arxiv: 2604.27286 · v1 · submitted 2026-04-30 · 🧮 math.NA · cs.NA

Thermodynamically Constrained Information Geometric Regularization for Compressible Flows

Seth Taylor , Raymond J. Spiteri , St\'ephane Gaudreault This is my paper

Pith reviewed 2026-05-07 10:13 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords compressible flowsinformation geometric regularizationthermodynamic constraintHessian manifoldcusp singularitiesgeodesic motionAmari-Chentsov tensorinviscid regularization

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The pith

Incorporating a thermodynamic constraint into information geometric regularization mitigates cusp singularities in compressible flow simulations.

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

The paper constructs a thermodynamic extension of information geometric regularization for modeling compressible flows. It uses the Shannon entropy to impose a thermodynamic length constraint on the regularized motion, derived through reformulation in mass and specific entropy variables that link the state to the diffeomorphism group. This leads to dynamics as constrained geodesics on a Hessian manifold with a dual affine connection, employing pullback geometry and the Amari-Chentsov tensor. The resulting model features an anisotropic stress tensor in the momentum equation and an additional elliptic equation. Simulations in one and two dimensions indicate reduced cusp singularities compared to other methods while retaining inviscid regularization advantages.

Core claim

The paper establishes a new compressible fluid model through a thermodynamic extension of information geometric regularization. Reformulating the equations in terms of mass and specific entropy connects the thermodynamic state to a position in the diffeomorphism group, enabling derivation of the regularized equations as constrained geodesic motion on a Hessian manifold using pullback geometry for the Levi-Civita connection and the cubic Amari-Chentsov tensor for the information geometric correction. This introduces an anisotropic stress tensor to the momentum equation that vanishes along isentropic directions along with an additional elliptic equation coupled to the barotropic regularization

What carries the argument

Constrained geodesic motion on a Hessian manifold using pullback geometry and the cubic Amari-Chentsov tensor, which generates the anisotropic stress tensor vanishing along isentropic directions.

Load-bearing premise

Reformulating the fluid equations in mass and specific entropy allows a valid derivation of constrained geodesic motion on a Hessian manifold using pullback geometry and the Amari-Chentsov tensor without unphysical effects or instabilities.

What would settle it

Numerical simulations in one or two dimensions that still exhibit cusp singularities or introduce new instabilities would falsify the claim that the thermodynamic constraint mitigates these issues effectively.

read the original abstract

We construct and analyze a thermodynamic extension of the recently proposed information geometric regularization of Cao and Sch\"afer. The construction extends their shock-mitigating Hessian metric geometry using the Shannon entropy to constrain the regularized motion based on a thermodynamic length. Reformulating the equations in terms of mass and specific entropy explicitly connects the thermodynamic state to a position in the diffeomorphism group, allowing for a derivation of the regularized equations using an information geometric mechanics formalism based on geodesics on a Hessian manifold with a dual affine connection. The dynamics are defined using a pullback geometry for the Levi--Civita connection, describing constrained geodesic motion, and the cubic Amari--Chentsov tensor describing the information geometric correction. This new compressible fluid model introduces an anisotropic stress tensor to the momentum equation that vanishes along isentropic directions and an additional elliptic equation coupled to the barotropic regularization. Numerical simulations in one and two spatial dimensions demonstrate that the geometrically consistent incorporation of a thermodynamic constraint mitigates cusp singularities previously observed in other approaches while still maintaining the benefits of an inviscid regularization.

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 adds a Shannon-entropy thermodynamic constraint to Cao and Schäfer's information-geometric regularization, derives an anisotropic stress tensor vanishing on isentropes plus a coupled elliptic equation, and reports that 1D/2D tests reduce cusp singularities while staying inviscid.

read the letter

This paper extends the information-geometric regularization of Cao and Schäfer by incorporating a thermodynamic length based on Shannon entropy. The authors reformulate the equations in mass and specific entropy to link the thermodynamic state to the diffeomorphism group, then use pullback geometry on a Hessian manifold with the Levi-Civita connection and Amari-Chentsov tensor to obtain the regularized dynamics. The result is an anisotropic stress tensor in the momentum equation that vanishes along isentropic directions together with an additional elliptic equation. Their 1D and 2D simulations indicate this constraint mitigates the cusp singularities seen in prior versions without introducing viscosity or free parameters. The construction is explicit and follows the established formalism without circular fitting. The numerical evidence is concrete for the cases shown and supports the claim that the thermodynamic constraint improves behavior on shocks. The main soft spot is the geometric step itself. The derivation assumes the reformulation preserves the Hessian metric and dual affine connection under pullback so that the resulting tensor and elliptic equation remain consistent with the original information-geometric mechanics. If that preservation does not hold exactly, the observed cusp mitigation could partly reflect discretization choices rather than the geometry. The paper presents the steps, but independent verification of entropy production and well-posedness would help. This is for people working on structure-preserving methods for compressible flows who already follow information-geometric or Hessian-manifold approaches. It deserves a serious referee because the construction is spelled out, the tests are reproducible in low dimensions, and the central claim is falsifiable with further runs. I would send it to peer review.

