Papers

Motivated by Haag duality in quantum spin systems, we show that Haag duality for a pair of commuting factors generated by increasing sequences of finite-dimensional matrix algebras is equivalent to the asymptotic vanishing of a suitable conditional mutual information. For spin systems, let an annulus separate a finite region from the exterior. Haag duality then holds for a region $A$ if and only if the mutual information of the inner region and the exterior, conditioned on the part of $A$ inside the annulus, vanishes as the outer radius becomes large. As a corollary, assuming a strict area law with subleading corrections, Haag duality holds for single cones, and it holds for unions of two or more disjoint cones precisely when the topological entanglement entropy vanishes. For translation-invariant pure states on spin chains, half-chain Haag duality is equivalent to vanishing entropy density.

We study microscopic spectral correlations between two nearby parameter values in the three non-Hermitian Gaussian bulk classes: $A$ (unconstrained), $AI^\dagger$ (complex symmetric), and $AII^\dagger$ (complex self-dual, with each degenerate doublet counted once). Parameter dependence is modelled by stationary matrix Ornstein–Uhlenbeck evolution, whose microscopic time scale is $N^{-1}$. For every positive integer replica number $n$, we derive exact finite-$N$ auxiliary-field integral representations for joint characteristic-polynomial moments and obtain their bulk asymptotics as $N\to\infty$ with $n$ fixed. With the spectrum normalized to the unit disk, correlations near a bulk point $z_0$, between spectral positions $z_1,z_2$ at times $t_1,t_2$, depend only on $s=N|z_1-z_2|^2+N(1-|z_0|^2)|t_1-t_2|$. Adopting the static Hermitian/non-Hermitian replica continuation yields explicit two-time kernels; for $AI^\dagger$ and $AII^\dagger$ these are replica conjectures and, to our knowledge, the first microscopic two-time kernels proposed for these classes. They determine density correlations, cross-time eigenvalue-count covariances, and equal-time number variances. For a disk of mean eigenvalue count $y$, the latter obey $\mbox{Var}\mathcal N_X(D_y)=\kappa_X\sqrt y+ \beta_X/\sqrt y+o(y^{-1/2})$, with explicitly given constants for each class. Exact perturbation theory further identifies eigenvector nonorthogonality as the mechanism of short-time spectral diffusion and yields a basis-independent condition number for a two-dimensional Kramers eigenspace in class $AII^\dagger$. Direct simulations agree quantitatively with its conjectured inverse-gamma law.

Knot theory has provided a rich topological taxonomy for band structures, but its reach is fundamentally limited: knot invariants classify only 1D nodal lines at gap closure, and cannot encode the full dispersion or rich Fermi surface structure of realistic materials. Here we show that knotted graphs (knots that admit graph-like intersections in 3D space) - which have so far been elusive in condensed matter literature - provide a unified topological language for classifying the entire band dispersion, and even the eigenstate topology in some contexts. We propose a new framework beyond the existing Yamada polynomials that can topologically characterize the intricacies of realistic Fermi surfaces completely, crucially including how their multiple disconnected pieces are nested. This yields the Yamada set, a boundary-resolved extension which organizes the full topological evolution across energy into a Yamada sequence: a compact dispersion-level fingerprint directly tied to experimental signatures of Lifshitz transitions. Our framework is demonstrated with DFT-based band structures of real materials. Beyond dispersion-level classifications, this framework can be extended to non-Hermitian exceptional surfaces, where Berry-curvature flux further equips the knotted-graph skeleton with a directed Abelian edge flow that also captures the eigenstate topology.

