Papers

The Rogers–Ramanujan identities equate a $q$-series whose exponents are governed by a quadratic form with an infinite product supported on two residue classes modulo~$5$. Identities of this shape are scarce, and a central problem is to identify the structures that produce them in families. Huang, Jiang, and Oblomkov have proposed a source of a new kind: to each pair of coprime integers $a,b>1$ they attach an infinite-rank $q$-series $Z_{a,b}(q)$, assembled from counts of commuting nilpotent matrix pairs $(A,B)$ with $A^a=B^b$ over finite fields, and they conjecture that it equals an explicit product of $(a-1)(b-1)/2$ modular units of level $a+b$. The $a=2$ cases are the Andrews–Gordon identities; no case with $a>2$ was known. We prove the conjecture for $(a,b)=(3,4)$, $(3,5)$, $(3,7)$, and $(3,8)$. Our proofs pass through a finer sum-to-sum identity, which we conjecture for all $b$ coprime to $3$ and establish for all $b$ when $q=1$. Lau and Ono have since proved that identity in general, and with it the full $a=3$ case. These identities have been formalized and verified in Lean by AxiomProver.

We consider the gain of entrainment (GOE)–the difference between the average steady-state output under a periodic input and the steady-state output under a constant input with the same mean–for an $n$-stage feedforward cascade of stable first-order filters interleaved with static nonlinearities. The main result is an exact decomposition of GOE as a weighted sum of local Jensen gaps, where each gap quantifies the mean shift generated by a nonlinearity, and each weight is a product of downstream incremental gains divided by linear time constants. We provide a Bregman-divergence interpretation of the decomposition, and a second-order small-amplitude of GOE separating local curvature, fluctuation energy, and differential gains. We demonstrate the theoretical results using a Michaelis-Menten cascade showing that any nonconstant periodic feeding strictly reduces the average terminal product relative to constant feeding with the same mean.

Iterative decoding has been central to the success of modern channel coding, where reliability information is repeatedly exchanged across decoding components to approach fundamental performance limits. This paper brings the same principle to semantic error correction by proposing iterative cross-layer semantic error correction (ICL-SEC), a framework that closes the loop between physical-layer soft channel decoder and application-layer language-model-empowered semantic decoder. In the proposed framework, a soft-input soft-output channel decoder first produces bit-level posterior probabilities, from which word-level reliabilities are derived. Words deemed reliable are exposed to a masked language model as semantic context, while unreliable words are masked. The language model then produces contextual word likelihoods, which are leveraged to generate extrinsic bit-level priors and fed back to the channel decoder for the next iteration. This iterative refinement progressively expands the set of confidently recovered words. A key contribution is our Confirm prior-update rule: once a word is judged reliable, its bits are assigned deterministic priors with probability one in subsequent iterations, making the word fully resolved side information for both the channel decoder and the language model. This successive-confirmation mechanism prevents oscillatory unmask-mask behavior and yields a reliability interpretation consistent across layers. Simulations over text transmission demonstrates that ICL-SEC substantially outperforms both conventional channel decoding and non-iterative CL-SEC. In particular, the proposed Confirm scheme reduces the bit error rate by more than two orders of magnitude relative to non-iterative CL-SEC, while also significantly improving the other five performance metrics.

In this paper we extend Hitchin-Kobayashi correspondence along holomorphic fibre space and we show that there exists a generalized Hermitian-Einstein metric which is twisted with Weil-Petersson current.In fact we generalise Song-Tian program for vector bundles

The covariance matrix of a basket of assets is a symmetric positive-definite object that evolves on a curved manifold, not on a flat vector space. Standard linear regression on its vectorised entries ignores that geometry and, in periods of market stress, can return fits that fail to be positive definite. We study geodesic regression on the symmetric positive-definite cone $\SPD(n)$ under the log-Euclidean and affine-invariant metrics, with the rolling sample covariance of a basket of log-returns as data. The fitted geodesic's velocity norm is proposed as a geometry-aware turbulence indicator. On daily data for four equity baskets- a ten-stock historical S&P sub-basket (2006-2024), a current S&P~100 mega-cap basket, the top ten listings of the Johannesburg Stock Exchange (JSE), and the ten most-traded Egyptian Exchange (EGX) tickers the indicator peaks cleanly on known stress episodes (the 2008 global financial crisis, the 2020 COVID crash, the 2022 Fed rate-hike cycle, and the October-November 2019 Egyptian political-economic tension episode) without any calibration. On hold-out evaluation in crash windows, log-Euclidean regression achieves 150 to 300 times smaller geodesic mean squared error than Euclidean ordinary least squares, because the latter produces non-SPD forecasts that blow up under the intrinsic metric. A naive velocity-aware de-risking strategy reduces maximum drawdown by 2.5 percentage points at the cost of 1.4 percentage points of annual return, consistent with a slow-moving indicator that signals stress reliably but translates imperfectly into tactical trading. All data, code, and experiments are reproducible from the companion repository.

