Papers

Let $p$ be a prime number and let $n\geq 1$ be an integer. We completely solve the exponential Diophantine equation $p^{x}+6^{y}=z^{2n}$ in positive integers $x,y,z$ for every prime $p$ such that the Legendre symbol $\leg{6}{p}$ equals $-1$, that is, for $p\equiv 7,11,13,17\pmod{24}$: the equation has no solution, with the single exception $(p,n,x,y,z)=(13,1,1,2,7)$. Since these residue classes contain half of all primes in the sense of Dirichlet density, this settles the equation for one prime out of two, uniformly in the exponent $n$. Two ingredients of the proof have independent interest. First, we show that the Ramanujan–Nagell type equation $p^{x}=2\cdot 6^{m}+1$ has no solution with $x\geq 2$, using Zsigmondy's theorem on primitive prime divisors; this removes the extra congruence hypotheses that appeared in earlier work of the author on the case $n=1$ and solves the open problems formulated there. Secondly, we prove that $13^{x}+6^{y}=z^{2}$ has $(x,y,z)=(1,2,7)$ as its unique solution, thereby completely resolving the exceptional prime. The Legendre condition is sharp: we exhibit solutions for primes with $\leg{6}{p}=+1$ realising each branch of the factorisation argument. All results are corroborated by extensive computational verification, and we state several open problems, including the complete classification of the solutions when $\leg{6}{p}=+1$.

While studying for a linear algebra final, the first named author prepared some test questions for herself to see how well she understood the material, and asked the second named author: \emph{If $A$ and $A^T$ have the same eigenvalues and eigenvectors, is $A$ a symmetric matrix?} We show how this excellent question is a great springboard to related questions, in particular when do equal eigenvalues and eigenvectors imply the matrices are, if not equal, at least closely related (such as similar or the transpose/complex conjugate transpose of each other)? The answer depends on how we interpret the question, and provides a great opportunity to talk about creating good questions. In particular, we characterize matrices $A$ for which the transpose or conjugate transpose shares the same eigenvectors (regardless of eigenvalues) and, for each eigenvalue, the same eigenpair (equivalently, the same eigenspace). Thus, a square matrix $A$ is Hermitian if and only if $A^*$ has the same eigenpairs as $A$; moreover, if $A$ is a real matrix with real eigenvalues and $A^T$ has the same eigenvectors as $A$, then $A$ is symmetric.

By utilizing two known hypergeometric summation identities, we extend two \mbox{well-known} Ramanujan series for $1/\pi$ by making each of them a member of an infinite family of series. We also find the corresponding families involving harmonic numbers and odd harmonic numbers. To further illustrate our method we derive families of \mbox{Ramanujan-like} series for $1/\pi$ associated with a hypergeometric series derived by Lavoie, two hypergeometric series from Bailey's book and a recent hypergeometric series found by Campbell. Thus, by leveraging known hypergeometric summation identities, our approach bypasses the more tedious, traditional limits or complex Fourier-Legendre expansions; offering a cleaner, unified template for generating Ramanujan-like series for $1/\pi$.

Can a LEGO model built from many kinds of rectangular bricks be rebuilt using only the familiar $2\times4$ brick? Enlarging every dimension by four makes the pieces fit, but they must also connect. We give a four-layer arrangement that joins every finite face-connected union of lattice cubes, using eight upright bricks per source cube, and show that no smaller universal enlargement works. A second question has an unexpected answer: if connectivity is set aside, the minimum integer scale is always $1$, $2$, or $4$, never $3$. The missing value follows from a checkerboard argument applied separately inside each enlarged cell.

