Papers

We study a stochastic Navier–Stokes system in which the spatial modes of the transport noise evolve dynamically and are coupled to the resolved velocity. The model couples a Navier–Stokes stochastic PDE to an infinite family of linearized Navier-Stokes equations for the noise correlation modes. These latter equations may contain a hyperviscosity. We prove existence of martingale solutions in two dimensions for hyperviscosity exponent $s\geq1$, and pathwise uniqueness for $s>1$ under additional regularity of the initial noise modes. In three dimensions, we establish existence of martingale solutions for $s>3/2$. The proof combines Galerkin approximations and compactness with estimates adapted to the coupled drift and stochastic terms.

In this paper we propose a universal way to define multivariate orthogonal polynomials for the positive branch of WLZZ matrix models. This allows us to generalize from the Gaussian case, previously considered in the literature, the strict superintegrability property and the Miwa deformation formulas.

We determine the maximum stable volume of bubbles attached to flat surfaces as a function of contact radius, contact angle, and capillary length. By solving the Young-Laplace equation using a shooting method, we calculate equilibrium bubble shapes and determine when bubbles detach through three distinct modes: necking at the cavity, sideways instability, and spreading followed by necking on the surrounding surface. Our predicted detachment volumes agree well with experimental measurements of electrolytic, boiling, and carbonated bubbles, as well as pendant drops. These results bring together and extend previous analytical estimates and, by symmetry, also apply directly to pendant drops.

This paper develops a staggered discontinuous Galerkin (SDG) scheme based on Cartesian grids for Stokes equations that is simple to implement, intrinsically pressure-robust, and superconvergent for all variables. Instead of the composite meshes used in standard SDG, we construct staggered quadrilateral meshes from Cartesian grids. The scheme takes velocity, pressure, and velocity gradient as unknowns, and employs piecewise-constant spaces with carefully designed staggered continuity. The additional gradient unknowns are locally eliminated via static condensation and can be further removed by mass lumping without loss of accuracy. An explicit pointwise formulation of the scheme is derived, which facilitates implementation and enables a detailed pointwise analysis. We rigorously prove pressure robustness and second-order superconvergence, which holds on general non-uniform Cartesian grids. The scheme is further extended to Navier-Stokes equations by introducing a novel discrete convection term with second-order consistency. Combined with the scalar auxiliary variable (SAV) approach and the Crank-Nicolson (CN) scheme, this yields an unconditionally energy-stable and second-order accurate scheme. Numerical experiments validate the theory and demonstrate accuracy and robustness.

We obtain a complete characterization of eventually entanglement breaking (EEB) unital channels in terms of some simple commutation properties of their eigenvectors. Specifically, the following properties are equivalent for a unital channel: (i) it is EEB; (ii) it is eventually positive partial transpose (PPT); (iii) it is eventually mixed twisted dephasing; and (iv) its eigenvectors corresponding to non-zero eigenvalues commute with all its peripheral eigenvectors. Similar results hold for one-parameter semigroups of unital quantum channels, where continuity forces such semigroups to become primitive. We employ a matrix integral technique inspired by a result of Watrous. We also characterize the extreme points of the convex set of unital EB channels and, using this characterization and the theory of equiangular tight frames (ETFs), identify several extreme points beyond the twisted dephasing channels.

We study the fixed-$q$ right-edge asymptotics of the ordinary $q$-Krawtchouk ensemble. For $0<q<1$, finite polynomial duality gives a direct proof of convergence of the correlation kernels to an explicit limiting kernel. The local Jacobi limit is a shifted continuous $q^{-1}$-Hermite expression with an indeterminate moment problem, so the limiting Jacobi expression does not determine a unique self-adjoint operator. A single spectral atom, asymptotically saturating the mass bound, selects the realization and yields strong-resolvent convergence. The limiting kernel is its positive spectral projection and admits spectral-series, Christoffel–Darboux, and partial-theta representations, and the associated dual operator agrees with a two-step spatial Markov operator of Borodin–Corwin. A companion $q>1$ limit is instead essentially self-adjoint, of Al-Salam–Carlitz~I type.

