Papers

Keith numbers are integers that reappear in sliding-window recurrences generated from their own digits. We introduce Keith mirrors, a subclass defined by saturation at the center of the recurrence's incidence profile. For an even index m, let C_d(m) denote the sum of the two central incidence counts. We prove that the central deficit m-C_d(m) is exactly the distance from m to the interval [2d,2d+2], which forces every Keith mirror of order d>=3 in base B>=3 to return at 2d+2. A digit identity and analytic capacity bounds then reduce the classification to finitely many cases. Exact enumeration, independently checked against the original recurrence and incidence definition, yields precisely 35 mirrors of order at least three, while order two consists of two explicit infinite families. We also classify exactly eight cross-base triple alignments satisfying R=C=x_1+x_d and record the conditional reconstruction property induced by incidence symmetry.

For multivariate polynomial bases of total degree at most n, a column of the Vandermonde matrix is identified by a multi-index of exponents. In this work, we give a direct, recursive map from the scalar column index to this multi-index, based on a prescribed ordering in which exponents of the last variable are increased at the block level and the same rule is applied recursively to the remaining variables. The map is first derived explicitly in the bivariate case, where it yields closed-form column indices for the univariate Vandermonde matrices, and is then extended recursively to arbitrary dimension by means of a binomial counting function, with a proof of correctness by induction. The procedure provides random access to any total-degree column without forming the full tensor-product matrix, and is particularly advantageous for streaming construction and matrix-free evaluation. It is basis independent and applies directly to Chebyshev–Vandermonde matrices. For the full dense matrix, nested loops over the exponent multi-indices achieve the same asymptotic cost; the gain of the present approach lies in exact indexing and the avoidance of superfluous intermediate arrays.

Let $M(N)$ be the number of similarity classes of positive rational perfect cuboids with ordered edges $a,b,c$, space diagonal $g$, and squarefree invariant $N=\operatorname{sf}(abcg/2)$. We prove that $M(N)$ is finite for every positive squarefree $N$, without any restriction on the rank of the associated congruent-number elliptic curve. More precisely, $M(N)\le C^{1+r_N}$ for an absolute constant $C>1$, where $r_N$ is the rank of $y^2=x^3-N^2x$. Using the established Paulsen-West rank obstruction and Smith's results on quadratic twists, we then prove \[ \lim_{X\to\infty}\frac1X \sum_{\substack{N\le X\\N\text{ positive squarefree}}}M(N)^q=0 \qquad\text{for every fixed }q>0. \] Thus the result counts cuboid classes with their full multiplicities at each $N$. A quantitative decay estimate is also obtained. The geometric step excludes every positive-dimensional real abelian coset through the positive part of the compatibility surface; quantitative Mordell-Lang then gives the required bound. The argument is unconditional and uses the cuboid correspondence and rank obstruction as prior results.

A set of integers has zero harmonic density if its elements, viewed as frequencies of continuous characters of the unit circle as a compact group, are measure-theoretically sparse enough that the values of Fourier transforms of Radon measures supported on small open arcs can interpolate those of any global Radon measure. In contrast, zero density quantifies the proportional size of a set of integers relative to expanding intervals on the real line. Although these notions arise from abstract harmonic analysis and elementary number theory, respectively, zero harmonic density is proven as a sufficient condition for zero density, resolving what has been stated in the literature as an open problem concerning the relation between these two notions. The proof makes use of classical gap theorems from complex analysis.

In this paper, we construct the matrix nth root of a multivariate polynomial with coefficients in a field K which possesses an nth root of unity. Moreover, we explain why this concept is interesting. Finally, we give several examples to illustrate our construction.

