Papers

Given a certain specific, $M$ by $M$ matrix form of the Feshbach's effective (i.e., energy-dependent) Hamiltonian $H^{(M)}_{e\!f\!f}(E)$, the inverse-problem reconstruction of the full-space, $N$ by $N$ matrix Hamiltonian $H^{(N)}$ is considered and reduced to the solution of a coupled set of polynomial algebraic equations. Using computer-assisted symbolic manipulations, an explicit algebraic reconstruction of $H^{(N)}$ is found feasible at not too large $K = N-M$.

We develop a finite-temperature formulation for a hypercomplex dissipative quantum field theory [1], using the imaginary-time path-integral approach and the idempotent structure of the hypercomplex algebra. The resulting partition function naturally separates into two conjugate complex sectors whose recombination preserves the hypercomplex Hermitian structure while generating a nontrivial thermal phase. From this construction, the standard thermodynamic observables are obtained consistently, and the conventional relativistic charged Bose gas is recovered in the vanishing-dissipation limit. Beyond this equilibrium correspondence, the hypercomplex formulation reveals a distinctive perturbative hierarchy: dissipative effects first appear in the complementary phase sector, while corrections to ordinary real thermodynamic quantities arise only at higher order. These results show that dissipation can be encoded through an enlarged algebraic thermal structure without abandoning the familiar framework of relativistic finite-temperature field theory, opening a path toward broader applications of hypercomplex methods in dissipative, open, and effectively non-Hermitian quantum systems.

Let $L_u$ be a symmetric Levy process representing an unobserved phase lift, coupling to a quantum system via charge $Q$. Averaging $e^{iL_uQ}$ gives a completely positive dephasing semigroup. If the characteristic exponent lacks an intrinsic charge scale, the Levy-Khintchine theorem forces $\eta(\xi)=D | \xi |^{\alpha}$ ($0<\alpha\le2$), so coherences decay at $\Gamma_{ab}=D | q_a-q_b |^{\alpha}$. For integer charge differences, the process descends to the wrapped variable $\Theta_u=L_u\bmod 2\pi$; for real charges, the lift is the relevant phase. Schoenberg's theorem ensures complete positivity for any finite real spectrum when $0<\alpha\le2$, while $\alpha>2$ yields an analytic obstruction. This boundary is exact, verified numerically here. A Gaussian integrated phase gives quadratic charge dependence, though colored Gaussian noise need not yield a Markov semigroup. Two idealized non-Gaussian mechanisms are analyzed: inverse-power Poisson shot noise ($\alpha=d/p$) and Bochner subordination of Brownian phase by an $\alpha/2$-stable clock. For a bounded reduction walk, we prove Born probabilities for every symmetric bounded proposal law. Simulations of truncated stable laws ($\alpha=0.5,1,1.5,2$) confirm this law-independence. In continuous time, $\alpha$-stable drivers fail to reduce: the exact-flow (Marcus) equation oscillates, and the naive Ito jump equation fails to preserve state. For Gaussian white noise, the Itô model collapses to Born probabilities, whereas the exact-flow (Stratonovich) model does not collapse without a threshold and yields non-Born exit probabilities. The quadratic case recovers Milburn's small-step limit, not his exact Poisson dynamics. Finally, a bi-temporal geometry is examined as a possible compact phase source, but closed timelike curves, nonunitary evolution, and an unstable mode tower prevent a consistent field-theoretic realization.