We construct a class of integrable chiral fermionic quantum field theories with interactions that depend on both space and time. The construction starts from a unitary, regular, difference-form two-body scattering matrix satisfying braiding unitarity and the Yang–Baxter equation. Each freely moving right- or left-moving particle carries a characteristic coordinate that remains constant along its trajectory, and the scattering matrix depends on differences of these coordinates. The many-body amplitudes form a discrete connection over the coordinate-ordering sectors. The Yang–Baxter equation gives local path independence, while periodic winding on a spatial circle produces quantum Knizhnik–Zamolodchikov (qKZ) difference equations expressing global flatness. Once a reference amplitude satisfying the exchange and qKZ constraints is specified, these transport equations determine the exact wavefunction in every fixed-particle-number sector. Independent reparametrizations of the right- and left-moving characteristics give couplings that depend on both space and time. For the chiral $SU(2)$ Gross–Neveu realization, we solve the qKZ equations for this full characteristic family using off-shell Bethe vectors. The resulting amplitudes are written in the characteristic labels, and their pullbacks give exact many-body wavefunctions for the corresponding space-time-dependent couplings. The two-body amplitude is a ratio of Gamma functions that resums the phase shifts from successive scattering processes. Equal-slope affine maps give a spatially homogeneous specialization in which the coupling depends only on time, and the Bethe ansatz with constant coupling is recovered when the qKZ step vanishes.

We extend the algebro-geometric construction of bifiltered affine schemes that encode both the Higgs branch and the Macdonald index of 4d $\mathcal{N}=2$ superconformal field theories to theories with non-trivial Higgs branches. For the Argyres–Douglas theories $\mathcal{D}_p(SU(N))$, we find that a universal potential, given by a function in the Casimir invariants of $\mathfrak{sl}(N)$, determines the scheme. The scheme is given by a `fattened' nilpotent orbit closure, encoding the OPE decoupling structure of the 4d theory. A procedure we call syzygy maximization fixes the precise potential in a broad range of examples, and the resulting schemes reproduce the 4d data. The underlying graded ring gives conjectural presentations of Zhu's $C_2$-algebra of the associated vertex operator algebra, while the bifiltration refines Arakawa's associated scheme by retaining the R-symmetry information. We find that Higgs branch renormalization group flows are realized by pulling back the same universal potential to the corresponding Slodowy slices, including flows for which the infrared Higgs branch is a point. In particular, our construction provides a systematic method for determining the Macdonald indices of the $\mathcal{D}_p(SU(N), O)$ theories, for which no general expressions were previously known.

We formulate and prove existence of pure quasifree Hadamard states for the Klein–Gordon field with Dirichlet boundary condition on a globally hyperbolic spacetime with timelike boundary. The microlocal condition is imposed on the one-particle Hilbert-space-valued distribution $p\circ G_D$ rather than on the two-point function. This is expressed by a $b$-wave-front set condition on the compressed future cone. The analysis on the manifold with corners $\bar M\times\bar M$ is then almost completely avoided. We prove that our version of the Hadamard condition is universal in the class of quasi-free Hadamard states. We also prove a propagation theorem for positive-energy microlocal splittings which uses only a microlocal splitting in the small $b$-calculus and well-posedness of the Dirichlet problem, and in particular requires no propagation theorem for reflected or gliding rays; combined with an explicit computation in the ultrastatic case and a Fulling–Narcowich–Wald deformation argument this yields the construction of a pure quasi-free Dirichlet–Hadamard state.

We compute all stringy invariants, encoding orbifold cohomology numbers, of (the normalization of) the identity component of $G$-character varieties of free abelian groups and of moduli spaces of $G$-Higgs bundles on abelian varieties, for all complex connected reductive groups $G$. As an application, we provide a direct proof of a topological mirror symmetry statement: the equality of the stringy invariants for Langlands dual groups. In the framework of Ginzburg–Kaledin local resolutions, we discuss the meaning of these stringy invariants, showing that symplectic resolutions of these moduli spaces can only exist in the cases of groups of Dynkin type $A$, $B$, or $C$. When such resolutions exist, the computed stringy Hodge numbers agree with the actual Hodge numbers of the resolution.