We derive exact finite-size nonperturbative representations of equilibrium modal occupations and related statistics for three representative nonlinear systems: the Majda-McLaughlin-Tabak dispersive-wave model, the Fermi-Pasta-Ulam-Tsingou beta anharmonic chain, and the discrete nonlinear Schrodinger lattice field. Independent simulations confirm the predictions from weak to strong nonlinearity. For DNLS, the theory remains accurate across the weak-coupling quasicondensation crossover, where large low-mode occupations and long-range coherence amplify interaction effects even when the bare nonlinear coefficient is small. The finite-ring DNLS occupations are further resolved into a positive sum of Rayleigh-Jeans channels with distinct correlation lengths, explaining when a single Rayleigh-Jeans law applies and why it fails near quasicondensation. In MMT and DNLS, the exact occupations also determine the mean modal frequencies even when the dynamical spectra broaden or split. The nonperturbative results allow a direct assessment of two representative perturbative approaches. Treating the mean interaction appropriately yields accurate low-order approximations, including at strong nonlinearity. At higher orders, however, the corrections cease to decrease and successive approximations oscillate with increasing amplitude; both finite-order approaches also fail near weak-coupling quasicondensation. Thus neither low-order agreement nor a small bare coupling guarantees a reliable perturbative description. The results establish nonperturbative equilibrium theory for widely used nonlinear wave and lattice models and provide a quantitative basis for modal distributions of energy, particles, and optical power in nonlinear optics, dispersive waves, anharmonic lattices, and cold-atom systems.

We show the height at time $t$ of the standard continuous-time ballistic deposition process has variance at most $C_d t, t \geq 1$ in every dimension $d \geq 1$. The crucial idea is to adapt a martingale argument due to Aldous, which gives tail bounds on first hitting times for monotone surface growth processes. Inverting these tail bounds and applying a duality theorem due to Penrose yields a linear variance bound for the height of ballistic deposition.

Preparing interacting quantum systems in low-energy or ground states is a fundamental task in quantum simulation and quantum information processing. In realistic settings, dissipation and active cooling can typically be engineered only on a limited subset of the system. We study the dissipative dynamics of excitation-number conserving quantum systems governed by a GKLS master equation with local jump operators acting on a subset of qubits. We establish sufficient conditions under which such localized dissipation drives the full system to a unique globally attractive zero-excitation state. In particular, we prove that if the Hamiltonian generates excitation transfer described by a graph for which the dissipative subsystem forms a zero forcing set, then the zero-excitation state is the unique globally attractive stationary state. When this state coincides with a ground state of the Hamiltonian, the same mechanism realizes ground-state cooling. Our results provide a graph-theoretic criterion for global state preparation from localized dissipation, which we illustrate using a nearest-neighbor Heisenberg spin chain. For this model, a reduction to the single-excitation sector further yields a scaling estimate with the length of the chain which indicates efficient cooling.

This article is the second part of a review devoted to spacetime singularities in classical General Relativity. While Primer I developed the mathematical framework for defining, classifying, and proving the existence of singularities, the present work examines their concrete realization in spatially homogeneous cosmological models. After reviewing the geometrical foundations of homogeneity and the Bianchi classification, we derive the field equations for orthogonal and tilted perfect fluid cosmologies and use the Kasner and Bianchi I solutions as elementary prototypes of anisotropic singular behaviour. Raychaudhuri focusing provides general conditions under which the homogeneous Cauchy development reaches a finite proper time boundary. We then distinguish matter singularities, characterized by divergent density and Ricci curvature, from more subtle non-scalar or whimper singularities associated with Cauchy horizons and extreme fluid tilt. The classical Ellis-King analysis is reconsidered in the light of later developments, including rigorous curvature-blow-up results for Bianchi VIII and IX models, kinematic singularities in geodesically complete tilted spacetimes, and the recent reappearance of whimper behaviour in dipole cosmology. The resulting picture shows that even spatially homogeneous cosmologies exhibit a remarkably rich hierarchy of singular boundaries. The detailed dynamical approach to these boundaries - Hamiltonian cosmology, Kasner transitions, Mixmaster dynamics and the BKL conjecture - is reserved for Primer III.

We give an explicit non-random example of nonadditivity of minimum output von Neumann entropy. The proof uses a finite-dimensional construction which imitates free Haar unitary behavior. The same channel gives a violation of additivity for the minimum output Rényi-$p$ entropy for any $p\in[1,\infty]$. Additionally, given any $p_0>0$ and $\mathfrak g>0$, we extend the construction to obtain an explicit non-random channel which has entropy gap $\ge\mathfrak g$ uniformly over $p\in[p_0,\infty]$.