Adding a uniform external magnetic field to a mean-field spin-glass model usually requires a new analysis specific to the model. The disordered system we consider corresponds to the Gaussian binary branching random walk (BRW) - in the spirit of Derrida and Spohn [21] and studied from the statistical-physics point of view by Jagannath [22] - and we prove that this is not the case : a single elementary observation - that the resulting Hamiltonian is still a BRW, now with independent, but non-identically distributed, displacements - allows us to use the available results for general BRW. Combining classical and recent results on general BRW (Biggins [3], Chauvin and Rouault [13], Mallein [25]), one obtains an essentially complete picture of the model in an external magnetic field : the ground state, the free energy, the one-step replica symmetry breaking (1-RSB) transition, and the limiting genealogical overlap distribution with Poisson-Dirichlet statistics for the Gibbs weights. We then prove a strong concentration result for the magnetization under the Gibbs measure at low temperature, giving its explicit optimal value. It turns out that this one-replica statement is not enough to control the classical Ising overlap between two independently sampled configurations : an elementary counterexample (Remark 4.1) shows that concentration of each replica's magnetization does not, by itself, determine their joint correlation. We address this by developing a two-replica large-deviation argument - resting on the classical method of types and the subadditivity of Shannon entropy, and taking the form of a uniform Chernoff bound over the joint empirical type of a pair of configurations, matched against a two-replica concentration estimate - which we use to obtain the distribution of a second (Ising) hypercube-type overlap.

We develop a real twistor-space formulation of four-dimensional CFT Wightman correlators. We write the conformal generators and little-group constraints in ambitwistor space. The quadric condition $Z\cdot W$=0 arises naturally from the little-group constraints. We then solve for both the parity-even and parity-odd correlators in the ambitwistor variables. We further establish the connection between the ambitwistor correlators and the ones in the helicity-basis Grassmannian via a half-Fourier transform, and also develop the Penrose transform to recover the corresponding position-space correlators. We notice that for correlators with multiple tensor structures, the Penrose transform yields the full space of conformally invariant structures, but regularizing the associated Schwinger integrals selects the physical correlator. Finally, starting from the covariant Grassmannian formulation, we construct massive twistors that make the little-group symmetries manifest and develop their Penrose transform, which again reproduces the correct position-space correlators.

Conformal symmetry strongly constrains correlation functions. In momentum space, the conformal Ward identities are solved for three-point functions by integrals of a product of three Bessel functions ("triple-$K$ integrals"); more generally, integrals of this type ("multiple-$K$ integrals"), serve as the building blocks of a wide class of higher-point conformal correlators. These integrals belong to the class of generalised hypergeometric functions, and while their series representations are known in principle, none converges throughout the entire physical region of kinematic space selected by momentum conservation. In this work, we construct series representations adapted to exactly this physical domain. Using the method of brackets, which turns the evaluation of definite integrals into solving a linear system of algebraic equations, we derive various Lauricella-type series representations for multiple-$K$ integrals. For triple-$K$ integrals, conventionally expressed in terms of the Appell $F_4$ function whose series converges only outside the triangle-inequality region, we find instead a compact, two-branch series that converges throughout the entire physical region and for arbitrary scaling dimensions. We then extend this construction to general multiple-$K$ integrals: by introducing a new set of kinematic variables, we build an iterative series representation that converges for all physical kinematic configurations.

Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six – the first dimension in which several continuous families coexist with an isolated solution – has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased $3 \times 3$ corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.

We study the supercritical regime $\beta>\beta_c$ of a large family of real-valued spin models on $\mathbb Z^d$, including the Blume-Capel and $P(\varphi)$ models. We consider their random cluster representations and prove that they are well-behaved, in the sense that local uniqueness of macroscopic clusters occurs with high probability, uniformly in the boundary conditions. This implies, among other things, a surface-order exponential bound for the (lower) large deviations of the empirical magnetisation. These results were previously known only in the cases of the Ising and $\varphi^4$ models.

We present a reformulation of four-dimensional chiral gravity in terms of a chiral gauge connection and a Clifford-valued coframe. The exterior algebra generated by $e^a \gamma_a$, together with the gauge curvature $F$, provides a natural and systematic construction of gauge-covariant differential forms. We show that suitable combinations of the chiral actions reproduce Einstein–Cartan–Holst gravity with a cosmological term. The original superconnection invariants yield only algebraic fermionic equations, so the fermionic one-forms do not propagate but instead generate a mass-like mixing between the two chiralities. Spinor propagation is supplied by a spinorial BF-type term, which under the matter ansatz reduces to the standard first-order Weyl kinetic action, including the torsion coupling. We also review the symmetry and variational properties of the matter ansatz, including its Lorentz equivariance, its relation to residual supersymmetry and twistor spinors, and the connection between the reduced and unrestricted field equations.

