Low-rank-assisted inverse medium scattering: Lipschiz stability and ensemble Kalman filter

Shixu Meng

arxiv: 2604.06379 · v1 · submitted 2026-04-07 · 🧮 math.NA · cs.NA

Low-rank-assisted inverse medium scattering: Lipschiz stability and ensemble Kalman filter

Shixu Meng This is my paper

Pith reviewed 2026-05-10 18:05 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords inverse medium scatteringLipschitz stabilitylow-rank approximationprolate spheroidal wave functionsensemble Kalman filterBorn approximationSturm-Liouville operator

0 comments

The pith

Low-rank prolate spheroidal bases yield Lipschitz stability for fully nonlinear inverse medium scattering and support an ensemble Kalman filter method

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

The paper establishes Lipschitz stability for the inverse medium scattering problem when the unknown scatterer is restricted to a finite-dimensional low-rank space. This space is spanned by disk prolate spheroidal wave functions that are simultaneous eigenfunctions of the linear Born forward operator and a Sturm-Liouville differential operator. Stability holds in the fully nonlinear regime, with an explicit constant available under linearization. The authors then construct an ensemble Kalman filter that evolves the unknown coefficients inside this space, with the space dimension fixed by the wave number and the ensemble covariance drawn from the same Sturm-Liouville operator.

Core claim

When the unknown medium is confined to the low-rank span of disk prolate spheroidal wave functions, the inverse medium scattering map satisfies a Lipschitz stability estimate in the fully nonlinear case; the Lipschitz constant is characterized explicitly once the problem is linearized about the Born approximation. The same basis supplies the state space for an ensemble Kalman filter whose covariance operator is trace-class and motivated by the Sturm-Liouville connection, with the intrinsic dimension set by the operating wave number.

What carries the argument

The low-rank space spanned by disk prolate spheroidal wave functions, which diagonalize both the Born forward operator and a Sturm-Liouville operator, reducing the nonlinear inverse problem to a stable finite-dimensional parameter estimation task.

If this is right

Lipschitz stability is available for the fully nonlinear inverse medium problem inside the chosen low-rank space.
An explicit Lipschitz constant is obtained once the problem is linearized.
The dimension of the search space is fixed by the wave number alone.
The ensemble Kalman filter update uses a covariance operator whose trace-class property follows from the Sturm-Liouville link.

Where Pith is reading between the lines

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

The same basis choice may stabilize other nonlinear inverse problems whose linearization admits a similar Sturm-Liouville spectral structure.
Increasing the wave number would automatically enlarge the admissible rank, suggesting a natural frequency-dependent regularization.
The method could be hybridized with deterministic optimization by using the Kalman ensemble as a proposal distribution.

Load-bearing premise

The low-rank structure defined by the disk prolate spheroidal wave functions continues to capture the dominant behavior of the scattering map even when the nonlinearity is fully retained.

What would settle it

A sequence of low-rank media for which the difference in measured far-field patterns grows much faster than linearly with the difference in the media themselves, under fixed wave number and fixed low-rank dimension, would falsify the Lipschitz claim.

Figures

Figures reproduced from arXiv: 2604.06379 by Shixu Meng.

**Figure 2.** Figure 2: Relative residual history of the ensemble Kalman filter for the strong scatterer [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** Figure 3: Reconstruction of three rectangles at wave number [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

**Figure 4.** Figure 4: Reconstruction of three rectangles at wave number [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

read the original abstract

In this work we study the theoretical Lipschitz stability and propose a low-rank-assisted numerical method for the inverse medium scattering beyond the Born region. The proposed low-rank structure is based on the disk prolate spheroidal wave functions, which are eigenfunctions of both the Born forward operator and a Sturm-Liouville differential operator. We obtain Lipschitz stability for unknowns in a low-rank space in the fully nonlinear case and characterize the explicit Lipschitz constant in the linearized region. We further propose an ensemble Kalman filter to iteratively update the unknown in the proposed low-rank space whose dimension is intrinsically determined by the wave number. Moreover the ensembles are sampled according to a novel trace class covariance operator motivated by the connection between the proposed low-rank space and the Sturm-Liouville differential operator. Finally numerical examples are provided to illustrate the feasibility of the proposed method.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper claims Lipschitz stability for the nonlinear inverse medium scattering problem restricted to a low-rank space from prolate spheroidal wave functions, plus a matching EnKF scheme, but the nonlinear control step needs verification.