Circularity Check

0 steps flagged

No significant circularity; derivation extends external formalism without self-reduction

full rationale

The paper's central construction reformulates the compressible Euler equations in mass and specific entropy variables, then invokes the established information-geometric mechanics framework (geodesics on Hessian manifolds, pullback of the Levi-Civita connection, and the Amari-Chentsov tensor) to derive an anisotropic stress term and an elliptic constraint equation. This step rests on the prior work of Cao and Schäfer (distinct authors) together with standard results in information geometry; no equation in the provided derivation chain is shown to be identical to its own input by definition, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness theorem is imported via self-citation. The numerical demonstrations of cusp mitigation are presented as empirical verification rather than as the sole justification for the geometric construction itself. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The central claim rests on the information geometric formalism applied to compressible flows and the assumption that entropy can define a thermodynamic length constraining geodesic motion in the diffeomorphism group.

axioms (3)

standard math Hessian manifold equipped with a dual affine connection
Basis for the information geometric mechanics formalism used in the derivation.
domain assumption Pullback geometry for the Levi-Civita connection describes constrained geodesic motion
Invoked to define the dynamics of the regularized equations.
standard math Cubic Amari-Chentsov tensor describes the information geometric correction
Part of the framework for the correction term in the model.

invented entities (2)

Anisotropic stress tensor no independent evidence
purpose: Added to the momentum equation to enforce the thermodynamic constraint
New term in the regularized model that vanishes along isentropic directions.
Thermodynamic length no independent evidence
purpose: Constrains the regularized motion using Shannon entropy
Extension of the Hessian metric geometry to incorporate thermodynamics.

pith-pipeline@v0.9.0 · 5493 in / 1602 out tokens · 99449 ms · 2026-05-07T10:13:46.591430+00:00 · methodology

discussion (0)

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Works this paper leans on

74 extracted references · 7 canonical work pages · 1 internal anchor

[1]

Springer206(1995)

Anderson, J.D., Wendt, J., et al.: Computational fluid dynamics. Springer206(1995)

1995
[2]

Annual review of fluid mechanics43(1), 163–194 (2011)

Pirozzoli, S.: Numerical methods for high-speed flows. Annual review of fluid mechanics43(1), 163–194 (2011)

2011
[3]

Nature525(7567), 47–55 (2015)

Bauer, P., Thorpe, A., Brunet, G.: The quiet revolution of numerical weather prediction. Nature525(7567), 47–55 (2015)

2015
[4]

Annual Review of Astronomy and Astrophysics52(1), 661–694 (2014)

Lehner, L., Pretorius, F.: Numerical relativity and astrophysics. Annual Review of Astronomy and Astrophysics52(1), 661–694 (2014)

2014
[5]

Matematičeskij sbornik47(3), 271–306 (1959)

Godunov, S.K., Bohachevsky, I.: Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Matematičeskij sbornik47(3), 271–306 (1959)

1959
[6]

Journal of computational physics43(2), 357–372 (1981)

Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of computational physics43(2), 357–372 (1981)

1981
[7]

SIAM review25(1), 35–61 (1983)

Harten, A., Lax, P.D., Leer, B.v.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM review25(1), 35–61 (1983)

1983
[8]

Springer, 315–344 (2009)

Toro, E.F.: The HLL and HLLC Riemann solvers. Springer, 315–344 (2009)

2009
[9]

Springer Science & Business Media (2013)

Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Science & Business Media (2013)

2013
[10]

Van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. Journal of computational Physics32(1), 101–136 (1979)

1979
[11]

Journal of computational physics131(1), 3–47 (1997)

Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. Journal of computational physics131(1), 3–47 (1997)

1997
[12]

Journal of computational physics115(1), 200–212 (1994)

Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. Journal of computational physics115(1), 200–212 (1994)

1994
[13]

Journal of computational physics126(1), 202–228 (1996) 33

Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted eno schemes. Journal of computational physics126(1), 202–228 (1996) 33

1996
[14]

Journal of computational physics87(2), 408–463 (1990)

Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. Journal of computational physics87(2), 408–463 (1990)

1990
[15]

Journal of computational physics160(1), 241–282 (2000)

Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. Journal of computational physics160(1), 241–282 (2000)

2000
[16]

Journal of applied physics21(3), 232–237 (1950)

VonNeumann, J., Richtmyer, R.D.: A method for the numerical calculation of hydrodynamic shocks. Journal of applied physics21(3), 232–237 (1950)

1950
[17]

In: 44th AIAA Aerospace Sciences Meeting and Exhibit, p

Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, p. 112 (2006)

2006
[18]

Courier Corporation, Mineola, New York (2001)

Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Corporation, Mineola, New York (2001)

2001
[19]

Journal of computational physics228(14), 4965–4969 (2009)

Bhagatwala, A., Lele, S.K.: A modified artificial viscosity approach for compressible turbulence simulations. Journal of computational physics228(14), 4965–4969 (2009)

2009
[20]

Journal of computational physics36(3), 281–303 (1980)

Wilkins, M.L.: Use of artificial viscosity in multidimensional fluid dynamic calculations. Journal of computational physics36(3), 281–303 (1980)

1980
[21]

Physical review letters101(14), 144501 (2008)

Frisch, U., Kurien, S., Pandit, R., Pauls, W., Ray, S.S., Wirth, A., Zhu, J.-Z.: Hyperviscosity, Galerkin truncation, and bottlenecks in turbulence. Physical review letters101(14), 144501 (2008)

2008
[22]

Journal of Computational Physics203(2), 379–385 (2005)

Cook, A.W., Cabot, W.H.: Hyperviscosity for shock-turbulence interactions. Journal of Computational Physics203(2), 379–385 (2005)

2005
[23]

arXiv preprint arXiv:2308.14127 (2023)

Cao, R., Schäfer, F.: Information geometric regularization of the barotropic Euler equation. arXiv preprint arXiv:2308.14127 (2023)

work page arXiv 2023
[24]

Proceedings of the Steklov Institute of Mathematics259(1), 73–81 (2007)

Khesin, B., Misiołek, G.: Shock waves for the Burgers equation and curvatures of diffeomorphism groups. Proceedings of the Steklov Institute of Mathematics259(1), 73–81 (2007)

2007
[25]

Springer194(2016)

Amari, S.-i.: Information geometry and its applications. Springer194(2016)

2016
[26]

arXiv preprint arXiv:2411.15121 (2024)

Cao, R., Schäfer, F.: Information geometric regularization of unidimensional pressureless Euler equations yields global strong solutions. arXiv preprint arXiv:2411.15121 (2024)

work page arXiv 2024
[27]

In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp

Wilfong, B., Radhakrishnan, A., Le Berre, H., Vickers, D., Prathi, T., Tselepidis, N., Dorschner, B., Budiardja, R., Cornille, B., Abbott, S.,et al.: Simulating many-engine spacecraft: Exceeding 1 quadrillion degrees of freedom via information geometric regularization. In: Proceedings of the International Conference for High Performance Computing, Network...

2025
[28]

arXiv preprint arXiv:2512.13948 (2025)

Barham, W., Tran, B.K., Southworth, B.S., Schäfer, F.: Hamiltonian information geometric regularization of the compressible Euler equations. arXiv preprint arXiv:2512.13948 (2025)

work page arXiv 2025
[29]

Bulletin of the American Mathematical Society58(3), 377–442 (2021)

Khesin, B., Misiołek, G., Modin, K.: Geometric hydrodynamics and infinite-dimensional newton’s equations. Bulletin of the American Mathematical Society58(3), 377–442 (2021)

2021
[30]

Springer Science & Business Media103 (2012)

Dolzhansky, F.V.: Fundamentals of geophysical hydrodynamics. Springer Science & Business Media103 (2012)

2012
[31]

Springer64(2017)

Ay, N., Jost, J., Vân Lê, H., Schwachhöfer, L.: Information geometry. Springer64(2017)

2017
[32]

Rao, C.R.,et al.: Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc37(3), 81–91 (1945)

1945
[33]

American Mathematical Society (1982)

Chentsov, N.N.: Statistical decision rules and optimal inference. American Mathematical Society (1982)

1982
[34]

Post RAAG Reports (1980)

Amari, S.-i.: Theory of information spaces: A differential geometrical foundation of statistics. Post RAAG Reports (1980)

1980
[35]

The Annals of Statistics, 357–385 (1982)

Amari, S.-I.: Differential geometry of curved exponential families-curvatures and information loss. The Annals of Statistics, 357–385 (1982)

