Abstract:

We study the structural and analytic aspects of the B\'{a}ez--Duarte approximation problem within the Nyman--Beurling framework, which furnishes a functional-analytic equivalent of the Riemann Hypothesis (RH). Our work studies structural features of this framework; it does not prove RH. First (Rank-one collapse and Hilbert-space theory). The integer-dilate Gram matrix \( G_M=\frac{1}{3}\textbf{dd}^\top \) is rank-one, giving \( span\{r_1,\ldots,r_M\}=span\{x\} \) and fixed distance \( d_M=\frac12 \) for all M. We give the explicit Moore–Penrose pseudoinverse \( G_M^+ \) and the one-dimensional collapse of the optimisation problem. Second (Exact Gram matrix formula). We prove a fully rigorous closed-form expression for the inner products of the correct sawtooth basis: using the Bernoulli polynomial representation of the fractional part, \( \int_0^1\{jx\}\{kx\}\,dx = \frac{\gcd(j,k)^2}{jk}\Bigl(\frac{1}{12} + \frac{B_2(0)}{2}\Bigr) + \frac{1}{4}\Bigl(1-\frac{\gcd(j,k)}{j}\Bigr)\Bigl(1-\frac{\gcd(j,k)}{k}\Bigr) + E_{jk}, \) where \( E_{jk} \) is an explicit correction from higher Bernoulli terms, expressed via the Hurwitz zeta function. The arithmetic role of \( \gcd(j,k) \) is made precise. Third (Hardy-space bounds). Using the \( H^2(\Pi^+) \) reproducing kernel and the Mellin isometry, we prove: (a) the distance identity \( d_M^2=\|1/s-F_M^*(s)\zeta(s)/s\|_{H^2}^2 \); (b) an explicit lower bound \( d_M^2\ge\sum_\rho\frac{|F_M^*(\rho)|^2|\zeta'(\rho)|^{-2}}{|\rho|^2}\cdot c(\rho) \) from the zeros of \( \zeta \); and (c) a pointwise Hardy-space inequality relating \( d_M \) to the supremum of \( |1-F_M^*({\tfrac12}+it)\zeta({\tfrac12}+it)/({\tfrac12}+it)| \) on the critical line. Fourth (Kalman filtration stability). Under the observation model \( z_M=d_M+\varepsilon_M \) with $\varepsilon_M$ sub-Gaussian of variance \( \sigma^2 \), the Kalman estimator satisfies a rigorous oracle inequality \( \mathbf{E}|d_M^{KF}-d_M|^2\le \sigma^2 K_\infty(2-K_\infty)^{-1} \), with an almost-sure bound \( |d_M^{KF}-d_M|\le CM^{-\alpha} \) whenever \( |d_M-d|=O(M^{-\alpha}) \). Fifth (Möbius sparsity). We prove \( |c_k^*|=O(k^{-1+\varepsilon}) \) via Dirichlet series techniques and show that the coefficient sequence is bounded in \( \ell^2 \), with connections to the Möbius function made precise through the optimality conditions. Sixth (Structural Mellin theorem). We identify a hidden structural observation in the Mellin identity: the Gram kernel \( K_G(s,w)=\zeta(s+\bar w)/(s+\bar w) \) appears as the reproducing kernel of the Hardy space \( H^2(\Pi^+) \) restricted to the approximation subspace \( W_M \), and its singularity at \( s+\bar w=1 \) encodes the pole of \( \zeta \) while the zeros of \( \zeta \) in the critical strip contribute exactly as spectral obstructions. Disclaimer. This paper does not prove RH. All results are structural, computational, and analytic observations within the equivalent framework.

Abstract: This work presents a complete arithmetic framework resolving the Collatz Conjecture by decomposing the odd–to–odd dynamics into two complementary structures: a local residue–phase automaton and a global affine counting system. The Inverse map R(n; k) = (2^kn − 1)/ 3 is shown to act on the live residues 1, 5 (mod 6) through a finite residue–phase state space, while every admissible exponent k = c + 2e induces an affine expansion factor 2^k whose inverse coincides exactly with the dyadic slice weight 2^−k. From this, every odd integer is seen to belong to a unique dyadic slice S_c,e, forming a disjoint partition of N_odd. Independently, the introduction of the normal–state lattice z(n) reveals a second, purely affine enumeration: each live odd n seeds a unique 4-adic rail m → 4m + 1 whose union also partitions the odd integers without overlap. We prove that these two partitions coincide exactly, yielding a unified global structure in which all odd integers arise bijectively from admissible lifts above minimal anchors produced by n ≡ 1, 5 (mod 6). The locked Forward–Inverse equivalence T(m) = (3m + 1)/ 2^ν2(3m+1) and R(T(n); k) = m then implies that Forward trajectories cannot branch or diverge: each Forward iterate lies on a single admissible rail descending toward its origin at 1. Because the residue–phase automaton is finite and every rail has a uniquely determined Forward parent, no infinite runaway is possible and no nontrivial odd cycle can exist. Together they provide a complete, closed arithmetic description of the Collatz dynamics and establish that every Forward trajectory converges to 1.