We construct Komlós signing laws with discrepancy below $6.84$, independent Gaussian reference blocks and exponentially many balanced signings. One law preserves hard constraints and exact conditional means while its reference controls every joint convex cost. We resolve both Reis–Rothvoss Schatten conjectures. For $n$ symmetric $n\times n$ matrices, one prescribed-mean signing law bounds all centered Schatten discrepancies with sharp powers of $n$ and a universal constant. One Gaussian event of measure $e^{-O(n)}$ controls every unitarily invariant norm at the square-function radius. We prove Reis's volume-ratio conjecture: every $r$-dimensional quotient of the $a\times b$ trace-class unit ball has volume ratio $O(\sqrt{1+\min(a,b)/r})$. We resolve Nutz–Wang–Zhang's directional and supermartingale conjectures and the higher-dimensional Bass-flow conjecture. For square-integrable laws, full convex reference support is the universal finite-potential criterion in Tschiderer's $q$-Bass problem. Every driver, even atomic, admits one extended potential and shift for all maximal-covariance martingale optimizers of each irreducible pair with full terminal affine span. For Talagrand's problem, five copies of a compact balanced Gaussian half-mass set contain compact convex output of mass at least $3/4$, also in Wiener space. Cheeger's bound is asymptotically exact throughout subexponential Dirichlet spectral ranges of mixed weighted products. For standard Borel sources, a Blackwell-greatest perfectly private full-data release exists exactly when, after discarding conditional point masses, the rest coincide or share a two-point support. Under conditional atomlessness, existence is equivalent to independence. Covariance-matched analytic uniformly log-concave sources can have a greatest useful-data-only private release with infinite information, while no greatest full-data release exists.

By eliminating the win condition in the game of Connect Four and extending the board to infinite height, a rich state space of positions is obtained. We investigate the number of positions reachable on an $n$-column board after $k$ color-alternating moves. For fixed $k$ we demonstrate polynomiality, derive a partial formula for the polynomial coefficients, and precisely characterize the asymptotic behavior as $n \to \infty$. We then turn our attention to the fixed-$n$ case and show that, under a natural addition operation, positions reachable in an even number of moves form a monoid with a highly symmetric finite generating set; by examining certain free submonoids, we bound the exponential growth rate as $k \to \infty$.

Gielis curves describe many natural shapes, and some of them have corners or cusps. We apply two recent ideas from the differential geometry of curves to Gielis curves: "regularizing" changes of metric and fractional curvatures based on the Caputo derivative. First, we show that the smoothness of a Gielis curve is controlled by the exponents $n_2$ and $n_3$, not by $n_1$. A change of coordinates, the Gielis chart, maps every Gielis curve that closes after one turn onto the unit circle. However, the corners and cusps of the curve reappear as a loss of regularity of the metric; they are transported, not removed. Second, for the fractional curvature defined by Caputo scaling, we distinguish two conventions that differ in the choice of parameter and derive their scaling laws under homotheties. We prove that the memory of the Caputo operator breaks the rotational symmetry: the fractional curvature takes different values at corresponding points of congruent arcs. For fractional order $\alpha<1$, the curves of constant fractional curvature are spirals, and the clothoid is a special case. We also introduce a fractional total turning for curves with corners, give closed forms for regular polygons, and show that fractional general and slant helices coincide with the classical ones. The Python library used in the computations is listed in the appendix.

Let $p$ be an integer $\geq2$, not necessarily prime, and let $K$ be a number field. In 2019, Tao showed that almost every positive integer (in the sense of logarithmic density) had an almost bounded trajectory under the Collatz map. In this paper, we generalize this result to a large class of Collatz-type maps ($p$-Hydras) acting on the ring of integers $\mathcal{O}_{K}$ of a given number field $K$ of dimension $d$ over $\mathbb{Q}$. Given an ideal $\Lambda\subseteq\mathcal{O}_{K}$ of index $p$, a $p$-Hydra map $H:\mathcal{O}_{K}\rightarrow\mathcal{O}_{K}$ is a collection of affine-linear maps $\left\{ H_{j}:K\rightarrow K\right\} _{j\in\mathcal{O}_{K}/\Lambda}$ of the form $z\mapsto r_{j}z+c_{j}$ for constants $c_{j}\in K$ and invertible $\mathbb{Q}$-linear maps $r_{j}:K\rightarrow K$ so that the map $1-r_{0}$ is invertible and so that $H\left(z\right)$ is defined to be $H_{\left[z\right]_{\Lambda}}\left(z\right)$, where $\left[z\right]_{\Lambda}$ is the projection of $z$ mod $\Lambda$. We show that the direct analogue of Tao's almost-boundedness result holds provided, in addition to several minor conditions, that the Dimension Constraint $\left(p-1\right)^{2}\rho_{H}^{d}<1$ is satisfied, where $\rho_{H}=\prod_{j=0}^{p-1}\left\Vert r_{j}\right\Vert _{\textrm{Mink}}$ is the product of the operator norms of the $r_{j}$s with respect to the Minkowski embedding of $K$. Unlike Tao's original argument, which uses a $3$-adic Fourier decay result for the Syracuse Random Variables, our approach employs analysis of an archimedean flavor, and does not require any Fourier decay estimates for our generalized Syracuse Random Variables. Our result includes Tao's original result as a special case.

