Computer Science and Mathematics

Abstract: We study real critical sets of wavefronts generated by the four third-order Zernike modes. Eliminating the spatial variables from the ideal generated by the two first derivatives and the Hessian determinant yields a principal ideal whose generator is a quartic sum-of-squares polynomial in coefficient space. Its real radical admits a prime decomposition with two projective components: a pure-trefoil line and a twisted cubic. Every nonzero pure-trefoil wavefront has a unique isolated degenerate critical point of elliptic-umbilic type \( D_4^- \), whereas the twisted cubic is precisely the projective coefficient locus of nonisolated critical sets, each consisting of two parallel Morse--Bott lines. The incidence map of the rotating family of critical lines has a nondegenerate Whitney fold, whose spatial caustic is the circle \( x^2+y^2=\frac16 \) in the normalized pupil plane and whose transverse generating family has type \( A_2 \). On the complementary nondegenerate locus, we derive explicit critical radii and obtain a complete classification of the number and Morse types of interior critical points: local maxima, local minima, and saddle points.

Abstract: We study graphs built from a Lie bracket and an invariant metric, subject to the local relations of the classical exceptional series with parameter a. Over K = ℚ(a), we prove that the Morrison–Snyder–Thurston candidate lists span all such diagrams with at most six external legs, regardless of the number of internal vertices. Closed diagrams form a commutative coefficient ring S. The four-, five- and six-boundary spaces are S-modules isomorphic to S⁵, S¹⁶ and S⁸⁰, respectively. The proof classifies the abstract algebra of boundary operations on a possible six-boundary remainder as ℚ × M₅(ℚ) × M₅(ℚ) × M₁₅(ℚ), where Mᵣ(ℚ) denotes the algebra of rational r × r matrices. Its central element ξ, corresponding to a(1 − a) in the diagram presentation, acts on these factors by −6/25, −2, −6, −20, respectively. We also identify the endomorphism algebra of an explicit three-strand summand as a product of seven copies of S, describe the possible remainders at E₆ and E₈, and construct Lie superalgebra evaluations detecting diagrams killed by the corresponding ordinary Lie-algebra evaluations at A₁, A₂ and D₄. The normal forms are relative to S; generic nonvanishing S ≠ 0 and the stronger identification S ≅ K remain open.

Abstract: Up to isomorphism there are 1, 3, 21, 479 and 40,791 GE-algebras of orders 2 to 6. Two separately written programs give the counts, checked with an SMT solver up to order 5. We give the counts for six named subclasses (transitive, commutative, left exchangeable, belligerent, antisymmetric and Hilbert; two of the columns were known) and, for every non-inclusion between them, the least order of a separating example together with the number of minimal examples. The nine patterns of properties that occur are exactly those allowed by the published inclusions, which therefore axiomatize the Boolean theory of the six properties. Finally, a GE-algebra fails the exchange identity exactly when two of its rows are different constant maps on a two-element set; hence every GE-algebra with at most four elements is left exchangeable, and the bound is sharp.

Abstract: We construct a division-free differential graded algebra for polynomial two-parameter differentiation with arbitrary parameters in a finite RIG path algebra. We characterize generation by functions and their differentials and classify isomorphisms fixing functions and external weights. The cohomology decomposes into finite Koszul blocks. At reflection resonance an explicit contraction removes all blocks with an odd weight and leaves a differential graded algebra near the identity parameter. Finite sampling integrals satisfy Stokes and integration by parts, while polynomial Hasse operators retain Taylor data when ordinary iterated derivatives vanish. An extension to ordinary internal Kähler differentials on the full mixed form module exists precisely when every unit-normalized parameter ratio is internally constant. Its connection correction is explicit and flat. Algebraic constructions retain the original localized integer ring; dimension statements specify real scalar extension.