We show that, over any algebraically closed field of any characteristic, the category of finite-dimensional projective modules over the preprojective algebra of generalized Dynkin type $\mathbb{L}_2$ has a pretriangulated category structure which is neither an algebraic nor a topological triangulated category structure.

The notion of Q-order convergence is arguably the most important tool for describing the asymptotic behavior of a convergent sequence. Loosely speaking, it captures the“speed”of convergence of an iterative method. The concept of Q-order convergence is not always well suited for sequences whose errors do not decrease monotonically at every step. In this paper, we introduce the notion of $p$-step Q-order convergence. It generalizes the classical notion of Q-order convergence by comparing errors that are $p$ iterations apart rather than errors of successive iterates. This definition recovers classical Q-order convergence as the special case $p=1$. We show that it extracts meaningful convergence information from certain non-monotonic sequences for which the classical Q-order either does not exist or assigns an overly pessimistic classification. We develop the basic theory of the new notion and locate it within the classical hierarchy by proving that $p$-step Q-order at least $\alpha$ implies R-order at least $\alpha$. Natural applications include iterative methods whose updates alternate or cycle over multiple steps.

We establish a duality between quotients and comprehensions in relational doctrines, a class of indexed posets modelling (a minimal fragment of) the calculus of relations. We show that every relational doctrine determines an “opposite" relational doctrine and that one has quotients if and only if the other has comprehensions. This duality extends to the 2-categorical level, yielding a 2-dual isomorphism between the 2-categories of relational doctrines with quotients and that with comprehensions, allowing to derive results on relational doctrines with comprehensions from their dual counterparts for quotients, and viceversa. We also study the interaction between quotients and comprehensions and their relative completions, giving sufficient conditions for the induced 2-monads to be composable, \ie sufficient conditions ensuring that applying both the completions, the second one preserves the properties added by the first one.

The Brauer–Fowler theorem bounds the order of a finite simple group in terms of the order of the centralizer of an involution. Hartley proved an automorphism version, bounding the order of a finite simple group in terms of the order of an automorphism and the order of its fixed-point subgroup. Strunkov asked whether, in the involution case, the order of the centralizer can be replaced by the number of involutions commuting with the given involution; this was recently answered affirmatively by Skresanov. Motivated by this question, and in contrast with Hartley's use of the order of a fixed-point subgroup, we study what can be deduced from counting elements of prescribed prime order inside such a subgroup. We show that the direct analogue of Skresanov's result fails for elements of odd prime order. We prove that if \(G\) is a finite simple group, \(x\in G\) has odd prime order \(p\), \(C_G(x)\) contains at most \(k\) elements of order \(p\), and the exponent of \(C_G(x)\) is at most \(e\), then \(|G|\) is bounded in terms of \(k\) and \(e\).

In this note we give an alternative derivation of Frobenius Character Formula using wedge product, the resulting formula is used to calculate the character table of $S_{2}$ and $S_{3}$ Group which reproduces their character table correctly, calculations for classes of $S_{3}$ Group are given in the appendix B. The calculations are given in terms Schur Functions. In the first section, a brief indicative review of Symmetric Group and related formulas is given. In Section second we derive the formula. A naive approach to computational complexity is also presented at the end.

We investigate structural properties of Lie affgebras, viewed as affine analogues of Lie algebras. We introduce the notion of an ideaf, the affine counterpart of an ideal, and establish characterizations of left and right ideafs. We further study centers, quotient structures, and product ideafs, proving, under suitable conditions, that the center of a Lie affgebra is an ideaf. We then develop the notion of affine nilpotency and establish connections between the nilpotency of Lie affgebras and that of their retracted Lie algebras. As an application, we prove an Engel-type theorem for Lie affgebras of the form $\mathfrak{a}(\mathfrak{g};\kappa=2\lambda,\lambda,s)$.

Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.