1982
[36]

Differential geometry in statistical inference10(2), 163–216 (1987)

Lauritzen, S.L.: Statistical manifolds. Differential geometry in statistical inference10(2), 163–216 (1987)

1987
[37]

The annals of statistics, 1543–1561 (1995)

Pistone, G., Sempi, C.: An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one. The annals of statistics, 1543–1561 (1995)

1995
[38]

Information geometry of diffeomorphism groups

Khesin, B., Misiołek, G., Modin, K.: Information geometry of diffeomorphism groups. arXiv preprint arXiv:2411.03265 (2024)

work page arXiv 2024
[39]

Calculus of Variations and Partial Differential Equations63(2), 56 (2024)

Bauer, M., Le Brigant, A., Lu, Y., Maor, C.: The l p-Fisher–Rao metric and Amari–Chentsovα- connections. Calculus of Variations and Partial Differential Equations63(2), 56 (2024)

2024
[40]

Differential geometry and its applications7(3), 277–290 (1997)

Shima, H., Yagi, K.: Geometry of hessian manifolds. Differential geometry and its applications7(3), 277–290 (1997)

1997
[41]

Advances in Mathematics137(1), 1–81 (1998) 34

Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler–Poincaré equations and semidirect products with applications to continuum theories. Advances in Mathematics137(1), 1–81 (1998) 34

1998
[42]

Oxford University Press12(2009)

Holm, D.D., Schmah, T., Stoica, C.: Geometric mechanics and symmetry: from finite to infinite dimensions. Oxford University Press12(2009)

2009
[43]

SIAM Review39(1), 152–152 (1997)

Marsden, J.E., Ratiu, T.S., Hermann, R.: Introduction to mechanics and symmetry. SIAM Review39(1), 152–152 (1997)

1997
[44]

Physical Review Letters99(10), 100602 (2007)

Crooks, G.E.: Measuring thermodynamic length. Physical Review Letters99(10), 100602 (2007)

2007
[45]

Physical Review E—Statistical, Nonlinear, and Soft Matter Physics79(1), 012104 (2009)

Feng, E.H., Crooks, G.E.: Far-from-equilibrium measurements of thermodynamic length. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics79(1), 012104 (2009)

2009
[46]

Reports on progress in physics75(12), 126001 (2012)

Seifert, U.: Stochastic thermodynamics, fluctuation theorems and molecular machines. Reports on progress in physics75(12), 126001 (2012)

2012
[47]

Physical Review X13(1), 011013 (2023)

Van Vu, T., Saito, K.: Thermodynamic unification of optimal transport: Thermodynamic uncertainty relation, minimum dissipation, and thermodynamic speed limits. Physical Review X13(1), 011013 (2023)

2023
[48]

Bulletin of the London Mathematical Society48(3), 499–506 (2016)

Bauer, M., Bruveris, M., Michor, P.W.: Uniqueness of the Fisher–Rao metric on the space of smooth densities. Bulletin of the London Mathematical Society48(3), 499–506 (2016)

2016
[49]

Numerische Mathematik84(3), 375–393 (2000)

Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the monge-kantorovich mass transfer problem. Numerische Mathematik84(3), 375–393 (2000)

2000
[50]

Nature Communications16(1), 10424 (2025)

Oikawa, S., Nakayama, Y., Ito, S., Sagawa, T., Toyabe, S.: Experimentally achieving minimal dissipation via thermodynamically optimal transport. Nature Communications16(1), 10424 (2025)

2025
[51]

Otto, F.: The geometry of dissipative evolution equations: the porous medium equation (2001)

2001
[52]

Annals of Mathematics, 903–991 (2009)

Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Annals of Mathematics, 903–991 (2009)

2009
[53]

Journal of Mathematical Physics62(3) (2021)

Li, W.: Hessian metric via transport information geometry. Journal of Mathematical Physics62(3) (2021)

2021
[54]

Springer132(1992)

LeVeque, R.J., Leveque, R.J.: Numerical methods for conservation laws. Springer132(1992)

1992
[55]

Shocks without shock capturing: Information geometric regularization of finite volume methods for Navier--Stokes-like problems

Radhakrishnan, A., Wilfong, B., Bryngelson, S.H., Schäfer, F.: Shocks without shock capturing: Informa- tion geometric regularization of finite volume methods for navier–stokes-like problems. arXiv preprint arXiv:2604.06546 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[56]

Foundations of Computational Mathematics21(4), 961–1001 (2021)

Gawlik, E.S., Gay-Balmaz, F.: A variational finite element discretization of compressible flow. Foundations of Computational Mathematics21(4), 961–1001 (2021)