Abstract: The positive integer solutions of the Markoff equation are generated by Vieta involutions and form the classical Markoff tree. This paper studies a layer-uniform asymptotic problem which is different from the uniqueness conjecture and from the endpoint Lyapunov spectrum: what is the typical logarithmic shape of a large-level Markoff triple, and what is the typical growth rate of its largest coordinate? For an ordered Markoff triple \((x,y,z)\) with \(x \le y \le z\), set \(u = (\log x)/(\log z)\) and \(v = (\log y)/(\log z)\). We prove the sharp shape-band estimate \[ u + v = 1 - \frac{c(x,y,z)}{\log z}, \qquad \log 2 \le c(x,y,z) \le \log 3. \] Thus every large-level weak shape limit is carried by the line \(u + v = 1\). On this line the one-dimensional layer shape converges, with a quantitative Wasserstein estimate, to the unique invariant measure of the two ordinary Möbius maps \[ a(u) = \frac{u}{1+u}, \qquad b(u) = \frac{1-u}{2-u}. \] Under the change of coordinate \(r = u/(1-u)\), this limiting shape measure is exactly the pushforward of Minkowski's question-mark measure. Hence it is continuous, non-atomic, and singular with respect to Lebesgue measure, and its distribution function is \(q(u/(1-u))\), where \(q\) denotes Minkowski's question-mark function. Finally, we prove an almost sure typical double-exponential growth law. If \(M_n\) is the largest coordinate on a Bernoulli-uniform branch at level \(n\), then \[ \frac{1}{n}\log\log M_n \longrightarrow l_{\mathrm{sh}} \] almost surely, where \[ l_{\mathrm{sh}} = \frac{1}{2}\int_0^{1/2} \log((1+u)(2-u)) \, d h_{\mathrm{sh}}(u) = \frac{1}{2}\int_0^1 \log\left(\frac{(1+2r)(2+r)}{(1+r)^2}\right) d h_q(r). \] Numerically, the shape mean is about \(0.32075\), the shape standard deviation is about \(0.09385\), and \(l_{\mathrm{sh}}\) is about \(0.39618\). The source package contains exact enumeration data and code reproducing these constants.

Abstract: Let \( N_R \) denote the smallest positive integer such that Ramanujan’s inequality \( \pi(x)^2 < \frac{ex}{\log x}\,\pi(x/e) \) holds for every real \( x \geq N_R \). Assuming the Riemann hypothesis, Dudek and Platt showed that \( N_R \) = 38358837683. In this paper we prove the same value unconditionally. For this purpose we close the range \( e^{103} < x < e^{3158.442} \), which was not covered by previous unconditional results, using an explicit formula for \( ψ(x) \) together with new explicit information on zeros of the Riemann zeta function slightly above the height \( 3 \cdot 10^{12} \), up to which the Riemann hypothesis has been verified.

Abstract: GE-algebras, introduced by Bandaru, Borumand Saeid and Jun as a generalization of Hilbert algebras, are implication-type algebras (X, ∗, 1) defined by three short axioms. This paper presents them in order of increasing difficulty and extends them to neutrosophic and NeutroAlgebraic settings. We start from the smallest examples (the two-element Boolean implication, the three-element chain, the Gödel implication on [0, 1] and a five-element algebra in which the induced relation is not antisymmetric), show by an explicit counterexample that the Łukasiewicz implication on [0, 1] is not a GE-algebra, and prove that a residuated implication gives a GE-algebra exactly when its conjunction is idempotent. We then prove a transport theorem and use it to build neutrosophic GE-algebras on triples (T, I, F), refined neutrosophic GE-algebras on (T₁, …, Tₚ; I₁, …, Iᵣ; N₁, …, N_q; F₁, …, Fₛ), and coupled versions in which the indeterminacy and falsity of an implication depend on the truth values; their orders are bounded distributive lattices. Finally, in the setting of Smarandache's NeutroAlgebra, we show that GE3 is never an AntiAxiom, so every groupoid satisfying GE1 and GE2 is either a GE-algebra or a NeutroGE-algebra; we bound the share of tuples satisfying each subclass identity (sharply for commutativity and antisymmetry), and we classify the instances of GE3 in partial Cayley tables as necessarily true, indeterminate or necessarily false, which yields a neutrosophic triple (T, I, F) for every partial table.