We show that, in $\MG^{-}$ — $\MG$, the Hilbert plane, deprived of the Side-Angle-Side axiom — a hypotenuse-angle criterion for right triangles, together with the existence of a perpendicular from an external point to a line and a ray correspondence principle governing the ordering of angles, suffices to reconstruct SAS in full — and hence, as standard consequences, SSS and SAA as well. This offers a more elementary alternative to Hilbert's own choice of SAS as an axiom, confined throughout to right triangles and provable without any continuity assumption. Several recent papers have proposed related alternatives on comparable grounds; we compare our approach to theirs in the concluding remarks. The result builds on, and is best read alongside, two earlier papers of ours on related reconstructions of SAS \cite{Volpe2026I,Volpe2026II}.

The Erdős–Sós theorem states that every graph of average degree greater than $t-2$ contains every tree on $t$ vertices. A short counting proof was discovered by GPT-6 Astra in 2026. We give a visual, reader-centered exposition of that argument whose presentation differs substantially from the original. We recast the counting objects as early neighbors in a reveal-and-stop procedure, organize the induction through explicit partitions and reversible swaps, and develop the proof through worked examples with consistent drawings. We also give direct and probabilistic conclusions, compare this formulation with other recent expositions, and record the classical consequence $R(T;q)\le q(t-2)+2$ for multicolor Ramsey numbers of trees.

As a new approach to search a counterexample of variants of Kurepa's conjecture, we investigate a parallel searching method of prime numbers $p$ satisfying \begin{eqnarray*} \sum_{k=0}^{p-1} c_k k! \equiv 0 \pmod{p} \end{eqnarray*} for a given integer sequence $(c_k)_{k=0}^{\infty}$. As an application, we prove that there are only finitely may prime numbers of the form $\sum_{k=0}^{n} (1+dk)!$ with $(d,n) \in (\mathbb{N}_{\leq 25} \setminus \{0,8,18,20,23,25\}) \times \mathbb{N}$.

We give a self-contained proof of the cubic case of Fermat's theorem through a signed form of Euler's descent, with the required extraction step proved directly in $\mathbb{Z}[\sqrt{-3}]$. A pigeonhole argument of Thue represents the relevant primes by $X^2+3Y^2$, and explicit division in $\mathbb{Z}[\sqrt{-3}]$ lifts primitive odd-power norms to powers in that order. For cubes, the resulting coordinate identities transform $2K(K^2+3M^2)=C^3$ into another equation of the same signed form with a strictly smaller non-zero integer height. The construction gives $|C'|\le |C|/39$. Half-sums and half-differences of the two odd entries place the sum and difference parity configurations in this single descent.

We establish a structural rigidity principle for the nontrivial zeros of the Riemann zeta function by analyzing the asymptotic interaction between discrete Dirichlet oscillations and their continuous analytic envelope. The starting point is a renormalized discrepancy operator, the C-transformation, which isolates the finite structural difference between the Dirichlet series and its integral approximation. Applied to the kernel x^(-s), this operator yields a decomposition of the zeta function into three parts: an oscillatory Dirichlet component X_N(s), an explicit analytic growth envelope Y_N(s), and a polynomially decaying remainder R_N(s). We show that the vanishing condition zeta(s)=0 forces an asymptotic compatibility between the two dominant components. This relation is formalized in a Bridge Lemma, which proves that zeros require asymptotic equality of the squared magnitudes |X_N(s)|^2 and |Y_N(s)|^2. A detailed asymptotic analysis then reveals that the ratio of these magnitudes exhibits three distinct regimes depending on the real part sigma=Re(s): divergence for sigma>1/2, collapse for sigma<1/2, and exact balance for sigma=1/2. This incompatibility of growth regimes produces a rigidity mechanism: the asymptotic balance required by the zero condition can occur only on the critical line. Consequently, every nontrivial zero must satisfy Re(s)=1/2. The result provides a deterministic explanation for the location of the zeros, showing that the critical line emerges from an intrinsic structural constraint of the Dirichlet representation of the zeta function.

We prove that the function $$ \log \frac{\Gamma(x+a+b)\Gamma(x)}{\Gamma(x+a)\Gamma(x+b)}-b\log\left(1+\frac{a}{x}\right) $$ is a generalized Stieltjes function of order $3$ for $a>0$ and $0<b<1$. A similar result is obtained when $a>0$ and $b>1$.