For a prime p >= 5 let f_I(p) and f_II(p) count the Type I and Type II solutions of the Erdos-Straus equation 4/p = 1/x + 1/y + 1/z, in the sense of Elsholtz and Tao. We do not address the Erdos-Straus conjecture itself; our subject is the number of solutions. We turn a bijection of Bradford into an exact counting identity: f_I(p) and f_II(p) are sums, over the denominators n coprime to p, of the number of divisors of n^2 in a single residue class modulo e_n = 4n - p. A dual identity puts both counts on the same integer and the same modulus, distinguished only by the classes -p and -1. Our main unconditional result holds for every numerator m: the number of Type II solutions of m/p = 1/x + 1/(pY) + 1/(pZ) with Y <= K is at most sum_{n <= K} tau(n) = K log K + O(K), uniformly in p and m; on average over p <= N it is only O_m((log K)^3 log log K). Hence Type II solutions concentrate at x/p -> 1/m, while Type I solutions cannot. The numerator carries arithmetic information in two independent ways. A real character chi modulo e_n that is trivial on the primes dividing n annihilates Type II when chi(-1) = -1, and Type I when chi(-1)chi(m_0) = -1, where m_0 is the squarefree kernel of m; these coincide exactly when m is a square, so for the numerator 4 no criterion modulo squares can separate the two types. Separately, f_I(n) = f_II(n) = 0 for every odd perfect square n whenever 4 divides m, generalising a theorem of Elsholtz and Tao. Finally we prove the sharp local inequality 2B(n) >= A(n) for e_n in {1,3,5}, with a construction showing it fails for every odd e with 7 <= e <= 2*10^4; we conjecture f_I(p) > f_II(p) for all p > 5 with p not congruent to 1 mod 8, verify this for p <= 10^6, and reduce it to a finite window. We also locate exactly the factor log log N in the Elsholtz-Tao upper bound for the sum of f_I(p).

For a finite group $G$ let $\psi(G)$ be the sum of the orders of its elements. The quotient $S_n/A_n$ has order two. Hence $\psi(S_n/A_n)=3$. The quantity to be compared with $\psi(S_n)$ is therefore $3\psi(A_n)$. Computation indicates that $3\psi(A_n)>\psi(S_n)$ for every $n\ge3$. We reduce this inequality to a question on the parity of square roots. Let $r(\beta)$ be the number of odd permutations $\sigma$ with $\sigma^{2}=\beta$. We show that the sum of the orders of the odd permutations equals $2\sum_{\beta\in A_n}r(\beta)o(\beta)$. We also show that $\sum_{\beta\in A_n}r(\beta)$ equals $|A_n|$. The inequality then says that $r$ and the order function are negatively correlated on $A_n$. It says equally that the average order of an odd permutation is less than twice the average order of an even one. We then rule out two natural approaches. No injection from the odd permutations into $A_n$ can halve the order at every point. The smallest failure occurs at $n=12$. A threshold argument through Landau's function fails as well. Computations up to $n=60$ are reported.

Starting from the Riemann–$\Xi$ specialization of Weil's explicit formula, we construct finite real-symmetric Prime–Weil matrices $(\mathbf S)$ from pole, archimedean, and finite prime-power data. This construction is a finite-dimensional arithmetic model within the Hilbert–Pólya program, which seeks a self-adjoint spectral realization of the nontrivial zeta-zero parameters. Their off-diagonal entries form a Loewner-type divided-difference matrix with a rank-two displacement identity. We formulate the spectral quotient as a Hermitian definite generalized eigenproblem on the fixed zero-mean contrast space. This realization is invariant under positive affine rescaling, avoids ground-vector normalization and an ill-conditioned oblique projector, and preserves the finite quotient spectrum. For a dimension-matched zero-side matrix built from $N$ distinct positive ordinates $\gamma_k$, rational interpolation gives the exact contrast-pencil spectrum $\{\pm\gamma_1,\ldots,\pm\gamma_N\}$ and positive-parity square spectrum $\{\gamma_1^2,\ldots,\gamma_N^2\}$. Since the ordinates are inputs, this is a reconstruction theorem. Assuming RH, the same interpolation vector proves $\lambda_{\min}(\mathbf S)\to0$. At $N=L=13$, Lemke's ground-state quotient and the contrast pencil agree numerically and reproduce the first three zeta ordinates to the reported precision. The remaining problem is a relative prime-to-zero perturbation theorem with uniform control of the compressed metric. No proof of RH is claimed.