Abstract: We study reflection-resonant two-parameter differentiation on finite commutative square-zero path algebras. For defects with independent coordinate linear parts and arbitrary higher terms, we reduce their joint Koszul cohomology to the squares of a finite implicit solution in the remaining path algebra. This reduction works over a localized integer ring without inverting 2. It gives a necessary and sufficient real stability criterion and a sharp threshold: perturbations starting in degree N cannot change the full-support block with fewer than 2N remaining positions. At exactly 2N positions, complementary path coefficients determine every Betti-number loss and the complete localized-integer torsion module. We identify monomial-defect blocks with augmented relative simplicial chains. Combining the two reductions realizes any finite simplicial homology in an explicitly graded reduced model, with arbitrarily high-order perturbations, independent linear parts, and pairwise conjugate single-direction operators of exact nilpotency order eight.

Abstract: For \( q≥1 \) and \( gcd(a,q)=1 \) we study the restricted weighted Goldbach sum \( R_{a,q}(N)=\sum_{p_{1}+p_{2}=N,\; p_{1}\equiv a\,(q)} (logp_{1})(logp_{2}) \), the sum being over primes. An elementary local obstruction shows that if an odd prime \( \ell \) divides \( q \), then on the set of even \( N≡a (mod \ell) \), of density \( 1/\ell \), the sum collapses to \( O_{q}(logN) \); hence a main term of size \( N/φ(q) \) can hold for almost all even \( N \) only for \( q=2^{k} \), and for general \( q \) the correct main term carries the admissibility condition \( gcd(N-a,q)=1 \). For \( q=2^{k} \) we prove, by the circle method with Vaughan’s minor-arc bound, the Siegel–Walfisz theorem on the major arcs and an exact evaluation of the singular-series coefficients, the mean-square estimate \( \sum_{N\leq X} |R_{a,q}(N)-M_{a,q}(N)|^{2}{\ll }_{A,q}X^{3}(logX)^{-A} \) for every \( A>0 \), an exceptional set of size \( O_{A,q}(X(logX)^{-A}) \), and an asymptotic formula with positive singular series for the restricted ternary sum, hence the representation of every large odd integer as a sum of three primes the first of which lies in a prescribed odd class modulo \( 2^{k} \). For general moduli we record the almost-all theorem for the corrected main term, conditional on a mean-value theorem of Tolev, and the local densities of the restricted ternary problem at every prime power dividing q. We then prove that the odd part \( {\mathfrak{S}}_{0}(p+h) \) of the singular series along shifted primes converges in distribution to the explicit random Euler product \( Y_{h}=\prod_{\mathrm{\ell }\gt 2,\; \mathrm{\ell }\nmid h} (\frac{\mathrm{\ell }-1}{\mathrm{\ell }-2})^{B_{\mathrm{\ell }}} \), with independent Bernoulli variables \( B_{\mathrm{\ell }} \) of mean \( 1/(\ell-1) \), by a self-contained argument based on the Brun–Titchmarsh inequality; all integer moments converge to absolutely convergent Euler products, the Mellin transform is entire of order at most one, the law has no atoms, unbounded support and a superpolynomial tail, and the moments are enclosed in rigorous interval arithmetic. Combining the two parts, we show that \( \varphi (q)R_{a,q}(p+h)/(2C_{2}(p+h)) \) converges in distribution to \( Y_{h} for q=2^{k} \) and odd \( h \). The relation to the additive richness of a prime-generation graph is established under a stated Hardy–Littlewood hypothesis, and an inverse richness–hub correlation of about -0.79 over the first 5000 primes is recorded as a computational finding. All implied constants coming from the Siegel–Walfisz theorem are ineffective, and the paper proves neither Goldbach’s conjecture nor finiteness of any exceptional set.

Abstract: For each Matsuo algebra in the Gorshkov–Mamontov–Staroletov classification, we prove that the Frobenius radical is the least ideal with Jordan quotient and identify the factor over ℚ. This solves both parts of their Problem 5.1 for the two infinite families and seven exceptional cases; scalar extension gives the factors over every field of characteristic zero. The argument uses a geometric characterization: a finite connected real Matsuo algebra at parameter 1/2 has Jordan quotient by its Frobenius radical if and only if its Fischer space is realized by a reflection-closed set of lines over ℝ, ℂ or ℍ, with pairwise angles 60° and 90°. The quotient map sends each axis to its rank-one orthogonal projector. Rational descent distinguishes the E₆ factor: it is the algebra of endomorphisms self-adjoint for ⟨1,1,1,1,1,3⟩, not Sym₆(ℚ). We give explicit radical generators and extend the classification to connected locally finite Fischer spaces. For the symplectic family, incidence with quadratic forms gives quotient coordinates and radical generators at both nontrivial degenerations of the Frobenius form, including non-Jordan quotients.

