Papers

Let $A$ be a finite-dimensional algebra over a field $\kk$. We show that the Auslander–Reiten bimodule $\ar_A:=\D A[-1]$ is central in the derived Picard group of $A$ and that, when $\gldim A<\infty$, it induces through the derived-invariance functor $\HB$ of \cite{Armenta19,ArmentaKeller17,ArmentaKeller19} a canonical automorphism $\sigma_A$ of the Tamarkin–Tsygan calculus of $A$; the pair $(\HB(A),\sigma_A)$ is invariant under derived equivalence. We then compute both components of $\sigma_A$. On the Hochschild homology of an elementary algebra, which is concentrated in degree zero, the matrix of $\sigma_A$ in the basis of idempotent traces is $-C_A^{-1}C_A^{\mathrm{T}}$, so its characteristic polynomial is the Coxeter polynomial; the enriched calculus strictly refines both the calculus and the Coxeter polynomial, as the path algebras of quivers of types $\mathbb{A}_4$ and $\mathbb{D}_4$ show, although it is not a complete derived invariant, as the smallest cospectral pair of trees shows. On Hochschild cohomology we prove that $\sigma_A$ is the identity: the left and right actions of $\HH^\bullet(A)$ on the bimodule $\D A$ coincide for every finite-dimensional $A$. This yields a short conceptual proof that the Nakayama automorphism of a Frobenius algebra acts trivially on Hochschild cohomology, recovering a recent theorem of Suárez-Álvarez. Finally we extend the construction to smooth and proper differential graded algebras, hence to perfect derived categories of smooth projective varieties; the enrichment degenerates precisely on Calabi–Yau categories, and on $\PP^n$ it is governed by the Coxeter polynomial $(x+(-1)^n)^{n+1}$ of the Beilinson algebra. Happel's trace formula and de la Peña's cyclotomicity theorem for fractionally Calabi–Yau algebras become statements internal to the enriched calculus.

A nonlinear Vlasov–Fokker–Planck equation is considered in a bounded domain, with specular reflection imposed on the boundary. It is proven that if the initial datum is sufficiently close, in a weighted $L^\infty$-space, to a radial and spatially homogeneous function that is regular enough, then a corresponding global-in-time weak solution exists, and this one decays with exponential rate to the global Maxwellian. The proof is split into two steps. In the first, the stability of the manifold of spatially homogeneous solutions to this equation is established. In the second part, the close-to-Maxwellian regime is considered. A careful regularization scheme is designed so that this mollified equation can be well-approximated by a linear system. Inspired by the recent preprint [Carrapatoso and Mischler, arxiv:2407.09031], for this linear system, $L^2$-hypocoercivity along with ultracontractivity properties are derived in order to change the space of functional decay to the $L^\infty$-framework. The properties of the original model are then recovered by means of fixed point arguments and carefully passing to the limit of the regularization parameter.

An even factor of a graph $G$ is a spanning subgraph in which every vertex has positive even degree. It is known that the minimum degree $\delta(G)\ge 2$ is a trivial necessary condition for $G$ to have an even factor. Recent spectral results for the existence of even factors used the certain complete-join graphs as exceptional extremal graphs. However, these graphs already contain $2$-factors and therefore are not genuine obstructions. This observation leads to the natural problem of determining the true sharp spectral threshold when the minimum degree is given. In this paper, we provide tight adjacency spectral radius conditions for a connected graph to contain an even factor, and characterize all extremal graphs, respectively. We also study the stronger requirement of a connected even factor, equivalently a spanning connected Eulerian subgraph. For this property, we also establish the corresponding sharp adjacency spectral radius condition and determine the unique extremal graph.

We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+\sigma^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $\sigma = o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(\Pi^\star)$, i.e., $\|X^\star-\Pi^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-\Pi^\star\|_F^2$ transitions from $o(n)$ for $\sigma = \tilde{o}(n^{-1/2})$ to $\Omega(n)$ for $\sigma = \tilde{\Omega}(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.

