Papers

We prove a boundary analogue of the rigidity theorem for Ricci-pinched three-manifolds under a superquadratic volume-growth assumption. More precisely, we show that a complete connected three-manifold with convex boundary, pinched Ricci tensor, and pinched second fundamental form is isometric to the Euclidean half-space. The proof is potential-theoretic. We construct and study mixed Dirichlet–Neumann $p$-capacitary potentials, establish monotonicity formulas for their free-boundary level sets, and use a weak Gauss–Bonnet formula adapted to this setting.

In this article, we establish central limit theorems for sample Fréchet means of stationary ergodic Markov chains taking values in manifolds, extending the asymptotic theory previously developed for independent observations to a class of dependent manifold-valued processes. Our results derive the asymptotic normality of the sample Fréchet mean from a central limit condition at the population Fréchet mean, under suitable local regularity conditions. We further provide sufficient geometric and probabilistic conditions under which these assumptions hold, formulated in terms of curvature bounds and a Wasserstein mixing condition. As an application, we establish a central limit theorem for sample Fréchet means for a class of random dynamical systems generated by contractive random maps.

We prove that every three-dimensional parallelohedron $P$ satisfies \[ \mathcal H^2(\partial P)\ge 12\sqrt2\,r(P)^2, \] where $r(P)$ is the inradius of $P$, with equality if and only if $P$ is similar to the regular rhombic dodecahedron.

Let $\Avar_n$ and $\VV_n$ be the varieties generated by all $n$-element additively idempotent semirings (ai-semirings) and all $n$-element semirings with commutative addition, respectively, in the signature $(+,\cdot)$ without constants. We prove that both varieties are finitely based if and only if $n\leq2$. The five-element case follows from cyclic identities and a structural analysis of products of squares. For $n\geq6$, each variety contains a finite inherently nonfinitely based ai-semiring. We also prove $\VV_n\cap\AI=\Avar_n$ for every $n$, where $\AI$ is the variety of all ai-semirings.

Let $\Delta$ be a reflexive four-polytope with primitive edges. We construct deformations of the toric fourfold defined by its face fan that smooth the singularities of a generic anticanonical Calabi-Yau hypersurface over a chosen two-dimensional face. The construction applies when the face cone is smoothable and the dual edge has an interior lattice point; its equations are trinomials in Cox coordinates. It gives a smoothing of the hypersurface when this is the only face whose cone is singular. If every singular face satisfies these conditions, simultaneous smoothing follows in the nodal case and, in general, under irreducibility of the semiuniversal deformation space. A seven-vertex example with one ordinary double point and no smoothing shows that the dual-edge condition is sharp. A scan of the Kreuzer–Skarke classification shows that the facewise criterion accounts for 3,774 of the 30,241 smoothings in the Batyrev–Kreuzer census; the other 26,467 require relations among exceptional curve classes from different faces.

We generalize de Rham's decomposition theorem to proper metric spaces: every proper metric space has a unique decomposition as the product of a Euclidean space and a finite or countable pointed $\ell^2$-product of nontrivial irreducible non-Euclidean factors. More generally, every complete metric space splits canonically into a Hilbert factor, a pointed $\ell^2$-product of irreducible non-Euclidean factors, and a factor with no nontrivial irreducible factors. These results partially answer a question of Ulam.

Given a finite preorder $\tau$, we study how its order relation determines the $h^*$-polynomial of the associated polar preorder polytope. Let $\mathcal R_{\tau}^{\vee}$ be the convex hull of the negative standard basis vectors and the indicator vectors of nonempty order ideals, and write $h_\tau^*(t)=h^*(\mathcal R_{\tau}^{\vee},t)$. Athanasiadis and Chapoton conjectured that this enumerative invariant takes ordinal sums to products: $$h_{\tau_1\oplus\tau_2}^*(t)=h_{\tau_1}^*(t)\,h_{\tau_2}^*(t).$$ We prove the conjecture for arbitrary finite preorders, including those with nontrivial equivalence classes. More precisely, we construct a positively homogeneous piecewise unimodular shear from the free sum $\mathcal R_{\tau_1}^{\vee}\oplus\mathcal R_{\tau_2}^{\vee}$ onto $\mathcal R_{\tau_1\oplus\tau_2}^{\vee}$, giving lattice-point bijections at every nonnegative real dilation scale. The key is a formula for the Minkowski functional in terms of the least nonnegative order-reversing majorant: after the shear, the functional becomes the sum of the two factor functionals. Together with the Ehrhart product formula for free sums, this proves the conjecture and the corresponding product formula for normalized volume.