Define a(n) as the number corresponding to the shortest computation using only addition, subtraction, or multiplication, as A173419 states on the related OEIS page. In a comment to the sequence, D. Kamenetsky conjectured that a(p) $\ge$ a(p -1) for every prime. We show that the conjecture is false through four counterexamples below 5000 that satisfy the condition a(p) < a(p -1). They are exactly 3359, 3623, 4909, and 4943.

Zadeh fuzzy logic is normally introduced by assigning scalar membership degrees and postulating standard complement, minimum conjunction, and maximum disjunction. We derive this calculus from a strictly positive atomless normalized Boolean measure algebra. Equality of measure sends Boolean complement uniquely to $1-x$, but it cannot send meet to a single-valued operation: the exact scalar image of conjunction is the overlap interval $[(x+y-1)^+,\min\{x,y\}]$. We quantify the information lost under scalarization, prove that no nontrivial measure-preserving compression can retain meet, and show that the overlap operation is associative as a hyperoperation. Zadeh conjunction $\min\{x,y\}$ is then forced as the unique nondecreasing idempotent selection and is realized globally by nested event chains; its De Morgan dual is Zadeh disjunction $\max\{x,y\}$. Thus the min–max–standard-negation calculus follows from measure theory once the nesting commitment is made explicit. The \Luk{} and product conjunctions arise as minimum-overlap and independent-factor selections, clarifying alternative scalarizations. Exact minimax results, an ordered-valued extension, and a three-dimensional realization obstruction delimit what scalar fuzzy logic retains from Boolean structure.

We formalize in Lean 4 Langlands's Second Main Lemma for local epsilon factors over nonarchimedean local fields. The lemma compares local constants of characters of distinct intermediate fields in a bicyclic Galois extension. The proof follows the author's companion mathematical paper and uses the formalization of the First Main Lemma. The First Main Lemma gives only a power relation, leaving a root-of-unity ambiguity. Resolving this ambiguity is the main part of the proof and requires further analysis according to ramification. The wild dyadic case is the mathematically new part of the companion proof, and its Lean formalization provides a machine-checked verification of this new case. Both mixed and equal characteristic are included. We give the exact Lean statement, identify the declarations used in the individual cases, and describe six simplifications of the mathematical proof suggested by the formalization. ChatGPT assisted with transferring formulas and citations from the mathematical manuscript and with adding references to the corresponding Lean source, and suggested improvements to the wording of the paper.

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.

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.

We develop mathematical principles for a generalized tonal practice. We begin with an algebraic theory of modes and scales, formalized as groups and torsors over cyclic groups, respectively. We identify three musically salient classes of mode isomorphism—translations, rotations, and deformations. Upon this basis we organize successive layers of harmonic structure into a hierarchy of Grothendieck fibrations, built from scale types (generalizing the diatonic scale), modes (generalizing the diatonic modes), orbit covers (generalizing their covering by tertian triads), deformation families (generalizing chromatic alteration, such as modal mixture and secondary chords), and harmonic-function assignments (generalizing the assignment of tonic, subdominant, and dominant function). Equipping the resulting structure with a generative syntax—generalizing Rohrmeier's generative grammar of tonal harmony—yields a formal definition of a harmonic practice: a harmonic vocabulary together with an assignment of harmonic functions and a generative syntax accommodating hierarchical harmonic organization. We show that harmonic practices form a category, with morphisms constituting an inheritance and enrichment of structure from one harmonic practice to another. The theory originates in the author's own compositional practice, developed initially through the series of compositions Op. 26–31.

A typical first-order ODE syllabus is often presented as a sequence of named methods, which can obscure the unity among them. We propose organizing a substantial part of the classical symbolic first-order syllabus around a single computational target: an exact representative. For a regular scalar equation represented by a one-form $\omega$, the reduction problem is to find, on an appropriate coordinate patch, a coordinate change $\Phi$ and a nonvanishing factor $\mu$ such that $\mu\Phi^*\omega=dG$, so that solution curves are level sets $G=C$. This yields a two-floor architecture. The structural floor records the target, transformations, and domain restrictions; the operational floor executes substitutions, normalizations, and integrations. Linear and separable equations arise from the simplest one-variable integrating factors, while homogeneous, Bernoulli, and related classes become reduction paths to the same target. In an AI- and CAS-assisted environment, this shifts emphasis toward recognition, justification, auditing, and interpretation. The extended version also develops reverse construction, a structural exercise laboratory, comparisons of reduction paths, and a higher-order outlook through first integrals, operator factorization, and structural normalization.