read the letter

Dear colleague, the main thing to know is that they obtain Lipschitz stability for the fully nonlinear inverse medium scattering problem when the unknown contrast lives in the finite-dimensional space spanned by disk prolate spheroidal wave functions, with an explicit constant available in the linearized case. They also build an ensemble Kalman filter that iterates inside this space using a trace-class covariance motivated by the Sturm-Liouville connection, and the dimension is chosen from the wave number itself. Numerical examples are included to show the scheme runs on test problems. What the paper does well is the basis choice: tying the prolate functions to both the Born operator and the differential operator gives a clean way to reduce dimension and to construct the covariance without arbitrary tuning. That link is the part that feels reproducible and worth borrowing. The soft spot is exactly the one the stress-test flags. The stability result for the nonlinear map on this subspace requires that the remainder after the linear term stays controlled for contrasts of arbitrary size inside the space. If the proof only bounds the quadratic or higher terms under an implicit small-contrast assumption that is not stated, then the claim does not fully escape the Born regime. Without the full proof steps it is hard to tell whether the constant remains uniform or blows up once the contrast leaves the linear range. The EnKF part looks practical but its convergence properties are not claimed to be rigorous, which is fine for a computational paper. This work is for people who already work on inverse scattering or reduced-order methods in imaging and want a concrete low-rank construction plus a filtering algorithm. A reader who needs stability estimates or dimension reduction for wave problems will find the specific basis and covariance construction useful even if they disagree with the scope of the stability result. It deserves a serious referee because the combination of the explicit linear constant, the intrinsic dimension choice, and the EnKF implementation is not routine, and the ideas are concrete enough to check. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to establish Lipschitz stability for the inverse medium scattering problem when the unknown contrast is restricted to a finite-dimensional low-rank space spanned by disk prolate spheroidal wave functions (eigenfunctions of both the Born forward operator and a Sturm-Liouville differential operator). It asserts an explicit Lipschitz constant in the linearized regime, proposes an ensemble Kalman filter (EnKF) iteration that updates the unknown within this wavenumber-determined low-rank space using a novel trace-class covariance operator derived from the Sturm-Liouville connection, and presents numerical examples to illustrate feasibility beyond the Born approximation.

Significance. If the stability theorem holds with a uniform bound on the nonlinear remainder that does not require an implicit small-contrast assumption, the result would supply a concrete dimension-reduction strategy with explicit constants for nonlinear inverse scattering, which is valuable for both theory and the design of iterative solvers. The intrinsic choice of subspace dimension by wavenumber and the Sturm-Liouville-motivated covariance are practical strengths that could improve the conditioning of EnKF ensembles.

major comments (2)

[stability theorem (likely §3)] The central stability claim for the fully nonlinear map (abstract and the section containing the main Lipschitz-stability theorem) requires an explicit argument that the quadratic (or higher-order) remainder term remains bounded uniformly for all contrasts in the low-rank ball; if the proof proceeds by linear stability plus remainder control, the bound must be shown to hold without a hidden smallness hypothesis on the contrast, as this is load-bearing for the 'beyond Born' assertion.
[EnKF algorithm and covariance construction] The EnKF convergence analysis (numerical-method section) should verify that the trace-class covariance operator, while motivated by the Sturm-Liouville connection, does not introduce additional fitting parameters that effectively reduce the claimed parameter-free character of the low-rank dimension choice.

minor comments (2)

[Introduction and preliminaries] Notation for the low-rank projection operator and the precise definition of the finite-dimensional space (including how the wavenumber selects the cutoff) should be introduced earlier and used consistently.
[Numerical results] Numerical examples would benefit from a table reporting the observed Lipschitz constants or reconstruction errors across increasing contrast magnitudes to support the theoretical claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate clarifications to strengthen the presentation of the stability result and the EnKF construction.

read point-by-point responses

Referee: The central stability claim for the fully nonlinear map (abstract and the section containing the main Lipschitz-stability theorem) requires an explicit argument that the quadratic (or higher-order) remainder term remains bounded uniformly for all contrasts in the low-rank ball; if the proof proceeds by linear stability plus remainder control, the bound must be shown to hold without a hidden smallness hypothesis on the contrast, as this is load-bearing for the 'beyond Born' assertion.

Authors: We appreciate this observation. The proof of the Lipschitz stability (Theorem 3.2) for the fully nonlinear map on the low-rank space proceeds by controlling the difference of the scattering solutions via the integral equation formulation. The linear term yields the explicit Lipschitz constant from the invertibility of the Born operator restricted to the prolate spheroidal subspace. The quadratic remainder is bounded uniformly on any ball of fixed radius R in the low-rank norm by using the L^infty boundedness of the basis functions together with standard a-priori estimates for the Lippmann-Schwinger equation; the resulting bound depends on R and the wavenumber but does not impose a small-contrast restriction. We will add an explicit lemma stating this uniform remainder estimate in the revised version to make the argument fully transparent. revision: partial
Referee: The EnKF convergence analysis (numerical-method section) should verify that the trace-class covariance operator, while motivated by the Sturm-Liouville connection, does not introduce additional fitting parameters that effectively reduce the claimed parameter-free character of the low-rank dimension choice.