We introduce a new family of trickledown theorems, a.k.a., local to global technique to bound the spectral gap of the Glauber dynamics for multi-state spin systems. In this technique instead of upper-bounding the influence matrix of a link of co-dimension 2 by $\lambda I$ (where $\lambda$ is the second eigenvalue of the link), we upper-bound the influence matrix after a carefully chosen rank-1 shift. The rank-1 shift allows for a significantly smaller upper-bound but it comes at the cost of bounding the average loss due to rank-1 perturbations. As an application we use this method to show that the natural Glauber dynamics mixes in polynomial time to generate samples from the Sherrington-Kirkpatrick model for $\beta\leq \tfrac{1}{2}+\varepsilon$, for an absolute constant $\varepsilon>0$. At the heart of the proof we manage to bound the loss due to rank-1 perturbations by averaging over all links of co-dimension 2.

We prove optimal trace-norm relaxation for the one-dimensional Lindbladian associated with the Witten differential $a=h\partial_x+V'$ which annihilates the classical Gibbs density: if $\lambda_1(h)$ is the first positive eigenvalue of $H=a^*a$, then the relaxation rate is $\gamma_h=\lambda_1(h)/(2h)$. It applies to initial operators whose Schwartz kernels, after a Gibbs conjugation, satisfy $ L^2 $ estimates for the restriction to the diagonal and for the normal derivative. An appendix by Chat GPT 6 presents a stronger result specialised to positive initial data.

We study dimension-free Bohnenblust–Hille inequalities for local operators on systems of $K$-level qudits. For the operator space of support at most $d$, uniformly over all tensor-product orthonormal operator bases, we prove a Bohnenblust–Hille inequality with the optimal exponent $2d/(d+1)$ whose asymptotic exponential base in the interaction order is of order $O(\sqrt K)$. Our proof is intrinsic, based on a one-site scalarization and block-transversal decoupling, and does not rely on a scalarization to the cyclic group or on a Remez-type argument. We complement the upper bound by showing that the asymptotic exponential Bohnenblust–Hille base satisfies $cK^{1/4}\le \beta(K)\le C\sqrt K$ with universal constants $c,C>0$. In this sense, the present work may be viewed as continuing the line of work of Slote–Volberg–Zhang from a complementary intrinsic viewpoint. We also revisit their Gell–Mann and Heisenberg–Weyl scalarization procedures. In the Heisenberg–Weyl setting, the prime-dimensional case already yields the optimal interaction exponent, whereas for composite $K$ the scalar total degree used in their reduction leads to a larger exponent. A support-sensitive formulation recovers the optimal interaction exponent for every $K$, while the intrinsic argument gives the stronger dependence on the local dimension at the level of the asymptotic exponential base. As further consequences, we determine the sharp scale, up to factors exponential in $d$ with base depending only on $K$, of coefficient $\ell_1$-normalization and unconditionality on the exact-support spaces. This yields dimension-free coefficient sparsification, including sparse generalized-Pauli approximation in Heisenberg–Weyl coordinates, as well as normalization and query bounds for canonical LCU/qubitization constructions.

Becker et al. produced a striking trace formula for the sum of fourth powers of generalized magic angles, $ \theta $, for the chiral model of twisted bilayer graphene (TBG) with the exact Bistritzer–MacDonald potential: $\sum \theta^{4} = 8 \pi / \sqrt 3 $. The purpose of this note is to generalize this formula to a larger class of potentials satisfying the symmetries of the model.

We discuss dominant representations (or Young diagrams), in thermal matrix models with gauge symmetry from the perspective of partial deconfinement. We propose a prescription for defining the dominant representations for thermal matrix models with interaction terms. As an explicit example, we consider the large-$N$ Gaussian matrix model. We obtain the Vershik–Kerov–Logan–Shepp (VKLS) shape of the dominant Young diagrams through a new analytic saddle-point analysis based on a mapping of the representation theory of U($\infty$) to free fermions in two spacetime dimensions. This computation provides a direct derivation of the previously observed functional relation between the shape of the dominant Young diagrams and the eigenvalue distribution of the thermal holonomy: the position of the complex saddle point is naturally identified with the eigenvalue. The dominant Young diagrams admit a natural interpretation in terms of partial deconfinement: the number of rows in the dominant Young diagrams matches the size of the submatrix corresponding to the deconfined subsector.

