Face-centered quad equations are five-point partial difference equations on the square lattice, with one field variable at a face centre and four at the corners of that face, introduced by Kels together with the multidimensional-consistency property CAFCC (consistency-around-a-face-centered-cube): an overdetermined system of fourteen five-point equations arranged on the face-centered cubic unit cell must be consistent. All known examples (of symmetry types A, B, C) originate from interaction-round-a-face solutions of the functional Yang-Baxter equation, have vanishing algebraic entropy, and admit Lax pairs derived from the CAFCC property itself. The conclusion of Kels (2022, Proc. R. Soc. A 478:20210892) explicitly poses the face-centered analogue of the Adler-Bobenko-Suris (ABS) classification problem: determine whether there are other equations that satisfy CAFCC, since a classification result for multilinear face-centered quad equations 'has not yet been explored'. The problem is to settle completeness of the known families within the class of multilinear face-centered quad equations satisfying CAFCC - either by an exhaustive classification theorem or by exhibiting inequivalent new equations; the results of Kels (2021, Nonlinearity) and Kels (2022) would imply the Lax pairs of any newly found equations.

Background

A central paradigm in the theory of discrete integrable systems is multidimensional consistency. A quad equation is a relation between the values of a field variable at the four corners of a quadrilateral of a quad-graph (a planar lattice whose cells are quadrilaterals), with lattice parameters attached to the two edge directions. The equation is consistent-around-a-cube (CAC), or multidimensionally consistent, when copies of it imposed simultaneously on the three faces of a cube that share a corner determine the opposite corner of the cube uniquely and independently of the order in which the faces are solved. Adler, Bobenko and Suris (2003) used this property to classify, up to Möbius transformations of the field variables (fractional-linear changes of the unknowns) and reparameterizations of the lattice parameters, all quad equations of sufficiently high symmetry - the ABS list, the foundational reference point of the field - and Boll (2011) completed the classification for the remaining symmetry types, at the cost of adjoining tetrahedral equations (additional relations among the four vertices of a cube corner, viewed as the alternating vertices of a tetrahedron) to the cube faces.

Kels (2021) introduced a five-point analogue. A face-centered quad equation is posed on the square lattice in the form A(x;xa,xb,xc,xd;α,β)=0A(x;x_a,x_b,x_c,x_d;\alpha,\beta)=0, where xx is the field value at the centre of a face, xa,xb,xc,xdx_a,x_b,x_c,x_d are the values at the four corners of that face, and α=(α1,α2)\alpha=(\alpha_1,\alpha_2), β=(β1,β2)\beta=(\beta_1,\beta_2) are two-component parameters attached to the horizontal and vertical lattice lines. The equation is required to be multilinear in the four corner variables - the natural condition for a well-defined lattice evolution - while the degree in the face variable is unrestricted; the general ansatz is a sixteen-term polynomial in the corner variables with coefficient functions κi(x,α,β)\kappa_i(x,\alpha,\beta), i=1,…,16i=1,\dots,16, denoted form (2.3) in Kels (2022). All known examples were derived from interaction-round-a-face (IRF) solutions of the functional Yang-Baxter equation - the star-star relations encoding integrability of two-dimensional lattice models of statistical mechanics, taken in a quasi-classical limit. The consistency property, CAFCC (consistency-around-a-face-centered-cube), is the CAC analogue on the face-centered cubic lattice (the three-dimensional lattice obtained by adding a site at the centre of every square face of the cubic lattice): an overdetermined system of fourteen five-point equations is arranged on the face-centered cubic unit cell, and CAFCC holds when this system is consistent, so that all values on the cell are determined compatibly from initial data regardless of the solution path. The known equations fall into three symmetry types - type-A, -B, and -C, distinguished by which of the square symmetries of the equation they satisfy - and all of them have vanishing algebraic entropy (the degrees of iterates of the lattice evolution grow at most linearly, a standard indicator of integrability; Gubbiotti and Kels 2021) and admit Lax pairs (a pair of matrices whose compatibility condition is equivalent to the equation, i.e. a zero-curvature representation) derived from the CAFCC property by comparing two evolution directions on the unit cell (Kels 2021, Nonlinearity).