We present a unified construction of all simply connected irreducible exceptional compact symmetric spaces as totally geodesic submanifolds of $E_{8}$. Using this construction, we obtain a unified description of totally geodesic embeddings between exceptional symmetric spaces by determining all inclusion relations among them. The resulting inclusion structure exhibits a remarkable and unexpected symmetry, suggesting a deeper geometric relationship among exceptional symmetric spaces.

We investigate Banach spaces generated by uniformly bounded families of functions taking values in a fixed finite set and establish a rigidity principle connecting the cardinality of the generating family with the geometry of its closed linear span. We prove that such a space is superreflexive precisely when the generating family is finite, or equivalently, when the resulting Banach space is finite-dimensional. The main ingredient is a finite-range spreading-model obstruction showing that an infinite family of finite-valued functions cannot generate a superreflexive space under the supremum norm. This principle is applied to Banach spaces generated by the characteristic functions of the left derivatives of formal languages. It yields geometric characterizations of regular languages in terms of finite dimensionality, superreflexivity, and the existence of an equivalent uniformly convex norm. We also obtain a canonical representation of the associated language space as a coordinate-function subspace of a space of continuous functions on a compact shift-orbit closure. The corresponding language Hankel operator is shown to be compact exactly for regular languages. In the nonregular case, we determine its exact distance from both the compact and finite-rank operators and compute all its nontrivial approximation numbers. Finally, we construct a single binary language whose associated Banach space contains an isometric copy of every separable real Banach space. These results reveal a sharp contrast between the geometric rigidity associated with regular languages and the universality that may occur in the nonregular setting.

When a linear differential problem is discretized by a linear numerical method characterized by a mesh fineness parameter $n$, the computation of the numerical solution reduces to solving a linear discrete problem identified by a matrix $A_n$ whose size grows with $n$. The sequence of discretization matrices $\{A_n\}_n$ often falls within the class of generalized locally Toeplitz (GLT) sequences, even when the numerical method belongs to the family of domain decomposition methods (DDMs), as illustrated herein through examples. Four widely used preconditioners for DDM discretization matrices are the block Jacobi (BJ), block Gauss–Seidel (BGS), additive Schwarz (AS), and multiplicative Schwarz (MS) preconditioners. In this paper, we provide formal definitions of the BJ/BGS/AS/MS preconditioners for arbitrary multilevel block matrices. These definitions and the associated notations are inspired by the theory of GLT sequences and are proposed as alternatives to those commonly used by the DDM community. We analyze the structure of the BJ/BGS/AS/MS preconditioners when applied to multilevel block matrices $A_n$ belonging to a GLT sequence $\{A_n\}_n$. Every GLT sequence $\{A_n\}_n$ is uniquely associated with a special function $\kappa$ called symbol. We prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $\kappa$, then the sequences of the BJ, BGS, and MS preconditioners are GLT sequences with symbol $\kappa$. For the AS preconditioner, we prove that $\{P_n^{AS}(A_n)\}_n$ is a GLT sequence with symbol $\kappa^{AS}\approx\kappa$, and $\kappa^{AS}=\kappa$ whenever the overlaps in the subdomains used for the construction of $P_n^{AS}(A_n)$ vanish as $n\to\infty$. A numerical validation of these results in the context of isogeometric DDMs is presented.

We establish an abstract storage theorem for projectively $C_0$ unitary representations admitting a pair of one-parameter subgroups with conjugation escape. It produces, for every $0<\varepsilon<1$, a single orbit whose sampling set is relatively separated modulo the projective kernel and whose frame bounds are $(1-\varepsilon)^2$ and $(1+\varepsilon)^2$. Two applications are obtained from the same mechanism. First, every nonabelian exponential Lie group admits a near-Parseval frame of left translates for its left regular representation. Consequently, a positive-dimensional exponential Lie group is an FT group if and only if it is nonabelian; in particular, the three-dimensional Heisenberg group admits such a frame. Second, every infinite-dimensional irreducible unitary representation of an exponential Lie group admits a near-Parseval discrete orbit frame. When the effective projective quotient is abelian, the orbit may be chosen to be an orthonormal basis arising from a transported Weyl lattice.