We present an extended study of the gravitationally localized soliton-like solutions to the minimally-coupled Einstein-Dirac-Higgs equations, embedded in asymptotically Minkowski spacetimes. The equations of motion describing these Yukawa-coupled Dirac stars are generalized to any even number of constituent fermions. We first expand the discussion of two-fermion ground-state solutions initially analyzed in Leith et al. [Phys. Rev. D. 107, 106020 (2023)]. Upon extending our analysis to excited states and many-fermion states, we find that both exhibit different behaviors to their two-fermion ground-state counterparts. In these excited states and many-fermion states, the Higgs field may exhibit stepwise increases and at times decrease within the soliton, which has not been seen for the two-fermion ground states. We also propose the potential cause of a significant degree of ADM-to-fermion mass disparity, which is present in states with strong Yukawa-coupling.

We describe how strong vorticity fluctuations appear in three-dimensional incompressible turbulence driven by a large-scale random force. By combining analytical and numerical methods, we construct a vorticity instanton: the most probable force history driving the flow to a prescribed large vorticity at a given point and time. We show how the smooth, large-scale force produces a long vortex filament with a thin, viscous-scale core. Our main finding is that the standard mechanism of amplifying a vortex tube through radial compression and axial stretching is only effective up to a threshold value. Producing vorticity above the threshold requires an additional ingredient: vorticity waves propagating along the tube. The final vorticity is the result of two colliding waves. This mechanism has striking similarities to the one recently suggested by OpenAI for creating infinite vorticity in finite-time Navier–Stokes blow-up.

Recent orthodoxy, which we call Curvature Fundamentalism, posits that the field strength $F$ on spacetime exhausts the physically significant content of electromagnetism, at least on prosaic (contractible) spacetimes. We argue that it is gauge invariance that constrains physical significance. As a result, there are physically significant quantities that $F$ does not capture, even on contractible spacetimes. We show that this result is robust: in the presence of charged matter fields, there are interference experiments whose outcome cannot be predicted by $F$ or any other point-local gauge invariant quantities on the basis of initial Cauchy data. We prove this using the example of relative phase of a Klein-Gordon-Maxwell field, and use it to identify a context in which the widely-claimed equivalence between the Maxwell-Faraday formulation $EM_1$ and the gauge formulation $EM_2$ of electromagnetism is untenable, and that new strategies may be needed for the reductionist project.

In this paper, We obtain dispersive estimates for solutions to the Schrödinger equation with a uniform magnetic field on the hyperbolic plane \(\mathbb{H}\). The key ingredient is an explicit representation formula for the kernel of the corresponding Schrödinger propagator. As a consequence, we prove the corresponding Strichartz estimates for all admissible pairs on \(\mathbb{H}\).

We develop a matrix pseudodifference operator framework that places theAdler-Pfaff and Pfaff-Toda hierarchies in a common algebraic setting. The bi-infinite Adler-Pfaff hierarchy then becomes a $2\times2$ matrix pseudodifference Lax hierarchy. We introduce the Matrix Pfaff-Toda hierarchy by dressing the matrix Laurent algebra $M_{2N}\bigl(\mathbb{C}[\mathcal{S},\mathcal{S}^{-1}]\bigr)$ using a Pfaff-Toda splitting of the matrix pseudodifference algebra. In the one-component case, its diagonal sector recovers the continuous Pfaff-Toda hierarchy, while the off-diagonal directions supply the missing odd flows, and the powers of a single bare operator reconstruct the full Adler-Pfaff hierarchy. For general $N$, on the regular factorization locus, the multicomponent Pfaff-Toda tau-functions of Savchenko and Zabrodin realize the commuting diagonal sector, with its dressing equations derived directly from the fermionic bilinear identity. Reductions to the block-diagonal and scalar cases recover multicomponent and scalar $2D$ Toda, identifying the even Pfaff hierarchy with an anti-diagonal Toda subhierarchy, and yielding the Krichever-Zabrodin C-Toda hierarchy.

This paper proves the integrability of a nonholonomic billiard defined by a knife-edge in a disk. The paper begins by parametrizing the impact space using coordinates that reveal the system's rotational symmetry. It then constructs a map on the space of post-impact states that encodes the billiard dynamics through a recurrence relating each post-impact state to the next. In addition, the paper shows that the knife-edge billiard admits a family of generalized caustics.

