Papers

We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned, or lack thereof, landscape. We focus on Calabi-Yau metrics specifically in a non-compact setting with a global potential, so defined geometrically rather than invoking the topological requirements of the Calabi conjecture. In non-compact settings, we can write the metric determinant with respect to a background in terms of a function, pluriharmonic or real. Using Yau's Liouville theorem, we get a partial eigenvalue blow-up effect. In an empirical setting, a Ricci-flat metric will not form, but a blow-up can be a local condition and can hold empirically on open sets even partially in generalized Kähler cases due to lower bounds on the Ricci potential. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist via geometric modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

Large Language Models (LLMs) are increasingly deployed in high-stakes applications, yet their tendency to generate toxic, harmful, or policy-violating content poses significant risks. Detecting these unsafe outputs efficiently in a black-box manner remains an open challenge. In this paper, we extend a recently proposed dynamical systems framework designed for hallucination detection to LLM safety classification. By projecting both prompts and responses into high-dimensional embedding spaces and fitting separate Koopman-based predictive models for safe and unsafe regimes, we classify new outputs using a new differential residual score that compares prediction errors of the safe and unsafe regimes. A key contribution is the incorporation of the prompt and response embedding dynamics, yielding fitted Koopman operators that capture crucial interaction patterns. We evaluate our black-box method across three safety benchmarks using three embedding models. Our results show that incorporating prompt embeddings yields consistent improvements, particularly for interaction-dependent violations when paired with causal decoders (e.g., in Llama-3), while response-only violations benefit more from dense semantic embedding representations. These findings opens the door for using dynamical systems to analyze AI systems rather than the dominant paradigm of using AI to model dynamical systems.

Let $X$ be a projective line bundle over a nonsingular toric surface which is a blowup the projective plane along at most 4 invariant points, or a blowup the product of the projective lines along at most 4 invariant points. Let $L$ be an ample line bundle on $X$. Then $L$ defines a projectively normal embedding to big projective space. We show the ideal of this embedded $X$ is generated by elements of degree two.

We establish marginal and functional convex order comparisons for graphon mean-field systems on $\mathbb{R}^d$, including the infinite-horizon setting. A key difficulty is that convex order cannot in general be transferred through the usual particle approximation, which requires us to work directly with Euler schemes for the graphon mean-field system. Under a suitable dissipativity condition, we establish Euler approximation estimates that are uniform in both time and the agent label, and combine them with forward-backward induction arguments to obtain the convex order comparisons. For the infinite-horizon problem, dissipativity provides the required uniform-in-time stability, while an exponentially weighted $L^2$-space enables trajectory-level convergence and leads to the functional convex order on $[0,\infty)$. As an application, we derive value-function comparisons for a class of one-dimensional linear-quadratic graphon mean-field games.

In this paper, we prove respectively that the disuccessor operations on the associative operad $\as$, the Lie operad $\lie$, and the pre-Lie operad $\prelie$ preserve Gröbner-Shirshov bases. This structural preservation enables the transfer of known bases to more complex operads. As a consequence, we introduce new methods for constructing Gröbner-Shirshov bases for the $\dend$ and $\prelie$ operads, and explicitly construct a Gröbner-Shirshov basis for the free L-dendriform algebra. This provides a conceptual and computationally efficient resolution of Madariaga's problem and offers new insights into the combinatorial structure of L-dendriform algebras.

We establish scale-invariant interior $C^{1,\alpha}$ estimates for bounded viscosity solutions of $(-\Delta)^su+b\cdot\nabla u=f$ for $s\in[1/2,1)$ with locally Hölder $b$ and $f$. The critical case uses Silvestre's parabolic theorem; the subcritical case uses Schauder estimates and interpolation. Applying this viscosity estimate to drifted Green sections, for finite Radon data above the critical order we obtain sharp solution and gradient potentials of orders $2s$ and $2s-1$, together with weak-*–to–strong local $W^{1,1}$ stability; hence the Green-potential SOLA is approximation-independent. For the normalized whole-space kernels, we identify the classical second-order limits as $s\uparrow1$, including the logarithmic kernel in dimension two; at the critical order, the whole-space gradient becomes a zero-order singular integral. For zero-exterior problems with compactly supported drift, we also prove $u/d^s\in C^{s-\varepsilon}(\overline\Omega)$ and identify the obstruction when the drift reaches the boundary.

