Papers

Let $K$ be a field, and $\mathcal{T}$ a $K$-linear essentially small triangulated category equipped with an extendable $t$-structure $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$ with respect to a good metric $\mathfrak{B}$. Given a torsion class $\mathcal{X}$ in the heart of $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$, we prove that lifting the HRS-tilt of $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$ at $\mathcal{X}$ along the completion of $\mathcal{T}$ coincides with the HRS-tilt of the lifted $t$-structure at the completion of $\mathcal{X}$. As an application, we provide the compatibility result between left silting mutation in $\mathcal{T}$ and HRS tilting in the ambient completion.

A switchback experiment alternates an entire system between treatment and control over time. It is especially useful when interactions between units can undermine standard unit-level experiments. Switchback experiments are commonly implemented on fixed temporal grids, which impose highly structured restrictions on when treatment can switch. We study a broader class of random-duration switchbacks, in which treatment and control alternate across runs whose durations are drawn from a common distribution. Under finite carryover, we show that the mean squared error of the Horvitz-Thompson estimator has a simple frequency-domain representation governed by how temporal outcome patterns align with design-induced imbalance and contamination patterns. This representation yields a tractable worst-case design criterion that can be computed directly from the duration distribution. Optimizing over even a simple two-rate family produces a run-age-dependent switching rule that reduces the asymptotic worst-case mean squared error by at least 32% relative to the standard independently randomized fixed-block switchback.

Dispersive and Strichartz estimates are fundamental tools for establishing the well-posedness and long-time behavior of solutions to nonlinear partial differential equations. While these estimates are well-understood in the boundaryless Euclidean setting, the presence of a geometric boundary introduces severe analytical complexities, such as the continuous formation of caustics. In this work, we establish sharp local-in-time dispersive estimates for the semiclassical Schrödinger equation inside a three-dimensional cylindrical domain $\Omega \subset \mathbb{R}^3$ subject to homogeneous Dirichlet boundary conditions. This paper provides the first comprehensive microlocal treatment for this anisotropic geometric setting, extending the optimal strictly convex boundary results of Ivanovici \cite{Ivanovici2023} to the parabolic setting. The primary analytical challenge in our cylindrical model stems from the fact that the boundary curvature is non-uniform, depending explicitly on the tracking angle of classical trajectories and vanishing identically along the flat longitudinal axis. Crucially, we demonstrate that the quadratic structure of the Schrödinger phase function ($\partial_\zeta^2 \Phi = 2t$) establishes global non-degeneracy, completely bypassing the arduous low-frequency trajectory ray-tracing mandatory in hyperbolic wave equations. By exploiting this structural advantage alongside a streamlined Littlewood-Paley dyadic block decomposition, we prove that the zero-frequency axial tail can be consistently integrated down to the flat limit $\eta=0$. This yields sharp global Strichartz estimates featuring an explicit derivative loss exponent of $\rho(q) = \frac{3}{2}\left(\frac{1}{2}-\frac{1}{q}\right)$.

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz–Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz–Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

We prove a Jacobian estimate underlying both the Heintze–Karcher comparison and the Alexandrov–Bakelman–Pucci method for submanifolds of arbitrary codimension. The estimate retains the contributions of ambient curvature along geodesics emanating from the submanifold and requires no curvature sign assumption. As applications, we obtain a quantitative Fenchel–Willmore inequality with explicit curvature remainders under nonnegative $n$-intermediate Ricci curvature, Michael–Simon Sobolev and isoperimetric inequalities under nonnegative $(n-1)$-intermediate Ricci curvature, and corresponding extensions under quadratic curvature decay. In each result, the required intermediate Ricci curvature condition depends only on the dimension of the submanifold and is independent of its codimension.

We give a new description of the differential calculus on the quantized irreducible flag manifold given by Heckenberger-Kolb. We also define a quantum analogue of the Dolbeault complex associated to an equivariant vector bundle on the irreducible flag manifold, and show that its cohomology groups coincide with the cohomology groups given by Andersen-Polo-Wen defined using the induction functor.

