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Fractal Uncertainty Theory (FUT) - Version 5

Jens Deutschmann

Published Feb 21, 2026• 25 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.18724415

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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.
The single-slit and double-slit experiments are among the most fundamental demonstrations of quantum behavior. Every interpretation of quantum mechanics must account for them, yet none has derived the mathematical structures used todescribe them from the experiments themselves. In this paper we show that Fractal Uncertainty Theory (FUT) achieves precisely this. The which-way measurement problem is resolved without wave-function collapse or many worlds.We argue that the slit experiments, properly understood through FUT, constitute the physical origin of arithmetic itself. 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.
We present the complete Emergence Chain of Fractal Uncertainty Theory (FUT): an eleven-stage derivation sequence (Stages 0–10) from a single qualitative relational primitive to the full structure of quantum mechanics and classical physics. Each stage is a necessary consequence of the previous one, with zero free parameters introduced at any stage. The chain proceeds: pure relation ∼f. This document establishes the Emergence Chain as a formal structure and serves as the index paper for the FUT/CoMath paper series.
Classical mathematics treats zero and one as absolute, observer-independent objects. Fractal Uncertainty Theory (FUT) and Coherent Mathematics (CoMath) challenge this at the foundational level. We show that 0 arises from perfect destructive interference and 1 from a single construc-tive detection event. Because every measurement denes its own interference condition, there are as many zeros and ones as there are uncertainty relations. Arithmetic structure emerges from the relational network between these many zeros and ones. Classical arithmetic is recovered as the high-coherence limit K → 1. The derivation requires no abstract axioms about numbers; it requires only wave physics and the principle that coherence is what survives.
We identify and formalise the minimal ontological primitive of Fractal Un-certainty Theory (FUT): the partial relational coherence relation ∼f, a non-distinguishability relation between observational view-vectors. All physical structure—space, time, quantum mechanics, the Standard Model—is derived from ∼f alone via successive coherence selections. The framework introduces four core domain concepts: Kohärenzregime (coherence regimes), Persistenzklassen (persistence classes), View-Vektoren (view-vectors), and Interferenzdynamik (interference dynamics). A key insight established in May 2026 is that ∼f is irreducibly preformal: any attempt to formalise it prematurely loses essential content. Axiom 0 is the first formal statement that can be made about ∼f, not a starting axiom.
The Heisenberg uncertainty principle constitutes one of the foundational pillars of quantum mechanics and represents a definitive rupture with the deterministic ideal of classical physics. This work develops a comprehensive analysis of the uncertainty principle from a mathematical, physical, and philosophical perspective, demonstrating that it should not be interpreted as an instrumental or epistemological limitation, but rather as an ontological property inherent to the formalstructure of quantum theory.Within the framework of Hilbert spaces, self-adjoint operators, and non-commutative algebras, it is shown that uncertainty emerges necessarily as a consequence of the structural incompatibility between conjugate observables. The study articulates the uncertainty principle with classical chaos theory and the sciences of complexity, establishing a fundamental distinction between epistemological unpredictability and ontological indeterminacy. While classical chaos preserves well-defined trajectories that are practically inaccessible, quantum mechanics eliminates the very notion of trajectory as a fundamental entity.This rupture manifests itself in the granularity of phase space, the impossibility of defining exact initial conditions, and the consequent reformulation of determinism. Contemporary experimental developments designed to exclude instrumental disturbances are analyzed, reinforcing the ontological interpretation of the uncertainty principle.Finally, the implications for simulation, predictability, and quantumOverall, this work argues that the uncertainty principle does not undermine scientific rationality, but rather redefines itsscope and deepens our understanding of physical reality. Keywords: uncertainty principle; quantum mechanics; mathematical structure; non-commutative operators; chaos theory; complexity; determinism; physical ontology.