Suppose $F$ is a nonarchimedean local field whose residue field is a proper extension of $\mathbb{F}_p$ with $p > 3$. Generalizing results of Ghate–Le–Sheth, we show that any split, connected, reductive group $G$ over $F$ which is not a torus admits a smooth, irreducible, non-admissible mod $p$ representation. We use this to show that the functor of pro-$p$-Iwahori invariants does not induce an equivalence between the category of smooth mod $p$ $G$-representations generated by their pro-$p$-Iwahori invariant vectors and modules over the pro-$p$-Iwahori–Hecke algebra, contrary to what happens for $\textrm{GL}_2(\mathbb{Q}_p)$.

In this paper, we prove a gap theorem for $\mathbb{F}$-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an $\varepsilon$-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).

Zhao and Zhao (2026) established a codimension-four Bogdanov–Takens singularity in a modified Leslie–Gower predator–prey model through a recursive normal form computation. We show that the nonlinear coefficients governing the restriction of the reduced system to the singularity's distinguished eigendirection are not independent. Instead, a single exact rational identity, already present in the transformed equations, generates the entire family of pure Taylor coefficients and proves that their simultaneous vanishing is an exact all orders property rather than a finite-order coincidence. We further show that this structural identity propagates through the planar Bogdanov–Takens normal form algorithm of Kuznetsov (2005). Although the pure coefficients vanish identically, the corresponding normal form coefficients are generated by mixed quadratic interactions. This leads to an explicit characterization of the degenerate Bogdanov–Takens locus and reduces the search for higher-order Bogdanov–Takens degeneracies to a single explicit algebraic condition.

We analyze the transient behavior of a Markovian queue with balking, conditional on exactly $K$ customers joining the system in a finite time interval $[0, T]$. A key quantity of interest is the cumulative number of balking customers, for which we consider the probability generating function (PGF) and derive an equation it satisfies. This approach circumvents the computational burden arising from dealing with high-dimensional Markovian system, which arises when attempting to directly compute the joint distribution of the cumulative number of balking customers together with the total number of joining customers and the number of customers in the system. To compute state transitions in $(t, T]$ efficiently, we examine the conditional state probability at time $T$ given the system state at time $T - u$, and show that it satisfies a linear differential equation in $u$. For piecewise-constant arrival rates, we develop a numerical procedure for evaluating the moment of the total number of balking customers, jointly with the cumulative number of joining customers and the number of customers in the system, under the condition that $K$ customers joined. We also present numerical examples that highlight counterintuitive behaviors arising from this conditioning, along with explanations of the underlying mechanisms.

Our previous papers [8, 9] are the first and second to discuss minimum numbers of “region” colors, while minimum numbers of arc colors such as Fox colors are well-studied. As the third installment, in this paper, we investigate the minimum number of Dehn $\mathbb{Z}$-colors. In particular, we show that the minimum number of Dehn $\mathbb{Z}$-colors of a nonsplittable $\mathbb{Z}$-colorable link is three.

Weak arcs are point sets in PG$(n-1,q)$ meeting every general hyperplane (those are the hyperplanes not going through one of the points given by the standard basis vectors) in at most $n-1$ points. In this paper, we study weak arcs together with balanced variants which are contained on the sides of the fundamental simplex. We give an upper bound on the size of weak arcs, characterise the largest balanced quasi-arcs in the plane and construct large balanced quasi-arcs in PG$(3, q)$. We then use these configurations to build point sets for the random-access problem in DNA-based storage. The constructions are explicit, work over small fields, and attain recovery expectations matching the best known asymptotic bounds.

