Papers

In this paper, we study the poset QIso(G) of classes of subgroups of a finite group G having isomorphic quotients. Particularly, we study it for finite abelian groups and for some classes of non-abelian metacyclic groups, namely dihedral groups Dn, the group U2np and affine groups Aff(F) over finite field F. In the case of abelian groups G, we show that there exists a duality from QIso(G) to Iso(G). We show that cyclic groups are the only groups for which QIso(G), up to duality is isomorphic to its subgroup lattice. We also show that cyclic groups are the only nilpotent groups for which QIso(G), up to duality is isomorphic to its lattice of normal subgroups. We characterize finite 2-groups all of whose subgroups are q-self dual with QIso(G) being a chain. We also characterize all p-groups G of order at most p4 whose QIso(G) is a chain.

We consider an optimal control problem with a general control constraint and a fixed initial state, for which convexity assumptions are absent. To establish the existence of optimal pairs, we introduce a control version of the Palais-Smale type condition adapted from critical point theory and prove that optimal pairs exist under this condition (see Theorem 2.1). Subsequently, we verify that a specific class of optimal control problems satisfies our Palais-Smale type condition (see Proposition 2.1). Furthermore, as an application, we present an example (Example 3.1) and show that, to the best of our knowledge, no existing result can be applied to establish the existence of an optimal pair in this case (see Section 3.1.1). Example 3.2 shows that finitely many Pontryagin maximum principle candidates do not guarantee the existence of an optimal control. Finally, Example 3.3 demonstrates that the proposed Palais-Smale type condition is sufficient, but not necessary, for the existence of optimal controls.

We study penalizations of the two-dimensional simple random walk by functions of direction-dependent excursion counts and edge-crossing counts. The former record completed excursions with prescribed pairs of departure and return directions at the origin, whereas the latter record crossings of a prescribed set of unoriented edges incident to the origin. We determine the distributions of their almost surely finite terminal values and derive explicit path decomposition formulas involving the random walk conditioned to avoid the origin. Moreover, both families of penalized measures can be represented by weighting and normalizing the same universal $\sigma$-finite measure associated with local-time penalization at the origin. Thus, direction-dependent excursion counts, edge-crossing counts, and local time at the origin give rise to penalizations belonging to a common universality class.

In this paper, we develop an abstract framework for the theory of uniform exponential attractors in non-autonomous dynamical systems. Our construction employs time-dependent absorbing sets while ensuring exponential attraction among the constructed sets. Moreover, we show that the resulting exponential attractors are uniform-that is, they remain stable under perturbations. As an application of this abstract theory, we study the weakly damped hyperbolic equation with sub-quintic nonlinearity. Using the proposed approach, we construct exponential attractors for the process generated by the Shatah-Struwe solution, which satisfies both energy and Strichartz estimates. Furthermore, we establish the Holder continuity of these exponential attractors with respect to the external forcing term.

Random spanning forests provide probability models for partitions and Monte Carlo methods on graphs. In these applications, the distribution produced by a sampler determines which forest functionals can be estimated. The Aldous–Broder algorithm constructs a spanning tree from the first-entrance edges of a covering random walk. Its distribution is known, but a direct calculation requires summing over all orders in which a fixed tree can be discovered. We evaluate this sum through an identity for rooted forests and real matrices with zero row sums. Under a nonsingularity condition on the principal submatrices encountered during growth, the sum of products of harmonic extension entries equals a principal cofactor of a matrix on the roots. With one root, the sum is one. The identity gives a direct proof of the extended Aldous–Broder theorem for every finite irreducible Markov chain started at a fixed state, without assuming reversibility or bidirectional transition support. It also gives the conditional distribution of the discovery order and the law of the forest obtained by stopping after all nonroot vertices have been visited. A completion weight describes the difference between this stopped-walk law and the weight-proportional forest law, including the support restriction needed for importance correction. Numerical examples illustrate the sampling implications. The identity, the Aldous–Broder theorem, and the stopped-forest law are formalized in Lean4.