Lee-Yang theorems are a powerful tool for studying many-body systems, with applications ranging from analyzing phase transitions to proving the efficiency of certain classical and quantum algorithms. In this work, we prove a Lee-Yang zero-freeness theorem for the partition function of a broad class of interacting fermion models, implying the existence of a provably efficient quantum algorithm for estimating their ground-state energies. This class includes several well-known models such as the attractive Hubbard model, repulsive Hubbard model on bipartite graphs, and the interacting Hofstadter model. Our results also rigorously establish the nonexistence of phase transitions in these models in the presence of a nonzero local external field.

We determine the electrostatic Dirichlet-to-Neumann map of an arbitrary four-dimensional Majumdar–Papapetrou spacetime. For a minimally coupled spectator Maxwell field, the potentials of the disconnected extremal horizons may be prescribed independently, and the resulting flux charges are related by an exact capacitance matrix. If the centers have masses $M_A$ and coordinate separations $R_{AB}$, its entries are $C_{AA}=M_A+\sum_{B\ne A}M_AM_B/R_{AB}$ and $C_{AB}=-M_AM_B/R_{AB}$. The matrix is a positive diagonal term plus the weighted Laplacian of the complete graph of horizons, giving an exact common-mode and differential-mode decomposition of the field energy. The grounded Dirichlet Green function shows that a point charge partitions its flux among infinity and the horizons according to elementary harmonic measures. A direct translation of the multicenter Green function of Frolov and Zelnikov gives the diagonal completion used in their published Eq.~(68), whose component-resolved Gauss flux preserves the total horizon charge but generally transfers charge between horizons. The symmetric Green function that preserves every horizon charge instead contains the inverse of the full capacitance matrix. For a binary, the difference gives a finite, negative-semidefinite shift of the electrostatic self-energy and a corresponding exact self-force difference without an additional ultraviolet subtraction. We finally extend the response matrix, flux partition, and fixed-charge Green function to $D=n+3$ dimensions, where the graph weights acquire the universal factor $\pi^{n/2}/\Gamma(n/2)$.

The Vlasov-Maxwell system with Newtonian particle transport is linearised about the Poisson equilibrium on $\mathbb{R}^3$, with the speed of light $c$ treated as a large parameter. For uniformly controlled initial data, with transverse fields that may remain of order one, the difference between the Vlasov-Maxwell perturbation and its linearised Vlasov-Poisson counterpart is bounded uniformly in time by their initial discrepancy plus a term of order $c^{-1}$. The same estimate holds for their scattering states, and the rate in $c$ is sharp. The longitudinal dynamics are governed by the same Volterra equation as in linearised Vlasov-Poisson and undergo Landau damping. The transverse fields exhibit Klein-Gordon-type dispersion, and control of their accumulated effect along free transport yields the uniform comparison.

On 7 and 8 September 2026 programmatic search produced singularities: a forced Navier-Stokes singularity at every fixed viscosity, statements (C) and (D) of the Clay problem, and two Euler singularities. The unforced problem, statements (A) and (B), stands open, and a search for it needs a target. This paper fixes one: the positive defect problem, that a Leray-Hopf solution from smooth data on the periodic cube loses energy on a finite window, at fixed viscosity, beyond what viscosity removes. A positive defect implies blowup and so a negative answer to statement (B); the converse is not known. The paper proves the target equivalent to a floor on the energy flux through the Littlewood-Paley shells, the Fourier-side form of the coarse-grained flux of the Onsager theory of turbulence, averaged over the window; states necessary conditions on a candidate: a singular time of Type II in velocity, energy concentrating on a set of zero length, a pressure outside L^2, a velocity outside the Onsager-critical class L^3_t B^{1/3}_{3,c_0}, an obstruction to collapse onto a fixed steady Euler profile; and states what cannot certify one: no finite computation witnesses a Galerkin-uniform ceiling, and selection and forcing return the question to a positive defect. A pseudo-spectral search at 128^3 and 256^3 shows which condition of the reduction binds: the fine-shell flux floor holds to within 1 to 6 percent of the ceiling for a third of a turnover time, and fails in scale at the Kolmogorov wavenumber, so a candidate must differ from generic turbulence in the depth of its cascade, not in its timing. Every implication not marked otherwise is a theorem in Lean 4 over Mathlib; the library contains no Navier-Stokes object, and the equation enters only through hypotheses.

