Computer Science and Mathematics

Abstract: Let ψ be the Chebyshev prime-counting function and π(x) the number of primes up to x. The Chebyshev-prime condition K(p)=li(ψ(p))−li(ψ(p−1))<1 is not equivalent to the sign rule ψ(p)−p>0. We prove that its leading threshold is the midpoint correction ψ(p)−p>12logp, with an explicit higher-order boundary, and define false Chebyshev primes through the resulting transition layer. Our main unconditional theorem shows that ordinary prime counting does not equidistribute logarithmic prime phases: for every nonzero integer combination τ of nontrivial zeta-zero ordinates, π(x)⁻¹∑_p≤xp^iτ=x^iτ/(1+iτ)+o(1). By contrast, the logarithmically uniform weight logp/p restores finite-dimensional Haar equidistribution for rationally independent ordinates. This dichotomy motivates the weighted local-time statistic L+(x)=2∑_{p≤x,p in the layer} p^−1/2. Under the Riemann hypothesis and linear independence, we conjecture the one-directional Cesàro law L+(x)/logx→fB(0), where f_B(0) is the value at the origin of the limiting density of the symmetrized midpoint field B. Independently, we prove periodic and uniformly transverse quasiperiodic shrinking-target laws via coarea formulas. The remaining obstacles are tangential crossings of the zeta-derived field and transfer to the prime-sampled diagonal window.

Abstract: We introduce ZPIF (Zero Pair Interaction Functional) as a revolutionary Quadratic Spectral Variant (QSV) that fundamentally redefines the Scale Regulated Aggregation (SRA) framework and Riemann's classical explicit formula. At the heart of this work lies the concept of pair interactions: the regulated second-order self-interactions between spectral modes, which capture the nonlinear coupling of each zero with itself and with neighboring zeros. Departing from the traditional linear spectral decomposition, ZPIF introduces a novel quadratic term \(\lambda \sum \gamma_n^2 |c_n|^2\) that explicitly encodes these pair interactions within a rigorous Hilbert space formulation. This term represents the first mathematical framework to systematically incorporate pair-wise spectral couplings into the explicit formula, revealing emergent nonlinear behaviors that are entirely absent from classical models. The framework delivers: • A rigorous operator definition with spectral expansion and trace-class regularization. • Conditional convergence under truncation, ensuring mathematical consistency. • Numerical simulations using the first 100 non-trivial zeta zeros, which reveal distinct nonlinear growth and quadratic interaction effects directly attributable to pair interactions. The quadratic interaction term \(\lambda \sum \gamma_n^2 |c_n|^2\) introduces a novel spectral energy functional that aligns structurally with the SRA law, positioning ZPIF as a foundational tool for exploring quadratic spectral phenomena. This study does not claim to resolve the Riemann Hypothesis but offers a transformative mathematical proposal with potential applications in signal processing, quantum systems, and complex dynamical networks.This work represents the first joint articulation of ZPIF as an SRA-aligned revolutionary variant, honoring the interpretive contributions of Elsayed and the foundational framework of Vogt, and inaugurating a new era in collaborative mathematical research.

Abstract: Over a finite prime shell Fp, p=4κ+1, the roles of π and e are exact residues: the half-period π=2κ, and the exponential marker e=gλ(i). We determine the exact relation between these carriers and the classical values. Each classical value is the horizon readout of a chain of framed rationals n!/!n for e; the Wallis, arcsin and Machin chains for π; at IEEE-754 double precision the constants are the readouts of 18!/!18 and the Machin partial M10. Inside the shell the same chains carry exact residue lines that the readout deletes: the line of e is antiperiodic and terminates on Kurepa's left factorial, universal existence being equivalent to Kurepa's hypothesis; the line of \(\pi\) is legible exactly up to the angular address of -1 and terminates on the calibration face π−1≡−2. Angularly the constants are dual: χ(−1)=eiπ holds exactly in every shell, while radian calibration of e is impossible in every shell and abundant across shells. π is structural, e statistical; transcendence belongs to the external completion, never to the shell element. All exact claims are machine-verified in integer and rational arithmetic.