Let $R$ be a ring. We organize the abstract elementary classes whose underlying class is the class of all $R$-modules and whose strong submodel relation lies between the submodule and direct summand relations into a lattice $\mathscr{L}_{R}$, ordered by reverse inclusion. We establish the basic lattice-theoretic properties of $\mathscr{L}_{R}$ and investigate its two natural sublattices, below and above purity. Below purity, we isolate relations defined by first-order pp-formulas for which amalgamation, tameness, and stability hold. Above purity, we introduce relations defined by infinitary pp-formulas and prove a broad stability result. Specializing to abelian groups, we show that the lattice $\mathscr{L}_{\mathbb{Z}}$ has the following properties: it has a strong submodel relation that is not positive syntactic, it contains an uncountable antichain and a strictly increasing proper-class-sized chain, and it has a broad region above purity where amalgamation fails.

For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlinear Sci., 2015) (resp. Chen et al. (J. London Math. Soc., 2026a)) when $F(x)$ has one fold point and no jump points (resp. two fold points). More recently, Chen et al. (J. London Math. Soc., 2026b) proved that the conjecture is correct when $F(x)$ has no fold points and one jump point. All other cases remain open. Here we verify that the lower bound for the maximum number of limit cycles of the system can be $n$ when $F(x)$ has only $n$ fold points, and $2n$ when $F(x)$ has only $n$ jump points, thereby confirming the lower bound part of Tonnelier's conjecture. Moreover, when $F(x)$ has $m$ jump points and $n-m$ fold points, $0\le m\le n$, we also show that the system can have $n+m=(n-m)+2m$ limit cycles. In addition, a complete classification of the {dynamics} near infinity for this class of systems is provided.

We construct a semistrict higher Chern–Simons (HCS) gauge theory in $2n+2$ dimensions associated with balanced 2-term $L_\infty$-algebras. Starting from the homotopy Maurer–Cartan theory, we first introduce 2-term $L_\infty$-algebra gauge theory, and show that there is a four-dimensional HCS construction. Then we extend invariant bilinear pairings to invariant multilinear forms of the appropriate degree, and define a $(2n+3)$-dimensional higher Pontryagin–Chern form, which is closed and invariant under infinitesimal gauge transformations. Its transgression yields an explicit $(2n+2)$-dimensional HCS form. We further establish a higher Chern–Weil theorem that generates higher transgression forms, and prove that the HCS theory is a distinguished instance of the higher transgression gauge theory. Finally, we apply the extended Cartan homotopy formula in this semistrict setting, and show that it is a common origin of both the higher Chern–Weil theorem and the associated triangle equation.

We establish sharp bilinear decoupling under shell-type Fourier support restrictions and show that its optimal coefficient need not be optimal for the corresponding periodic bilinear Strichartz estimate. For $N_1\geq N_2$, the sharp decoupling coefficient is $N_2^{(2d-7)/4}(1+N_2^2/N_1)^{1/4}$, up to an $N_2^\varepsilon$ loss. This yields a shell-type bilinear Strichartz estimate with the same coefficient on every fixed rectangular torus, including irrational tori. On the standard square torus, we improve this Strichartz estimate for $d\geq6$, obtaining the optimal coefficient $(N_2^{d-4}+N_2^{d-2}/N_1)^{1/2}$, up to an $N_2^\varepsilon$ loss. The proof combines localization in spatial frequency and energy, a circle-method estimate for the localized high-frequency factor, and a linear estimate on arbitrarily oriented strips. As applications, we obtain estimates for toral eigenfunctions, mixed additive energy, and separated nonlinear smoothing for the periodic Zakharov system.

In this paper, we study the structure of the value function in constrained infinite-time optimal control (CITOC) problems of piecewise affine (PWA) systems, with $\ell_1$ or $\ell_\infty$ stage cost. Existing works, such as [1], establish that the resulting value function is PWA in the state. However, existing results do not analyze whether the value function is a proper PWA function, i.e., with a finite number of affine pieces over compact sets, or whether the number of pieces can be infinite. We show that the latter case is indeed possible by means of an explicit example, which is also instrumental in establishing rigorous and easily verifiable sufficient conditions that ensure that the resulting value function is a proper PWA function. Our theoretical findings complement well-known results, e.g., the linear-quadratic case, and serve as support for recent learning-based control schemes for PWA systems. Throughout the paper, the proposed results are illustrated by means of a numerical example.