Abstract: The Riemann Hypothesis, one of the most celebrated open problems in mathematics, addresses the location of the non-trivial zeros of the Riemann zeta function and their profound connection to the distribution of prime numbers. Since Riemann’s original formulation in 1859, countless approaches have attempted to establish its truth, often by examining the asymptotic behavior of arithmetic functions such as Chebyshev’s function θ(x). In this work, we introduce a new criterion that links the hypothesis to the comparative growth of θ(x) and primorial numbers. By analyzing this relationship, we demonstrate that the Riemann Hypothesis follows from intrinsic properties of θ(x) when measured against the structure of primorials. This perspective highlights a striking equivalence between the distribution of primes and the analytic behavior of ζ(s), reinforcing the deep interplay between multiplicative number theory and analytic inequalities. Beyond its implications for the hypothesis itself, the result offers a fresh framework for understanding how prime distribution governs the analytic landscape of the zeta function, thereby providing new insight into one of mathematics’ most enduring mysteries.

Abstract: This study aims to prove the Riemann Hypothesis and the Generalized Riemann Hypothesis by ex-tending the Riemann zeta function and Dirichlet L -functions to the elliptic complex domain, based ona newly constructed system of elliptic complex numbers Cλ(λ < 0) . The core challenge addressed is theinherent difficulty in resolving these conjectures within the traditional ”circular complex domain” frame-work (λ = −1); the author posits that a complete proof is unattainable strictly within this conventionalsetting.The primary innovation of this work lies in the formulation of the theory of elliptic complex numbers,specifically identifying the limiting case as λ → 0− as the key to the proof. Through rigorous deduction,a bijective correspondence between zeros across different complex planes is established. By employingproof by contradiction and leveraging the correspondence between Cλ (as λ → 0) and the circle complexplane C, the Riemann Hypothesis and the Generalized Riemann Hypothesis are ultimately proven. Thispaper is organized into three parts:(1) Construction and Geometric Properties: The first part details the construction of elliptic complexnumbers and their fundamental geometric properties, laying the necessary foundation for subsequentanalysis and the proof of the conjectures.(2) Analytic Extension: The second part introduces elliptic complex numbers into mathematical anal-ysis, deriving numerous results analogous to those in classical complex variable function theory.(3) Proof of Conjectures: The final part presents the formal proofs of the Riemann Hypothesis and theGeneralized Riemann Hypothesis.

Abstract: What are the numbers made of? More precisely, what are prime numbers made of? I posed this question to myself on the evening of August 19, 2025, which prompted prolonged introspection and profound contemplation. Then, I began constructing a numerical pyramid with prime numbers. The number one took the place of the central axis. Therefore, it is possible that large prime numbers could be surrounded by prime numbers on either side of one. However, this property extends to all even and odd non-prime numbers, but without one. The Goldbach ternary conjecture, which was proven by Harald Helfgott and is now recognized as the Goldbach-Helfgott theorem, is applicable to the observation that all odd non-prime numbers can be expressed as a sum of at least three prime numbers. This is due to the fact that non-prime numbers are a subset of all numbers greater than five. Once Goldbach's binary conjecture is proven, it will likely lead to the proof of Riemann's conjecture because we will be able to detect the structure of even numbers preceding prime numbers. For now, we can visualize this in the numerical structure of the first one trillion numbers and even further up to the largest known prime number. Let 3 203 431 780 337 be our number, which is verified as prime. If we subtract another prime number, 3 333 977 , from it, we obtain 3 203 428 446 360$. Subtracting one from the product verifies that 3 203 428 446 359 is prime. If so, then the sum of the two prime numbers plus one equals the proposed prime number above. This study has two objectives. First, it aims to present prime numbers as more than just their primality property. Second, it seeks to define the numbers 2 and 3 as a set of authentic prime numbers.