We construct two word metrics on the same seven-dimensional torsion-free nilpotent lattice whose asymptotic cones are isometric, but for which every quasi-isometry has multiplicative constant at least $1+\varepsilon$, for one fixed $\varepsilon>0$. We use a convex invariant of a fixed quasi-isometry $\Psi:N\to M$. The displacement cocycle of $\Psi$ and the right-invariant means on $N$ determine a compact convex set $\mathcal A(\Psi)\subseteq\operatorname{Hom}(V_N,V_M)$. If $\mathscr G_{\mathrm{pt}}(\Psi)$ denotes the first-layer maps of the graded homomorphisms that occur as blow-downs of $\Psi$ with varying basepoints, then $$ \mathcal A(\Psi)=\operatorname{conv}\mathscr G_{\mathrm{pt}}(\Psi), \qquad\operatorname{Ext}\mathcal A(\Psi)\subseteq\mathscr G_{\mathrm{pt}}(\Psi). $$ For every extreme point $L\in\mathcal A(\Psi)$, the dual map $L^*$ is also the degree-one part of an isomorphism of the full real cohomology rings. This cohomological compatibility, together with the realization of first-layer Pansu differentials by invariant means, yields the multiplicative constant gap stated above.

We prove that, for $2\leq p<n$, the optimal $L^p$-Sobolev constant on every complete $n$-dimensional Euclidean minimal immersion without boundary is the Euclidean Aubin–Talenti constant. The proof combines optimal transport to a suitably chosen target with a two-kernel comparison linking the tangential transport moment to deficits in the normal-fibre and Jacobian estimates.

We prove that, for every fixed $d\geq 2$ and $\varepsilon>0$, the $\mathbb{F}_q$-points of a geometrically irreducible reflexive plane curve of degree $d$ over $\mathbb{F}_q$ can be covered by at most $\left(\frac{1}{d}+\varepsilon\right)q$ many $\mathbb{F}_q$-lines, provided that $q$ is sufficiently large. The same conclusion holds for $\mathbb{F}_q$-irreducible curves that are not geometrically irreducible, since they have only $O(d^2)$ $\mathbb{F}_q$-points. In particular, for sufficiently large $q$, the bound applies to every $\mathbb{F}_q$-irreducible plane curve of degree $d$ when $\operatorname{char}(\mathbb{F}_q)>d$. Our bound is asymptotically sharp. The proof combines Entin's Chebotarev density theorem for varieties over finite fields with the Pippenger–Spencer theorem on covers of nearly regular hypergraphs.

We study the relationship between strong data-processing inequalities for Rényi divergences and hypercontractivity. For Rényi divergences of order greater than one, we characterize the contraction of Rényi divergences through a property of the dual Markov operator that we name power contraction. Using the contraction coefficient, we derive an upper bound on rectangle probabilities in terms of their marginals that is at least as strong as the corresponding bound obtained via hypercontractivity. Combining this result with inversion of the binary Rényi divergence, we derive communication lower bounds for one-way communication-assisted agreement distillation. Examples involving binary symmetric and rare-event Z-channels illustrate the improvements over standard hypercontractive converses.

In this paper, we propose a linear and decoupled fully discrete scheme that preserves the entropy production property for the dynamic van der Waals model. This model provides a thermodynamically consistent diffuse-interface framework for describing non-isothermal phase transitions in single-component liquid-vapor systems, containing the coupled conservation laws of mass, momentum and energy. We first introduce an equivalent reformulation of the governing equations that avoids the use of chain-rule identities, which generally fail under discrete differentiation. Further, a component-wise convex-splitting method is employed to calculate the discrete chemical potential and temperature. This treatment provides a discrete entropy-stable inequality without a time step restriction. Based on the explicitly treated velocity, the scheme allows the three conservation equations to be solved linearly and sequentially in a decoupled manner. Since the momentum equation does not directly affect the entropy-stability analysis, we use a classical linear semi-implicit scheme to solve it. Numerical experiments verify the expected numerical accuracy. Simulations of droplet evaporation and two- and three-dimensional Leidenfrost effects further demonstrate the proposed method can capture complex non-isothermal phase-transition dynamics while numerically reproducing the discrete entropy production behavior.