Abstract: We introduce the novel concept of weighted EP elements with respect to two elements in a ring to establish a unified framework that naturally generalizes classical EP elements. We then characterize the fundamental properties of weighted EP elements, demonstrating that many classical properties extend to this general setting with two weights. The relations among the weighted EP element, the weighted group inverse, the weighted core inverse, and the weighted Moore-Penrose inverse are thereby established.

Abstract: A self-stirring sequence appends the base-b expansion of a current digit count, followed by that digit, and immediately updates all counts. We develop its exact dynamics in every base and determine how its statistical constraints change when the readings are perturbed. Every exact policy produces a normal sequence, with uniform quantitative bounds for growing words. For the cyclic policy we obtain an explicit logarithmically periodic length expansion, natural query moments, a digital-path invariance principle, and a sharp prefix-resolution threshold. The Brownian limit in digit depth coexists with freezing over subexponential lags in generation time. Carries force macroscopic local accelerations and an infinite sequence of logarithmic-power discrepancy lower bounds. In base two, an exact quotient exhibits arbitrarily large merging families near balance, while the cumulative drift reduces to an explicit Takagi term and a precisely defined low-digit remainder. For readings with error K L^α, 0 ≤ α < 1, every empirical limit has a forced uniform Bernoulli component of mass 1 − α; each invariant residual measure η is attainable and its fiber has Hausdorff dimension αh(η)/log b. No such output is strongly normal, whereas every positive error L^1+o(1) permits a full-dimensional strongly normal set from each seed. Finally, an exact series test determines the Lebesgue size of realizable outputs for every eventually nondecreasing error function. All statements are proved for the count-dependent updates themselves; no independence assumption is imposed on the feedback orbit.

Abstract: In this paper, we characterize the two-weight core-EP inverse using the weighted core decomposition and establish its novel representations through the weighted g-Drazin inverse. Furthermore, we define and investigate the weighted core-EP orders within this general setting. These results subsume and extend the properties of the core-EP inverse to a broader algebraic setting.

Abstract: We disprove the conjecture of Camacho, Khudoyberdiyev and Omirov that \(x_i^2 = \sum_j b_{ij}x_j\) has a solution with every coordinate nonzero for every nonsingular complex matrix \(B\). Dimension three is the smallest in which this conjecture fails. In that dimension, all counterexamples are equivalent under simultaneous coordinate permutations and nonzero diagonal rescalings. The exceptional locus in \(\mathrm{GL}_3(\mathbb{C})\) is the disjoint union of six smooth, closed, three-dimensional algebraic tori, each a complete intersection of codimension six. Every exceptional matrix has exactly seven nonzero entries, the minimum possible for a counterexample. A regular three-dimensional evolution algebra with at most six nonzero structure constants has a full-support idempotent, and this bound is sharp. We also give polynomial equations characterizing the exceptional locus. The proof uses elimination, traces in an Artinian complete intersection, and local multiplicities.

Abstract: We formulate the structural equivalents of the classical constants \( \pi \) and \( e \) as exact residues over finite fields: the half-period \( \piP \), \( \gen^{\piP}=-1 \), and the exponential unit \( \eP=\gen^{\lambda(\im)} \), where \( \gen \) is a primitive root. We determine the exact relation between these carriers and their corresponding classical values, as the horizon readouts of a chain of framed rationals (\( n!/\dr{n} \) for \( e \); the Wallis, arcsin and Machin chains for \( \pi \)); at IEEE-754 double precision the constants are the readouts of \( 18!/\dr{18} \) and the Machin partial \( M_{10} \). Inside the shell the same chains carry exact residue lines that the readout deletes: the derangement line of \( e \) is antiperiodic and its residue line 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 \( \piA^{-1}\equiv-2 \). Angularly the constants are dual: \( \chi(-1)=e^{\imR\pi} \) holds exactly in every shell, while radian calibration of \( \eP \) is impossible in every shell and abundant across shells. \( \pi \) is a structural constant and \( e \) a statistical one; transcendence belongs to the external completion and not to any element of the shell. All major claims are machine-verified in integer arithmetic and formalised in Lean 4.