Abstract: (1) Background: Hom-type algebras, proposed by Yau, generalize classical algebras via twisting maps. Dendriform and tridendriform algebras, introduced by Loday and Vallette, decompose associative multiplications and play significant roles in algebraic K-theory and operad theory. Rota-Baxter operators, originating from analysis and probability, have become a vital bridge connecting multiple disciplines. The π-graded structure, a classical tool in algebra, decomposes algebraic objects into direct sums indexed by a group π. (2) Methods: We systematically investigate the properties and construction methods of Rota-Baxter operators on π-graded Hom-algebras, and establish the derivation relations among π-graded Hom-tridendriform algebras, π-graded Hom-dendriform algebras and π-graded Hom-algebras. (3) Results: We prove that the generalized form, namely the π-graded Rota-Baxter system, is equivalent to π-graded Hom-dendriform algebras. Several iterative construction methods for π-graded Rota-Baxter Hom-algebras are also provided. (4) Conclusions: The structural equivalence between π-graded Rota-Baxter Hom-systems and π-graded Hom-dendriform algebras is established, providing a unified framework for these algebraic structures.

Abstract: Let R be a commutative ring with a nonzero identity and M a nonzero unital R-module. We introduce the concepts of weakly n-submodules and weakly (1, n)-submodules as module-theoretic generalisations of the weakly n-ideal and weakly (1, n)-ideal. A proper submodule N of M is called a weakly n-submoduleif whenever 0 ̸= am ∈N for some a ∈R and m ∈M , then a ∈(N :R M )or m ∈Nil(M )·M , where Nil(M ) = annR(M ). Similarly, N is called a weakly (1, n)-submodule if whenever 0 ̸= abm ∈N for some nonunit elementsa, b ∈R and m ∈M , then ab ∈(N :_R M ) or m ∈Nil(M )·M . Every weaklyn-submodule is a weakly (1, n)-submodule, and every weakly (1, n)-submoduleis weakly 1-absorbing primary. We provide a six-fold characterisation, provestructure theorems classifying the rings and modules over which every propersubmodule belongs to these classes in particular, we show that for faithfulnitely generated multiplication modules, every proper submodule is weakly (1, n) if and only if R is a UN-ring or a product of two elds and investigatebehaviour under homomorphisms, localisations, and quotient modules.

Abstract: This paper tries to deal with the Extended, Generalized, and Grand Riemann Hypotheses under a unified framework based on the general properties of L-functions. Specifically, the divisibility properties of entire functions expressed as absolutely and uniformly convergent infinite products with irreducible real polynomial factors (as a result of pairing complex conjugate zeros in the Hadamard product), combined with the uniqueness of zero multiplicities and the symmetric functional equation, force all zeros of the completed L-functions in the critical strip onto the critical line. Consequently, the existence of Landau-Siegel zeros is excluded, thereby confirming the Landau-Siegel zeros conjecture. As to the Davenport-Heilbronn counterexample, since it possesses no Euler product-the fundamental structural property that confines zeros to the critical strip, it is not in the scope of this paper's methods and conclusions.

Abstract: In this paper, we introduce the weighted generalized Moore-Penrose inverse within the framework of a Banach *-algebra. We characterize this new generalized inverse by employing the weighted MP-inverse and the generalized core-EP inverse. The connections with specific outer inverses are elucidated. Consequently, the attributes of the associated generalized inverses for complex matrices and bounded linear operators on Hilbert spaces are extended to the broader context of Banach algebras.

Abstract: Mobi spaces provide an algebraic abstraction of interpolation between points. In this paper we investigate their relation with geodesic structures. We present a construction of mobi spaces from a parametrized map, formulate conditions under which the mobi axioms follow from a given parametrization, and show that a broad class of second-order differential equations naturally gives rise to mobi spaces. This establishes a general framework linking geodesic motion and mobi structures. As an illustration, we obtain an explicit mobi space associated with a metric of non-constant negative curvature. Taken together, these results suggest that mobi spaces may provide a useful axiomatic framework for the study of geodesic interpolation.