We show that the Coulomb branches of 5d $\mathcal{N}=1$ necklace quiver gauge theories on $\mathbb{R}^3 \times T^2$ are identified with the phase spaces of elliptic spin Ruijsenaars-Schneider models with dynamical inhomogeneities. This identification resolves the long-standing problem of determining a Poisson structure and a quantization of these models. We further show that the resulting quantum system is the integrable system governing supersymmetric indices of 4d $\mathcal{N}=1$ theories of class $\mathcal{S}_k$.

We argue for the universality of quantum theory using a dynamical consistency argument, within a specific Hamiltonian setting. We analyse two different types of coupling between simple quantum harmonic oscillators. Each illustrates an aspect of the free and interacting quantum fields and shows the inadequacy of semiclassical models. In particular, we establish that requiring the canonical algebra to be preserved under joint unitary dynamics rules out specific hybrid classical-quantum models. We apply our reasoning to the gravitational field in the linear regime, coupled to the quantised electromagnetic field and, separately, to quantised matter. We conclude with a comparison to DeWitt's analysis of quantum measurement, in which the apparatus, if classical, must be at least stochastic to preserve the Heisenberg Uncertainty Principle. We also note that stochastic models are inconsistent with the strict version of conservation principles, even if they comply with a probabilistic (on average) conservation.

It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n^2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.

We prove the absence of nontrivial local conserved quantities in a class of $U(1)$-symmetric spin-$1$ chains with nearest-neighbor interactions in which some of the quadrupolar couplings vanish, a class that is not covered by previous studies. Applying the technique of Shiraishi to these systems, we show that, for every model in this class on a periodic chain of $N$ sites, there is no $k$-local conserved quantity for any $3\le k\le N/2$. In particular, for a frustration-free spin-$1$ chain that exhibits spontaneous $U(1)$ symmetry breaking at zero temperature in one spatial dimension, we prove that every local conserved quantity with support up to half of the system size is a linear combination of the identity, the total magnetization $S^z$, and the Hamiltonian itself. This rigorously establishes that, unlike the Heisenberg ferromagnet, the model admits no local order parameter commuting with the Hamiltonian, so that its continuous symmetry breaking is enabled by the frustration-free structure rather than by a conserved order parameter. We also prove the absence of $k$-local conserved quantities for $3\le k\le N/2$ in the periodic Motzkin chain, a frustration-free spin-$1$ chain closely related to the original Motzkin chain, for which spontaneous $U(1)$ symmetry breaking at zero temperature has also been reported.

Let $\tilde{\Gamma}(x)$ denote the modified gamma function recently introduced by the authors as an alternating analogue of the classical Euler gamma function. In this paper, we establish a complete alternating generalization of Mizuno's celebrated product formula: \begin{equation*} \prod_{m=0}^{\infty}\left(\prod_{j=1}^{n}(m+x_{j})^{(-1)^{m}}\right) =\frac{\left(\sqrt{\frac{\pi}{2}}\right)^n}{\prod_{j=1}^{n}\tilde{\Gamma}(x_{j})} =\prod_{j=1}^{n}\left(\prod_{m=0}^{\infty}(m+x_{j})^{(-1)^{m}}\right). \end{equation*} This identity, which we refer to as the alternating Mizuno formula, replaces the classical gamma function $\Gamma$ and the constant $\sqrt{2\pi}$ by their natural alternating counterparts $\tilde{\Gamma}$ and $\sqrt{\pi/2}$. As immediate consequences, we recover the alternating Lerch formula and, by specializing to $x=2$ and multiplying by $2$, a remarkably concise derivation of Wallis' famous product \begin{equation*} \frac{2\cdot2}{1\cdot3}\cdot\frac{4\cdot4}{3\cdot5}\cdot\frac{6\cdot6}{5\cdot7}\cdot\cdots=\frac{\pi}{2}. \end{equation*} More generally, by exploiting polynomial factorizations, we derive Kurokawa–Wakayama type formulas for alternating products over cyclotomic fields. Beyond these product identities, we investigate the multiple alternating gamma functions $\bar\Gamma_N(x)$, obtaining explicit factorizations in terms of Barnes' multiple gamma functions, closed-form evaluations involving the Glaisher–Kinkelin constant, and a Gauss–Legendre type multiplication formula. Finally, we introduce an alternating analogue of Shintani's double sine function and establish a clean arithmetic dichotomy: for algebraic $\tau$, its special values at integer points are algebraic precisely when $\tau$ is rational, and transcendental otherwise.