We study two closely related topics over algebraic surfaces: stability of extensions of stable sheaves by stable sheaves, and maximal subsheaves of a given stable sheaf. For the former, we provide a construction of complete families of sheaves via extensions and prove the stability for some cases. The construction enables us to prove certain cases of Weak Brill–Noether over rational surfaces. The second topic leads to a refinement of the Segre stratification of the moduli of sheaves. We obtain the expected dimension of certain refined Segre strata over rational surfaces and K3 surfaces. Over the projective plane, we prove our main result that all refined Segre strata corresponding to line bundles are irreducible of the expected dimension and are nested under taking closures. As a consequence, we obtain more cases of the stability of extensions. Our results suggest that refined Segre strata behave much better than Brill–Noether strata. Our study of the refined Segre strata relies crucially on Bridgeland stability conditions and the work of Li and Zhao.

Modern real application problems involve matrix-valued parameters, yet conventional optimizers treat them as vectors, thereby motivating matrix-aware methods that exploit input-output geometry, such as Muon which orthogonalizes the momentum matrices before parameter updates. Its empirical success raises a conceptual question: can orthogonalized momentum remain effective beyond smooth optimization? This paper studies this question for locally Lipschitz functions using a generalized derivative framework compatible with backpropagation. Our first contribution is to identify a key limitation: for every fixed momentum factor $\beta\in[0,1)$, Muon can fail to approach the global optimal solution of a convex Lipschitz objective from almost every initialization, when step sizes adapt to the full gradient history. The failure can occur even along bounded iterates. Our example is inspired by the one of Parshakova et al. which only covers $\beta\in[0,\frac{1}{2})$. Then, we show that the obstruction lies in fixed momentum rather than orthogonalization. Indeed, when the momentum factor is adaptive and approaches 1 together with a vanishing step size, Muon recovers asymptotic convergence for nonconvex nonsmooth optimization under the boundedness and regularity conditions. Moreover, we propose MAGD, which combines orthogonalized momentum with gradient, weighted based on their relative progress. MAGD retains asymptotic convergence in nonconvex settings and achieves an $O(\min\{m,n\}\epsilon^{-2})$ rate in convex settings. A lower bound shows the optimal dimension dependence. Experiments on synthetic problems, image classification, and LLM pretraining show MAGD is a simple and practical alternative to Muon. Together, our results characterize when orthogonalized momentum fails without smoothness and how it can be made reliable and we hope that the analysis may be useful more broadly.

For fixed nonnegative integers $m_1,m_2$, let $A_n=(a_{j-i})_{i,j=1}^n$ be the leading $n\times n$ section of a Toeplitz matrix with lower and upper semibandwidths $m_1$ and $m_2$. We give two constructive Laplace-expansion methods for scalar recurrences of $\det A_n$ and $\perm A_n$. The increasing-rows method eliminates a fixed family of boundary cofactors and gives recurrence order at most $d=\binom{m_1+m_2}{m_1}$ for both sequences. The row-column method closes normalized boundary minors recursively and packages them in a sparse transfer matrix. Its reachable states are classified exactly: level $j$ is indexed by a pair of $j$-subsets of $[m_1]$ and $[m_2]$. Hence the transfer dimension is $d$, and we obtain an explicit formula for the number of nonzero transitions. For determinants, the complementary cofactors of the increasing-rows construction are coordinates of the classical compound companion representation. The independently constructed row-column transfer has the Widom characteristic polynomial and is generically similar to the compound transfer. Thus the order $d$ recurrence is generically minimal for the unrestricted fixed-band determinant family. For permanents the same state graph gives the binomial upper bound, without a general minimality claim. The pentadiagonal case recovers Sweet's order-six determinant recurrence and its permanent analogue, while the one-superdiagonal family gives closed scalar recurrences and rational generating functions. Position-dependent band weights preserve the finite state graph but replace the constant transfer by a cocycle. For cyclic determinants, Fourier diagonalization produces all subset products of the symbol roots and a generically minimal annihilator of degree $2^{m_1+m_2}$, corresponding to the passage from one exterior degree to the full exterior algebra.

The basic reproduction number $R_0$ is the central dimensionless quantity in mathematical epidemiology, characterizing the threshold for disease outbreak and the early growth rate of an epidemic. The algebraic form of $R_0$ varies widely across models of distinct transmission mechanisms, and its interpretation can yield further biological insight into transmission dynamics and disease intervention. We present a structural taxonomy of $R_0$ for compartmental ordinary differential equation models spanning a range of disease transmission features, including staged progression, differential infectivity, competing strains, vector-borne transmission, sexually transmitted infection, recurrent infection, maternal transmission, and gene drive inheritance. For each structural class, we provide a worked example, deriving $R_0$ via the next-generation matrix approach and interpreting the resulting algebraic expression in terms of the underlying transmission pathway. By unifying disparate derivations under a single structural framework, we clarify when and why particular algebraic forms of $R_0$ arise. This review serves both as a research synthesis and as a self-contained pedagogical reference for students and researchers entering the field.