We broaden Buzano-type inequalities to provide novel numerical radius bounds for operators of the type $AXB$, thereby generalizing the results obtained by Sababheh et al. For the $\lambda$-mean transform $M_\lambda(T)$, we provide a counterexample demonstrating that $r_\sigma(M_\lambda(T)) \le r_\sigma(T)$ fails to hold in general for $\lambda \in (0, 1)$, establish that $(r_\omega(M_\lambda(T)))^n$ and $r_\omega(T^n)$ are typically incomparable for $n \ge 2$, and confirm that $M_\lambda(T^*) = (M_\lambda(T))^*$ is valid for $\lambda \in [0, 1)$ if and only if $T$ is a member of a newly established $\sigma$-class. Furthermore, we examine the transformation characteristics of $T$ and the tensor products $T \otimes S$, refine Zamani's inequalities, and unify operator modulus bounds $|\widetilde{T}| \le |\widehat{T}| \le |T|$. In this application, we demonstrate that the conditions for norm preservation, $\|\widetilde{T}\| = \|T\|$ and $\|M_\lambda(T)\| = \|T\|$, are equivalent to the statement $\|T^2\| = \|T\|^2$, and we offer precise norm estimates for $2 \times 2$ off-diagonal block operator matrices under $\lambda$-mean transformation.

The paper contains An elementary proof of Marton and Shields' obstruction to finitary coding.

In \cite{zhang2025dimension}, the author discusses the applicability of a nonexistence result for self-similar profiles to the Hou–Luo scenario and raises questions about the approximate self-similar profile constructed in \cite{ChenHou2023a,ChenHou2023b}. In this note, we clarify two main points. First, the far-field discrepancy identified in \cite{zhang2025dimension} is the raw nonlocal Poisson error, not the profile residual error used in the stability proof of \cite{ChenHou2023a,ChenHou2023b}. The relevant nonlocal contribution to the profile residual contains the nonlocal velocity error multiplied by additional decaying factors, which provide crucial smallness in the far field estimates. Second, we examine the figure-based evidence used in \cite[Remark 2.4]{zhang2025dimension} to support the applicability of the assumption in \cite[Proposition 2.3(i)]{zhang2025dimension}, and show that the same grid-point data do not support that interpretation. Thus the comparisons and figure-based evidence in \cite{zhang2025dimension} neither invalidate the residual estimates in \cite{ChenHou2023a,ChenHou2023b} nor justify the claimed applicability of the angular-increase assumption to the Hou–Luo scenario.

In apparel retail and other applications, when a customer's preferred size is unavailable, demand may shift to nearby sizes. This substitution creates new assortment and inventory optimization challenges by coupling product availability and demand across sizes. We introduce the consider-fit-then-choose (CFTC) model to capture such size-dependent choice behavior. Products may be offered in multiple sizes, which affect customer preferences and consideration sets through fit, measured by distance from the customer's ideal size. We study assortment optimization and show-all inventory selection, in which the retailer chooses initial inventory and subsequently offers every in-stock product. We show that assortment optimization under the CFTC model is NP-hard and develop a PTAS when customers deviate by at most $O(1)$ sizes from their ideal size. Combined with the recent black-box framework of Fu et al. (2026), this yields a nearly $0.272$-approximation for show-all inventory selection. We next exploit the specific choice dynamics of the CFTC model. By stocking only every other size, we decouple demand across stocked sizes and reduce CFTC to a special class of mixed multinomial logit models that we prove satisfies the convex chain decomposition (CCD) property of Goyal et al. (2023). For the fluid problem, we develop a polynomial-time $(1/2-\epsilon)$-approximation under adjacent-size substitution and a mild condition on preference weights. For the stochastic problem, we establish an asymptotic $1/2$-approximation using a new coupling argument connecting the stochastic inventory process to its fluid counterpart. Numerical experiments calibrated using footwear data show small optimality gaps across a broad range of substitution patterns and problem settings.

In this paper we show how to extend quantile-stratified sampling to produce simulations from various multivariate distributions. These simulations have desirable space-filling and coverage properties relative to simulation using IID sampling. We examine the coverage performance of these simulations against IID sampling by looking at plots of ordered log-density values from the simulations.

This paper extends a one-shot (finite-blocklength) information-theoretic framework built on a single primitive: the pairwise error probability (PEP) of a randomized, dither-broken decoding rule, and the error spectrum it induces. A companion paper developed the framework for point-to-point channel coding – uniformity of the PEP, the error-spectrum representation of achievability and converse, and the linear-programming form of the prior-optimized minimax meta-converse. Here we show that the same primitive, read on an enlarged candidate space, governs four further settings: lossy source coding under average distortion, joint source-channel coding with list decoding, channel coding with an erasure/undetected-error option, and the two-user multiple-access channel. In each case a single spectrum yields a random-coding achievability bound and exact fixed-code identities, and we indicate how the convex – indeed linear-programming – prior optimization of the channel-coding case extends under matched decoding. The development recovers the one-shot lossy bound of Matsuta-Uyematsu and the joint source-channel bounds in the Csiszar tradition, complements the lossy bounds of Kostina-Verdu, and connects the multiuser case to the comparable achievability/converse pair through three pairwise error events.