Let $p$ be a prime number. In 1986, David Lampert asked if one expands a $\mathbf{Q}_p$-algebraic number as a $p$-adic Hahn series, then what is the possible order type of its support? In 2001, Kedlaya made a vague conjecture that the possible ordinals can only be finite ordinals, $\omega$ and $\omega^\omega$, where $\omega$ is the first infinite ordinal. In this article, we solve Lampert's problem by confirming Kedlaya's intuition. Moreover, we completely characterize the $\mathbf{Q}_p$-algebraic numbers with order type of support at most $\omega$. As an application, we determine the bi-algebraic set of the $p$-adic shadow map.

Let $X$ be an irreducible reduced compact complex space of dimension $n$, and let $\omega$ be a Hermitian metric on $X$. For each $1\le k\le n-1$, we prove that there exist a constant $c_k$ and a smooth positive function $\rho$ on $X_{\mathrm{reg}}$ such that $\ii\partial\bar\partial(\rho\omega^k)\wedge\omega^{n-k-1}=c_k\rho\omega^n$. The function $\rho$ is bounded above and bounded away from zero. The constant $c_k$ is unique, and $\rho$ is unique up to a positive multiple among bounded smooth positive solutions. We also prove that the full sign set $\{-1,0,1\}$ of the generalized Gauduchon constants $c_k$ on a compact complex manifold is preserved by finite sequences of blow-ups with smooth centers.

We study a strong pure infiniteness property for extensions, and give a classification result for extensions which contains many (and possibly all) previous cases, under standard and mild assumptions. We also prove a Voiculescu-type theorem decomposition theorem for certain C*-subalgebras of a corona algebra, for which the inclusion map has this strong pure infiniteness property.

We study stochastic dynamic matching with abandonment, a fundamental model in which agents of different types arrive over time, may abandon the market while waiting, and can be matched in pairs for rewards that depend on their types. No online policy can guarantee more than $0.5$ of the omniscient benchmark, which knows all arrival and departure times. Whether this guarantee is achievable has been a central open problem, even in special cases. We resolve this problem by showing that, on every instance, a simple static priority policy, which is computable in polynomial time, earns at least $0.5$ of the omniscient benchmark. This establishes that the optimal competitive ratio is exactly $0.5$, improving the previous best guarantee of 0.192 for the general model [Li et al. 2026; Collina et al. 2020] Our analysis rests on a key stochastic result, which is a new lower bound on the probability that some agent from a given set of types is available. Building on this bound, we develop two approaches: a primal approach that reads the priorities from an optimal solution of a novel linear program, and a dual approach that assigns a “value” to each type, and prioritizes matches with high “surplus” (reward minus value of involved agents). These values are the unique solution of a fixed-point equation derived from linear programming duality. We also show that the two approaches are closely connected.

We study Fourier–Laplace transforms on null phases arising in passive inverse problems. Densities with codimension-one dependence in oblique cylinders are first considered. Integration in the distinguished direction reduces the data to an entire transform on a real-analytic maximally totally real submanifold. Holomorphic uniqueness then gives injectivity. A fixed transverse polytope swept along a polygonal curve, forming a polygonal cylinder, is next considered. The density is assumed to be polynomial in the material transverse variables on each swept piece. A weighted Brion–Barvinok expansion separates the vertex terms and the successive pieces. The same representation gives a finite-dimensional Lipschitz estimate when the polynomial degree is fixed. We apply these transform results to passive wave and heat equations in arbitrary dimension. The initial state is zero outside the observation domain. Equality of the Laplace-domain boundary data determines the quantity of the initial data divided by the speed squared in the acoustic model and the quantity of the initial data divided by the coefficient of thermal conductivity in the parabolic model. The question of what remains true when only weighted generalized boundary data on a finite interval \([0,T]\) are available is then studied. These data determine the harmonic moments of the quotient up to the weighted terminal residual. This identity gives a Hölder estimate in an oblique cylinder and a Lipschitz estimate in a polygonal cylinder. Finally, a continuous generating curve is approximated by a polygonal curve and a continuous material density by piecewise transverse polynomials. The resulting reconstruction estimate separates measurement noise, density approximation error, geometric error, and the condition number of the finite sampling matrix.