We prove quantitative polynomial Szemerédi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial. For nonlinear configurations of length at least three, these results provide the first quantitative versions of such theorems. In the linear case, our results improve on work by the last two authors. Our density bounds are strongest in the case of distinct degree polynomials, where they give polylogarithmic bounds, of the same shape as recent bounds by Shao and Wang with integer shifts. The proofs combine recent quantitative results for polynomial configurations in the integers with quantitative Gowers uniformity bounds of the primes. For multiples of a fixed polynomial, we adapt a comparison argument of Altman and Sawhney to obtain uniformity over the polynomial families produced by the $W$-trick. For distinct degree progressions, we establish a comparison between prime-weighted and unweighted polynomial counts that is uniform throughout the density increment argument and accounts for a possible Siegel zero.

It is well established that in time-dependent quantum systems, integrability preserving time-dependent interaction strengths are identical to the renormalization group (RG) trajectories of the corresponding static model when time `$t$' in the driven model is identified with the logarithm of the cutoff `$\log\Lambda$' of the static model. We refer to this integrability preserving driving as the RG protocol. In this work we extend the class of time-dependent integrable models to include non-Hermitian quantum models with time-dependent interaction strengths. Using the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that the constraints imposed by integrability are more general: The interaction strengths of the static model that flow in the RG follow the respective RG trajectories in the corresponding time-dependent model as described above. In addition, the interaction strengths of the static model that are RG invariant can either be constant or have a specific time-dependence in the corresponding time-dependent model which is constrained by integrability. Thus we establish that in the context of time-dependent non-Hermitian systems, the set of integrability preserving time-dependent strengths is larger than the set corresponding to the RG protocol.

We introduce Lagrange functions associated with height and distance-squared functions on implicit plane curves. We show that degeneracies of these functions characterize geometric properties of the curves, such as inflections and vertices, and give geometric interpretations of the corresponding Lagrange multipliers in terms of contact with lines and circles.

We consider the Bessel operator defined by \[ B_\lambda =-\frac{d^2}{dx^2}+\frac{\lambda^2-1/4}{x^2}, \] on $(0,\infty)$, with $\lambda>-1$. We study the fractional power $B_\lambda^s$, $s\in (-1,1)$, $s\neq 0$, and the logarithm $\log^kB_\lambda $, $k\in \mathbb N$, of $B_\lambda$. We obtain pointwise representations of these operators and asymptotic Taylor expansions of the operators $B_\lambda^s$ in terms of logarithmic operators $\log^kB_\lambda $. We also obtain $\log B_\lambda$ as the solution of an extension problem.

For given $m \times n$ data matrices $X, Y$, we investigate the Normal Procrustes Problem—the least squares optimization problem that aims to minimize $\|AX-Y\|_F^2$, where $A$ is constrained to be a normal $m \times m$ matrix. As far as the author of this article is aware, no other method that attempts to solve the Normal Procrustes Problem exists in the literature; we thus propose what is, to our knowledge, the first such method. We, furthermore, adapt our approach to address the Real Normal Procrustes Problem, where $A$ must be real. In our treatment of these problems, we first reduce our complex and real objective functions to be purely optimizable over the Riemannian manifolds of the unitary and real orthogonal matrices, respectively. This reduction enables us to apply techniques in Riemannian manifold optimization to approximate solutions to both. The Closest Normal Matrix and Real Closest Normal Matrix Problems are both special cases of their respective Procrustes Problems (by setting $m = n$ and $X = I_m$) and have been previously studied in the literature. Our approach thus recovers a novel Riemannian optimization method for approximating solutions to both these problems. We further numerically test the performance of our method across all such problems (including against previously developed algorithms on the Closest Normal Matrix Problems) and obtain competitive residuals and favorable scaling in wall-clock time.