In this paper we study the Borel summability and quantum modularity of the free energy computed by refined topological recursion on the Weber-type refined spectral curve. On the one hand, we verify expected properties, namely, Stokes rays come in a pair and the Borel-Laplace sum of the free energy coincides with the Barnes double gamma function. Somewhat unexpectedly, on the other hand, we prove that the generating function of the Stokes constants along each ray is a Jacobi quantum modular function. Our proof suggests that quantum modularity may be a general feature of refined topological recursion, applicable beyond the Weber-type refined spectral curve.

We address the problem of determining the degeneracy of a particular energy eigenvalue corresponding to a quantum particle trapped in a cubic box. In other words, we aim to find the number of ways a specific positive integer can be written as the sum of three natural number squares. This is a classic problem in number theory. We provide a closed-form, exact solution to the problem. We do not claim to provide proofs, as we have none; instead, we build up the formulae starting from simple cases where the answer could be deduced by studying the list of degeneracies. We present ample examples that could be verified against the degeneracy calculated numerically via the brute-force method. We arrived at the solution by meticulously studying the list of degeneracies for the first five million numbers, generated by Mathematica$^{\circledR}$.

A recent pre-print [arXiv:2609.19214] proposed a three-dimensional (3D) strongly aperiodic monotile: a shape that tiles Euclidean space only non-periodically and which admits no symmetry of infinite order. The proof takes the 3D Chair tile identified previously by Lee and Moody, and adds geometric decorations to the faces so as to force non-periodicity (without these decorations The Chair also admits periodic tilings). Here we establish general requirements on face decorations to achieve the same end, in order to facilitate the search for physical realisations. We find that the requirements are minimal. We provide matching rules using three colours of arrow that are equivalent to the original rules, in that they force the same local and global configurations. These rules force Chairs to compose into `Superchairs' with doubled linear dimensions. In this process the matching rules themselves compose uniquely. We find that the same global structure can be forced using simpler rules based on the colours of squares, regardless of orientation. Any physical system encoding these rules (geometrically or otherwise) will force the same strongly aperiodic monotilings. We provide simple examples.

We classify all smooth local conservation laws of three-dimensional homogeneous incompressible Euler flow in velocity and pressure, allowing explicit time and position and derivatives of arbitrary finite order. Modulo currents vanishing on solutions and currents with identically zero divergence, the classical mass, momentum, angular-momentum, kinetic-energy and helicity balances, together with generalised time-dependent momentum, exhaust the classification. We also classify all smooth finite-order local symmetry characteristics and adjoint symmetries: differential expressions satisfying the linearised Euler equations and their formal adjoint on solutions, respectively. Every symmetry characteristic is equivalent on solutions to one induced by a point transformation. Every adjoint symmetry has a multiplier representative, whose product with the Euler equations is identically a divergence. These representatives exhaust the restrictions of all local conservation-law multipliers to solutions. The conservation-law proof has three stages: classification of adjoint symmetries using their determining equations, verification of explicit multiplier identities, and the correspondence between multiplier restrictions and current classes. A common analysis of highest-derivative responses reduces both symmetry problems to order at most one.

We study complex geodesic actions and heat-kernel resurgence on isotropic Heisenberg groups. For real endpoints with $r>0,y\ne0$, the nonzero finite singularities reached by analytic continuation of the minimizing-saddle Borel germ are exactly the finite complex action differences. For nonresonant endpoints, we construct row-finite directional Stokes operators and compute the positive-ray Stokes matrix and its logarithm explicitly; the primitive alien support can be strictly smaller than the full Borel singular support. Near the vertical axis, we obtain uniform two-term large-order asymptotics from the nearest Borel singularity. Cubic resonances admit uniform $A_2$ confluence formulas. We also identify the combination of resurgent sectors selected by the original Fourier contour. Positive Borel summation is exact at horizontal endpoints. At nonzero isotropic complex endpoints, no finite normal trajectories exist, but the signed vertical action lattice is the complete limiting spectrum. The action and Stokes structures are independent of the dimension on isotropic $H^n$.