We study full-level Hilbert Hecke eigenform product identities over totally real fields of degree at least two, allowing arbitrary compatible narrow class characters. Combining our results with previous work, we classify products of two Eisenstein eigenforms of positive parallel integral weights and products of an Eisenstein eigenform of weight at least two with a cuspidal eigenform of positive parallel integral weight. The unique identity in each of these two classes is given by $E_{4}=60 E_{2}^{2}$ and $h_{8}=120 E_{2} \cdot h_{6}$, respectively, both over $\mathbb{Q}(\sqrt{5})$. The proofs use constant-term comparisons, indecomposable Fourier exponents, integrality of reciprocal Eisenstein constant terms, and a Rankin-Selberg formula for Petersson inner products. In the remaining case of a weight-one Eisenstein factor and a cuspidal factor, we also establish $h_5=12E_1(1,\varepsilon)\cdot h_4$ over $\mathbb Q(\sqrt3)$, where $\varepsilon$ is the nontrivial narrow class character. All eigenforms in these identities have unit-ideal coefficient one.

The minimum principle states that every real-valued uniformly continuous function f : X -> R on a compact metric space X attains its minimum. Working in the base system EL0 of constructive reverse mathematics, we show that the minimum principle is equivalent to weak Konig's lemma and to the Cantor intersection theorem (i.e., the contrapositive of the Heine-Borel theorem). Moreover, the above equivalence naturally extends to the equivalence between the corresponding principles with unique existence assumptions, which are then shown to be equivalent to the Fan theorem for detachable bars and the positivity property for a real-valued uniformly continuous function on a compact space. These results sharpen the equivalence among these principles obtained by Ishihara [Math. Log. Q. (1990), 237-240] and Berger and Ishihara [Math. Log. Q. (2005), 360-364] in Bishop's constructive mathematics.

In 2003, Héthelyi and Külshammer proved that if $G$ is a finite solvable group and $p$ is a prime such that $p^2$ divides $|G|$, then $k(G)\geq (49p+1)/60$, and they constructed an infinite family of groups for which this bound was claimed to be attained. We point out that the groups in this family are not Frobenius groups, so that their class numbers are considerably larger than stated, and that in fact the bound is never attained. We then determine the best possible bound of this shape: we show that $k(G)\geq (49p+7)/60$ for all primes $p$, which is sharp for $p=17$, and that $k(G)\geq (49p+451)/60$ for all primes $p>2719$, which is sharp for infinitely many primes $p$. Thus the slope $49/60$ found by Héthelyi and Külshammer is best possible, but the correct additive constant is $451/60$ rather than $1/60$.

Matching repair after utility changes can reuse valid observations, but new evaluations must respect matching capacity and may return out of order. We study announced invalidations with bounded, possibly reward-dependent reporting delays and pending evaluations that do not occupy future matching capacity. Residualload repair (RLR) combines sound outcome certificates with batched matching schedules for unequal sample deficits. Its complete dyadic quota schedule admits a sharp issue-round bound with coefficient cb < 1.607 relative to the terminal residual endpoint load, together with separate delay and evidence-lifetime bounds. RLR-P monitors complete issue-order prefixes without changing the sampling plan before acceptance. With a deterministic pure oracle, a single-repair coupling from the same drained initial state guarantees no increase in certification time, issued observations, or time to a reusable checkpoint. We also derive an information-load lower bound. On fixed priority-star markets with unknown Bernoulli gaps, PrioritySearch attains the history-adjusted first-order constant for expected certification time as {\delta} {\sim} 0, under hypothesis-independent linear history scaling and sublogarithmic reporting delays. Paired simulations and measured-feedback replay quantify the trade-offs: in the replay, RLR-P has lower mean capped update-certification time than a monitored width-based baseline, at higher oracle CPU cost.

Let $b(n)$ denote the minimum number of perfect matchings in a simple brace of order $n$. De~Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 27(2): (2013) 940-958) proved that $b(n)\ge(n-2)^2/8$ and conjectured that $b(n)=(n-2)^2/4$ for all sufficiently large even $n$. We prove this conjecture.

After centering and scaling, the length of the longest weakly increasing subsequence of a finite-state Markov random word converges to a Brownian last-passage functional with dependent components. We investigate when this functional, and more generally the limiting RSK Young-diagram shape, admits a Gaussian random-matrix representation. We introduce Hermitian Gaussian matrices with correlated diagonal entries and show that isotropy of the Brownian covariance on the trace-zero hyperplane yields such a representation. Within this natural class of Hermitian models with independent Gaussian off-diagonal entries, we prove a rigidity theorem showing that isotropy, together with equality of the off-diagonal variances, characterizes the conditional largest-eigenvalue representation. Explicit three-letter non-cyclic Markov chains demonstrate that this representation is not universal: the limiting Brownian functional may fail to have the law of the largest eigenvalue of any centered jointly Gaussian $3\times 3$ Hermitian matrix. Finally, we extend the random-matrix identification to chains with non-uniform stationary distributions and to countable state spaces.