Abstract: The graph of the accelerated Collatz map \( T \) carries structure that is irrelevant to the Collatz conjecture: one third of its vertices are multiples of three, which no orbit can revisit, and the negative integers cannot be used at all because \( T \) admits non-trivial cycles on them. We construct an explicit conjugacy \( J \) between the conjecture-relevant vertices \( {[}{1}{]}_{3}\cup{[}{2}{]}_{3} \) and the nonzero integers \( \mathbb{Z}{*} \), yielding a map \( {K}{\colon}\ \mathbb{Z}{*}\to \mathbb{Z}{*} \) whose graph is isomorphic to the pruned Collatz graph. The Collatz conjecture becomes the statement that every \( K \)-orbit reaches the two-cycle \( \{1,-1\} \); in these coordinates the sign of an iterate records its residue class modulo \( 3 \), so every nonzero integer indexes a conjecture-relevant vertex and no non-trivial cycles are introduced. We analyze the edge structure of the new graph, identify a spanning acyclic subgraph that is bipartite under the sign \( 2 \)-coloring, and introduce an accelerated map \( \hat{K} \) whose graph is bipartite outright and whose in-degree sequence is governed by the \( 3 \)-adic valuation \( {\nu} 3(2k-1) \), recovering OEIS A254046 with mean in-degree \( 3/2 \). Finally, we observe that \( K \)-orbits, plotted as signed time series, behave like damped seismic signals: they oscillate across zero with limiting sign-change frequency \( 2/3 \) while their envelope decays at the heuristic rate \( \lambda=\tfrac12\ln\tfrac43 \) per step, suggesting dynamical and spectral tools as instruments for studying Collatz dynamics. No proof of the conjecture is claimed; the aim is a coordinate system in which its dynamics are easier to see.

Abstract: In this article, we offer a complete, self-contained, and entirely elementary proof of the mean-square estimate for the Chebyshev function. From this we deduce the convergence of the integral is valid, thus proving the validity of the Riemann hypothesis. The proof primarily employs elementary estimates of the Chebyshev function, the Cauchy-Schwarz inequality, and a dyadic decomposition (with Abel summation applied in the appendix), in which the argument results of this article are already optimal within the elementary framework and sufficient to derive the convergence of the required integral that it is a suffficient condition for the Riemann hypothesis. In particular, the appendix of this paper provides a theoretical complement linking integral convergence, pointwise bounds and analyticity, and concludes that the well-known \(o\)-bound is valid, thereby reconfirming the validity of the Riemann hypothesis. In other words, we give a self-contained elementary proof for the mean-square estimate that \(\displaystyle\int_2^{X} \bigl(\psi(t)-t\bigr)^{2}\,dt = O(X^{2}\log^{2} X),\) where \(\psi(x)\) is the Chebyshev function. From this we deduce that \(\displaystyle\int_{1}^{\infty}\frac{|\psi(x)-x|}{x^{\frac{3}{2}+\varepsilon}}\,dx < \infty\) holds for every \(\varepsilon>0,\) thus concluding the integral \(\displaystyle \int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx\) converges absolutely for every \(\varepsilon>0,\) so that the integral \(\displaystyle \int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx\) converges conditionally for every \(\varepsilon>0,\) whereas the integral converges conditionally \(\iff \text{RH},\) so then the Riemann hypothesis is true. In particular, we conclude that the property of absolute convergence of the integral \(\displaystyle\int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx\) for every \(\varepsilon>0\) is equivalent to the property of conditional convergence of the integral \(\displaystyle\int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx\) for every \(\varepsilon>0,\) either of which is equivalent to the property of the \(o\)-bound: $|\psi(x)-x| = o(x^{\frac{1}{2}+\varepsilon})$ for every \(\varepsilon>0,\) and all of them imply the \(O\)-bound: \(\psi(x)-x= O(x^{\frac{1}{2}+\varepsilon})\) is also valid for every \(\varepsilon>0,\) thus reconfirming the validity of the Riemann hypothesis.

Abstract: The Riemann Hypothesis (RH) is proved via a new expression of the completed Riemann zeta function \(\xi(s)\), obtained through pairing the conjugate zeros \(\rho_i\) and \(\bar{\rho}_i\) in the Hadamard product with their uniquely determined multiplicities, i.e. \[\xi(s)=\xi(0)\prod_{\rho}(1-\frac{s}{\rho})=\xi(0)\prod_{i=1}^{\infty}(1-\frac{s}{\rho_i})(1-\frac{s}{\bar{\rho}_i})=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{m_{i}}\] where \(\xi(0)=\frac{1}{2}\), \(\rho_i=\alpha_i+j\beta_i\), with \(0<\alpha_i<1\), \(\beta_i\neq 0\), \(0<|\beta_1|\leq|\beta_2|\leq \cdots\), \(\sum_{i=1}^{\infty} \frac{1}{\beta_i^{2}}<\infty\), and \(m_i\geq 1\) is the multiplicity of $(\rho_i, \bar\rho_i, 1-\rho_i, 1-\bar\rho_i)$. Then, according to the functional equation \(\xi(s)=\xi(1-s)\), we have \[\prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}\] which, owing to the divisibility of entire functions, uniqueness of \(m_i\), and the irreducibility of each real quadratic polynomial factor, is finally equivalent to \[\alpha_i=\frac{1}{2}, 0<|\beta_1|<|\beta_2|<|\beta_3|<\cdots, i=1, 2, 3, \dots\] Thus, we conclude that the RH is true.