For a fixed integer $k\ge 2$, let $W_{2k+1}=K_1\vee C_{2k}$ be an odd wheel graph. The fixed-size spectral extremal problem aims to determine \[ \operatorname{spex}(m,W_{2k+1}):=\max\{\rho(G): e(G)=m,\ G \text{ is } W_{2k+1}\text{-free}\}, \] where $\rho(G)$ denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [18] proposed the following conjecture: For large $m$, every $W_{2k+1}$-free graph of size $m$ satisfies $\rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2}$ with equality precisely for $K_k\vee qK_1$ and $m-\binom{k}{2}=kq$. Very recently, Fang, Zhai and Zhang [7] confirmed the Yu–Li–Peng conjecture. When $m$ is large, $k\ge 2$ and $m-\binom{k}{2}$ is not divisible by $k$, the exact solution for the above problem is still open. Regarding this problem, Yu, Zhang, and Zhang [19] proposed the following conjecture: Let $r$ be a nonzero remainder when $m-\binom{k}{2}$ is divided by $k$. Then $S_{k,m}$ is the unique graph among $W_{2k+1}$-free graphs of size $m$ having maximum spectral radius, where $S_{k,m}$ is obtained from $K_k\vee qK_1$ by adding a vertex $z$ and joining it to exactly $r$ vertices of the $K_k$. In this paper we address this problem in each nonzero residue class. Our result completely settles the Yu-Zhang-Zhang conjecture for $k\ge 3$.

What can a finished text reveal about the process that produced it? Drawing on Turing's imitation game and statistical decision theory, this article examines the limits of AI-text detection as an inference from a completed object to an unobserved history. For two known source distributions with equal prior probabilities, a standard identity gives the minimum average classification error as half their probability overlap. This overlap is one minus their total variation distance. Perfect detection therefore requires nonoverlapping distributions; useful discrimination does not. In practice, the problem is harder: human and machine writing form changing families of distributions, posterior probabilities depend on base rates, and “AI-written” becomes ambiguous when people and software contribute to the same text. Turing's interrogator can ask another question; a detector restricted to finished prose cannot. Distinguishability, source attribution, authentication, and compliance are different inferential tasks. The paper's own documented human-AI provenance shows what a production history can reveal that a binary label cannot.

Let $\mathrm{RM}_q(r,m)$ denote the generalized Reed–Muller code of order $r$ and length $q^m$ over the finite field $\mathbb{F}_q$. We completely determine the weight spectra of $\mathrm{RM}_3(2m-i,m)$ for $i=3,4$, $\mathrm{RM}_4(3m-i,m)$ for $i=2,3$, $\mathrm{RM}_5(4m-4,m)$, and $\mathrm{RM}_7(6m-4,m)$, in the ranges of $m$ specified in the corresponding results. The proofs proceed by induction on $m$, using a sumset inclusion for GRM weight spectra together with results on low-weight codewords. The required base cases are established by computations in Magma and, for certain ternary cases, exact constraint solving with Z3.

In this paper, we study the boundary regularity of harmonic map heat flows from a bounded domain in a metric measure space with generalized Ricci curvature bounded from below into a non-positively curved metric space.

This work addresses decentralized online Riemannian optimization on Hadamard manifolds. Prior work under geodesic convexity (g-convexity) may require curvature information in the optimization analysis, typically through a finite lower bound on the sectional curvature. Curvature may also enter the step size or contraction factor of tangent-space Riemannian consensus schemes. In this work, we relax the curvature dependence for a narrower class of horospherical convex (h-convex) functions. We study Distributed Riemannian Online Gradient Descent (D-ROGD), which combines local Riemannian h-subgradient updates with an implicit Fréchet-mean consensus. For h-convex and strongly h-convex local objectives, we establish $O(\sqrt{T})$ and $O(\log T)$ static regret, respectively, matching the corresponding Euclidean rates with respect to $T$, with network dependence governed solely by the spectral gap. To our knowledge, these are the first curvature-independent regret guarantees for decentralized online optimization on Hadamard manifolds. Experiments on hyperbolic embeddings corroborate the predicted rates, with no observable degradation due to curvature.