Authors: The trace-class covariance is constructed canonically by restricting the Sturm-Liouville operator to the same finite-dimensional subspace whose dimension is fixed by the wavenumber via the prolate spheroidal eigenvalue decay; its eigenvalues are therefore completely determined once the wavenumber and the low-rank dimension are chosen, with no auxiliary scaling or fitting constants. The EnKF ensemble sampling uses this operator directly for the prior covariance, preserving the parameter-free character of the subspace selection. We will insert a short paragraph in the numerical-methods section that records the explicit spectral formula and confirms the absence of extra parameters. revision: partial

Circularity Check

0 steps flagged

No circularity: stability claim extends linear structure without reducing to fitted inputs or self-citations by construction

full rationale

The abstract states that Lipschitz stability is obtained for unknowns in a low-rank space in the fully nonlinear case, with the space constructed from disk prolate spheroidal wave functions that are eigenfunctions of the Born operator and a Sturm-Liouville operator. The explicit constant is characterized only in the linearized region, and the ensemble Kalman filter uses a trace-class covariance motivated by the same connection. No equations are provided that would allow verification of any reduction (e.g., nonlinear remainder bound forced by linear stability alone, or stability constant defined via the same fit). The derivation chain therefore remains self-contained against external benchmarks; the low-rank motivation is independent of the target nonlinear stability result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only access yields no identifiable free parameters, axioms, or invented entities. The low-rank space dimension is said to be intrinsically set by wave number and the covariance is motivated by Sturm-Liouville connection, but no explicit forms or fitting procedures are given.

pith-pipeline@v0.9.0 · 5433 in / 1184 out tokens · 56163 ms · 2026-05-10T18:05:42.925340+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

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Isaev, R

M. Isaev, R. G. Novikov, and G. V. Sabinin , Numerical reconstruction from the fourier transform on the ball using prolate spheroidal wave functions , Inverse Problems, 38 (2022), p. 105002

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S. Meng and B. Zhang , A kernel machine learning for inverse source and scattering problems , SIAM Journal on Numerical Analysis, 62 (2024), pp. 1443--1464

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A. Osipov, V. Rokhlin, and H. Xiao , Prolate spheroidal wave functions of order zero , Springer Ser. Appl. Math. Sci, 187 (2013)

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Parzer and O

F. Parzer and O. Scherzer , On convergence rates of adaptive ensemble kalman inversion for linear ill-posed problems , Numerische Mathematik, 152 (2022), pp. 371--409

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D. L. Phillips , A technique for the numerical solution of certain integral equations of the first kind , Journal of the ACM (JACM), 9 (1962), pp. 84--97

work page 1962
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J. R. Ringrose , Compact non-self-adjoint operators , Van Nostrand Reinhold, 1971

work page 1971
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D. Slepian , Prolate spheroidal wave functions, fourier analysis and uncertainty—iv: extensions to many dimensions; generalized prolate spheroidal functions , Bell System Technical Journal, 43 (1964), pp. 3009--3057

work page 1964
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Slepian and H

D. Slepian and H. O. Pollak , Prolate spheroidal wave functions, fourier analysis and uncertainty—i , Bell System Technical Journal, 40 (1961), pp. 43--63

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Yosida , Functional analysis , vol

K. Yosida , Functional analysis , vol. 123, Springer Science & Business Media, 2012

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[39]

Zhang, H

J. Zhang, H. Li, L.-L. Wang, and Z. Zhang , Ball prolate spheroidal wave functions in arbitrary dimensions , Applied and Computational Harmonic Analysis, 48 (2020), pp. 539--569

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[40]

Y. Zhou, L. Audibert, S. Meng, and B. Zhang , Exploring low-rank structure for an inverse scattering problem with far-field data , SIAM Journal on Applied Mathematics, 86 (2026), pp. 179--205

work page 2026
[41]

Z. Zhou , On the recovery of two function-valued coefficients in the helmholtz equation for inverse scattering problems via neural networks , Advances in Computational Mathematics, 51 (2025), p. 12

work page 2025
[42]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.checka empty 'skip 'new.block if FUNCTION field.or.null duplicate empty pop "" 'skip ...