Abstract: We develop the central identities of the theory of automorphic forms centering on the Jacobi theta constants \( \vartheta_2, \vartheta_3, \vartheta_4 \), the weight-4 Eisenstein series \( E_4 \), the discriminant \( \Delta \), the j–invariant, and the modular \( \lambda \)–function. The classical theory is organized around a single minimality theorem: the pair \( (\vartheta_3(\tau), \vartheta_3(2\tau)) \) suffices to recover every primary automorphic invariant at level \( \leq 2 \) as an explicit polynomial or rational function.Building on this foundation, we derive four new structural observations. \( \textbf{(I)} \) The shifted invariant \( J(\tau) := j(\tau) - 744 \) satisfies \( J(2\tau) = J(\tau)^2 - 2 \cdot 196884 + O(q^2) \) (and in fact \( J(2\tau)=J(\tau)^2-2\cdot 196884-2\cdot 21493760\,q^2+O(q^4) \)), placing the first Monster moonshine coefficient as the \emph{leading deviation from perfect squaring} under the doubling isogeny; the corresponding quadratic fixed-point polynomial has discriminant 1575073. \( \textbf{(II)} \) The sequence \( \vartheta_3(2^n\tau)^2 \) is the arithmetic-mean sequence of the arithmetic-geometric mean (AGM) iteration initialized at \( (\vartheta_3(\tau)^2, \vartheta_4(\tau)^2) \); the unique AGM fixed-point symmetry \( \vartheta_3(\tau) = \vartheta_4(\tau) \) identifies \( j(\tau) = 1728 \) (\( \tau = i \)) as the self-dual elliptic curve. \( \textbf{(III)} \) The \( \lambda \)-ODE \( d\lambda/dt = -\pi\lambda(1-\lambda)\vartheta_3(it)^4 \) approaches a logistic regime for \( t\gg 1 \); matching the exact midpoint value \( \lambda(i)=1/2 \) produces the explicit sigmoid approximation \( \lambda(it) \approx (1 + e^{\pi(t-1)})^{-1} \) for large t. \textbf{(IV)} The quantity \( R(\tau) := 2\vartheta_3(2\tau)^2-\vartheta_3(\tau)^2 = \vartheta_4(\tau)^2 \) satisfies the square-root recursion \( R(2\tau) = \vartheta_3(\tau)\sqrt{R(\tau)} \) under doubling; equivalently, \( \vartheta_4(2^n\tau) \) lies in a nested-radical (generically quadratic) extension tower over the dyadic \( \vartheta_3 \)-field, growing by one quadratic layer at each step---an algebraic obstruction distinct from the polynomial j-isogeny ladder.

Abstract: Define the perrank of an m×n matrix M to be the size of a largest square submatrix with nonzero permanent; if this is equal to m or n, then we say M has full perrank. We derive numerous results on the perrank of a matrix by methods of commutative algebra.

Abstract: We introduce an abelian group structure on the positive real numbers via the operation a ⊗κ b = exp(κ ln a ln b) for a parameter κ > 0. The transformation Tκ (x) = ln(κ ln x) establishes a group iso- morphism (M^>1_κ , ⊗κ ) ∼= (R, +), enabling harmonic analysis on the scale group. We define generalized zeta functions ζκ (s) = ∑ n^{−⊗κ s} and prove ζκ (s) = ζ(κ ln s) [11 , 13]. The zeros of ζκ (s) are given by sn = exp(ρn/κ) where ρn are the zeros of ζ(s). Under the Riemann hypothesis, these zeros lie on the circle |s| = e^1/(2κ). Scale prime numbers arise naturally as irreducible elements, with correspondence p = exp(ep/κ) to ordinary primes [8]. All results hold for any κ > 0 and are verified numerically with errors below 10−¹⁴. The complete verification code and figures are provided as supplementary material.

Abstract: This paper proposes a geometric propagation model on the plane formed by two alternating integer fields placed on parallel layers y=n. Odd layers carry an expansion field and even layers carry a collapse field. Local directions are specified through explicit gradient (tangent-slope) laws for the fields Ψ, yielding parallel corridors, trajectory merging, and event-driven switching at inter-layer boundaries. This gradient field is connected to the Riemann–P (three-singularity Fuchsian) differential equation: choosing a half-integer local exponent produces square root scaling, so dilations of the independent variable generate multiplicative amplitude updates. When integers are embedded as a half-integer leaf u=n+1/2 and a first-return-to-leaf rule selects dyadic contraction depth, the induced return map is exactly the Collatz map. We provide vector-field examples, switching rules, a formal equivalence, and a numerical propagation example illustrating why this reformulation is useful in the digamma form. This model naturally leads to the Ψ-function and a new Collatz constant, β=0.93982.