We study quantum-assisted balanced rendezvous on the $d$-dimensional hypercube in a one-step two-agent quantum-walk framework, where the two agents move the same Hamming distance and meet at a common midpoint. The key structure is a projective $t$-sphere resolution, which governs exact finite-dimensional quantum attainment. Its scalar analogue recovers the corresponding classical characterization. We characterize finite-dimensional quantum attainment of the no-signalling benchmark, construct quantum strategies from projective sphere resolutions, and establish a rigidity theorem for optimal finite-dimensional strategies. A marked difference appears between nearest-neighbor and higher-cardinality updates: quantum advantage can occur in the former, whereas in the latter the geometry of the relevant Hamming spheres rules out exact finite-dimensional attainment of the no-signalling value. Together, these results connect quantum-walk dynamics, Hamming-sphere geometry, and quantum rendezvous within the quantum Bernoulli noises (QBN) framework.

Permutons can be thought of as "continuum limits" of permutations of diverging size, which encode information about the number of occurrences of any given finite subpattern. We construct a natural composition operation on permutons, analogous to composition of permutations, and explore its properties via a correspondence between permutons, Markov kernels, and Markov operators. We show that no extension of permutation composition to permutons can be (jointly) continuous with respect to the usual "weak" topology on permutons, but we construct a "strong" topology with respect to which composition is jointly continuous. Moreover, we characterize strong convergence of sequences of permutations of diverging size: a sequence of permutations converges strongly if and only if it converges weakly and its limit is the transpose of the graph of a measure-preserving function. The key insight behind these results is to apply functional analysis to certain operators associated to permutons; with this shift in perspective, our results have elegant and relatively "soft" proofs. As an application, we describe the permuton limit of a product of finitely many independent uniformly sampled separable permutations and give a formula for the expectation of this limit. We also show that in this case, as the number of compositions goes to infinity, the aforementioned expectation tends to the uniform permuton.

For a non-singular quadratic form $F$ in four variables, we study integer solutions to $F=0$ with an arbitrary weight attached to one of the variables. This allows us to investigate genuinely sparse subsets of the corresponding variety. Our main input is to obtain cancellation in sums of character sums over the moduli arising in the application of the DFI's delta method to the quadratic hypersurface. This cancellation relies crucially on the equidistribution of quadratic roots and the Weil bound for Kloosterman sums.

Conformal prediction provides finite-sample, distribution-free coverage guarantees that can support decision-making in specific settings. In multistage problems, earlier decisions determine the populations relevant to later calibration. When later scores are refit using earlier calibration results, reusing those cases for ranking need not preserve exchangeability with a surviving test case. We study sequential conformal calibration with a fresh fold at each stage: conditional on earlier folds, the selected fold scores and the surviving test score are exchangeable. This yields finite-sample coverage guarantees for every prefix of the decision sequence. For atomless survivor score distributions, we derive a universal Binomial-thinned Beta chain for the realized coverage process. Its law depends only on fold sizes and target levels, even under predictable refitting, and yields exact coverage slack, abstention probabilities, and calibration-conditional guarantees. We further characterize the tradeoff between freshness and data reuse. For fixed scores, strict-survivor reuse is exactly valid. For adaptive scores refit using past calibration data, reusing survivors can incur worst-case rank loss, while blocked designs provide an intermediate option between full reuse and stagewise fresh folds. As a case study, we apply the framework to serial multi-echelon inventory problem and propose a sequential conditional-quantile formulation. Each stock level is learned as a context-dependent quantile of cumulative demand among scenarios satisfying earlier no-backlog constraints. We connect these targets to common-shift stationarity and derive decision-level implications for the resulting calibrated policies. In a three-echelon simulation, fresh-fold refitting achieves 81.98% final-prefix coverage against an 81.82% target while reducing mean cost by 5.5% relative to the Bonferroni baseline.