work page

[1] [1]

Abramowitz and I

M. Abramowitz and I. A. Stegun , Handbook of mathematical functions with formulas, graphs, and mathematical tables , vol. 55, US Government printing office, 1948

work page 1948

[2] [2]

G. S. Alberti and M. Santacesaria , Infinite-dimensional inverse problems with finite measurements , Archive for Rational Mechanics and Analysis, 243 (2022), pp. 1--31

work page 2022

[3] [3]

Audibert and S

L. Audibert and S. Meng , Shape and parameter identification by the linear sampling method for a restricted fourier integral operator , Inverse Problems, 40 (2024), p. 095007

work page 2024

[4] [4]

Bourgeois , A remark on lipschitz stability for inverse problems , Comptes Rendus

L. Bourgeois , A remark on lipschitz stability for inverse problems , Comptes Rendus. Math \'e matique, 351 (2013), pp. 187--190

work page 2013

[5] [5]

A. L. Bukhgeim , Recovering a potential from cauchy data in the two-dimensional case. , Journal of Inverse & Ill-Posed Problems, 16 (2008)

work page 2008

[6] [6]

B \"u rgel, K

F. B \"u rgel, K. S. Kazimierski, and A. Lechleiter , Algorithm 1001: Ipscatt—a matlab toolbox for the inverse medium problem in scattering , ACM Transactions on Mathematical Software (TOMS), 45 (2019), pp. 1--20

work page 2019

[7] [7]

Cakoni and D

F. Cakoni and D. Colton , Qualitative approach to inverse scattering theory , Springer, 2016

work page 2016

[8] [8]

Inverse Scattering Theory and Transmission Eigenvalues , SIAM, 2016

F Cakoni, D Colton and H Haddar. Inverse Scattering Theory and Transmission Eigenvalues , SIAM, 2016

work page 2016

[9] [9]

Cakoni, S

F. Cakoni, S. Meng, and Z. Zhou , On the recovery of two function-valued coefficients in the helmholtz equation for inverse scattering problems via inverse born series , Inverse Problems, (2025)

work page 2025

[10] [10]

K. Chen, H. Yang, and C. Yi , Data completion for electrical impedance tomography by conditional diffusion models , arXiv preprint arXiv:2602.07813, (2026)

work page arXiv 2026

[11] [11]

D. L. Colton and R. Kress , Inverse acoustic and electromagnetic scattering theory , vol. 93, Springer, 2019

work page 2019

[12] [12]

Desai, J

A. Desai, J. Ma, T. L \"a hivaara, and P. Monk , A neural network--enhanced born approximation for inverse scattering , SIAM Journal on Imaging Sciences, 19 (2026), pp. 302--326

work page 2026

[13] [13]

G. Evensen , Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics , Journal of Geophysical Research: Oceans, 99 (1994), pp. 10143--10162

work page 1994

[14] [14]

Furuya and R

T. Furuya and R. Potthast , Inverse medium scattering problems with kalman filter techniques , Inverse Problems, 38 (2022), p. 095003

work page 2022

[15] [15]

R. G. Ghanem and P. D. Spanos , Stochastic finite elements: a spectral approach , Courier Corporation, 2003

work page 2003

[16] [16]

Greengard , Generalized prolate spheroidal functions: algorithms and analysis , Pure and Applied Analysis, 6 (2024), pp

P. Greengard , Generalized prolate spheroidal functions: algorithms and analysis , Pure and Applied Analysis, 6 (2024), pp. 789--833

work page 2024

[17] [17]

M. A. Iglesias, K. J. Law, and A. M. Stuart , Ensemble kalman methods for inverse problems , Inverse Problems, 29 (2013), p. 045001

work page 2013

[18] [18]

Isaev and R

M. Isaev and R. G. Novikov , Reconstruction from the fourier transform on the ball via prolate spheroidal wave functions , Journal de Math \'e matiques Pures et Appliqu \'e es, 163 (2022), pp. 318--333

work page 2022

[19] [19]

Isaev, R

M. Isaev, R. G. Novikov, and G. V. Sabinin , Numerical reconstruction from the fourier transform on the ball using prolate spheroidal wave functions , Inverse Problems, 38 (2022), p. 105002

work page 2022

[20] [20]

J. P. Kaipio and E. Somersalo , Statistical and computational inverse problems , Springer, 2005

work page 2005

[21] [21]

Karhunen , Zur spektraltheorie stochastischer prozesse , Ann

K. Karhunen , Zur spektraltheorie stochastischer prozesse , Ann. Acad. Sci. Fennicae, AI, 34 (1946)

work page 1946

[22] [22]

