…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.