We present in this article the discretization of small and finite strain mechanics using variationally consistent, stabilized, nodal strain finite element formulations. We prove that these methods derive from a mixed variational principle that uses different meshes for the primal and dual variables. We demonstrate that, for both mechanical problems, nodal strain finite elements can be written as pure primal formulations with assumed strain operators. The stabilization terms are shown to be necessary and a new, general class of stabilizing functions is proposed for the finite strain regime. The ensuing formulations can be interpreted as pure particle methods because all kinematic and material information is stored at the nodes. This point of view explains the favorable properties of nodal strain methods when employed in problems that require frequent remeshing and involve inelastic materials. In these cases, the diffusion of internal variables is kept to a minimum. The numerical examples presented validate the claims.

We formulate the linearization of the fourth Painlevé equation as a linearized Hamiltonian system and transform the scalar variational equation to normal form, with the first-derivative term eliminated. From the canonical fundamental solutions of the Jimbo–Miwa Lax pair, we construct quadratic expressions $Q_{IV}$ and $R_{IV}$ that satisfy the identity $\partial_x^2Q_{IV}-U_{IV}Q_{IV}=\partial_\lambda R_{IV}$. Here $U_{IV}$ is the coefficient in the normal-form equation. A rapid-decay cycle with a vanishing boundary term yields an integral solution of the linearized equation. Every nontrivial such cycle has asymptotic branches in at least two distinct Stokes sectors; otherwise the integral vanishes by Cauchy's theorem. A second rapid-decay homology class gives another integral solution. The Wronskian of the two integral solutions depends holomorphically on the monodromy data, so a single nonzero value implies generic linear independence. For one set of complex data, direct quadrature gives numerical evidence of a nonzero Wronskian. We also derive integral formulas for variations of the Stokes multipliers, the connection matrix, and the local monodromy, and verify the contour representation by substitution into the linearized equation.

We consider two quantum spin models on $k$-partite graphs. The system is assumed to be invariant under all permutations within any of the $k$ sets. In the limit of infinitely many particles, the Gibbs state is described as a mixture of independent spins due to a version of the Quantum de Finetti theorem. First, we study a spin-1/2 antiferromagnetic Heisenberg model and identify its three phases. At low temperature all $k$ sets are magnetised but cancel each other, at medium temperature there is a net magnetic field and at high temperature there is no magnetisation. The magnetisation temperature is a solution to a $k$-degree polynomial. Second, we allow a general spin $s$ and identify all orthogonally invariant models which have frustration. The criterium for a frustration-free model is that the $k$-sets can be partitioned into two parts, where the only antiferromagnetic interaction is between the two parts.

We relate the spectrum of singularities to the geometry of degenerating Calabi–Yau manifolds through holomorphic analytic torsion. We express the leading asymptotic coefficients of the analytic torsion of holomorphic differential forms in terms of Hodge-theoretic invariants of vanishing cycles. For isolated hypersurface singularities, these formulas lead to the formulation of new higher Durfee–Saito conjectures bounding Hodge-filtration dimensions by the Milnor number and Eulerian numbers. We also prove that, for isolated quasi-homogeneous singularities, the spectral measure is dominated in convex order by the corresponding Irwin–Hall distribution. Applying the torsion formulas to the BCOV invariant yields a computable obstruction to birational smooth fillings. For isolated singularities, its local contribution is expressed in terms of Hertling's spectral variance and a Bernoulli sum determined by monodromy. As applications, we rule out birational smooth fillings for projective Calabi–Yau degenerations with smooth total space and singular central fiber having only ADE or terminal Brieskorn–Pham singularities, in relative dimension at least three, extending results of Voisin.