A central challenge in mesh-free space–time approximation for time-dependent partial differential equations is to represent evolving temporal scales while keeping residual minimization computationally tractable. Random feature methods simplify this algebraic problem by freezing nonlinear trial functions and fitting only a linear readout, but standard static space–time activations provide no explicit relaxation-scale mechanism, making temporal-scale resolution a finite-dimensional bottleneck in stiff, dispersive, or multi-scale regimes. We introduce liquid random feature methods (L-RFM), which replace static temporal activations by closed-form liquid time-constant responses with sampled relaxation scales. The resulting frozen features form temporally structured local or global trial spaces with analytic space–time derivatives for residual least-squares assembly. A density theorem proves density of the deterministic trial spaces in the continuous space–time function class, and a temporal-rank calculation clarifies the role of sampled relaxation scales. Ablation and finite-feature tests identify the liquid temporal response as the primary source of the observed accuracy improvement. Across stiff reaction–diffusion, nonlinear transport, dispersive, complex-valued, and multidimensional benchmarks, L-RFM improves finite-feature accuracy in regimes where temporal-scale representation controls the approximation. By embedding relaxation scales directly into frozen trial functions, L-RFM provides a route to high-accuracy continuous space–time surrogates for evolutionary PDEs while preserving the simplicity of linear least-squares solvers.

Test-time training (TTT) adapts a pretrained model to each prompt via parameter updates, improving accuracy under pretraining-to-test distribution shifts. Yet, its performance often suffers from instability and sensitivity to hyperparameters such as update steps and subspace. We explain this behavior through a decision-theoretic lens, treating TTT as implicit Bayesian inference in the kernel regime. Under a Gaussian process benchmark, we show that TTT reduces prediction error when updates are spectrally matched to the prompt's signal-to-noise ratio and aligned with query-relevant eigen-directions. This perspective underpins the following results: (1) we show when fixed update steps and subspaces fail under distribution shifts, motivating adaptive strategies; (2) we prove that selecting update steps via prompt evidence admits a PAC-Bayes guarantee against overfitting; and (3) we characterize the Bayes-optimal update subspace under a linear-Gaussian correction model, yielding a scoring rule for selecting Transformer blocks and heads. Our theory helps explain the empirical instability of TTT, taking a step toward principled guidance for when, how far, and which directions to adapt.

We study topological edge solitons in a nonlinear Su–Schrieffer–Heeger (SSH) lattice subject to parametric driving and linear damping. Starting from a vertically driven pendulum chain, we derive an effective driven–damped nonlinear SSH model and investigate its stationary edge-localized states. Analytical calculations reveal the existence of two phase-locked dissipative edge-soliton families that emerge from the nonlinear continuation of the topological edge mode. Using numerical continuation and spectral stability analysis, we construct the corresponding nonlinear branches and determine their stability properties. We show that parametric driving and damping fundamentally modify the conservative edge-state family by generating two dissipative branches with markedly different stability characteristics: one branch remains predominantly unstable, whereas the other develops substantially larger stability regions and significantly weaker instability growth rates. Direct numerical simulations further demonstrate that the robust branch can remain strongly localized over long time intervals even when weakly unstable. Simulations of the full driven–damped Klein–Gordon pendulum chain confirm the persistence of the edge-localized dynamics predicted by the reduced model. These results identify parametric driving and damping as an effective mechanism for enhancing the robustness and persistence of nonlinear topological localization in active lattice systems.

This paper focuses on a class of fully nonlinear elliptic equations with variable double phase type degeneracy law and Hamiltonian terms. We obtain improved gradient Hölder regularity results at points where the Hamiltonian coefficients and source terms vanish. Furthermore, we establish a Schauder-type estimate at local extrema, which is sharp with respect to the vanishing rate of the Hamiltonian coefficient and source term. Our approach adapts compactness and dichotomy arguments to capture the interplay between the degeneracy rate and the growth of the Hamiltonian term.

Let $C$ be a compact Riemann surface of genus at least two, and let $G$ be a connected complex reductive group. We study quadratic one-forms associated to logarithmic $G$-Higgs bundles on a pointed curve $(C,D)$ with nilpotent residues. We use the elementary pole cancellation for invariant polynomials, where nilpotency of the residue removes the leading pole term. In degree two this places $B(\Phi,\Phi)$ in the cotangent space of pointed Teichmuller space, and hence gives a logarithmic quadratic one-form. We relate this one-form to the variation of the energy for tame nilpotent harmonic bundles. The energy formula is proved under a positive decay assumption near the punctures. We also write the corresponding compact formulae for classical groups and standard real forms.