The "Riemann-type integrals" of the title are not a single new integral, but the family of classical constructions – Riemann, Riemann-Stieltjes, Young-Kondurar, complex-analytic, and Itô – unified by one elementary method. We give a simple argument computing the Riemann integral of a polynomial directly, establishing the fundamental theorem of calculus for polynomials and then continuous functions. The same idea extends to the real line, the complex plane, the multidimensional case, the Riemann-Stieltjes integral, and the Itô integral of a polynomial of a Wiener process (more generally, a continuous semimartingale), recovering the classical Itô formula. The contribution is methodological: a single algebraic telescoping identity, paired with an estimate on a quadratic remainder term, drives every case. What changes is only the remainder's fate: it vanishes identically for polynomials on $\mathbb{R}$; it vanishes in the limit under a quadratic-variation condition $Q(\pi)\to0$ on $\mathbb{C}$ and for Hölder paths; and for a continuous semimartingale it survives as the quadratic variation producing the Itô correction – except for a planar Brownian motion, or more generally a conformal martingale, where an algebraic cancellation inside the complex square removes it again. We regard this last point – that the Itô correction and its cancellation in the conformal case are two instances of the same quadratic remainder, not unrelated facts – as the paper's most distinctive observation. The manuscript has three parts: Part I develops the core method on $\mathbb{R}$ and $\mathbb{C}$; Part II extends it deterministically to the multidimensional, Riemann-Stieltjes, and Hölder (Young-Kondurar) cases; Part III extends it to the stochastic setting.

Whether the constant $$G=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}=\frac1{1^2}-\frac1{3^2}+\frac1{5^2}-\frac1{7^2}+\cdots$$ introduced by Catalan in the nineteen century is irrational, is a long-standing open problem. In this paper we prove the irrationality of $G$ via using suitable weights.

For a positive integer \(n\), let \(M_n=\min_{x\in\mathbb R}\sum_{k=1}^n |\cos(kx)|\). We determine \(M_n\) exactly. Apart from the exceptional values \(M_2=1/\sqrt2\), \(M_4=1+\sqrt3/2\), and \(M_6=(-1+3\sqrt5+2\sqrt{5+2\sqrt5})/4\), one has \(M_n=\lfloor n/2\rfloor\). The minimizers are also classified: for \(n\notin\{2,4,6\}\), equality is attained only at \(x\equiv \pi/2\pmod{\pi}\), while the exceptional cases are attained at \(x\equiv\pm\pi/4\), \(\pm\pi/6\), and \(\pm\pi/10\pmod{\pi}\), respectively. The proof is elementary and uses piecewise concavity, a permutation modulo \(2q\), and two finite trigonometric estimates.

In this study, we analyze the number of pairs (i.e., two cards of the same rank) in a set of cards randomly selected from a deck of playing cards. While the standard deck of playing cards comprises 52 cards excluding the joker with 4 suits and 13 ranks, our analysis considers a generalized deck where the numbers of suits and ranks can be set arbitrarily. We derive the exact formulas for the mean and variance of the number of pairs in a given number of cards randomly selected from this generalized deck, expressed using the Gauss hypergeometric function. Moreover, we derive the asymptotic behavior of the mean and variance as the number of ranks tends to infinity.