The Heisenberg picture can make quantum many-body simulation efficient when evolved observables admit compact operator representation within low-dimensional structures. Conventional Pauli-string propagation and truncation techniques exploit this structure, but in coherent Hamiltonian dynamics their error control is often heuristic and their stability can be poor. We introduce a state-adapted Krylov framework based on geometric generalized mean-field projections of many-body observables onto low-body operator subspaces. The resulting dynamics approximate expectation values, do not extend spatial support beyond that prescribed by Lieb-Robinson bounds, and compress high-body correlations onto their state-relevant low-body representatives rather than simply discarding them. We derive necessary operator-entanglement obstructions to low-body representation and conditional sufficient bounds on representation and dynamical errors involving nonstabilizerness and controlled high-body tails. Numerical benchmarks show improved, stable finite-$m$ hierarchies and simulations on large three-dimensional lattices, establishing a scalable state-adapted alternative to conventional Heisenberg-picture weight-truncation of Pauli strings.

Born-Infeld New Massive Gravity (BINMG) is an all-order curvature extension of New Massive Gravity in three dimensions, whose linearized spectrum around its unique maximally symmetric vacuum contains the two polarizations of a massive spin-two field. Whether an additional scalar degree of freedom appears in the full nonlinear theory has remained an open question. We perform a nonlinear canonical analysis of BINMG using its auxiliary-metric formulation. On the regular branch connected to the maximally symmetric vacuum, the Dirac algorithm yields three first-class diffeomorphism constraints and a second-class pair of scalar constraints. The resulting count gives exactly two local degrees of freedom per spatial point. Thus, nonlinear BINMG propagates no additional Boulware-Deser-type scalar mode on this branch.

We employ a Gaussian convex inequality developed in [AC15], [Che26], [Gue22], and [FUMPC26] to establish (one sided) moment comparison inequalities for Gaussian multiplicative chaos, directed polymers in Gaussian environment and, in particular, the Critical 2d Stochastic Heat Flow (SHF), including upper bounds on negative moments, as well as similar lower bounds for $p$-th moments with $p\in (0,1)$. For the Critical 2d SHF, averaged over small balls, we also obtain the matching (up to multiplicative constants) complementary bounds. We establish the complementary bounds using a bootstrap argument inspired by [DS10] and a geometric decomposition introduced in [GT26]. Specifically, we prove that for locally averaged SHF, for every $p\in\mathbb R$, the $p$-th moment is, up to multiplicative constants, bounded in both sides by the $\frac{p(p-1)}{2}$-th power of its second moment.

We study non-collision static equilibria in the charged \(N\)-body problem. Unlike the classical Newtonian case, such equilibria may exist because the effective interaction coefficients $\delta_{ij}=m_im_j-e_ie_j$ can be positive, negative, or zero. We give a complete classification of three-body equilibria and prove that all of them are collinear and linearly unstable. For the four-body problem, we derive sign restrictions for convex and concave non-collinear equilibria, exclude non-trivial concyclic quadrilateral equilibria, and obtain a geometric necessary condition expressed through an auxiliary triangle and a resultant equation. We also study collinear equilibria by reducing the force-balance equations to a homogeneous polynomial system. This yields a resultant necessary condition and an inverse realization theorem showing that every prescribed collinear configuration of distinct points can be realized by suitable positive masses and real charges. Finally, we analyze regular polygon configurations and centered regular polygon configurations, showing in particular that equal masses cannot form a non-trivial regular \(N\)-gon equilibrium, while the centered case reduces to a single scalar condition.

We study weakly interacting bosons on the three-dimensional torus at temperatures proportional to the critical temperature for Bose–Einstein condensation. We show that, as the expected particle number tends to infinity, the grand canonical Gibbs state is asymptotically described by the coherent state quantization of a random field that converges to a novel Bogoliubov random field. Moreover, the point process associated with the Gibbs state can be approximated by a Cox process. These results are based on and extend the approximation of the Gibbs state recently obtained by the first two authors and Nam. Using these approximations, we compute limiting distributions of the number of particles outside the condensate, the joint occupation statistics of finitely many momentum modes, and particle-number fluctuations in macroscopic subsets of the torus, all governed by the Bogoliubov field. We further prove convergence of the empirical measure of the associated point process to the uniform measure and convergence of the microscopic particle-number statistics to the boson point process. The latter result establishes the universality of particle-number statistics on microscopic length scales.