Kels (2022) reformulated CAFCC so that type-C equations are centered at faces of the unit cell rather than corners, established that the ten combinations of type-A and type-C equations listed in its Table 1 satisfy CAFCC (Theorem 2.1, checkable by direct computation), and derived the Lax pairs of type-C equations; type-B equations satisfy the original CAFCC formulation of Kels (2021). Its conclusion poses the problem addressed here: to determine whether there are other equations that satisfy CAFCC - a question explicitly not yet explored for multilinear face-centered quad equations of the form (2.3) - ideally through a classification result analogous to ABS, with the Lax pairs of any new equations then following from the established mechanism. The same conclusion records further, softer open directions: the integrability of the equations involving six to nine vertices that arise as algebraic consequences on assemblies of adjacent CAFCC cubes in higher-dimensional lattices (their explicit forms are not known; only for type-A systems is a relation to Adler's discrete Toda equations on planar graphs expected); a possible relation between CAFCC and the consistency-around-a-cuboctahedron property for ordinary quad equations of Joshi and Nakazono; and possible reductions to discrete Painlevé equations (second-order nonlinear difference equations that are discrete analogues of the Painlevé transcendents and commonly arise as reductions of multidimensionally consistent lattice systems, as Joshi and Nakazono 2021 demonstrated for cuboctahedron systems).

Problem Statement

Determine whether the known face-centered quad equations exhaust the class of multilinear face-centered quad equations satisfying CAFCC; as suggested in Kels (2022), this might be done by establishing a classification of that class analogous to the Adler-Bobenko-Suris classification. Concretely: find all equations of the general form (2.3) of Kels (2022), namely

A(x;xa,xb,xc,xd;α,β)=κ1xaxbxcxd+κ2xaxbxc+⋯+κ16=0,A(x;x_a,x_b,x_c,x_d;\alpha,\beta)=\kappa_1 x_a x_b x_c x_d+\kappa_2 x_a x_b x_c+\dots+\kappa_{16}=0,

that is, polynomial expressions multilinear in the four corner variables xa,xb,xc,xdx_a,x_b,x_c,x_d with coefficient functions κi(x,α,β)\kappa_i(x,\alpha,\beta), i=1,…,16i=1,\dots,16, depending on the face variable xx and the two-component lattice parameters α=(α1,α2)\alpha=(\alpha_1,\alpha_2), β=(β1,β2)\beta=(\beta_1,\beta_2), that satisfy the consistency-around-a-face-centered-cube property (CAFCC). CAFCC is the property that the overdetermined system of fourteen five-point equations arranged on the face-centered cubic unit cell is consistent; it is a property of the combined system, and an individual equation of the form (2.3) satisfies CAFCC when it participates in such a consistent system. The admissible class covers both formulations: the new formulation of Kels (2022), whose fourteen-equation system involves type-A and type-C equations centered at both faces and corners of the cell, and the original formulation of Kels (2021), in which type-A and type-B equations are centered at faces and type-C equations only at corners. The classification is to be stated up to the natural equivalences of the classification program - Möbius transformations of the five field variables x,xa,xb,xc,xdx,x_a,x_b,x_c,x_d, reparameterizations of the lattice parameters, and the square symmetries - exactly as in the comparable Adler-Bobenko-Suris and Boll classifications. A complete answer either (a) proves that the known type-A, type-B, and type-C families are exhaustive within this class - where the ten CAFCC combinations of Theorem 2.1 of Kels (2022) comprise type-A and type-C equations, and type-B equations satisfy the original CAFCC formulation of Kels (2021) - or (b) exhibits additional equations satisfying CAFCC that are inequivalent to all of them; the results of Kels (2021, Nonlinearity 34:7064) and Kels (2022) would imply the Lax pairs of any newly found equations. In either case the answer should make explicit the status of the known ten-combination list and the consequences for integrability of the classified equations (Lax pairs, algebraic entropy).