We introduce a new class of magic figures defined on a finite triangular region of the trihexagonal tiling. The vertices of the region are labeled with the integers $1,2,\dots,3n(n+1)/2$, each used exactly once. The labeling is called \textbf{trihexagonal magic figure of order $n$} if all triangular faces have the same vertex-sum and all hexagonal faces have the same vertex-sum, with the latter equal to twice the former. We note three immediate structural features. First, the ratio between the number of independent constraints and the number of variables approaches $1$ as $n\to\infty$, indicating that the system is asymptotically tight. Second, a necessary parity condition arises: since the labeling uses the integers $1,2,\dots,3n(n+1)/2$, the total number of vertices must be odd, which holds if and only if $n \equiv 1$ or $2 \pmod{4}$. Third, the vertex set forms a perfectly regular triangular boundary, so the configuration has an exact polygonal symmetry without boundary irregularities.

An Euler magic matrix is an integer matrix $M$ satisfying $MM^{\mathsf{T}}=\gamma I$ together with two diagonal square-sum conditions; it is proper when its entry squares are pairwise distinct. Müller proved that Euler magic matrices exist in every order other than $3$, and that no Euler magic matrix of order $3$ exists at all, while proper examples are considerably more restrictive. We describe a tensor construction whose factors are only required to satisfy the orthogonality equation $AA^{\mathsf{T}}=\gamma I$: for a linear reindexing $L$ of the row index group $\mathbb{F}_3^k$ obeying an explicit support condition, the reindexed Kronecker product of $k$ such $3\times3$ factors satisfies the full Euler magic conditions in order $3^k$. Choosing factors whose entry squares have pairwise distinct products, we obtain proper Euler magic matrices of orders $9$, $27$, $81$ and $243$. The construction therefore produces proper examples in powers of three even though order three admits no Euler magic matrix, and the passage from the factors to the product is exactly where the Euler conditions are created rather than inherited. We give an explicit support condition on $L$ and show that it characterises the linear reindexings forcing the two Euler diagonal identities for every tuple of semi-magic factor arrays; over $\mathbb{F}_p$ with two factors, such a reindexing exists only when $p\le3$. The four existence results are formalised in Lean 4, as are the two instances of the construction used to obtain them; the order-$243$ certificate is taken from the archived development and was not rebuilt in preparing this paper, although its witness was reproduced here by exact integer arithmetic. Further proper examples of orders $729$ and $2187$ are verified by exact integer computation only.

Analytic summability of real and complex functions was introduced in 2016. In the topic, the Bernoulli numbers and polynomials were used for defining analytic summand of a given function with a power series on an open domain $D$. In this paper, we study derivatives and integrals of the analytic summand functions and show that the derivatives are the same Bernoulli polynomials series $\sum_{n=0}^{\infty}c_nB_n(z)$ up to a unit forward shift. Therefore, the topic of the analytic summability is a platform for studying Bernoulli polynomials series. In the way, we obtain many new results for the Bernoulli polynomials series such as some related series convergent tests and upper bounds for the Bernoulli polynomials and the mentioned series. For instance, we observe that $\sum_{n=0}^{\infty}c_nB_n(z)$ is absolutely convergent on $\mathbb{C}$, if the numerical series $\sum_{n=0}^{\infty}\frac{n!}{\pi^n}c_n$ is absolutely convergent. Also, we present some applications and various examples of the topic such as the inequality $$|\sum_{n=1}^{\infty}\frac{\pi^n}{n!n^p}B_n(z)|\leq 2e^{\pi|z|}\zeta(p)$$ held for every fixed real number $p>1$ and all $z\in \mathbb{C}$.

This work studies modifications of the Gromov–Hausdorff distance in which the infimum is taken over pairs of morphisms of a given class $\FF$ — the class of morphisms of a subcategory of the category of metric spaces (the $\FF$-Gromov–Hausdorff distance). We prove that the $\FF$-distance depends Lipschitz-continuously on the class $\FF$ in the Hausdorff metric on morphisms. As a consequence, for second countable $C^k$-manifolds equipped with a metric compatible with the topology, the smooth distance coincides with the continuous one: $\dGH{C_k} = \dGH{C}$. We also introduce the partially continuous and partially locally constant distances (continuity is weakened to continuity on an open subset of the support of full measure) and prove that on the class of metric spaces they coincide with the classical Gromov–Hausdorff distance.