We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial $Q$-flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao–Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.

We study the eigenvector-weighted spectral statistics \[ T_n(\varphi,g)=p^{-1}tr\{\varphi(\boldsymbol{S}_n)g(\boldsymbol{\Sigma})\} \] of large-dimensional sample covariance matrices \(\boldsymbol{S}_n\) with population covariance \(\boldsymbol{\Sigma}\). These statistics combine functions of sample eigenvalues with sample–population eigenvector overlaps and arise in statistical losses that depend on both. As the dimension and sample size grow proportionally, we establish a joint central limit theorem for \(\{T_n(\varphi_a,g_a)\}_{a=1}^r\) with multiple analytic test functions and population weights, centered by finite-dimensional deterministic equivalents. A key intermediate step is a joint Gaussian process limit for the weighted resolvent traces \[ m_n^g(z)=p^{-1}tr\{(\boldsymbol{S}_n-z\boldsymbol{I}_p)^{-1}g(\boldsymbol{\Sigma})\}. \] The limiting means and cross-covariances are expressed in terms of the population spectral distribution. We apply these results in two settings: one concerns the variance component of minimum-norm prediction risk, and the other concerns the Frobenius loss of fixed rotationally invariant covariance estimators. For fixed affine shrinkage rules, the loss means, variances, and cross-covariances have explicit formulas in terms of population spectral moments.

Topology optimization typically involves a geometric volume/mass constraint due to the limited use of material. As a popular technique for topology optimization, the level-set method typically employs the Lagrange method, the penalty method, or the augmented Lagrangian approach to address the volume constraint. This requires careful parameter tuning and may lead to non-monotonic changes in the objective. In this paper, we present a new volume-controlled level-set method for topology optimization. By imposing a scalar weighted volume?compatibility constraint in a modified Stokes-type extension-regularization problem, the velocity is made volume compatible at the beginning of each frozen transport interval. At the continuous level, we prove that the initial rate of change of the regularized volume vanishes and that the subsequent frozen-velocity drift is of order $T^2$; under a narrow-band assumption, the corresponding constant is independent of the smoothing parameter. For the discontinuous Galerkin spatial semi-discretization of the level-set transport equation, we derive a quantitative regularized-volume drift estimate that displays the dependence on the smoothing parameter, spatial mesh size, transport time, and discrete velocity. The effectiveness of the present algorithm is demonstrated through the minimum compliance problem for linear elasticity, the eigenfrequency maximization for structural vibration, and the minimum energy dissipation problem for Stokes flows.

We compare the $p$-adic GKZ hypergeometric complex with the hypergeometric arithmetic $\mathcal D$-module defined via six functors. Under the $p$-adic semi-nonresonance condition, the natural comparison morphism is a quasi-isomorphism. We prove concentration of the $p$-adic GKZ hypergeometric complex in degree $n$, establish overholonomicity under another non-Liouville hypothesis. When $(q-1)\gamma\in\mathbb Z^n$, we construct a Frobenius structure on its $n$-th cohomology and prove that its restriction to the nondegenerate locus is an overconvergent $F$-isocrystal of rank $n!\operatorname{vol}(\Delta)$.

AI shopping assistants increasingly influence which products consumers consider. We study retail assortment planning under AI mediation, where a retailer chooses a catalog, an AI intermediary forms a customer-facing shortlist, and the customer chooses from that shortlist according to a multinomial logit model. We consider both deterministic mediation, in which the AI follows a fixed segment-specific ranking, and stochastic mediation based on the Plackett-Luce model. Empirically, we construct a dataset of product rankings from leading AI models, estimate Plackett-Luce ranking models, and evaluate their predictive performance on held-out assortments. Combining these estimates with synthetic customer preferences, we find that accounting for AI mediation can materially change optimal assortments and revenue. Across the tested capacity configurations, ignoring mediation yields mean revenue losses between 2.52% and 11.37% relative to the PL optimum. A much simpler deterministic plug-in, based only on the ordering of fitted AI utilities, performs substantially better, with mean losses of at most 0.53%. Motivated by this performance, we study assortment optimization under deterministic mediation. We characterize its computational complexity and show that, for any fixed number of customer segments, the problem admits a fully polynomial-time approximation scheme.