We address a problem concerning cinema audiences: “A cinema has $n$ seats numbered from $1$ to $n$, and there are $n$ people with tickets numbered from $1$ to $n$. People enter the cinema in order. If someone has the ticket number $i$, they can choose seats whose numbers are multiples of $i$. They should exit the cinema if the permitted seats are occupied by previous audience members. In how many ways can they be seated under these conditions?" We give an algorithm to create the list of situations that meet these conditions. We also focus on finding the number of situations in two special cases: when exactly one seat is unoccupied whose total number is denoted by $\omega(n)$, and when all audiences $1, \ldots, n-1$ are seated, whose total number is denoted by $\psi(n)$. Giving the recursive formula $\psi(n)=1+\sum_{d|n, d\neq n}\psi(d)$ with the initial value $\psi(1)=1$, we provide an explicit formula for $\psi(p^\alpha q^\beta)$, where $p$ and $q$ are distinct prime numbers. Furthermore, we show that $\omega(n)=-n+\sum_{i=1}^n\psi(i)$.

Understanding eco-evolutionary dynamics presents significant mathematical challenges, particularly regarding predictability, equilibrium, and the mathematical representation of fitness. Using Open System Dynamics and Stochastic Analysis, we develop a theoretical framework to characterize fitness change and its relationship with predictability. We explore the limiting behavior of biomass abundances modeled by stochastic processes, deriving continuous-time diffusion counterparts from discrete-time models. Focusing on neutrality and the search for equilibrium, we analyze diffusions in two distinct state spaces. For relative abundances on a d-dimensional simplex, we demonstrate the central role of orthogonal polynomials in model construction. For non-proportional abundances in the positive orthant of d-dimensional space, we detail two specific families of models. Finally, we address statistical inference by presenting numerical examples for relative abundances, illustrating how orthogonal polynomials facilitate maximum likelihood parameter estimation.

We study the numerical evaluation of contour integral representations for solutions of the linearized second and fourth Painlevé equations. The construction combines a nonlinear background solution, continuation of canonical Lax-pair columns, normalization matching, and quadrature over cycles with decaying branches in distinct sectors. Accuracy is assessed using a reference fundamental matrix, differential-equation residuals, and variations of monodromy data computed independently from the spectral problem. An exact identity describes the propagation of initial-basis errors. A stagewise a posteriori refinement criterion uses increments in monodromy variations to allocate numerical accuracy. The contour algorithm is compared with the Dormand–Prince method on a regular background, in a region of rapid solution growth, and through hard loss of stability followed by approximately regular oscillations. Accumulated offsets in monodromy variations and subsequent drift are measured separately. The experiments show how drift depends on spectral conditioning, quadrature accuracy, and error allocation between initial and subsequent parts of the computation. Five additional initial-data and parameter cases demonstrate the dependence of comparative performance on the nonlinear background.

Fractional operator algebras provide a natural setting for linking algebraic normal forms to causal inversion, but arbitrary smooth multipliers and Volterra kernels destroy finite fractional rewriting. We construct an algebra generated by polynomial multipliers, fractional powers indexed by an additive subgroup $\Gamma\subset\mathbb{R}$ containing $\mathbb{Z}$, and flat causal convolution operators. Its finite polynomially weighted diagonal-flat convolution operators form the smallest two-sided ideal containing the flat generators, polynomial multipliers are intrinsic, and distributional separation yields a unique finite normal form with an additive order filtration. For monic normal forms, the leading power selects a canonical causal right factor that extends from the flat core to the natural range $J^\alpha C[0,T]$. Completing the negative-order convolution sector in $L^1$ admits integrable Sonine and Prabhakar kernels, while an independently prescribed continuous bivariate Volterra term remains external. Under injectivity and power-type kernel bounds, variable lower-order terms and this bivariate perturbation reduce to one weakly singular second-kind equation, whose classical resolvent exists on every finite interval without smallness assumptions. A finite boundary-jet block incorporates prescribed initial data without changing the reduced kernel. Sonine, distributed-order, and regularized-Prabhakar realizations follow. In the noninteger Prabhakar regime, an exact range criterion characterizes the classical $AC^m$ subdomain, and an explicit Riemann–Liouville example proves that it is strictly smaller than the continuous natural range.

