arXiv subjects

Ramon Moya

Publications and source records attributed to Ramon Moya.

5 recordsLinked to original sources

arXiv2026

Unified Nilpotent Operational Framework: Foundations, Algebraic Exactness, and Complexity

A unified algebraic framework is developed to study nilpotency as a structural mechanism for exactness in operational, combinatorial, and computational problems. The central object is the Nilpotent Operational System (SON), formalized as a tuple (R, N, m, M_R), where R is a C-algebra, N satisfies N^{m+1}=0, and M_R(m) is the arithmetic cost of a product in R. The basic result is the Exact Termination Lemma: every formal series evaluated at N collapses to an exact finite sum of m+1 terms. Three complexity regimes are obtained: truncated series (quasi-linear bound via Newton iteration), nilpotent operators (linear bound via Horner evaluation), and incidence algebras (quasi-quadratic bound). Applications include classical and free cumulants, Appell sequences, orthogonal polynomials, Stirling numbers, M\"obius inversion, Witt vectors, and local holonomic functions. In all cases except Stirling numbers, strict complexity improvements over classical algorithms are obtained. For classical cumulants, equivalence with polynomial multiplication is achieved: L(C_n)=Theta(M(n)). For free cumulants, the complete equivalence T_{FC}(n)=Theta(M(n)) is established via the Voiculescu functional equation and compositional reversal.

math.GM↗

arXiv2026

Vector Determinant in the ARE Framework: From Scalar to Vector-Valued

The ARE (Action, Rectification, and Structure) method is presented as a framework for reorganizing the Leibniz expansion of the determinant through the action of the cyclic group C_n on S_n. This action partitions the permutations into orbital classes and naturally leads to a finite Fourier decomposition of the determinant structure. The central object of the work is the vector determinant, whose components are Fourier modes associated with orbital sums of Leibniz terms. The classical determinant appears exactly as the fundamental mode G_0(A)=det(A), while the remaining modes are multilinear spectral observables associated with the orbital organization of the expansion. The manuscript establishes the fundamental properties of this formalism, including exact determinant recovery through a linear readout functional, multilinearity, Hermitian symmetry for real matrices, vector Jacobi and Laplace formulas, orbital Parseval identities, a vector Hadamard inequality, and finite trigonometric interpolation. The framework extends naturally to continuous frequencies through an associated orbital polynomial. It does not reduce the factorial complexity of determinant computation, but reveals an orbital-Fourier structure not visible in the classical scalar determinant. The work also clarifies the relation between orbital modes and circulant structures, proving that the crude orbital modes cannot generally coincide with circulant eigenvalues because of polynomial-degree incompatibility.

math.RA↗

arXiv2026

ARE Method: Orbital Decompositions and Dihedral Cancellations for Determinants

We develop the ARE method (Action-Rectification-Expansion), a structural framework for the organization of Leibniz terms in determinants through cyclic group actions and orbital decompositions. The symmetric group S_n is partitioned into (n-1)! disjoint orbits of size n under right composition by the cyclic group C_n. Each orbit admits a canonical representative and generates a family of determinant terms related by cyclic rotation. We prove explicit sign laws for orbital rotations, establish a rectification theorem transforming orbital polylines into parallel-line configurations through a single block permutation, and characterize companion orbitals through dihedral symmetries. The framework yields an exact reorganization of the Leibniz expansion preserving all n! terms while exposing hidden geometric and combinatorial structure. We further prove an impossibility theorem showing that no fixed-width direct extension of the classical Sarrus rule can capture all determinant terms for n >= 4. The method provides three equivalent visualizations: polylines, parallel rectified lines, and total-line representations. Deterministic orbital generation algorithms and computational verification against standard determinant methods are also presented. Although the approach does not reduce factorial complexity, it provides a systematic geometric and algebraic interpretation of determinant structure extending the conceptual spirit of Sarrus to arbitrary dimension.

math.RA↗

arXiv2026

Exact Nilpotent Collapse of Born-Neumann Expansions in Finite Quantum Systems: A SON Formulation for Exact Algebraic Closures of Scattering Series

We identify a class of finite quantum systems, namely, acyclic systems whose transition graph is a directed acyclic graph (DAG), for which the Born series collapses into an exact algebraic identity with finitely many terms and strictly zero truncation error. The sufficient condition is the nilpotency of the transfer operator T = G_0(E)V. If T^{m+1} = 0, then the exact solution of the Lippmann-Schwinger equation is the finite sum |psi> = sum_{k=0}^{m} T^k |phi>, with no condition on ||T||. We prove that the acyclicity of the transition graph implies the nilpotency of T (Theorem 19), and that the nilpotency index coincides with the maximal path length of the graph (Proposition 21). The main result (Theorem 23) concerns the four-level quantum system with diamond-graph structure. In this case, the transition amplitude toward the final state is A_4 = t_{42}t_{21} + t_{43}t_{31}, an exact algebraic identity encoding constructive interference, exact destructive interference (dark state formation), and partial interference. The first-order Born approximation predicts identically zero amplitude in all regimes, thereby failing quantitatively in 100% of the cases. The Born-SON framework additionally provides the exact full resolvent, the exact T-matrix, explicit error control in the quasi-nilpotent regime, and a scalar structural metric, the Born-SON depth, quantifying the intrinsic complexity of an acyclic quantum system.

quant-ph↗

arXiv2026

Hypergeometric Functions of Nilpotent Operators: Functional Collapse and Structural Depth at Exceptional Points

We study hypergeometric functions of nilpotent operators in finite-dimensional settings, motivated by the algebraic structure of exceptional points in non-Hermitian quantum mechanics. Our starting point is the following exact result: if N is a nilpotent operator of index m+1 in an associative algebra over C, then every generalized hypergeometric function pFq evaluated at N reduces to a finite polynomial in N of degree at most m, without any analytic convergence requirement. This "functional collapse" is distinct from the classical parameter-termination mechanism and arises purely from the nilpotent structure of the argument. The main result is a "nilpotent depth criterion" (Theorem 2): if the first non-constant coefficient of a formal series F appears in degree r >= 1, then the nilpotent part F(N) - F(0)I has nilpotency index bounded above by ceil((m+1)/r). We apply this criterion to Hamiltonians at exceptional points, where H = lambda I + N with N^{m+1} = 0. Theorem 3 establishes that a function F analytic at lambda reduces the Jordan depth of the exceptional point from m+1 to at most ceil((m+1)/r), where r is the contact order of F at lambda. As consequences: the time evolution operator e^{tH} preserves the full Jordan depth for all t != 0; a function with a zero of order m+1 at lambda annihilates the entire Jordan structure; and the order of the pole of the modified resolvent is reduced from m+1 to at most m+1-r. Results are illustrated with explicit 3x3 Jordan block computations for 1F1, 2F1, and the time evolution operator, confirming sharpness of the bounds.

math-ph↗