We prove that there are exactly $2^\mathfrak{c}$ simultaneously bireflective and bicoreflective subconstructs of $\mathbf{FTS}$. By the Lowen–Wuyts bijection, these correspond to the total fuzzy topologies contained in $M$, the fuzzy topology consisting of all nondecreasing left-continuous self-maps of the unit interval. The upper bound follows from a cardinality estimate. The matching lower bound is obtained by encoding $2^{\mathfrak c}$ ultrafilters on $\mathbb N$ into total fuzzy topologies contained in $M$.

The moving sofa problem asks for the largest planar set that can pass through a unit-width right-angled hallway. Gerver's 1992 construction remains the largest explicit example, but its defining switching parameters, continuum of supporting hallways, and contact topology involve transcendental equations and global inequalities. We give a reproducible exact-rational certification of the construction. A strict four-dimensional Krawczyk inclusion encloses the reduced switching system, while an independent twenty-two-dimensional Krawczyk calculation verifies Romik's full matching and contact equations in an explicit rational box. Exact interval propagation through the five analytic phases proves the support identities, global half-plane constraints, inner-wall separation, niche topology, oriented contacts, injectivity, and area bounds. From the certified support functions we identify every supporting hallway as L_t=x(t)+R_tL. For G=K_0\setminus\mathcal N(K_0) we prove directly that G is nonempty, closed, and connected and that \Phi_t(q)=R_{-t}(q-x(t)) is a continuous path in SE(2) carrying G through the hallway, with the correct initial and final arms. We further identify G with Romik's reconstruction of Gerver's classical eighteen-piece sofa. The deterministic companion verifier uses only Python's standard library and exact rational arithmetic. As a consequence, the Gerver-specific geometric hypotheses in Baek's recent variational framework are rigorously certified; we also clarify the one-sided tangent-speed convention at two contact-curve junctions.

We study the initial run of equal adjacent parts in an integer composition. For a composition alpha, let P(alpha) be the common value of the parts in the initial run, let K(alpha) be the number of parts in that run, and set S(alpha)=P(alpha)K(alpha). We obtain exact finite formulas for the joint parameters (P,K,S) and the corresponding limiting laws. In the unrestricted model, each fixed event (P,K)=(p,k) reaches its limiting probability once n is above an explicit threshold. The limiting distribution of the total size is P(S=s)=2^{-s} sum_{d|s}(1-2^{-d}). Pointing a unit cell in the initial run gives M_1(z)=((1-z)/(1-2z)) sum_{m>=1} sigma_1(m) z^m. For every r>=1, the r-th power-moment generating function has coefficient function A_r(n)=sum_{j=1}^r (-1)^{j+1} binom(r,j) n^{r-j} sigma_j(n), so higher moments involve explicit finite combinations of generalized divisor sums. We also give factorial-moment formulas for the initial-run length and extend the construction to compositions with parts in a prescribed set A, where the corresponding formulas involve restricted divisor sums and the dominant root of the part generating function. Copy

A normal–inverse-gamma (NIG) latent hierarchy has four parameters, but its induced latent law does not identify all four. For $\sigma^2\sim\mathrm{InvGamma}(\alpha,\beta)$, $\mu\mid\sigma^2\sim\mathcal{N}(\gamma,\sigma^2/\nu)$, and $z\mid\mu,\sigma^2\sim\mathcal{N}(\mu,\sigma^2)$, the marginal law of $z$ depends on $(\nu,\beta)$ only through $c=\beta(1+1/\nu)$. Hence the reconstruction-visible parameter space is the three-dimensional quotient $(\gamma,\alpha,c)$, with a one-dimensional fiber degree of freedom. For a fixed hierarchical variational objective, exact partial minimization of the forward KL divergence to a complete NIG prior selects a unique prior-relative representative on each fiber, yielding an exact three-coordinate reduction with the same optimum as the four-coordinate objective. Writing $\rho_0=2\beta_0/\nu_0$ and $T=c/\{\alpha[(\gamma-\gamma_0)^2+\rho_0]\}$, we show that inverse canonical allocation $1/\nu_{\rm can}$ is an explicit strictly increasing function of $T$. For $\alpha>1$, the ratio $u_{\rm epi}/u_{\rm var}=1/\nu_{\rm can}$ is therefore determined by the quotient state and prior; for rank-only use under a common calibration, $T$ contains the same coordinatewise ordinal information. The residual prior gauge is characterized rather than eliminated: $(\gamma_0,\rho_0)$ govern ordinal dependence, while $(\nu_0,\alpha_0)$ determine numerical calibration and the analytic ceiling of $1/\nu_{\rm can}$.