We study a Brownian time change on the unit square whose speed measure is constructed from a Dirichlet eigenfunction expansion with independent non-Gaussian coefficients. For $0<\gamma<\sqrt2$, the measure is obtained by a second-moment martingale argument. Finite coefficient translations induce coherent exponential tilts of the speed measure, and conditioning on the complementary coefficients gives positive Lebesgue densities on every finite-dimensional orbit. Unitary transport along these orbits gives a common-domain analytic family and explicit first-order cluster derivatives. A first-order splitting argument proves almost-sure simplicity, while the local eigenfunction-square identity and a Vandermonde argument give joint densities for all finite vectors of ordered positive eigenvalues.

We determine the exact spectral gaps for both the finite-particle and infinite-particle unlabelled Ginibre interacting Brownian motions. In every finite-particle system, the spectral gap is equal to one, whereas for the infinite-particle system it is equal to two. The infinite-particle dynamics therefore has a strictly larger spectral gap than any of its finite-particle counterparts, revealing an enhancement of spectral gaps in the passage from finite to infinite particle systems.

We consider a Su-Schrieffer-Heeger(SSH)-type trimer model with next-nearest-neighbor(NNN) interaction and balanced loss-gain(BLG) to study the combined effect of lattice symmetries, topology, non-hermiticity and general boundary conditions(GBC)on the existence of flat band and the nature of Bulk-Boundary correspondence(BBC). We derive the necessary and sufficient conditions for the existence of an entirely real spectrum under the periodic boundary condition(PBC). The exact expressions for the compact localized states(CLS) and energy eigenvalues corresponding to flat bands are derived analytically under the PBC. We establish topological phase transitions(TPT) for PT-symmetry and pseudo-chiral symmetry through the computation of the Zak phase and sub-lattice Zak phase, respectively. The Hamiltonian under the open boundary condition(OBC) is studied numerically, and edge states are observed in the topologically non-trivial phase, thereby establishing the non-hermitian BBC. The CLS exists in both bulk and the boundary for systems having only pseudo-chiral symmetry, and an additional PT-symmetry destroys the CLS at the boundary. We generalize a known formalism to study the same Hamiltonian under GBC, and derive analytic expressions for the energy and eigenstates for a class of boundary conditions in parametric ranges which admit flat band under the PBC. The edge states for these boundary conditions, including the OBC, are obtained analytically in the topologically non-trivial phase, thereby establishing BBC. The non-hermitian skin effect(NHSE) is seen in the model with reciprocal bulk interaction and strongly non-reciprocal boundary terms. The winding number based on spectral topology is computed analytically.

We classify left-invariant Rarita-Schwinger fields on 2- and 3-dimensional Lie groups with left-invariant Riemannian metrics. The left-invariant Rarita-Schwinger equation is reduced to an algebraic system on the associated metric Lie algebra. We show that, in dimension 2, non-trivial examples occur only in the flat abelian case. In dimension 3, apart from the flat abelian case, the only non-trivial examples occur on SU(2) equipped with a special left-invariant metric, for which the space of left-invariant Rarita-Schwinger fields has complex dimension 2. This metric is a Berger metric on $S^3$.

In this work, we apply supersymmetric transformations to an optical medium with quadratic index distributions on both transversal coordinates defining an elliptic profile. In particular, we implement first- and second-order transformations to construct new multi-parametric gradient index patterns that may host one or two subsidiary guided modes with a wide range of spectral properties and transversal intensity and phase distributions. These transformations may be continued up to an arbitrary order enabling the possibility to control the spectral and propagation properties of light beams.