We construct gauge-origami systems on the toric Calabi–Yau fourfold $\mathcal{C}\times\mathbb{C}$ from its brane brick model. Brick matchings and phase boundaries determine elementary D6- and D4-brane framings, respectively, realizing the magnificent-four, tetrahedron-instanton, and spiked-instanton sectors in a unified framed quiver quantum mechanics. We define their partition functions by Jeffrey–Kirwan residues and describe the fixed points in terms of colored BPS crystals. General gauge-origami configurations are obtained by combining these elementary sectors through their common fractional D0-brane quiver. This construction provides a first step toward gauge origami on general toric Calabi–Yau fourfolds.

We identify a directional tangency mechanism responsible for anomalous far-field decay in Schrödinger equations. For constant-coefficient operators, we introduce a directional decomposition of the limiting resolvent based on the geometry of the Fermi surface and the observation direction. In two dimensions, we show that an analytic source produces a super-algebraically small tangency contribution, whereas a transverse non-analyticity located on the directional tangency set generates an algebraic contribution of order \(R^{-(2\alpha+1)}\). Shifting the singularity away from the tangency set restores super-algebraic decay, showing that geometric alignment is essential to the anomalous behavior. We further construct a bounded, decaying real-valued potential for which the same mechanism occurs in the first Born correction through the effective source \(Vu_0\). In a suitable long-range regime, the resulting tangency contribution decays more slowly than the conventional two-dimensional far-field term.

We establish a resolvent factorization of the hyperbolic projection associated with the finite-range dual operators, by a kernel with exponential off-diagonal decay. As an application, we prove that, for every irrational frequency and every analytic potential, each subcritical Type I energy satisfying the gap-labelling condition is an endpoint of an open spectral gap. This confirms the conjecture of Ge–Jitomirskaya–You [GJY,You] in the subcritical regime.

Consider an $N\times N$ random band matrix $H$ with bandwidth $W$ with centered real Gaussian entries. We prove that, under the (almost) sharp assumption $W^2\ll N/\log N$, the bulk local statistics of $H$ are asymptotically a Poisson point process. Combined with the bulk universality results from arXiv:2501.01718 and arXiv:2506.06441, this establishes the Poisson-RMT transition of the local statistics of random band matrices, and complements the results about the localization-delocalization transition of the bulk eigenvectors previously established in arXiv:2501.01718 and arXiv:2506.06441, and arXiv:2508.05802. The key technical inputs are: (1) the fractional moment estimates from arXiv:2508.05802; (2) a novel estimate for the density of states, obtained from the local laws in arXiv:2501.01718 and arXiv:2506.06441.

We study the instability of the naked singularities arising in the spherically symmetric self-similar collapsing of the Einstein–Euler system under gravitational perturbations. We show that small $C^{1,\alpha}$ perturbations (without any symmetries) of the initial conformal metric lead to trapped surface formation. One key point is that the speed of sound is strictly slower the the speed of light, so the gravitational radiation can become sufficiently concentrated to form a trapped surface before the fluid develops any singularity.

We develop an effective radial framework for analyzing new field-modified potential function of a charged spinless particle in collinear electric and magnetic fields. Using the weak- magnetic field approximation, we neglect the diamagnetic term and other angular projections under an explicit smallness condition $\ll \Delta E$, where $E$ is the electric field, and the replacement of the angular factor by $C=\cos\theta$ in the spherical coordinates. The resulting equation is therefore a field-aligned radial model whose validity is limited to angularly localized states. Within this setting, we apply the indirect mode (or search mode) of Tridiagonal representation approach in the Laguerre and Jacobi bases. In the Laguerre basis, the tridiagonal constraint generates linear-plus-inverse and linear-plus-quadratic radial potentials, and the coefficient recurrences are connected with Meixner–Pollaczek and Meixner polynomial type structures, therefore yielding the analytic spectra and expansion wavefunctions. In the Jacobi basis, the mapping of the radial half-line to a finite interval produces a singular confining effective potential and a three-term recurrence that is treated through a truncated generalized eigenvalue problem. The formulation provides analytically and numerically tractable benchmark models for field-modified potential functions, while keeping the assumptions, parameter restrictions, endpoint behavior, and convergence tests explicit. All energy spectra formulae obtained under the present models are new and are coupled with magnetic fields analytically and numerically when using the Laguerre bases and the Jacobi bases respectively.