We study a genus-three hyperelliptic holomorphic null-curve family in $\mathbb{C}^4$, modelled on the algebraic data of the classical Schwarz P/D family, whose real parts define minimal immersions into $\mathbb{R}^4$. For the order-four automorphism $\omega\mapsto i\omega$ of the underlying Schwarz curve, we compute explicitly its action on the four holomorphic Weierstrass $1$-forms and derive the resulting identities for all real period vectors. Consequently, for every lattice invariant under the induced target rotation, torus-period closure can be checked on one representative from each orbit of a symmetry-stable homology generating set. We also identify precisely when the additional parameter produces a nondegenerate codimension-two deformation. The paper does not claim the construction of a new embedded periodic minimal surface in $\mathbb{R}^4$; rather, it provides an explicit symmetry reduction of the period problem associated with this family.

We discuss Reeb spaces of non-proper smooth functions. (Non-)proper functions are maps between topological spaces the preimages of compact sets by which are compact (resp. may not be compact). Their Reeb spaces are the spaces of connected components of their level sets and with the natural quotient topologies. In proper cases, explicit theory are developing since the 1950s. This is closely related to Morse(-Bott) function theory and in such cases we have graphs naturally. Recently, related general topological studies with combinatorial ones are actively developing, mainly due to Gelbukh and Saeki, and they are at most $1$-dimensional in tame cases. We extend a theorem by Saeki in the 2020s: the Reeb space of a smooth function on a closed manifold is naturally a graph if and only if its critical value set is finite. The author has been a pioneer of the non-proper case and previously investigated examples.

For coprime $a+b=c$, the parity class in which $a$ and $b$ are odd and $c$ is even admits an exact treatment. Triples are measured by the radical excess $E_{\varepsilon}=\log c-(1+\varepsilon)\log\operatorname{rad}(abc)$, and called transgressive at fixed $\varepsilon>0$ when it is non-negative. The $abc$ conjecture asserts that such triples are finite in number. The excess is written exactly in terms of the defect, the mass of repeated primes the radical discards, and rearranges into a linear threshold in which the smaller summand $s=\min\{a,b\}$ appears explicitly. Either $s$ stays bounded along a subsequence, or the defect must overshoot the threshold and rejoin it at an amplified exponent. For the Mersenne family $\mathcal{V}_m = (1, 2^m-1, 2^m)$ transgression at $\varepsilon=0$ holds precisely when $2^m-1$ fails to be squarefree, on a set of exponents of density $47/210$. The defect obeys the exact law $\Delta_m = \Omega_m + \log G_m$, where $\Omega_m$ collects the Wieferich primes dividing $2^m-1$ and $G_m$ is the largest divisor of $m$ whose prime factors divide $2^m-1$. This divisor is trivial on prime-power exponents and largest on $m_k = \operatorname{lcm}(1,\dots,k)$, where the margin $q(\mathcal{V}_{m_k}) - 1 \geq (1-o(1))\log m_k / (m_k\log 2)$ is the most any Wieferich-free mechanism can give. An $abc$ counterexample family along the Mersenne line would require infinitely many Wieferich primes with exponential order–defect growth. A final section separates two elliptic curves attached to the class: the Frey curve, whose minimal discriminant expresses the total defect with a bounded correction at $2$, and a congruent-number Jacobian, whose Szpiro quotient stays below $3$.

We show that $S^2\times S^4$ admits a complex structure. Starting from a modular family of complex two-tori associated with the $(3,4,\infty)$ triangle group and the compactification constructed in [Alpöge 2026], we replace the period lattice by its unique monodromy-invariant index-two superlattice and compactify the resulting family. In the integral Mayer–Vietoris calculation, a primitive local class whose double is the class of a cusp component and the unique nonzero torsion class contributed by the multiplicity-four fibre restrict to the same class of order two on the common boundary. We then prove that the resulting compact complex threefold is diffeomorphic to $S^2\times S^4$. The proof is discovered by the Odin Automatic AI Research Agent.