In this paper we study on infinite graphs the Leibenson equation $$ \partial_t u = \Delta_p u^q, $$ where $p>1$, $q>0$ and $\Delta_p$ denotes the discrete $p$-Laplacian. We prove, for any integrable initial data $u_0$, the existence of a global solution, which is unique for a certain range of $p$ and $q$. Assuming a Faber–Krahn inequality, we obtain sharp $\ell^1$-$\ell^\infty$ smoothing estimates and quantitative bounds on the propagation of solutions with initially finite support. Under certain assumptions on $p$ and $q$, we also prove finite-time extinction results for solutions when the graph satisfies an isoperimetric inequality. In particular, on Cayley graphs with polynomial volume growth, we establish the optimal large-time decay rate of the $\ell^\infty$-norm for nonnegative finite-mass solutions when $q(p-1)>1$, and demonstrate a sharp dichotomy regarding the finite-time extinction of exhaustion solutions.

This work investigates the class of SIR/SEIR models and their extension to an ODE-based SVEIR system, whose parameters are calibrated and validated using real influenza outbreak data. The theoretical part establishes existence and uniqueness of solutions for the SVEIR model via the Picard-Lindelof theorem and analyzes invariance of the admissible region. To describe spatiotemporal infection spread in an urban setting, a diffusion-chemotaxis extension of the SIR framework is formulated as a PDE system; an additional term is introduced to represent fear/avoidance effects in the system dynamics. A numerical workflow is proposed, including grid discretization, Neumann boundary conditions enforced by mirror extension, a semi-implicit IMEX time-stepping scheme, and sparse linear solves; computational correctness is monitored, in particular, by checking conservation of the total population.

A recent interpolation-based Quantum Signal Processing (QSP) framework by Alase bypasses the phase-finding procedures required in conventional QSP, allowing for a direct encoding of the target polynomial into a quantum circuit. However, this approach assumes access to a diagonal block encoding of function values without providing an explicit circuit construction. In this work, we address this gap by developing an explicit circuit construction method for diagonal block encodings. The resulting algorithm achieves a computational cost of $\mathcal{O}(d\log d)$ for explicitly constructing block encodings of matrix polynomials, improving upon the best-known theoretical bounds of previous methods. Numerical results confirm this scaling, demonstrating that circuit parameters for polynomial degrees up to $10^7$ can be computed in about a minute on a standard CPU.

As is known, the $A_2$-continued representation of numbers is not topologically equivalent to the classical binary representation; therefore, the digit projector of such representations is a discontinuous function. In this paper, we introduce a continuous function that serves as an analogue of the digit projector of the classical binary representation of numbers into the digits of the $A_2$-continued representation with zero redundancy, namely a function of the form \[f(\Delta^2_{\alpha_1\alpha_2...\alpha_{2n-1}\alpha_{2n}...})= \Delta^{A_2}_{(\frac{1}{2})^{1-\alpha_1}(\frac{1}{2})^{\alpha_2}... (\frac{1}{2})^{1-\alpha_{2n-1}}(\frac{1}{2})^{\alpha_{2n}}...}, \alpha_n\in \{0,1\}.\] It is proved that the function $f$ is well-defined, continuous, and monotone. Using the normal properties of numbers with respect to their binary representation and Lebesgue's theorem asserting the existence of a finite derivative for a continuous monotone function almost everywhere, the singularity of the function $f$ is established. The paper also establishes a relationship between the considered function, the right-shift operator on digits, and the inversor of the continued representation of numbers. This relationship is then used to establish the singularity of the inversor.

We investigate integrability of Euler-Lagrange equations associated with 3D second-order Lagrangians of the form \begin{equation*} \int f(u_{xy},u_{xt},u_{yt})\ \text{d}x\text{d}y\text{d}t. \end{equation*} It is demonstrated that there are exactly four different types of such Lagrangian densities $f$: the first one is given by the formula $f=\sqrt{u_{xy}u_{xt}u_{yt}}$, the second and the third are more complicated (although still representable in elementary functions), whereas the most generic fourth one is expressible in terms of the Lobachevsky function, revealing unexpected links to spherical/hyperbolic trigonometry and Schläfly-type formulas. Dispersionless Lax pairs and integrable dispersive deformations of the corresponding Euler-Lagrange equations are also constructed.Remarkably, dispersive deformation of the first Lagrangian density coincides with the Lagrangian of the classical Darboux system arising in the theory of triply-orthogonal coordinate systems in $\mathbb{R}^3$. Dispersive deformations of the three other cases provide Lagrangian formulation of semi-discrete and fully discrete versions of the Darboux system, with one, two and three discrete variables, respectively.