2021
[57]

Acta Numerica32, 291–393 (2023)

Cotter, C.J.: Compatible finite element methods for geophysical fluid dynamics. Acta Numerica32, 291–393 (2023)

2023
[58]

Journal of Computational Physics268, 17–38 (2014)

Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial–boundary-value problems. Journal of Computational Physics268, 17–38 (2014)

2014
[59]

arXiv preprint arXiv:2511.20529 (2025)

Bach, D., Rueda-Ramírez, A.M., Sonnendrücker, E., Fernández, D.C., Gassner, G.J.: SBP-FDEC: Summation-by-Parts Finite Difference Exterior Calculus. arXiv preprint arXiv:2511.20529 (2025)

work page arXiv 2025
[60]

Journal of Computational Physics226(1), 379–397 (2007)

Hou, T.Y., Li, R.: Computing nearly singular solutions using pseudo-spectral methods. Journal of Computational Physics226(1), 379–397 (2007)

2007
[61]

SIAM Journal on Scientific Computing36(4), 2076–2099 (2014)

Kopriva, D.A., Gassner, G.J.: An energy stable discontinuous galerkin spectral element discretization for variable coefficient advection problems. SIAM Journal on Scientific Computing36(4), 2076–2099 (2014)

2076
[62]

In: Handbook of Numerical Methods for Hyperbolic Problems vol

Tadmor, E.: Entropy stable schemes. In: Handbook of Numerical Methods for Hyperbolic Problems vol. 17, pp. 467–493. Elsevier, Amsterdam (2016)

2016
[63]

Journal of Computational Physics242, 169–180 (2013)

Hu, X.Y., Adams, N.A., Shu, C.-W.: Positivity-preserving method for high-order conservative schemes solving compressible Euler equations. Journal of Computational Physics242, 169–180 (2013)

2013
[64]

Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. i. Mathematics of Computation49(179), 91–103 (1987)

1987
[65]

ACM Transactions on Mathematical Software 49(4), 1–30 (2023)

Ranocha, H., Schlottke-Lakemper, M., Chan, J., Rueda-Ramírez, A.M., Winters, A.R., Hindenlang, F., Gassner, G.J.: Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous galerkin methods for conservation laws. ACM Transactions on Mathematical Software 49(4), 1–30 (2023)

2023
[66]

Journal of Scientific Computing90(3), 79 (2022)

Gassner, G.J., Svärd, M., Hindenlang, F.J.: Stability issues of entropy-stable and/or split-form high-order schemes: analysis of linear stability. Journal of Scientific Computing90(3), 79 (2022)

2022
[67]

application to entropy conservative and entropy dissipative schemes

Abgrall, R.: A general framework to construct schemes satisfying additional conservation relations. application to entropy conservative and entropy dissipative schemes. Journal of Computational Physics 372, 640–666 (2018)

2018
[68]

arXiv preprint arXiv:2501.16529 (2025) 35

Chan, J.: An artificial viscosity approach to high order entropy stable discontinuous galerkin methods. arXiv preprint arXiv:2501.16529 (2025) 35

work page arXiv 2025
[69]

Reviews of Modern Physics 67(3), 605 (1995)

Ruppeiner, G.: Riemannian geometry in thermodynamic fluctuation theory. Reviews of Modern Physics 67(3), 605 (1995)

1995
[70]

American Journal of Physics78(11), 1170–1180 (2010)

Ruppeiner, G.: Thermodynamic curvature measures interactions. American Journal of Physics78(11), 1170–1180 (2010)

2010
[71]

Physical Review Letters133(5), 057102 (2024)

Zhong, A., DeWeese, M.R.: Beyond linear response: Equivalence between thermodynamic geometry and optimal transport. Physical Review Letters133(5), 057102 (2024)

2024
[72]

Information geometry7, 441–483 (2024)

Ito, S.: Geometric thermodynamics for the Fokker–Planck equation: stochastic thermodynamic links between information geometry and optimal transport. Information geometry7, 441–483 (2024)

2024
[73]

Entropy19(10), 518 (2017)

Leok, M., Zhang, J.: Connecting information geometry and geometric mechanics. Entropy19(10), 518 (2017)

2017
[74]

Journal of Geometry and Physics, 105854 (2026) 36

Gay–Balmaz, F., Abella, Á.R., Yoshimura, H.: Infinite-dimensional Lagrange–Dirac systems with bound- ary energy flow ii: Field theories with bundle-valued forms. Journal of Geometry and Physics, 105854 (2026) 36

2026