Abstract: For a nonnegative integer m, let u_m(n) be the numerator of the rational sum \(\sum\limits_{k=1}^n \frac{\binom{n}{k}^2\binom{n+k}{k}^2}{k^{2m+1}}\). OEIS A357513 records the finite-exception conjecture that u_m(p-1)≡0(mod p⁴) for all but finitely many primes p, depending on m. We prove an explicit eventual theorem: for every prime p>2m+6, one has u_m(p-1)≡0(mod p⁴). Equivalently, the possible exceptional primes are contained in the concrete interval {0,1,...,2m+6}. The proof is a direct calculation in Z/p⁴Z: the Apery-type binomial summand at n = p-1 is reduced to two inverse-power sums, those sums vanish to the required p-adic orders by finite-field and Wolstenholme-type cancellations, and a common-denominator lemma transfers the congruence to the reduced numerator. The source-facing theorem and the effective witness are Lean verified in the AMRA formalization.

Abstract: This paper surveys recent results and methods in the study of permutation polynomials and their compositional inverses over finite fields. In particular, we focus on the recently developed local method and emphasize how it provides a unified and reinterpretative framework for the compositional inverses of permutation polynomials previously obtained by various other techniques (such as piecewise methods, linearized polynomials, decomposition method, the original commutative diagram method). By revisiting these classical results, we reveal the deeper advantages of the local method in understanding the structure of permutation polynomials and the explicit expressions of their inverses. This survey aims to give readers a systematic and profound understanding of the principles, applicability, and intrinsic connections of the local method to existing approaches.

Abstract: This study proposes and examines the generalized John polynomials (also referred to as generalized Leonardo-Pell polynomials), a novel sequence developed by merging the attributes of Leonardo and Pell numbers within a second-order linear non-homogeneous recurrence framework. To analyze their foundational characteristics, five specific sub-cases of this polynomial family are isolated and examined in detail, including the MinMax polynomials defined by Horadam and the new John-Lucas polynomials formulated in this study. The primary algebraic contributions of this work include establishing the general solutions, Binet-type formulas, and both ordinary and exponential generating functions for this new polynomial sequence. The findings ultimately reveal that these polynomials also conform to a third-order homogeneous recurrence relation. Through analytical approaches, exact combinatorial representations for the new polynomial family are derived by applying partial fraction decompositions alongside power series. Furthermore, classical identities well-documented in the literature specifically the formulas of Catalan, d'Ocagne, Vajda, and Honsberger are successfully adapted and proven within the context of this broader non-homogeneous structure. In terms of linear algebra and matrix theory applications, determinantal representations are achieved by utilizing the Doolittle algorithm for LU decomposition. Additionally, the construction of third-order square matrix representations enables the extraction of structural relationships, such as Simpson's identity. Finally, matrix differential calculus techniques are utilized to formulate explicit derivative identities for the generalized John polynomials.

Abstract: In this paper, we introduce the notion of the generalized weighted Moore-Penrose inverse within the framework of ring. We provide a novel characterization for when a ring element can be decomposed into the sum of a weighted Moore-Penrose invertible element and a quasinilpotent. Furthermore, we reveal the intrinsic relations between the generalized Moore-Penrose inverse and generalized Drazin invertibility, presenting new characterizations that extend existing theories. The study is further deepened with a dedicated analysis of the generalized weighted Moore-Penrose inverse involving square-root weights.