We study root-of-unity sections of quotients formed from dilated Euler products. At the boundary point \(q=1\), odd sections tend to nonzero cyclotomic constants, whereas even sections have infinitely many simple real poles. We determine the locations, spacing, and residues of all sufficiently late poles. In the cubic case we find the leading oscillatory error in an elementary approximation and prove that its exponential scale is sharp. At the origin we determine exact contact, including a gap-dependent parity law, and integral divisibility with the first nonzero coefficient determined for every modulus \(|k-1|\ge2\). For Euler powers \(k\ge2\), a moving formal pole controls the section error at small fixed nomes; a real zero-free interval gives a further convergence regime. For the twenty-fourth power we prove explicit CM approximation bounds and a signed asymptotic as the dilation increases.

Let $$ M_{r,p}^{(d)}(g)=\int_{\mathbb R^d}|x|^r|g(x)|^p\,dx, \qquad 1<p<\infty. $$ Write $p'=p/(p-1)$ and $\omega_d=|B(0,1)|$. A direct step in the Wigderson–Wigderson moment method asks for a single-function estimate $$ \lVert g\rVert_1\le K\lVert g\rVert_q^{\,1-p\eta} \bigl(M_{r,p}^{(d)}(g)\bigr)^\eta, \qquad \eta=\frac{d(1-1/q)}{r+d-dp/q}. $$ We give a self-contained proof that, in its scale-invariant regime, this estimate holds exactly when $r>d(p-1)$. We then quantify the transition: the best constant blows up like $[r-d(p-1)]^{-1/p'}$ at the critical line, while on the annulus $1\le |x|\le R$ the endpoint loss is exactly $(d\omega_d\log R)^{1/p'}$. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls $L^{1,p}$, $$ \lVert g\rVert_{L^{1,p}}\le \omega_d^{1/p'}M_{d(p-1),p}^{(d)}(g)^{1/p}, $$ and strengthening the weighted norm from $L^p$ to $L^{p,1}$ restores the strong target, $$ \lVert g\rVert_1\le \omega_d^{1/p'} \lVert |x|^{d/p'}g\rVert_{L^{p,1}}. $$ More generally, every power moment has a sharp target Lorentz space, and its secondary index is optimal. These results identify the endpoint lost by the direct same-$q$ reduction. They do not obstruct auxiliary-norm versions of the Wigderson framework, which reach the Cowling–Price range by a different route.

Let $\mathbb{A}=\mathbb{R}1_{\mathbb{A}}\oplus\mathfrak{m}$ be a Weil algebra with $\mathfrak{m}^{N+1}=0$. We study the exponential geometry of the proper orthogonal group $\mathrm{SO}(3,\mathbb{A})$ through finite Weil prolongation. An exact Rodrigues formula is obtained for every element of $\mathfrak{so}(3,\mathbb{A})$, and the exponential map $\mathfrak{so}(3,\mathbb{A})\to\mathrm{SO}(3,\mathbb{A})$ is proved to be surjective. Although every proper orthogonal Weil tensor is therefore exponential and has an unnormalized Rodrigues representation, a factorization of a generator into one scalar angle and one unit axis may fail above the identity; the obstruction is established for every logarithmic branch, including scalar angles $2k\pi$. The analytic mechanism is an explicit finite-regularity rigidity theorem: an $\mathbb{A}$-holomorphic map on $U+\mathfrak{m}$ with scalar-valued real trace is uniquely its finite Weil jet. Together with functoriality, this yields exact transfer of functional, matrix, orthogonality, and composition identities. The results provide a rigorous basis for dual, multidual, hyper-dual, and hyper-multidual calculations in theoretical and computational kinematics.

In this work, we establish a criterion for classifying elliptic curves that have no solutions in the set of positive integers. In particular, we prove that if $E: y^2 = Ax^3 + Bx^2 + Cx + D$ is an elliptic curve where $A$, $B$, $C$, and $D$ are coefficients of the curve, and $p$ is a prime number where $p \equiv 1 \pmod{4}$, and $(x,y)$ is a positive integer point, then if $$ -3A + C \equiv -1 \pmod{4}, $$ $$ A - B + C \equiv -1 \pmod{4}, $$ and $$ p^2 = A - B + C - B, $$ then there are no positive solutions for the curve $E$, where $E(\mathbb{Z}^{+}) = \varnothing$. Furthermore, we generalize this criterion by incorporating techniques from linear algebra. Specifically, we introduce a matrix construction to establish the nonexistence of positive integer solutions for a family of elliptic curves.