This paper focuses on a class of fully nonlinear elliptic equations with general double phase degeneracy/singularity law and Hamiltonian terms of the form $$\Phi(|Du|,x)F(D^2 u, x)+H(Du,x) =f(x) \quad \text{in} \quad B_{1},$$ where $\Phi$ takes one of two typical forms: $$\Phi(|Du|,x)=\sigma_{1}(|Du|)+a(x)\sigma_{2}(|Du|)\quad {\rm or}\quad \Phi(|Du|,x)=\frac{\sigma_{1}(|Du|)}{|Du|}+a(x)\frac{\sigma_{2}(|Du|)}{|Du|}.$$ Under suitable assumptions on the operator $F$, Hamiltonian term $H$, source term $f$ and modulating coefficient $a$, we establish $C^{1}$ regularity for viscosity solutions, provided that $\sigma_{1},\sigma_{2}$ are moduli of continuity and their inverses are Dini continuous. Our argument is based on a tangential analysis via approximating hyperplanes combined with a new recursive renormalization algorithm adapted to the present framework. It is noteworthy that our results are new even for the case $a(x)\equiv 0$.

In large language model (LLM) serving, each request accumulates persistent graphics processing unit (GPU) memory during service as its key-value cache grows with every generated token. Under high concurrency, aggregate memory usage therefore increases endogenously over time: the service process itself creates future capacity pressure. When memory capacity is exceeded, systems evict active requests, discarding cached state and restarting them later, which wastes computation and reduces throughput. We develop a discrete-time dynamical model of memory-constrained LLM inference that captures admission, memory growth, and eviction under continuous batching. In the saturated-input regime, the system admits both eviction-free fixed points and limit cycles with evictions. For homogeneous workloads, we show that the eviction-free equilibrium is unstable and that, except for a Lebesgue-measure-zero exact-capture set, the system converges to a unique worst-case limit cycle that is asymptotically stable outside this exceptional set, with throughput losses as large as 50%. For heterogeneous workloads, we prove a stability criterion in the two-class common-input setting and explain how the survival-polynomial mechanism generalizes to multiple classes and heterogeneous-input lengths. Under an input-dominated scaling regime, coprime decoding lengths stabilize the eviction-free equilibrium, while non-coprime lengths create synchronized modes that drive instability. These results characterize when workload heterogeneity desynchronizes completions and helps stabilize memory-constrained serving. More broadly, we identify service-induced congestion as a structural instability mechanism and derive scheduling design principles for sustaining high throughput.

In this paper, we prove that a linear cocycle with a set of strongly invariant cones admits a decomposition of quasi-all invariant bundles. We further obtain the certain location-relations between the fibres of positive invariant bundles and these cones, as well as we get the order-relations among characteristic exponents of the linear cocycle on the positive invariant bundles. The key of proofs is to analyze relations among positive invariant bundles with respect to a linear cocycle that admits both of a $k$-exponential separation and a $k^{\prime}$-exponential separation. These results are applied to a class of semilinear equations on a Hilbert space.

In this paper, we study transitivity for endomorphisms. Our results are related to a conjecture of F. Abdenur, C. Bonatti and L. Díaz, concerning the relationship between transitivity and the existence of a non-wandering set with nonempty interior. We obtain transitivity under the assumption that there exists a hyperbolic non-wandering set with non-empty interior, and in the setting of accessible partially hyperbolic endomorphisms.

The classical theorem of Livšic and Sinai states that a transitive $C^2$ Anosov diffeomorphism whose Jacobian along every periodic orbit equals one admits an invariant volume form. Recently, it was observed that the periodic Jacobian condition alone already implies transitivity, rendering the transitivity assumption unnecessary. We extend this rigidity phenomenon to the non-invertible setting. We prove that if a $C^2$ Anosov endomorphism satisfies the natural periodic Jacobian condition $J(f^n(p)) = \deg(f)^n$ for every periodic point $p,$ such that $f^n(p) = p,$ then the system is automatically transitive and preserves a $C^1$ volume form. As a key ingredient, we establish a $C^1$ version of the Livšic cohomological theorem for hyperbolic endomorphisms.

In this paper, we prove that the 2D Boussinesq equations are strongly ill-posed in the supercritical Besov spaces $B^s_{p,q}$ and Sobolev spaces $W^{s,p}$ with $(p,q)\in(1,\infty)\times [1,\infty]$ and $s\in(0,1+2/p)$ by constructing an initial data with arbitrarily small norm for which the solution of the system exhibits norm inflation almost instantaneously. As a further application, we prove the instability of perturbations near the hydrostatic equilibrium for the 2D Boussinesq equations in the same $B^s_{p,q}$ and $W^{s,p}$.