A saying attributed to Bruce Lee unfavorably contrasts a martial artist who has practiced ten thousand kicks once each with another who has practiced a single kick ten thousand times. We read the saying as implicitly raising a question about how to allocate a fixed practice budget among several skills, and we show that the answer is governed by two ingredients: the shape of the learning curve that converts practice into skill, and the rule by which separate skills are aggregated into overall effectiveness. Neither ingredient alone settles the matter. Our central result reduces the multivariable allocation problem to the maximization of a single scalar efficiency function \(\eff(x)=f(x)^{p}/x\); under a simple uniqueness and divisibility condition, every optimal practice schedule is balanced, dividing the budget equally among a definite number of skills. The optimal number of skills is then determined by a transparent criterion equating the elasticity of the learning curve to the reciprocal of the aggregation parameter. Hard-saturation models, heterogeneous learning rates, a sharp specialization threshold, and a genuinely intermediate optimal repertoire all follow as consequences.

Constrained particle trajectories along lower-dimensional world sheets constitute a foundational domain in relativistic kinematics and submanifold theory. This paper establishes a comprehensive geometric and kinematic framework for third-order particle dynamics constrained to timelike surfaces in Minkowski 3-space. By utilising the localised Darboux frame, we explicitly derive the relativistically invariant parameters, namely the geodesic curvature, normal curvature, and geodesic torsion and analyse the non-linear coupling between the intrinsic surface topology and the extrinsic spacetime geometry. Primary focus is dedicated to formulating the relativistic "jerk" vector, which accounts for the instantaneous time rate of change of acceleration and captures the high-order structural stresses of the motion. Furthermore, we generalise Siacci's theorem to this Lorentzian setting, providing a non-perpendicular geometric decomposition of both the acceleration and jerk fields into explicit tangential and radial components relative to a fixed coordinate origin. Our formulations explicitly unveil the hidden central-force dynamics and dimensional compactification mechanisms under planar constraints, showing that out-of-plane torsional shocks vanish identically when the modified geodesic torsion is suppressed. These results offer novel analytical insights for structural stability and conserved angular momentum-like quantities governing constrained physical systems in non-Euclidean spacetimes.

We develop a Dirichlet-beta counterpart and a structural extension of the coefficient-extraction framework for hyperbolic and logarithmic tangent integrals. For integers $m\geq n\geq1$ with $m+n$ even, the zeta-type formula \[ \int_0^\infty\frac{\tanh^{m+1}x}{x^{n+1}}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} \binom{2p}{n}(2^{2p+1}-1) \frac{\zeta(2p+1)}{\pi^{2p}} [u^{m+n-2p}](u\cot u)^{m+1} \] admits the beta-type analogue \[ \int_0^\infty \frac{\tanh^m x}{x^n\cosh x}\,dx = (-1)^{(m-n)/2} \sum_{p=\lceil n/2\rceil}^{(m+n)/2} 2^{2p}\binom{2p-1}{n-1} \frac{\beta(2p)}{\pi^{2p-1}} [u^{m+n-2p}] \frac{u}{\sin u}(u\cot u)^m. \] The proof uses derivative polynomials, a beta kernel, and a coefficient-collapse argument. We then establish strict sign alternation for several coefficient families. In the boundary shifted-hyperbolic case the sign is controlled by generalized Bernoulli polynomials, while the arctanh and logarithmic tangent families are identified with continuous dual Hahn and continuous Hahn polynomials. Their zero distributions turn the observed alternation into structural statements valid for all admissible indices. Finally, with $\lambda(s)=(1-2^{-s})\zeta(s)$, we show that the parallel zeta–lambda and beta identities are the two binary specializations of a single Lerch-transcendent scheme with $\varepsilon\in\{0,1\}$. The same convention $\varepsilon=0$ for the beta case and $\varepsilon=1$ for the zeta–lambda case is used throughout. Besides the shifted hyperbolic, odd-$\sinh$, logarithmic tangent, and $\tanh$ families, this framework yields a further unified pair of reciprocal-arctanh integral formulae on $(0,1)$.