We disprove a weighted generalization of the union-set conjecture [Fra95] raised in [Gow16]. We construct nonnegative weights on subsets of a finite ground set satisfying $w(\varnothing)=0$ and $w(A\cup B)\ge\min\{w(A),w(B)\}$, while every coordinate belongs to an arbitrarily small fraction of the total weight.

We introduce Latin Eulerian numbers $\LE{n\atop k_1,\dots,k_n}$, a multivariate refinement of classical Eulerian numbers counting order-$n$ Latin squares by column ascents. We establish their fundamental symmetries, univariate marginals, and an exact multiplicity divisibility property. For the total ascent statistic $\Sigma(L)=\sum_i k_i(L)$, we prove the sharp, isolated bounds $n-1 \le \Sigma(L) \le (n-1)^2$, demonstrating that the adjacent values $n$ and $(n-1)^2-1$ are strictly unattainable. To analyze intermediate values, we show that symbol permutations typically ignored in classical enumeration directly govern $\Sigma$ via an exact formula. This insight yields explicit constructions for the interior range, motivates a unimodality conjecture for the total ascent distribution.

Inspired by the polymath Abraham Ibn Ezra (1089-1167) who approximated {\pi} by 20/9 times the square-root of 2, and contemporary scholar, RSJ Reddy, who gave (14-sqrt(2))/4, we point out that both of these approximations are optimal in some sense, and we generate many other ones that are even more optimal, albeit not as nice.

In this short note, we provide a simple proof of the irrationality of Euler's number $e$, without using the classical estimates based on continued fractions or Taylor series expansions.

We revisit the combinatorial loop-equation method for computing planar disc amplitudes in multi-matrix models with polynomial potentials. We develop a procedure which organises the Schwinger-Dyson constraints by the level of their boundary insertions and, when the resulting system closes, produces an algebraic equation for the planar disc function. Using this method we recover the disc partition functions associated with fixed, mixed, and free boundary conditions in the three-state Potts model, previously obtained by saddle-point methods. As a novelty we derive an algebraic equation for the fixed-boundary resolvent of the three-state Potts model with unequal quadratic and cubic couplings. We further show one can compute the disc partition function for several unsolved two- and three-matrix models. These examples demonstrate that loop equations provide a practical route to identifying and solving sectors of multi-matrix models beyond the standard one- and two-matrix cases.

This paper studies disjointness and \(\mathscr{F}\)-product recurrence with respect to zero-entropy systems. First, we prove that, for a point whose orbit closure has zero topological entropy, \[ \text{distality}\rightleftharpoons \Finf\PRzero\rightleftharpoons \Fpubd\PRzero \rightleftharpoons \Fps\PRzero \rightovernotleft \Fs\PRzero. \] We also construct a minimal topological dynamical system with uniform positive entropy which is disjoint from all zero-entropy \(M\)-systems but is not disjoint from some zero-entropy \(E\)-system, i.e., \(\Ezero^{\perp}\subsetneq \Mzero^{\perp}\), giving an affirmative answer to a question of \cite[W.~Huang, K.~K. Park and X.~Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France 135 (2007), 259–282]. Moreover, the same construction shows that there exists an \(\Fps\PRzero\) point which is not \(\Fpubd\PRzero\), i.e., $\Fps\PRzero$ $\nRightarrow \Fpubd\PRzero$, giving a negative answer to a question of \cite[P.~Oprocha and G.~H. Zhang, On weak product recurrence and synchronization of return times, Adv. Math. 244 (2013), 395–412].