References

  1. A. P. Kels, 'Discrete integrable equations on face-centred cubics: consistency and Lax pairs of corner equations', Proc. R. Soc. A 478 (2022) 20210892. DOI: 10.1098/rspa.2021.0892 (arXiv:2112.12211). https://doi.org/10.1098/rspa.2021.0892
  2. A. P. Kels, 'Interaction-round-a-face and consistency-around-a-face-centered-cube', Journal of Mathematical Physics 62 (2021) 033509. DOI: 10.1063/5.0024630 (arXiv:2003.08883). https://doi.org/10.1063/5.0024630
  3. A. P. Kels, 'Lax matrices for lattice equations which satisfy consistency-around-a-face-centered-cube', Nonlinearity 34 (2021) 7064-7094. DOI: 10.1088/1361-6544/ac1f76 (arXiv:2007.01196). https://doi.org/10.1088/1361-6544/ac1f76
  4. G. Gubbiotti and A. P. Kels, 'Algebraic entropy for face-centered quad equations', J. Phys. A: Math. Theor. 54 (2021) 455201. DOI: 10.1088/1751-8121/ac2aeb (arXiv:2105.12404). https://doi.org/10.1088/1751-8121/ac2aeb
  5. A. P. Kels, 'Integrable systems on hexagonal lattices and consistency on polytopes with quadrilateral and hexagonal faces', arXiv:2205.02720 (2022). https://arxiv.org/abs/2205.02720
  6. G. Gubbiotti, A. P. Kels and C.-M. Viallet, 'Algebraic entropy for hex systems', Nonlinearity 37 (2024) 125007. DOI: 10.1088/1361-6544/ad88cd (arXiv:2311.01359). https://doi.org/10.1088/1361-6544/ad88cd
  7. A. P. Kels, 'Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations', Proc. R. Soc. A 481 (2025) 20250386. DOI: 10.1098/rspa.2025.0386 (arXiv:2412.21096). https://doi.org/10.1098/rspa.2025.0386
  8. V. E. Adler, A. I. Bobenko and Yu. B. Suris, 'Classification of Integrable Equations on Quad-Graphs. The Consistency Approach', Communications in Mathematical Physics 233 (2003) 513-543. DOI: 10.1007/s00220-002-0762-8. https://doi.org/10.1007/s00220-002-0762-8
  9. R. Boll, 'Classification of 3D Consistent Quad-Equations', Journal of Nonlinear Mathematical Physics 18 (2011) 337-365. DOI: 10.1142/S1402925111001647. https://doi.org/10.1142/S1402925111001647
  10. N. Joshi and N. Nakazono, 'Classification of quad-equations on a cuboctahedron', arXiv:1906.06650 (2019). https://arxiv.org/abs/1906.06650
  11. N. Joshi and N. Nakazono, 'Reduction of quad-equations consistent around a cuboctahedron I: additive case', Proc. Amer. Math. Soc. Ser. B 8 (2021) 320-335. DOI: 10.1090/bproc/96. https://doi.org/10.1090/bproc/96
  12. V. E. Adler, 'Discrete equations on planar graphs', J. Phys. A: Math. Gen. 34 (2001) 10453-10460. DOI: 10.1088/0305-4470/34/48/310. https://doi.org/10.1088/0305-4470/34/48/310
  13. A. Doliwa, 'Non-autonomous multidimensional Toda system and multiple interpolation problem', J. Phys. A: Math. Theor. 55 (2022) 505202. DOI: 10.1088/1751-8121/acad4d. https://doi.org/10.1088/1751-8121/acad4d
  14. A. Doliwa and A. Siemaszko, 'Integrability and geometry of the Wynn recurrence', Numerical Algorithms 92 (2022) 571-596. DOI: 10.1007/s11075-022-01344-5. https://doi.org/10.1007/s11075-022-01344-5
  15. P. Sun and C. Zhang, 'Classification of Integrable Boundary Equations for Integrable Systems on Quad-Graphs', International Mathematics Research Notices 2023 (2023) 13601-13657. DOI: 10.1093/imrn/rnac207. https://doi.org/10.1093/imrn/rnac207
  16. E. Catak and M. Mullahasanoglu, 'Flipping relation as a reduced star-star relation', European Physical Journal C 85 (2025) 1176. DOI: 10.1140/epjc/s10052-025-14931-2. https://doi.org/10.1140/epjc/s10052-025-14931-2