Polygon equations form a family of equations generalizing the pentagon equation. In this paper, we construct semi-simplicial sets of permitted colorings associated with set-theoretic solutions of polygon equations and use them to define the corresponding (co)homology groups. We investigate several properties of these groups and establish an equivalence of categories between set-theoretic solutions of polygon equations and higher Segal semi-simplicial sets satisfying certain conditions. As a special case, our result recovers the correspondence between bijective set-theoretic solutions of the pentagon equation and $2$-Segal semi-simplicial sets proved by Dyckerhoff–Kapranov.

This paper considers robust nonadaptive regulation for general nonlinear systems in an output-feedback setting with arbitrarily high relative degree. We develop a nonadaptive design that combines an input-driven filter and a generic internal model with a recursive backstepping law, thereby recasting the regulation problem as the robust input-to-state stabilization of an augmented error system. Unlike adaptive schemes, the proposed method does not rely on linearly parameterized regressors and does not require the construction of Lyapunov functions having merely nonpositive derivatives. Under standard assumptions on the exosystem, including purely imaginary and simple eigenvalues, together with a minimum-phase input-to-state stability condition on the internal dynamics, we establish global asymptotic regulation and derive explicit, verifiable inequalities for selecting the design gains. The resulting nonadaptive framework guarantees convergence of the estimation and tracking errors even when the controlled-system dynamics are complex or only partially known. The effectiveness of the theoretical results is demonstrated using a benchmark controlled Duffing system.

We study the monic Romanovski (pseudo-Jacobi) polynomials with degree dependent parameters $\beta_K=-K$ and $\alpha_K=c/(K+1)$. We write out the parameter preserving first order ladder relations at fixed $(\alpha,\beta)$, with real constants. Combining them returns the three-term recurrence; iterating them builds $R_n^{\alpha,\beta}$ from the constant polynomial. We solve in triangular form the connection problem with the $\alpha=0$ family, identified with the Gegenbauer polynomials, and transfer the ladder relations to the $(n,\ell)$ lattice. Under $x=\cot\chi$ the in-level operators depend on $\ell$ alone. The dressed functions carry intrinsic lowest weight $\mathrm{su}(1,1)$ modules along columns of fixed $\ell$; rows need a boundary prescription and a rescaling. The deformed functions carry the same modules through higher order operators, which yield relations between Romanovski polynomials with different parameter pairs. Algebraically the family lies in the Jacobi class, and we indicate throughout which statements are inherited from that framework by transformation or specialization and which are not.

We study resolution-independent relative logarithmic source complexes for AKSZ-BV-BFV theory on face-oriented Joyce manifolds with generalized corners. Under the geometric hypotheses (G1)-(G3), the Dupont-Panzer-Pym total relative logarithmic complexes of admitted smooth monoidal resolutions represent $j_!\Omega_{X^\circ}^\bullet$ and carry the same compactly supported derived trace. The comparison is deliberately unfiltered: it identifies total relative cocycle classes through the common interior, but not individual resolved faces, separate BFV descendants, nonlinear mapping spaces, or absolute regularized integrals. For the intrinsic $b$-source ${}^{b}T[1]X$ we formulate the additional Stokes and transfer data needed to recover strict facewise structures. A multiplicative regularized Stokes trace system on Joyce's face category gives the presymplectic intrinsic BV-BFV identity and incidence descent. The finite datum (G4) is a linear subdivision/aggregation transfer criterion; (G5) is a separate cyclic field-level criterion for classical abelian BF theory. Neither is a general existence theorem, and no general strict multiplicative intrinsic trace is constructed here. We prove the interval contractions needed for codimension-two collars and analyze the positive real conifold through the common star refinement of its two diagonal resolutions. On an explicit finite product-Whitney logarithmic class the exceptional square satisfies (G4) after normal-face totalization, and the transferred differential is exactly the signed intrinsic incidence differential. Its finite algebraic BF dual gives a cyclic logarithmic-cellular shadow, but not the full continuum intrinsic $b$-de Rham datum (G5). General nonlinear continuum pushforward, existence of the intrinsic multiplicative trace, and loop-level logarithmic graph integrals remain open.