The AES S-box is constructed from finite field inversion followed by a fixed affine transformation. Since inversion possesses intrinsic Frobenius symmetries among its coordinate realizations, we study how these basis symmetries are altered by outer affine transformations. We first develop a deterministic rigidity criterion for transformed inversion and apply it to the AES S-box. This shows that the linear part of the AES S-box affine transformation alone makes the transformed inversion map basis rigid. We then investigate the corresponding generic problem when the outer invertible linear transformation varies. The existence of a nontrivial linear stabilizer is reduced to a conjugacy problem for semilinear candidates arising from two sided linear equivalences of inversion, which we characterize in terms of relative norms and Frobenius orbits. We also determine the dimensions of the associated centralizer algebras exactly. These structural results imply that, for a uniformly chosen outer linear transformation, the probability that the linear stabilizer is nontrivial is bounded by $2^{-\Omega(n^2)}$, with sharper finite dimensional bounds obtained from the exact conjugacy condition. Computational experiments independently verify the AES rigidity result, the conjugacy and centralizer formulas, and the finite dimensional estimates in small dimensions.

We classify the polyominoes obtained by adjoining four straight arms to a single square, allowing zero arm lengths, according to their ability to tile rectangles, half-strips, bent strips, quadrants, strips, half-planes, and the plane. We also classify their ability to tile an integer enlargement of themselves. Tiles occupy whole square-grid cells; translations, rotations, and reflections are permitted. Exactly five capability profiles occur. For the family $P(n,1,1,0)$, the rectangle profile holds for $n\le3$ and the bent-strip profile, with no half-strip or rep-tiling, for every $n\ge4$. A cross with four positive arms tiles the plane precisely when two opposite arms have length one; it never tiles a half-plane. Explicit periodic constructions and geometric obstructions are combined with finite symbolic case certificates. A Lean 4 development verifies the full classification for every natural four-tuple, including the interpretation of the certificates as statements about arbitrary infinite tilings. The account incorporates the author's 2020–2021 L- and T-polyomino work, reconstructs Dahlke's gun argument, and documents the subsequent AI-assisted proof development and formalization.

We prove sharp dispersive $L^1 \to L^\infty$ estimates for the three-dimensional attractive Coulomb operator $H_Z=-\Delta-Z|x|^{-1}$, where $Z>0$. The absolutely continuous part of the Schrödinger evolution decays at the free rate for short times, whereas its leading contribution decays like $|t|^{-1}$ for long times, with amplitude proportional to $Z$. This slower decay is driven by the threshold and is sharp when $Z^2|t|\gg1$.

We study the spherical statistics of all the points in a thin shell of a high-dimensional random lattice. The exact relations between lattice points make it unclear when predictions based only on spherical geometry should hold. We answer this question for several statistics, including the number of shell points, the balance of their directions, and the occurrence, repetition, and distribution of differences between them. We identify sharp thresholds as the shell radius grows and show that these statistics undergo several distinct phase transitions. A shell can already agree with one geometric prediction while still differing strongly from another. Our results hold for a single sampled lattice, with probability tending to one as the dimension grows. They include estimates that hold across a complete shell, bounds close to the transition thresholds, and extensions to randomly shifted lattices. A Lean formalization verifies the main results, assuming the classical formulas and probability model stated in the code.

Let ${R^\ast}(n)$ be the least radius of a central circle to which nonoverlapping circles of radii $1,\ldots,n$ are externally tangent. We prove that ${R^\ast}(n)={C_\ast} n^2+o(n^2)$ and characterize ${C_\ast}$ by finite linear programs with an explicit error tending to zero. The reduction preserves arbitrary orders and all pairwise constraints: the limiting problem places marked points on a line at pairwise separation at least the geometric mean of their marks. Concatenation with a bounded boundary cost proves existence, and balanced finite-word programs supply matching effective upper and lower bounds. Their certified gap is $(1/k+1/r)/\pi$, before directed arithmetic error, for $k$ mark types and words of length $r$. A quantitative reflected-block recovery theorem supplies genuine permutations and full ring geometry, with a countable extension and a strict four-block improvement. The explicit interval is $C_{\mathrm{term}}+\eta_{\mathrm{width}}\le{C_\ast}\le U_4$; neither endpoint is asserted sharp. In particular, the coefficient $1/8$ proposed in the preceding finite study is false. An elementary expression for ${C_\ast}$, efficient high-precision evaluation and global floating-circle structure remain open.