In this paper, we introduce a two-variable parity polynomial invariant for virtual knotoids, defined on oriented virtual knotoid diagrams. The construction is based on the parity of classical crossings, where each crossing is classified as even or odd and treated accordingly in the definition of the invariant. We study several fundamental properties of this invariant. We demonstrate that the parity polynomial can distinguish pairs of virtual knotoids that are not distinguished by the odd writhe, prove that it is equivalent to the affine index polynomial, and establish that it is a Vassiliev invariant of order one. Finally, we give its relationship with the Petit gluing invariant.

In this study, we investigate the Jaynes–Cummings model (JCM) using Lie symmetry analysis and conservation-law theory. The dynamics is formulated as a system of partial differential equations by projecting the von Neumann equation onto the atomic degrees of freedom and representing the field mode through its characteristic function. We determined the admitted point and generalized symmetries and constructed invariant solutions by imposing quantum-mechanical constraints. The conventional dressed-state dynamics is recovered, while a second class of solutions with radial dependence expressed through Heun polynomials is obtained for coupled atom–field configurations. We also applied a generating-function methodology to derive the local conservation laws of the JCM differential system in Fourier phase space. In addition to recovering the standard excitation-number constant of motion, we obtained additional conserved currents involving atomic populations, coherence, reduced-state purity, and derivatives of the field characteristic functions. One of these currents yields, at the Fourier-space origin, a balance equation for a combination of atomic purity and coherence whose evolution is driven by atom–field coupling and joint atom–field moments. For globally pure states, the purity contribution provides an indirect connection with the development of atom–field entanglement. The symmetry structure generates generalized symmetries and an infinite hierarchy of conservation laws.

Simple-minded collections in the bounded derived category $\mathsf{D}^\mathrm{b}(\operatorname{\mathsf{mod}} A)$ are necessary tools in tilting theory of finite dimensional algebras, as well as silting complexes in the perfect derived category $\mathsf{K}^\mathrm{b}(\operatorname{\mathsf{proj}} A)$. In this paper, we give an explicit mutation formula of simple-minded collections at arbitary direct summands, which is compatible with that of silting complexes. Though this mutation formula may be known to experts, we provide a thorough proof for completeness and future reference. Then we use our formula to revisit $\tau$-tilting theory in terms of mutations of 2-term simple-minded collections, including their relationship with maximal/minimal completions of 2-term presilting complexes and $\tau$-tilting reduction.

In this paper, we investigate the pseudoconvexity of locally trivial holomorphic ball bundles over compact Riemann surfaces of genus $\geq 2$, as well as the intermediate pseudoconvexity of their complements in the associated projective space bundles. Inspired by Brunella's work, we prove that any such ball bundle is $1$-convex, while its complement is $n$-convex, where $n$ denotes the dimension of the ball fiber, provided that the bundle admits a harmonic section with a regular point.

This paper traces, with explicit numerical values, how PyTorch's automatic differentiation (AD) engine computes gradients for Physics-Informed Neural Network (PINN) training – a setting that requires two levels of differentiation: computing the physics derivative $\hat{y}'(t)=d\hat{y}/dt$ through the network, and computing parameter gradients $\nabla_\theta L$ of a loss that itself depends on $\hat{y}'(t)$. Using a 1-3-3-1 multilayer perceptron and the initial value problem $y'(t)+y(t)=0$, $y(0)=1$, we trace the complete pipeline at every node: the computational graph built during the forward pass, the reverse-mode backward traversal that computes all 22 parameter gradients in a single pass, and the graph-on-graph mechanism by which create\_graph=True enables correct differentiation through the physics-informed residual. Every adjoint value is verified against the hand derivations of Tahimi (2026), connecting the $P/Q$ sensitivity framework to the vector–Jacobian products used by PyTorch's autograd engine.

