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.
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.
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.
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.
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.
As opposed to the classical logic of true and false, viewed as an axiomatised theory, ordinary arithmetic conveys the three logical values: provable, negatable and logically independent. This research proposes the hypothesis that Axioms of Arithmetic are the fundamental foundation running arithmetical processes in Nature, upon which physical processes rest. And goes on to show, in detail, that under these axioms, quantum mathematics derives and initiates logical independence, agreeing with indeterminacy in quantum experiments. Supporting arguments begin by explaining logical independence in arithmetic, in particular, independence of the square root of minus one. The method traces all sources of information entering arithmetic, needed to write mathematics of the free particle. Wave packets, prior to measurement, are found to be the only part of theory logically independent of axioms; the rest of theory is logically dependent. Ingress of logical independence is via uncaused, unprevented self-reference, sustaining the wave packet, but implying unitarity. Quantum mathematics based on axiomatised arithmetic is established as foundation for the 3-valued logic of Hans Reichenbach, which reconciles quantum theory with experimental anomalies such as the Einstein, Podolsky & Rosen paradox.
The quasi-classical world of modern physics is a class of classical worlds superposed, literally a 'metaworld'. The measurement problem is itself evidence for the two different types of world, at the different levels of logical type. The enactment of linear dynamics in the classical worlds alters the class invariant, and thus induces physical collapse of the metaworld to a subclass. No new physics is required. The complex formalism that seems deeply counterintuitive is class arithmetic. Superposition and interference: properties of classes of worlds. Measurement: set-selection of a subclass compatible with the observed outcome. The uncertainty principle: incompatible set-selections. Fourier transforms: switching set-selection. Operators: enactment of set-selection. Decoherence: loss of coherence between worlds. Normalisation is required because all worlds are equally real, thus the probability of an outcome is essentially the proportion of worlds that instantiate it. True probability arises through phase, which is the degree to which the rest of each world aligns internally with this outcome. This is the second-logical-type phenomenon of interference: the degree to which worlds cohere in the classical actions that give rise to the outcome.This transforms the familiar structures of the formalism from postulates to inherent attributes of the class structure. The Born rule, unitary evolution, the complex structure of amplitudes, the probabilistic character of prediction, and the wave-mechanical form of the theory emerge as outputs of the ontology rather than independent axioms. The wavefunction is real. It is the ontological architecture of the metaworld and the reality of the quasi-classical world: this is the quantum realm. The classical world, quantum mechanics and relativistic physics are unified in logical types.