We use stationary common-neighborhood properties to study highly connected Ramsey relations and partition hypotheses. For weakly compact $\kappa$, $\operatorname{Coll}(\omega_1,{<}\kappa)$ forces $\omega_2\to_{\mathrm{hc},<5}(\omega_2)^2_\omega$ and $\operatorname{PH}_1(\omega_2)$. If $\kappa$ is $T^{\kappa^+}_{\omega_1}$-Ramsey, the same collapse forces that every countable coloring of $[\omega_2]^2$ has a stationary set $X\subseteq\omega_2$ and a color $i$ such that every finite subset of $X$ has stationarily many color-$i$ common neighbors in $X$. From one weakly compact cardinal, we obtain a model of the ${<}5$-edge relation at $\omega_3$ and $\operatorname{PH}_1(\omega_3)$, in which $\check H^2(\omega_3,A_d)\ne0$ for every nontrivial abelian group $A$. This separates $\operatorname{PH}_1(\omega_3)$ from $\operatorname{PH}_2(\omega_3)$, with the exact consistency strength of one weakly compact cardinal. We also show that $\operatorname{PH}_1(\omega_2\times\omega_5)$ is equiconsistent with two weakly compact cardinals.

This paper introduces and investigates a non-classical positional numeral system constructed on the normalized Mersenne basis $M_k = (2^k-1)/3$, structurally linking the classical Mersenne sequence (OEIS A000225) with the integer sequence of alternating binary bitmasks (OEIS A002450). The main objective of this study is to provide a rigorous algebraic proof of the system's completeness—specifically, theorems on the existence and absolute uniqueness of a canonical representation for any natural number. We show that the internal dynamics of the basis are deterministically governed by the splitting identity $3M_k = M_{k+1} + 2M_{k-1}$, which strictly restricts the valid alphabet of coefficients to the set $\{0, 1, 2\}$. Based on the lemma concerning the supremum of lower-order digits and a constructive greedy algorithm, it is proven that the proposed basis forms a closed and unambiguous algebraic space, entirely precluding the overlap of positional masses.

High-order isogeometric discretizations produce a small number of non-physical outlier modes at the top of the discrete spectrum. These modes can pollute the numerical solution in wave propagation and structural dynamics. Most existing remedies modify the spatial discretization through outlier-free bases, mass lumping, subspace projection, and related constructions. We introduce MIISO (Modal Integrators for Isogeometric Stabilization of Outliers), a family of exponential Rosenbrock-Krylov methods of orders two through six that suppress the outliers in the time integrator and leave the spatial discretization unchanged. We construct a non-holomorphic amplification map by blending a Padé approximation that preserves physical oscillatory modes with a Pade approximation that damps high frequencies, with the blend placed at the known spectral location of the outliers. For orders three and higher the damping approximant is repaired so that the method is unconditionally A-stable. Physical modes receive no numerical dissipation and the outliers are driven to zero. The outlier handling and the method steps are fully matrix-free, and the method extends to nonlinear problems. Numerical experiments recover the predicted orders, match spatial outlier removal without changing the discretization, show lower cost than generalized-alpha at matched accuracy and in scaling, and demonstrate the method on nonlinear and three-dimensional problems.

We reconstruct the generalized hexagon of order $(2,8)$ from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of $F_4(\mathbb R)$ and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of $N(A_1^{24})$ with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which $L_3(2)\cong L_2(7)$ acts as on $\mathbf P^1(\mathbf{F}_7)$; two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as ${}^3D_4(2):3$, with a central factor $C_2$ for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.