We establish the first pointwise convergence result for ergodic averages with iterates along explicit and deterministic sequences of intermediate growth, that is, growing faster than any polynomial but slower than any exponential. In particular, we show that the sequence $(\lfloor \exp((\log n)^c)\rfloor)_{n\in\mathbb{Z}+}$, with $c\in(1,8/7)$, is universally $L^p$-good for every $p\in(1,\infty]$. This gives an affirmative answer to an open problem dating back to the mid 1980s and contributes to Bellow's program, initiated in the earlier part of the same decade, on the characterization of $L^p$-good sequences in pointwise ergodic theorems. The proof combines the so-called one-frequency circle method with a delicate application of Vinogradov's method for estimating exponential sums whose phases involve $\big(\lfloor \exp((\log n)^c)\rfloor\big)_{n\in\mathbb{Z}_+}$. An interesting feature of our analysis, reminiscent of estimates arising in the study of the zero-free region of the Riemann zeta function, is that the argument relies on the classical Vinogradov method, in the sense that it necessitates estimates on the number of solutions for the Vinogradov system of Diophantine equations with explicit dependence on the system's parameters.

We construct a projective log-smooth surface pair $(X,\Delta)$ and a morphism $f:X\to E$ to an elliptic curve such that $f_*\OO_X(K_X+\Delta)$ is Atiyah's indecomposable rank-two bundle $F_2$ of degree zero. This gives a negative answer to Question 3.4 of Jiang \cite{Jiang}. We also construct a log canonical pair $(X,\Delta)$ and a morphism $f$ to an abelian surface such that $f_*\OO_X(m(K_X+\Delta))$ does not admit a Chen–Jiang decomposition for any integer $m\ge2$. Products yield counterexamples in every higher direct image degree and over abelian bases of arbitrary positive dimension.

This paper studies joint sampling, route activation, and preemption control over $N$ heterogeneous transmission routes under a long-run objective combining Age of Information (AoI) with sampling and transmission costs. We formulate the problem as an average-cost impulse-control model, derive a vector-form integral average-cost optimality equation over a hybrid state space, and establish structural reductions on the state. We consider a two-route case and compute optimized policies using discretized policy iteration. Surprisingly, the results show that preemption can reverse routing preferences: the optimized policy may exclusively use a route with both a larger mean delay and a larger delay variance while avoiding the seemingly superior route. Relative to baselines that do not allow preemption, the proposed policy reduces average AoI by up to $96\%$.

We introduce standard-norm generalized gyrovector spaces: generalized gyrovector spaces whose norm-value line has the ordinary real operations and whose fixed scalar maps are continuous. We prove that every bijective gyrometric isometry between nonempty connected open subsets of two such spaces extends uniquely to a global bijective gyrometric isometry. The proof combines gyromidpoints, point reflections, continuation of local isometry germs, and monodromy; joint continuity of scalar multiplication is derived, rather than assumed. The class includes all nonzero real normed spaces and standardized Möbius, Einstein, and proper-velocity models, and is stable under finite and countable $\ell^p$-products. We also obtain bounded mapping-space examples and, under natural continuity assumptions, standard-norm function-space GGVs $C(K,G)$ for compact $K$. These yield non-Hilbertian and hybrid examples. For every unital JB-algebra, the positive invertible cone carries a one-parameter family of standard-norm GGV structures whose gyrometrics are conjugate to Thompson's metric by power maps. The projective positive cone has a parallel family, based on the variation norm modulo the unit, conjugate to Hilbert's projective metric. The projective structures are contractible. Together with global JB-cone isometry classifications, the abstract theorem yields local-to-global extension and classification results for Thompson and Hilbert geometries. Full-rank density matrices and normalized state spaces of finite-dimensional Euclidean Jordan algebras, including the Albert algebra, give concrete projective examples.

We present an analytical solution for 1D atmospheric pollutant dispersion in a bounded domain. Using coordinate transformation and spectral decomposition, we obtain a Fourier series solution for time-dependent wind velocity. For constant wind, Laplace inversion yields dual representations via residue calculus and Poisson summation. The solution extends to semi-infinite domains, recovering classical error-function profiles.