Wirsching (2003) reduces uniform positive predecessor density for the $3n+1$ map to a chain of five conditions, organized into three conjectures. We prove two of them. Conjecture 1 concerns his path-counting generators: in the convolution carrying them to his Elka functions the partition weights grow subexponentially while the binomial ratio decays geometrically, so a window of width $O(\sqrt{\ell})$ dominates and its radius fits inside the hypothesis. Conjecture 3 concerns the asymptotics near 0 of an invariant density $\varphi$, a base-3 analogue of the Fabius density, against an explicit $\varphi_0$ due to Berg and Krüppel. The exact log-Laplace transform of $\varphi$ splits into a smooth part, a log 3-periodic correction $H$, and a doubly exponentially small remainder. Berg and Krüppel represented that correction as an infinite product in 1998; their analysis did not determine whether it is constant. We give $H$ as a Fourier series with coefficients in closed form in $\Gamma$ and $\zeta$; a classical zero-free theorem for $\zeta$ shows it is not constant, and we enclose its oscillation rigorously. Wirsching's comparison class fixes one phase of $H$, and there Conjecture 3 holds with limit $e^{H(0)}$, certified to lie in $(0.53412203666478,0.53412203666479)$. Off that class the phase sweeps a full period, so the unrestricted asymptotic $\varphi(t)\sim\kappa\varphi_0(t)$ fails. Condition $(\star4)$ follows, at every window radius, with $\mu=1/3$: what Wirsching's argument needs is weaker than Conjecture 3 itself, and the same saddlepoint chain settles it directly. With Conjecture 1 the chain reduces to the single condition $(\star3)$. Conjecture 2 is his route to it and remains open.

Which distinctions in a present state must be retained to predict a specified future exactly? We study this question on a finite reachable set of a deterministic five-agent ABCW model in which binary agent actions and strategies coevolve with a weighted directed influence field. The dataset is generated from four initial-field families and all 4,096 action-strategy initial conditions, yielding 56,536 transitions and 2,562 distinct current fields. We retain the current action and seek the coarsest field-only partition that uniquely determines the complete one-step-ahead field on every observed transition. Nine natural feature families and all 511 of their nonempty combinations provide strong approximate predictors but no exact nontrivial field compression. We therefore construct an incompatibility graph whose vertices are current fields and whose edges join fields that produce different next fields under a shared observed action. A proper 692-coloring gives a constructive upper bound. An independent anchor-action decomposition, followed by exact solution of the 50 nontrivial induced subgraphs, gives the matching lower bound. Hence the minimum number of field classes is exactly 692. This number applies only to the specified finite reachable set, field-only compression, retained current actions, one-step target, and exact-prediction requirement; it is not a universal macrostate count for ABCW systems.