Let $M$ be a closed connected Riemannian manifold. A continuous semimartingale segment can be closed by a Borel family of paths of uniformly bounded length, producing a singular homology class $H_t\in H_1(M;\R)$. For every linear choice of smooth closed representatives of $H^1(M;\R)$, the corresponding Stratonovich homology differs from $H_t$ by a uniformly bounded term, almost surely and uniformly in time. The classes $H_t$ are additive under time shift up to a uniformly bounded error. For Brownian motion this comparison yields the Gaussian central limit theorem with covariance given by the normalized Hodge inner product, a functional central limit theorem for the polygonal interpolation of the closed homology classes, and the almost sure limit $H_t/t\to0$. For elliptic diffusions, the large-deviation principle of Galkin–Mariani passes to $H_T/T$ with the same rate function. The construction uses Schwartzman's closing procedure and is independent, at these asymptotic scales, of the chosen bounded closing family.

We prove the cyclic-product conjecture of Holmes, Holroyd and Ramírez: among all circular arrangements of a decreasing positive sequence, the circular-symmetric “greedy” arrangement maximises every consecutive cyclic product sum $P_r$.

We study the local metric completion of positive singular conformal metrics near a codimension-two corner when the conformal factor is generated by a compact positive measure on exponent space. The Newton support function determines noncritical weighted-ray accessibility, but at the critical power order support geometry alone is insufficient. We identify the missing invariant as a Laplace transform measuring how exponent mass approaches the critical total-degree face. If the maximal total degree exceeds 2, the local completion fibre over the corner is empty. When all total degrees are at most 2, the fibre is empty or a singleton according as the square root of the critical Laplace profile is nonintegrable or integrable at infinity. We give support-identical measures with opposite critical completion behaviour, sharpness examples, monotone truncation phenomena, intrinsic Lipschitz completion coordinates, and a bi-Lipschitz transfer principle for Newton-comparable densities.

Let $P$ be an irreducible reversible Markov kernel on an $m$-state space $\Omega$, and denote its right spectral gap by $\Gap(P) =1-\lambda_2(P)$. For a stationary trajectory, let $K_t$ be the occupation-count vector of the length-$n$ window beginning at time $t$. The stationary pair $(K_0,K_1)$ defines a reversible projected count kernel $\widetilde P_n$. For each fixed $m\ge2$, define \[ c_m^\star= \inf_{\substack{|\Omega|=m,\ P\ \mathrm{irreducible\ and\ reversible} n\ge2}} \frac{n\Gap(\widetilde P_n)}{\Gap(P)}. \] We prove \[ \frac1{1080m}\le c_m^\star\le q_{m-2}, \qquad q_0=\frac14,\quad q_{r+1}=q_r(1-q_r). \] We also show $q_{m-2}=(m+\log m+O(1))^{-1}$, which implies that $c_m^\star=\Theta(m^{-1})$. Thus, the reciprocal optimal comparison coefficient $C_m^{\mathrm{opt}}=(c_m^\star)^{-1}$ is of the order $m$. In the two-state case, we determine the exact values $c_2^\star=1/4$ and $C_2^{\mathrm{opt}}=4$. %the exact constants for $m\ge3$ remain open. The lower bound also holds for $n=1$ and is uniform in $P$, including sparse and periodic kernels. The proof combines short-window decorrelation with an averaged anchor-excursion decomposition, a Green-kernel hitting estimate, and a stopped Carleson–Hardy inequality. A nested rare-state construction produces finite $m$-state witnesses whose normalized Rayleigh quotients approach $q_{m-2}$ through an ordered sequence of limits. For every fixed finite irreducible reversible aperiodic kernel on at least two states, $\Gap(\widetilde P_n)=\Theta_P(n^{-1})$.