Data assimilation aims to infer the state of complex dynamical systems based on observational data. However, accurate inference of the multimodal posteriors induced by nonlinear or non-injective observation operators remains a key challenge under high-dimensional and sparse observation conditions. Ensemble filters scale to high dimensions but are confined by restrictive distributional assumptions, while training-free generative filters (e.g., EnSF, EnFF) alleviate this limitation but may introduce structural errors and hinder information propagation under sparse observations. To address these issues, we propose a training-free, asymptotically exact posterior transport method. Firstly, a covariance-aware posterior transport scheme is designed, which embeds the forecast cross-covariance into flow-based transport and accurately recovers unobserved states while preserving the non-Gaussian posterior structure. Furthermore, the method combines a tractable observation-adaptive proposal with posterior correction, ensuring accurate approximation of the nonlinear posterior distribution. Finally, we establish the corresponding posterior flow theory, from which the asymptotic exactness of the proposed method relative to finite-ensemble surrogates and the Wasserstein error bound are derived. Experiments on Double-Well, Lorenz-96, and Kolmogorov flow show that the proposed method captures complex posterior structure and remains accurate under sparse, nonlinear, and non-injective observations. In the sparse non-injective setting, it reduces RMSE by 56% relative to the best baseline.

We construct finite-state Markov arrival-rate processes for which faster modulation increases the mean stationary workload of a stable single-server queue, disproving the universal ordering in Leskelä's formulation of Ross's second conjecture. A 64-state example has positive transition rates between every pair of states, independent service times uniform on $[0.99,1.01]$, and an explicit positive workload gap. A 128-state example additionally has strictly positive arrival-rate autocovariance at every lag. More generally, finite-state examples can exhibit any prescribed finite number of separated workload rises and falls. The proofs use a second-order expansion with a nonnegative, speed-uniform remainder under only a finite second service moment. We characterize the service distributions for which the second-order workload coefficient is nonincreasing for every bounded stationary environment: the function $s\mapsto\mathbb{E}(1-\cos(sB))/s^2$ must be nonincreasing. Mixtures of exponential service distributions satisfy this criterion. An exact Palm identity transfers the workload reversals to customer waiting times.

Self-calibrating monitors adapt their threshold online to guarantee a prescribed long-run false-alarm rate under arbitrary drift. We compute the price of that guarantee, stating every law with its exact domain of validity. First, the guarantee is an accounting identity, insensitive to what the monitor is meant to detect. Two evidence identities make the cost exact for the online quantile tracker: a persistent step of height $\delta$ yields excess alarm mass within one alarm of $\delta/\eta$, and exactly $\delta/\eta$ pathwise when $\delta$ is a lattice multiple of the gain $\eta$; a ramp of slope $c$ yields a stationary excess rate of exactly $c/\eta$, independent of accumulated size, up to a boundary $c=\eta(1-\alpha)$ coinciding with the alarm-rate cap. Second, the certificate's own fluctuation obeys an exact law: the windowed alarm rate has standard deviation of order $1/L$, not the binomial $1/\sqrt{L}$, since the windowed mass telescopes to a difference of a tight internal state; the closed-form constant is validated with no fitted parameter. Detectors calibrated on the binomial scale are miscalibrated by $\sqrt{\eta\varphi(q_0)L}$, and correct calibration turns detection windows from quadratic to linear in the inverse fault speed. Third, any monitor required to tolerate a drift class $\mathcal{D}$ is blind, at any horizon and for any rule, to every fault in $\mathcal{D}-\mathcal{D}$; the proof is a deliberately elementary two-point argument and the contribution is the object it identifies: for speed-bounded classes the blind set is exactly the doubled-speed class, and the tracker absorbs a speed class fixed by its own gain, so that under a certification regime declaring absorbed drift normal, the monitor manufactures $\mathcal{D}$. An exact Gaussian projection bound, sharper than Pinsker and never vacuous, quantifies power outside it.

In work of the first author, the geometric Satake equivalence was (non-rigorously) reinterpreted as an equivalence between two algebraically-defined $2$-categories, one built from representations of a Lie algebra, and one built using singular Soergel bimodules. It was then explained how to $q$-deform this equivalence in type $A$, replacing the special linear Lie algebra with its quantum group, and using singular Soergel bimodules for a deformed reflection representation. Both the algebraic reformulation and its $q$-deformation were only proven in types $A_1$ and $A_2$. In this paper, we prove the result in type $A_{n-1}$ for $n \ge 4$, while working generically: more precisely, we work over a field of characteristic zero, where $q$ is not a root of unity, and having adjoined an $n$-th root of $q$. Along the way we generalize certain results in the literature (e.g. the Soergel-Williamson categorification theorem, the Soergel conjecture for spherical elements, various symmetries) to the deformed reflection representation.