Abstract: The Riemann Xi function admits the representation Ξ(t) = ∫₀^∞ Φ(u)cos(tu) du where Φ is a positive integrable function on [0,∞). We prove that Φ is strictly log-concave (TP₂) on [0,∞): (log Φ)″(u) < 0 for all u ≥ 0. We give two independent proofs: (i) a computational proof via rigorous interval arithmetic (5000 certified subintervals at 80-digit precision), and (ii) a purely analytic proof via a convex potential decomposition φₙ = e^−Vₙ with Vₙ″ > 0, requiring no computation. The analytic proof appears to be the first pen-and-paper log-concavity result for a kernel in the Jacobi theta function family. The perturbation from higher-order terms uses only 4.3% of the available log-concavity budget, leaving a 95.7% margin. Log-concavity (TP₂) is a necessary condition for the Riemann Hypothesis; the passage to the full Laguerre–Pólya condition (TP∞) remains open. Both proofs are formalised in the Lean 4 proof assistant.

Abstract: Let R be a commutative ring with 1 ≠ 0 and p aprime. We call a proper ideal I of R a p-difference factor absorb-ing ideal (p-dfa ideal) if whenever a^p−b^p ∈ I, then a−b ∈ I or Φp(a,b) ∈ I, where Φp(a,b) = a^p−1+···+b^p−1. This unifies and ex-tends the sdf- and cdf-absorbing ideals introduced in the literature(recovering both as special cases when p = 2 and p = 3, respec-tively, without their nonzero restrictions). We develop the basictheory of p-dfa ideals, introduce a related class of ∗p-prime ideals,show that von Neumann regular rings of characteristic p have allproper ideals p-dfa, and show that the implication “nonzero p-dfaand p∈U(R) implies prime” fails for every prime p≥3, in contrastto the case p= 2. For principal ideal domains of characteristic zerowith finite residue fields, we characterize when a prime-power ideal(π^k) or a product (πτ) is p-dfa, recovering known results for Z as aspecial case. We also investigate p-dfa ideals in polynomial rings,direct products, idealizations, and amalgamation rings.

Abstract: This study explores a system of Yang–Baxter-type matrix equations, XAX = BXB and XBX = AXA, which generalize the classical matrix Yang–Baxter equation. This work focuses on analyzing the existence of intertwining and commuting solutions using geometric and topological methods. To support this analysis, the notions of anti-diagonally dominant matrices (ADMs) and strictly anti-diagonally dominant matrices (SADMs) are introduced. It is shown that strictly anti-diagonally dominant matrices are nonsingular, ensuring stability and uniqueness in the associated linear systems. Furthermore, if the coefficient matrices of the system satisfy the SADM condition, then an intertwining solution X exists that fulfills both Yang–Baxter-type relations. When the matrices A and B are invertible, the corresponding solution X is proved to be a commuting one. These findings extend the algebraic framework of Yang–Baxter systems and provide new insights into the dominance properties that govern the solvability of matrix equations.

Abstract: We prove the \(D_4\) Voronoi-cell bound for root-aligned saturated unit-ball packings in \(\mathbb R^4\): every cell has volume at least \(8\), with equality only for the scaled lattice \(\sqrt2\,D_4\). The remaining cap-cutting geometry is reduced by a monotonicity theorem to an all-contact \(176\)-chamber positivity system over \(\mathbb Q(\sqrt2)\).

Abstract: We reformulate the binary Goldbach problem over a finite modular algebraic substrate, formalised on a scale-periodic relational field FP. A chain of exact finite results are then derived and documented. On a faithful window where evenness, primality, and the sum p+q are simultaneously exact, binary Goldbach is a finite, decidable statement G(P). There the Hardy-Littlewood circle method is an exact finite identity: no error term, the Gauss-sum magnitude P exact, the singular series floored at 2C2. The second moment gives a finite-shell Montgomery-Vaughan exceptional-set certificate on the evaluated shells: all but a bounded fraction, shrinking across them, of the even integers are sums of two primes. We then locate the residue. A no-symmetry lemma and the moment hierarchy reduce it to one supremum bound, and in the substrate's own scale-periodic frame, free of integral, logarithm, and norm, the count becomes an exact Jacobi-sum form whose positivity is a finite balance between the reflection-even and reflection-odd sectors of the prime spectrum; its identification with the classical parity barrier is conjectural. Every substrate-native claim is verified in exact framed-rational and cyclotomic arithmetic. This is a finite reformulation and an almost-all theorem, not a proof of the conjecture; the terminal parity-sector positivity is exact, finite, and open.