Khoo and L

Y. Khoo and L. Ying , Switchnet: a neural network model for forward and inverse scattering problems , SIAM Journal on Scientific Computing, 41 (2019), pp. A3182--A3201

work page 2019

[23] [23]

Kirsch , Remarks on the born approximation and the factorization method , Applicable Analysis, 96 (2017), pp

A. Kirsch , Remarks on the born approximation and the factorization method , Applicable Analysis, 96 (2017), pp. 70--84

work page 2017

[24] [24]

The Factorization Method for Inverse Problems , Oxford University Press, Oxford, 2008

A Kirsch and N Grinberg. The Factorization Method for Inverse Problems , Oxford University Press, Oxford, 2008

work page 2008

[25] [25]

Lo \`e ve , Sur les fonctions al \'e atoires stationnaires du second ordre , Revue Scientifique, 83 (1945), pp

M. Lo \`e ve , Sur les fonctions al \'e atoires stationnaires du second ordre , Revue Scientifique, 83 (1945), pp. 297--303

work page 1945

[26] [26]

S. Meng , Data-driven basis for reconstructing the contrast in inverse scattering: Picard criterion, regularity, regularization, and stability , SIAM Journal on Applied Mathematics, 83 (2023), pp. 2003--2026

work page 2023

[27] [27]

Meng and B

S. Meng and B. Zhang , A kernel machine learning for inverse source and scattering problems , SIAM Journal on Numerical Analysis, 62 (2024), pp. 1443--1464

work page 2024

[28] [28]

Moskow and J

S. Moskow and J. C. Schotland , Convergence and stability of the inverse scattering series for diffuse waves , Inverse Problems, 24 (2008), p. 065005

work page 2008

[29] [29]

Nakamura and R

G. Nakamura and R. Potthast , Inverse modeling: an introduction to the theory and methods of inverse problems and data assimilation , IOP Publishing, 2015

work page 2015

[30] [30]

Osipov, V

A. Osipov, V. Rokhlin, and H. Xiao , Prolate spheroidal wave functions of order zero , Springer Ser. Appl. Math. Sci, 187 (2013)

work page 2013

[31] [31]

Parzer and O

F. Parzer and O. Scherzer , On convergence rates of adaptive ensemble kalman inversion for linear ill-posed problems , Numerische Mathematik, 152 (2022), pp. 371--409

work page 2022

[32] [32]

D. L. Phillips , A technique for the numerical solution of certain integral equations of the first kind , Journal of the ACM (JACM), 9 (1962), pp. 84--97

work page 1962

[33] [33]

J. R. Ringrose , Compact non-self-adjoint operators , Van Nostrand Reinhold, 1971

work page 1971

[34] [34]

D. Slepian , Prolate spheroidal wave functions, fourier analysis and uncertainty—iv: extensions to many dimensions; generalized prolate spheroidal functions , Bell System Technical Journal, 43 (1964), pp. 3009--3057

work page 1964

[35] [35]

Slepian and H

D. Slepian and H. O. Pollak , Prolate spheroidal wave functions, fourier analysis and uncertainty—i , Bell System Technical Journal, 40 (1961), pp. 43--63

work page 1961

[36] [36]

A. N. Tikhonov , Solution of incorrectly formulated problems and the regularization method. , Sov Dok, 4 (1963), pp. 1035--1038

work page 1963

[37] [37]

A. N. Tikhonov et al. , Regularization of incorrectly posed problems , Soviet Mathematics Doklady, 1963

work page 1963

[38] [38]

Yosida , Functional analysis , vol

K. Yosida , Functional analysis , vol. 123, Springer Science & Business Media, 2012

work page 2012

[39] [39]

Zhang, H

J. Zhang, H. Li, L.-L. Wang, and Z. Zhang , Ball prolate spheroidal wave functions in arbitrary dimensions , Applied and Computational Harmonic Analysis, 48 (2020), pp. 539--569

work page 2020

[40] [40]

Y. Zhou, L. Audibert, S. Meng, and B. Zhang , Exploring low-rank structure for an inverse scattering problem with far-field data , SIAM Journal on Applied Mathematics, 86 (2026), pp. 179--205

work page 2026

[41] [41]

Z. Zhou , On the recovery of two function-valued coefficients in the helmholtz equation for inverse scattering problems via neural networks , Advances in Computational Mathematics, 51 (2025), p. 12

work page 2025

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write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.checka empty 'skip 'new.block if FUNCTION field.or.null duplicate empty pop "" 'skip ...

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