We establish a concentration–compactness theory for a one-dimensional singular Moser-type functional with logarithmic weight function $$\sup\limits_{v\in E }\int_0^1 e^{\left(\log\frac{e}{s}+\gamma\log\log\frac{e}{s}\right)\frac{v^2(s)}{s}}ds<\infty,~\gamma\in[0,1],$$ where $$E:= \left\{u \in H^{1}(0, 1): u(0) = 0, \int_0^1 |u'|^2dx=1\right\}.$$ In particular, we prove compactness for all normalized sequences in the subcritical range $\gamma<1$, thereby resolving an open problem proposed in [J. M. do Ó, B. Ruf and P. Ubilla, CVPDE 2023]. We also study the critical Bliss-Moser inequality: \begin{equation}\nonumber \sup\limits_{v\in E }I^c(v)< +\infty,\quad I^c(v)=\int_0^1 e^{\left(\log\frac{e}{s}+\log\log\frac{e}{s}+cs\right)\frac{v^2(s)}{s}}ds. \end{equation} For any $c\in(-\infty,+\infty)$, we determine the sharp concentration level $$\sup_{v_j\in E,v_j~\text{is~NCS}}\limsup\limits_{j\to\infty} I^c(v_j)=1+e^2,$$ and obtain explicit sufficient conditions for attainability under $c>c^*$ and non-attainability under $c<-c_*$. This results describe how linear perturbations govern the transition between compactness, concentration, and attainability at the critical level.

The Fresnel wave surface governs the propagation of light in a transparent biaxial crystal. It is a special Kummer quartic, and we identify it exactly. Over the complex numbers the wave surface of a crystal is the Kummer surface of the Jacobian of an explicit genus-two curve branched at the signed square roots of the three principal permittivities. This Jacobian is isogenous, by an isogeny with kernel of order four, to a product of two elliptic curves. One elliptic curve carries the three permittivities, and the other carries the optic-axis angle. The physical family is Zariski dense in the locus of genus-two curves with an extra involution, and its automorphism strata are explicit. The identification instantiates a task-aware quotient, which identifies parameters that differ by a nuisance transformation and carries invariant coordinates and explicit strata. For biaxial crystals, two ratios of the permittivities form a complete invariant of the wave surface up to rotation and rescaling, and the four real nodes are given in closed form. At an interface the candidate transmitted waves are the roots of a quartic of exact degree four. Its real-root count, root order, and repeated-root events are decided by exact algebraic predicates, and along the generic single-node encounters of the paper its discriminant vanishes to second order. Floating-point solvers drop forward transmitted modes near the optic axes, and the certified solver does not. On exact equivalence classes, learned models on quotient coordinates are invariant and more accurate than models on raw tensors, while learned root-count predicates fail near the optic axes.

We study numerical observables relevant to delocalization in the three-dimensional Manhattan mirror model, also known as the K-lattice. Our principal experiment separates the locations of transmitting boundary channels from the routing between them. We derive an exact conditional comparison that preserves all transmission and reflection aperture sets, and find a statistically resolved deficit of physical transmission relative to this comparison. Nearby transmitting channels are themselves suppressed relative to a uniform subset with the same cardinality. A second experiment resolves the contribution of repeated vertices by their return age. The squared norm of the signed old-memory remainder is consistent with an inverse-square-root decay over the measured range. Annular experiments quantify decreasing continuation losses, while an initially apparent history dependence fails independent replication. Finally, a periodic configuration has a connected auxiliary contact graph but only finite routing cycles. These results identify estimates on reflection routing and adaptively selected revisit charges that may assist a rigorous proof. They do not establish an infinite-volume delocalized phase.

We relate the monodromy group of a branched cover to the monodromy groups of its restrictions. We show that the presence of a transposition in a restricted monodromy group significantly constrains the monodromy group of the original branched cover. Using this result, we complete the sparse trace test algorithm, which numerically verifies whether a set of solutions to a sparse polynomial system is complete. Along the way, we develop a generalization of the notion of a trace test in the setting of branched covers.