Abstract: Let X be an n-dimensional idempotent evolution algebra over a field k, and let ρ_X: Aut(X) → S_n be the natural permutation representation determined by a natural basis. Costoya, Mayorga, and Viruel proved that, for n ≥ 3, 2-transitivity of the image of ρ_X implies that ρ_X is faithful, whereas, for n ≥ 5, 4-transitivity forces the image to be the full symmetric group. We classify the proper 3-transitive case left open by these results.If ρ_X(Aut(X)) ≤ S_n is 3-transitive, ρ_X(Aut(X)) ≠ S_n, and n ≥ 5, then n = q + 1 for a prime power q = p^m ≡ 1 (mod 3), the field k contains a nontrivial cube root of unity, and char(k) ≠ p. Moreover, after a monomial change of natural basis, the structure matrix of X is a cubic-character matrix C_θ = (c_xy)x,y∈P¹(F_q), c_xx = 0, c_xy = θ(det(v_x,v_y)) for x ≠ y, where v_x = (x,1)^T for x ∈ F_q, v_∞ = (1,0)^T, and θ: F_q× → k^× is a nontrivial character of order 3. The key rigidity comes from a compatibility relation derived from the evolution-algebra automorphism equation. Its exponent 2 reflects the squaring of a natural basis element: if χ is the character induced by the monomial lifts on a point stabilizer and s interchanges two points, then χ(s^-1ks) = χ(k)² for k in their common stabilizer. In the projective action this combines with inversion to force χ³ = 1.Conversely, every such cubic-character matrix defines an idempotent evolution algebra precisely when char k≠ p; indeed C_θC_θ^-1 = qI. If X_θ denotes the resulting algebra, then Aut(X_θ) ≅ PGL₂(q) ⋊ Gal(F_q/F_p)_θ, where the subscript denotes the subgroup of field automorphisms fixing θ. Finally, for every nontrivial character η: F_q× → k^× of order 3, \( X_\theta\cong X_\eta \quad\Longleftrightarrow\quad \theta \) and η lie in the same Frobenius orbit. Thus the 4-transitivity threshold of Costoya–Mayorga–Viruel cannot be lowered to 3 in general. For q = 4, the construction yields the unique five-dimensional isomorphism class with automorphism group A₅, and dimension 5 is minimal.

Abstract: This paper investigates the algebraic and combinatorial structures governing the exponentiation of generalized hybrid numbers, a non-commutative ring that unifies complex, hyperbolic, and dual number systems. Hybrid rings arise naturally when extending sequences, such as Fibonacci-type or Leonardo-type numbers, into algebraic structures with multiple components. We rigorously demonstrate that every hybrid number satisfies a unique quadratic characteristic equation dictated strictly by its real trace and hybrid character. This foundational property establishes a surjective, non-injective mapping between hybrid numbers and classical second-order linear recurrence sequences, specifically defining the fundamental and fundamental-Lucas characteristic sequences. The study formalizes the anti-involutive property of the hybrid conjugate and provides comprehensive, structural proofs for the multiplicative character identity across all positive, negative, and degenerate integer powers. The core contribution of this work is the development of four distinct, mathematically unified frameworks for evaluating arbitrary integer powers of a hybrid number: a sequential recursive framework, a decoupled Lucas-equivalence formulation, an analytic structural root expansion utilizing generalized Binet formulas, and a purely real combinatorial expansion that entirely bypasses complex algebraic arithmetic. Finally, these established architectures are extended to resolve the complex non-commutative mixed products of arbitrary hybrid powers, providing exact, closed-form combinatorial identities for both positive and negative exponent combinations.