What constitutes depth in a mathematical proof? This paper addresses this foundational question through a close case study of Andrew Wiles's proof of Fermat's Last Theorem. We propose a five-criteria framework: Difficulty, Originality, Fruitfulness, Unity, and Explanatory Power. By critically examining the limitations of each criterion and the intricate interrelations among them, we show that they do not merely coexist but capture genuinely complementary aspects of proof depth. Applying this framework to Wiles's proof, we then demonstrate that its celebrated depth does not reside in any single virtue, but rather emerges from a synthesis of all five criteria. Our analysis thus offers not only a novel lens for understanding Wiles's achievement, but also a flexible framework for evaluating depth across mathematical proofs more broadly.

Satellite gravimetry has become essential in many areas of earth science. However, the resolution of satellite gravitational models remains low at scales of a few hundreds km and no gravity recovery methods can take full advantages of unprecedented high accuracy of satellite tracking measurements. We first provide a unified theoretical framework of parameter estimation in differential equations for satellite gravimetry and then briefly review the mathematical methods to compute the gravity field of the Earth from satellite tracking. We focus on the collocation method, Kaula linear perturbations, two-point boundary value problems and orbit-energy-based methods. The numerical integration method is also included in this review, though it has been proved to be mathematically incorrect and physically not permitted. The reason is that it has become the standard method to routinely produce global gravitational models from satellite tracking data, which have been widely applied in many different areas of earth science. Because it is not clear how the incorrect foundation would affect gravity products from satellite tracking, we do not review any applications of these products. We then present a measurement-based perturbation theory to estimate the gravity field of the Earth, which can fully utilize both precise satellite orbits of arbitrary length and unprecedented high accuracy of satellite and inter-satellite tracking. The method is theoretically free of modeling errors, is capable of extracting any small forces from satellite and inter-satellite tracking data and provides a guarantee for high-precision and high-resolution global gravity models. Finally, we assume a reference gravity model and derive local solutions to the Newton's nonlinear governing differential equations of satellite motion for scattered tracking data that can still be important in some applications.

We uncover the resolvent origin of the spectral duality governing reset-neutral distributions in Markov chains with geometric resetting. Starting from the abstract conditions of Paper~III, we show that the spectral duality $B_\nu(z)=\kappa(z)\,A_\nu(\sigma(z))$ is equivalent to a single symmetry of the resolvent $R(\gamma)=(I-(1-\gamma)P)^{-1}$: the intertwining relation $[\Delta^2\mathcal{R},R(\gamma)]=0$, where $\mathcal{R}$ is the reflection operator of an involution $\sigma$ and $\Delta=\operatorname{diag}(\sqrt{\kappa(z)})$; equivalently, $\widetilde{\mathcal{T}}=K^{-1/2}\Delta^2\mathcal{R}$ is an involution. This symmetry determines the universal critical value $C^*=1/(1+\sqrt{K})$, with $K=\kappa(z)\kappa(\sigma(z))$, which depends only on the scalar $K$ — not on the resetting rate $\gamma$, the reset distribution, or the particular chain. We characterize the class of $(\sigma,\kappa)$-reversible chains, encompassing both the biased random walk and genuinely non-homogeneous dynamics sharing the same $C^*$; a Doob $h$-transform realizes the duality $K\mapsto1/K$, hence $C^*\mapsto1-C^*$, with fixed point $C^*=1/2$. The orientation field admits the explicit resolvent representation $\psi(\gamma)=R(\gamma)(b^{(0)}-C^*b)$: its gauge-normalized form $\Delta^{-1}\psi(\gamma)$ is antisymmetric under $\sigma$, it has an exact node at the fixed point of $\sigma$, and it governs the exact sign law $\operatorname{sgn}(C(\pi,\gamma)-C^*) =\operatorname{sgn}\langle\pi,\psi(\gamma)\rangle$. Numerical experiments confirm the theory to machine precision. These results establish the operator-theoretic foundation of the spectral duality of Paper~III and provide the bridge to the information-geometric framework of Paper~V.