Fofana spaces, introduced on the basis of Wiener amalgam spaces, have attracted some interest recently and have found applications in the field of partial differential equations. In this paper, we study the normed Köthe structure of the predual spaces $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},\alpha^{\prime})$ of Fofana spaces with variable exponent $(L^{p(\cdot)}, \ell^{q})^{\alpha}$, where $1\leq p(\cdot) \leq \alpha \leq q\leq \infty$. As an application of this investigation, we identify the Köthe duals and Köthe biduals of these two spaces, and we obtain a predual of $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},\alpha^{\prime})$. Finally, we prove that neither $(L^{p(\cdot)}, \ell^{q})^{\alpha}$ nor $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},\alpha^{\prime})$ is reflexive.

We prove the lower bounds \[ B_\infty>0.51,\qquad L>0.51, \] where $B_\infty$ is the locally univalent Bloch constant and $L$ is Landau's constant, improving the bound $\frac{1}{2}+2\cdot10^{-8}$ of Chen and Shiba. The argument passes to the logarithm $g=\log f'$ of a normalized locally univalent Bloch function, where the Bloch condition becomes a one-sided linear constraint on $\operatorname{Re} g$; positive Toeplitz moment matrices then provide a finite-dimensional outer relaxation for the low logarithmic coefficients, and any non-negative dual vector for a linear program over that relaxation certifies a bound. Floating-point linear programming is used only to discover such dual vectors; the same duality also certifies the second-derivative bounds that control the Taylor allowances and the coefficient caps that contract the cover. An independent arbitrary-precision interval-arithmetic program reconstructs every witness and checks a finite cover of the normalized coefficient body by $1179$ boxes, the smallest certified radius being $0.51003$. All numerical premises were verified with Arb at $80$ decimal digits.

We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space $\mathcal{V}$ (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming–Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over $\mathcal{V}$.

We establish a comprehensive algebraic, geometric, and physical classification of the wave closure space generated from primitive plane wave sequences on the discrete spatial lattice $\mathbb{Z}$. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group $\mathcal{G}$ under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to $(\mathbb{Q}/\mathbb{Z}) \times (\mathbb{Q}/\mathbb{Z})$, with canonical decomposition into Prüfer $p$-groups and maximal cyclotomic value field $\mathbb{Q}^{\mathrm{ab}}$. Extending to coordinate powers yields the polynomial phase group $\mathcal{G}_{\mathrm{poly}}$, classified by integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. Adjoining addition yields the group algebra $\mathbb{C}[\mathcal{G}_{\mathrm{poly}}]$, for which we establish the Permutation-Symmetric Phasor Superposition Theorem, factoring superpositions into collective barycentric carriers and closed Born probability envelopes. Using algebraic sieves over roots of unity, we construct idempotent projectors and complementary Not-sieve notch filters, achieving exact algebraic synthesis of both momentum and localized Kronecker position bases. Generalizing to non-abelian quaternions $\mathcal{V}_\mathbb{H}$, we prove a polar decomposition into scalar envelopes and $\mathrm{SU}(2)$ spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime $s^2 = c^2 t^2 - \|\mathbf{x}\|^2$ and the Lorentz group $\mathrm{SO}^+(1,3)$. We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.

We study the large-time behavior of the displacement for wave equations with non-integrable speeds and zero initial displacement. For initial velocities in $L^2\cap L^1$, we derive an $L^2$ upper bound under a weighted Poincaré inequality and symbol estimates up to order two on the propagation speed. The rate is determined by the speed and its time integral. For these speeds, an additional symbol estimate for the third derivative and the nonzero mean condition on the initial velocity give a lower bound of the same order, obtained without using an explicit formula for the Fourier multiplier. At the endpoint $a(t)=(1+t)^{-1}$, the sharp rate is $t(\ln t)^{-\frac{n}{4}}$. The estimates apply to polynomial, logarithmically modified, oscillating, and exponential-type speeds.