We revisit the combinatorial loop-equation method for computing planar disc amplitudes in multi-matrix models with polynomial potentials. We develop a procedure which organises the Schwinger-Dyson constraints by the level of their boundary insertions and, when the resulting system closes, produces an algebraic equation for the planar disc function. Using this method we recover the disc partition functions associated with fixed, mixed, and free boundary conditions in the three-state Potts model, previously obtained by saddle-point methods. As a novelty we derive an algebraic equation for the fixed-boundary resolvent of the three-state Potts model with unequal quadratic and cubic couplings. We further show one can compute the disc partition function for several unsolved two- and three-matrix models. These examples demonstrate that loop equations provide a practical route to identifying and solving sectors of multi-matrix models beyond the standard one- and two-matrix cases.

We study numerical observables relevant to delocalization in the three-dimensional Manhattan mirror model, also known as the K-lattice. Our principal experiment separates the locations of transmitting boundary channels from the routing between them. We derive an exact conditional comparison that preserves all transmission and reflection aperture sets, and find a statistically resolved deficit of physical transmission relative to this comparison. Nearby transmitting channels are themselves suppressed relative to a uniform subset with the same cardinality. A second experiment resolves the contribution of repeated vertices by their return age. The squared norm of the signed old-memory remainder is consistent with an inverse-square-root decay over the measured range. Annular experiments quantify decreasing continuation losses, while an initially apparent history dependence fails independent replication. Finally, a periodic configuration has a connected auxiliary contact graph but only finite routing cycles. These results identify estimates on reflection routing and adaptively selected revisit charges that may assist a rigorous proof. They do not establish an infinite-volume delocalized phase.

The classical theory of the wind-driven surface layer of a rotating ocean closes the momentum balance with a local flux-gradient law, in which the turbulent stress at a given depth is proportional to the shear at that depth. It predicts a surface current deflected forty-five degrees from the wind, which exceeds most direct measurements. This study asks what follows when locality is relaxed. Beginning from the exact integral relation between turbulent stress and mean shear, and requiring the memory kernel to carry no preferred vertical scale, we obtain a power-law kernel and therefore a stress law of fractional order. The resulting equation cannot be posed on the velocity, because the fractional derivative of a bounded profile vanishes at the surface, so the wind stress cannot be applied, while the alternative definition of the derivative leaves the surface current unbounded. Posed on the stress instead, the problem is solvable in closed form in Mittag-Leffler functions at every order between zero and one. The surface deflection then depends on the closure order alone and is smaller than forty-five degrees throughout, whereas the depth-integrated transport stays exactly normal to the wind, because that constraint follows from the momentum balance and not from the closure. The far field decays as a power law rather than exponentially, and its amplitude vanishes in the local limit, so that limit is singular. Under a suddenly applied stress the surface transient decays algebraically. Three independent algorithms agree closely, and no observational or model data are used.

We construct boundary observables that retain memory for a quantified time window in quantum chains whose basis configurations carry a conserved ordered product of matrix labels. Under strong-irreducibility and proximality hypotheses on the label matrices, projective contraction of random matrix products localizes a conserved charge in the root-mean-square over uniform basis configurations. A finite-time inequality then converts readout overlap and reset leakage into a correlation bound: for a conserved reference observable $H$, a readout $F$, accumulated squared leakage $\mathcal B$, and the normalized Hilbert-Schmidt inner product, $\langle F,\Phi(F)\rangle\ge 2\langle H,F\rangle^2/(\|H\|_2^2+\mathcal B)-\|F\|_2^2$. Here $\Phi$ is a sequence of conserving channels and partial resets. The bound holds for each such circuit, without averaging its gates. It yields an exponential memory window in the distance from the noisy region, with quantitative corrections for spatially distributed noise and imperfect conservation. For a thirteen-state elementary-matrix model, exact polynomial certificates give root-mean-square localization error $e^{-r/1400}$ between the charge and its restriction to the last $r$ sites, and limiting variance at least $1/19$ for a rational charge. A related normalized-Gram charge gives an eight-site certificate retaining more than half the specified readout correlation through two boundary-reset rounds. Uniform depolarization imposes an inverse-noise-rate lifetime ceiling. A two-qubit IBM experiment illustrates the general reset inequality; it does not realize the matrix-product chain. We provide proofs, exact certificate programs, and an independent recount of the archived experimental outcomes.