From Cantor's Infinities to Hierarchical Superluminal Particles in Ferrion Theory

This preprint presents a novel theoretical framework that bridges Georg Cantor's transfinite hierarchy of infinities with the physics of superluminal particles. Starting from the author's original proposal of ferrions — hypothetical PT-symmetric particles that achieve faster-than-light propagation without purely imaginary mass, stabilized via Higgs-field coupling [Sánchez Ferra, 2026] — the work extends this concept into an infinite hierarchical ladder inspired by Cantor's cardinals (ℵ₀ < 2^ℵ₀ < 2^(2^ℵ₀) < …). The core innovation is the introduction of a hierarchical d'Alembertian operator □n = □ + ξ_n m²c²/ℏ², where the level-dependent factor ξ_n = i^(2^ℵ{n-1}) (with ℵ_{-1} = 0) is rigorously derived through iterated complex phases and transfinite exponents. This operator generalizes the Klein-Gordon equation to accommodate escalating superluminal regimes, with PT symmetry ensuring real energy spectra at each level. The framework is further extended to: The Dirac equation for spin-1/2 fermions, The Proca equation for massive vector bosons, Kaluza-Klein theory, where extra-dimensional mode towers are superluminalized, String theory, incorporating infinite vibrational modes with transfinite mass modifications. Detailed step-by-step derivations of ξ_n are provided for all cases, including a particularly thorough treatment for string modes that links transfinite exponents to iterated dualities (T-duality, S-duality) and dense phase structures. The paper concludes by exploring potential applications in quantum gravity, such as resolutions to the black hole information paradox (via transfinite information channels), UV completion of quantum field theory (through hierarchical regularization), dark energy modeling (from vacuum contributions of infinite superluminal towers), and unification across forces and dimensions. This work is highly speculative yet mathematically consistent, offering a new interdisciplinary bridge between set theory and high-energy physics. It proposes that nature may permit an infinite escalation of superluminal phenomena, analogous to Cantor's boundless infinities, challenging traditional causality constraints while preserving physical consistency through PT symmetry and hierarchical stabilization.