Let $P$ and $Q$ be independent points chosen uniformly from the interior of a nondegenerate triangle $ABC$, and let $I$ be its incenter. We study the probability that the closed disk with diameter $PQ$ contains $I$. In the language of multivariate statistics, this is the spherical depth of $I$ with respect to the uniform distribution on the triangle. We first give an elementary planar form of the normalized cone-measure construction. If $O$ is an interior point of a convex polygon, then the direction from $O$ to a uniformly distributed interior point has the same law as the direction from $O$ to a boundary point whose density on each side is proportional to the distance from $O$ to that side. Consequently, this boundary point is uniform in arclength if and only if the polygon is tangential with incircle center $O$; for a triangle, this characterizes the incenter. Using this transfer principle, we obtain the closed formula \[\mathrm{SphD}(I) =\left(\frac r s\right)^2 \left[ \frac{8R}{r}-1 -\Gamma(\cos A)-\Gamma(\cos B)-\Gamma(\cos C) \right], \] where $r,R,s$ are the inradius, circumradius, and semiperimeter, and \[ \Gamma(t)=\frac1t-\frac{1-t^2}{t^2}\mathrm{arctanh}\ t \] with continuous values $\Gamma(0)=0$ and $\Gamma(\pm1)=\pm1$. Finally we prove the sharp inequality \[\mathrm{SphD}(I)\le \frac13+\frac{\log 3}{6}, \] with equality if and only if $ABC$ is equilateral.

Let $R(K)$ and $s(K)$ denote the Ramsey number and the stick number of a knot $K$. Johnson proved that $R-s$ is unbounded on $(p-1,p)$ torus knots and asked whether other such families exist. We give an affirmative answer. We construct an explicit eight-stick alternating knot $J$ with $c(J)=6$ and $\operatorname{br}(J)=2$. For $K_n=\#^nJ$, the connected-sum formulas yield $R(K_n)\ge 7n+3$ and $s(K_n)\le 5n+3$.

In this paper, we introduce the RLC-fractal derivative along with the fractal Laplace and fractal Sumudu transforms of it. Then we apply these transformations to solve economic models involving the RLC-fractal derivative.

Conditional aggregation operators evaluate profiles relative to admissible contexts whose active sources, criteria, or rules may vary. For a fixed context, such an operator is indeed an ordinary monotone functional on the restricted profile. The nontrivial structure appears at the family level, where local functionals on different profile spaces must be compared and possibly represented by one global rule. We develop a fibered representation theory for families of conditional aggregation operators. After separating fiberwise properties from cross-context compatibility, we characterize global representability without assuming that the full universe is itself an admissible context. Such a family is generated by a monotone global functional if and only if its local values respect the pointwise order after canonical zero extension; when this holds, explicit least and greatest global generators are available. This yields a hierarchy whose inclusions can be strict between arbitrary, zero-extension-consistent, and globally representable families. For finite systems, we then establish an extremal extension theorem for partial capacities and derive an exact overlap criterion for gluing local capacities into global ones. Finally, calibrated representation-preserving integral schemes unify the Choquet, Sugeno, and Shilkret cases: zero-extension consistency is equivalent to projective compatibility of local set functions, while overlap-compatible local capacities yield sharp lower and upper fuzzy-integral scores. Evidence-fusion and multi-criteria applications interpret these results as robustness under inactive sources, context-dependent interaction, and structural uncertainty under incomplete cross-context information.

In measure and integration theory, the themes of extending premeasures to outer measures and extending Daniell linear functionals to Daniell integrals are governed by the foundational extension theorems of Caratheodory and Daniell. These are proven to be equivalent to Littlewood's Three Principles, which are known to be equivalent alternatives to a completeness axiom for an ordered field. The continuity theorems of measure and integration theory are consequently shown to be included in the list of equivalences.

Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, $\textit{canonically}$ stems from a $\textit{single}$ definitional primitive $-$ the ($\mathbb R^{\vcenter{\hbox{$\scriptscriptstyle+$}}}\!$-quantitative) invariant $\mathcal Q$-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared $p = |\mathfrak a|^2$ $-$ the quantum Born rule $-$ and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the $\textit{well-defined}$ interpretable turns out to be, up to gauge $\mathcal Q {\,\to\,} \mathrm{const} {\,\vcenter{\hbox{$\scriptstyle\times$}}\,} \mathcal Q$, the unique gauge-invariant measure $\mathcal Q=|\hspace{-0.18em}| {\cdot}{\cdot}{\cdot} |\hspace{-0.18em}|^2$; independently of the field $\mathbb R$ or $\mathbb C$.