Multi-step retrieval-augmented generation (RAG) has been widely deployed as LLM-powered web services for complex question answering, where iterative retrieval-reasoning rounds deliver strong multi-hop accuracy. However, this paradigm causes historical documents and reasoning traces to accumulate across rounds, inflating cumulative input tokens approximately as $O(N^2)$ with progressively increasing noise density. In API-based service architectures, such growth directly amplifies per-request billing cost, network payload, and response latency. Existing compression approaches rely on pretrained modules or GPU-level KV cache access, introducing model hosting overhead incompatible with API-native, Serverless, and edge-side deployments. To address this issue, this paper proposes ConCise, a training-free state-layer protocol that restructures cross-round context transmission for multi-step RAG services. Specifically, ConCise replaces raw-text accumulation with an append-only chain of structured conclusions, compressing cumulative context growth from $O(N^2)$ to approximately $O(N)$. Furthermore, a fused generation mechanism is introduced to jointly emit reasoning and conclusions in a single API call, eliminating repeated input billing from serial dual-invocation overhead. Extensive experiments across twelve paired configurations spanning three models, two datasets, and two representative frameworks demonstrate that ConCise achieves 64.63% average token savings while maintaining acceptable accuracy, providing a plug-and-play, deployment-friendly solution for cost-efficient multi-step RAG service optimization.

We develop an algebraic invariant calculus for complex hyperbolic two-space $\mathbb{C}\mathrm{H}^2$ in its Hermitian projective model. Starting from a three-dimensional Hermitian vector space of signature $(2,1)$, we treat $\mathbb{C}\mathrm{H}^2$ as the projectivized negative cone and its boundary as the projectivized null cone. The resulting calculus replaces metric and angular quantities by projectively invariant algebraic data: Hermitian pair moduli, determinant quadrance, polar spread, triple-product phase, cross-ratio invariants, contact boundary forms, normalized ball kernels, and finite Hermitian incidence laws. We prove denominator-cleared identities for three-point Gram determinants, triangle trigonometry, complex geodesics, bisectors, spinal spheres, projective-unitary covariance, CR/contact atlas transitions, and kernel boundary limits. The triple-product phase supplies the algebraic source of the Cartan angular invariant, while pair invariants recover the usual Bergman distance and polar-spread data over $\mathbb{C}$. The same formulas are also organized over starred fields, including finite Hermitian geometries, giving a field-uniform projective calculus for complex hyperbolic geometry.

For every integer $g\ge 2$, we determine exactly which integers occur in the greedy 3-sumfree sequence that starts with $1$, $g$, and $g+1$. This gives a direct proof of a conjecture of Bosma, Bruin, Fokkink, Grube, Reuijl, and Tromp. We also obtain an explicit eventual periodic description, including both the preperiod and the repeating block.

For a monic polynomial $p$, its filled unit lemniscate is the planar set ${z: |p(z)|<1}$. Let $\kappa_n(K,1)$ denote the least possible area of this set among monic polynomials of degree $n$ whose zeros lie in a compact set $K$. We prove that there are absolute constants $c,C>0$ such that $c/\log n \leq \kappa_n(\overline{\mathbb{D}},1) \leq \kappa_n(\mathbb{T},1) \leq C/\log n$. Thus the recently established lower bound has the correct order, even when all zeros are required to lie on the unit circle. The upper bound is obtained by combining a quantitative Faber-polynomial separator for a thin keyhole domain with an equal-weight midpoint discretization that preserves the degree exactly. We also deduce that the critical boundary-zero minimizers form a normal family in $\mathbb{D}$.

In this paper, we investigate the Gateaux differentiability of some duality mappings in the uniformly convex and uniformly smooth Banach space, which includes the normalized duality mapping as a special case, and it is denoted by J. We will introduce a general duality mapping and a generalized duality mapping. After the differentiability is proved, we find the explicit Gateaux and coderivative of J, and the duality mappings. By using coderivatives, we find the covering constants for these duality mappings, respectively. We also prove that the Gateaux derivative operator of the normalized duality mapping J has Lipchitz property.

In part I, \cite{Frauenfelder:2026c}, we showed that collisional periodic orbits of twisted Zeeman systems can be detected variationally by a non-local Hamiltonian action functional. In this part II we show that the linearized gradient flow of this non-local functional is a Fredholm operator and prove a non-local elliptic regularity result. These results are obtained with the theory of almost extendability of weak Hessian fields introduced in \cite{Frauenfelder:2025g}.