We give an explicit extremal function for one step of gradient descent when the gradient error is bounded relative to the true gradient. The function is the sum of a quadratic in one coordinate and a Huber function in the other. It attains, for every admissible relative error level and every stepsize in the intermediate regime, the proposed worst-case constants for both the final squared gradient divided by the initial objective gap and the final squared gradient divided by the objective decrease. This proves Conjecture 2.6 of Vernimmen and Glineur, first stated in June 2025. A change of variable reduces the parameter description to a scalar cubic with a rigorously specified root. We prove all parameter inequalities, derive a nonnegative-square certificate for the upper bound, and verify the matching construction directly. We also characterize the exact one-dimensional worst-case value and prove that two dimensions are necessary in the open intermediate regime.

Products of positive semidefinite matrices arise when successive steps of a randomized optimization method use different data points. Their average can have a negative eigenvalue even though every individual matrix is positive semidefinite. Lai and Lim proved at ICML 2020 that $\sum_{i\ne j}A_iA_j\succeq-\frac{1}{4}n(n-1)I$ whenever $A_i\succeq0$ and $\sum_i A_i\preceq nI$, and conjectured that the constant is sharp for every $n\ge2$. We give an explicit family of $n$ rank-one matrices in dimension $n$ that attains this bound for every $n$, proving that sharpness statement. The construction compresses one direction of an orthonormal basis by a factor $1/\sqrt{2}$. We also give a short, self-contained proof of the known inequality and classify all equality cases: every extremizer contains a common invariant $n$-dimensional block equivalent to the construction. Consequently, dimension $n$ is necessary for equality, and positive definite matrices approach but never attain the lower endpoint. Finally, strongly convex quadratic objectives realize the extremal negative eigenvalue as a two-step expected gradient-descent update.

We study relative global dimensions of extensions $B \subseteq A$ of Artin algebras over a perfect field. A controllable extension is introduced as one for which the relative global dimension is determined by the ordinary global dimension of the quotient $A/AJ(B)A$. General inequalities and sufficient conditions for controllability are established, including the case in which $J(B)$ is a two-sided ideal of $A$. The interaction between relative global dimensions and relative projective dimensions of the regular bimodule is then studied. The class of homologically controllable extensions is introduced, and it is shown to be closed under tensor products. Consequently, for homologically controllable extensions $B \subseteq A$ and $D \subseteq C$, we obtain \[\operatorname{gldim}(A \otimes C, B \otimes D)=\operatorname{gldim}(A,B) + \operatorname{gldim}(C,D).\] The proof relies on a relative Künneth-type result for tensor products of complexes, which may be of independent interest. Finally, controllable extensions with prescribed relative global dimension are constructed, along with families of non-controllable extensions for which the relative global dimension differs arbitrarily from the global dimension of $A/AJ(B)A$.

Scalar rounding decisions interact through matrix multiplication. We study deterministic product-aware rounding after scales, clipping bounds, and grids are fixed, with each active scalar choosing between adjacent levels. For dynamic activation rounding, null-space reduction preserves the relaxed product while leaving at most $r$ fractional decisions, where $r$ is the rank of the active gap-weighted weight block. Conditional-expectation completion gives a deterministic polynomial-time algorithm with squared product error at most $ \mathrm{OPT}_{\mathrm{dyn}}+r\nu_{\max}^2/4$, where $\mathrm{OPT}_{\mathrm{dyn}}$ is the best admissible error and $\nu_{\max}$ is the largest row norm of that block. For reusable static weights, the exact expected product-loss metric is the uncentered input second moment with fixed output bias; free bias recalibration yields the centered covariance. Exact optimization is NP-hard even at rank one. In balanced blocks with $K=1024$ and $r=16$, conditional- expectation completion attains a dither-normalized median error of $0.010$, compared with $0.899$ for round-to-nearest. Clipping-aware initialization reduces median normalized error by a factor of $43.4$ at ten-percent clipping. Held-out Digits experiments show that retaining the input mean or correcting the output bias improves median product error over round-to-nearest in all four tested bit-width and calibration-size settings.

Modern design optimization frameworks aim first and foremost for models with the most accurate predictions without balancing computational overhead. It remains a reason why scaled architecture and multidisciplinary design optimization problems are difficult to address, even with sample-efficient Bayesian optimizers. In this paper, a metric quantifying the computational energy footprint is introduced within a Bayesian optimization framework to guide the parameter setting of a model towards configurations that balance both performance and frugality. The computer experiments highlighted existing tradeoffs between optimum convergence and the underlying energy footprint, and sometimes resulted in both a better-found optimum and lower energy consumption.