Dr. Pedro Jesús Cáceres Candea

Publicaciones

• Quantum Data Statistics for Financial Systems: Operator-Valued Bayesian Learning, QDS Test Statistics, and Regime Bifurcation with Applications to Enterprise Markets and Crypto Finance

  QDS es un nuevo marco matemático para analizar sistemas dinámicos complejos mediante operadores de densidad en lugar de estadísticas clásicas. QDS captura estructura, interacción y cambios de régimen ocultos que los métodos tradicionales no detectan. Su aplicación permite identificar transiciones tempranas, mejorar la toma de decisiones bajo incertidumbre y desarrollar sistemas predictivos avanzados en finanzas, inteligencia artificial, operaciones, automatización y otros entornos complejos.

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• Asymptotic Balance and Structural Rigidity in the Riemann Zeta Function

  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 Z into three parts

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• The Riemann Hypothesis from Additive Decomposition, Interaction Rigidity, and Critical-Line Geometry

  This work reformulates the Riemann Hypothesis as a problem of additive rigidity arising from structured interactions within finite approximations of the zeta function. By isolating and analyzing the interaction mechanism responsible for cancellation, the problem is reduced to a deterministic geometric structure governed by logarithmic oscillations. The analysis reveals three distinct regimes, with stable cancellation possible only on the critical line. Consequently, the Riemann Hypothesis is transformed from an analytic question into a finite-scale geometric rigidity problem.

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• A BAIRE-CATEGORY OBSTRUCTION FOR AN ERDŐS PROBLEM ON LAGRANGE INTERPOLATION

  This work addresses an open problem posed by Paul Erdős concerning Lagrange interpolation and the behavior of unstable interpolation points. Using Baire category methods, it establishes a fundamental obstruction showing that if the set of unstable points is countable, there always exists a continuous function for which the interpolation process diverges at every such point. As a consequence, any positive solution to the problem requires an uncountable set of instability points, revealing deep structural limitations in polynomial interpolation.

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• QUANTUM DATA STATISTICS: AN OPERATOR-THEORETIC FRAMEWORK FOR STRUCTURED UNCERTAINTY

  Quantum Data Statistics (QDS) introduces a new mathematical framework for analyzing uncertainty in dynamic systems. By representing empirical data as density operators in a Hilbert space, QDS captures hidden structure, regime interactions, and latent dynamics that remain invisible to traditional statistical methods. The framework provides novel indicators of instability, bifurcations, and structural change through measures such as purity, entropy, coherence, and dual uncertainty. This work establishes the theoretical foundations of QDS as a general approach for studying complex time-evolving systems and detecting transitions before they become observable through conventional statistics.

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• Asymptotic Balance and Structural Rigidity in the Riemann Zeta Function

  This work develops a structural approach to the nontrivial zeros of the Riemann zeta function by examining the interaction between discrete Dirichlet oscillations and their continuous analytic counterpart. Using the C-transformation, a new operator that isolates the finite structural difference between a series and its integral approximation, the zeta function is decomposed into oscillatory, analytic, and decaying components. This framework reveals underlying rigidity properties governing cancellation mechanisms and provides a new perspective on the structure of zeta zeros and the Riemann Hypothesis.

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• CONDITIONAL ASYMPTOTIC RIGIDITY IN THE C-DECOMPOSITION OF THE RIEMANN ZETA FUNCTION

  This work introduces the C-transformation, a new analytical framework that decomposes the Riemann zeta function into an oscillatory component and an explicit growth component. The analysis reveals a fundamental structural distinction between regions of the complex plane, showing that the asymptotic behavior of these components is compatible only on the critical line. On this line, a rigid cancellation mechanism emerges from logarithmic phase interactions, producing coherent and stable behavior absent elsewhere. Numerical results support the predicted scaling laws and suggest a deep structural rigidity underlying the distribution of the nontrivial zeros of the zeta function.

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