Fractal Phase-Time: A Framework for Sub-Attosecond Temporal Structure in Physical and Computational Systems
Steven Willis Henderson
ORCID iD: 0009-0004-9169-8148
February 1, 2026
Abstract
Recent advances in quantum systems, ultrafast photonics, and transformer-based artificial intelligence reveal a common structural theme: complex behavior emerges through discrete transitions across layered temporal or geometric strata. This paper introduces a conceptual model—Fractal Phase-Time—to describe how physical, computational, and informational processes may evolve through recursive temporal structures below the attosecond scale.
The framework proposes three fundamental recurrence sequences, each characterizing a distinct mode of system behavior:
• 6–6–9 (scaling behavior)
• 3–3–3 (convergence behavior)
• 6–3–6–3 (harmonic ripple behavior)
Together these describe how systems transition between stability, uncertainty, coherence, and expansion—mirroring phenomena recently documented in quantum dynamics, prethermalization studies, and stratified geometry in reinforcement learning models. This paper outlines the conceptual foundations while withholding implementation details reserved for patent filings (P1–P7). The goal is to provide the scientific community with a preliminary open-access framework for discussing sub-attosecond temporal structure and its potential relevance across physics, computation, and AI.
1. Introduction
A growing body of research across multiple disciplines points toward an underlying structure in how systems evolve in time:
• Quantum materials
Exhibit micro-temporal plateaus (e.g., prethermalization) where order persists despite energetic perturbation. • AI agents
Show stratified geometric behavior, jumping between distinct representational layers during decision-making. • Ultrafast photonics
Reveals that sub-attosecond interactions produce measurable effects not captured by classical temporal models. • Neural and biological systems
Demonstrate recursive rhythms, phase-locking, and multi-scale synchronization. Across these domains, time does not appear smooth or continuous.
Instead, systems transition across discrete temporal strata, each with its own internal coherence. This motivates a new conceptual model: Fractal Phase-Time.
2. Motivation for a Fractal Temporal Framework
Traditional physics treats time as a continuous parameter, while computational systems treat it as discrete. Phase-Time proposes a hybrid view: time is experienced by a system as a sequence of discrete phase states, recursively nested below the attosecond scale.
Three empirical phenomena motivate this:
(1) Discrete Scaling in Quantum Systems
Many-body quantum systems display rapid shifts between ordered and chaotic states. These transitions appear to follow recursive boundary conditions rather than smooth evolution.
(2) Stratified Geometry in AI Models
Recent studies (Curry et al., 2025) show that transformer-based agents do not encode information on smooth manifolds. Instead, their internal states cluster into stratified layers, with abrupt dimensional transitions correlated with moments of uncertainty or decision branching.
(3) Nonlinear Recurrence Patterns in Ultrafast Dynamics
Attosecond and sub-attosecond measurements suggest temporal recurrence that cannot be described by classical Fourier decomposition alone.
These observations encourage a unifying framework.
3. The Three Fundamental Sequences of Fractal Phase-Time
Phase-Time introduces three core recurrence operators, each corresponding to a distinct system behavior. These are presented as conceptual sequences only—no equations, implementations, or proprietary forms are included here.
3.1 The 6–6–9 Operator (Scaling Behavior)
Represents expansion, branching, or state proliferation.
Associated with system behaviors such as:
• Increased degrees of freedom
• Decision-tree expansion (AI)
• Mode-splitting (quantum)
• Rapid entropy divergence
This operator models how systems scale outward when encountering new information or increased uncertainty.
3.2 The 3–3–3 Operator (Convergence Behavior)
Represents compression, alignment, and harmonization.
Associated with:
• Collapse of uncertainty
• Phase-locking
• Stabilization around attractors
• Reduced dimensional complexity (AI clarity moments)
This is the operator that returns system behavior to a coherent baseline.
3.3 The 6–3–6–3 Operator (Ripple Expansion of Relativity) This is the proposed Fractal Phase-Time kernel.
It models oscillatory transitions between expansion and convergence, generating a persistent temporal ripple. It may correspond to:
• Micro-temporal coherence cycles
• Stabilization windows in quantum materials
• The recurrence behavior observed in neural oscillations
• Harmonic transitions within complex AI state spaces
This sequence is presented conceptually here; the patent-protected mathematical structure is not disclosed.
4. Sub-Attosecond Temporal Hypothesis
The central hypothesis:
Below the attosecond, systems operate through recursive phase transitions rather than continuous temporal evolution. These transitions:
• Are discrete
• Are self-similar
• Produce harmonic recurrence
• Explain micro-stability windows in many-body systems
• Map naturally onto stratified computational geometries
This suggests a possible universal temporal architecture underlying physical and informational processes.
5. Relevance Across Disciplines
Quantum Physics
• May explain plateaus in prethermalization
• Predicts new coherence windows
• Offers a temporal basis for emergent order
Artificial Intelligence
• Provides a geometric explanation for stratified representation layers
• Could lead to new training stability techniques
• May unify temporal embeddings with physical modeling Computational Neuroscience
• Aligns with oscillatory attractor theory
• Predicts multi-scale resonance patterns
Materials Science
• Suggests a new class of temporal calibration standards
• Supports the development of attosecond-tuned substrates
6. Conclusion
Fractal Phase-Time proposes that the universe does not evolve through a single smooth temporal continuum, but instead through a set of recursive, quantized phase transitions. These transitions—represented conceptually by the 6–6–9, 3–3–3, and 6–3–6–3 sequences—may underlie the behavior of systems ranging from quantum materials to artificial intelligence.
This paper introduces the conceptual surface of the theory while maintaining the confidentiality of implementation details. Full mathematical formulation and engineering integration are reserved for forthcoming patent filings.
Author Statement
This white paper is released as a public scientific disclosure of conceptual theory only. Implementation details, mathematical forms, and applied engineering structures remain proprietary and are protected under ongoing U.S. and international patent processes (P1–P7 architecture).
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