State, Explain and Develop the Fractal Uncertainty Principle

Enrique Vidal Silvente

Sundaram Thangavelu

Enrique Vidal Silvente

We develop a fractal-spectral reformulation of the Heisenberg uncertainty principle within a nonlinear phase-space framework motivated by the Quantum Fractal-Logical Unified Field architecture. Instead of working on the standard Hilbert space L 2 (R) with Lebesgue measure and linear position/momentum operators, we consider a fractal Hilbert space H F = L 2 (F, µ H) built on a Cantor-type metric-measure substrate with Hausdorff measure µ H and a self-adjoint fractal Hamiltonian. Within this setting we define effective position and momentum observables (X F , P F) adapted to the fractal geometry and derive a generalized uncertainty relation

Jean-Sebastien Beaulieu

This trilingual research and scientific-exposition article develops a method for keeping mathematical objects, physical observables, scales, operators, and chains of inference distinct when discussing fractals, Fourier uncertainty, neutrinos, microtubules, anesthesia, consciousness, and proton spin. It presents the fractal uncertainty principle as an operator bound rather than an ontological certificate, incorporates a bounded Richardson-Mandelbrot scaling contribution by Hector Fernando Aguilar, and separates measured energy, statistical information, biological code, behavior, and conscious experience. The article proposes no unified physical theory and makes no claim that an intrinsic fractal geometry of neutrinos or a microtubule mechanism of consciousness has been established.

Jens Deutschmann

Fractal Uncertainty Theory (FUT) is a fundamental reformulation of mathematics and physics that redefines the classical concepts of "0" and "1" as observer-dependent, fuzzy states. The theory introduces view-vectors and perspective-dependent null-zones as the primary ontological objects, from which physical laws emerge. Thisdocument presents the complete axiomatic foundation of FUT (Version 5.0). The Fractal Cancellation Theorem (FCT) is part of Fractal Uncertainty Theory (FUT). Note on status. This document represents the current state of FUT development as of February 2026.Predictions may be refined or extended as the theoretical framework develops and experimental capabilities advance.