The bosonic thermal-loss channel is a fundamental model for quantum communication. This Gaussian channel models the transmission of bosonic systems subject to both loss and thermal noise. It describes many practical systems, including optical fibers, waveguides and free-space links, where background thermal noise is significant. Determining the channel's two-way-assisted quantum capacity, the maximum rate at which quantum information can be transmitted reliably through the channel with two-way classical assistance, remains an open problem. Here, we improve upon the best known lower bound for this capacity for a wide range of channel parameters. We achieve this by combining an improved qubit-bosonic distribution technique with a recently introduced technique for discovering entanglement distillation protocols.

How a quantum state responds to small parameter changes underlies state distinguishability, many-body response, and metrological sensitivity. The Pauli probability spectrum provides a natural description of many-body quantum states, but it is not evident how much of this response survives in the corresponding classical distribution. Here we show that, for pure states, the complete labeled Pauli distribution retains the full local metric of the pure-state manifold, reproducing the quantum Fisher metric up to a fixed normalization. This correspondence has a direct measurement realization: preparing a state together with its complex conjugate and performing pairwise Bell measurements between corresponding degrees of freedom converts the quantum response into an ordinary classical distribution without losing any of the sensitivity available in the two inputs. The readout is fixed, factorizes into local pairwise measurements even for strongly entangled many-body states, and preserves the response along all parameter directions simultaneously. We trace this correspondence to the reality of the underlying operator amplitudes and show that the same mechanism extends beyond qubits. When the correspondence fails, we identify explicitly where the missing information resides: part of the response can remain hidden in phases for identical-copy and complex-coordinate readouts, while mixed states develop transverse operator-space motion that is invisible to the Bell probabilities. These results provide a direct link between Pauli/Bell statistics, many-body response, nonstabilizerness, and quantum geometry.

The quantum tunneling process is a fundamental concept that presents the unconventional nature of quantum dynamics. The main anomaly of this effect revolves around the choice of imposing the classical 4-momentum to become imaginary during the barrier traversal. This paper addresses this paradox by introducing a complex-time parametrization, which uses a complex temporal coordinate, $\tau = t + i\kappa$. Here, the real axis $(t)$ represents standard thermodynamic time, while the imaginary axis $(i\kappa)$ is correlated to spatial evanescence. We demonstrate that evaluating this extended complex-time framework yields a hyperbolic section of the causal lightcone, proving that the traversal is strictly bounded by the speed of light and preserves causal disconnection. Applied to quantum tunneling, we resolve the anomaly of imaginary momentum through a continuous geometric signature flip of the effective complex-time momentum norm from a timelike to a spacelike state. This theoretical model finds a physically phenomenological analogy in microwave scattering experiments using open, non-Hermitian systems, which link complex transmission time delay to spatial wave decay.

Motivated by the geometric properties of curved tubes $T\left(P,a\right)$ defined by closed curves $P$, we compute the Cheeger constant $h\left(T\left(P,a\right)\right)$ in the real space forms of arbitrary dimensions with constant sectional curvature. The structure and properties of the system of Fermi coordinates allows us to parametrize the curved tube and straightforwardly compute the upper bound of $h\left(T\left(P,a\right)\right)$ using the exact formulas for the area and volume of $T\left(P,a\right)$. Next, we derive the lower bound by combining the geometry of tubes with a calibration-type argument. The key idea is to describe the tube using geodesic spheres moving along the underlying curve $P$, which provides a natural outward direction and makes the estimate geometrically transparent. This allows us to compute the lower bound via the divergence theorem. Finally, for the class of unbounded curved tubes in noncompact real space forms, we also compute the Cheeger constant and prove that there is no finite-volume Cheeger set.