Abstract: Let \( C=\{m^3:m\in Z\} \). We prove that there is a set \( A\subseteq Z \) for which the addition map \( A\times C\to Z \) is a bijection. If \( k \) is a fixed noncube and \( [E_k(N)=\{n\in[N,2N)\cap Z:n^3-k\in C-C\}, \) we prove the quantitative estimate \( \#E_k(N)\ll_{k,\delta} N^{18/19+\delta} \qquad(\delta>0). \) The small-gap contribution is handled by a square sieve on the progression polynomial attached to the gap \( a=y-x \); all local branches, including sieve primes dividing \( a \) and the zero values of the polynomial, are treated explicitly. For larger gaps, a theorem of Heath–Brown on fixed levels of diagonal ternary forms gives the required power saving after a direct height calculation and an exclusion of every special solution. Density zero then gives simultaneous avoidance for every finite collection of noncubes, and a two-sided greedy construction produces \( A \). Independently of this source-level proof, a complete Lean 4 development checks the exact theorem and a self-contained determinant–conic–binary-sextic route to the coarser bound \( O_k(N^{99/100}) \). The formal sources, dependency lock, and axiom audit are supplied as a separate companion archive; they are not used as a substitute for the mathematical arguments in the paper.

Abstract: We read the primes and the zeros of the Riemann zeta function through three exact structures: the residue classes of the divisor in the rebuild law \( \sum_{r\le x}M(x/r)=1 \), the squares of the primes, and the reversal of the Euler product. Split by \( r\bmod3 \) the rebuild law is \( (1,a(x),-a(x)) \). Its zero class reproduces the whole law at \( x/3 \), so the classes modulo \( 3^\kappa \) form a self-similar tower whose leaf energy is \( \ll x^{1+\varepsilon} \) for every \( \varepsilon>0 \) if and only if the Riemann Hypothesis (RH) holds; at every zero of \( \zeta \) the three classes stand as \( (0,a,-a) \). Each prime square enters the logarithm of the Euler product with weight \( \frac12 \), and at that weight the lean of the prime count vanishes: the weight fitted from the exact counts to \( 10^{10} \) is \( 0.4994 \). The squares part of \( \zeta \) is zero-free for \( \sigma>\frac12 \) and has a half-order pole at \( s=\frac12 \), the square-root horizon, matched by a half-order zero of the prime factor, which carries every zero of \( \zeta \) in \( \sigma>\frac12 \). Reversing the sign of every prime turns \( \zeta(s) \) into \( \zeta(2s)/\zeta(s) \) and every zero into a pole; the Liouville-weighted sawtooths rebuild the square staircase exactly, and if they approximate the odd-square staircase in \( L^2(0,\infty) \) then RH holds. On the critical line every forward term of the approximate functional equation is balanced by its reversed partner, and off it no two pairs are balanced at once; the pair truncated without its remainder has zeros off the line, at each of which \( \zeta \) itself is far from zero. The Jensen polynomials of \( \Xi \), read as Jacobi chains, have every coupling positive to degree \( 400 \), and each prime pulls the zeros toward the reversal of its clock, as Landau's formula prescribes. We state the axiom of harmonic independence: the prime waves are related only through the mechanism of their construction, and a zero off the critical line would be a relation without a mechanism. The Davenport--Heilbronn function, whose zeros leave the line, adds two Euler products with the root number as weight and couples distinct primes in its logarithm, as the axiom predicts.

Abstract: The distinctive algebraic properties of tropical semirings and their connections to computationally difficult problems have made tropical cryptography a subject of growing interest. However, a growing body of cryptanalytic results has shown that many such schemes are vulnerable to efficient attacks. In this paper, we develop a unified mathematical framework for analyzing this phenomenon through the notion of structural collapse. Structural collapse occurs when the reconstruction problem on protocol-generated instances reduces to a problem that is polynomial-time solvable on the re-sulting instances and whose solutions yield a valid reconstruction of the cryptographic target. We examine recurring structural reduction mechanisms, including linearization, periodicity and mono-tonicity, optimization and inequality-based reformulations, and residual separability, and distin-guish structural reduction from structural collapse. We further formalize the complexity gap be-tween established worst-case hardness of unrestricted algebraic problems and polynomial-time re-construction on structured protocol-generated families. We show that structural vulnerabilities identified in a broad class of tropical and semiring-based cryptographic protocols are not caused by insufficient algebraic richness, but by structural properties that permit reductions to computationally simpler problems. The framework provides a basis for analyzing the computational structure of semiring-based cryptographic protocols and for identifying structural properties relevant to future protocol design.