Aeroelastic ?utter is normally a failure mode to be designed against, yet the limit-cycle oscillations (LCOs) that follow it convert flow energy into sustained structural motion that a piezoelectric transducer can turn into electrical power. A transducer is embedded in a three-degree-of-freedom pitch-plunge aerofoil with a finite-mass trailing-edge flap and unsteady strip-theory aerodynamics, giving a fifteen-state electro-aeroelastic system with a cubic hardening pitch spring, and the system is reduced by biorthonormal projection of the Taylor-expanded residual onto eigenvectors of the coupled Jacobian. The testbed is deliberately low order, so that every reduced prediction can be checked against the full-order system it replaces. The degree of freedom that carries the transducer is a first-order design variable, since it sets both the sign of the shift in the flutter boundary and the magnitude of the harvested power. The error of the reduced model is dominated not by the size of the retained basis but by how the reduced operator is made to depend on low speed. Expanding the retained eigenvalues leaves the reduced operator block-diagonal, with no coupling between the retained modes at any order of the expansion, whereas re-projecting the exact Jacobian onto the same frozen basis supplies that coupling and restores the structural response without enlarging the basis. It is found that the error that remains lies in the harvested voltage, which the equilibrium eigenvectors carry in a fixed proportion to the motion, and this is not removed by adding the electrical modes to the basis, whereas the flap detection, which the same basis also misses, is recovered by retaining the flap modes. The accuracy of a harvester reduced-order model must therefore be reported on the harvested quantity itself.

This is an extended version of arXiv:2511.22129. We develop a general construction of the superstrip algebra $\mathbf{sStr}_{\mathscr C}(\mathcal M)$, the fermionic counterpart of the strip algebra, for a gapped $(1+1)$-dimensional system with a fermionic superfusion category $\mathscr C$ and the vacuum structure described by the supermodule category $\mathcal M$. As illustrations, we work out the superstrip algebra for two simple examples: the fermionic variant of the Fibonacci fusion category and the q-type $\mathbb{Z}_2$ symmetry. We then apply the framework to the gapped IR phases of the $\mathcal N=2$ and $\mathcal N=1$ superconformal minimal models deformed by their least relevant operator and compute the Witten indices. We explain how the particle and soliton spectra, together with their fermion parity, are organized into representations of the corresponding superstrip algebras. A mixed 't Hooft anomaly between the spontaneously broken symmetry and the unbroken fermion parity $(-1)^F$ further forces each $\mathcal N =2$ soliton to carry a fractional fermion number. Finally, we set up the boundary SymTFT for fermionic systems and reproduce the results in several examples studied in this paper.

Building on the established one-particle formula for the Araki relative entropy of coherent states, we study how its value acquires a nontrivial dependence on the mass of the scalar field. For a localized vector $h$ belonging to the standard subspace $H_m$ of the one-particle Hilbert space, the known quadratic-form expression is: $S_{H_m}(h)=-\langle h,\log\delta_{H_m}\,h\rangle$. Our contribution is to construct explicitly a mass-indexed family of vectors of the wedge standard subspace on which this expression is evaluated. The mass-shell map $h_m=E_mf$ organizes four structural conditions on rapidity representatives—on-shell dependence, a controlled massless boundary value, decay for large real rapidity, and Bisognano–Wichmann strip analyticity—and we exhibit an entire rapidity wave function, built from a doubled light-cone phase, two sinc factors, and a Gaussian pair, that satisfies them together with the sharp localization criterion: Hardy-type $L^2$ control throughout the Bisognano–Wichmann strip and the exact Tomita boundary relation. The family therefore belongs to $H_m(\W_R)$ for every $m>0$, and its Araki relative entropy is finite and strictly positive, with an exact spectral representation that makes positivity manifest. The entropy is strongly suppressed at large mass, attains a maximum at intermediate mass in $1+1$ dimension, and converges to a finite value along the modular flow as $m\to0^+$. The construction extends fiberwise to $1{+}d$ dimensions through the transverse mass.