Abstract: Around 1637, Pierre de Fermat famously wrote in the margin of a book that he had a proof showing the equation aⁿ + bⁿ = cⁿ has no positive integer solutions for exponents n greater than 2. This statement, now known as Fermat’s Last Theorem, remained unproven for centuries despite the efforts of countless mathematicians. Andrew Wiles’s work in 1994 finally provided a rigorous proof of Fermat’s Last Theorem. However, Wiles’s proof relied on advanced mathematical techniques far beyond the scope of Fermat’s time, raising questions about whether Fermat could have truly possessed a proof using only the methods available to him. Wiles’s achievement was widely celebrated, and he was awarded the Abel Prize in 2016; the citation described his proof as a “stunning advance” in mathematics. Combining short and elementary tools, we prove the Beal conjecture, a well-known generalization of Fermat’s Last Theorem. The present work potentially offers a solution closer in spirit to Fermat’s original idea.

Abstract: The Riemann Reciprocal Sum (formula) for all nontrivial zeros of the Riemann zeta function is a constant, namely, 1/2 (2 + γ – ln4π) [1], and also a supernatural result. Our paper provides a proof of this formula based on research findings since the Riemann Hypothesis was proposed.

Abstract: An N(2,2,0)-algebra (abbreviated as NA-algebra) is an algebraic structure equipped with two binary operations, $\ast$ and $\bigtriangleup$, satisfying specific axioms. This paper investigates a special class of NA-algebras where the operation "$\ast $" exhibits nilpotent properties. We study several fundamental concepts within NA-algebras, including ideals, congruence decomposition, congruence kernels, and multiplicative stabilizers. A notion of NA-morphism is introduced, and a corresponding NA-morphism theorem is established. Furthermore, we explore the relationships between NA-algebras and other related logical algebraic structures, such as quantum B-algebras, Q-algebras, CI-algebras, pseudo-BCH-algebras, and RM-algebras. Notably, we prove that any nilpotent NA-algebra forms a quantum B-algebra. These results lay a foundation for further research into the structure and potential applications of NA-algebras.

Abstract: We prove that every even integer 2N ≥ 8 is the sum of two distinct primes. This variant of the classical Goldbach conjecture is established through three components: (1) a novel geometric equivalence reformulating the problem in terms of nested squares with semiprime areas, (2) a theoretical proof for all N ≥ 3275 using Dusart’s refinement on prime distribution, and (3) direct computational verification for 4 ≤ N ≤ 3274. The geometric framework reveals that the conjecture is equivalent to finding, for each N ≥ 4, an integer M ∈ [1, N − 3] such that the L-shaped region N2 − M2 between nested squares has area P · Q where P = N − M and Q = N + M are both prime. We define DN = {(Q − P)/2 | 2 < P < N < Q < 2N, both prime} to be the set of achievable half-differences from straddling prime pairs. The conjecture becomes equivalent to proving that DN ∩ {N − p | 3 ≤ p < N, p prime} ̸= ∅ for all N ≥ 4. Our gap function G(N) = log²(2N) − ((N − 3) − |DN|) measures the margin by which this condition holds. Computational analysis for N ∈ [4, 2¹⁴] reveals that G(N) > 0 universally, with minima strictly increasing across dyadic intervals. For N ≥ 3275, we prove theoretically that G(N) > 0 by showing that Dusart’s prime distribution theorem guarantees |DN| > (N − 3) − log²(2N). The pigeonhole principle then ensures existence of valid Goldbach partitions: since there are π(N −1)−1 > log²(2N) candidate primes P < N, and fewer than log²(2N) “bad” M-values, at least one candidate yields both P and Q = 2N − P prime. This completes the proof of the distinct-prime Goldbach variant and demonstrates the power of geometric reformulation combined with modern analytic number theory.

Abstract: The Prime Representing Constant infinitely generates prime numbers. We discovered how to compute the Prime Representing Constant using the Hadamard product expansion. We also modified the classical Riemann–von Mangoldt explicit formula. While mathematically equivalent to the classical formula, the cosine-phase form is novel in its computational and structural presentation, enabling faster, memory efficient, and intuitive computation of ψ(x). Combining all the properties will give us a trace-type oscillatory operator that infinitely generates prime numbers.

Abstract: This paper presents a family of No-Sum (NS) sequences defined by read-once arithmetic derivations, as introduced in a previous paper, and introduces the scalable Hybrid NS algorithm, which maintains strong combinatorial hardness while enabling long-range generation. In Phase 1, we construct a strict NS sequence under (+,−,×) to establish the existence, uniqueness, and finiteness of a governing derivable set for greedy progression. The paper then introduces a prefix-lock, the positive derivability closure of the strict prefix, which is employed in subsequent phases to avoid collisions with previous exclusions. Phase 2 applies a relaxed NS rule (e.g., (+,−) or bounded read-once derivations), and Phase 3 applies an efficient sum-free rule (+), with all phases prefix-locked to preserve the definitional integrity of the phases and the greedy-minimality of the entire sequence. In this paper, we present a formal sequence construction with provable uniqueness and a scalable hybrid extension, and discuss the complexity of the sequence motivated by cryptographic hardness.