We study integrable variable-mass sine-Gordon model (vmSG) with point defects. Using the Lax and Bäcklund-gauge formulations, we construct type-I and type-II defect matrices, derive the corresponding sewing conditions, and generate the bulk and defect contributions to an infinite hierarchy of conserved charges. The lowest members reduce to the standard sine-Gordon energy and momentum in the homogeneous limit, while for inhomogeneous backgrounds they define integrability-generated energy- and momentum-type quantities. Defect compatibility imposes matching conditions on the variable-mass functions across the defect. An analytical transmission factor $z$ characterizes soliton transmission, topological conversion, and absorption/emission processes. We then deform the type-I sewing conditions by parameters $\alpha$ and $\beta$ and derive the associated defect anomalies. Consistent one-soliton transmission is recovered for $\alpha \beta =1$, whereas for $\alpha \beta \neq 1$ parity-centered kink-kink and kink-antikink transmissions display vanishing lowest order integrated anomalies but no generic vanishing at higher order. Numerical simulations reproduce the integrable transmission/conversion regimes and show that non-integrable defects generate weak radiative tails and lowest order quasi-conserved charges, while quasi-conservation of the higher order charges remains unestablished. These results elucidate how exact defect integrability deforms into charge-dependent quasi-conservation and establish a framework for defect-controlled soliton transport in inhomogeneous media, with potential applications to nonuniform Josephson junctions, magnetic and nonlinear-optical systems, and effective molecular and DNA models.

Contact Hamiltonian dynamics gives dissipative mechanics an intrinsic action variable, but explicit contact splittings reach only kinetic energies whose terms are exactly integrable: frozen-coordinate diagonal metrics (the spherical pendulum, a torus particle) are included, while dense metrics with momentum cross terms, with the double pendulum as flagship, are not. We introduce a projected Pihajoki-contact integrator for this non-separable setting, combining phase-space duplication, symmetric projection onto the physical diagonal, and constant-friction damping half-steps, with the action factor carried by an exact Herglotz update. As in the projected extended-phase-space framework it builds on, the construction needs no binding parameter, returns the copies to the diagonal at every step, and confines the nonlinear solve to the $2n$ projection variables. For constant friction the step rescales $\omega=d\eta$ by the exact factor $e^{-\gamma\tau}$ when the projection is solved exactly (a classical conformally symplectic identity, realized here for this class), while time-symmetry, consistency, and smoothness yield an $O(\tau^3)$ one-step contact-form residual, a bound not specific to the contact form. On the damped double pendulum, spherical pendulum, and torus particle the method is second-order accurate, reproduces the contact decay law, and controls long-time energy and contact drift in coarse or stiff regimes where the Tao baseline and the unprojected average lose the solution. A head-to-head with exact-contactomorphism splittings delimits the niche: where a frozen-coordinate splitting exists it preserves the contact form exactly and wins at matched cost; for the dense double-pendulum metric the realizable alternative is first-order with a prohibitive constant and the projected method prevails. The contact-form estimate is local, one-step, and constant-friction.

Let $\mathsf{d}\geq1$, $0<\mathsf{s}<\mathsf{d}$, and $N\geq1$. We prove that the optimal fixed-particle-number constant $\Lambda_N(\mathsf{s},\mathsf{d})$ in the Riesz Lieb–Oxford inequality is attained and that these constants are strictly increasing in $N$. The proof combines grand-canonical concentration–compactness with a strict one-particle extension. After recentering, a limiting plan arising from a maximizing sequence may assign positive probability to several particle numbers and hence be grand-canonical. A strict $N$-particle completion excludes this case, while the inequalities $\Lambda_N>\Lambda_k$ for $k<N$ exclude limits with a fixed lower particle number. Once attainment at particle number $N$ is known, the compact-support theorem of Di Marino and Lelotte arXiv:2607.11440, valid for all $0<\mathsf{s}<\mathsf{d}$, permits a non-product one-particle extension and yields $\Lambda_{N+1}(\mathsf{s},\mathsf{d})>\Lambda_N(\mathsf{s},\mathsf{d})$. Together, these implications close an induction beginning at $N=1$.