We introduce a simple iterative geometric construction on the integer lattice $\mathbb{Z}^2$ consisting of paired unit squares that share a single corner. At each step, a new square is constructed outwardly on the hypotenuse of the isosceles right triangle formed by the outer edges adjacent to the shared corner. This process generates two interlocking affine spiral orbits on $\mathbb{Z}^2$: a positive orbit {P_n} starting at (0,0) and a negative orbit $\{N_n\}$ starting at (-1,2). Both sequences satisfy linear recurrences driven by multiplication by the Gaussian integer $1+i$. We show that the paired points remain symmetric with respect to the fixed midpoint $M=(-1/2,1)$ for all $n$, satisfying the invariant $P_n + N_n = (-1,2)$. Extending the iteration backward under the associated inverse maps produces an iterated function system whose attractor is similar to the Twindragon fractal, providing a concrete lattice-based viewpoint on its geometry. In addition, the paired points yield the normalized integer sequence $a(n) = (|P_n|^2 + |N_n|^2)/5 = 2^(n+1) + 1 - 2*Re((1+i)^n)$, which is always an integer and appears as A396151 in the OEIS.

Discovering the governing equations of a physical system from data is a central goal across the sciences, yet in most experiments only a few states are accessible while the rest stay hidden. Existing approaches treat this partial observability as an obstacle to be removed by first reconstructing the hidden state—a step that is ill-posed under noise and that discards the physical constraints, such as energy conservation, that the true dynamics obey. We show that for conservative (Hamiltonian) systems no reconstruction is needed: projecting the dynamics onto the measured coordinates yields a memory kernel that we prove to be a lossless positive-real rational matrix, whose poles are the hidden natural frequencies and whose positive-semidefinite residues encode the couplings. From this kernel we recover a closed, interpretable governing equation for the observed dynamics—identified from output data alone, passive by construction, and validated by out-of-sample forecasting. Under stronger conditions—an equipartitioned measure with position coupling, or a forced input–output experiment—the bare hidden frequencies and couplings of the underlying Hamiltonian are additionally recoverable. We test the method on linear, nonlinear, and chaotic systems under realistic noise. Because it returns energy-conserving equations of motion from partial measurements, it offers a common tool for problems spanning mechanics, fluid and plasma physics, and beyond.

Linear-programming (LP)-based primal-dual methods are fundamental for designing and analyzing algorithms in adversarial (prior-free) online resource allocation. This chapter provides a tutorial on two modern primal-dual frameworks, emphasizing recent developments and contemporary models in operations research. Part~I develops an LP-based convex-programming framework where solving a regularized convex program at each arrival captures the tradeoff between greediness and hedging, yielding a dual certificate via Karush-Kuhn-Tucker (KKT) conditions. Because standard LP relaxations can be weak or intractable for stochastic outcomes, Part~II introduces a complementary LP-free framework that provides a universal certificate system for evaluating competitive ratios under such uncertainty. Covering a wide array of models – including online vertex-weighted bipartite matching, edge-weighted online matching with free disposal, online matching with stochastic rewards, reusable resources, two-sided assortment optimization, configuration allocation (whole-page optimization), AdWords, and costly cancellations – the tutorial equips readers with versatile proof templates to analyze existing algorithms and develop new solutions for emerging applications.

Assembly theory is an experimental and theoretical framework that introduces a metrological approach to detecting life, with potential applications across diverse substrates. Its two central observables are assembly index and copy number. The assembly index is the minimum number of joining operations required to construct an object from its elementary parts; for molecules, it can be measured using mass spectrometry, infrared spectroscopy, and NMR. Copy number is the abundance of a given distinguishable object within a sample. A key empirical result of the theory is that high assembly index combined with high copy number constitutes a signature that cannot arise abiotically, and this has been validated experimentally in application to molecular biosignatures. The foundational theoretical concept underlying these results is the assembly space, which encodes the causal possibilities determinable from observed objects, with the assembly index the shortest path to them given the physical constraints of a given substrate. Here, we provide a generalized formalism to describe assembly spaces and tools for assembly index approximations. We begin by reviewing the applications of assembly theory across molecules, minerals and atmospheres, and then introduce a general, substrate-independent formal definition of assembly spaces and assembly indices. We develop a unified path hierarchy framework to clarify relationships among the various representations of assembly spaces and assembly paths that appear in the literature on molecular assembly. Finally, we show how formal grammar algorithms can be adapted to efficiently bound assembly index calculations and provide clarification on the utility of such approximations, with the goal to increase the accessibility of tools to explore this emerging area for a broader group of researchers across chemistry, biology, and complexity science.