Abstract: This paper analyzes the Binary Goldbach Conjecture (bGC) through a deterministic structural lens, employing a Failure Mode Analysis (FMA) framework to map prime and composite inventories onto the Left-Right Partition Table (LRPT). We establish structural identities governing the conservation of partition elements, demonstrating that the count of Prime-Prime (PP) pairs functions as a necessary deterministic residual. The analysis identifies tiered inadmissible failure states where, in each Tier, the exhaustion of composite inventories mathematically forces prime-prime partitions into existence to preserve information conservation. Numerical analysis for N up to 10⁶ validates these findings, showing that the boundary of failure admissibility, parameterized by the ratio \( \hat{\lambda}_L(N) \), converges toward a global structural ceiling. Furthermore, by leveraging the midpoint symmetry of Goldbach primes, the FMA approach yields a ``Mirror Search'' mechanism for distal primes that demonstrates superior discovery efficiency compared to sequential scanning methods guided by the Prime Number Theorem. The analysis also reveals that the failure state (PP(N)=0) precipitates an information-theoretic paradox: it implies that the global prime counting function π(2N) can be fully reconstructed from the local modular geometry of a subset of composites, violating the established algorithmic irreducibility of the prime sequence.

Abstract: A simple proof of Fermat’s Last Theorem (FLT) for the cube is obtained using the binomial expansion formula. It is shown that the difference between two natural numbers raised to the same natural power must be represented by an incomplete binomial formula. It is proven that the cube of a natural number cannot be represented as an incomplete binomial, which means a simple proof of FLT for n=3 has been obtained.

Abstract: We develop a complete operator-theoretic and spectral framework for the Collatz map by analyzing its backward transfer operator on weighted Banach spaces of arithmetic functions. The associated Dirichlet transforms form a holomorphic family that isolates a zeta-type pole at s=1, while on a finer multiscale space adapted to the dyadic-triadic geometry of the Collatz preimage tree we establish a two-norm Lasota-Yorke inequality with an explicit contraction constant, yielding quasi-compactness, a spectral gap, and a Perron-Frobenius theorem in which the eigenvalue 1 is algebraically and geometrically simple, no other spectrum meets the unit circle, and the unique invariant density is strictly positive. The fixed-point relation is converted into an exact multiscale recursion for the block averages c_j, revealing a rigid second-order coupling with exponentially small error terms and asymptotic profile c_j~ 6^-j. This spectral classification forces every weak* limit of the Cesàro averages derived from any hypothetical infinite forward orbit to be either 0 or a scalar multiple of the Perron-Frobenius functional, with convergence to 0 occurring precisely under the Block-Escape Property. Since the forward map satisfies an unconditional exponential upper bound, whereas Block-Escape combined with linear block growth along a subsequence would impose an incompatible exponential lower bound, all analytic and spectral components needed for such a contradiction are complete, reducing the Collatz conjecture to excluding infinite orbits exhibiting Block-Escape without the supercritical linear block growth prohibited by the spectral theory.

Abstract: A revision mainly in the appendix.

Abstract: This paper investigates the final coalgebra for the endofunctor \( F(X) = \mathbb{Z}_3 \times X^3 \) on the category Z₃-Set of sets with a Z₃-action. We call the resulting F-coalgebras Helical Triadic Coalgebras (HTCs). The factor Z₃ records an observable phase that makes distinct cyclic positions distinguishable. We develop the notion of Z₃-bisimulation, which generalizes standard bisimulation by allowing cyclic shifts. Our main results concern a natural HTC structure on the srs lattice (Laves graph). The canonical morphism from the srs coalgebra S to the final coalgebra Ω is not injective: translations induce a bisimulation collapsing S onto a 12-element quotient \( \mathcal{Q} \cong K_4 \times \mathbb{Z}_3 \). The \( V_4 \)-symmetry of srs further collapses Q onto a 3-element image I. A symmetry analysis reveals that I is symmetric while Ω is not. We also define orbital invariants (binding index, degeneracy, multiplicity) and establish that every regular coalgebra is chiral. Finally, we prove that among sub-coalgebras of the final coalgebra, symmetry and connectivity alone characterize srs uniquely (up to chirality). These results bridge coalgebraic methods with graph theory and crystallography.